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	# Globular Clusters in Virgo Ellipticals: Unexpected Results for Giants, Dwarfs, and Nuclei from ACS Imaging Jay Strader, Jean P. Brodie, Lee Spitler, Michael A. Beasley UCO/Lick Observatory, University of California, Santa Cruz, CA 95064 strader@ucolick.org, brodie@ucolick.org, lees@ucolick.org, mbeasley@ucolick.org ###### Abstract We have analyzed archival _Hubble Space Telescope_/Advanced Camera for Surveys images in \(g\) and \(z\) of the globular cluster (GC) systems of 53 ellipticals in the Virgo Cluster, spanning massive galaxies to dwarf ellipticals (dEs). Several new results emerged: (i) In the giant ellipticals (gEs) M87 and NGC 4649, there is a correlation between luminosity and color for _individual_ metal-poor GCs, such that more massive GCs are more metal-rich. A plausible interpretation of this result is self-enrichment, and may suggest that these GCs once possessed dark matter halos. (ii) In some gEs (most notably M87), there is an "interloping" subpopulation of GCs with intermediate colors (1.0 \(<g-z<\) 1.25) and a narrow magnitude range (0.5 mag) near the turnover of the GC luminosity function. These GCs look otherwise identical to the classic metal-poor and metal-rich GC subpopulations. (iii) The dispersion in color is nearly twice as large for the metal-rich GCs than the metal-poor GCs. However, there is evidence for a nonlinear relation between \(g-z\) and metallicity, and the dispersion in metallicity may be the same for both subpopulations. (iv) Very luminous, intermediate-color GCs are common in gEs. These objects may be remnants of many stripped dwarfs, analogues of \(\omega\) Cen in the Galaxy. (v) There is a continuity of GC system colors from gEs to some dEs: in particular, many dEs have metal-rich GC subpopulations. We also confirm the GC color-galaxy luminosity relations found previously for both metal-poor and metal-rich GC subpopulations. (vi) There are large differences in GC specific frequency among dEs, independent of the presence of a nucleus and the fraction of metal-rich GCs. Over \(-15<M_{B}<-18\), there is little correlation between specific frequency and \(M_{B}\) (in contrast to previous studies). But we do find evidence for two separate \(S_{N}\) classes of dEs: those with \(B\)-band \(S_{N}\sim 2\), and dEs with populous GC systems that have \(S_{N}\) ranging from \(\sim 5-20\) with median \(S_{N}\sim 10\). Together, these points suggest multiple formation channels for dEs in the Virgo Cluster. (vii) The peak of the GC luminosity	[]
	function (GCLF) is the same for both gEs and dEs. This is contrary to expectations of dynamical friction on massive GCs, unless the primordial GCLF varies between gEs and dEs. Among gEs the GCLF turnover varies by a surprising small 0.05 mag, an encouraging result for its use as an accurate standard candle. (viii) dE,Ns appear bimodal in their nuclear properties: there are small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. The role of dynamical evolution in shaping the present-day properties of dE GC systems and their nuclei remains ambiguous. globular clusters: general -- galaxies: star clusters -- galaxies: formation ## 1 Introduction It is increasingly apparent that globular clusters (GCs) offer important constraints on the star formation and assembly histories of galaxies. Recent spectroscopic studies of GCs in massive early-type galaxies (e.g., Strader _et al._ 2005) indicate that the bulk of star formation occurred at relatively high redshift (\(z\gtrsim 2\)) in high density environments (as environmental density and galaxy mass decrease, the fraction of younger GCs may increase; see Puzia _et al._ 2005). These findings allow the age-metallicity degeneracy to be broken and lead to the conclusion that the bimodal color distributions seen in most nearby luminous galaxies are due primarily to two old GC subpopulations: metal-poor (blue) and metal-rich (red). The metallicities of these peaks correlate with host galaxy luminosity (Larsen _et al._ 2001, Strader, Brodie, & Forbes 2004; see also Lotz, Miller, & Ferguson 2004 for dwarfs). Most recent photometric studies of GC systems in ellipticals have used the Wide Field and Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). Compared to ground-based imaging, this strategy gains photometric accuracy and minimizes contamination at the expense of small spatial coverage. Among the larger HST studies of early-type galaxies utilizing deep imaging are Larsen _et al._ (2001) and Kundu & Whitmore (2001). These studies found bimodality in many of their sample galaxies (extending down to low-luminosity ellipticals) and a nearly uniform log-normal GC luminosity function (GCLF) with a peak at \(M_{V}\sim-7.4\). However, the GC systems of dwarf ellipticals (dEs) are more poorly understood. The large HST surveys to date (primarily of Virgo and Fornax) have been limited to relatively shallow snapshot imaging; this precluded the study of color and luminosity distributions in detail. Among the suggestions of this initial work are a correlation of increasing specific frequency (\(S_{N}\)) with decreasing galaxy luminosity, a dichotomy in	[]
	the GC systems of nucleated and non-nucleated dEs (dE,N and dE,noN, respectively), and the difficulty of making dE nuclei as observed through dynamical friction of GCs (Miller _et al._ 1998, Lotz _et al._ 2001). Data from the Advanced Camera for Surveys (ACS) Virgo Cluster Survey (HST GO 9401, P. I. Cote) offers an important step forward in understanding the detailed properties of the GC systems of ellipticals over a wide range in galaxy mass. The use of F475W and F850LP filters (henceforth called \(g\) and \(z\) for convenience, though the filters do not precisely match the Sloan ones) allows a much wider spectral baseline for metallicity separation than \(V\) and \(I\). Though only a single orbit is used per galaxy, the increased sensitivity of ACS (compared to WFPC2) allows one to reach \(\sim 3\) mag beyond the turnover of the GCLF, encompassing most of the GCs in a given galaxy. More accurate photometry for the brighter GCs is also possible. Finally, the field of view of ACS is twice that of WFPC2. Together, these attributes allow a study of the color and luminosity distribution of GCs in a large sample of galaxies in much more detail than previously possible. In what follows, elliptical (unabbreviated) refers to all galaxies in our sample and denotes no specific luminosity. The three brightest galaxies are described as giant ellipticals (gEs); these have \(M_{B}\leq-21.4\). Galaxies of intermediate or high luminosity are called Es. Faint galaxies with exponential surface brightness profiles are dEs; the transition from E to dE occurs traditionally at \(M_{B}\sim-18\) (Kormendy 1985). We include two galaxies with \(M_{B}=-18.1\) (VCC 1422 and VCC 1261) under this heading, since these galaxies have nuclei similar to those commonly found among dEs. The E/dE classifications have been taken from the literature and we do not perform independent surface photometry in this paper (though we note the increasing debate in the literature about whether this dichotomy is real, e.g., Graham & Guzman 2003). We have updated the nucleation status of a dE if appropriate; the vast majority of the dEs in our study have nuclei. ## 2 Data Reduction and Analysis All data were taken as part of the ACS Virgo Cluster Survey (Cote _et al._ 2004); this survey includes both ellipticals and S0s. We used all galaxies classified as ellipticals, excepting a few dwarfs quite close to luminous Es whose GC systems could not be isolated. This left a final sample of 53 galaxies. Images were first processed through the standard ACS pipeline. _Multidrizzle_ was utilized for image combining and cosmic ray rejection. GC candidates were selected as matched-filter detections on 20 \(\times\) 20 pixel median-subtracted images. Using DAOPHOT II (Stetson 1993), aperture photometry was performed in a 5-pixel aperture and adjusted to a 10-pixel aperture using corrections of \(-0.09\) in \(g\) and \(-0.15\)	[]
	in \(z\). These are median corrections derived from bright objects in the five most luminous galaxies in the Virgo Cluster Survey: VCC 1226, VCC 1316, VCC 1978, VCC 881, and VCC 798. These 10-pixel magnitudes were then corrected to a nominal infinite aperture using values of \(-0.10\) in \(g\) and \(-0.12\) in \(z\) (Sirianni _et al._ 2005; this paper describes the photometric calibration of ACS). Finally, the magnitudes were transformed to the AB system using zeropoints from Sirianni _et al._ (26.068 and 24.862 for \(g\) and \(z\), respectively), and corrected for Galactic reddening using the maps of Schlegel Finkbeiner, & Davis (1998). Most GCs at the distance of Virgo are well-resolved in ACS imaging. Half-light radii (\(r_{h}\)) for GC candidates were measured on \(g\) images (since the \(g\) PSF is more centrally concentrated) using the _ishape_ routine (Larsen 1999). For each object, King models with fixed \(c=30\) (for \(c=r_{tidal}/r_{core}\)) and varying \(r_{h}\) were convolved with a distant-dependent empirical PSF derived from bright isolated stars in the images to find the best-fit \(r_{h}\). This \(c\) is typical of non core-collapsed GCs in the Milky Way (Trager, King, & Djorgovski 1995). We experimented with allowing \(c\) to vary, but it was poorly constrained for most GCs. However, the adopted \(c\) in _ishape_ has little effect on the derived \(r_{h}\) (Larsen 1999). To convert these measured sizes into physical units, galaxy distance estimates are required. We used those derived from surface brightness fluctuation measurements in the literature when possible: these were available from Tonry _et al._ (2001) for the bright galaxies and from Jerjen _et al._ (2004) for several dEs. For the remainder of the galaxies we used a fixed distance of 17 Mpc, which is the mean of the ellipticals in Tonry _et al._ Due to the depth of the images (\(z\gtrsim 25\)), some of the fields suffer significant contamination from foreground stars and especially background galaxies. Using the gEs and several of the more populous dEs as fiducials, we chose the following structural cuts to reduce interlopers: 0.55 \(<\) sharp \(<\) 0.9, \(-0.5<\) round \(<0.5\), and \(1<r_{h}\) (pc) \(<13\), where the sharp and round parameters are from DAOPHOT. A large upper limit for \(r_{h}\) is used since the size measurements skew systematically larger for fainter GCs. We further applied a color cut of \(0.5<g-z<2.0\) (\(>0.3\) mag to each blue and red of the limiting metallicities expected for old GCs; Jordan _et al._ 2004) and an error limit \(<\) 0.15 mag. In practice, this magnitude limit excluded most GCs within the innermost few arcsec of the brightest galaxies (whose GC systems are quite populous). Finally, we visually inspected all GC candidates, and excluded those which were obviously background galaxies. Our criteria are illustrated visually in Figure 1 for the bright dE VCC 1087, which displays a good mix of actual GCs and contaminants. These cuts remove nearly all foreground stars. However, compact galaxies (or compact star-forming regions within larger galaxies) with the appropriate colors can masquerade as GCs. In some images, clusters of galaxies are clearly visible. The increasing numbers of	[]
	background sources at \(z\gtrsim 23\), combined with the difficulty of accurate size measurements below this magnitude, makes efficient rejection of contaminants challenging. For gEs this is a minimal problem, due to the large number of GCs within the ACS field of view (hundreds to \(\sim 1700\) for M87). But with dEs and even low-luminosity Es, contaminants can represent a large fraction of the GC candidates. Due to these concerns, we chose only to use GCs brighter than \(z=23.5\) to study the colors and total numbers of GCs for the remainder of the paper. However, we used minimal cuts to study GC luminosity functions; this is described in more detail in SS4. Basic data about the 53 galaxies in our sample are given in Table 1, along with GC system information as discussed below. ## 3 Color Distributions ### Massive Ellipticals In Figures 2 and 3 we show the color-magnitude diagrams (CMDs) for the three most luminous galaxies in our sample: NGC 4472 (\(M_{B}=-21.9\)), M87 (\(M_{B}=-21.5\)), and NGC 4649 (\(M_{B}=-21.4\)). Figure 4 is a plot of magnitude vs. photometric error in \(g\) and \(z\) for M87. The CMDs in Figures 2 and 3 contain considerable structure only apparent because of the large number of GCs; we have chosen to discuss them in some detail. All three gEs clearly show the bimodality typical of massive galaxies, with blue and red peaks of \(g-z\sim 0.9\) and \(\sim 1.4\), respectively. This separation is twice as large as is typical of studies of GC systems in \(V-I\) (e.g., Larsen _et al._ 2001; Kundu & Whitmore 2001), due to the larger metallicity sensitivity of the \(g-z\) baseline. However, a new result is that the red peak is clearly broader than the blue peak; at bright magnitudes (\(z<22\)) there is little photometric error so this must be due to real color differences. To gauge the size of this effect, we fit a heteroscedastic normal mixture model to the M87 colors in the range \(21<z<22\). Subtracting a median photometric error of 0.02 mag in quadrature, we find \(\sigma_{blue}\sim 0.07\) and \(\sigma_{red}\sim 0.14\). These \(\sigma\) values may be overestimates because of the presence of contaminants in the tails of the color distributions, but provide first-order estimates for investigation. Given the lack of evidence for significant age differences among bright GCs in massive early-type galaxies (Strader _et al._ 2005), it is reasonable to attribute the dispersion in \(g-z\) entirely to metallicity. To convert these dispersions into metallicity, we must find a relation between [M/H] and \(g-z\). Jordan _et al._ (2004) used Bruzual & Charlot (2003) models to find a linear relationship in the range \(-2.3\leq\) [M/H] \(\leq+0.4\), however, the relation may be nonlinear for	[]
	low metallicities. We fit a quadratic relation for [M/H] and \(g-z\) using Maraston (2005) model predictions for \(g-z\) for four metallicities (\(-2.25,-1.35,-0.33,0\)) and Bruzual & Charlot (2003) predictions at five metallicities (\(-2.3,-1.7,-0.7,-0.4,0\)). Both sets of models assume a 13 Gyr stellar population and a Salpeter initial mass function. The resulting fit is: [M/H] \(=-8.088+9.081(g-z)-2.524(g-z)^{2}\). Using this relation, the blue and red GC dispersions correspond to \(1\sigma\) metallicity ranges of (\(-2.0\), \(-1.4\)) and (\(-0.7\), \(-0.1\)), respectively. Thus, despite the wider color range of the red GCs, their logarithmic metallicity range appears no wider than that of the blue GCs due to the nonlinear relationship between \(g-z\) and metallicity. A caveat is that this conclusion depends critically upon our assumed relation, which is likely to be most uncertain in the metal-poor regime where the stellar libraries of the models have few stars. In M87, there is a clear enhancement of GCs at \(z\sim 22.5\) with _intermediate_ colors, giving the CMD the appearance of a "cosmic H". This is illustrated more clearly in Figure 5, which shows color histograms of GCs in the regions \(22.2<z<22.7\) and \(22.8<z<23.3\), just below. Such a subpopulation of "H" clusters is also present, albeit less clearly and slightly fainter, in NGC 4472. Since this is near the turnover of the GCLF (with the largest number of GCs per magnitude bin), it is difficult to acertain whether the enhancement is present at all colors or only in a narrow range. However, this subpopulation appears normal in all other respects. Defining a fiducial sample as lying in the range: \(1.0<g-z<1.25\) and \(22.2<z<22.7\), the sizes and radial distribution of these GCs lie between those of the blue and red GCs, though perhaps more similar to the blue ones. Visually they are indistinguishable from GCs of similar luminosity. With current data we cannot say how common these "H" GCs are in massive ellipticals, though their presence in NGC 4472 suggests that in the Virgo Cluster the phenomenon is not limited to M87. Here only 34 GCs fall into the limits defined above (though this is unlikely to define a complete sample); if this subpopulation scales with GC system richness, 10 or fewer might be expected in other luminous galaxies, rendering their detection difficult. At \(z\sim 22.5\) spectroscopy of these GCs is feasible (though difficult), and could help establish whether their intermediate colors are due principally to metallicity or age, and whether they have kinematics distinct from the blue or red GCs. There may be a tail of these objects that extend to brighter magnitudes, but it is difficult to tell whether these are just outlying GCs in the normal blue or red subpopulations. Also of interest is a group of anomalously bright GCs (\(z\lesssim 20\)), which have a wide range in color (\(0.9<g-z<1.5\)) and in some galaxies are separated from the bulk of the GC system by 0.1 mag or more. In M87, these GCs are on average \(15\%\) larger (with mean \(r_{h}=2.7\) pc) than GCs in the rest of the system, and have median galactocentric distances \(\sim 10\%\) smaller (\(\sim 5\) kpc) than the GC system as a whole. The dispersions in these properties appear consistent with the GC system as a whole, but with few bright	[]
	clusters this is difficult to constrain. Some of these luminous GCs are likely in the tail of the normal blue and red subpopulations, but given the wide range in colors (including many with intermediate colors), small galactocentric radii, and the larger-than-average sizes, a portion may also be the stripped nuclei of dwarf galaxies--analogues of \(\omega\) Cen in the Galaxy (Majewski _et al._ 2000). The color distribution of dE nuclei in this sample (see below) peaks at \(g-z\sim 1.0-1.1\), consistent with the blue end of the intermediate-color objects. The surface brightness profiles of these objects resemble those of other GCs and do not have the exponential profiles seen in some ultra-compact dwarf galaxies (de Propris _et al._ 2005), though we note our size selection criterion for GCs would exclude some Virgo UCDs (Hasegan _et al._ 2005). Similar bright, intermediate-color GCs have also been found in the NGC 1399 (Dirsch _et al._ 2003), NGC 4636 (Dirsch, Schuberth, & Richtler 2005), and NGC 1407 (Cenarro _et al._ 2005); they appear to be a common feature of gEs. #### 3.1.1 The Blue Tilt A feature present in the CMDs of M87 and NGC 4649 is a _tilt_ of the color distribution of blue GCs, in the sense that the mean color of the blue GCs becomes redder with increasing luminosity. No such trend is apparent for the red GCs. A precise measure of this observation is challenging; due to the multiple subpopulations and "H" GCs, a direct linear fit is not viable. Instead, we divided the M87 GC candidates into four 0.5 mag bins in the range \(20<z<22\) and one 0.4 mag bin (\(22.8<z<23.2\), avoiding the "H" GCs). To each of these bins we fit a heteroscedastic normal mixture model, and then fit a weighted linear model to the resulting blue peaks. This model is \(g-z=-0.043\,z+1.848\); the slope is \(4\sigma\) significant. A fit to the corresponding red peaks is consistent with a slope of zero. These fits, as well as the binned values, are overplotted on the M87 CMD in Figure 2. Including a bin with the "H" GCs (\(22<z<22.8\)) gives a slope which is (unsurprisingly) slightly more shallow (\(-0.037\)) but still significant. NGC 4649 has fewer GCs than M87 and appears to have no "H" GCs, so for this galaxy we fit three 1.0 mag bins in the range \(20<z<23\). The resulting blue GC model is \(g-z=-0.040\,z+1.817\), which agrees very well with that of M87, and there is again no significant evidence for a nonzero red GC slope. The smoothness of the change argues against stochastic stellar population changes (e.g., horizontal branch stars, blue stragglers) as the cause of the trend. If due to age, its size--\(\sim 0.12-0.13\) mag in \(g-z\) over \(\sim 3\) mag in \(z\)--would require an unlikely age spread of \(\sim 7-8\) Gyr at low metallicity using Maraston (2005) models. If due to metallicity, the color-metallicity relation derived above indicates the trend corresponds to a mean slope of \(\sim 0.15-0.2\) dex/mag. For blue GCs in these galaxies, _metallicity correlates with mass_.	[]
	Interpretations of this surprising finding are discussed below; first we consider whether a bias in observation or analysis might be the cause. Given that the correlation extends over a large range in GC luminosity, and is not seen for red GCs, selection bias (choosing redder GCs at bright magnitudes and bluer GCs at faint magnitudes) seems unlikely to be a factor. There is no significant correlation between GC luminosity and galactocentric radius, ruling out a radial variation in any quantity as a cause (e.g., dust). Together these facts also suggest that a systematic photometric error cannot be blamed. To physically produce the observed correlation, either more massive GCs must have formed from more enriched gas, or individual GCs must have self-enriched. In the former picture, we could imagine blue GCs forming in proto-dwarf galaxies with varying metal enrichment. The essential problem is that there is no evidence that the GCLF varies strongly among dEs, as we would need the most metal-rich dEs to have few or no low-mass GCs to preserve the relation. GC self-enrichment might explain the correlation, as more massive GCs could retain a larger fraction of supernovae (SNe) ejecta. The self-enrichment of GCs has been studied in some detail as a possible origin to the chemical inhomogeneities observed among stars in Galactic GCs (e.g., Smith 1987). Early works (e.g., Dopita & Smith 1986) argued that only the most massive GCs could retain enough gas to self-enrich, but this depends critically on the assumed initial metal abundance of the proto-GC cloud and on the details of the cooling curve. Morgan & Lake (1989) found that a more accurate cooling curve reduced the critical mass to \(\geq 10^{5}M_{\odot}\) in a "supershell" model, as suggested by Cayrel (1986). In the model of Parmentier _et al._ (1999), proto-GC clouds are confined by a hot protogalactic medium, and this model in fact predicts an _inverse_ GC mass-metallicity relation, in which the most massive GCs are the most metal-poor (Parmentier & Gilmore 2001). Clearly a wide range of models exist, and it is possible that with the appropriate initial conditions and physical mechanism a self-enrichment model of this sort can be made to work. Another possibility is that the blue GCs formed inside individual dark matter (DM) halos. This scenario was first proposed by Peebles (1984), but fell into disfavor (Moore 1996) after studies of Galactic GCs found low mass-to-light ratios (Pryor _et al._ 1989) and tidal tails were observed around several GCs (e.g., Pal 5; Odenkirchen _et al._ 2003). Recently, Bromm & Clarke (2002) and Mashchenko & Sills (2005a,b) have used numerical simulations to argue that GCs with primordial DM halos could lose the bulk of the DM through either violent relaxation at early times or subsequent tidal stripping. If true, then a present-day lack of DM does not necessarily imply that GCs never had DM halos. It seems qualitatively plausible to produce the correlation in this context, but whether it could be sustained in	[]
	detail requires additional simulation. Any such model need also be compared to the rather stringent set of other observations of blue GCs (some of which are not usually considered), including the lack of GC mass-radius and metallicity-galactocentric radius relations and the presence of a _global_ correlation between the mean metallicity of blue GCs and parent galaxy mass. In addition, since the Galaxy itself (and perhaps NGC 4472) show no obvious blue GC mass-metallicity relationship, variations among galaxies are needed. It is also important to explain why the red GCs do not show such a relation. If the mass-metallicity relation was in terms of _absolute_ metallicity, then a small increase in metallicity (\(0.01-0.02Z_{\odot}\)) could be visible among the blue GCs but not among the red GCs. Even if no weak relation exits, one cannot rule out a metallicity-dependent process that results in a relation only for the blue GCs even if both subpopulations formed the same way. Many other physical properties of the blue and red GCs are similar enough (e.g., GC mass functions, sizes) that it may be challenging to invoke completely different formation mechanisms. Some of the similarities could be due to post-formation dynamical destruction of low-mass or diffuse GCs, which might act to erase initial variations in some GC system properties. No consensus exists in the literature on the effectiveness of GC destruction in shaping the present-day GC mass function (e.g., Vesperini 2001; Fall & Zhang 2001). ### Subpopulation Colors and Numbers The GC color distributions were modeled using the Bayesian program Nmix (Richardson & Green 1997), which fits normal heteroscedastic mixture models. The number of subpopulations is a free parameter (ranging to 10). For nearly all of the bright galaxies, two subpopulations were preferred; in no such galaxy was there a strong preference for a uni- or trimodal color distribution. Thus we report bimodal fits for these galaxies. While many of the dE color distributions visually appear bimodal, they generally had too few GCs to constrain the number of subpopulations with this algorithm. We adopted the following solution: we fit one peak to galaxies which only had GCs with \(g-z<1\); to the remaining galaxies we fit two peaks. A histogram of the dE GC colors (Figure 6) shows bimodality, which suggests that this approach is reasonable. Of course, some of the dEs have few GCs, so the peak locations may be quite uncertain. Linear relationships have previously been reported between parent galaxy luminosity and the mean colors (peak/mode of a Gaussian fit) of the red (Larsen _et al._ 2001; Forbes & Forte 2001) and blue (Strader, Forbes, & Brodie 2004; Larsen _et al._ 2001; Lotz, Miller & Ferguson 2004) subpopulations. Except for the massive Es, the individual subpopulations in this study have few GCs. Thus, errors on peak measurements are significant for most	[]
	galaxies. Nevertheless, Figure 7 shows that clear linear relationships are present for both the blue and red GCs over the full \(\sim 6-7\) mag range in parent galaxy luminosity. These weighted relations are \(g-z=-0.014\,M_{B}+0.642\) and \(g-z=-0.053\,M_{B}+0.225\) for the blue and red GCs, respectively. The plotted error bars are standard errors of the mean. The fits exclude the gE NGC 4365, whose anomalous GC system has been discussed in detail elsewhere (Larsen, Brodie, & Strader 2005, Brodie _et al._ 2005, Larsen _et al._ 2003, Puzia _et al._ 2002). For the red GCs there is a hint that the slope may flatten out for the faintest galaxies, but a runs test on the red residuals gave \(p=0.33\) (and \(p=0.45\) for the blue GCs), suggesting reasonable model fits. Many possible systematic errors could affect the faintest galaxies, e.g., the larger effects of contamination, and the uncertainty in the distances to individual galaxies, which could change their \(M_{B}\) by \(\sim 0.2-0.3\) mag. Thus, one cannot conclude that the GC color-galaxy luminosity relations are well-constrained at the faint end of our sample. However, they are consistent with extrapolations from brighter galaxies. Our results are also consistent with previous slope measurements: Larsen _et al._ (2001) and Strader _et al._ (2004) found that the \(V-I\) red:blue ratio of slopes is \(\sim 2\), while we find \(\sim 3.7\) in \(g-z\). This is consistent with \(g-z\propto 2\,(V-I)\), a rough initial estimate of color conversion (Brodie _et al._ 2005). There is at least one ongoing program to study Galactic GCs in the Sloan filter set which should improve this (and similar) conversions considerably. It does appear that the residuals of the blue and red peak values are correlated; this is probably unavoidable when fitting heteroscedastic mixture models to populations which are not well-separated. Since the red and blue GC subpopulations clearly have different dispersions where this can be tested in detail, fitting homoscedastic models does not make sense. We experimented with fitting two-component models with the variances fixed to the mean value for the brightest galaxies, for which the large number of GCs (at least partially) breaks the degeneracy between peak location and dispersion. The slopes of the resulting blue and red relations are similar to those found using the above approach: \(-0.012\) and \(-0.057\), respectively. However, the blue peak values for many of the galaxies appear to be artificially high--this may be because galaxies less massive than gEs have smaller intra-subpopulation metallicity spreads. Thus we have chosen to leave the original fits as our final values. The very existence of red GCs in faint galaxies with \(M_{B}\sim-15\) to \(-16\) is an interesting and somewhat unexpected result. In massive early-type galaxies and many spiral bulges, the number of red GCs normalized to spheroid luminosity is approximately constant (Forbes, Brodie, & Larsen 2001). This suggests that red GCs formed along with the spheroidal field stars at \(\sim\) constant efficiency. However, many properties of dEs (e.g., surface brightness profiles, M/L ratios, spatial/velocity distribution, stellar populations) suggest that their formation mechanism was different from massive Es (e.g., Kormendy 1985, though see Graham & Guzman 2003 for a different view). A continuity of red GC properties between Es and at	[]
	least some of the dEs in our sample could imply either that their formation mechanisms were more similar than expected or that red GCs are formed by a self-regulating, local process that can occur in a variety of contexts. The mixture modeling also returns the number of GCs in each subpopulation. In Figure 8 we plot the fraction of blue GCs vs. galaxy luminosity. There is a general trend (with a large scatter) for an increasing proportion of blue GCs with decreasing galaxy luminosity. The average fraction of blue GCs for the dEs is \(\sim 0.7\), with many galaxies having no red GCs. The fraction asymptotes to \(\sim 0.4-0.5\) for the gEs, but since these data cover only the central, more red-dominated part of their GC systems, the global fraction is likely higher (e.g., in a wide-field study of three gEs by Rhode & Zepf 2004, the blue GC fraction ranged from 0.6-0.7.). The GC systems of the dEs fall entirely within the ACS field of view (see discussion below), so their measured blue GC fractions are global. These results show clearly that the classic correlation between GC metallicity/color and galaxy luminosity (Brodie & Huchra 1991) is a combination of two effects: the decreasing ratio of blue to red GCs with increasing galaxy luminosity, and, more importantly, the GC color-galaxy luminosity relations which exist for _both_ subpopulations. There does not seem to be a single "primordial" GC color-galaxy luminosity relation, as assumed in the accretion scenario for GC bimodality (e.g., Cote, Marzke, & West 2002) ## 4 Luminosity Functions and Nuclei Many previous works have found (e.g., Harris 1991, Secker 1992, Kundu & Whitmore 2001, Larsen _et al._ 2001) that the GCLF in massive galaxies is well-fit by a Gaussian or \(t_{5}\) distribution with similar properties among well-studied galaxies: \(M_{V}\sim-7.4\), \(\sigma_{V}\sim 1.3\). However, the shape among dwarf galaxies is poorly known. Individual galaxies have too few GCs for a robust fit, and thus a composite GCLF of many dwarfs is necessary. Using ground-based imaging, Durrell _et al._ (1996) found that the turnover of the summed GCLF of 11 dEs in Virgo was \(\sim 0.4\pm 0.3\) mag fainter the M87 turnover. Lotz _et al._ (2001) presented HST/WFPC2 snapshot imaging of 51 dEs in Virgo and Fornax; the summed GCLF in Lotz _et al._ is not discussed, but a conference proceeding using the same data (Miller 2002) suggests \(M_{V}\sim-7.4\) and \(-7.3\) for Virgo and Fornax, respectively. We study the dE GCLF through comparison to gEs (VCC 1316-M87, VCC 1226-NGC 4472, and VCC 1978-NGC 4649) previously found to have "normal" GCLFs (Larsen _et al._ 2001). For dwarfs we constructed a summed GCLF of all 37 dEs (\(M_{B}<-18.2\) in our sample). Since the individual distance moduli of the dEs are unknown, this GCLF could	[]
	have additional scatter because of the range of galaxy distances; we find below that this appears to be quite a small effect. The individual GC systems of dEs are quite concentrated: most GCs are within \(30-40\arcsec\), beyond which the background contamination rises sharply (size measurements are unreliable at these faint magnitudes). Thus for the dEs we selected only those GCs within \(30\arcsec\) of the center of the galaxy. In the previous sections we used photometric and structural cuts to reduce the number of background galaxies and foreground stars interloping in our GC samples. However, these same cuts cannot be directly used to fit GCLFs, since these measurements become increasingly inaccurate for faint GCs (which may then be incorrectly removed). For example, applying only the color cut (\(0.5<g-z<2.0\)) vs. the full selection criteria to M87 changes the number of GCs with \(z<23.5\) by 16% (1444 vs. 1210) and shifts the peak of the GCLF by \(\sim 0.4\) mag, which is quite significant. We directly fit \(t_{5}\) distributions to the three gE GCLFs using using the code of Secker & Harris (1993), which uses maximum-likelihood fitting and incorporates photometric errors and incompleteness (Gaussian fits gave similar results within the errors; note that \(\sigma_{gauss}=1.29\,\sigma_{t5}\)). These GCLFs only have a color cut applied: \(0.5<g-z<2.0\) for M87 and NGC 4472 and \(0.85<g-z<2.0\) for NGC 4649 (since star-forming regions in the nearby spiral NGC 4647 strongly contaminate the blue part of the CMD). The results are given in Table 2. We fit both total populations and blue and red GCs separately, using the color cuts given in the table. Similar to what is commonly seen in \(V\)-band GCLFs (e.g., Larsen _et al._ 2001), the \(g\) turnovers are \(\sim 0.3-0.4\) mag brighter for the blue GCs than the red GCs. This is predicted for equal-mass/age turnovers separated by \(\sim 1\) dex in metallicity (Ashman, Conti & Zepf 1995). However, in \(z\) the blue GCs are only slightly brighter than the red GCs (the mean difference is negligible, but the blue GCs are brighter in M87 and NGC 4472 and fainter in NGC 4649, which may be biased slightly faint because of contamination). This difference between \(g\) and \(z\) is qualitatively consistent with stellar population models: Maraston (2005) models predict that equal-mass 13 Gyr GCs with [M/H] = \(-1.35\) and \(-0.33\) will have \(\Delta g=0.6\) and \(\Delta z=0.2\); the \(z\) difference is one-third of the \(g\) difference. This effect is probably due to the greater sensitivity of \(g\) to the turnoff region and the larger number of metal lines in the blue. The errors for the blue GCLF parameters may be slightly larger than formally stated because of the presence of the "H" GCs discussed previously. Variations in this feature and in the blue tilt among galaxies could represent a fundamental limitation to the accuracy to using _only_ the blue GCLF turnover as a standard candle (see, e.g., Kissler-Patig 2000). However, the total peak locations themselves are quite constant, with a range of only 0.03 mag in \(g\) and 0.05 mag in \(z\). At least for the well-populated old GC systems of gEs, the GCLF turnover appears to be an accurate distance indicator whose primary limitation is accurate photometry.	[]
	Unfortunately, we cannot directly fit the composite dE GCLF as for the gEs--it is far too contaminated. Instead, we must first correct for background objects. As noted above, the GC systems of the dEs are very centrally concentrated. Thus we can use the outer regions of the dE images as a fiducial background. We defined a background sample in the radial range \(1.1-1.25\arcmin\) (corresponding to 5.4-6.2 kpc at a distance of 17 Mpc), then subtracted the resulting \(z\) GCLF in 0.1 mag bins from the central \(30\arcsec\) sample with the appropriate areal correction factor. We confined the fit to candidates with \(0.7<g-z<1.25\) because of the small color range of the dEs. The resulting fit (performed as above) gave a turnover of \(M_{z}=-8.14\pm 0.14\), compared to the weighted mean of \(M_{z}=-8.19\) for the gEs. These are consistent. Since the \(z\) GCLFs are being used, the overall color differences between the GC systems should have little effect on the peak locations. The dispersion of the dEs (\(\sigma_{z}=0.74\)) is less than that of the gEs (\(\sigma_{z}=1.03\)); this is probably partially due to the smaller color range of the dEs. It also indicates that the range of galaxy distances is probably not significant, consistent with the projected appearance of many of the dEs near the Virgo cluster core (Binggeli _et al._ 1987). That the peak of the GCLF appears to be the same for both gEs and dEs is perhaps puzzling. Even if both galaxy types had similar primordial GCLFs, analytic calculations and numerical simulations of GC evolution in the tidal field of the dEs suggest that dynamical friction should destroy high-mass GCs within several Gyr (Oh & Lin 2000, Lotz _et al._ 2001). Tidal shocks (which tend to destroy low-mass GCs) are more significant in massive galaxies, so it is unlikely that the loss of high-mass GCs in dEs could be balanced by a corresponding amount of low-mass GC destruction to preserve a constant turnover. _ab initio_ variations in the GCLF between dEs and gEs are another possibility, but cannot be constrained at present. If we see no evidence in the dE GCLF turnover for dynamical friction, does this provide significant evidence against this popular scenario for forming dE nuclei? In order study the relationship between the nuclei and GCs of our sample galaxies, we performed photometry and size measurements on all of the dE nuclei. The procedures were identical to those described above for the GCs, except that aperture corrections were derived through analytic integration of the derived King profile, since many of the nuclei were large compared to typical GC sizes. The properties of the nuclei are listed in Table 3. In Figure 9 we plot parent galaxy \(M_{B}\) vs. \(M_{z}\) of the nucleus. Symbol size is proportional to nuclear size. The overplotted dashed line represents Monte Carlo simulations of nuclear formation through dynamical friction of GCs from Lotz _et al._(2001). These assume 5 Gyr of orbital decay (implicitly assuming the GCs are \(\sim 5\) Gyr old); we have converted their \(M_{V}\) to \(M_{z}\) using stellar populations models of Maraston (2005). Lotz _et al._ found that the expected	[]
	dynamical friction was inconsistent with observations, since the observed nuclei were fainter than predicted by their simulations. They argued that some additional process was needed to oppose dynamical friction (e.g., tidal torques). At first glance, this is supported by Figure 9--for galaxies in the luminosity range \(-15<M_{B}<-17\), there is a \(\sim 3\) mag range in nuclear luminosity. But most of the bright nuclei are quite small, while those much fainter than predicted from the simulations are generally larger (by factors of 2-4). In addition, the small nuclei tend to be redder than the large ones; some of the large faint nuclei are blue enough that they require some recent star formation. While not a one-to-one relation, it seems plausible that these luminosity and size distributions reflect two different channels of nuclear formation: small bright red nuclei are formed by dynamical friction, while large faint blue nuclei are formed by a dissipative process with little or no contribution from GCs. The red nuclei have typical colors of \(g-z\sim 1.1\), consistent with those of the red GCs in dEs. This implies that red GCs (rather than blue GCs) would need to be the primary "fuel" for nuclei built by dynamical friction, at least among the brigher dEs. The formation of blue nuclei could happen as the dE progenitor entered the cluster, or during tidal interactions that drive gas to the center of the galaxy (i.e., harassment, Mayer _et al._ 2001). Present simulations do not resolve the central parts of the galaxy with adequate resolution to estimate the size of a central high-density component, but presumably this will be possible in the future. It is unclear how to square the hypothesis that some nuclei are built by dynamical friction with the GCLF results from above. Perhaps the small \(\sigma_{z}\) of the composite dE (compared to the gEs) is additional evidence of GC destruction. Our background correction for the dE GCLF is potential source of systematic error in the turnover determination. Deeper ACS photometry for several of the brighter Virgo dEs could be quite helpful, since it could give both better rejection of background sources and more accurate SBF distances to the individual dEs. ## 5 Specific Frequencies For the Es in our sample, the ACS FOV covers only a fraction of their GC system, and no robust conclusions about their total number of GCs can be drawn without uncertain extrapolation. The GC systems of fainter Es and dEs are less extended and fall mostly or entirely within ACS pointings. As previously discussed, \(S_{N}\) is often used as a measure of the richness of a GC system. In fact, Harris & van den Bergh (1981) originally defined \(S_{N}\) in terms of the _bright_ GCs in a galaxy: the total number was taken as the number brighter than the turnover doubled. The justification for this procedure was that the faint end of the	[]
	GCLF was often ill-defined, with falling completeness and rising contamination. In addition, for a typical log-normal GCLF centered at \(M_{V}=-7.4\sim 3\times 10^{5}M_{\odot}\), 90% or more of the total GC system mass is in the bright half of the LF. Many of the dEs in this study have tens or fewer detected GCs, making accurate measurements of the shape of the LF impossible. Thus counting only those GCs brighter than the turnover is unwise; small variations in the peak of the GCLF among galaxies (either due to intrinsic differences or the unknown galaxy distances) could cause a large fraction of GC candidates to be included or excluded. As discussed in SS2, we chose \(z=23.5\) as the magnitude limit for our study. This is \(\sim 0.5\) mag beyond the turnover of the composite dE GCLF discussed in the previous section (\(M_{z}\sim-8.1\), \(\sigma_{z}\sim 0.7\)) at a nominal Virgo Cluster distance of 17 Mpc. For such galaxies our magnitude cutoff corresponds to \(\sim 75\)% completeness. Thus, to get total GC populations for the dEs, we divided the number of GCs brighter than \(z=23.5\) by this factor (for the few dEs with individual distance moduli, we integrated the LF to find the appropriate correction factor). Finally, as discussed in SS4, the photometric and structural cuts used to reduce contaminants also remove real GCs. Using M87 as a standard, 16.2% of real GCs with \(z<23.5\) were falsely removed; we add back this statistical correction to produce our final estimate of the total GC population. Due to the small radial extent of the dE GC systems (primarily within a projected galactocentric radius of \(\sim 30-40\arcsec\)) no correction for spatial coverage is applied. We calculated \(B\)-band \(S_{N}\) using the absolute magnitudes in Table 1. Unless otherwise noted, all \(S_{N}\) refer to \(B\)-band values. These \(S_{N}\) can be converted to the standard \(V\)-band \(S_{N}\) by dividing our values by a factor \(10^{0.4(B-V)}\); a typical dE has \(B-V=0.8\), which corresponds to a conversation factor \(\sim 2.1\). \(S_{N}\) is plotted against \(M_{B}\) in Figure 10. In this figure the dE,N galaxies are filled circles and the dE,noN galaxies are empty circles. The symbol size is proportional to the fraction of blue GCs. Miller _et al._ (1998) found that dE,noNs had, on average, lower \(S_{N}\) than dE,Ns, and that for both classes of dE there was an inverse correlation between \(S_{N}\) and galaxy luminosity. We do not see any substantial difference between dE,noN and dE,N galaxies. There does appear to be a weak correlation between \(S_{N}\) and \(M_{B}\) in our data (with quite large \(S_{N}\) for some of the faintest galaxies), but there is a large spread in \(S_{N}\) for most luminosities. The \(S_{N}\) does not appear to depend strongly on the fraction of blue GCs, so differing mixes of subpopulations cannot be responsible for the \(S_{N}\) variations. Thus the observed values correspond to total blue GC subpopulations which vary by a factor of up to \(\sim 10\) at a given luminosity. There appear to be two sets of galaxies: one group with \(S_{N}\sim 2\), the other group ranging from \(S_{N}\sim 5-20\) and a slight enhancement at \(S_{N}\sim 10\). We only have one galaxy in common with Miller _et al._ (VCC 9), but our \(S_{N}\) for this single galaxy is consistent with theirs within the	[]
	errors. We note again that background contamination remains as a systematic uncertainty for total GC populations. It is tempting to argue that these groups represent different formation channels for dEs, e.g., the fading/quenching of dIrrs, harassment of low-mass spirals, or simply a continuation of the E sequence to fainter magnitudes. In some scenarios for blue GC formation (e.g., Strader _et al._ 2005; Rhode _et al._ 2005), galaxies with roughly similar masses in a given environment would be expected to have similar numbers of blue GCs per unit mass of their DM halos. Variations in \(S_{N}\) at fixed luminosity would then be due to differences in the efficiency of converting baryons to stars--either early in the galaxy's history (due to feedback) or later, due to galaxy transformation processes as described above. Two pieces of observational evidence should help to differentiate among the possibilities. First, GC kinematics can be used directly to constrain dE mass-to-light ratios (e.g., Beasley _et al._ 2005). It will be interesting to see whether GC kinematics are connected to the apparent dichotomy of rotating vs. non-rotating dEs (Pedraz _et al._ 2002, Geha _et al._ 2002, 2003, van Zee _et al._ 2004). Second, more detailed stellar population studies of dEs are needed, especially any method which (unlike integrated light spectroscopy) can break the burst strength-age degeneracy. A rather ad hoc alternative is that the efficiency of blue GC formation varies substantially among Virgo dEs, but this cannot be constrained at present. It is interesting that the fraction of blue GCs appears unrelated both to the nucleation of the dE and the \(S_{N}\) variations; this suggests that whatever process leads to red GC formation in dEs is independent of these other factors. ## 6 Summary We have presented a detailed analysis of the GC color and luminosity distributions of several gEs and of the colors, specific frequencies, luminosity functions, and nuclei of a large sample of dEs. The most interesting feature in the gEs M87 and NGC 4649 is a correlation between mass and metallicity for individual blue GCs. Self-enrichment is a plausible interpretation of this observation, and could suggest that these GCs once possessed dark matter halos (which may have been subsequently stripped). Among the other new features observed are very luminous (\(z\gtrsim 20\)) GCs with intermediate to red colors. These objects are slightly larger than typical GCs and may be remnants of stripped dwarf galaxies. Next, we see an intermediate-color group of GCs which lies near the GCLF turnover and in the gap between the blue and red GCs. Also, the color spread among the red GCs is nearly twice that of the blue GCs, but because the relation between \(g-z\) and metallicity appears to be nonlinear, the \(1\sigma\) dispersion in metallicity (\(\sim 0.6\) dex) may be the same for	[]
	UCO/Lick Observatory, University of California, Santa Cruz, CA 95064 strader@ucolick.org, brodie@ucolick.org, lees@ucolick.org, mbeasley@ucolick.org ###### Abstract We have analyzed archival _Hubble Space Telescope_/Advanced Camera for Surveys images in \(g\) and \(z\) of the globular cluster (GC) systems of 53 ellipticals in the Virgo Cluster, spanning massive galaxies to dwarf ellipticals (dEs). Several new results emerged: (i) In the giant ellipticals (gEs) M87 and NGC 4649, there is a correlation between luminosity and color for _individual_ metal-poor GCs, such that more massive GCs are more metal-rich. A plausible interpretation of this result is self-enrichment, and may suggest that these GCs once possessed dark matter halos. (ii) In some gEs (most notably M87), there is an "interloping" subpopulation of GCs with intermediate colors (1.0 \(<g-z<\) 1.25) and a narrow magnitude range (0.5 mag) near the turnover of the GC luminosity function. These GCs look otherwise identical to the classic metal-poor and metal-rich GC subpopulations. (iii) The dispersion in color is nearly twice as large for the metal-rich GCs than the metal-poor GCs. However, there is evidence for a nonlinear relation between \(g-z\) and metallicity, and the dispersion in metallicity may be the same for both subpopulations. (iv) Very luminous, intermediate-color GCs are common in gEs. These objects may be remnants of many stripped dwarfs, analogues of \(\omega\) Cen in the Galaxy. (v) There is a continuity of GC system colors from gEs to some dEs: in particular, many dEs have metal-rich GC subpopulations. We also confirm the GC color-galaxy luminosity relations found previously for both metal-poor and metal-rich GC subpopulations. (vi) There are large differences in GC specific frequency among dEs, independent of the presence of a nucleus and the fraction of metal-rich GCs. Over \(-15<M_{B}<-18\), there is little correlation between specific frequency and \(M_{B}\) (in contrast to previous studies). But we do find evidence for two separate \(S_{N}\) classes of dEs: those with \(B\)-band \(S_{N}\sim 2\), and dEs with populous GC systems that have \(S_{N}\) ranging from \(\sim 5-20\) with median \(S_{N}\sim 10\). Together, these points suggest multiple formation channels for dEs in the Virgo Cluster. (vii) The peak of the GC luminosity function (GCLF) is the same for both gEs and dEs. This is contrary to expectations of dynamical friction on massive GCs, unless the primordial GCLF varies between gEs and dEs. Among gEs the GCLF turnover varies by a surprising small 0.05 mag, an encouraging result for its use as an accurate standard candle. (viii) dE,Ns appear bimodal in their nuclear properties: there are small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. The role of dynamical evolution in shaping the present-day properties of dE GC systems and their nuclei remains ambiguous. globular clusters: general -- galaxies: star clusters -- galaxies: formation ## 1 Introduction It is increasingly apparent that globular clusters (GCs) offer important constraints on the star formation and assembly histories of galaxies. Recent spectroscopic studies of GCs in massive early-type galaxies (e.g., Strader _et al._ 2005) indicate that the bulk of star formation occurred at relatively high redshift (\(z\gtrsim 2\)) in high density environments (as environmental density and galaxy mass decrease, the fraction of younger GCs may increase; see Puzia _et al._ 2005). These findings allow the age-metallicity degeneracy to be broken and lead to the conclusion that the bimodal color distributions seen in most nearby luminous galaxies are due primarily to two old GC subpopulations: metal-poor (blue) and metal-rich (red). The metallicities of these peaks correlate with host galaxy luminosity (Larsen _et al._ 2001, Strader, Brodie, & Forbes 2004; see also Lotz, Miller, & Ferguson 2004 for dwarfs). Most recent photometric studies of GC systems in ellipticals have used the Wide Field and Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). Compared to ground-based imaging, this strategy gains photometric accuracy and minimizes contamination at the expense of small spatial coverage. Among the larger HST studies of early-type galaxies utilizing deep imaging are Larsen _et al._ (2001) and Kundu & Whitmore (2001). These studies found bimodality in many of their sample galaxies (extending down to low-luminosity ellipticals) and a nearly uniform log-normal GC luminosity function (GCLF) with a peak at \(M_{V}\sim-7.4\). However, the GC systems of dwarf ellipticals (dEs) are more poorly understood. The large HST surveys to date (primarily of Virgo and Fornax) have been limited to relatively shallow snapshot imaging; this precluded the study of color and luminosity distributions in detail. Among the suggestions of this initial work are a correlation of increasing specific frequency (\(S_{N}\)) with decreasing galaxy luminosity, a dichotomy in the GC systems of nucleated and non-nucleated dEs (dE,N and dE,noN, respectively), and the difficulty of making dE nuclei as observed through dynamical friction of GCs (Miller _et al._ 1998, Lotz _et al._ 2001). Data from the Advanced Camera for Surveys (ACS) Virgo Cluster Survey (HST GO 9401, P. I. Cote) offers an important step forward in understanding the detailed properties of the GC systems of ellipticals over a wide range in galaxy mass. The use of F475W and F850LP filters (henceforth called \(g\) and \(z\) for convenience, though the filters do not precisely match the Sloan ones) allows a much wider spectral baseline for metallicity separation than \(V\) and \(I\). Though only a single orbit is used per galaxy, the increased sensitivity of ACS (compared to WFPC2) allows one to reach \(\sim 3\) mag beyond the turnover of the GCLF, encompassing most of the GCs in a given galaxy. More accurate photometry for the brighter GCs is also possible. Finally, the field of view of ACS is twice that of WFPC2. Together, these attributes allow a study of the color and luminosity distribution of GCs in a large sample of galaxies in much more detail than previously possible. In what follows, elliptical (unabbreviated) refers to all galaxies in our sample and denotes no specific luminosity. The three brightest galaxies are described as giant ellipticals (gEs); these have \(M_{B}\leq-21.4\). Galaxies of intermediate or high luminosity are called Es. Faint galaxies with exponential surface brightness profiles are dEs; the transition from E to dE occurs traditionally at \(M_{B}\sim-18\) (Kormendy 1985). We include two galaxies with \(M_{B}=-18.1\) (VCC 1422 and VCC 1261) under this heading, since these galaxies have nuclei similar to those commonly found among dEs. The E/dE classifications have been taken from the literature and we do not perform independent surface photometry in this paper (though we note the increasing debate in the literature about whether this dichotomy is real, e.g., Graham & Guzman 2003). We have updated the nucleation status of a dE if appropriate; the vast majority of the dEs in our study have nuclei. ## 2 Data Reduction and Analysis All data were taken as part of the ACS Virgo Cluster Survey (Cote _et al._ 2004); this survey includes both ellipticals and S0s. We used all galaxies classified as ellipticals, excepting a few dwarfs quite close to luminous Es whose GC systems could not be isolated. This left a final sample of 53 galaxies. Images were first processed through the standard ACS pipeline. _Multidrizzle_ was utilized for image combining and cosmic ray rejection. GC candidates were selected as matched-filter detections on 20 \(\times\) 20 pixel median-subtracted images. Using DAOPHOT II (Stetson 1993), aperture photometry was performed in a 5-pixel aperture and adjusted to a 10-pixel aperture using corrections of \(-0.09\) in \(g\) and \(-0.15\) in \(z\). These are median corrections derived from bright objects in the five most luminous galaxies in the Virgo Cluster Survey: VCC 1226, VCC 1316, VCC 1978, VCC 881, and VCC 798. These 10-pixel magnitudes were then corrected to a nominal infinite aperture using values of \(-0.10\) in \(g\) and \(-0.12\) in \(z\) (Sirianni _et al._ 2005; this paper describes the photometric calibration of ACS). Finally, the magnitudes were transformed to the AB system using zeropoints from Sirianni _et al._ (26.068 and 24.862 for \(g\) and \(z\), respectively), and corrected for Galactic reddening using the maps of Schlegel Finkbeiner, & Davis (1998). Most GCs at the distance of Virgo are well-resolved in ACS imaging. Half-light radii (\(r_{h}\)) for GC candidates were measured on \(g\) images (since the \(g\) PSF is more centrally concentrated) using the _ishape_ routine (Larsen 1999). For each object, King models with fixed \(c=30\) (for \(c=r_{tidal}/r_{core}\)) and varying \(r_{h}\) were convolved with a distant-dependent empirical PSF derived from bright isolated stars in the images to find the best-fit \(r_{h}\). This \(c\) is typical of non core-collapsed GCs in the Milky Way (Trager, King, & Djorgovski 1995). We experimented with allowing \(c\) to vary, but it was poorly constrained for most GCs. However, the adopted \(c\) in _ishape_ has little effect on the derived \(r_{h}\) (Larsen 1999). To convert these measured sizes into physical units, galaxy distance estimates are required. We used those derived from surface brightness fluctuation measurements in the literature when possible: these were available from Tonry _et al._ (2001) for the bright galaxies and from Jerjen _et al._ (2004) for several dEs. For the remainder of the galaxies we used a fixed distance of 17 Mpc, which is the mean of the ellipticals in Tonry _et al._ Due to the depth of the images (\(z\gtrsim 25\)), some of the fields suffer significant contamination from foreground stars and especially background galaxies. Using the gEs and several of the more populous dEs as fiducials, we chose the following structural cuts to reduce interlopers: 0.55 \(<\) sharp \(<\) 0.9, \(-0.5<\) round \(<0.5\), and \(1<r_{h}\) (pc) \(<13\), where the sharp and round parameters are from DAOPHOT. A large upper limit for \(r_{h}\) is used since the size measurements skew systematically larger for fainter GCs. We further applied a color cut of \(0.5<g-z<2.0\) (\(>0.3\) mag to each blue and red of the limiting metallicities expected for old GCs; Jordan _et al._ 2004) and an error limit \(<\) 0.15 mag. In practice, this magnitude limit excluded most GCs within the innermost few arcsec of the brightest galaxies (whose GC systems are quite populous). Finally, we visually inspected all GC candidates, and excluded those which were obviously background galaxies. Our criteria are illustrated visually in Figure 1 for the bright dE VCC 1087, which displays a good mix of actual GCs and contaminants. These cuts remove nearly all foreground stars. However, compact galaxies (or compact star-forming regions within larger galaxies) with the appropriate colors can masquerade as GCs. In some images, clusters of galaxies are clearly visible. The increasing numbers of background sources at \(z\gtrsim 23\), combined with the difficulty of accurate size measurements below this magnitude, makes efficient rejection of contaminants challenging. For gEs this is a minimal problem, due to the large number of GCs within the ACS field of view (hundreds to \(\sim 1700\) for M87). But with dEs and even low-luminosity Es, contaminants can represent a large fraction of the GC candidates. Due to these concerns, we chose only to use GCs brighter than \(z=23.5\) to study the colors and total numbers of GCs for the remainder of the paper. However, we used minimal cuts to study GC luminosity functions; this is described in more detail in SS4. Basic data about the 53 galaxies in our sample are given in Table 1, along with GC system information as discussed below. ## 3 Color Distributions ### Massive Ellipticals In Figures 2 and 3 we show the color-magnitude diagrams (CMDs) for the three most luminous galaxies in our sample: NGC 4472 (\(M_{B}=-21.9\)), M87 (\(M_{B}=-21.5\)), and NGC 4649 (\(M_{B}=-21.4\)). Figure 4 is a plot of magnitude vs. photometric error in \(g\) and \(z\) for M87. The CMDs in Figures 2 and 3 contain considerable structure only apparent because of the large number of GCs; we have chosen to discuss them in some detail. All three gEs clearly show the bimodality typical of massive galaxies, with blue and red peaks of \(g-z\sim 0.9\) and \(\sim 1.4\), respectively. This separation is twice as large as is typical of studies of GC systems in \(V-I\) (e.g., Larsen _et al._ 2001; Kundu & Whitmore 2001), due to the larger metallicity sensitivity of the \(g-z\) baseline. However, a new result is that the red peak is clearly broader than the blue peak; at bright magnitudes (\(z<22\)) there is little photometric error so this must be due to real color differences. To gauge the size of this effect, we fit a heteroscedastic normal mixture model to the M87 colors in the range \(21<z<22\). Subtracting a median photometric error of 0.02 mag in quadrature, we find \(\sigma_{blue}\sim 0.07\) and \(\sigma_{red}\sim 0.14\). These \(\sigma\) values may be overestimates because of the presence of contaminants in the tails of the color distributions, but provide first-order estimates for investigation. Given the lack of evidence for significant age differences among bright GCs in massive early-type galaxies (Strader _et al._ 2005), it is reasonable to attribute the dispersion in \(g-z\) entirely to metallicity. To convert these dispersions into metallicity, we must find a relation between [M/H] and \(g-z\). Jordan _et al._ (2004) used Bruzual & Charlot (2003) models to find a linear relationship in the range \(-2.3\leq\) [M/H] \(\leq+0.4\), however, the relation may be nonlinear for low metallicities. We fit a quadratic relation for [M/H] and \(g-z\) using Maraston (2005) model predictions for \(g-z\) for four metallicities (\(-2.25,-1.35,-0.33,0\)) and Bruzual & Charlot (2003) predictions at five metallicities (\(-2.3,-1.7,-0.7,-0.4,0\)). Both sets of models assume a 13 Gyr stellar population and a Salpeter initial mass function. The resulting fit is: [M/H] \(=-8.088+9.081(g-z)-2.524(g-z)^{2}\). Using this relation, the blue and red GC dispersions correspond to \(1\sigma\) metallicity ranges of (\(-2.0\), \(-1.4\)) and (\(-0.7\), \(-0.1\)), respectively. Thus, despite the wider color range of the red GCs, their logarithmic metallicity range appears no wider than that of the blue GCs due to the nonlinear relationship between \(g-z\) and metallicity. A caveat is that this conclusion depends critically upon our assumed relation, which is likely to be most uncertain in the metal-poor regime where the stellar libraries of the models have few stars. In M87, there is a clear enhancement of GCs at \(z\sim 22.5\) with _intermediate_ colors, giving the CMD the appearance of a "cosmic H". This is illustrated more clearly in Figure 5, which shows color histograms of GCs in the regions \(22.2<z<22.7\) and \(22.8<z<23.3\), just below. Such a subpopulation of "H" clusters is also present, albeit less clearly and slightly fainter, in NGC 4472. Since this is near the turnover of the GCLF (with the largest number of GCs per magnitude bin), it is difficult to acertain whether the enhancement is present at all colors or only in a narrow range. However, this subpopulation appears normal in all other respects. Defining a fiducial sample as lying in the range: \(1.0<g-z<1.25\) and \(22.2<z<22.7\), the sizes and radial distribution of these GCs lie between those of the blue and red GCs, though perhaps more similar to the blue ones. Visually they are indistinguishable from GCs of similar luminosity. With current data we cannot say how common these "H" GCs are in massive ellipticals, though their presence in NGC 4472 suggests that in the Virgo Cluster the phenomenon is not limited to M87. Here only 34 GCs fall into the limits defined above (though this is unlikely to define a complete sample); if this subpopulation scales with GC system richness, 10 or fewer might be expected in other luminous galaxies, rendering their detection difficult. At \(z\sim 22.5\) spectroscopy of these GCs is feasible (though difficult), and could help establish whether their intermediate colors are due principally to metallicity or age, and whether they have kinematics distinct from the blue or red GCs. There may be a tail of these objects that extend to brighter magnitudes, but it is difficult to tell whether these are just outlying GCs in the normal blue or red subpopulations. Also of interest is a group of anomalously bright GCs (\(z\lesssim 20\)), which have a wide range in color (\(0.9<g-z<1.5\)) and in some galaxies are separated from the bulk of the GC system by 0.1 mag or more. In M87, these GCs are on average \(15\%\) larger (with mean \(r_{h}=2.7\) pc) than GCs in the rest of the system, and have median galactocentric distances \(\sim 10\%\) smaller (\(\sim 5\) kpc) than the GC system as a whole. The dispersions in these properties appear consistent with the GC system as a whole, but with few bright clusters this is difficult to constrain. Some of these luminous GCs are likely in the tail of the normal blue and red subpopulations, but given the wide range in colors (including many with intermediate colors), small galactocentric radii, and the larger-than-average sizes, a portion may also be the stripped nuclei of dwarf galaxies--analogues of \(\omega\) Cen in the Galaxy (Majewski _et al._ 2000). The color distribution of dE nuclei in this sample (see below) peaks at \(g-z\sim 1.0-1.1\), consistent with the blue end of the intermediate-color objects. The surface brightness profiles of these objects resemble those of other GCs and do not have the exponential profiles seen in some ultra-compact dwarf galaxies (de Propris _et al._ 2005), though we note our size selection criterion for GCs would exclude some Virgo UCDs (Hasegan _et al._ 2005). Similar bright, intermediate-color GCs have also been found in the NGC 1399 (Dirsch _et al._ 2003), NGC 4636 (Dirsch, Schuberth, & Richtler 2005), and NGC 1407 (Cenarro _et al._ 2005); they appear to be a common feature of gEs. #### 3.1.1 The Blue Tilt A feature present in the CMDs of M87 and NGC 4649 is a _tilt_ of the color distribution of blue GCs, in the sense that the mean color of the blue GCs becomes redder with increasing luminosity. No such trend is apparent for the red GCs. A precise measure of this observation is challenging; due to the multiple subpopulations and "H" GCs, a direct linear fit is not viable. Instead, we divided the M87 GC candidates into four 0.5 mag bins in the range \(20<z<22\) and one 0.4 mag bin (\(22.8<z<23.2\), avoiding the "H" GCs). To each of these bins we fit a heteroscedastic normal mixture model, and then fit a weighted linear model to the resulting blue peaks. This model is \(g-z=-0.043\,z+1.848\); the slope is \(4\sigma\) significant. A fit to the corresponding red peaks is consistent with a slope of zero. These fits, as well as the binned values, are overplotted on the M87 CMD in Figure 2. Including a bin with the "H" GCs (\(22<z<22.8\)) gives a slope which is (unsurprisingly) slightly more shallow (\(-0.037\)) but still significant. NGC 4649 has fewer GCs than M87 and appears to have no "H" GCs, so for this galaxy we fit three 1.0 mag bins in the range \(20<z<23\). The resulting blue GC model is \(g-z=-0.040\,z+1.817\), which agrees very well with that of M87, and there is again no significant evidence for a nonzero red GC slope. The smoothness of the change argues against stochastic stellar population changes (e.g., horizontal branch stars, blue stragglers) as the cause of the trend. If due to age, its size--\(\sim 0.12-0.13\) mag in \(g-z\) over \(\sim 3\) mag in \(z\)--would require an unlikely age spread of \(\sim 7-8\) Gyr at low metallicity using Maraston (2005) models. If due to metallicity, the color-metallicity relation derived above indicates the trend corresponds to a mean slope of \(\sim 0.15-0.2\) dex/mag. For blue GCs in these galaxies, _metallicity correlates with mass_. Interpretations of this surprising finding are discussed below; first we consider whether a bias in observation or analysis might be the cause. Given that the correlation extends over a large range in GC luminosity, and is not seen for red GCs, selection bias (choosing redder GCs at bright magnitudes and bluer GCs at faint magnitudes) seems unlikely to be a factor. There is no significant correlation between GC luminosity and galactocentric radius, ruling out a radial variation in any quantity as a cause (e.g., dust). Together these facts also suggest that a systematic photometric error cannot be blamed. To physically produce the observed correlation, either more massive GCs must have formed from more enriched gas, or individual GCs must have self-enriched. In the former picture, we could imagine blue GCs forming in proto-dwarf galaxies with varying metal enrichment. The essential problem is that there is no evidence that the GCLF varies strongly among dEs, as we would need the most metal-rich dEs to have few or no low-mass GCs to preserve the relation. GC self-enrichment might explain the correlation, as more massive GCs could retain a larger fraction of supernovae (SNe) ejecta. The self-enrichment of GCs has been studied in some detail as a possible origin to the chemical inhomogeneities observed among stars in Galactic GCs (e.g., Smith 1987). Early works (e.g., Dopita & Smith 1986) argued that only the most massive GCs could retain enough gas to self-enrich, but this depends critically on the assumed initial metal abundance of the proto-GC cloud and on the details of the cooling curve. Morgan & Lake (1989) found that a more accurate cooling curve reduced the critical mass to \(\geq 10^{5}M_{\odot}\) in a "supershell" model, as suggested by Cayrel (1986). In the model of Parmentier _et al._ (1999), proto-GC clouds are confined by a hot protogalactic medium, and this model in fact predicts an _inverse_ GC mass-metallicity relation, in which the most massive GCs are the most metal-poor (Parmentier & Gilmore 2001). Clearly a wide range of models exist, and it is possible that with the appropriate initial conditions and physical mechanism a self-enrichment model of this sort can be made to work. Another possibility is that the blue GCs formed inside individual dark matter (DM) halos. This scenario was first proposed by Peebles (1984), but fell into disfavor (Moore 1996) after studies of Galactic GCs found low mass-to-light ratios (Pryor _et al._ 1989) and tidal tails were observed around several GCs (e.g., Pal 5; Odenkirchen _et al._ 2003). Recently, Bromm & Clarke (2002) and Mashchenko & Sills (2005a,b) have used numerical simulations to argue that GCs with primordial DM halos could lose the bulk of the DM through either violent relaxation at early times or subsequent tidal stripping. If true, then a present-day lack of DM does not necessarily imply that GCs never had DM halos. It seems qualitatively plausible to produce the correlation in this context, but whether it could be sustained in detail requires additional simulation. Any such model need also be compared to the rather stringent set of other observations of blue GCs (some of which are not usually considered), including the lack of GC mass-radius and metallicity-galactocentric radius relations and the presence of a _global_ correlation between the mean metallicity of blue GCs and parent galaxy mass. In addition, since the Galaxy itself (and perhaps NGC 4472) show no obvious blue GC mass-metallicity relationship, variations among galaxies are needed. It is also important to explain why the red GCs do not show such a relation. If the mass-metallicity relation was in terms of _absolute_ metallicity, then a small increase in metallicity (\(0.01-0.02Z_{\odot}\)) could be visible among the blue GCs but not among the red GCs. Even if no weak relation exits, one cannot rule out a metallicity-dependent process that results in a relation only for the blue GCs even if both subpopulations formed the same way. Many other physical properties of the blue and red GCs are similar enough (e.g., GC mass functions, sizes) that it may be challenging to invoke completely different formation mechanisms. Some of the similarities could be due to post-formation dynamical destruction of low-mass or diffuse GCs, which might act to erase initial variations in some GC system properties. No consensus exists in the literature on the effectiveness of GC destruction in shaping the present-day GC mass function (e.g., Vesperini 2001; Fall & Zhang 2001). ### Subpopulation Colors and Numbers The GC color distributions were modeled using the Bayesian program Nmix (Richardson & Green 1997), which fits normal heteroscedastic mixture models. The number of subpopulations is a free parameter (ranging to 10). For nearly all of the bright galaxies, two subpopulations were preferred; in no such galaxy was there a strong preference for a uni- or trimodal color distribution. Thus we report bimodal fits for these galaxies. While many of the dE color distributions visually appear bimodal, they generally had too few GCs to constrain the number of subpopulations with this algorithm. We adopted the following solution: we fit one peak to galaxies which only had GCs with \(g-z<1\); to the remaining galaxies we fit two peaks. A histogram of the dE GC colors (Figure 6) shows bimodality, which suggests that this approach is reasonable. Of course, some of the dEs have few GCs, so the peak locations may be quite uncertain. Linear relationships have previously been reported between parent galaxy luminosity and the mean colors (peak/mode of a Gaussian fit) of the red (Larsen _et al._ 2001; Forbes & Forte 2001) and blue (Strader, Forbes, & Brodie 2004; Larsen _et al._ 2001; Lotz, Miller & Ferguson 2004) subpopulations. Except for the massive Es, the individual subpopulations in this study have few GCs. Thus, errors on peak measurements are significant for most galaxies. Nevertheless, Figure 7 shows that clear linear relationships are present for both the blue and red GCs over the full \(\sim 6-7\) mag range in parent galaxy luminosity. These weighted relations are \(g-z=-0.014\,M_{B}+0.642\) and \(g-z=-0.053\,M_{B}+0.225\) for the blue and red GCs, respectively. The plotted error bars are standard errors of the mean. The fits exclude the gE NGC 4365, whose anomalous GC system has been discussed in detail elsewhere (Larsen, Brodie, & Strader 2005, Brodie _et al._ 2005, Larsen _et al._ 2003, Puzia _et al._ 2002). For the red GCs there is a hint that the slope may flatten out for the faintest galaxies, but a runs test on the red residuals gave \(p=0.33\) (and \(p=0.45\) for the blue GCs), suggesting reasonable model fits. Many possible systematic errors could affect the faintest galaxies, e.g., the larger effects of contamination, and the uncertainty in the distances to individual galaxies, which could change their \(M_{B}\) by \(\sim 0.2-0.3\) mag. Thus, one cannot conclude that the GC color-galaxy luminosity relations are well-constrained at the faint end of our sample. However, they are consistent with extrapolations from brighter galaxies. Our results are also consistent with previous slope measurements: Larsen _et al._ (2001) and Strader _et al._ (2004) found that the \(V-I\) red:blue ratio of slopes is \(\sim 2\), while we find \(\sim 3.7\) in \(g-z\). This is consistent with \(g-z\propto 2\,(V-I)\), a rough initial estimate of color conversion (Brodie _et al._ 2005). There is at least one ongoing program to study Galactic GCs in the Sloan filter set which should improve this (and similar) conversions considerably. It does appear that the residuals of the blue and red peak values are correlated; this is probably unavoidable when fitting heteroscedastic mixture models to populations which are not well-separated. Since the red and blue GC subpopulations clearly have different dispersions where this can be tested in detail, fitting homoscedastic models does not make sense. We experimented with fitting two-component models with the variances fixed to the mean value for the brightest galaxies, for which the large number of GCs (at least partially) breaks the degeneracy between peak location and dispersion. The slopes of the resulting blue and red relations are similar to those found using the above approach: \(-0.012\) and \(-0.057\), respectively. However, the blue peak values for many of the galaxies appear to be artificially high--this may be because galaxies less massive than gEs have smaller intra-subpopulation metallicity spreads. Thus we have chosen to leave the original fits as our final values. The very existence of red GCs in faint galaxies with \(M_{B}\sim-15\) to \(-16\) is an interesting and somewhat unexpected result. In massive early-type galaxies and many spiral bulges, the number of red GCs normalized to spheroid luminosity is approximately constant (Forbes, Brodie, & Larsen 2001). This suggests that red GCs formed along with the spheroidal field stars at \(\sim\) constant efficiency. However, many properties of dEs (e.g., surface brightness profiles, M/L ratios, spatial/velocity distribution, stellar populations) suggest that their formation mechanism was different from massive Es (e.g., Kormendy 1985, though see Graham & Guzman 2003 for a different view). A continuity of red GC properties between Es and at least some of the dEs in our sample could imply either that their formation mechanisms were more similar than expected or that red GCs are formed by a self-regulating, local process that can occur in a variety of contexts. The mixture modeling also returns the number of GCs in each subpopulation. In Figure 8 we plot the fraction of blue GCs vs. galaxy luminosity. There is a general trend (with a large scatter) for an increasing proportion of blue GCs with decreasing galaxy luminosity. The average fraction of blue GCs for the dEs is \(\sim 0.7\), with many galaxies having no red GCs. The fraction asymptotes to \(\sim 0.4-0.5\) for the gEs, but since these data cover only the central, more red-dominated part of their GC systems, the global fraction is likely higher (e.g., in a wide-field study of three gEs by Rhode & Zepf 2004, the blue GC fraction ranged from 0.6-0.7.). The GC systems of the dEs fall entirely within the ACS field of view (see discussion below), so their measured blue GC fractions are global. These results show clearly that the classic correlation between GC metallicity/color and galaxy luminosity (Brodie & Huchra 1991) is a combination of two effects: the decreasing ratio of blue to red GCs with increasing galaxy luminosity, and, more importantly, the GC color-galaxy luminosity relations which exist for _both_ subpopulations. There does not seem to be a single "primordial" GC color-galaxy luminosity relation, as assumed in the accretion scenario for GC bimodality (e.g., Cote, Marzke, & West 2002) ## 4 Luminosity Functions and Nuclei Many previous works have found (e.g., Harris 1991, Secker 1992, Kundu & Whitmore 2001, Larsen _et al._ 2001) that the GCLF in massive galaxies is well-fit by a Gaussian or \(t_{5}\) distribution with similar properties among well-studied galaxies: \(M_{V}\sim-7.4\), \(\sigma_{V}\sim 1.3\). However, the shape among dwarf galaxies is poorly known. Individual galaxies have too few GCs for a robust fit, and thus a composite GCLF of many dwarfs is necessary. Using ground-based imaging, Durrell _et al._ (1996) found that the turnover of the summed GCLF of 11 dEs in Virgo was \(\sim 0.4\pm 0.3\) mag fainter the M87 turnover. Lotz _et al._ (2001) presented HST/WFPC2 snapshot imaging of 51 dEs in Virgo and Fornax; the summed GCLF in Lotz _et al._ is not discussed, but a conference proceeding using the same data (Miller 2002) suggests \(M_{V}\sim-7.4\) and \(-7.3\) for Virgo and Fornax, respectively. We study the dE GCLF through comparison to gEs (VCC 1316-M87, VCC 1226-NGC 4472, and VCC 1978-NGC 4649) previously found to have "normal" GCLFs (Larsen _et al._ 2001). For dwarfs we constructed a summed GCLF of all 37 dEs (\(M_{B}<-18.2\) in our sample). Since the individual distance moduli of the dEs are unknown, this GCLF could have additional scatter because of the range of galaxy distances; we find below that this appears to be quite a small effect. The individual GC systems of dEs are quite concentrated: most GCs are within \(30-40\arcsec\), beyond which the background contamination rises sharply (size measurements are unreliable at these faint magnitudes). Thus for the dEs we selected only those GCs within \(30\arcsec\) of the center of the galaxy. In the previous sections we used photometric and structural cuts to reduce the number of background galaxies and foreground stars interloping in our GC samples. However, these same cuts cannot be directly used to fit GCLFs, since these measurements become increasingly inaccurate for faint GCs (which may then be incorrectly removed). For example, applying only the color cut (\(0.5<g-z<2.0\)) vs. the full selection criteria to M87 changes the number of GCs with \(z<23.5\) by 16% (1444 vs. 1210) and shifts the peak of the GCLF by \(\sim 0.4\) mag, which is quite significant. We directly fit \(t_{5}\) distributions to the three gE GCLFs using using the code of Secker & Harris (1993), which uses maximum-likelihood fitting and incorporates photometric errors and incompleteness (Gaussian fits gave similar results within the errors; note that \(\sigma_{gauss}=1.29\,\sigma_{t5}\)). These GCLFs only have a color cut applied: \(0.5<g-z<2.0\) for M87 and NGC 4472 and \(0.85<g-z<2.0\) for NGC 4649 (since star-forming regions in the nearby spiral NGC 4647 strongly contaminate the blue part of the CMD). The results are given in Table 2. We fit both total populations and blue and red GCs separately, using the color cuts given in the table. Similar to what is commonly seen in \(V\)-band GCLFs (e.g., Larsen _et al._ 2001), the \(g\) turnovers are \(\sim 0.3-0.4\) mag brighter for the blue GCs than the red GCs. This is predicted for equal-mass/age turnovers separated by \(\sim 1\) dex in metallicity (Ashman, Conti & Zepf 1995). However, in \(z\) the blue GCs are only slightly brighter than the red GCs (the mean difference is negligible, but the blue GCs are brighter in M87 and NGC 4472 and fainter in NGC 4649, which may be biased slightly faint because of contamination). This difference between \(g\) and \(z\) is qualitatively consistent with stellar population models: Maraston (2005) models predict that equal-mass 13 Gyr GCs with [M/H] = \(-1.35\) and \(-0.33\) will have \(\Delta g=0.6\) and \(\Delta z=0.2\); the \(z\) difference is one-third of the \(g\) difference. This effect is probably due to the greater sensitivity of \(g\) to the turnoff region and the larger number of metal lines in the blue. The errors for the blue GCLF parameters may be slightly larger than formally stated because of the presence of the "H" GCs discussed previously. Variations in this feature and in the blue tilt among galaxies could represent a fundamental limitation to the accuracy to using _only_ the blue GCLF turnover as a standard candle (see, e.g., Kissler-Patig 2000). However, the total peak locations themselves are quite constant, with a range of only 0.03 mag in \(g\) and 0.05 mag in \(z\). At least for the well-populated old GC systems of gEs, the GCLF turnover appears to be an accurate distance indicator whose primary limitation is accurate photometry. Unfortunately, we cannot directly fit the composite dE GCLF as for the gEs--it is far too contaminated. Instead, we must first correct for background objects. As noted above, the GC systems of the dEs are very centrally concentrated. Thus we can use the outer regions of the dE images as a fiducial background. We defined a background sample in the radial range \(1.1-1.25\arcmin\) (corresponding to 5.4-6.2 kpc at a distance of 17 Mpc), then subtracted the resulting \(z\) GCLF in 0.1 mag bins from the central \(30\arcsec\) sample with the appropriate areal correction factor. We confined the fit to candidates with \(0.7<g-z<1.25\) because of the small color range of the dEs. The resulting fit (performed as above) gave a turnover of \(M_{z}=-8.14\pm 0.14\), compared to the weighted mean of \(M_{z}=-8.19\) for the gEs. These are consistent. Since the \(z\) GCLFs are being used, the overall color differences between the GC systems should have little effect on the peak locations. The dispersion of the dEs (\(\sigma_{z}=0.74\)) is less than that of the gEs (\(\sigma_{z}=1.03\)); this is probably partially due to the smaller color range of the dEs. It also indicates that the range of galaxy distances is probably not significant, consistent with the projected appearance of many of the dEs near the Virgo cluster core (Binggeli _et al._ 1987). That the peak of the GCLF appears to be the same for both gEs and dEs is perhaps puzzling. Even if both galaxy types had similar primordial GCLFs, analytic calculations and numerical simulations of GC evolution in the tidal field of the dEs suggest that dynamical friction should destroy high-mass GCs within several Gyr (Oh & Lin 2000, Lotz _et al._ 2001). Tidal shocks (which tend to destroy low-mass GCs) are more significant in massive galaxies, so it is unlikely that the loss of high-mass GCs in dEs could be balanced by a corresponding amount of low-mass GC destruction to preserve a constant turnover. _ab initio_ variations in the GCLF between dEs and gEs are another possibility, but cannot be constrained at present. If we see no evidence in the dE GCLF turnover for dynamical friction, does this provide significant evidence against this popular scenario for forming dE nuclei? In order study the relationship between the nuclei and GCs of our sample galaxies, we performed photometry and size measurements on all of the dE nuclei. The procedures were identical to those described above for the GCs, except that aperture corrections were derived through analytic integration of the derived King profile, since many of the nuclei were large compared to typical GC sizes. The properties of the nuclei are listed in Table 3. In Figure 9 we plot parent galaxy \(M_{B}\) vs. \(M_{z}\) of the nucleus. Symbol size is proportional to nuclear size. The overplotted dashed line represents Monte Carlo simulations of nuclear formation through dynamical friction of GCs from Lotz _et al._(2001). These assume 5 Gyr of orbital decay (implicitly assuming the GCs are \(\sim 5\) Gyr old); we have converted their \(M_{V}\) to \(M_{z}\) using stellar populations models of Maraston (2005). Lotz _et al._ found that the expected dynamical friction was inconsistent with observations, since the observed nuclei were fainter than predicted by their simulations. They argued that some additional process was needed to oppose dynamical friction (e.g., tidal torques). At first glance, this is supported by Figure 9--for galaxies in the luminosity range \(-15<M_{B}<-17\), there is a \(\sim 3\) mag range in nuclear luminosity. But most of the bright nuclei are quite small, while those much fainter than predicted from the simulations are generally larger (by factors of 2-4). In addition, the small nuclei tend to be redder than the large ones; some of the large faint nuclei are blue enough that they require some recent star formation. While not a one-to-one relation, it seems plausible that these luminosity and size distributions reflect two different channels of nuclear formation: small bright red nuclei are formed by dynamical friction, while large faint blue nuclei are formed by a dissipative process with little or no contribution from GCs. The red nuclei have typical colors of \(g-z\sim 1.1\), consistent with those of the red GCs in dEs. This implies that red GCs (rather than blue GCs) would need to be the primary "fuel" for nuclei built by dynamical friction, at least among the brigher dEs. The formation of blue nuclei could happen as the dE progenitor entered the cluster, or during tidal interactions that drive gas to the center of the galaxy (i.e., harassment, Mayer _et al._ 2001). Present simulations do not resolve the central parts of the galaxy with adequate resolution to estimate the size of a central high-density component, but presumably this will be possible in the future. It is unclear how to square the hypothesis that some nuclei are built by dynamical friction with the GCLF results from above. Perhaps the small \(\sigma_{z}\) of the composite dE (compared to the gEs) is additional evidence of GC destruction. Our background correction for the dE GCLF is potential source of systematic error in the turnover determination. Deeper ACS photometry for several of the brighter Virgo dEs could be quite helpful, since it could give both better rejection of background sources and more accurate SBF distances to the individual dEs. ## 5 Specific Frequencies For the Es in our sample, the ACS FOV covers only a fraction of their GC system, and no robust conclusions about their total number of GCs can be drawn without uncertain extrapolation. The GC systems of fainter Es and dEs are less extended and fall mostly or entirely within ACS pointings. As previously discussed, \(S_{N}\) is often used as a measure of the richness of a GC system. In fact, Harris & van den Bergh (1981) originally defined \(S_{N}\) in terms of the _bright_ GCs in a galaxy: the total number was taken as the number brighter than the turnover doubled. The justification for this procedure was that the faint end of the GCLF was often ill-defined, with falling completeness and rising contamination. In addition, for a typical log-normal GCLF centered at \(M_{V}=-7.4\sim 3\times 10^{5}M_{\odot}\), 90% or more of the total GC system mass is in the bright half of the LF. Many of the dEs in this study have tens or fewer detected GCs, making accurate measurements of the shape of the LF impossible. Thus counting only those GCs brighter than the turnover is unwise; small variations in the peak of the GCLF among galaxies (either due to intrinsic differences or the unknown galaxy distances) could cause a large fraction of GC candidates to be included or excluded. As discussed in SS2, we chose \(z=23.5\) as the magnitude limit for our study. This is \(\sim 0.5\) mag beyond the turnover of the composite dE GCLF discussed in the previous section (\(M_{z}\sim-8.1\), \(\sigma_{z}\sim 0.7\)) at a nominal Virgo Cluster distance of 17 Mpc. For such galaxies our magnitude cutoff corresponds to \(\sim 75\)% completeness. Thus, to get total GC populations for the dEs, we divided the number of GCs brighter than \(z=23.5\) by this factor (for the few dEs with individual distance moduli, we integrated the LF to find the appropriate correction factor). Finally, as discussed in SS4, the photometric and structural cuts used to reduce contaminants also remove real GCs. Using M87 as a standard, 16.2% of real GCs with \(z<23.5\) were falsely removed; we add back this statistical correction to produce our final estimate of the total GC population. Due to the small radial extent of the dE GC systems (primarily within a projected galactocentric radius of \(\sim 30-40\arcsec\)) no correction for spatial coverage is applied. We calculated \(B\)-band \(S_{N}\) using the absolute magnitudes in Table 1. Unless otherwise noted, all \(S_{N}\) refer to \(B\)-band values. These \(S_{N}\) can be converted to the standard \(V\)-band \(S_{N}\) by dividing our values by a factor \(10^{0.4(B-V)}\); a typical dE has \(B-V=0.8\), which corresponds to a conversation factor \(\sim 2.1\). \(S_{N}\) is plotted against \(M_{B}\) in Figure 10. In this figure the dE,N galaxies are filled circles and the dE,noN galaxies are empty circles. The symbol size is proportional to the fraction of blue GCs. Miller _et al._ (1998) found that dE,noNs had, on average, lower \(S_{N}\) than dE,Ns, and that for both classes of dE there was an inverse correlation between \(S_{N}\) and galaxy luminosity. We do not see any substantial difference between dE,noN and dE,N galaxies. There does appear to be a weak correlation between \(S_{N}\) and \(M_{B}\) in our data (with quite large \(S_{N}\) for some of the faintest galaxies), but there is a large spread in \(S_{N}\) for most luminosities. The \(S_{N}\) does not appear to depend strongly on the fraction of blue GCs, so differing mixes of subpopulations cannot be responsible for the \(S_{N}\) variations. Thus the observed values correspond to total blue GC subpopulations which vary by a factor of up to \(\sim 10\) at a given luminosity. There appear to be two sets of galaxies: one group with \(S_{N}\sim 2\), the other group ranging from \(S_{N}\sim 5-20\) and a slight enhancement at \(S_{N}\sim 10\). We only have one galaxy in common with Miller _et al._ (VCC 9), but our \(S_{N}\) for this single galaxy is consistent with theirs within the errors. We note again that background contamination remains as a systematic uncertainty for total GC populations. It is tempting to argue that these groups represent different formation channels for dEs, e.g., the fading/quenching of dIrrs, harassment of low-mass spirals, or simply a continuation of the E sequence to fainter magnitudes. In some scenarios for blue GC formation (e.g., Strader _et al._ 2005; Rhode _et al._ 2005), galaxies with roughly similar masses in a given environment would be expected to have similar numbers of blue GCs per unit mass of their DM halos. Variations in \(S_{N}\) at fixed luminosity would then be due to differences in the efficiency of converting baryons to stars--either early in the galaxy's history (due to feedback) or later, due to galaxy transformation processes as described above. Two pieces of observational evidence should help to differentiate among the possibilities. First, GC kinematics can be used directly to constrain dE mass-to-light ratios (e.g., Beasley _et al._ 2005). It will be interesting to see whether GC kinematics are connected to the apparent dichotomy of rotating vs. non-rotating dEs (Pedraz _et al._ 2002, Geha _et al._ 2002, 2003, van Zee _et al._ 2004). Second, more detailed stellar population studies of dEs are needed, especially any method which (unlike integrated light spectroscopy) can break the burst strength-age degeneracy. A rather ad hoc alternative is that the efficiency of blue GC formation varies substantially among Virgo dEs, but this cannot be constrained at present. It is interesting that the fraction of blue GCs appears unrelated both to the nucleation of the dE and the \(S_{N}\) variations; this suggests that whatever process leads to red GC formation in dEs is independent of these other factors. ## 6 Summary We have presented a detailed analysis of the GC color and luminosity distributions of several gEs and of the colors, specific frequencies, luminosity functions, and nuclei of a large sample of dEs. The most interesting feature in the gEs M87 and NGC 4649 is a correlation between mass and metallicity for individual blue GCs. Self-enrichment is a plausible interpretation of this observation, and could suggest that these GCs once possessed dark matter halos (which may have been subsequently stripped). Among the other new features observed are very luminous (\(z\gtrsim 20\)) GCs with intermediate to red colors. These objects are slightly larger than typical GCs and may be remnants of stripped dwarf galaxies. Next, we see an intermediate-color group of GCs which lies near the GCLF turnover and in the gap between the blue and red GCs. Also, the color spread among the red GCs is nearly twice that of the blue GCs, but because the relation between \(g-z\) and metallicity appears to be nonlinear, the \(1\sigma\) dispersion in metallicity (\(\sim 0.6\) dex) may be the same for both subpopulations. The peak of the GCLF is the same in the gEs and a composite dE, modulo uncertainties in background subtraction for the dEs. This observation may be difficult to square with theoretical expectations that dynamical friction should deplete massive GCs in less than a Hubble time, and with the properties of dE nuclei. There appear to be two classes of nuclei: small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. Though dominated by blue GCs, many dEs appear to have bimodal color distributions, with significant red GC subpopulations. The colors of these GCs form a continuity with those of more massive galaxies; both the mean blue and red GC colors of dEs appear consistent with extrapolations of the GC color-galaxy luminosity relations for luminous ellipticals. We confirm these relations for both blue and red GC subpopulations. While previous works found an inverse correlation between dE \(S_{N}\) and galaxy luminosity, we find little support for such a relation. 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The lower panel shows the normal gap between the blue and red GC subpopulations, while the upper panel shows how the “H” GCs have filled the gap in.", "captionBoundary": { "x1": 96, "x2": 720, "y1": 846, "y2": 896 }, "figType": "Figure", "imageText": [], "name": "5", "regionBoundary": { "x1": 102, "x2": 692, "y1": 221, "y2": 810 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508001-Figure5-1.png", "source": "fig" } ]
	# Keck/HIRES Spectroscopy of Four Candidate Solar Twins Jeremy R. King Department of Physics and Astronomy, 118 Kinard Laboratory, Clemson University, Clemson, SC 29634-0978 jking2@ces.clemson.edu Ann M. Boesgaard1 Footnote 1: affiliation: Visiting Astronomer, W.M. Keck Observatory, jointly operated by the California Institute of Technology and the University of California. Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822 boes@ifa.hawaii.edu Simon C. Schuler Department of Physics and Astronomy, 118 Kinard Laboratory, Clemson University, Clemson, SC 29634-0978 sschule@ces.clemson.edu ###### Abstract We use high S/N, high-resolution Keck/HIRES spectroscopy of 4 solar twin candidates (HIP 71813, 76114, 77718, 78399) pulled from our _Hipparcos_-based Ca II H & K survey to carry out parameter and abundance analyses of these objects. Our spectroscopic \(T_{\rm eff}\) estimates are some \({\sim}100\) K hotter than the photometric scale of the recent Geneva-Copenhagen survey; several lines of evidence suggest the photometric temperatures are too cool at solar \(T_{\rm eff}\). At the same time, our abundances for the 3 solar twin candidates included in the Geneva-Copenhagen survey are in outstanding agreement with the photometric metallicities; there is no sign of the anomalously low photometric metallicities derived for some late-G UMa group and Hyades dwarfs. A first radial velocity determination is made for HIP 78399, and \(UVW\) kinematics derived for all stars. HIP 71813 appears to be a kinematic member of the Wolf 630 moving group (a structure apparently reidentified in a recent analysis of late-type _Hipparcos_ stars), but its	[]
	metallicity is 0.1 dex higher than the most recent estimate of this group's metallicity. While certainly "solar-type" stars, HIP 76114 and 77718 are a few percent less massive, significantly older, and metal-poor compared to the Sun; they are neither good solar twin candidates nor solar analogs providing a look at the Sun at some other point in its evolution. HIP 71813 appears to be an excellent solar analog of age \({\sim}8\) Gyr. Our results for HIP 78399 suggest the promise of this star as a solar twin may be equivalent to the "closest ever solar twin" HR 6060; follow up study of this star is encouraged. stars: abundances -- stars: activity -- stars: atmospheres -- stars: evolution -- stars: fundamental parameters -- stars: late-type ## 1 Introduction The deliberate search for and study of solar analogs has been ongoing for nearly 30 years, initiating with the seminal early works of Hardorp (e.g., Hardorp 1978). Cayrel de Strobel (1996) gives an authoritative review of this early history, many photometric and spectroscopic results, and the astrophysical motivations for studying solar analogs. As of a decade ago, these motivations were of a strong fundamental and utilitarian nature, seeking answers to such questions as: (a) what is the solar color? b) how well do photometric indices predict spectroscopic properties? c) how robust are spectral types at describing or predicting the totality of a stellar spectrum? d) are there other stars that can be used as exact photometric and/or spectroscopic proxies for the Sun in the course of astrophysical research programs? While these important questions remain incompletely answered and of great interest, the study of solar analogs and search for solar twins has taken on renewed importance. Much of this has been driven by the detection of planetary companions around solar-type stars; the impact of these detections on solar analog research was foreshadowed with great prescience by Cayrel de Strobel (1996). Precision radial velocity searches for exoplanets are most robust when applied to slowly rotating and inactive stars; solar analogs are thus fruitful targets-metal-rich ones apparently even more fruitful (Fischer & Valenti, 2005). The appeal in searching for elusive terrestrial exoplanets around solar analogs remains a natural one given the existence of our own solar system. Solar analogs of various age also provide a mechanism to examine the past and future evolution of the Sun without significant or total recourse to stellar models. Such efforts looking at the sun in time (Ribas et al., 2005) now appear to be critical complements to studying	[]
	the evolution of planets and life surrounding solar-type stars. For example, it has been suggested that solar-type stars may be subject to highly energetic superflare outbursts, perhaps induced by orbiting planets, that would have dramatic effects on atmospheres surrounding and lifeforms inhabiting orbiting planets (Rubenstein & Schaefer, 2000; Schaefer, King & Deliyannis, 2000). It also seems clear that the nominal non-stochastic gradual evolution of solar-type chromospheres has important implications for a diversity of planetary physics (in our own solar system and others): the structure and chemistry of planetary atmospheres, the water budget on Mars, and even the evolution of planetary surfaces (Ribas et al., 2005); such issues are critical ones to understand in the development and evolution of life. The utilitarian importance of studying solar analogs has also persisted. For example, there should be little argument that differential spectroscopic analyses performed relative to the Sun are most reliable when applied to stars like the Sun- early G dwarfs. Happily, such objects can be found in a large variety of stellar populations having an extreme range of metallicity and age. The development of large aperture telescopes and improved instrumentation such as multi-object spectrographs and wide field imagers over the next decade or so mean that the stellar astronomy community is poised to undertake abundance surveys of tens or hundreds of thousands of Galactic stars. Critical questions confronting such ambitious but inevitable initiatives include: a) how reliable are photometric metallicities? b) can low-resolution spectroscopy yield results as robust as those from high-resolution spectroscopy? c) will automated spectroscopic analyses needed to handle such large datasets yield reliable results? All these questions can be addressed well by comparison with the results of high-resolution spectroscopy of solar analogs. Despite the importance of carrying out high-resolution spectroscopic analyses of solar analogs, efforts at doing so have been deliberate in pace. Recent exceptions to this include the solar analog studies of Gaidos, Henry & Henry (2000) and Soubiran & Triaud (2004). Here, we present the first results from a small contribution aimed at remedying this pace of study. Using the results of Dr. D. Soderblom's recent chromospheric Ca II H & K survey of nearby (\(d{\leq}60\) pc) late-F to early-K dwarfs in the _Hipparcos_ catalog, we have selected a sample of poorly-studied solar twin candidates having \(0.63{\leq}(B-V){\leq}0.66\), Ca II chromospheric fluxes within a few tenths of a dex of the mean solar value, and \(M_{V}\) within a few tenths of a magnitude of the solar value; there are roughly 150 such objects accessible from the northern hemisphere. These objects have been or are being observed as time allows during other observing programs. Here, we present echelle spectroscopy of 4 candidates obtained with Keck/HIRES. The objects are HIP 71813, 76114, 77718, and 78399. ## 2	[]
	Data and Analysis ### Observations and Reductions Our 4 solar twin candidates were observed on UT July 8 2004 using the Keck I 10-m, its HIRES echelle spectrograph, and a Tektronix \(2048{\times}2048\) CCD detector. The chosen slit width and cross-disperser setting yielded spectra from 4475 to 6900 A at a resolution of \(R{\sim}45,000\). Exposure times ranged from 3 to \({\sim}6\) minutes, achieving per pixel S/N in the continuum near 6707 A of \({\sim}400\). A log of the observations containing cross-identifications is presented in Table 1. Standard reductions were carried out including debiasing, flat-fielding, order identification/tracing/extraction, and wavelength calibration (via solutions calculated for an internal Th-Ar lamp). The H\(\alpha\) and H\(\beta\) features are located at the blue edge of their respective orders; the lack of surrounding wavelength coverage with which to accomplish continuum normalization thus prevented us from using Balmer profile fitting to independently determine \(T_{\rm eff}\). Samples of the spectra in the \({\lambda}6707\) Li I region can be found later in Figures 3 and 4. + Footnote †: margin: Fig. 5 ### Parameters and Abundances Clean, "case a" Fe I and Fe II lines from the list of Thevenin (1990) were selected for measurement in our 4 solar twin candidate spectra and a similarly high S/N and \(R{\sim}45,000\) Keck/HIRES lunar spectrum (described in King et al. (1997)) used as a solar proxy spectrum. Equivalent widths were measured using the profile fitting routines in the 1-d spectrum analysis software package SPECTRE(Fitzpatrick & Sneden, 1987). Line strengths of all the features measured in each star and our solar proxy spectrum can be found in Table 2. Abundances were derived from the equivalent widths using the 2002 version of the LTE analysis package MOOG and Kurucz model atmospheres interpolated from ATLAS9 grids. Oscillator strengths were taken from Thevenin (1990); the accuracy of these is irrelevant inasmuch as normalized abundances [x/H] were formed on a line-by-line basis using solar abundances derived in the same manner. The solar model atmosphere was characterized by \(T_{\rm eff}=5777\) K, log \(g=4.44\), a metallicity of [m/H]=0., and a microturbulent velocity of \({\xi}=1.25\); the latter is intermediate to the values of \({\xi}\) from the calibrations of Edvardsson et al. (1993) and Allende Prieto et al. (2004). An enhancement factor of 2.2 was applied to the van der Waals broadening coefficients for all lines. + Footnote †: margin: Fig. 5 Stellar parameters were determined as part of the Fe analysis in the usual fashion. \(T_{\rm eff}\) and \({\xi}\) were determined by requiring zero correlation coefficient between the _solar normalized_ abundances (i.e., [Fe/H]; again, accomplished on a line-by-line ba	[]
	sis) and the lower excitation potential and reduced equivalent width, respectively. This approach leads to unique solutions when there is no underlying correlation between excitation potential and reduced equivalent width. We show in Figure 1 that there is no such underlying correlation in our Fe I sample. Figure 2 displays the Fe I-based line-by-line [Fe/H] values versus both lower excitation potential (top) and reduced equivalent width (bottom) using our final model atmosphere parameters for the case of HIP 76114; the linear correlation coefficients in both planes are \({\sim}0.00\). Our abundance analysis is thus a purely differential one, and the derived parameters do not depend on the rigorous accuracy of the \(gf\) values. The 1\(\sigma\) level uncertainties in \(T_{\rm eff}\) and \({\xi}\) were determined by finding the values of these parameters where the respective correlation coefficients became significant at the 1\({\sigma}\) confidence level. Gravity estimates were made via ionization balance of Fe. The error estimates for log \(g\) include uncertainties in both [Fe I/H] and [Fe II/H] due to measurement uncertainty, \(T_{\rm eff}\) errors, and \({\xi}\) errors. The final parameters and their uncertainties can be found in the summary of results in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 Abundances of Al, Ca, Ti, and Ni were derived in a similar fashion using the line data in Table 2 and model atmospheres characterized by the parameters determined from the Fe data. Abundances of a given species were normalized on a line-by-line basis using the values derived from the solar spectrum, and then averaged together. Typical errors in the mean are only 0.01-0.02 dex, indicative of the quality of the data. The sensitivity of the derived abundances to arbitrarily selected fiducial variations in the stellar parameters (\({\pm}100\) K in \(T_{\rm eff}\); \({\pm}0.2\) dex in log \(g\); and \({\pm}0.2\) km s\({}^{-1}\) in microturbulence) are provided for each element in Table 3. Coupling these with the parameter uncertainties and the statistical uncertainties in the mean yielded total uncertainties in the abundance ratio of each element. The mean abundances and the \(1{\sigma}\) uncertainties are given in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 ### Oxygen Abundances O abundances were derived from the measured equivalent widths of the \({\lambda}6300\) [O I] feature (Table 2) using the blends package in MOOG to account for contamination by a Ni I feature at 6300.34. Isotopic components (Johansson et al., 2003) of Ni were taken into account with the \(gf\) values taken from Bensby, Feltzing & Lundstrom (2004); the [O I] \(gf\) value (-9.717) is taken from Allende Prieto, Lambert, & Asplund (2001). The assumed Ni abundances were taken as [Ni/H]=0.00, -0.04, -0.16, and -0.01 for HIP 71813, 76114, 77718, and 78399 respectively. Abundances are given in Table 4. Uncertainties in [O/H] are dominated by those in the equivalent widths (\(0.5\) mA) measurements of the stars and the Sun, and that in log \(g\) (\(0.12\) dex). These uncertainties from these 3 sources were added in	[]
	quadrature to yield the total uncertainties associated with the [O/H] values given in Table 4. ### Lithium Abundances Li abundances were derived from the \({\lambda}6707\) Li I resonance features via spectrum synthesis. Utilizing the derived parameters, synthetic spectra of varying Li abundance were created in MOOG using the line list from King et al. (1997). No contribution from \({}^{6}\)Li was assumed, a reasonable assumption given that the Li abundances in our objects are well-below meteoritic (log \(N\)(Li)=3.31; \({}^{6}\)Li/\({}^{7}\)Li\(=0.08\)). Smoothing was carried out by convolving the synthetic spectra with Gaussians having FWHM values measured from clean, weak lines measured in our spectra. Comparison of the syntheses (solid lines) and the Keck/HIRES spectra in the \({\lambda}6707\) region are shown in Figures 3 and 4. Total uncertainties include those due to uncertainties in the \(T_{\rm eff}\) value (Table 3) and in the fit itself. The Li results are listed in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 ### Rotational Velocity and Chromospheric Emission The same FWHM values measured for each star and used to smooth the syntheses were assumed to be the quadrature sum of components due to spectrograph resolution and (twice the projected) rotational velocity. The resulting \(v\) sin \(i\) values are listed in Table 4. Inasmuch as we assume no contribution from macroturbulent broadening mechanisms, we present these estimates as upper limits to the projected rotational velocity. The Ca II H&K chromospheric emission indices of our objects are listed in Table 4 and come from the low-resolution (\(R{\sim}2000\)) KPNO coude' feed-based survey of D. Soderblom. ### Masses and Ages Masses and ages of the Sun and our four solar twin candidates were estimated by placing them in the \(M_{V}\) versus \(T_{\rm eff}\) plane using our temperature estimates and the Hipparcos-based absolute visual magnitudes. Comparison of these positions with isochrones and sequences of constant mass taken from appropriate metallicity Yonsei-Yale Isochrones (Yi, Kim & Demarque, 2003) (as updated by Demarque et al. (2004)) yielded the mass and age estimates in Table 4. The uncertainties in mass and age are calculated assuming the influence of uncertainties in our \(T_{\rm eff}\) and \(M_{V}\) values; including the uncertainty in our metallicity esti	[]
	mates (\({\sigma}{\sim}0.04\) dex) has a negligible effect on the uncertainty of our estimated masses, but would contribute an additional 0.4 Gyr uncertainty in the age estimates. The HR diagrams containing our objects and these isochrones are shown in Figure 5. + Footnote †: margin: Fig. 5 ## 3 Results and Discussion ### Comparison with Previous Results HIP 71813 is included in the recent Geneva-Copenhagen solar neighborhood survey of Nordstrom et al. (2004). Their photometric metallicity determination of [Fe/H]\(=+0.01\) is in outstanding agreement with our Al, Ca, Ti, Fe, and Ni abundances, which range from \(-0.02\) to \(+0.02\). Their photometric \(T_{\rm eff}\) estimate of 5662 K is some 90 K lower than our spectroscopic value. If the solar color, \((B-V)_{\odot}=0.642\), adopted in Table 4 is to be believed, then our \(T_{\rm eff}\) value would seem to be more consistent with the nearly indistinguishable \((B-V)\) index (0.644) of HIP 71813. HIP 76114 is also included in the Geneva-Copenhagen survey. The Nordstrom et al. (2004) photometric metallicity of [Fe/H]\(=-0.05\) is also in outstanding agreement with our Al, Ca, Ti, Fe, and Ni abundances, which range from \(-0.06\) to \(-0.02\). The photometric \(T_{\rm eff}\) difference between HIP 71813 and 76114 (former minus latter) of 52 K is in excellent agreement with our spectroscopic difference of 40 K. HIP 77718 has a photometric metallicity, [Fe/H]\(=-0.19\) from the Nordstrom et al. (2004) solar neighborhood survey that is in good agreement with our Al, Ti, Fe, and Ni determinations, which range from \(-0.15\) to \(-0.22\); our [Ca/H] abundance of \(-0.09\) appears only mildly anomalous in comparison. The HIP 77718 minus 71813 photometric \(T_{\rm eff}\) difference of 92 K is in outstanding agreement with the 90 K spectroscopic difference. Gray et al. (2003) have determined the parameters and overall abundance of HIP 77718 via the analysis of low resolution blue spectra as part of their NStar survey. The independent spectroscopic \(T_{\rm eff}\) estimate, made via different comparisons of different spectral features in a different part of the spectrum, of 5859 K is only 19 K larger than our own and 105 K larger than the photometric value. The Gray et al. (2003) metallicity of [m/H]\(=-0.15\) is indistinguishable from our own result. HIP 78399 has not been subjected to any published abundance or high-resolution spectroscopic analysis that we are aware of. Accordingly, it lacks a radial velocity determination. We remedied this by determining a radial velocity relative to HIP 76114 via cross-correlation of the spectra in the 6160-6173 A range. We assumed the precision radial velocity of \(-35.7\) km/s from Nidever et al. (2002) for HIP 76114. Cross-correlation of the telluric B-Band	[]
	spectra in the 6880 A region revealed a 9.9 km/s offset between the spectra. While larger than anticipated, this intra-night drift was confirmed by comparison of telluric water vapor features in the 6300 A region. Accounting for this drift and the appropriate relative heliocentric corrections, we find a radial velocity of \(-24.7{\pm}0.7\) km/s for HIP 78399. ### HIP 71813 and the Wolf 630 Moving Group Eggen (1969) included HIP71813 as a member of the Wolf 630 moving group. Membership in this putative kinematic population was defined by Eggen in a vast series papers as traced in the work of Mcdonald & Hearnshaw (1983). Regardless of one's view on the reality of these kinematic assemblages, it is likely that the recent passing of O. Eggen has meant that a wealth of modern data (in particular _Hipparcos_ parallaxes and precision radial velocities) has not yet been brought to bear on the reality, properties, and detailed membership of the Wolf 630 group. A notable exception is the work of Skuljan, Hearnshaw & Cottrell (1999), who find a clustering of late-type stars at \((U,V)=(+20,-30)\) km s\({}^{-1}\) that is absent in the kinematic phase space of their early-type stellar sample; this is highly suggestive of an old moving group at Eggen's suggested position of the Wolf 630 group in the Bottlinger (\(U\),\(V\)) diagram. The salient characteristics identified by Eggen for the Wolf 630 group are a) a kinematically old disk population, b) a characteristic Galactic rotational velocity of \(V=-33\) km/s, and c) a color-luminosity array similar to M 67. It is beyond the scope of this paper to revisit or refine characteristics of the Wolf 630 group. However, several notes can be made. First, our 8 Gyr age estimate for HIP 71813 is certainly consistent with an old disk object. Second, if the estimate of Taylor (2000) of [Fe/H]\(=-0.12\) for the Wolf 630 group metallicity is accurate, then HIP 71813 would not seem to be a member. Third, using _Hipparcos_ parallaxes and proper motions, and modern radial velocity determinations (Nordstrom et al., 2004; Tinney & Reid, 1998), the \(UVW\) kinematics of HIP 71813 can be compared with those of Wolf 629, a Wolf 630 group defining member according to Eggen. The heliocentric Galactic velocities of all our objects are listed in Table 4. The (U,V)=(+21.3\({\pm}1.5\),-36.3\({\pm}1.3\)) results for HIP 71813 are in excellent agreement with those for Wolf 629 (+21.0\({\pm}1.3\),-33.4\({\pm}1.0\)), and consistent with the canonical Wolf 630 group values (26, -33) given by Eggen (1969). None of our other candidate solar twins has kinematics, which are listed in Table 4, consistent with those of the Wolf 630 group.	[]
	### Solar Twin Status Evaluation _HIP 71813_. The \(T_{\rm eff}\) value, light metal-abundances, and chromospheric Ca II emission of HIP 71813 are indistinguishable from solar values. The Li abundance, however, appears to be depleted by a factor of \({\geq}2\) compared to the Sun. More importantly, however, the star appears significantly more evolved than the sun. The \(M_{V}\) and log \(g\) values are significantly lower than the solar values, and our estimated age is a factor of 2 older than the Sun's. While clearly an inappropriate solar twin candidate, the star would appear to be an excellent solar analog of significantly older age. _HIP 76114_. HIP 76114 is marginally cooler than the Sun, \({\Delta}T_{\rm eff}=-67\) K. While any of the light element abundances alone are indistinguishable from solar, taken together they suggest a metallicity some 0.04 dex lower than solar; this is confirmed by the photometric metallicity of Nordstrom et al. (2004). The Ca II emission and Li abundance is solar within the uncertainties, but the star appears marginally evolved relative to the Sun as indicated by its slightly lower \(M_{V}\) and log \(g\) values; table 4 suggests that HIP 76114 is \({\geq}1.5\) Gyr older than the Sun. This object may be a suitable solar analog of slightly older age, albeit of likely slightly lower metallicity, that can be included in studies looking at solar evolution. _HIP 77718_. While the Ca II chromospheric emission and age determination of HIP 77718 are observationally indistinguishable from the Sun, our analysis indicates this star is clearly warmer (\({\Delta}T_{\rm eff}=63\) K) and some 0.16 dex metal-poor relative to solar; both the warmer temperature and slightly metal-poor nature are independently confirmed by the spectroscopic analysis of Gray et al. (2003). The Li abundance is some 20 times higher than solar. This difference may be related to reduced PMS Li depletion due to lower metallicity or reduced main-sequence depletion due to a younger age; our observations can not distinguish between these possibilities. Regardless, this star is not a good solar twin candidate, nor an optimal metal-poor or younger solar analog. _HIP 78399_. The poorly-studied HIP 78399 appears to hold great promise as a solar twin candidate. Its \(T_{\rm eff}\), luminosity, mass, age, light metal abundances, and rotational velocity are all indistinguishable from solar values. The only marked difference seen is the Li abundance, which is a factor of \({\sim}6\) larger than the solar photospheric abundance. While the evolution of Li depletion in solar-type stars is a complex and still incompletely understood process subject to vigorous investigation, this difference may suggest a slightly younger age for HIP 78399, which is allowed by our age determination and may be consistent with a slightly larger Ca II chromospheric flux. Currently, the "closest ever solar twin" title belongs to HR 6060 (Porto De Mello & Da Silva, 1997). Several spectroscopic analyses of this star have been carried out (Luck & Heiter	[]
	, 2005; Allende Prieto et al., 2004; Gray et al., 2003; Porto De Mello & Da Silva, 1997). \(T_{\rm eff}\) estimates range from 5693 to 5835 K, and [Fe/H] estimates from -0.06 to +0.05; the precision \(T_{\rm eff}\) analysis using line ratios (Gray, 1995) indicates a \(T_{\rm eff}\) difference with respect to the Sun of 17 K. The Ca II H&K emission index (-5.00, Gray 1995) and rotational velocity (\({\leq}3\) km/s, De Mello & Da Silva 1997) are indistinguishable from solar values. The _Hipparcos_-based absolute magnitude strongly suggests the mass and age of HR 6060 are virtually identical to the Sun's (Porto De Mello & Da Silva, 1997). Just as for HIP 78399, the only glaring outlying parameter is Li abundance, which is a factor of \({\sim}4\) larger than the solar photospheric Li abundance (Stephens, 1997). The work of Jones, Fischer & Soderblom (1999) on 1 \(M_{\odot}\) stars in the solar-age and -abundance cluster M67 suggests that we can expect such objects to exhibit a \({\sim}1\) dex range in Li; thus, the Sun may not be an especially good Li "standard". Based on our analysis, we believe there is a case to be made that HIP 78399 share the stage with HR 6060 as the closest ever solar twin. For those engaged in studies of solar twins or the Sun in time, HIP 78399 is certainly worthy of closer follow-up study. Particularly valuable would be: a) refining its T\({}_{\rm eff}\) and luminosity estimates relative to the Sun via Balmer line profile fitting, analysis of line ratios, etc. b) analysis of the \({\lambda}7774\) O I lines to confirm whether its O abundance is truly subsolar, c) performing an independent check on its relative age via the [Th/Nd] ratio (Morell, Kallander, & Butcher, 1992), and d) determining a \({}^{9}\)Be abundance, which is more immune to the effects of stellar depletion and also contains embedded information about the "personal" integrated Galactic cosmic-ray history of matter comprised by the star. ## 4 Summary We have carried out high S/N high-resolution Keck/HIRES spectroscopy of four candidate solar twins drawn from a _Hipparcos_-defined Ca II H&K survey. Parameters, abundances, masses, ages, and kinematics have been derived in a differential fine analysis. Comparisons suggests that the _relative_ photometric \(T_{\rm eff}\) values of Nordstrom et al. (2004) and our spectroscopic temperatures are indistinguishably robust; however, the photometric \(T_{\rm eff}\) values are typically 100 K cooler. There are several lines of evidence that suggest the photometric scale is misanchored (at least near solar \(T_{\rm eff}\)). First, if the solar color of Cayrel de Strobel (1996) is nearly correct, then our spectroscopic \(T_{\rm eff}\) values are in outstanding accord with the colors of HIP 71813 and 78399. Second, the independent analysis of HIP 77718 by Gray et al. (2003) using different spectral features in the blue yields a spectroscopic \(T_{\rm eff}\) in outstanding agreement with our own. Third, the Nordstrom et al. (2004) photometric \(T_{\rm eff}\) estimate for the "closest ever solar twin" HR 6060 is 5688 K, some \(100\) K lower than the precision	[]
	\(T_{\rm eff}\) estimate of Gray (1995). At the same time, our light metal-abundances are in excellent agreement with the photometric metallicity estimates for the 3 of our objects in Nordstrom et al. (2004), differing by no more than a few hundredths of a dex. There is no sign of the abnormally low photometric metallicity values seen for some very cool Pop I dwarfs in the Hyades and UMa group as noted by King & Schuler (2005). As these authors note, anomalous photometric estimates may be restricted to late G dwarfs. Our spectroscopic metallicity for HIP 77718 is in nearly exact agreement with that derived from low-resolution blue spectra by Gray et al. (2003). We present the first abundances and radial velocity estimate for HIP 78399. Using the radial velocities and _Hipparcos_ proper motions and parallaxes, we derive the \(UVW\) kinematics of our four solar twin candidates. The position of HIP 71813 in the \((U,V)\) plane is consistent with membership in Eggen's Wolf 630 moving group, a kinematic structure of late-type _Hipparcos_ stars apparently verified by Skuljan, Hearnshaw & Cottrell (1999). Our metallicity for HIP 71813, [Fe/H]\(=-0.02\), is 0.1 dex higher than the Wolf 630 estimate of Taylor (2000), however. Revisiting the characteristic metallicity via identification of assured Wolf 630 group members using _Hipparcos_ data and new precision radial velocities, and follow-up high resolution spectroscopy to determine abundances would be of great value. HIP 77718 is \({\sim}70\) K warmer than the Sun, significantly more metal-poor ([m/H]\({\sim}-0.16\)), significantly more Li-rich (log \(N\)(Li)\({\sim}2.3\)) and a few percent lass massive than the Sun; we deem it neither a suitable solar twin nor solar analog to trace the evolution of the Sun. The light-metal and Li abundances of HIP 76114 are much closer to solar. However, HIP 76114 does appear to be slightly metal-poor ([m/H]\(=-0.04\)), cooler \({\Delta}T_{\rm eff}=67\) K, older \({\Delta}{\tau}{\geq}3\) Gyr, and a few percent less massive compared to the Sun. HIP 71813 appears to be an excellent solar analog of solar abundance, mass, and \(T_{\rm eff}\), but advanced age-\(M_{V}=4.45\) and \({\tau}{\sim}8\) Gyr; the more evolved state of this star is likely reflected in the subsolar upper limit to its Li abundance. Finally, our first ever analysis of HIP 78399 suggests this object may be a solar twin candidate of quality comparable to the "closest ever solar twin" HR 6060 (Porto De Mello & Da Silva, 1997). The \(T_{\rm eff}\), mass, age, and light metal abundances of this object are indistinguishable from solar given the uncertainties. The only obvious difference is that which characterizes HR 6060 as well- a Li abundance a factor of a few larger than the solar photospheric value. This object merits additional study as a solar twin to refine its parameters; of particular interest will be confirming our subsolar O abundance derived from the very weak \({\lambda}6300\) [O I] feature. We are indebted to Dr. David Soderblom for the use of his nearby star activity catalog from which our objects were selected and for his valuable comments on the manuscript. It is a	[]
	# Initial Populations of Black Holes in Star Clusters Krzysztof Belczynski12 Aleksander Sadowski3 , Frederic A. Rasio4 , Tomasz Bulik 5 Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 Footnote 2: The number of single BHs formed out of binary systems may be inferred by comparing the numbers of binary BHs with the single BHs listed under “binary disruption” and “binary mergers” in Tables 2–5, 7 and 8 Footnote 3: affiliationmark: Footnote 4: affiliationmark: Footnote 5: affiliationmark: \({}^{1}\) New Mexico State University, Dept of Astronomy, 1320 Fregner Mall, Las Cruces, NM 88003 \({}^{2}\) Tombaugh Fellow \({}^{3}\) Astronomical Observatory, Warsaw University, Al. Ujazdowskie 4, 00-478, Warsaw, Poland \({}^{4}\) Northwestern University, Dept of Physics and Astronomy, 2145 Sheridan Rd, Evanston, IL 60208 \({}^{5}\) Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland kbelczyn@nmsu.edu, oleks@camk.edu.pl, rasio@northwestern.edu, bulik@camk.edu.pl ###### Abstract Using an updated population synthesis code we study the formation and evolution of black holes (BHs) in young star clusters following a massive starburst. This study continues and improves on the initial work described by Belczynski, Sadowski & Rasio (2004). In our new calculations we account for the possible ejections of BHs and their progenitors from clusters because of natal kicks imparted by supernovae and recoil following binary disruptions. The results indicate that the properties of both retained BHs in clusters and ejected BHs (forming a field population) depend sensitively on the depth of the cluster potential. In particular, most BHs ejected from binaries are also ejected from clusters with central escape speeds \(V_{\rm esc}\lesssim 100\,{\rm km}\,{\rm s}^{-1}\). Conversely, most BHs remaining in binaries are retained by clusters with \(V_{\rm esc}\gtrsim 50\,{\rm km}\,{\rm s}^{-1}\). BHs from single star evolution are also affected significantly: about half of the BHs originating from primordial single stars are ejected from clusters with \(V_{\rm esc}\lesssim 50\,{\rm km}\,{\rm s}^{-1}\). Our results lay a foundation for theoretical studies of the formation of BH X-ray binaries in and around star clusters, including possible "ultra-luminous" sources, as well as merging BH-BH binaries detectable with future gravitational-wave observatories. Subject headings: binaries: close -- black hole physics -- gravitational waves -- stars: evolution ## 1. INTRODUCTION ### Black Holes in Star Clusters Theoretical arguments and many observations suggest that BHs should form in significant numbers in star clusters. Simple assumptions about the stellar initial mass function (IMF) and stellar evolution indicate that out of \(N\) stars formed initially, \(\sim 10^{-4}-10^{-3}\,N\) should produce BHs as remnants after \(\sim 20\,\)Myr. Thus any star cluster containing initially more than \(\sim 10^{4}\) stars should contain at least some BHs; large super star clusters and globular clusters should have formed many hundreds of BHs initially, and even larger systems such as galactic nuclei may contain many thousands to tens of thousands. Not surprisingly, observations are most sensitive to (and have provided constraints mainly on) the most massive BHs that may be present in the cores of very dense clusters (van der Marel 2004). For example, recent observations and dynamical modeling of the globular clusters M15 and G1 indicate the presence of a central BH with a mass \(\sim 10^{3}-10^{4}\,M_{\odot}\) (Gerssen et al. 2002, 2003; Gebhardt et al. 2002, 2005). However, direct \(N\)-body simulations by Baumgardt et al. (2003a,b) suggest that the observations of M15 and G1, and, in general, the properties of all _core-collapsed_ clusters, could be explained equally well by the presence of many compact remnants (heavy white dwarfs, neutron stars, or \(\sim 3-15{\,M_{\odot}}\) BHs) near the center without a massive BH (cf. van der Marel 2004; Gebhardt et al. 2005). On the other hand, \(N\)-body simulations also suggest that many _non-core-collapsed_ clusters (representing about 80% of globular clusters in the Milky Way) could contain central massive BH (Baumgardt et al. 2004, 2005). In any case, when the correlation between central BH mass and bulge mass in galaxies (e.g., Haring & Rix 2004) is extrapolated to smaller stellar systems like globular clusters, the inferred BH masses are indeed \(\sim 10^{3}-10^{4}\,M_{\odot}\). These are much larger than a canonical \(\sim 10\,M_{\odot}\) stellar-mass BH (see, however, SS3.1.6), but much smaller than the \(\sim 10^{6}-10^{9}\,M_{\odot}\) of supermassive BHs. Hence, these objects are often called _intermediate-mass black holes_ (IMBHs; see, e.g., Miller & Colbert 2004). Further observational evidence for IMBHs in dense star clusters comes from many recent _Chandra_ and XMM-_Newton_ observations of "ultra-luminous" X-ray sources (ULXs), which are often (although not always) clearly associated with young star clusters and whose high X-ray luminosities in many cases suggest a compact object mass of at least \(\sim 10^{2}\,M_{\odot}\) (Cropper et al. 2004; Ebisuzaki et al. 2001; Kaaret et al. 2001; Miller et al. 2003). In many cases, however, beamed emission by an accreting stellar-mass BH may provide an alternative explanation (King et al. 2001; King 2004; Zezas & Fabbiano 2002). One natural path to the formation of a massive object at the center of any young stellar system with a high enough density is through runaway collisions and mergers of massive stars following gravothermal contraction and core collapse (Ebisuzaki et al. 2001; Portegies Zwart & McMillan 2002; Gurkan, Freitag, & Rasio 2004). These runaways occur when massive stars can drive core collapse _before they evolve_. Alternatively, if the most massive stars in the cluster are allowed to evolve and produce supernovae, the gravothermal contraction of the cluster will be reversed by the sudden mass loss, and many stellar-mass BHs will be formed. The final fate of a cluster with a significant component of stellar-mass BHs remains highly uncertain. This is because realistic dynamical simulations for such clusters (containing a large number of BHs _and_ ordinary stars with	[]
	a realistic mass spectrum) have yet to be performed. For old and relatively small systems (such as small globular clusters), complete evaporation is likely (with essentially all the stellar-mass BHs ejected from the cluster through three-body and four-body interactions in the dense core). This is expected theoretically on the basis of simple qualitative arguments based on Spitzer's "mass-segregation instability" applied to BHs (Kulkarni et al. 1993; Sigurdsson & Hernquist 1993; Watters et al. 2000) and has been demonstrated by dynamical simulations (Portegies Zwart & McMillan 2000; O'Leary et al. 2005). However, it has been suggested that, if stellar-mass BHs are formed with a relatively broad mass spectrum (a likely outcome for stars of very low metallicity; see Heger et al. 2003), the most massive BH could resist ejection, even from a cluster with low escape velocity. These more massive BHs could then grow by repeatedly forming binaries (through exchange interactions) with other BHs and merging with their companions (Miller & Hamilton 2002; Gultekin, Miller, & Hamilton 2004). However, as most interactions will probably result in the ejection of one of the lighter BHs, it is unclear whether any object could grow substantially through this mechanism before running out of companions to merge with. A single stellar-mass BH remaining at the center of a globular cluster is very unlikely to become detectable as an X-ray binary (Kalogera, King, & Rasio 2004). In addition to its obvious relevance to X-ray astronomy, the dynamics of BHs in clusters also plays an important role in the theoretical modeling of gravitational-wave (GW) sources and the development of data analysis and detection strategies for these sources. In particular, the growth of a massive BH by repeated mergers of stellar-mass BHs spiraling into an IMBH at the center of a dense star cluster may provide an important source of low-frequency GWs for LISA, the Laser-Interferometer Space Antenna (Miller 2002; Will 2004). Similarly, dynamical hardening and ejections of binaries from dense clusters of stellar-mass BHs could lead to greatly enhanced rates of BH-BH mergers detectable by LIGO and other ground-based interferometers (Portegies Zwart & McMillan 2000; O'Leary et al. 2005). A crucial starting point for any detailed study of BHs in clusters is an accurate description of the initial BH population. Here, "initial" means on a timescale short compared to the later dynamical evolution timescale. Indeed most \(N\)-body simulations of star cluster dynamics never attempt to model the brief, initial phase of rapid massive star evolution leading to BH formation. The goal of our work here is to provide the most up-to-date and detailed description of these initial BH populations. This means that we must compute the evolution of a large number of massive stars, including a large fraction of binaries, all the way to BH formation, i.e., on a timescale \(\sim 10-100\,\)Myr, taking into account a variety of possible cluster environments. ### Previous Work In a previous study (Belczynski et al. 2004; hereafter Paper I) we studied young populations of BHs formed in a massive starburst, without explicitly taking into account that most stars are formed in clusters. For many representative models we computed the numbers of BHs, both single and in various types of binaries, at various ages, as well as the physical properties of different systems (e.g., binary period and BH mass distributions). We also discussed in detail the evolutionary channels responsible for these properties. In this follow-up study, we consider the possible ejection of these BHs from star clusters with different escape speeds, taking into account the recoil imparted by supernovae (SNe) and binary disruptions. During SNe, mass loss and any asymmetry in the explosion (e.g., in neutrino emission) can impart large extra speeds to newly formed compact objects. If a compact object is formed in a binary system, the binary may either _(i)_ survive the explosion, but its orbital parameters are changed and the system (center-of-mass) speed changes, or _(ii)_ the binary is disrupted and the newly formed compact object and its companion fly apart on separate trajectories. The secondary star in a binary may later undergo a SN explosion as well, provided that it is massive enough. The effects of this second explosion are equally important in determining the final characteristics of compact objects. In Paper I we included the effects of SNe, both natal kicks and mass loss, on the formation and evolution of BHs (single and in binaries), but we did not keep track of which BHs and binaries would be retained in their parent cluster. Starbursts form most of their stars in dense clusters with a broad range of masses and central potentials (and hence escape speeds; see, e.g., Elmegreen et al. 2002; McCrady et al. 2003; Melo et al. 2005). Smaller clusters of \(\sim 10^{4}\,M_{\odot}\) (open clusters or "young populous clusters," such as the Arches and Quintuplet clusters in our Galactic center) could have escape speeds as low as \(V_{\rm esc}\lesssim 10\,{\rm km}\,{\rm s}^{-1}\) while the largest "super star clusters" with much deeper potential wells could have \(V_{\rm esc}\gtrsim 100\,{\rm km}\,{\rm s}^{-1}\). On the other hand the natal kick velocities could be relatively high, \(\sim 100-500\,{\rm km}\,{\rm s}^{-1}\) for low-mass BHs, so that a large fraction of BHs might leave the cluster early in the evolution. Here we repeat our study of young BH populations taking into account ejections from star clusters. We perform our calculations with a slightly updated version of our population synthesis code StarTrack (SS2) and we present results for both the retained cluster BH populations and the ejected BHs, which will eventually become part of the field BH population surrounding the surviving clusters. Our models and assumptions are discussed in SS2, with particular emphasis on the updates since Paper I. In SS3 we present our new results and in SS4 we provide a summary and discussion. ## 2. MODEL DESCRIPTION AND ASSUMPTIONS ### Population Synthesis Code Our investigation is based on a standard population synthesis method. We use the StarTrack code (Belczynski, Kalogera & Bulik 2002, hereafter BKB02), which has been revised and improved significantly over the past few years (Belczynski et al. 2006). Our calculations do not include any treatment of dynamical interactions (collisions) between binaries and single stars or other binaries1. In particular, the star clusters we consider are assumed to have avoided the 'runaway collision instability" that	[]
	can drive rapid collisions and mergers of massive main-sequence stars during an early episode of cluster core collapse (Freitag et al. 2006a,b). Instead, our results can provide highly realistic initial conditions for dynamical simulations of dense star clusters in which the early phase of massive star evolution proceeded 'normally," without significant influence from cluster dynamics. Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 All stars are evolved based on the metallicity- and wind-mass-loss-dependent models of Hurley, Pols & Tout (2002), with a few improvements described in BKB02. The main code parameters we use correspond to the standard model presented in SS2 of BKB02 and are also described in Paper I. Each star, either single or a binary component, is placed initially on the zero-age main sequence (ZAMS) and then evolved through a sequence of distinct phases: main sequence (MS), Hertzsprung Gap (HG), red giant branch (RG), core He burning (CHeB), asymptotic giant branch (AGB); if a star gets stripped of its H-rich envelope, either through wind mass loss or Roche lobe overflow (RLOF) it becomes a naked helium star (He). The nuclear evolution leads ultimately to the formation of a compact object. Depending on the pre-collapse mass and initial composition this may be a white dwarf (WD), a neutron star (NS) or a BH. The population synthesis code allows us to study the evolution of both single and binary stars. Binary star components are evolved as single stars while no interactions are taking place. We model the following processes, which can alter the binary orbit and subsequent evolution of the components: tidal interactions, magnetic braking, gravitational radiation, and angular momentum changes due to mass loss. Binary components may interact through mass transfer and accretion phases. We take into account various modes of mass transfer: wind accretion and RLOF; conservative and non-conservative; stable or dynamically unstable (leading to common-envelope evolution). The mass transfer rates are calculated from the specific binary configurations and physical properties (masses, evolutionary stages, etc.) of the stars involved. Binary components may loose or gain mass, while the binary orbit may either expand or shrink in response. Moreover, we allow for binary mergers driven by orbital decay. In this study, we evolve binary merger products assuming that they restart on the ZAMS. An exception is made when a BH takes part in the merger, in which case we assume the remnant object to be a BH again. The mass of the merger product is assumed equal to the total parent binary mass for unevolved and compact remnant components; however we assume complete envelope mass loss from any evolved star (HG, RG, CHeB, or evolved He star) involved in a merger. A few additions and updates to StarTrack since Paper I are worth mentioning here (see Belczynski et al. 2006 for more details). System velocities are now tracked for all stars (single and binaries) after SNe (see SS2.3). The new magnetic braking law of Ivanova & Taam (2003) has been adopted, although this has minimal impact on our results for BHs. Two new types of WDs have been introduced: hydrogen and "hybrid" (these are possible BH donors in binaries). An improved criterion is adopted for CHeB stars to discriminate between those with convective (\(M<7{\,M_{\odot}}\)) and radiative (\(M\geq 7{\,M_{\odot}}\)) envelopes; this affects the stellar response to mass loss. We have also added a new tidal term for RLOF rate calculations. Some minor problems in the calculations of Paper I were also identified and are corrected in this study. The evolution of a small fraction of BH RLOF systems with donors at the end of the RG stage was terminated when the donor contracted and detached after entering a CHeB phase. However, the donor may restart RLOF during expansion on the AGB, which is now properly accounted for. Another small fraction of systems, evolving through the rapid RLOF phase with HG donors, were previously classified as mergers and subsequently evolved as single stars (merger products). However, the RLOF at that stage may be dynamically stable and in some cases a binary system may survive and continue its evolution, which is now also properly taken into account. None of these corrections affect the results of Paper I significantly. ### Black Hole Formation Black holes originate from the most massive stars. The formation time is calculated for each star using the stellar models of Hurley et al. (2000) and Woosley (1986). For intermediate-mass stars the FeNi core collapses and forms a hot proto-NS or a low-mass BH. Part of the envelope falls back onto the central object while the rest is assumed to be ejected in a SN explosion. We use the results of Fryer (1999) and Fryer & Kalogera (2001) to determine how much matter is ejected. In general, for the highest masses (\(>30\,M_{\odot}\) for low-metallicity models) total fall-back is expected, with no accompanying SN explosion. Motivated by the large observed velocities of radio pulsars we assume significant asymmetries in SN explosions. Here we adopt the kick velocity distribution of Arzoumanian, Cordes & Chernoff (2002), taking into consideration more recent observations (e.g., White & Van Paradijs 1996; Mirabel & Rodrigues 2003). NSs receive full kicks drawn from the bimodal distribution of Arzoumanian et al. (2002). Many BHs form through partial fall back of material initially expelled in a SN explosion, but then accreted back onto the central BH. For these the kick velocity is lowered proportionally to the mass of accreted material (for details see BKB02). For the most massive stars, the BH forms silently through a direct collapse without accompanying SN explosion, and in this case we assume _no_ BH natal kick. The mass loss and kick velocity together determine whether a binary hosting the BH progenitor is disrupted by the SN explosion. Our calculated initial-to-final mass relation for various metallicities is discussed in detail in Paper I, where it is also demonstrated that (within our BH kick model) for solar metallicity many BHs are formed with lowered kicks through fall back. This occurs for single stars with initial masses in the range \(20-42M_{\odot}\) and \(50-70M_{\odot}\). For metallicity \(Z=0.001\), BHs receive a kick in the narrower ranges \(18-25M_{\odot}\) and \(39-54M_{\odot}\), while for \(Z=0.0001\) only BHs formed from stars of \(18-24M_{\odot}\) receive kicks, with others forming silently. ### Spatial Velocities All stars, single and binaries, are assumed to have zero initial velocities. This means we are neglecting their orbital speeds within the cluster. Indeed, for a variety of reasons (e.g., relaxation toward energy equipartition, formation near the cluster center), massive stars (BH progenitors) are expected to have lower velocity dispersions	[]
	given system. Masses of single stars and binary primaries (more massive components) are drawn from the three-component, power-law IMF of Kroupa, Tout, & Gilmore (1993) (see also Kroupa & Weidner 2003) with slope \(\alpha_{1}=-1.3\) within the initial mass range \(0.08-0.5{\,M_{\odot}}\), \(\alpha_{2}=-2.2\) for stars within \(0.5-1.0{\,M_{\odot}}\), and \(\alpha_{3}=-2.35\) within \(1.0\,M_{\odot}-M_{\rm max}\). The binary secondary masses are generated from an assumed flat mass ratio distribution (\(q=M_{\rm a}/M_{\rm b}\); \(M_{\rm a},\ M_{\rm b}\) denoting the mass of the primary and secondary, respectively). The mass ratio is drawn from the interval \(q_{\rm min}\) to 1, where \(q_{\rm min}=0.08{\,M_{\odot}}/M_{\rm a}\), ensuring that the mass of the secondary does not fall below the hydrogen burning limit. The only exception is model B, in which both the primary and the secondary masses are sampled independently from the assumed IMF (i.e., the component masses are not correlated). This IMF is easily integrated to find the total mass contained in single and binary stars for any adopted \(\alpha_{1},\alpha_{2},\alpha_{3}\) values. The particular choice of low-mass slope of the IMF (\(\alpha_{1},\alpha_{2}\)) does not change our results, since low-mass stars do not contribute to the BH populations. However, as most of the initial stellar mass is contained in low-mass stars, a small change in the IMF slope at the low-mass end can significantly change the mass normalization. In our simulations, we do not evolve all the single stars and binaries described above since the low-mass stars cannot form BHs. Out of the total population described above we evolve only the single stars with masses higher than \(4{\,M_{\odot}}\) and the binaries with primaries more massive than \(4{\,M_{\odot}}\) (no constraint is placed on the mass of the secondary, except that it must be above \(0.08{\,M_{\odot}}\)). All models were calculated with \(10^{6}\) massive primordial binaries. We also evolved \(2\times 10^{5}\) massive single stars but then scaled up our results to represent \(10^{6}\) single stars1. The mass evolved in single stars and binaries was then calculated and, by extrapolation of the IMF (down to hydrogen burning limit), the total initial cluster mass was determined for each model simulation. Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 In the discussion of our results we assume an initial (primordial) binary fraction of \(f_{\rm bin}=50\%\), unless stated otherwise (i.e., tables and figures usually assume equal numbers of single stars and binaries initially, with 2/3 of stars in binaries). However, our results can easily be generalized to other primordial binary fractions \(f_{\rm{bin}}\) by simply weighing differently the numbers obtained for single stars and for binaries. Our assumed distribution of initial binary separations follows Abt (1983). Specifically, we take a flat distribution in \(\log a\), so that the probability density \(\Gamma(a)\propto{1\over a}\). This is applied between a minimum value, such that the primary's initial radius (on the zero-age main sequence) is half the radius of its Roche lobe, and a maximum value of \(10^{5}\,{\rm R}_{\odot}\). We also adopt a standard thermal eccentricity distribution for initial binaries, \(\Xi(e)=2e\), in the range \(e=0-1\) (e.g., Heggie 1975; Duquennoy & Mayor 1991). #### 2.4.4 Cluster Properties The only cluster parameter that enters directly in our simulations is the escape speed \(V_{\rm esc}\) from the cluster core. All single and binary BHs are assumed immediately ejected from the cluster if they acquire a speed exceeding \(V_{\rm esc}\). We do not take into account ejections from the cluster halo (where the escape speed would be lower) as all BHs and their progenitors are expected to be concentrated near the cluster center. In Tables 2, 3, 4, 5 we present results of simulations for our standard model corresponding to four different values of the escape speed: \(V_{\rm esc}=10,~{}50,~{}100,~{}300{\,{\rm km}\,{\rm s}^{-1}}\). For any assumed cluster model the escape speed can be related to the total mass \(M_{\rm cl}\) and half-mass radius \(R_{\rm h}\): \[V_{\rm esc}=f_{\rm cls}\,\left(\frac{M_{\rm cl}}{10^{6}{\,M_{\odot}}}\right)^{ 1/2}\,\left(\frac{R_{\rm h}}{1\,{\rm pc}}\right)^{-1/2}.\] (18) For example, for a simple Plummer sphere we have \(f_{\rm cls}=106{\,{\rm km}\,{\rm s}^{-1}}\), while for King models with dimensionless central potentials \(W_{0}=3,~{}5,~{}7,~{}9\) and 11, the values are \(f_{\rm cls}=105.2\), 108.5, 119.3, 157.7, and \(184.0{\,{\rm km}\,{\rm s}^{-1}}\), respectively. For our four considered values of the escape speed, \(V_{\rm esc}=10\), 50, 100, and \(300{\,{\rm km}\,{\rm s}^{-1}}\), in a \(W_{0}=3\) King model with \(R_{\rm h}=1\,\)pc (typical for a variety of star clusters), the corresponding cluster masses are \(M_{\rm cl}=0.009\), 0.226, 0.904, and \(8.132\ \times 10^{6}{\,M_{\odot}}\), respectively. In each table we present the properties of BH populations at five different cluster ages: 8.7, 11.0, 15.8, 41.7 and \(103.8\,\)Myr. These correspond to MS turnoff masses of 25, 20, 15, 8 and \(5\,M_{\odot}\), respectively. The tables include information on both the BHs retained in the clusters (with velocities \(<V_{\rm esc}\)) and those ejected from clusters. ## 3. RESULTS ### Standard Reference Model #### 3.1.1 Black Hole Spatial Velocities In Figure 2 we show distributions of spatial velocities for _all_ single and binary BHs shortly after the initial starburst (at \(8.7\,\)Myr). The distribution shows a rather broad peak around \(\sim 30-300{\,{\rm km}\,{\rm s}^{-1}}\), but also includes a large fraction (\(\sim\) 2/3) of BHs formed with no kick. The peak originates from a mixture of low-velocity binary BHs and high-velocity single BHs. The no-kick single and binary BHs originate from the most massive stars, which have formed BHs silently and without a kick. All the no-kick systems (with zero velocity assumed) were placed on the extreme left side of all distributions in Figure 2 to show their contribution in relation to other non-zero velocity systems (the bin area is chosen so as to represent their actual number, although the placement of the bin along the velocity axis is arbitrary). Binary stars hosting BHs survive only if the natal kicks they received were relatively small, since high-magnitude kicks tend to disrupt the systems. We see (middle panel of Fig. 2) that most BH binaries have spatial velocities around \(50{\,{\rm km}\,{\rm s}^{-1}}\), which originate from the low-velocity side of the bimodal Arzoumanian et al. (2002) distribution. Single BHs originating from single stars follow closely the bimodal distribution of natal kicks, but the final BH velocities are slightly lower because of fall-back and direct BH formation (see SS2.3). The low- and high-velocity single BHs have speeds around \(50{\,{\rm km}\,{\rm s}^{-1}}\) and \(250{\,{\rm km}\,{\rm s}^{-1}}\), respectively. Single BHs originating from binary disruptions gain high speeds (\(\sim 100-400{\,{\rm km}\,{\rm s}^{-1}}\)), since binaries	[]
	are disrupted when a high-magnitude kick occurs. Finally, the single BHs formed through binary mergers have the lowest (nonzero) velocities (\(\sim 10-100{\,{\rm km}\,{\rm s}^{-1}}\)), since they are the most massive BHs and therefore most affected by fall-back. In Figure 3 we show the velocity distributions at a later time (\(103.8\,\)Myr) when essentially all BHs have formed, and no more SNe explosions are expected, so the velocity distribution is no longer evolving (the MS turnoff mass for that time is down to \(5{\,M_{\odot}}\)). The velocities have now shifted to somewhat higher values (with a single peak at \(\simeq 200{\,{\rm km}\,{\rm s}^{-1}}\) for non-zero velocity BHs), while the relative contribution of no-kick systems drops to around 1/3. At this later time the population is more dominated by single BHs. Most of the non-zero velocity single BHs come from binary disruptions (see middle panel of Fig. 3) and therefore they have received larger kicks, shifting the overall distribution toward slightly higher velocities. Also, at later times, lower-mass BH progenitors go through SN explosions, and they receive on average larger kicks (since for lower masses there is less fall back). We note that most of the non-zero-velocity BHs gain speeds of \(50-200{\,{\rm km}\,{\rm s}^{-1}}\) in SNe explosions. Depending on the properties of a given cluster they may be ejected or retained, and will either populate the field or undergo subsequent dynamical evolution in the cluster. We now discuss separately the properties of the retained and ejected BH populations. #### 3.1.2 Properties of Retained (Cluster) BH Populations Retained BHs in clusters could be found either in binaries or as single objects. Binary BHs are found with different types of companion stars, while single BHs may have formed through various channels, which we also list in Tables 2 - 5. Shortly after the starburst the most frequent BH companions are massive MS stars, but, as the population evolves, these massive MS companions finish their lives and form additional BHs. Double BH-BH systems begin to dominate the binary BH population after about \(15\,\)Myr. At later times less massive stars evolve off the MS and start contributing to the sub-population of BHs with evolved companions (CHeB stars being the dominant companion type, with a relatively long lifetime in that phase) or other remnants as companions (WDs and NSs). Once the majority of stars massive enough to make BHs end their lives (around \(10-15\,\)Myr) we observe a general decrease in the total number of BHs in binaries. The number of BHs in binaries is depleted through the disruptive effects of SNe and binary mergers (e.g., during CE phases). Both processes enhance the single BH population. This single population is dominated by the BHs formed from primordial single stars (assuming \(f_{\rm bin}=50\%\)). The formation along this single star channel stops early on when all single massive stars have finished evolution and formed BHs (at \(\simeq 10-15\,\)Myr). In contrast, the contribution of single BHs from binary disruptions and mergers is increasing with time, but eventually it also saturates (at \(\simeq 50-100\,\)Myr), since there are fewer potential BH progenitor binaries as the massive stars die off. In general, the single BHs are much more numerous in young cluster environments than binary BHs. At early times (\(\simeq 10\,\)Myr) they dominate by a factor \(2-4\), but later the ratio of single to binary BHs increases to almost 10 (after \(\sim 100\,\)Myr), as many binaries merge or are disrupted (adding to the single population). #### 3.1.3 Properties of Ejected (Field) BH Populations Tables 2 - 5 show also the properties of BHs ejected from their parent clusters, assuming different escape speeds. Significant fractions (\(\gtrsim 0.4\)) of single and binary BHs are likely to be ejected from any cluster with escape speed \(V_{\rm esc}\lesssim 100{\,{\rm km}\,{\rm s}^{-1}}\). In general, single BHs are more prone to ejection since they gain larger speeds in SNe explosions (compared to heavier binaries). Early on the number of ejected BHs increases with time as new BHs of lower mass (and hence receiving larger kicks) are being formed. At later times (after \(\simeq 15\,\)Myr), the number of fast BHs remains basically unchanged. Ejected binaries consist mostly of BH-MS and BH-BH pairs in comparable numbers. Rare BH-NS binaries are ejected more easily than other types since they experience two kicks. Single ejected BHs consist mostly of BHs originating from single stars which have received large kicks and from the components of a disrupted binary (the involved kicks were rather large to allow for disruption). #### 3.1.4 Dependence on Cluster Escape Velocity and Initial Binary Fraction In Table 6 we list fractions of retained BHs at \(103.8\,\)Myr after the starburst. The results are presented for initial cluster binary fractions of \(f_{\rm bin}=0,\ 50,\ 100\%\), and can be linearly interpolated for the desired \(f_{\rm bin}\). For our standard model the results are shown for the four considered escape velocities. For an initial cluster binary fraction \(f_{\rm bin}=50\%\) we find that the retained fraction can vary from \(\sim 0.4\) for low escape velocities (\(V_{\rm esc}=10{\,{\rm km}\,{\rm s}^{-1}}\)) to \(\sim 0.9\) for high velocities (\(V_{\rm esc}=300{\,{\rm km}\,{\rm s}^{-1}}\)). For escape velocities typical of globular clusters or super star clusters (\(V_{\rm esc}\sim 50{\,{\rm km}\,{\rm s}^{-1}}\)), retained and ejected fractions are about equal. The retained fractions for various types of systems are plotted as a function of \(V_{\rm esc}\) in Figure 4. All curves are normalized to total number of BHs, both single and binaries. Results are listed in Table 6 for different binary fractions. In particular, these can be used to study the limiting cases of pure binary populations (\(f_{\rm bin}=100\%\)) and pure single star populations (\(f_{\rm bin}=0\%\)). Note that even an initial population with all massive stars in binaries will form many single BHs through binary disruptions and binary mergers. We also note the decrease of the retained fraction with increasing initial binary fraction. Clusters containing more binaries tend to lose relatively more BHs through binary disruptions in SNe compared to single star populations. #### 3.1.5 Orbital Periods of Black Hole Binaries Figure 5 presents the period distribution of BH binaries for our standard model (for the characteristic escape velocity \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\)). We show separately populations retained and ejected from a cluster. The distributions for different values of the escape velocity are similar. In Paper I we obtained a double-peaked period distribution for BHs in field populations: tighter binaries were found around \(P_{\rm orb}\sim 10\,\)d while wider systems peaked around \(P_{\rm orb}\sim 10^{5}\,\)d. The shape of this distribution comes	[]
	from the property that tighter BH progenitor systems experienced at least one RLOF/CE episode leading to orbital decay, while wider systems never interacted and stayed close to their initial periods. The two peaks are clearly separated with a demarcation period \(P_{\rm s}\sim 10^{3}\,\)d. It is easily seen here that slow and fast BH populations add up to the original double-peaked distribution of Paper I. Only the shortest-period and hence most tightly bound systems (\(P_{\rm orb}<P_{\rm s}\)) survive SN explosions and they form a population of fast, short-period BH binaries (see bottom panel of Fig. 5). In contrast, systems retained in clusters have again a double-peaked orbital period distribution. The slowest systems have rather large periods (\(P_{\rm orb}\sim 10^{5}\,\)d) and they will likely get disrupted through dynamical interactions in the dense cluster core. The short-period cluster binaries (\(P_{\rm orb}\sim 10-100\,\)d) are much less numerous, since most of the short-period systems gained high post-SN velocities and contributed to the ejected population. Compared to Paper I we note that the inclusion of ejections further depletes the cluster _hard_ binary BH population. Only about 1/3 of systems are found with periods below \(P_{\rm s}\), half of which are retained within a cluster with \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\). For a cluster with \(V_{\rm esc}=100{\,{\rm km}\,{\rm s}^{-1}}\) about 80% of the short-period systems are retained. #### 3.1.6 Black Hole Masses Black hole mass distributions are presented in Figures 6, 7 for \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\). With few exceptions the models for different escape velocity values are very similar. The retained and ejected populations are shown in separate panels. The retained populations of BHs shown in the top panel of Figure 6 have a characteristic triple-peaked mass distribution: a first peak at \(M_{\rm BH}\sim 6-8{\,M_{\odot}}\), a second one at \(M_{\rm BH}\sim 10-16{\,M_{\odot}}\), and third at \(M_{\rm BH}\sim 22-26{\,M_{\odot}}\); beyond this it steeply falls off with increasing mass. The shape of the distribution is determined by the combination of IMF and initial-to-final mass relation for single BHs (presented and discussed in detail in Paper I): the most massive stars (\(\geq 50{\,M_{\odot}}\)) form BHs with masses in the range \(\sim 10-16{\,M_{\odot}}\); stars within an initial mass range \(25-35{\,M_{\odot}}\) form BHs of mass \(\sim 25{\,M_{\odot}}\); stars of initially \(40-50{\,M_{\odot}}\) tend to form \(7{\,M_{\odot}}\) BHs. Both single and binary BHs contribute significantly to the second and third peaks. However, only single stars are responsible for a first narrow peak corresponding to a pile up of BHs in the initial-to-final mass relation around \(6-8{\,M_{\odot}}\). This characteristic feature is a result of a very sharp transition in single star evolution, from H-rich to naked helium stars, which is caused by wind mass loss and the more effective envelope removal for single stars above a certain initial mass. In binary stars, removal of the envelope can happen not only through stellar winds but also through RLOF, and so it is allowed for the entire mass range and the first peak is washed out. The ejected populations, shown in the bottom panel of Figure 6, are dominated by single BHs (due to their high average speeds) with masses \(\sim 3-30{\,M_{\odot}}\). The distributions have one sharp peak at \(M_{\rm BH}\sim 6-8{\,M_{\odot}}\), corresponding to the first low-mass peak in the distribution for retained populations. The high-mass BHs are very rare in the ejected populations since the kick magnitudes decrease with increasing BH mass (because of significant fall-back or direct BH formation at the high-mass end). BHs in binary systems reach a maximum mass of about \(30{\,M_{\odot}}\) for both cluster and ejected populations. Most single BHs have masses below \(30{\,M_{\odot}}\). However, the tail of the single BH mass distribution extends to \(\sim 50{\,M_{\odot}}\) for ejected populations and to about \(80{\,M_{\odot}}\) for cluster populations. This is shown in Figure 7 (note a change of vertical scale as compared to Figure 6). The highest-mass BHs are always retained in the clusters and they are formed through binary mergers. These mergers are the result of early CE evolution of massive binaries. The most common merger types are MS-MS, HG-MS and BH-HG mergers. During mergers involving HG stars we assume that the envelope of the HG star is lost, while the BH/MS star and the compact core of the HG star merge to form a new, more massive object. The merger product is then evolved and it may eventually form a single BH. Even with significant mass loss through stellar winds and during the merger process, a small fraction of BHs reach very high masses, up to about \(80{\,M_{\odot}}\). With a less conservative assumption, allowing some fraction of the HG star envelope to be accreted onto the companion in a merger, the maximum BH mass could then reach even higher values \(\gtrsim 100{\,M_{\odot}}\). In Figure 8 we show the results of a calculation with the merger product's mass always assumed equal to the total binary mass. Although the amount of mass loss in a merger is rather uncertain, the two models above (with and without mass loss) indicate that binary star evolution could lead to the formation of single \(\sim 100{\,M_{\odot}}\) BHs. These most massive BHs form very early in the evolution of a cluster (first \(\sim 5-10\,\)Myr) since they originate from the most massive and rapidly evolving stars. These BHs are retained in clusters (direct/silent BH formation with no associated natal kick) and they may act as potential seeds for building up intermediate-mass BHs through dynamical interactions during the subsequent cluster evolution (Miller & Hamilton 2002; O'Leary et al. 2005). ### Parameter Study #### 3.2.1 Black Hole Spatial Velocities For most alternative models the velocity distributions are similar to those found in the reference model (see Fig. 2 and 3). These distributions are generally characterized by the same wide, high-velocity peak (tens to hundreds of \({\,{\rm km}\,{\rm s}^{-1}}\)) and a rather large population of zero-kick BHs. In particular, for models B, F, G2, J and H, the distributions are almost identical to those of the reference model at all times. For models D, G1, and I the distributions show slight differences. With lowered CE efficiency (model D) it is found that there are fewer fast binary BHs, and most surviving binaries do not gain higher velocities at early times. Basically, many tight binaries that survived SN explosions in the reference model have now merged in a first CE phase, even before the first SN explosion occurred. In model G1, in which we consider only primordial stars up to \(M_{\rm max}=50\,{\,M_{\odot}}\), the population of massive BHs formed through direct collapse (with no kick) is significantly reduced. This results in a velocity distribution similar to that of the reference model for non-zero velocity systems, but with a much lower number of zero-kick BHs.	[]
	The model I distribution is slightly different, especially at early times when most BHs form with no kick, since in this model we consider only the most massive BHs formed mainly through direct collapse. A few models show more significant differences. Different metallicities lead to changes in BH velocities, especially at early times. For very low metallicity (model C1) almost all BHs are formed with no kick, while for high, solar-like metallicity (model C2) most BHs have non-zero velocities in a wide range (\(\sim 10-1000{\,{\rm km}\,{\rm s}^{-1}}\)). Metallicity strongly affects the wind mass loss rates, which are most important for the evolution of the most massive stars (i.e., at early times). In particular, for low-\(Z\) values, the wind mass loss rates are smaller (hence more high-mass pre-SN stars and direct collapses), while for high \(Z\) the winds are very effective in removing mass from BH progenitors (hence smaller mass pre-SN stars, and more fall-back BH formation). The most significant difference is found in model E, where we allow for full BH kicks. All BHs are formed with rather high (\(\geq 100{\,{\rm km}\,{\rm s}^{-1}}\)) velocities. The distribution, shown in Figures 15 and 16, is double-peaked both for early and late times. The single stars dominate the population, forming the low- (\(\sim 100{\,{\rm km}\,{\rm s}^{-1}}\)) and high-velocity component (\(\sim 500{\,{\rm km}\,{\rm s}^{-1}}\)), a direct result of the adopted bimodal natal kick velocity distribution. Binary stars are found at lower velocities (\(\sim 100{\,{\rm km}\,{\rm s}^{-1}}\)) but they are only a minor contributor to the overall BH population since most of them are now disrupted at the first SN explosion. #### 3.2.2 Properties of Retained (Cluster) BH Populations In Table 7 we present the properties of cluster BH populations \(11\,\)Myr after the starburst. Results for the various models may be easily compared with our reference model. Binary BHs for different model assumptions are still in general dominated by BH-MS and BH-BH binaries. These systems appear in comparable numbers in most models. Only for models B and E do we find a smaller contribution of BH-BH binaries (\(\sim\) 5% and almost zero for models E and B, respectively). In model B the independent choice of masses produces systems with extreme mass ratios, so that massive primordial binaries with two BH progenitors are very rare. Obviously for model E, in which the two BHs receive full kicks, the BH-BH binary formation is strongly suppressed by binary disruptions. The highest number of binaries containing BHs is found in our model with the lowest tested metallicity (C1). For low metallicities BHs form preferentially with high masses (low wind mass loss rates) through direct collapse with no kick. In contrast model E, assuming full BH kicks, results in the lowest number of BH binaries. Many models (D, G2, H, I, J) result in very similar contents to our reference model. It is worth noting in particular that the CE treatment (either lowered efficiency in model D, or different prescription in model J) does not appear to play a significant role in determining cluster initial binary BH populations. For all models the single BHs dominate the population even at very early times (as early as \(11\,\)Myr). Single BHs originate predominantly from primordial single stars, with smaller contributions from disrupted binaries and binary mergers. The basic general trends seen in our reference model are preserved in other models. Also most models (B, C1, D, G2, H, I, J) form similar numbers of single BHs as our reference model. It is found, as in the binary populations, that the highest number of single BHs is seen in our model with lowest metallicity (C1), while the model with full BH kicks (E) generates the lowest number of single BHs retained in a cluster. At \(103.5\,\)Myr (see Table 8), when no more BHs are being formed, single BHs strongly dominate (by about an order of magnitude) over binary BHs. Single BHs still originate mostly from primordial single stars, but there is an increased contribution from binary mergers and disruptions. The binary population remains dominated by BH-BH and BH-MS systems in most models, but with an increased contribution from other evolved systems (BH-WD and BH-NS) compared to earlier times. Note that only in model E does the number of systems other than BH-BH and BH-MS end up dominating the binary population. #### 3.2.3 Properties of Ejected (Field) BH Populations In Tables 7 and 8 we also characterize the populations of ejected (field) BHs for various models. Results for both times are comparable for binary BHs, but with significantly more single BHs being ejected at later times. The BH-MS and BH-BH binaries, which dominate the total populations, are also found to be most effectively ejected from clusters. However, BH-NS systems, receiving two natal kicks, are also found to be easily ejected. Indeed, in many models (C1, C2, E, F, G1, G2, H, J), they constitute a significant fraction of ejected systems. Contrary to our intuitive expectation, evolution with the full BH kicks (model E) does not generate a particularly large population of fast BH binaries. In fact, the ejected population is smaller than in the reference model. Higher kicks are much more effective in binary disruption than in binary ejection. The numbers of fast single BHs are comparable in most models (A, B, D, G1, G2, H, J), with the ejected populations usually consisting equally of BHs coming from binary disruptions and primordial single stars, with a smaller contribution from merger BHs. For models with massive BHs (C1 and I) which receive small kicks there are fewer single BHs in the ejected population (by a factor \(\sim 2\)). On the other hand, for the model with full BH kicks (E), the ejected single BH population is larger (by a factor of \(\sim 3\)) compared to the reference model. #### 3.2.4 Dependence on Cluster Escape Velocity and Initial Binary Fraction Retained fractions for different evolutionary models follow in general the same trends as in our reference model, i.e., retained fractions decrease with increasing initial binary fraction. The exception to that trend is for models with full BH kicks (E), increased metallicity (C2) or uncorrelated binary component masses (B). Also, independent of the escape velocity, it is found that at least \(\sim 40\%\) of BHs are retained simply because of no-kick BHs (for an initial binary fraction of 50%), with the obvious exception of the model with full BH kicks (E). In particular, for models C1, D, F, G1, G2, H, I, and J, the dependence of the retained fraction on \(V_{\rm esc}\) is very similar to that seen in the reference model (see Fig. 4). In model B, th secondary mass is on average very small compared to the BH mass (due to our choice of initial conditions for this model). Therefore, BHs in binary systems	[]
	gain similar velocities (almost unaffected by their companions) as single BHs, and this leads to almost constant fraction of retained systems (\(\sim 0.64\)) independent of the initial binarity of the cluster. In models C2 and E the fraction of retained systems may be as small as 0.4 and 0, respectively. In our model with high metallicity (C2), as discussed above (SS 3.2.1), high wind mass loss rates lead to higher BH kicks and hence smaller retained fractions. The most dramatic change is observed for model E, with full BH kicks. The retained fractions for this model are shown in Figure 17. Here, we also normalize all curves to the total number of BHs (single and in binaries). The retained fraction increases from 0 to \(\sim 0.9\), approximately proportional to the escape velocity, with no apparent flattening up to \(V_{\rm esc}\sim 1000{\,{\rm km}\,{\rm s}^{-1}}\) as a result of the high speeds BHs receive at formation. The total retained fraction does not reach unity, since there is still a small number of BHs with velocities over 1000 km s\({}^{-1}\). Larger BH kicks (switching from standard lowered kicks to full kicks) decrease the retained fraction from 0.6 to 0.2 for \(V_{\rm esc}\simeq 50{\,{\rm km}\,{\rm s}^{-1}}\) and \(f_{\rm bin}50\%\). A summary of retained and ejected fractions for different initial cluster binary fractions is presented in Table 6 for \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\). In particular, we show results for pure single star populations (\(f_{\rm bin}=0\%\)) and for all binaries (\(f_{\rm bin}=100\%\)). Note that single star populations will obviously form only single BHs, while the binary-dominated clusters will form both BH binaries and single BHs (through disruptions and mergers)2. Footnote 2: The number of single BHs formed out of binary systems may be inferred by comparing the numbers of binary BHs with the single BHs listed under “binary disruption” and “binary mergers” in Tables 2–5, 7 and 8 For the standard \(f_{\rm bin}=50\%\) it is found that the retained fraction of BHs varies from 0.4 - 0.7 across almost all models. The only exception is model E with full BH kicks for which the retained fraction is only 0.2. For more realistic and higher initial cluster binary fractions (\(f_{\rm bin}=75-100\%\); see Ivanova et al. 2005) the retained BH fraction is found in an even narrower range 0.4 - 0.6 (again with the exception of model E). Therefore, despite the number of model uncertainties, the initial BH cluster populations, as far as the numbers are concerned, are well constrained theoretically. The issue of BH kicks is not resolved yet, but both observational work (e.g., Mirabel & Rodrigues 2003) and theoretical studies (e.g., Willems et al. 2005) are in progress. #### 3.2.5 Orbital Periods of Black Hole Binaries In Figures 9 and 10 we show the dependence of the period distributions of BH binaries on model assumptions. In general, the period distribution remains bimodal in most of the models (B, C1, C2, D, F, G1, G2, H) for retained BH binaries, while only short-period binaries tend to be ejected from clusters, as explained in SS 3.1.5. Most of the retained binaries are formed with rather large orbital periods (with the exception of model E, see below) and they will be prone to dynamical disruption in dense cluster environments. The major deviations from the reference model are found for model E, with full BH kicks. The retained population is rather small as compared to the other models and consists mostly of short-period binaries, since all of the wide BH systems were disrupted by SN natal kicks. The majority of short-period binaries which survived gained significant velocities (\(\geq 50{\,{\rm km}\,{\rm s}^{-1}}\); see Fig. 16) and the ejected population is the most numerous in this model. In several other models we find smaller variations from the reference period distribution. Models with different CE efficiency and treatment (D and J), in which most close binaries merge, have very small numbers of short-period binaries. Also, the model in which we consider only the most massive BHs (descendants of wide primordial binaries) is characterized by a smaller short-period binary population. #### 3.2.6 Black Hole Masses In Figures 11 and 12 we present the BH mass distributions from all the models in our study, for both single and binary BH populations. The shape of the distribution for the retained and ejected BH populations is not greatly affected by different choices of parameter values, with the exception of metallicity and BH kicks (see Fig. 11). This is easily understood, as the highest-mass BHs are formed only at low metallicity (Models C1 and A) and the lightest BHs are formed at high metallicity (Model C2). For full BH kicks (model E) the majority of BHs gain high speeds, and the mass distribution for the ejected population is similar to the combination of ejected and retained populations in the reference model. Most of the BHs do not exceed \(\sim 25{\,M_{\odot}}\). However, a small fraction of single BHs in many models reach very high masses around \(80\,{\,M_{\odot}}\). Figures 13 and 14 show the mass distributions of single BH subpopulations. In all the models with high-mass BHs the most massive BHs are formed through binary mergers. In all the calculations presented here we have assumed mass loss during the merger process if an evolved star was involved (as discussed in SS 3.1.6). The highest maximum BH masses are found in the lowest metallicity environments (models A and C1), in larger systems (with high \(M_{\rm max}\), model G2), and for binaries formed with full BH kicks (model E), quite independent of other evolutionary parameters. We find \(\sim 10-100\) BHs with masses over \(60{\,M_{\odot}}\) in models C1, C2, E, G2 and J (and fewer in other models), for a total starburst mass of \(\sim 10^{8}{\,M_{\odot}}\) (see Table 1). The highest mass BHs are retained in clusters with the exception of the model incorporating full BH kicks, in which they are found both with high and low speeds. Only in a few models, with uncorrelated initial binary component masses (B), low CE efficiency (D), or low \(M_{\rm max}\) (G1), does the maximum BH mass stay below \(\sim 60{\,M_{\odot}}\). And in particular, in the model B, the maximum BH mass stay below \(\sim 30{\,M_{\odot}}\). This is due to the fact, that in this model BHs are accompanied by relatively low mass companions and therefore there is no mass reservoir to increase substantially initial (formation) BH mass. ## 4. SUMMARY AND DISCUSSION Using the population synthesis code StarTrack we have studied the formation of single and binary BHs in young star clusters. Our study continues and improves on the initial work described in Paper I by taking explicitly into account the likely ejections of BHs and their progenitors from	[]
	star clusters because of natal kicks imparted by SNe or recoil following binary disruptions. The results indicate that the properties of both retained BHs in clusters and ejected BHs (forming a field population) depend sensitively on the depth of the cluster potential. For example we find that most BHs ejected from binaries are also ejected from clusters with central escape speeds \(V_{\rm esc}\lesssim 100\,{\rm km}\,{\rm s}^{-1}\), while most BHs remaining in binaries are retained by clusters with \(V_{\rm esc}\gtrsim 50\,{\rm km}\,{\rm s}^{-1}\). Also, approximately half of the single BHs originating from the primordial single star population are ejected from clusters with \(V_{\rm esc}\lesssim 50\,{\rm km}\,{\rm s}^{-1}\). The overall BH retention fraction increases gradually from \(\sim 0.4\) to 0.7 as the cluster escape speed increases from \(\sim 10\) to \(100{\,{\rm km}\,{\rm s}^{-1}}\) (Fig. 4). Tables 2-5 give the numbers of BHs in different kinds of systems, both retained in and ejected from clusters with different escape speeds. Their main properties are illustrated in Figures 5-8. Single BH masses can become as large as \(\sim 100\,M_{\odot}\) (as a consequence of massive binary mergers, especially if mass loss during mergers is small). These "intermediate-mass" BHs are almost always retained in clusters. If they were to acquire a new binary companion through dynamical interactions in the dense cluster environment, they could become ULXs. However, it was recently demonstrated that although massive BHs easily acquire binary companions, it is rather unlikely to find such a binary at high ultraluminous X-ray luminosity (Blecha et al. 2006). BH-BH binaries (rather than double NSs), are probably the most promising GW sources for detection by ground-based interferometers (Lipunov, Postnov, & Prokhorov 1997; Bulik & Belczynski 2003). Merging BH-BH systems therefore are important sources for present projects to detect astrophysical GW sources (e.g., GEO, LIGO, VIRGO). The properties of BH-BH binaries in much larger stellar systems with continuous star formation (e.g., disk galaxies) were studied extensively by Bulik & Belczynski (2003; Bulik, Belczynski & Rudak 2004a; Bulik, Gondek-Rosinska & Belczynski 2004b). We find that the properties of BH-BH binaries in starbursts are not too different from those found in previous studies. Most BH-BH systems are characterized by rather equal masses, with a mass ratio distribution peaking at \(q\simeq 0.8-1.0\) (cf. the Pop II models of Bulik et al. 2004b). For most models only a small fraction (a few per cent; e.g., 5% for Model A) of the BH-BH systems are tight enough to merge within a Hubble time and produce observable GW signals. For models which tend to produce tighter BH-BH binaries (D and E), the fraction can be significantly higher (\(\sim\) 10-40 %). However, in Model E there are almost no BH-BH binaries, and the higher fraction of coalescing systems does not mean a higher BH-BH merger rate. Models C1 and D are the most efficient in producing merging BH-BH binaries: 2035 and 1370, respectively (for a total starburst mass of \(\sim 10^{8}{\,M_{\odot}}\); Table 1), while for most of the other models (including the reference model) we find \(\sim 300-600\) merging BH-BH systems. This work was supported in part by KBN grants PBZ-KBN-054/P03/2001 and 1 P03D 022 28 at the Copernicus Center, Poland, and by NSF Grant PHY-0245028 and NASA Grants NAG5-12044 and NAG5-13236 at Northwestern University. 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	# How Far Away are Gravitational Lens Caustics? Wrong Question Sun Hong Rhie ###### Abstract It has been a persistent question at least for a decade where the gravitational lens caustics are in the radial direction: whether in front of the lensing mass, behind the lensing mass, or on the plane normal to the line of sight that passes through the lensing mass, the radiation source, or the observer. It is a wrong question. And, the truth angers certain referees who somehow possess the ability to write lengthy rubbish referee reports and delay certain papers indefinitely. General relativity is a metric theory, particularly of Riemannian geometry, which is characterized by the existence of an inner product - or, the invariance of the proper time. According to Einstein field equations, a compact mass defines a spherical geometry around it and focuses photons from a distant source to an observer with the source and observer as the two focal points. When the mass is spherically symmetric, the two dimensional lens equation that relates the angular positions of a source and its images defines a point caustic at the angular position of the lensing mass. The third (radial) position of the point caustic is not defined. For an arbitrary mass, the caustic extends into a web of piecewise smooth curves punctuated by cusps and again its notion exists only within the context of the lens equation. We point out a few errors in a couple of papers, published in the Astrophysical Journal, which may be influential. : _Subject headings:_ gravitational lensing ## 1 Spherical Geometry and Focal Points If we consider a 2-sphere, the geodesics ("straight lines") are the great circles and any two great circles intersect in two places that are known as antipodal points in geography. There are no parallel geodesics on the sphere, and that is the well-known break-down of the fifth axiom (or postulate; see section 1.1) of plane geometry (or Euclidean geometry). Given a pair of antipodal points such as north pole and south pole, there are infinitely many geodesics that diverge from one antipodal point and converge to the other. The antipodal points are the foci of the geodesics.	[]
	The 2-sphere (\(S^{2}\)) is characterized by a constant Gaussian curvature, and Gaussian curvature is nothing but the Riemann curvature scalar where the non-unity constant between them is just a matter of historical conventions. In two dimensions, Riemann curvature tensor has only one independent component, and the Einstein tensor is given in terms of the Riemann curvature scalar and "trivial tensor structure" built from metric components. For the homogeneous space \(S^{2}\), the scalar curvature is constant - Gauss curvature. (Weinberg1, p.144; Weinberg hereafter) Footnote 1: _Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity_, Steven Weinberg, John Wiley & Sons, Inc., 1972; Weinberg henceforth. Just for a moment, let's stipulate for the purpose of illustration that the longitudes on \(S^{2}\) are photon paths between a radiation source at the north pole and an observer at the south pole. A image of an object an observer sees in its detector (say an array of light receptors including counters with \(100\%\) efficiency, perfect homogeneity, and infinite resolution) is a distribution of photons on the detector that last-scattered the object (the last-scattering surface either of a radiation emission source that generates the photons, or of a reflecting object such as a planet, etc) whose image the observer seeks. Thus, if we assume that the radiation emission source at the north pole emits photons isotropically, the image of the radiation emission source (of the radius of a "point") that the observer at the south pole sees in its detector will be a circle (unless the detector is set exactly at the focal point and is burned; if the emission is directional such as in lasing, the image will not be a circle because most of the geodesics that connect the source and observer will not be followed by the photons from the "point" source.) When the observer visits the north pole, the observer will find through meticulous measurements around the north pole that the radiation source has the size of a "point" and the shape of the radiating surface is not a circle. If we allow the observer and source for the third dimension around the north pole for the sake of imagination, the observer will be able to take a mugshot of the radiation surface which will show up as a "point" in its detector. In fact, the observer will most likely bring a lab-prepared isotropically emitting "point" light source to set up at the north pole and confirm that the "point source" indeed produces a circular image when measured at the south pole. This dimensional change from a zero-dimensional point source to a one-dimensional circular image results in the point caustic of a spherically symmetric lensing mass in the context of the lens equation as we shall review in the following section. It should be useful to note that focal points are also referred to as conjugate points. For example, two antipodal points are a pair of conjugate points; Or, the south pole is the	[]
	conjugate point of the north pole. ### Axioms of Euclid Geometry We list the five axioms of Euclid which can be found in Euclid's textbook _Elements_ written around 300 BC and had been used well into the 20th century. 1. 1.A straight line can be drawn from any point to any point. 2. 2.In a straight line, a finite straight line can be produced continuously. 3. 3.A circle with any center and radius can be described by introducing a straight line segment as the radius and one of its end points as the center. 4. 4.All right angles equal one another. 5. 5.If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. In order to avoid causing unnecessarily rigid authoritative impression of the five postulates (or axioms) of Euclid, we remark that the axioms of plane geometry have evolved. For example, seek the axioms by George Birkhoff who is perhaps better known for Birkhoff theorem (that the metric of a spherically symmetric mass is static and given by Schwarzschild metric). We should further note that Euclid was reluctant to use the fifth axiom and his postulates are sometimes known as 4+1 postulates. Considering the length of the fifth axiom, it is tempting to jump to a "fuzzy axiom" that _simplicity is the beauty of truth_. ## 2 The Lens Equation of a Spherically Symmetric Mass ### Newtonian Deflection Angle In 1911, following up on his 1907 article on the gravitational influence on the propagation of light,2 Einstein published a calculation of the gravitational deflection angle by the Sun in an article titled "_Uber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes_," _Annalen	[]
	der Physik_, 35. 3 Einstein got the deflection angle too small by factor 2 by considering the gravitational effect on the clock correctly but omitting the effect on the measuring rod. In other words, he got Newtonian deflection angle. It was before the systematic exposition of the general relativity culminating in what are now referred to as Einstein field equations (1916). We take it as a historical lesson that we resort to the Einstein field equations whenever in doubt. Footnote 2: A. Einstein, Jahrbuch für Radioact. und Elektronik, 4, 1907. Footnote 3: “On the influence of gravitation on the propagation of light” (translation) in “The Principle of Relativity: A collection of original memoirs on the special and general theory of relativity,” H.A. Lorentz, A. Einstein. H. Minkowski, and H. Weyl, Methuen & Co, LTD., 1923. The masslessness of the light particles or \(U(1)\) gauge invariance was not known in the era of Newton, but Newton's equation of motion of two particles under the mutual gravitation in the center of mass coordinates can be "naturally" extended to describe the motion of a massless photon under the gravitational influence of a mass \(M\). If the masses of the two particles are \(m_{1}\) and \(m_{2}\), and their reduced mass is \(m:\ m^{-1}=m_{1}^{-1}+m_{2}^{-1}\), the motion of the reduced mass \(m\) (with relative position vector \(x=r\hat{r}\)) is given by two equations in spherical coordinates where the scattering plane is chosen by \(\theta=\pi/2\). \[m\,E={m\over 2}\dot{r}^{2}+{m\over 2}{J^{2}\over r^{2}}-{m\,GM\over r}\ ; \qquad m\,J=m\,r^{2}\dot{\varphi}\ ,\] (1) where \(E\) and \(J\) are the (conserved) energy and angular momentum per unit reduced mass, and \(M\) is the total mass. For the motion of photons, divide the equations with the reduced mass \(m\neq 0\) and consider the resulting equations in the limit of \(m\to 0\). If impact parameter is denoted \(b\) (as usual), \[b\equiv\lim_{t\rightarrow-\infty}r\sin\,(\varphi-\varphi_{-})\] (2) where \(\varphi_{-}\equiv\varphi(t=-\infty)\). The energy and angular momentum of the photon can be determined from \(b\) and eq. (1) with \(t=-\infty\). \[E={1\over 2}\ ;\qquad J=b\ .\] (3) The stationary condition \(\dot{r}=0\) has one solution because \(E>0\), and that is the distance (\(r_{0}\)) of the closest approach of the photon to the mass \(M\). \[r_{0}=-GM+(G^{2}M^{2}+J^{2})^{1/2}\] (4) Since \(J>0\), the azimuthal angle increases with time (\(\dot{\varphi}>0\)).	[]
	the equation of the orbit. The procedure is similar to that of the Newtonian scattering in the previous subsection (except that the integral can not be expressed in terms of elementary functions) and can be found in any textbooks on general relativity (e.g., Weinberg, Chap. 8). In the linear regime \(r>>GM\), we obtain the Einstein deflection angle \[\delta\varphi={4GM\over r_{0}}\ ,\] (20) where \(r_{0}\) is the distance of the periastron of the photon trajectory. It is worth noting that \(g_{tt}=-B\) and \(g_{rr}=A^{-1}\) equally contribute to the Einstein deflection angle, and the Newtonian deflection angle amounts to the contribution from \(g_{tt}\). The factor 2 discrepancy between the Newtonian and Einstein deflection angles are the well-known general relativistic factor 2, which was tested in 1919 during the eclipse notably by Eddington and his crew and numerously since. It should be useful to note that the variation of the action \(S\) in eq. (12) leads to the "standard" geodesic equation, \[0=\ddot{x}^{\lambda}+\Gamma^{\lambda}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\ ,\] (21) where \(\Gamma^{\lambda}_{\mu\nu}\) is the affine connection. Hence the path parameter \(p\) is a so-called affine parameter which is a linear function of the proper time. If we choose an arbitrary parameter, the equation develops extra terms as one can check easily. ### The Lens Equation The source star and the observer are far away from the lensing mass \(M\), and there the metric is effectively flat. The coordinates have been chosen such that \(B,\,A\to 1\) as \(r\rightarrow\infty\), hence the observer should feel relaxed to use the familiar flat space coordinate systems to make local measurements or to chart the sky knowing that the coordinate systems are valid all the way from the observer's neighborhood to the neighborhood of the source star except inside the star. In the asymptotically flat coordinate system, the photon arriving at the observer's detector after a long flight along a null geodesic would seem to come from a position in the sky that differs from the position of the source star where the latter is determined by hypothetically turning off the gravity by setting the Newton's constant \(G=0\). The relation between a source position and its images in the observer's sky is the lens equation. Fig. 1 shows an overlay of the observer's sky on the orbital plane \(\theta=\pi/2\) and a photon trajectory (null geodesic) that connects the source and the observer. The position angles of the lens, source star, and image are depicted as \(\gamma\), \(\beta\), and \(\alpha\), and they are related to \(r_{0}\) and	[]
	\(\delta\varphi\) by the followings. \[b=D_{\ell}\sin(\alpha-\gamma)\ ;\] (22) \[D_{s}\sin(\alpha-\beta)=\tan\delta\varphi\left(D_{s}\cos(\alpha- \beta)-D_{\ell}\cos(\alpha-\gamma)-b\tan(\delta\varphi/2)\right)\ ,\] (23) where \(D_{\ell}\) and \(D_{s}\) are the (radial) distances to the lens mass \(M\) and the source star from the observer. In the linear order small angle approximation, the equations (22) and (23) become \[b=D_{\ell}(\alpha-\gamma)\ ;\qquad D_{s}(\alpha-\beta)=\delta\varphi(D_{s}-D_{ \ell})\ .\] (24) Using the Einstein deflection angle \(\delta\varphi=4GM/r_{0}\) and \(r_{0}=b-GM\approx b\), the familiar single lens equation is obtained. \[\alpha-\beta={4GM\over\alpha-\gamma}\left({1\over D_{\ell}}-{1\over D_{s}} \right)\equiv{\alpha_{E}^{2}\over\alpha-\gamma}\] (25) Given the angular positions of a source (\(\beta\)) and a lens (\(\gamma\)), there are usually two solutions for \(\alpha\); the source, lens, and two images are collinear in the sky. If the periastron distance \(r_{0}=R_{E}=D_{1}\alpha_{E}\), however, the lens and source are aligned along the line of sight (\(\beta=\gamma\)), and the image forms on a ring due to the symmetry around the axis connecting the observer and the mass \(M\). The angular radius of the ring image is given by \(\alpha_{E}\). (There are two solutions \(\alpha=\pm\alpha_{E}\) for \(\theta=\pi/2\), and the axial symmetry allows two solutions for each value of \(\theta=[0,\pi]\) resulting in solutions of two half circles of the same radius \(\alpha_{E}\). See section 2.4.1 for a way to understand the transition from two images to two half-circle images.) The circular image is known as Einstein ring, hence the subscript \(E\) in \(\alpha_{E}\). The gravity of the mass \(M\) makes the geometry of the space around it spherical and focuses photons from the source to the observer; The source and observer are the two focal points of the geodesics that connect them. There are usually two geodesics that connect the source and observer, hence two images of a given source. When the source and lens are aligned (as seen from the observer with \(G=0\)), infinitely many null geodesics connect the source and the observer, similarly to the longitudes connecting the south and north poles on \(S^{2}\) discussed in section 1; The source and observer are the two focal points of the infinitely many photon trajectories, and the observer sees a circular image of the source. In a following section, we shall see that the Einstein ring is also the critical curve. The caustic is by definition the curve onto which the critical curve is mapped, and the caustic of the single lens is a point caustic since the entire critical curve is mapped to one point under the lens equation. In fact, the point caustic is a degenerate cusp. A cusp is by definition a point onto which a precusp - a critical point where the critical eigenvector is tangent to the critical curve - is mapped	[]
	. #### 2.3.1 A Diversion: Einstein Ring or Chwolson Ring? We find in monograph "Gravitational Lenses", P. Schneider, J. Ehlers, and E. E. Falco, 1993, p. 4 that Chwolson _remarked_ in "_Uber eine mogliche Form fiktiver Doppelsterne_" of a ring-shaped image of the background star centered on the foreground star. Schneider et al. states that the circular image should be called Chwolson ring instead of Einstein ring. Chwolson's main concern in the short article was a spectroscopic double star made of the foreground star and one (faint) image of the background star near the foreground lensing star. We recognize that the title "On a possible form of fictitious double stars" may invoke in the minds of today's readers of a spectroscopic double star made of the foreground star and the images of the aligned background star instead, which may be more practically referred to as "spectroscopic differentiation of blending". Chwolson did not consider the lensing effects on the fluxes of the images and also concluded with a sentence "Whether the case of a fictitious double star considered here actually occurs, I can not judge." Then, it is very curious what Chwolson might have imagined for the flux of the ring image of which he did not make any statements. If he presumed the same flux as that of the unlensed background star as he did in his article for the (faint) image, what must he have thought of the flux density around the ring? Distribute two background star fluxes around the ring because two images turn into a ring image? Distribute infinitely many point source stars around the ring? Which will result in an infinitely bright point star to an observer with a poor resolution detector such as human eyes? Or, then, distribute finitely many background stars around the ring because the stars have finite sizes? Then, what was the size of the ring? What if the ring size is not an integer multiple of the size of the stars - which in fact will be mostly the case apart from very rare coincidences? We do not have access to Chwolson's article and can only guess from Schneider et al. that Chwolson must have recognized the axial symmetry but did not have enough details or interest to determine (or conjecture) the radius of the ring image. And, Chwolson was not aware of the lensing effect on the image fluxes. Then, it is possible that Chwolson's remark on the ring image was mainly to point out the deviated situation from the case of a spectroscopic double star of which the latter was of his interest. It seems to be quite reasonable to use the term _Einstein ring radius_ since Einstein was the first author to calculate the radius of the circular image. Einstein wrote in 1936 5, "An observer will perceive ... instead of a point-like star A, a luminous circle of the angular	[]
	radius \(\beta\) around the center of B, where \(\beta=\sqrt{\alpha_{0}R_{0}/D}\)." Einstein calculated the ring radius in an approximation where the distance \(D\) to the lensing star (_e.g._, lensing by the Sun) is much smaller than the distance to the source star. (\(\alpha_{0}\) is the deflection angle of a photon trajectory grazing the surface of the foreground star and \(R_{0}\) is the radius of the foreground star.) Einstein calculated the total magnification of the images (the second equation in the article) and knew that the ring image is luminous. He wrote, "This apparent amplification \(q\) ... is a most curious effect, not so much for its becoming infinite, ... ", and didn't seem to have concerned himself with the formal divergence. Physical objects are never a point and physical quantities are never exactly a delta function. In fact, Dirac delta function exists as a distribution which is defined within the context of integration. The third equation in the Science article shows \(q=1/x\,(x\to 0)\) where \(x\) is the angular distance between the stars A and B. Integration of the apparent amplification over a small disk \({\cal D}\) centered at \(x=0\) Footnote 5: “Lens-Like Action of a Star by the Deviation of the Light in the Gravitational Field”, Science, 84 (1936), p.506. \[\int_{\cal D}\Sigma(x_{1},x_{2})\,q\,dx_{1}dx_{2}=\int\Sigma(x_{1},x_{2})\,q\, xdx2\pi\] (26) is well-defined and finite as far as the stellar surface flux density \(\Sigma(x_{1},x_{2})\) is regular, which indeed is the case. Then, how do we recognize that Chwolson was aware of and wrote in 1924 that the image would be a ring centered at the foreground star when the foreground and background stars are aligned (always meaning as seen from the observer with \(G=0\))? Should we make the recognition by calling the circular image Chwolson ring instead of Einstein ring? We consider a few aspects before casting an intellectually reasonable vote. * Chwolson ring refers to a circular image. Einstein ring also refers to a circular image. (Precisely speaking, two half-circle images which should be distinguishable by putting trace markers on the background star radiation pattern.) Chwolson mentioned it in 1924, and Einstein calculated it 12 years later in 1936. Chwolson ring is an image ring with unspecified radius and unspecified characteristics. Einstein ring is an image ring with a specified radius and flux characteristics determined from the involved physical variables. *It seems most reasonable to call \(\alpha_{E}\) the Einstein ring radius because Einstein is the one who first recognized the importance of the radius and calculated it. Now if we choose to call a ring image a Chwolson ring, the radius of the Chwolson ring may be most naturally called Chwolson ring radius. If Chwolson had studied the characteristics of the image ring (such as the flux) establishing its physical nature and simply left out the radius in his article to be supplemented by, say, Einstein 12 years later, it would	[]
	be reasonable to call the image ring Chwolson ring and its radius Chwolson-Einstein radius. But Chwolson did not. Considering the historical facts above, then, it may be reasonable to call the image ring Chwolson-Einstein ring and the ring radius Einstein ring radius. * For general lenses, Einstein ring and Einstein ring radius of a mass do not refer to the shape and radius of an image (or conjoined two images) but refer to an imaginary circle and its radius that function as a scale disk where the functionality derives from the Einstein ring formula Einstein derived for the first time (albeit for a case where the distance to the lensing star is small). Since Chwolson ring is an image ring without physical functionality, it seems too big a leap, for example, to call the Einstein ring of the total mass of a binary lens Chwolson ring. *Thus, we conclude that it is most reasonable to call Einstein ring and Einstein ring radius Einstein ring and Einstein ring radius in their most general contextual significances as is the current practice. On the other hand, it seems reasonable to call an observational circular image Chwolson-Einstein ring and its ring radius Einstein ring radius. ("Circular images" are being found in cosmological lensings by galaxies - extended mass distributions whose surface densities are not necessarily circularly symmetric, and the "circular images" are not exactly circular even when the emission source is centered at the center of the caustic.) ### The Linear Differential Behavior of the Lens Equation The lens equation in eq. (25) is written in terms of the variables defined in the orbital plane \(\theta=\pi/2\). In order for the lens equation to describe the lensing corresponding to photon trajectories in an arbitrary orbital plane, the lensing variables should be expressed in terms of the two dimensional variables defined in the observer's sky. If \(x\), \(s\), and \(y\) are the two dimensional angular positions of an image, its source, and the lens, the lens equation is given by \[x-s=\alpha_{E}^{2}{x-y\over(x-y)^{2}}\ .\] (27) In terms of complex coordinates, \[\omega=z-{\alpha_{E}^{2}\over\bar{z}-\bar{x}_{1}}\ ;\qquad(\alpha_{E}=1)\] (28) where \(z\), \(\omega\), and \(x_{1}\) are the complexifications of \(x\), \(s\), and \(y\). It is convenient and customary to normalize the lens equation so that \(\alpha_{E}=1\) (as is indicated in the parenthesis).	[]
	The lens equation is an explicit function from a two-dimensional image variable to a two-dimensional source variable. In other words, the lens equation is a mapping from a complex plane to itself. The Jacobian matrix of the lens equation describes its linear differential behavior, and when the Jacobian determinant vanishes, the dimension of the vector space of the mapped decreases. In other words, an infinitesimal two-dimensional image is mapped to an infinitesimal source of one-dimension. (The trace of the Jacobian matrix is non-zero, and when one eigenvector vanishes, the other is non-zero - in fact, 2.) The set of points where the Jacobian determinant \(J\) vanishes is called the critical curve. \[J=1-\|\kappa\|^{2}=1-{1\over\|z-x_{1}\|^{4}}\ ;\qquad\kappa\equiv{\partial\bar{ \omega}\over\partial z}\] (29) The critical curve (\(J=0\)) is a circle of radius 1 (\(1=\alpha_{E}\) or \(R_{E}\), usually, depending on how to normalize the lens equation) centered at \(x_{1}\). In other words, the critical curve coincides with the circular image of the Einstein ring. The critical curve is mapped to \(\omega=x_{1}\), hence the lens position is the position of the point caustic. Inversely, a point source at the lens position produces the circular image on the critical curve. It is useful to define linear Einstein ring with radius \(R_{E}\equiv D_{1}\alpha_{E}\) ( mentioned above). The photon trajectories that form the circular image passes through the (linear) Einstein ring at their closest approaches to the mass \(M\); in other words, their periastron distances is \(r_{0}=R_{E}\). The Jacobian matrix \[{\cal J}=\pmatrix{1\ \bar{\kappa}\cr\kappa\ 1}\] (30) has eigenvalues \(\lambda_{\pm}=1\pm\|\kappa\|\), and the eigenvectors are \[e_{+}=\pmatrix{E_{+}\cr\bar{E}_{+}}\ ;\quad e_{-}=\pmatrix{E_{-}\cr\bar{E}_{-}}\] (31) where \[\kappa=\|\kappa\|e^{-i2\theta}\ ;\qquad E_{+}\equiv e^{i\theta}\ ;\qquad E_{-} \equiv iE_{+}\ .\] (32) On the critical curve, \(\|\kappa\|=1\) and \(\lambda_{-}=0\), and the critical eigenvector \(\pm E_{-}\) is tangent to the critical curve at every critical point. (Eigenvectors are not assigned the senses. Thus, \(\pm E_{-}\).) In other words, every critical point is a precusp, and the point caustic is a degenerate cusp. #### 2.4.1 Breaking the Degeneracy with a Small Constant Shear We break the degeneracy of the point caustic of a point mass by introducing a small constant shear to understand the degeneracy as a limit. Recall how to produce a constant	[]
	electric field using a dipole where two large opposite charges are separated by a large distance. We can introduce a large mass at a large distance such that \(mass/distance^{2}\) is constant, to a similar effect. If the point mass is at \(z=x_{1}\) where \(x_{1}=0\) is real, and the large mass is on the negative real axis, the lens equation is given by \[\omega=z+\epsilon\bar{z}-{1\over\bar{z}}\;\qquad\epsilon\geq 0\ .\] (33) For our purpose, we need small \(\epsilon\). The lens equation (33) known as Chang-Refsdal model 6 has a bifurcation of the critical curve (hence also caustic curve)7 at \(\epsilon=1\), hence we assume \(\epsilon<1\) so as to take a smooth limit of \(\epsilon\to 0\). Footnote 6: K. Chang and S. Refsdal, 1979, Nature, 282, 561. Footnote 7: _Ibid._, A & A, 132, 168. \[\kappa=\epsilon+{1\over z^{2}}\equiv\|\kappa\|\,e^{-2i\theta}\] (34) and the critical curve is where \(\|\kappa\|=1\). If \(z\) is real, then \(\kappa\) is real and positive, hence \(\kappa=1\) where the critical curve intersects the real axis (or the "dipole axis"). The critical points on the "dipole axis" are \[z=\pm(1-\epsilon)^{-1/2}\ ,\] (35) and they exist for \(\epsilon<1\). The critical eigenvector at the critical points are \(\pm E_{-}=\pm ie^{i\theta}=\pm i\) since \(\kappa=1\ (\theta=0,\ \pi)\). The tangent to the critical curve at the critical points, \[{dz\over d\theta}=\pm i\,(1-\epsilon)^{-3/2}\ \propto\ \pm E_{-}\ ,\] (36) are parallel to the critical eigenvector \(\pm E_{-}\), and so the critical points are precusps. The corresponding cusps are at \[\omega=2\epsilon z=\pm 2\epsilon(1-\epsilon)^{-1/2}\ .\] (37) The lens equation has only one singularity (pole) at \(z=0\) and its topological charge is 1. There are two limit points where \(\kappa=0\), namely, at \(z=\pm i\epsilon^{-1/2}\). Thus, the critical curve is made either of one loop enclosing the pole (<1) or of two loops each enclosing a limit point (>1). One-loop critical curve has topological charge 1 and produces a 4-cusp caustic. Each loop of the two-loop critical curve has topological charge \(1/2\) and produces a triangular caustic. The bifurcation from one quadroid to two trioids occurs at \(\infty\). Here we are interested in the quadroid because it is the quadroid that contracts to a point in the limit of a point mass lens \(\epsilon\to 0\). The quadroid has two cusps on the real axis given in eq. (37) and it is easy to guess correctly from the symmetry that the other two are on the imaginary	[]
	axis, reflection symmetric with respect to the "dipole axis". The quadroid is bisected by the real axis. Now consider \(\omega\) on the real line. There are two real solutions for \(z\). \[0=(1+\epsilon)z^{2}-\omega z-1\] (38) For \(\|\omega\|>2\epsilon(1-\epsilon)^{-1/2}\), one solution is negative (\(J\leq 0\)) and the other is positive (\(J\geq 0\)). For \(\|\omega\|>2\epsilon(1-\epsilon)^{-1/2}\), both are negative (\(J<0\)). The lens equation (33) can be embedded into a fourth order polynomial equation for \(z\), hence there can be two more solutions. Substitute \(z=re^{i\phi}\) and find that the solutions are on a circle centered at the position of the point mass lens element. \[r^{2}={1\over 1-\epsilon}\ ;\qquad\omega=2\epsilon r\cos\phi\] (39) The solutions exist for \(\|\omega\|\leq 2\epsilon(1-\epsilon)^{-1/2}\), which defines the real line segment contained by the two cusps; Inside the quadroid, there are two extra images for each \(\omega\) and they are on the circle of radius \((1-\epsilon)^{-1/2}\). They are positive images (\(J\geq 0\)). The two extra images satisfy the following quadratic equation, which can be found by dividing the fourth order polynomial by the quadratic equation for the real solutions. \[0=z^{2}-\omega\epsilon^{-1}z+(1-\epsilon)^{-1}\] (40) where \(\omega\) is real and inside the quadroid caustic. If we consider \(\omega\) moving on the real line in the positive direction and crossing a cusp \(\omega=-2\epsilon(1-\epsilon)^{-1/2}\), the positive image on the real line crosses the critical point \(z=-(1-\epsilon)^{-1/2}\) into the area enclosed by the critical curve and turns into a negative image. The other image on the real line moves in the positive direction maintaining its (negative) parity until \(\omega\) crosses the other cusp \(\omega=2\epsilon(1-\epsilon)^{-1/2}\). In the mean time, the two extra images trace the circle: one image, the half-circle in the upper half plane, and the other image, the half-circle in the lower half plane. Figure 2 is a depiction of the Einstein ring as the critical curve of the single point mass lens \(M\). Let's consider the ring, for a moment, as the circle of the two extra images (and imagine the quadroid centered at \(M\)): the arrow near the point mass lens depicts the motion of the source \(\omega\); two arrows on the real line depict the images moving on the real line; two arrows on the ring depict the two extra images that start at one precusp and end at the other precusp. Now take the limit \(\epsilon\to 0\), and one can visualize two images tracing the Einstein ring instantaneously at the crossing of the point caustic. We find the above a comfortable (usually involving continuity or traceability) way to think of the transition from two point images to two half-circle images. The criticality of	[]
	a lens equation in general is related to formation or disappearance of the two extra images (or higher even number of images). In the case of the single point mass lens, the two extra images trace a ring instantaneously forming a ring image due to the degeneracy, and the dimensional change of the linear differential vector spaces of the lens equation is manifested in a global manner. ## 3 Caustics of Lensing by Slowly Moving Matter The exquisite symmetry of the Schwarzschild lens (spherically symmetric matter) that led to the caustic of degenerate cusp is readily broken in a more general lens, and the caustic is generally a one-dimensional curve as is the critical curve. The lens equation obtained with the small angle approximation is a mapping of the complex plane (extension of a small neighborhood of the observer's sky where the small angle approximation is valid) to itself. The criticality condition \(J=0\) imposes one constraint reducing the dimension of the variable space by one. Thus, the critical curve is a one dimensional curve which may be a disjoint sum of many loops, and they are usually smooth. Also, the lens equation is usually smooth almost everywhere except at the poles due to the point mass lens elements (\(z=x_{1}\) in the case of the single point mass lens discussed above). At the poles, \(J=-\infty\), hence it is reasonable to assume that the lens equation is smooth on the critical curve for gravitational lenses in general. A smooth curve is mapped to a smooth curve by a smooth mapping except at the stationary points of the mapping. Recall that one of the two eigenvectors vanishes on the critical curve. If we consider a small deviation \(dz\) from a critical point, \[dz=dz_{+}E_{+}+dz_{-}E_{-}\qquad\rightarrow\qquad d\omega=d\omega_{+}E_{+}\ ,\] (41) hence \(\omega\) changes only in the direction of \(E_{+}\). Now if \(dz_{+}=0\), then \(d\omega=0\). So, if we imagine tracing the critical curve (or integrating the tangent to the critical curve) and mapping to draw the caustic curve, the procession of the caustic curve stops momentarily where \(dz_{+}=0\) and turns around. In other words, the caustic curve develops a cusp where the tangent to the critical curve is parallel to the critical eigenvector \(\pm E_{-}\). The critical point where the tangent has \(dz_{+}=0\) is called precusp. Thus, the caustic curve of a lens equation is a piecewise smooth curve punctuated by cusps which may be a disjoint sum of cuspy loops that may intersect themselves. As the point caustic at the lens position of the single lens is defined within the context of the lens equation, the caustic curve of a general gravitational lens is defined only within the context of the lens equation. ## 4	[]
	Comments on Two Papers We discuss a couple of papers found in the Astrophysical Journal that may play an unfortunate role of perpetuating misunderstandings. ### "Superluminal Caustics" in 2002 The paper entitled "Superluminal Caustics of Close, Rapidly-Rotating Binary Microlenses", Zheng Zheng and Andrew Gould, ApJ 541 (2002), 728, considers the caustic curve of the binary lens equation as an object of the size given by the multiplication of its angular size and distance \(D_{1}\) and of an (unspecified) inertia whose velocity has to be compared with the speed of light and be concerned of for its tachyonic nature. As we discussed in this article, the caustic is defined only within the context of the lens equation, and especially, it is not an object of an inertia. There is nothing wrong with thinking of the caustic curve as a large linear object by projecting it to the normal plane passing through the source or passing through the lens, or at any distance along the line of sight if that serves to understand or apply the lens equation for some purpose as far as the contextual existence and characteristics of the caustics as defined within the lens equation are valid. Certainly, inertia is not a quality of the caustics defined in the lens equation; There does not exist a tachyonic caustic. If we stipulate for a moment for the purpose of clearing up another conceptual mistake that the caustics be endowed with inertias, what the caustics can newly acquire is the tachyonic nature but not a superluminal phenomenon known in astrophysics as we discussed in astro-ph/0002414. Zheng and Gould speculate to detect their "superluminal caustics" using very large telescopes such as of 100m aperture. We repeat that tachyonic caustics do not exist; The nomenclature of "superluminal caustic" is a misidentification even within the context of their (mistaken) idea about the motion of the caustics. There does not exist a physically perceivable notion of "superluminal caustic" nor a tachyonic caustic, and it would be an unnecessary waste of resources to consider detecting "superluminal caustics" with say LHT (Larger than Huge Telescope). Other serious problems of the paper by Zheng and Gould can be found elsewhere. How is such a paper published in the Astrophysical Journal in 2002? Perhaps, in the same way other worthy papers are smothered in the referee system	[]
	. ### "Fermat's Principle" in General Relativity in 1990 The paper entitled "Fermat Principle in Arbitrary Gravitational Fields", Israel Kovner, ApJ 351 (1990), 114, is cited in the monograph on lensing titled "Gravitational Lenses" by Schneider et al. for Fermat's principle. Thus, it is likely that the Kovner's article is influential. Among others, we discuss why "the least proper time principle", a generic variational principle in general relativity, should not be referred to as Fermat's principle. #### 4.2.1 Focal Points of a Gravitational Spherical Geometry are not Caustics Kovner seems to misidentify the source and observer as the caustics (and "past caustics") as he states in the second paragraph of section IV, p. 118, "The caustics are mergings of \(\geq 2\) extremals of the emission time, ... the _past_-caustics of the past light cone emanating from the observer ... ." We discussed above hat the source and observer are the two focal points of a particular set of null geodesics, namely the null geodesics that connect the source and the observer. An observer can see a photon from the source only if the photon arrives at the detector (or the eye) of the observer, and it is imperative that there exist some null geodesics that connect the source and the observer if the observer will be able to image the source photonically. In plane geometry, there is only one (null) geodesic between the two points defined by the source and the observer. In spherical geometry, the geodesics cross, resulting in multiple geodesics connecting two points. There is where the notion of the focal points comes in. They are the focal points of the geodesics of the spherical geometry. Distinct null geodesics produce distinct images, the spherical geometry is responsible for multiple images, and the source and observer are the two foci of the null geodesics corresponding to the multiple images. The misidentification of the source and observer as caustics may derive from the definition of caustics in electromagnetic lensing. See, for example, "The Classical Theory of Fields", L. D. Landau and E. M. Lifshitz, 1975, p.133. #### 4.2.2 Least Action Principle in Non-Relativistic Mechanics "Fermat's principle" in general relativity is a four-dimensional version of the least (abbreviated) action principle8 of the non-relativistic mechanics where the lagrangian has only	[]
	the kinetic terms. Footnote 8: The least action principle seems to referred to as Maupertuis’ principle in certain literature even though the formulation was due to Euler and Lagrange. Goldstein (chap. 7) writes, “However, the original statement of the principle by (Pierre de) Maupertuis (1747) was vaguely theological and could hardly pass muster today. The objective statement of the principle we owe to Euler and Lagrange.” We gather that calling the principle of least action Maupertuis’ principle may amount to calling Fermat’s principle Cureau’s principle. The principle of least action in non-relativistic mechanics states that the "abbreviated" action \[S_{0}=\int dt\,P\dot{Q}\] (42) of a system for which the Hamiltonian is conserved is the extremum along the equation of the motion. \(Q\) is the coordinate variable of the particle and \(P\) is its conjugate. Recall that the equations of motion are obtained as the extremum of the action \[S=\int dt\,L=\int PdQ-Hdt\ .\] (43) In a system with conserved Hamiltonian, one can consider only the paths that satisfy the conservation of the Hamiltonian by constraining by the equation of the motion and allowing a variation of the final time \(t\) (or final time and initial time: See Goldstein or Landau and Lifshitz9), and the (abbreviated) action \(S_{0}\) is the action that has an extremum under the variation. If we denote the variation \(\Delta\) variation after Goldstein (chap. 7), Footnote 9: _Mechanics_, L. D. Landau and E. M. Lifshitz, Pergamon press, 1969. \[\Delta S=\left(L-{\partial L\over\partial\dot{Q}}\dot{Q}\right)\Delta t=-H \Delta t\ .\] (44) From the second equality of eq. (43) and conservation of \(H\), \[\Delta S=\Delta S_{0}-H\Delta t\ ,\] (45) hence \(\Delta S_{0}=0\). Now if there is no external force and the kinetic energy is conserved, \(S_{0}=2H\int dt\) and the particle follows a path such that the time it takes is the extremum (usually minimum, hence least time principle). \[0=\Delta S_{0}=2H\Delta t\] (46) It recalls Fermat's principle in geometric optics which is also called _the principle of least time_. In order to see the structural parallel (as a functional analysis) with the four-dimensional case of the general relativity, note that Hamiltonian and time variable are a pair of canonical conjugates.	[]
	#### 4.2.3 Least Action Principle in General Relativity The Hamiltonian of the system of a freely falling particle described in eq. (12) is nothing but the (half the) momentum square of the particle10 and is conserved along the equation of motion. The Hamiltonian \({\cal H}\) and the path parameter \(p\) are canonical conjugates. There is no external force for a particle freely falling in a curved space time as defined by the metric, hence this is an exact parallel with the non-relativistic case discussed above where the kinetic energy is conserved. The resulting variational principle is a principle of extremum path parameter \(p\), or equivalently a principle of extremum proper time \(\tau\). Recall the action in eq. (12) and define \(S_{0}\), Footnote 10: The scalar product of the four momenta \(P^{2}=M^{2}\) where \(M\) is the mass of the particle, and Kovner refers to the set of \(\{P_{\mu}\}\) that satisfies the condition the mass shell as is customary in particle and nuclear physics. Off the mass shell states are known as virtual particles. Feynman diagrams are a web of interactions of virtual and real particles that pictorially describe combinatoric calculations of Feynman path integrals. \[S=\int{\cal L}dp\ ;\qquad S_{0}=\int P_{\mu}\dot{x}^{\mu}dp\ ,\] (47) and the \(\Delta\) variation of the action \(S\) is given by \[\Delta S=-{\cal H}\Delta p\ .\] (48) Since the Hamiltonian is conserved, \[\Delta S=\Delta S_{0}-{\cal H}\Delta p\ ,\] (49) hence \(\Delta S_{0}=0\). From \(S_{0}=\int 2{\cal H}dp\), \(0=\Delta S_{0}=2{\cal H}\Delta p\). Expressing in terms of the \(\Delta\) variation of the proper time, \[0=\Delta S_{0}=2{\cal H}\Delta p=-{d\tau^{2}\over dp^{2}}\Delta p=-{d\tau\over dp }\Delta\tau=-(-g_{mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})^{1/2}\Delta\tau\ .\] (50) The space-time path taken by the particle is such that the proper time elapsed is in extremum. \[0=\Delta\int d\tau=\Delta(\tau-\tau_{-})=\Delta\tau\] (51) If we consider photons leaving a light source at \(\tau=\tau_{-}\), the light paths are determined such that the arrival time \(\tau=\tau_{+}\) ( for a given \(\tau_{-}\)) or the travel time \(\delta\tau=\tau_{+}-\tau_{-}\) is an extremum. One potential pitfall in eq. (51) is the case where \(d\tau^{2}=0\) because, then, \(0=\Delta S_{0}\) does not necessarily imply \(0=\Delta\tau\). We may take the limit \(d\tau^{2}\to 0\) as the case for photons	[]
	and assume that \(0=\Delta\tau\) is valid. Then, we need to confirm that the new action \(\int d\tau\) does generate the equations of motion as its extremum. \[S_{\tau}=\int d\tau=\int{d\tau\over dp}dp=\int(-g_{\mu\nu}\dot{x}^{\mu}\dot{x} ^{\nu})^{1/2}dp\] (52) Under the variation of the path \(x^{\mu}(p)\), the variation of the action is (Weinberg, p.76) \[\delta S_{\tau}=-\int\left({d^{2}x^{\alpha}\over d\tau^{2}}+\Gamma^{\alpha}_{ \mu\nu}{dx^{\mu}\over d\tau}{dx^{\nu}\over d\tau}\right)g_{\alpha\beta}\, \delta x^{\beta}\,d\tau\ .\] (53) If a particle follows the equations of free fall, the variation of the action vanishes. In other words, the physical space-time path a particle follows is such that the proper time elapsed is an extremum. Now, there is no ambiguity related to \(d\tau^{2}=0\); Photons indeed follow paths for which the travel times (or arrival times for given departure times) are extrema. 1. 1.The last paragraph in p.100 of the monograph "Gravitational Lenses" by Schneider et al. states, " ... does not refer to the 'time' a light ray needs to travel from the source to the observer ... but a stationary property of the (invariant) _time of arrival_ at the observer ... ." Since the arrival time \(\tau=\tau_{+}\) and the travel time \(\delta\tau=\tau_{+}-\tau_{-}\) are effectively the same variables once given the time of the departure \(\tau=\tau_{-}\), it is a self-contradiction. If the arrival time can be defined, so can the travel time. If the arrival time along a geodesic is an extremum, so is the travel time along the same geodesics. Then, why do Schneider, Ehlers, and Falco denounce the notion of travel time while accepting arrival time? It is unclear within the section on "Fermat's principle" in the book. The source of the error may be \(\delta\lambda_{obs}=0\) in the Kovner's article where Kovner does not elaborate the new action \(S_{0}\). (\(\lambda\) is the path parameter in Kovner's.) It is apparent that the error has propagated unchecked. 2. 2.What is conspicuous is that eq. (52) is just another commonly used path integral and variation in general relativity to derive the equations of motion from a variational principle (Weinberg, p.76). The freely falling particle can be massive or massless. For the latter, \(d\tau^{2}=0\), and for the former, \(d\tau^{2}\neq 0\) which we may normalize such that \(d\tau^{2}=1\). Now, should we pluck out, of the continuum (at least classically) of mass spectrum, the massless subset given by \(d\tau^{2}=0\) and call it Fermat's principle? Most certainly not. 3. 3.The evolution of the fundamental physics has been in the direction of unifications. Not only the light but a car is a wave according to quantum mechanics except that the latter has much shorter matter (or de Broglie) wave length. In the limit of zero wavelength, the classical equations are recovered either for massless particles (photons, phonons,	[]
	etc) or massive particles. Maxwell equations for light lead to the eikonal equation (or geometric optics) in the limit of zero wavelength that is suitable to describe the propagation of the wave front of the light bundle in a medium whose electromagnetic properties vary slowly - usually, wrapped up as a slowly varying refraction index. The eikonal equation can be derived from an action and its variation. It corresponds to a case where the kinetic energy is conserved. Thus, Fermat's principle, or the principle of least time. (The historical account of Fermat's principle can be found in a separate article.) In general relativity, unlike in Newtonian gravity, the gravitation is not given as an external force but is integrated into the general covariance. It is exactly in the same manner as the electromagnetic forces are integrated into \(U(1)\) gauge covariance, for example, in quantum electrodynamics. In this framework of "geometrization", the forces are integrated into dynamic variables. As a result, the lagrangian \({\cal L}\) in eq. (12) describes a free particle, has only the kinetic terms, which is conserved, and allows simple variations. 4. 4.Geometric optics (or eikonal equation) is exactly valid in the limit of zero wavelengths and can be described by the geodesic equation when under gravity. Huygens principle is about diffraction phenomenon where the wavelengths of the light with respect to the aperture matter: "Huygens wavelets" propagate out as small spherical waves from every point of the aperture (with a subtle understanding of the nature of the approximation that the point is sufficiently smaller than the aperture and is sufficiently larger than the wavelength). It is unclear how the "Huygens wavelets" are related to light cones11 and "Fermat's principle" as Kovner states in p.116, "Another way to regard the Fermat principle for light is provided by the Huygens principle as illustrated in Figure 2." Does Kovner imply that diffractions are also described by geometric optics? Footnote 11: Light cones of a space-time point \(x_{0}\) is the set \(\{x^{\mu}\}\) that satisfies \((x-x_{0})^{2}=0\). In a sufficiently small neighborhood of \(x_{0}\), the set is made of two back-to-back “cones” connected at \(x_{0}\). For a massive particle, the future and past hyperboles are separated by the mass gap.	[]
	# The Largest Scale We Can Detect in the Universe and the Inflation Wen Zhao, Yang Zhang Astrophysics Center University of Science and Technology of China Hefei, Anhui, China Abstract From the damping of the Cosmic Microwave Background Radiation (CMB) anisotropy power spectrum at large scale and the recent accelerating expansion of the Universe, we find that, there may be a largest scale which we can detect in the Universe. From this, we can get the inflation parameters as spectrum index \(n_{s}\) of the initial scalar spectrum, e-fold \(N\), Hubble parameter \(H\), the ratio of tensor and scalar \(r\), the lasting time of reheating stage \(\alpha\) for special inflation models. We do them in three inflation models, and find that all the results fit fairly well with the observations and the inflation theory. PACS numbers: 98.80.-k, 98.80.Es, 04.30.-w, 04.62+v, Key words: inflation, dark energy, CMB e-mail: wzhao7@mail.ustc.edu.cn	[]
	The release of the high resolution full sky Wilkinson Microwave Anisotropy Probe (WMAP) data shows that the data is consistent with the predictions of the standard inflation-\(\Lambda\)CDM cosmic model, expect for several puzzles[1][2][3][4][5][6][7]. One of those is that the power spectrum shows that there is a much damping at the large scale \(l<10\)[2][6][7], which has been much deeply discussed[8]. The simplest explanation is that there is a cut at the very small wavenumber (IR cut-off) for the initial scalar perturbation. We show the power spectrum of CMB in figure (1), where the dots are the WMAP observation result, and the red line is the spectrum without IR cut-off, the green line with cut-off at 1/k=4000Mpc, the blue line with cut-off at 1/k=3000Mpc, the magenta line with cut-off at 1/k=2000Mpc. We have set the scale factor of now \(a_{0}=1\), and chosen the cosmological parameters as \(\Omega_{m}=0.047\), \(\Omega_{\Lambda}=0.693\), \(\Omega_{dm}=0.29\), \(h=0.72\), \(n_{s}=0.99\)[5], and without consider the reionization and the running of \(n_{s}\). From this figure, we find that the cut of the initial perturbation is nearly at \(1/k\simeq 2000-4000Mpc\), but why this cut-off exist? We all know that there is an inflation stage at the very early Universe[9][10]in the inflation-\(\Lambda CDM\) cosmic model. The attraction of this paradigm is that it can set the initial conditions for the subsequent hot big bang, which otherwise have to be imposed by hand. One of these is that there be no unwanted relics (particles or topological defects which survive to the present and contradict observation). Another is that the initial density parameter should have the value \(\Omega=1\) to very accuracy, to ensure that its present value has at least roughly this value. There is also the requirement that the Universe be homogeneous and isotropic to high accuracy. But the most important is that the scale-invariant initial scalar perturbation power spectrum which predicted, has been detected from CMB and LSS, especially the recent WMAP[2][5][7] and SDSS[11] observation. In this paradigm, the scale factor expanded much more rapidly in the initial inflation stage than the horizon1. From the sketch figure (2), we find that the scale of \(a/k_{1}\) goes out the horizon at inflation, and reenteres the the horizon at radiation (dust)-dominating stage. Where \(k_{0}\) is the smallest wavenumber, which had been in the horizon, when \(k<k_{0}\), the wave had never been in the horizon, so the initial power spectrum should only at \(k>k_{0}\). So for the scale with \(l_{2}=2\pi a_{i}/k_{2}\) is larger that \(l_{H}=2\pi a_{i}/k_{0}\) ( \(a_{i}\) is the scale factor at \(t_{i}\), \(k_{0}\) satisfies that \(a_{i}/k_{0}=1/H\) , where \(H\) is the Hubble parameter of inflation), the Universe will never be homogeneous and isotropic. if this cut of \(k_{0}\) is just nearly \(1/k_{0}\sim 3000Mpc\), it will naturally answer the damping of the CMB power spectrum. The discussion in below will all base on this idea. If the Universe was dominated by dust, the scale factor \(\propto t^{2/3}\), but the horizon is nearly \(\propto ct\) (c is the speed of light), so there must be a time \(t_{m}\), \(2\pi a_{m}/k_{0}\sim t_{m}\), after which, the Universe will not be homogeneous and isotropic. But that was not the true Universe. Footnote 1: The horizon in this paper all denotes the particle horizon. A wide range of observational evidence indicates that our Universe may be accelerating expansion[12][11][13][2][5][15][14]. If we assume that long-range gravity obeys Einstein's General R	[]
	so we can get the inflation parameters in the same way: \[n_{s}=0.97,~{}~{}N=64.35,~{}~{}r=0.12,~{}~{}\alpha=105.55,~{}~{}H=3.86\times 1 0^{13}GeV<H_{m};\] (14) (3) \(V(\phi)=\Lambda^{4}(\phi/\mu)^{4}\) inflation model: In this model, the two simple relations are \[N=\frac{3}{1-n_{s}},~{}~{}~{}~{}~{}~{}r=\frac{16}{3}(1-n_{s}),\] (15) so the inflation parameters are: \[n_{s}=0.95,~{}~{}N=64.53,~{}~{}r=0.25,~{}~{}\alpha=88.42,~{}~{}H=5.46\times 10 ^{13}GeV<H_{m};\] (16) From the observation, one gets the index of the initial spectrum of \(n_{s}=0.99\pm 0.04\) from WMAP only, and \(n_{s}=0.96\pm 0.02\) from WMAPext+2dFGRS+Lyman \(\alpha\)[5]. The index from these three models are all fit fairly well with the observation. The constraint on \(r\) form WMAP and SDSS is \(r<0.36\)[7][17], which is also consistent with the calculation results. Naturally, we accept that \(N\geq 60\), \(H\sim 10^{13}GeV\)[7], which are all consistent with the models results. We also get the \(\alpha\) is nearly \(100\) (independent on the reheating models), which tell us that the reheating process is a very quick process, at this stage, the scale factor only expanded nearly \(100\) times. From the before calculation, we find that, there inflation parameters are only dependent on the two relations of \(N\) & \(n_{s}\), and \(r\) & \(n_{s}\). The models of (1) and (3) have the same relations, so they give the same inflation parameters. In summary, from the damping of the CMB anisotropy power spectrum at large scale, we think it is for that the initial power spectrum has a cut-off at \(k<k_{0}\), which generated for the inflation must began at some time. One find that this cut-off is nearly equal to the largest scale which can reenter the horizon in the accelerating Universe. From this we elicit a simple Assumption: \(k_{0}\equiv k_{a}\) so the largest scale, which can be in the particle horizon in the inflation-accelerating expansion Universe is \(\sim 1/k_{a}\). This Assumption will keep that the Universe will always be homogeneous and isotropic. To check the rationality of this Assumption, we calculated the inflation parameters od \(n_{s}\), \(N\), \(H\),\(\alpha\) in three inflation models from this Assumption, and find that they are all consistent with the observation and the inflation theory. The accelerating expansion of the recent Universe is a very puzzle for cosmologist. Here we consider this question in another point: for the early inflation stage must have a beginning at some time, which make that there exist a largest scale \(1/k_{0}\), when the scale is larger than it, the Universe will not be homogeneous and isotropic. The Universe began to decelerating expansion after inflation, if this deed was kept for all time, which will make the Universe not be homogeneous and isotropic at large scale at some time, for the expansion of the horizon is much quick that the scale factor. But to our surprise is that the Universe began to accelerating expansion at redshift \(z\simeq 0.653\), which exactly elegantly make the Universe always be homogeneous and isotropic. Usually, we always ask the question: why the accelerating expansion exist? but here, we ask another question: why it is needed? Our answer is that: to keep the cosmological principle being right for all time. About why the Universe is this? which however, need to research. But from the discussion above, we think the below viewpoint is rational: the recent accelerating	[]
	must have some relation with the early inflation, these two accelerating expansion stages make Universe always be homogeneous and isotropic. ACKNOWLEDGMENT: We thank B. Feng for helpful discussion, and also thank T.Padmanabhan for his advise and his advised paper[18]. We acknowledge the using of CMBFAST program[19]. Y. Zhang's research work has been supported by the Chinese NSF (10173008) and by NKBRSF (G19990754). W.Zhao's work is partially supported by Graduate Student Research Funding from USTC. ## References * [1] http://lambda.gsfc.nasa.gov/; * [2] C. L. Bennett, _et al_.,Astrophys.J.Suppl. 148 (2003) 1; * [3] E. Komatsu, _et al_.,Astrophys.J.Suppl. 148 (2003) 119-134; * [4] L. Verde, _et al_.,Astrophys.J.Suppl. 148 (2003) 195; * [5] D. N. Spergel, _et al_., Astrophys.J.Suppl. 148 (2003) 175; * [6] G. Hinshaw, _et al_. Astrophys.J.Suppl. 148, (2003) 135; * [7] H.V.Peiris, _et al_. Astrophys.J.Suppl. 148, (2003) 213; * [8] C.Contaldi, _et al_. JCAP 0307,(2003) 002; B. Feng, X.M. Zhang, Phys.Lett. B 570, (2003) 145; S.L.Bridle, _et al_. Mon.Not.Roy.Astron.Soc.342,(2003) L72; G.Efstathiou, Mon.Not.Roy.Astron.Soc.343, (2003) L95; A.Lindle, JCAP 0305,(2003) 002; J.M.Cline, _et al_. JCAP 0309 (2003) 010; S.Tsujikawa, _et al_. Phys.Lett.B 574, (2003) 141; M.Baskero-Gil, _et al_. Phys.Rev.D 68 (2003) 123514; M.Liguori, _et al_. JCAP 0408 (2004) 011; J.P.Luminet, _et al_. Nature. 425,(2003) 593; Y.-S.Piao, _et al_. Phys.Rev.D 69, (2004) 103520; * [9] Edward W.Kolb and Michael S. _Turner, The Early Universe_ Addison-Wesley Publishing Company (1990); * [10] Andrew R. Liddle Davis H. Lyth _Cosmological Inflation an Large-Scale Structure_ Cambridge University Press (2000); * [11] M Tegmark _et al_.,Astrophys.J. 606 (2004) 702-740; * [12] A. G. Riess _et al._, Astron. J. 116 (1998) 1009; S. Perlmutter _et al._, Astrophys. J. 517 (1999) 565; J. L. Tonry _et al._, Astrophys. J. 594 (2003) 1; R. A. Knop _et al._, astro-ph/0309368; * [13] M Tegmark _et al_., Phys.Rev. D69 (2004) 103501; Adrian C. Pope _et al_.,Astrophys.J. 607 (2004) 655-660; Will J. Percival _et al_., MNRAS 327 (2001); 1297; * [14] B. Feng, X.L. Wang, X.M. Zhang, Phys.Lett. B607 (2005) 35;	[]
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	# Perturbations of the Yang-Mills field in the universe Wen Zhao Wen.Zhao@astro.cf.ac.uk Department of Physics, Zhejiang University of Technology, Hangzhou, 310014, China School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom ###### Abstract It has been suggested that the Yang-Mills (YM) field can be a kind of candidate for the inflationary field at high energy scales or the dark energy at very low energy scales, which can naturally give the equation of state \(-1<\omega<0\) or \(\omega<-1\). We discuss the zero order and first order Einstein equations and YM field kinetic energy equations of the free YM field models. From the zero order equations, we find that \(\omega+1\propto a^{-2}\), from which it follows that the equation of state of YM field always goes to \(-1\), independent of the initial conditions. By solving the first order Einstein equations and YM field equations, we find that in the YM field inflationary models, the scale-invariant primordial perturbation power spectrum cannot be generated. Therefore, only this kind of YM field is not enough to account for inflationary sources. However, as a kind of candidate of dark energy, the YM field has the 'sound speed' \(c_{s}^{2}=-1/3<0\), which makes the perturbation \(\phi\) have a damping behavior at the large scale. This provides a way to distinguish the YM field dark energy models from other kinds of models. pacs: 98.70.Vc, 98.80.Cq, 04.30.-w ## I Introduction It is well known that the physics of inflationary fields and dark energy are two of the most important problems in modeling cosmology. They all need a kind of matter with negative pressure \(p\simeq-\rho\) (Kolb & Turner inf; Bennett et al. map; Riess et al. sn; Tegmark et al. sdss; Feng et al. age), where \(p\) and \(\rho\) are the pressure and energy density of matter respectively. People always describe them with a scalar field, which can naturally give an equation of state of \(-1\leq\omega\leq 0\) (Kolb & Turner inf; Wetterich de; Bharat & Peebles peebles). In particular, when the potential of the scalar field is dominant, \(\omega\) will go to \(-1\). Thus, the expansion of the Universe is close to the de Sitter expansion. Recently, the observations of the cosmic microwave background radiation (CMB) temperature and polarization anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP) shows that the spectral index of the primordial density perturbation \(n_{s}=1.20^{+0.12}_{-0.11}\) at wavenumber k=0.002Mpc\({}^{-1}\) (Bennett et al. map). Observations of Type Ia Supernova (SNeIa), CMB and large scale structure (LSS) (Bennett et al. map; Riess et al. sn; Tegmark et al. sdss) also suggest that the equation of state of the dark energy may be \(\omega<-1\) (Corasaniti et al.[<-1]). These are all very difficult to obtain from single scalar field models. So it is necessary to look for a new candidate for the inflationary field and dark energy. Recently, a number of authors have considered using a vector field as the candidate of the inflationary field or dark energy (Ratra vec). We have advised the effective YM condensate (Zhang Zhang; Zhang Zhang2; Zhao & Zhang ymc1; Zhang et al. ymc2) as a kind of candidate, which can be used to describe the inflation at high energy scales and dark energy at very low energy scales. In our models, a quantum effective YM condensate is used as the source of inflation or dark energy, instead of a scalar field. This model has the desired interesting feature: the YM field is an indispensable cornerstone to any particle physics model with interactions mediated by gauge bosons, so it can be incorporated into a sensible unified theory of particle physics. Besides, the equation of state of this field is different from that of general matter as well as scalar fields, and the state \(\omega<-1\) can be naturally realized. In this paper, we shall discuss the evolution of the equation of state of the YM field and cosmic perturbations by solving the zero and first order Einstein equations and kinetic energy equations. From the zero order Einstein equations, we find that the YM field can easily give a state of homogeneity and isotropy, and it can naturally give an equation of state \(\omega<-1\) or \(\omega>-1\). We also find that \(\omega+1\propto a^{-2}\) from the zero order equations. It follows that \(\omega\) naturally goes to \(-1\) with the expansion of the Universe, independent of the initial condition. By considering the evolution of cosmic perturbations, we investigate the first order Einstein equations and kinetic energy equations. In the simplest condition with only an electric field, we find that this YM field has the sound speed \(c_{s}^{2}=-1/3<0\), which is very different from scalar field models. We also find that a scale-invariant primordial perturbation power spectrum cannot be generated, which shows that an electric YM field alone cannot be a candidate for the inflationary field. However, as a candidate of dark energy, YM field makes the cosmic fluctuation \(\phi\) have a damping at large scales. This is helpful for answering large scale damping of the CMB anisotropy power spectrum. ## II The Quantum Effective Yang-Mills Field In the quantum effective YM field dominated Universe, the effective YM field Lagrangian is given by (Adler L2) \[\mathcal{L}_{\rm eff}=\frac{1}{2}bF\ln\|{F}/{e\kappa^{2}}\|,\] (1) where \(\kappa\) is the renormalization scale with dimension of squared mass, \(F\equiv-\frac{1}{2}F_{\mu\nu}^{a}F^{a\mu\nu}\) plays the role of the order parameter of the YM field. The Callan-Symanzik coefficient \(b=11N/24\pi^{2}\) for \(SU(N)\) when the fermion's contribution is neglected. For the gauge group \(SU(2)\) considered in this paper, one has \(b=2\times 11/24\pi^{2}\). For the case of \(SU(3)\), the effective Lagrangian in Eq.(1) leads to a phenomenological	[]
	description of the asymptotic freedom for the quarks inside hadrons (Adler L2). It should be noted that the \(SU(2)\) YM field is introduced here as a model for a cosmic inflationary field or dark energy, and it may not be directly identified as QCD gluon fields, nor the weak-electromagnetic unification gauge fields. An explanation can be given for the form in Eq.(1) as an effective Lagrangian up to a 1-loop quantum correction (Pagels & Tomboulis L1; Adler L2). A classical \(SU(N)\) YM field Lagrangian is \(\mathcal{L}=\frac{1}{2g_{0}^{2}}F\), where \(g_{0}\) is the bare coupling constant. As is known, when the 1-loop quantum corrections are included, the bare coupling \(g_{0}\) will be replaced by a running \(g\) as used in the following (Gross&Wilczez Gross; Pagels & Tomboulis L1; Adler L2), \(g_{0}^{2}\to g^{2}=\frac{4\times 12\pi^{2}}{11N\ln(k^{2}/k^{2}_{0})}= \frac{2}{b\ln(k^{2}/k_{0}^{2})}\), where \(k\) is the momentum transfer and \(k_{0}\) is the energy scale. To build up an effective theory (Pagels & Tomboulis L1; Adler L2), one may just replace the momentum transfer \(k^{2}\) by the field strength \(F\) in the following manner, \(\ln(k^{2}/k_{0}^{2})\to 2\ln\|F/e\kappa^{2}\|\), yielding Eq.(1). The expression of \(1/g^{2}\) is based on the renormalization group estimates. As emphasized in references (Adler L2; Adler & Piran Adl), this estimate is formally valid whenever the running coupling \(g^{2}\) is small in magnitude, which is true both when \(F/\kappa^{2}\gg 1\) (given \(g^{2}\) is small and positive) and when \(F/\kappa^{2}\ll 1\) (given \(g^{2}\) is small and negative). The attractive features of this effective YM model include the gauge invariance, the Lorentz invariance, the correct trace anomaly, and the asymptotic freedom (Pagels & Tomboulis L1). With the logarithmic dependence on field strength, \(\mathcal{L}_{\rm eff}\) has a form similar to the Coleman-Weinberg scalar effective potential (Coleman & Weinberg Col), and the Parker-Raval effective gravity Lagrangian (Parker & Raval Par). The dielectric constant is defined by \(\epsilon=2\partial\mathcal{L}_{\rm eff}/\partial F\), and in the 1-loop order it is given by \[\epsilon=b\ln\|{F}/{\kappa^{2}}\|.\] (2) As analyzed in (Adler L2), the 1-loop model is a universal, leading semi-classical approximation. Thus, depending on whether the field strength \(\|F\|\geq\kappa^{2}\) or \(\|F\|\leq\kappa^{2}\), the YM condensate belongs to the family of forms whose dielectric constant \(\epsilon\) can be positive or negative. The properties mentioned above are still true even if 2-loop order corrections are taken into account, an essential feature of the effective model (Adler L2; Adler & Piran Adl). It is straightforward to extend the model to the expanding Robertson-Walker (R-W) spacetime. For simplicity we shall work in a spatially flat R-W spacetime with a metric \(ds^{2}=a^{2}(\tau)(d\tau^{2}-\gamma_{ij}dx^{i}dx^{j})\), where we have set the speed of light \(c=1\), \(\gamma_{ij}=\delta^{i}_{j}\) denoting background space is flat, and \(\tau=\int(a_{0}/a)dt\) is the conformal time. The dominant matter is assumed to be the quantum YM condensate, whose effective action is \(S=\int\mathcal{L}_{eff}~{}a^{4}(\tau)~{}d^{4}x\), and the Lagrangian \(\mathcal{L}_{eff}\) is defined in Eq.(1). By variation of \(S\) with respect to \(g^{\mu\nu}\), one obtains the energy-momentum tensor, \[T_{\mu\nu}=-g_{\mu\nu}\frac{b}{2}F\ln\|{F}/{e\kappa^{2}}\|+\epsilon F^{a}_{\mu \sigma}F^{a\sigma}_{\nu},\] (3) where the energy-momentum tensor is the sum of \(3\) energy-momentum tensors of vectors, \(T_{\mu\nu}=\sum_{a}T^{a}_{\mu\nu}\). In order to keep the total energy-momentum tensor homogeneous and isotropic, we assume that the gauge fields are only functions of time \(t\), and \(A_{\mu}=\frac{i}{2}\sigma_{a}A_{\mu}^{a}(t)\) (Zhao&Zhang ymc1). YM field tensors are defined as usual: \[F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+\epsilon^{ abc}A_{\mu}^{b}A_{\nu}^{c}.\] (4) This tensor can be written in the form with electric and magnetic fields as \[F^{a\mu}_{~{}~{}\nu}=\left(\begin{array}[]{cccc}0&E_{1}&E_{2}&E_{3}\\ -E_{1}&0&B_{3}&-B_{2}\\ -E_{2}&-B_{3}&0&B_{1}\\ -E_{3}&B_{2}&-B_{1}&0\end{array}\right).\] (5) From the definition in Eq.(4), we can find that \(E_{1}^{2}=E_{2}^{2}=E_{3}^{2}\) and \(B_{1}^{2}=B_{2}^{2}=B_{3}^{2}\). Thus \(F\) has a simple form \(F=E^{2}-B^{2}\), where \(E^{2}=\sum_{i=1}^{3}E_{i}^{2}\) and \(B^{2}=\sum_{i=1}^{3}B_{i}^{2}\). Inserting (4) in (3), we can obtain the energy density and pressure of the YM field given by (Zhao&Zhang, ymc1), \[\rho=\frac{1}{2}\epsilon(E^{2}+B^{2})+\frac{1}{2}b(E^{2}-B^{2}),\] (6) \[p=\frac{1}{6}\epsilon(E^{2}+B^{2})+\frac{1}{2}b(B^{2}-E^{2}),\] (7) and \[\rho+p=\frac{2}{3}\epsilon(E^{2}+B^{2}).\] (8) Eq.(8) follows a conclusion of this study: the weak energy condition can be violated: \(\rho+p<0\), by the effective YM condensate matter in a family of quantum states with the negative dielectric constant \(\epsilon<0\). The whole range of allowed values of \(F\) is divided into two domains with \(\|F\|>\kappa^{2}\) and \(\|F\|<\kappa^{2}\), respectively. In the domain where \(\|F\|>\kappa^{2}\), one always has \(\epsilon>0\), so the WEC is still satisfied. The other domain is \(\|F\|<\kappa^{2}\) in which \(\epsilon<0\), so that the WEC is now violated. ## III The Zero Order Equations ### The Zero Order Einstein Equations Let us firstly investigate the Friedmann equations, which can be written as \[\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho,~{}~{}~{}~{}~{}\frac{ \ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p),\] where "dot" denotes \(d/dt\). If we consider the simplest case with only an "electric" field, as used in the previous works (Zhang et al. ymc2; Zhao&Zhang ymc1), the Friedmann equations can be reduced to \[\frac{d\rho}{da^{3}}=-\frac{4}{3}\frac{\beta\rho}{a^{3}},\] (9) where we have defined \(\beta\equiv\epsilon/b\). If \(\|\beta\|\ll 1\), one gets \[a^{2}\beta={\rm constant},\] (10) which follows the relation of the equation of state : \[\omega+1\propto a^{-2}.\] (11) From this relation, we find that \(\omega\) will run to the critical condition with \(\omega=-1\) with the expansion of the Universe. In the next subsectioin, we will find that from the zero order YM field kinetic equation, we can get the same result, which means that the YM field equation and the Einstein equations are self-rational.	[]
	### The Zero Order Yang-Mills Field Kinetic Energy Equations By variation of \(S\) with respect to \(A_{\mu}^{a}\), one obtains the effective YM equations (Zhao&Zhang ymc1) \[\partial_{\mu}(a^{4}\epsilon~{}F^{a\mu\nu})-f^{abc}A_{\mu}^{b}(a^{4}\epsilon~{ }F^{c\mu\nu})=0,\] (12) which can be simplified as (Zhao&Zhang, ymc1), \[a^{2}E\epsilon={\rm constant}.\] (13) If the Universe is dominated by an "electric" YM field, the kinetic energy equation follows as \[a^{2}\beta e^{\beta/2}={\rm constant},\] (14) When \(\|\beta\|\ll 1\), then this equation becomes \[a^{2}\beta(1+\beta/2)\simeq a^{2}\beta={\rm constant}.\] which also follows as \[\omega+1\propto a^{-2}.\] (15) This is exactly the same as that of the zero order Einstein equations. ## IV The First Order Equations ### The First Order Einstein Equations As a kind of candidate of inflationary field or dark energy, it is very important to study the evolution of perturbations in the YM field and cosmic fluctuations. In this section, let us consider the flat R-W metric with the scalar perturbation in the conformal Newtonian gauge \[ds^{2}=a^{2}(\tau)[(1+2\phi)d\tau^{2}-(1-2\psi)\gamma_{ij}dx^{i}dx^{j}].\] (16) The gauge-invariant metric perturbation \(\psi\) is the Newtonian potential and \(\phi\) is the perturbation to the intrinsic spatial curvature. The first order Einstein equations become (Mukhanov et al. Muk): \[-3\mathcal{H}(\mathcal{H}\phi+\psi^{\prime})+\nabla^{2}\psi=4\pi Ga^{2}\delta T _{0}^{0},\] (17) (Hph+ps'),i=4pGa2dTi0, (18) [(2H'+H2)ph+Hph'+ps''+2Hps' \[+\] 122D]dji-12gikD\|kj (19) \[= -4\pi Ga^{2}\delta T_{j}^{i},\] where \(\mathcal{H}=a^{\prime}/a\), \(D=\phi-\psi\) and the prime denotes \(d/d\tau\). From the definition of the energy-momentum tensor (3) and the metric of (16), one can get the first order energy-momentum tensor: \[\delta T_{0}^{0}=(\epsilon-b)(B\delta B-E\delta E)+2\epsilon E\delta E+{2\phi} (\epsilon-b)B^{2}+\frac{B^{2}+E^{2}}{2}\delta\epsilon,\] (20) \[-\delta T_{i}^{i} = (\epsilon-b)(-B\delta B+E\delta E)-{2\phi}(\epsilon-b)B^{2}\] (21) \[+ \epsilon[(2B_{2}\delta B_{2}+2B_{3}\delta B_{3}-2E_{1}\delta E_{1 })+4(B_{2}^{2}+B_{3}^{2})\phi]\] \[+ \frac{1}{2}(E^{2}-B^{2})\delta\epsilon+\frac{2B^{2}-E^{2}}{3} \delta\epsilon,\] and others are all zero, where \(E\delta E=E_{1}\delta E_{1}+E_{2}\delta E_{2}+E_{3}\delta E_{3}\), and similar for \(B\delta B\). To obtain the gauge-invariant equations of motion for cosmological perturbations in a Universe dominated by this kind of YM field, we insert the general equations of the energy-momentum tensor into \(\delta T_{\nu}^{\mu}\). First of all, from the \(i-j\)\((i\neq j)\) equation it follows that we can set \(\phi=\psi\), since \(\delta T_{j}^{i}=0\)\((i\neq j)\). Substituting the energy-momentum tensor \(\delta T_{\nu}^{\mu}\) into the general equations and setting \(\psi=\phi\), we find: -3H(Hph \[+\] ph')+2ph (22) \[=\] 4pGa2{(-b)(BdB-EdE)+2EdE \[+\] 2ph(-b)B2+B2+E22d}, (Hph+ph'),i=0, (23) [(2H' \[+\] H2)ph+Hph'+ph''+2Hph'] (24) \[=\] 4pGa2{(-BdB+EdE)(-b)-2phB2(-b) \[+\] 3[4BdB-2EdE+8phB2]+E2+B26d}. where \(\delta\epsilon=(2E\delta E-2B\delta B-4B^{2}\phi)/(E^{2}-B^{2})\), the relation between \(\delta E\) and \(\delta B\) depends on the YM field equations as below. ### The First order kinetic equations of the YM field The metric as before with \(\psi=\phi\) is \[ds^{2}=a^{2}(\tau)[(1+2\phi)dt^{2}-(1-2\phi)\gamma_{ij}dx^{i}dx^{j}],\] (25) then \(\sqrt{-g}=a^{4}(1-2\phi)\). The equation kinetic energy equation is \[\partial_{\mu}(a^{4}(1-2\phi)\epsilon~{}F^{a\mu\nu})-f^{abc}A_{\mu}^{b}(a^{4}( 1-2\phi)\epsilon~{}F^{c\mu\nu})=0,\] (26) from which, one can get the first order perturbation equations: \[\partial_{\mu}[a^{2}(\epsilon\partial_{\tau}\delta A+\delta\epsilon\partial_{ \tau}A-2\epsilon\phi\partial_{\tau}A)]=0,~{}~{}~{}(\mu=0,1,2,3)\] (27) \[\partial_{i}[\delta\epsilon A^{2}+2\epsilon\phi A^{2}+2\epsilon A\delta A]=0,~ {}~{}~{}(i=1,2,3)\] (28) which also can be written as \[\partial_{i}(B\delta\epsilon+2\epsilon\phi B+2\epsilon\delta B)=0,\] (29) \[\partial_{i}(\epsilon\delta E+\delta\epsilon E-2\epsilon\phi E)=0,\] (30) \[\partial_{\tau}[a^{2}(\epsilon\delta E+\delta\epsilon E-2\epsilon\phi E)]=0.\] (31) From these equations, we can immediately get the simple relation between \(\delta B\) and \(\delta E\): \[a^{2}(\epsilon\delta E+\delta\epsilon E-2\epsilon\phi E)={\rm constant}.\] (32) which is useful when solving the first order Einstein equations (22)-(24).	[]
	### The Solution of the Perturbations. Here we only discuss the simplest case with \(B\equiv 0\), the first order energy-momentum tensor becomes very simply \[\delta T_{0}^{0}=(\epsilon+2b)E\delta E,\] (33) \[-\delta T_{i}^{i}=(\epsilon-2b)E\delta E/3,\] (34) \[\delta T_{i}^{0}=-2\epsilon[E_{1}(\delta B_{3}-\delta B_{2})]=0,\] (35) where we have used \(\delta\epsilon=2bE\delta E/E^{2}\) when \(B\equiv 0\). From Eq.(33)-(35), we find that \(\delta T^{\mu}_{\nu}\) is independent of the metric perturbation of \(\phi\), which is different from the scalar field models (Weller & Lewis Wel; Armendariz-Picon et al. Arm; DeDeo et al. Deo). The first order Einstein equations become \[-3\mathcal{H}(\mathcal{H}\phi+\phi^{\prime})+\nabla^{2}\phi=4\pi Ga^{2}E\delta E (\epsilon+2b),\] (36) (Hph+ph'),i=0, (37) \[[(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\phi+\phi^{\prime\prime}+3\mathcal{H} \phi^{\prime}]=4\pi Ga^{2}E\delta E(\epsilon-2b)/3.\] (38) From this we obtain the main equation, which describes the evolution of the metric perturbation \(\phi\) with time: \[\phi^{\prime\prime}+3\mathcal{H}(1-\gamma)\phi^{\prime}+\gamma\nabla^{2}\phi+( 2\mathcal{H}^{\prime}+\mathcal{H}^{2}-3\mathcal{H}^{2}\gamma)\phi=0,\] (39) where \(\gamma\equiv\frac{2b-\epsilon}{6b+3\epsilon}\). From this equation, one can easily find that the evolution of \(\phi\) only depends on \(\gamma\) and \(\mathcal{H}\), but not on the first order YM field kinetic equations. This is because we have only considerd the "electric" YM field. If we also consider the \(B\) components, the YM equation (38) will be used to relate \(\delta E\) and \(\delta B\). For the inflationary field, the most important prediction is that the inflation can generate a scale-invariant primordial scalar perturbation power spectrum, which has been found in the CMB power spectrum and large scale structure. Now we shall firstly consider whether a Universe dominated by this YM field can also generate a scale-invariant spectrum using the general scalar inflationary models. If we consider the YM field as a kind of candidate of inflationary field, the YM field should satisfy the following constraints: 1) Firstly, we should require that the inflation can exist, which means that the YM field has a state with \(\omega<-1/3\). Using the \(p\) and \(\rho\) in Eq.(12), one can easily get a constraint on the YM field: \[\epsilon<b.\] (40) 2) The energy density of the YM field should be positive, which induces the second constraint on the YM field: \[\epsilon>-b.\] (41) 3) Due to equation (39), if we can define an adiabatic vacuum state at the very high frequency \((k\rightarrow\infty)\), which requires that \(\gamma<0\), this would require the third constraint on the YM field: \[\epsilon>2b,~{}~{}{\rm or}~{}~{}\epsilon<-2b.\] (42) We find these three simple constraints on the YM field cannot be satisfied at the same time, so this model cannot generate a scale-invariant primordial power spectrum as a general scalar field inflationary model. However, it is necessary to notice that this does not mean that the YM field cannot be the source of the inflation. Recently, some authors have discussed a kind of curvaton reheating mechanism in non-oscillatory inflationary models (Feng & Li cur). In this kind of model, the primordial spectrum and the reheating can be generated by the other curvaton field. So, although the YM field cannot generate a scale-invariant primordial spectrum, the YM field can also be the background field in the curvaton field inflationary models, which will be discussed in a future work. However, the YM field can be a good candidate of dark energy (Zhao & Zhang ymc1; Zhang et al. ymc2). Now let us discuss the evolution of the cosmic scalar fluctuations \(\phi\) in the YM field dark energy models. Neglecting anisotropic stress, the potential \(\phi\) evolves as Weller & Lewis Wel; Ma & Bertschinger Ma; Gorden & Hu Chr) \[\phi^{\prime\prime}+3\mathcal{H}\left(1+\frac{p^{\prime}}{\rho^{ \prime}}\right)\phi^{\prime} - \frac{p^{\prime}}{\rho^{\prime}}\nabla^{2}\phi+\left[\left(1+3 \frac{p^{\prime}}{\rho^{\prime}}\right)\mathcal{H}^{2}+2\mathcal{H}^{\prime} \right]\phi\] (43) \[= 4\pi Ga^{2}\left(\delta p-\frac{p^{\prime}}{\rho^{\prime}}\delta \rho\right),\] where \(p=\sum_{i}p_{i}\) and \(\rho=\sum_{i}\rho_{i}\), which should include the contributions of the baryon, photon, neutron, cold dark matter, and the dark energy. Here we consider the simplest case with only the YM field dark energy, and it has the equation of state\(\omega_{de}=-1\). The 'sound speed' is defined by \(c_{s}^{2}=\delta p/\delta\rho\). From the Eqs. (39) and (40), one finds that \(c_{s}^{2}=-1/3\). So then, the equation (43) becomes: \[\phi^{\prime\prime}+2\mathcal{H}\phi^{\prime}+\frac{1}{3}\nabla^{2}\phi+2 \mathcal{H}^{\prime}\phi=0,\] (44) which is the same as equation (39) with \(\gamma=1/3\). Defining \(u\equiv a\phi\), one gets \[u_{k}^{\prime\prime}-\frac{k^{2}}{3}u_{k}=0,\] (45) where \(u_{k}\) is the Fourier component with wavenumber \(k\), we have used \(a\propto 1/\tau\), which has the solution of \[u_{k}\propto e^{\pm\frac{k\tau}{\sqrt{3}}}.\] (46) For the large wavelength case (\(k\tau\ll 1\)), we have \[\phi\propto a^{-1}\] (47) and when \(k\tau\gg 1\), one has the growing solution \[u\propto e^{-\frac{k\tau}{\sqrt{3}}},~{}~{}{\rm and}~{}~{}\phi\sim a^{-1}e^{- \frac{k\tau}{\sqrt{3}}}.\] (48) We find in this universe, the evolution of the cosmic fluctuation \(\phi\) is very different from that in the scalar field dark energy models. For the fluctuation with large wavelength \(k\tau\ll 1\), it is always damped with the expansion of the Universe. However for the fluctuation with small wavelength, it has a rapid growth with time, which is because of the negative sound speed \(c_{s}^{2}=-1/3<0\). This should have an important effect on the large scale integrated Sachs-Wolfe effect of CMB. We should mention that the result in this subsection is only qualitative since we have not considered other effects, such as the baryons, photons, neutrons, dark matter and so on, which are also important for the evolution of the cosmic fluctuation \(\phi\), especially at small scales. ## V Conclusion and Discussion In this paper, we have investigated the effective YM field as a kind of candidate for an inflationary field or dark energy.	[]
	From the zero order Einstein equations and YM field kinetic energy equations, we find that this field can naturally give an equation of state \(-1<\omega<0\) and \(\omega<-1\). This is one of the most important feature of this model. Also, we get \(\omega+1\propto a^{-2}\), which suggests that \(\omega\) will go to \(-1\) with the expansion of the Universe. This makes the Universe dominated by this field have a de Sitter expansion. When considering the evolution of the perturbations, we solved the first order Einstein equations and YM field kinetic energy equations. We find that the model with only an electric field cannot generate a scale-invariant primordial scalar perturbation power spectrum. This means that a model that only used the YM field is not a good candidate for describing the inflationary field. However, as a kind of candidate of dark energy, we obtained the equation of the YM perturbation, and found that this field is very different from the general scalar field dark energy models. This YM field has a negative sound speed, which makes the cosmic perturbations dampen at large scales. This is a very important source of the integrated Sachs-Wolfe effect of the CMB power spectrum. This damping of \(\phi\) at large scale may be helpful for answering the very small quadrupole problem of the CMB temperature anisotropy power spectrum. We should mention that, in this paper we have not considered the possible interaction between the YM field dark energy and the other components, especially the dark matter (Zhang et al. ymc2). We leave this topic to future work. Acknowledgement: W.Zhao appreciates useful discussions with Prof.Y.Zhang. This work is supported by Chinese NSF grants No.10703005 and No.10775119. ## References * (1) Adler S., 1981, Phys.Rev.D 23, 2905 * (2) Adler S. and Piran T., 1982, Phys.Lett.B 117, 91 * (3) Armendariz-Picon C., Mukhanov V. and Steinhardt P.J., 2000, Phys.Rev.Lett. 85, 4438 * (4) Bennett C.L. et al., 2003, Astrophys.J.Suppl. 148, 1 * (5) Bharat R. and Peebles P.J., 1988, Phys.Rev.D 37, 3406 * (6) Coleman S. and Weinberg E., 1973, Phys.Rev.D 7, 1888 * (7) Corasaniti P.S., Kunz M., Parkinson D., Copeland E.J. and Bassett B.A., 2004, Phys.Rev.D 70, 083006 * (8) DeDeo S., Caldwell R.R., and Steinhardt P.J., 2003, Phys.Rev.D 67, 103509 * (9) Feng B. and Li M.Z., 2003, Phys.Lett.B 564, 169 * (10) Feng B., Wang X.L. and Zhang X.M., 2005, Phys.Lett.B 607, 35 * (11) Gordon C. and Hu W., 2004, Phys.Rev.D 70, 083003 * (12) Gross D.J. and Wilczez F., 1973, Phys.Rev.Lett. 30, 1343 * (13) Kolb E.W. and Turner M.S., 1990, _The Early Universe_ Addison-Wesley Publishing Company * (14) Ma C.P. and Bertschinger E., 1995, Astrophys.J. 455, 7 * (15) Mukhanov V.F., Feldman H.A., and Brandenberger R.H., 1992, Phys.Rep. 215, 203 * (16) Pagels H. and Tomboulis E., 1987, Nucl.Phys.B 143, 485 * (17) Parker L. and Raval A., 1999, Phys.Rev.D 60, 063512 * (18) Politzer H., 1973, Phys.Rev.Lett. 30, 1346 * (19) Ratra B., 1992, Astrophys.J. 391, L1 * (20) Riess A.G. et al., 1998, Astron. J. 116, 1009 * (21) Tegmark M. et al., 2004, Astrophys.J. 606, 702 * (22) Weller J. and Lewis A.M., 2003, Mon.Not.Roy.Astron.Soc. 346, 987 * (23) Wetterich C., 1995, Astron.Astrophys. 301, 321 * (24) Zhang Y., 1994, Phys.Lett.B 340, 18 * (25) Zhang Y., 2003, Chin.Phys.Lett.20, 1899 * (26) Zhang Y., Xia T.Y. and Zhao W., 2007, Class. Quantum Gravity, 24, 3309 * (27) Zhao W. and Zhang Y., 2006, Class. Quantum Gravity, 23, 3405; Zhao W. and Zhang Y., 2006, Phys.Lett.B 640, 69; Zhao W. and Xu D.H., 2007, Int.J.Mod.Phys.D 16, 1735; Zhao W., 2008, Int.J.Mod.Phys.D 17, 1245	[]
	# GRB050223: A faint Gamma-Ray Burst discovered by _Swift_ K.L. Page\({}^{1}\), E. Rol\({}^{1}\), A.J. Levan\({}^{1}\), B. Zhang\({}^{2}\), J.P. Osborne\({}^{1}\), P.T. O'Brien\({}^{1}\) A.P. Beardmore\({}^{1}\), D.N. Burrows\({}^{3}\), S. Campana\({}^{4}\), G. Chincharini\({}^{4,5}\), J.R. Cummings\({}^{6}\) G. Cusumano\({}^{7}\), N. Gehrels\({}^{6}\), P. Giommi\({}^{8}\), M.R. Goad\({}^{1}\), O. Godet\({}^{1}\), V. Mangano\({}^{6}\) G. Tagliaferri\({}^{4}\) & A.A. Wells\({}^{1}\) \({}^{1}\) X-Ray and Observational Astronomy Group, Department of Physics & Astronomy, University of Leicester, LE1 7RH, UK \({}^{2}\) University of Nevada, Box 454002, Las Vegas, NV 89154-4002, USA \({}^{3}\) Department of Astronomy & Astrophysics, 525 Davey Lab, Pennsylvania State University, University Park, PA 16802, USA \({}^{4}\) INAF - Osservatorio Astronomico di Brera, Via Bianchi 46, 23807 Merate, Italy \({}^{5}\) Universita degli studi di Milano-Bicocca, Dipartimento di Fisica, Piazza delle Scienze 3, I-20126 Milan, Italy \({}^{6}\) NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA \({}^{7}\) INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica Sezione di Palermo, Via Ugo La Malfa 153, 90146 Palermo, Italy \({}^{8}\) ASI Science Data Center, via Galileo Galilei, 00044 Frascati, Italy ###### Abstract GRB050223 was discovered by the _Swift_ Gamma-Ray Burst Explorer on 23 February 2005 and was the first Gamma-Ray Burst to be observed by both _Swift_ and _XMM-Newton_. At the time of writing (May 2005), it has one of the faintest GRB afterglows ever observed. The spacecraft could not slew immediately to the burst, so the first X-ray and optical observations occurred approximately 45 minutes after the trigger. Although no optical emission was found by any instrument, both _Swift_ and _XMM-Newton_ detected the fading X-ray afterglow. Combined data from both of these observatories show the afterglow to be fading monotonically as 0.99\({}^{+0.15}_{-0.12}\) over a time frame between 45 minutes to 27 hours post-burst. Spectral analysis, allowed largely by the higher through-put of _XMM-Newton_, implies a power-law with a slope of \(\Gamma\) = 1.75\({}^{+0.19}_{-0.18}\) and shows no evidence for absorption above the Galactic column of 7 \(\times\) 10\({}^{20}\) cm\({}^{-2}\). From the X-ray decay and spectral slopes, a low electron power-law index of \(p\) = 1.3-1.9 is derived; the slopes also imply that a jet-break has not occured up to 27 hours after the burst. The faintness of GRB050223 may be due to a large jet opening or viewing angle or a high redshift. keywords: gamma-rays:bursts Received / Accepted ## 1 Introduction The _Swift_ Gamma-Ray Burst Explorer (Gehrels et al. 2004) was launched on 20th November 2004. It is a multi-wavelength observatory, covering the gamma-ray, X-ray and UV/optical bands. The observatory is designed to slew rapidly and autonomously to point narrow-field instruments (the X-ray and Ultra-Violet/Optical Telescopes - XRT and UVOT, respectively) towards any Gamma-Ray Bursts (GRBs) detected by the Burst Alert Telescope (BAT). This allows prompt observations of the afterglow on a timescale of minutes, much more quickly than was previously feasible on a regular basis. The on-board instruments are described in detail in Barthelmy (2004, 2005; BAT), Burrows et al. (2004, 2005; XRT) and Roming et al. (2004, 2005; UVOT). _Swift_ is significantly more sensitive to detecting GRBs than previous instruments capable of providing rapid, accurate (to within a few arcmin) localisations (e.g., _HETE-2_ and _BeppoSAX_). Thanks to its rapid repointing capability, _Swift_ is also able to observe afterglows at early times. Since GRB afterglows fade rapidly, this ensures they are observed at their brightest, allowing _Swift_ to detect fainter afterglows and thus look further down the GRB afterglow luminosity function than has previously been possible. Investigating the faint end of this function is of particular importance in understanding the structure of the bursts themselves. Faint	[]
	the individual spectra. No change in spectral shape is found between the _XMM-Newton_ spectra. Note that the spectrum is shown in detected counts s\({}^{-1}\) keV\({}^{-1}\) for each of the instruments. Thus, while the _XMM-Newton_ spectrum may have a higher count-rate, due to the higher throughput, this does not correspond to an increased flux. The unabsorbed fluxes (0.5-10 keV) for observations one, three and four (as named in Table 1) were found to be (8.18\({}^{+3.32}_{-2.74}\)) \(\times\) 10\({}^{-13}\), (1.18\({}^{+0.19}_{-0.36}\)) \(\times\) 10\({}^{-13}\) and (5.42\({}^{+0.98}_{-1.44}\)) \(\times\) 10\({}^{-14}\) erg cm\({}^{-2}\) s\({}^{-1}\) respectively. ### UV and Optical data Neither the _Swift_-UVOT (Gronwall et al. 2005) nor the _XMM-Newton_ Optical Monitor (Blustin et al. 2005) detected a source at the position of the X-ray afterglow. As mentioned above, the UVOT observation started about 46 minutes after the BAT trigger, due to the delayed slew; the _XMM-Newton_-OM data were collected 11 hours after the trigger. No new sources were identified by _ROTSE-III_ (to a limiting unfiltered magnitude of 18 from approximately a minute after the burst; Smith 2005), the Mount John University Observatory (to R = 20.5, 10 hours after the burst; Gorosabel et al. 2005) or the _PROMPT_ robotic telescope array (limiting magnitude of \(\approx\) 21 for Rc, V and Ic filters, with the mean time for these observations being 4-5 hours after the trigger; Nysewander et al. 2005). ## 3 Discussion GRB050223 has, at the time of writing (May 2005), one of the faintest GRB X-ray afterglows observed by _Swift_; comparison with figure 1 of Piro (2004) shows the 11 hour flux of GRB050223 to be below all those detected by _BeppoSAX_. ### Afterglow models Three GRB afterglow models are initially considered, as summarised by Zhang & Meszaros (2003). The 'ISM' model has a fireball expanding into the (homogeneous) interstellar medium (Sari, Piran & Narayan 1998), while, in the 'Wind' model, the fireball expands into a wind environment,	[]
	# Measuring Fundamental Galactic Parameters with Stellar Tidal Streams and SIM PlanetQuest Steven R. Majewski1 , David R. Law2 , Allyson A. Polak1 & Richard J. Patterson1 Footnote 1: Samples should be unbiased with respect to spread perpendicular to that plane, but this should be trivial to achieve. Footnote 2: We do not model the possibly complex interaction between the Small and Large Magellanic Clouds since we are interested in testing a hypothetical stream with desirable properties. Footnote 1: Samples should be unbiased with respect to spread perpendicular to that plane, but this should be trivial to achieve. Footnote 1: Samples should be unbiased with respect to spread perpendicular to that plane, but this should be trivial to achieve. srm4n, aap5u,rjp0i@virginia.edu, drlaw@astro.caltech.edu ###### Abstract Extended halo tidal streams from disrupting Milky Way satellites offer new opportunities for gauging fundamental Galactic parameters without challenging observations of the Galactic center. In the roughly spherical Galactic potential tidal debris from a satellite system is largely confined to a single plane containing the Galactic center, so accurate distances to stars in the tidal stream can be used to gauge the Galactic center distance, \(R_{0}\), given reasonable projection of the stream orbital pole on the \(X_{GC}\) axis. Alternatively, a tidal stream with orbital pole near the \(Y_{GC}\) axis, like the Sagittarius stream, can be used to derive the speed of the Local Standard of Rest (\(\Theta_{\rm LSR}\)). Modest improvements in current astrometric catalogues might allow this measurement to be made, but NASA's Space Interferometry Mission (SIM PlanetQuest) can definitively obtain both \(R_{0}\) and \(\Theta_{\rm LSR}\) using tidal streams. Subject headings: Milky Way: structure - Milky Way: dynamics - Sagittarius dwarf galaxy ## 1. Distance to the Galactic Center With the assumption that globular clusters trace the general shape and extent of the Milky Way (MW), Shapley (1918) first showed how they can be used to estimate the distance (\(R_{0}\)) to the Galactic center (GC), expected to lie at the center of the cluster distribution. Though Shapley's first execution of this experiment exaggerated \(R_{0}\) due to cluster distance scale problems, the overall scheme of mapping an extended distribution of Galactic tracer objects to determine the location of its center remains a valid, if traditionally underutilized, strategy. The globular cluster sample is relatively small and concentrated to the GC, where dust effects introduce large distance uncertainties and a likely still incomplete and lop-sided cluster census. Population II tracers like RR Lyrae, blue horizontal branch (BHB) or giant stars are much more plentiful outside of the MW bulge. Unfortunately, the current census for these tracers is even more incomplete than for globulars. Though this situation may be remedied by currently planned wide angle surveys, several inherent problems remain with exploitation of these tracers as GC benchmarks. As with the clusters, the MW Zone of Avoidance (ZA) will always result in biased sample distributions and potential \(R_{0}\) underestimates -- exacerbated if surveys do not reach the far side of the MW. Even more challenging is that the global distributions of halo stars are far from dynamically mixed: Recent surveys of the above tracers reveal a halo streaked with substructure, likely originating as satellite disruption debris (e.g., Vivas et al. 2001, Newberg et al. 2002, Majewski 2004) and eroding simple halo axisymmetry. The very existence of numerous tidal streams motivates the present contribution. _Individual_ tidal streams actually possess a relatively simple spatial configuration. Within spherical potentials, tidal debris arms from a disrupting satellite will lie along the satellite orbital plane, which contains the GC. A sufficiently extended tidal debris arc defines that plane, which intersects the MW \(X_{GC}\) axis at the GC. This simpler, almost two-dimensional geometry of tidal stream arcs removes the need for sample completeness: In principle, \(R_{0}\) should be derivable from the (l,b,distance) distribution of only a large enough sample of tidal stream stars to define their orbital plane.1 Footnote 1: Samples should be unbiased with respect to spread perpendicular to that plane, but this should be trivial to achieve. In reality, non-spherical potentials precess tidal streams. Fortunately, this is a relatively small effect in the MW, at least for \(R_{GC}\)'s of tens of kiloparsecs. Johnston et al. (2005; "J05") showed the Sagittarius (Sgr) tidal stream precession is sufficiently small to conclude the MW potential is only slightly oblate within the Sgr orbit (peri:apo-Galacticon of 13:57 kpc). Moreover, as pointed out also by Helmi (2004), that part of the Sgr trailing arm arcing across the southern MW hemisphere (see Majewski et al. 2003, "MSWO") is so dynamically young that it hasn't had _time_ to precess (see Fig. 5 of J05). Unfortunately, as noted by MSWO, Sgr is in almost the worst possible orientation to undertake the proposed \(R_{0}\)-gauging: With a virtually negligible angle between the Sgr orbital plane and \(X_{GC}\) axis, small errors in the definition of the orbital plane (due to small residual precession, and the finite width of the debris plane) lead to substantial uncertainties in derivation of \(R_{0}\). The ideal tidal debris configuration for estimating \(R_{0}\) has a _pole_ closer to the \(X_{GC}\) axis. Given the pace of discovery, such a stream may soon be found. Based on the nearly polar orientation of the HI Magellanic Stream and the typically measured proper motions (\(\mu\)'s) for the Magellanic Clouds (Gardiner & Noguchi 1996, van der Marel et al. 2002 and references therein), it is clear that a stellar counterpart to the Magellanic Stream would have almost the perfect orientation for gauging \(R_{0}\). Systematic errors in a tracer distance scale translate to estimates of \(R_{0}\). However, because streams contain different stellar types (e.g., giant stars, RR Lyrae, BHB), uncertainties from photometric/spectroscopic parallaxes can be cross-checked. In most cases, reddening and crowding effects can be of negligible concern. Alterna	[]
	tively, with NASA's Space Interferometry Mission (SIM), direct _trigonometric_ parallaxes will be well within reach: For a putative Magellanic stellar stream orbiting at \(\sim 50\) kpc radius, the \(\sim 10\)-20 \(\mu\)as parallaxes are well above the SIM wide-angle astrometric accuracy goal of 4\(\mu\)as, assuming K giant star tracers (\(V\sim 18\)). As a test of what might be achieved, we ran N-body simulations of different mass satellites disrupting for 5 or 10 Gyr (whatever was needed to produce \(>270^{\circ}\)-long tails) in the Galactic potential that best fitted the Sgr debris stream in Law et al. (2005; "L05" hereafter). The orbit was constrained to match the current position, radial velocity (RV) and \(\mu\) (Gardiner & Noguchi 1996) of the Small Magellanic Cloud (SMC), with orbital pole \((l,b)=(196,-5)^{\circ}\).2 All other model parameters were similar to those in L05. Each simulation was "observed" outside a \(\|b\|>15^{\circ}\) ZA, with \(\sim 10^{3}\), \(10^{4}\) and \(10^{5}\) tracer stars (apportioned with \(\sim 90\%\) of these in the satellite core and \(\sim 10\%\) in the tidal tails), and with 0, 10 and 20% random Gaussian distance errors imposed. The simplest (though not best!) analysis of these data is simply to fit a plane and measure its intersection with the \(X_{GC}\) axis. With this 0th-order method, even for large samples of stars in dynamically cold streams (i.e., not that from a 10\({}^{10}\) M\({}_{\sun}\) progenitor) and no distance errors, relatively large (\(<7\%\)) systematic errors in \(R_{0}\) can remain (Fig. 1) because plane-fitting does not account for the residual precessional twisting of the debris arms. The direction of precession (determined by the direction of the stream angular momentum vector) drives the sense of the imposed systemic \(R_{0}\) error (i.e. closer or farther), and random distance errors add additional uncertainties depending on details of the stream orientation relative to the ZA. A better, now proven method (e.g. L05) is to use N-body modeling to reconstruct a given stellar stream; such modeling can precisely account not only for precession but also for stream dispersion and other higher order uncertainties, which would permit a more accurate identification of the center of the MW potential for an appropriately oriented stream. Footnote 2: We do not model the possibly complex interaction between the Small and Large Magellanic Clouds since we are interested in testing a hypothetical stream with desirable properties. Recent measurements of stellar motions around Sgr A\({}^{}\) have led to dynamical parallaxes good to 5% (\(7.94\pm 0.42\) kpc; Eisenhauer et al. 2003), a measurement sure to improve with longer Sgr A\({}^{}\) field monitoring campaigns. Few percent quality trigonometric parallaxes of stars near the GC will be measured as part of a SIM Key Project. In either method, the target stars are reasonably expected to lie at the assumed center of the MW potential. The proposed use of tidal streams to measure \(R_{0}\) will provide an interesting test of this hypothesis, since tidal streams orbit the _true_ dynamical center of the _integrated_ potential over tens of kiloparsec scales. A comparison of this center to the Sgr A\({}^{}\) distance could reveal whether the MW may be a lop-sided spiral (e.g., Baldwin et al. 1980, Richter & Sancisi 1994, Rix & Zaritsky 1995). Such lop-sidedness can, in fact, be induced by mergers of large satellites (Walker, Mihos & Hernquist 1997). In principle, three well-measured tidal streams can verify whether the GC lies along \((l,b)=(0,0)\), since the true GC should lie at a mutual intersection of the three corresponding stream orbital planes. ## 2. Velocity of the Local Standard of Rest Despite decades of effort, the local MW rotation rate remains poorly known, with measurements varying by 25%. Hipparcos \(\mu\)'s (Feast & Whitelock, 1997) suggest that the Local Standard of Rest (LSR) velocity is \(\Theta_{\rm LSR}=(217.5\pm 7.0)(R_{0}/8)\) km s\({}^{-1}\) -- i.e. near the IAU adopted value of 220 km s\({}^{-1}\). But a more recent measurement of \(\mu\) for Sgr A\({}^{}\) (Reid & Brunthaler 2004) yields a higher \((235.6\pm 1.2)(R_{0}/8)\) km s\({}^{-1}\), whereas direct HST measurements of the \(\mu\)'s of bulge stars against background galaxies in the same field yield \((202.4\pm 20.8)(R_{0}/8)\) km s\({}^{-1}\) (Kalirai et al. 2004) and \((220.8\pm 13.6)(R_{0}/8)\) km s\({}^{-1}\) (Bedin et al. 2003). Of course, these measures (as well as any of those depending on the Oort constants) rely on an accurate measure of \(R_{0}\) (SS1). The solar peculiar motion must also be known, but is a smaller correction (e.g., \(5.3\pm 0.6\) km s\({}^{-1}\); Dehnen & Binney 1998). On the other hand, considerations of non-axisymmetry of the disk yield corrections to the measurements that suggest \(\Theta_{\rm LSR}\) may be as low as \(184\pm 8\) km s\({}^{-1}\)(Olling & Merrifield, 1998) or lower (Kuijken & Tremaine 1994). Independent methods to ascertain \(\Theta_{\rm LSR}\) are of great value because it is fundamental to establishing the MW mass scale. Eventually, as part of a Key Project of SIM, \(\Theta_{\rm LSR}\) will be measured directly by the absolute \(\mu\) of stars near the GC. Here we describe an independent method for ascertaining \(\Theta_{\rm LSR}\) using halo tidal streams that overcomes several difficulties with working in the highly dust-obscured, crowded GC, and one also insensitive to \(R_{0}\) (for all reasonable values of the latter). The ideal tidal stream for this method is one with an orbital pole lying near the \(Y_{GC}\) axis. The Sgr tidal stellar stream not only fulfills this requirement, but its stars, particularly its trailing arm M giants, are ideally placed for uncrowded field astrometry at high MW latitudes, and at relatively bright magnitudes for, and requiring only the most modest precisions from, SIM. Indeed, as we show, this method is even within the grasp of future high quality, ground-based astrometric studies. It is remarkable that the Sun presently lies within a kiloparsec of the Sgr debris plane (MSWO). The pole of the plane, \((l_{p},b_{p})=(272,-12)^{\circ}\), means that the line of nodes of its intersection with the MW plane is almost coincident with the \(X_{GC}\) axis. Thus (Fig. 2) the motions of Sgr stars _within_ this plane are almost entirely contained in their Galactic \(U\) and \(W\) velocity components, whereas the \(V\) motions of stars in the Sgr tidal tails almost entirely reflect _solar motion_. To the degree that its \(V\) distribution is not completely flat in Figure 2 is due to the slight amount of streaming motion projected onto the \(V\) motions from the 2\({}^{\circ}\) Sgr orbital plane tilt from \(X_{GC}\), compounded by (1) Keplerian variations in the space velocity of stars as a function of orbital phase, as well as (2) precessional effects that lead to \(\Lambda_{\sun}\)-variable departures of Sgr debris from the nominal best fit plane to all of the debris. The latter is negligible for trailing debris but is much larger for the leading debris, which is on average closer to the GC and dynamically older compared to the trailing debris when viewed near the Galactic poles (J05). In addition, because the leading debris gets arbitrarily close to the Sun (L05), projection effects make it more complicated to use for the present purposes. Addi	[]
	tional problems with the leading arm debris, which suggest that more complicated effects have perturbed it are discussed in L05 and J05. In contrast, the Sgr trailing tail is beautifully positioned fairly equidistantly from us for a substantial fraction of its stretch across the Southern MW hemisphere (MSWO). This band of stars arcing almost directly "beneath" us within the \(X_{GC}\)-\(Z_{GC}\) plane provides a remarkable zero-point reference against which to make direct measurement of the solar motion _almost completely independent of the Sun's distance from the GC._ The extensive mapping (MSWO) of the Sgr tails with Two Micron All-Sky Survey (2MASS) M giants provides an ideal source list for individual stellar targets from this \(>360^{\circ}\)-wrapped, MW polar ring. L05 used M giant spatial (MSWO) and RV data (Majewski et al. 2004) to constrain models of Sgr disruption, best fitting when a 3.5 \(\times 10^{8}\) M\({}_{\sun}\) Sgr of 328 km s\({}^{-1}\) space velocity orbits with period 0.85 Gyr and apo:peri-Galactica of 57:13 kpc. These models fit what appears to be \(\sim\)2.5 orbits (2.0 Gyr) of Sgr mass loss in M giants. The adopted MW potential is smooth, static and given by the sum of a disk, spheroid and halo described by the axisymmetric function \(\Phi_{\rm halo}=v_{\rm halo}^{2}\ln(R^{2}+[z/q]^{2}+d^{2})\) where \(q\) is the halo flattening, \(R\) and \(z\) are cylindrical coordinates and \(d\) is a softening parameter. Additional model details are given in L05. Model fits to the Sgr spatial and velocity data allow predictions of the 6-D phase space configuration of Sgr debris. Figure 2 shows predicted \(U,V,W\) velocity components of debris as a function of longitude, \(\Lambda_{\sun}\), in the Sgr orbital plane (see MSWO). The debris is shown assuming \(q=0.9\), \(R_{0}=7\) kpc, and a characterization of the total potential whereby the LSR speed is \(\Theta_{\rm LSR}\) = 220 km s\({}^{-1}\). L05 explores how variations in \(q\) affect primarily the \(U\) and \(W\) (through projection along RV). Figure 3 (green, yellow, and magenta points) shows how variations in the _scale_ of the potential, expressed through variations in adopted \(\Theta_{\rm LSR}\), affect \(V\). Clearly, \(\Theta_{\rm LSR}\) ranging from 180 to 260 km s\({}^{-1}\) translates to obvious variations in observed \(V\) for trailing arm stars. This effect is easily separable from any residual uncertainty in the shape of the potential or \(R_{0}\): Figure 3 (red and blue points respectively) illustrates negligible \(V\) changes produced by holding \(\Theta_{\rm LSR}\) fixed at 220 km s\({}^{-1}\) but varying \(q\) from 0.9 to 1.25 (i.e. oblate to prolate) and \(R_{0}\) from 7 to 9 kpc. Figure 3 is the basis for the proposed use of Sgr to measure \(\Theta_{\rm LSR}\). Ideally, to execute the experiment requires obtaining \(V\) from the observed \(\mu\) and RVs of Sgr arm stars. However, because of the particular configuration of Sgr trailing arm debris, almost all of \(V\) is reflected in the \(\mu\) of these stars, and, more specifically, the reflex solar motion is contained almost entirely in the \(\mu_{l}\cos(b)\) component of \(\mu\) for Sgr trailing arm stars away from the MW pole. Working in the observational, \(\mu\) regime means that vagaries in the derivation of _individual_ star distances can be removed from the problem, as long as the system is modeled with a proper _mean_ distance for the Sgr stream as a function of \(\Lambda_{\sun}\). Figure 4 shows three general regimes of the trailing arm \(\mu_{l}\cos(b)\) trend: (1) \(\Lambda_{\sun}\gtrsim 100^{\circ}\) where \(\mu_{l}\cos(b)\) is positive and roughly constant, (2) the region from \(100^{\circ}\gtrsim\Lambda_{\sun}\gtrsim 60^{\circ}\) where \(\mu_{l}\cos(b)\) flips sign as the debris passes through the South Galactic Pole to shift the Galactic longitudes of the trailing arm by \(\sim 180^{\circ}\), and (3) \(\Lambda_{\sun}\lesssim 60^{\circ}\), where \(\mu_{l}\cos(b)\)is negative and becomes smaller with decreasing \(\Lambda_{\sun}\) (because the Sgr stream becomes increasingly farther). The sign flip in \(\mu_{l}\cos(b)\) is a useful happenstance in the case where one has \(\mu\) data not tied to an absolute reference frame but which is at least robust to systematic zonal errors: In this case the peak to peak amplitude of \(\mu_{l}\cos(b)\) for the trailing arm stars yields (two times) the reflex motion of the Sun3. Footnote 3: We find that these peaks lie at \(\Lambda_{\odot}=60-65^{\circ}\) and \(115-120^{\circ}\); note that technically \(V\propto(\mu/d)_{60-65}+(\mu/d)_{115-120}\). The intrinsic RV dispersion of the Sgr trailing arm has been measured to be \(\sim\)10 km s\({}^{-1}\) (Majewski et al. 2004); assuming symmetry in the two transverse dimensions of the stream gives an intrinsic \(\mu\) dispersion of the Sgr trailing arm of \(\sim\)0.1 mas yr\({}^{-1}\) (see Fig. 4a). Thus, until SIM-quality proper motions exist, the measurement of the reflex solar motion by this method will be dominated by the error in \(\mu\). To quantify the accuracy of the proposed method, we introduce artificial random errors into the proper motions of the five models shown in Figure 3 and calculate the accuracy with which we expect to recover the solar reflex motion. Simply applying the formalism described above, we recover \(\Theta_{\rm LSR}\) values of4 212, 255, and 279 km s\({}^{-1}\) in models for which input \(\Theta_{\rm LSR}=180,220,\) and 260 km s\({}^{-1}\) respectively. This indicates that the method systematically overpredicts \(\Theta_{\rm LSR}\) by about 30 km s\({}^{-1}\); this is because (see Fig. 3) the trend of \(V\) with \(\Lambda_{\odot}\) is not perfectly flat but changes by \(\sim 30\) km s\({}^{-1}\) between the peaks at \(\Lambda_{\odot}=60-65^{\circ}\) and 115-120\({}^{\circ}\). Correcting for this systematic bias, we perform 1000 tests where we randomly draw particles from the model debris streams in these ranges with artificially added random scatter in the \(\mu\), and find that recovering the solar velocity to within 10 km s\({}^{-1}\) requires a sample of approximately 200 stars with \(\mu\) measured to about 1 mas yr\({}^{-1}\) precision with no zonal systematics. Using the models with \(\Theta_{\rm LSR}=\) 180/220/260 km s\({}^{-1}\), these tests recover mean values of 182/225/249 km s\({}^{-1}\) respectively with a dispersion of results between the tests of 10 km s\({}^{-1}\). As expected from Figure 3, varying \(q\) and \(R_{0}\) has negligible effect: Tests on models in both of these MW potentials (where \(\Theta_{\rm LSR}=220\) km s\({}^{-1}\)) recover mean values of 225 and 228 km s\({}^{-1}\). Footnote 4: Correcting for the assumed 12 km s\({}^{-1}\) speed of the Sun with respect to the LSR. Present astrometric catalogs are just short of being able to do this experiment: Hipparcos is not deep enough, the Southern Proper Motion Survey (Girard et al. 2004) has not yet covered enough appropriate sky area, and UCAC2 (Zacharias et al. 2004) has several times larger random errors than useful as well as comparably-sized zonal systematic errors at relevant magnitudes (N. Zacharias, private communication). However, to demonstrate how only modest advances in all-sky \(\mu\) precisions are needed to make a definitive measurement, Figure 4 includes a direct comparison of the \(\mu_{l}\cos(b)\) trend for Sgr M giants using UCAC2 \(\mu\)'s for 2MASS M giants. Impressively, the overall expected \(\mu_{l}\cos(b)\) trends can be seen, but the large scatter and systematic shifts in the trailing arm motions belie the limits of UCAC2 accuracies at \(V\sim 15\). Even a factor of two improvement in UCAC2 random errors _and_ elimination of zonal errors might lead to a useful measurement of \(\Theta_{\rm LSR}\). It is not unreasonable to expect advances in all-sky \(\mu\) catalogues at this level	[]
	soon (e.g., from the Origins Billion Star Survey or Gaia), but in any case SIM PlanetQuest will _easily_ obtain the necessary \(\mu\) (and parallaxes) of selected Sgr trailing arm giants. We appreciate funding by NASA/JPL through the Taking Measure of the MW Key Project for SIM PlanetQuest, NSF grant AST-0307851, the Packard Foundation, and the F.H. Levinson Fund of the Peninsular Community Foundation. SRM appreciates the hospitality of the Carnegie Observatories during the writing of this paper. ## References * Baldwin et al. (1980) Baldwin, J. E., Lynden-Bell, D., & Sancisi, R. 1980, MNRAS, 193, 313 * Bedin et al. (2003) Bedin, L. R., Piotto, G., King, I. R., & Anderson, J. 2003, AJ, 126, 247 * Dehnen & Binney (1998) Dehnen, W., & Binney, J. J. 1998, MNRAS, 298, 387 * Eisenhauer et al. (2003) Eisenhauer, F., Schodel, R., Genzel, R., Ott, T., Tecza, M., Abuter, R., Eckart, A., & Alexander, T. 2003, ApJ, 597, L121 * Feast & Whitelock (1997) Feast, M. W. & Whitelock, P. 1997, MNRAS, 291, 683 * Gardiner & Noguchi (1996) Gardiner, L. T., & Noguchi, M. 1996, MNRAS, 278, 191 * Girard et al. (2004) Girard, T. M., Dinescu, D. I., van Altena, W. F., Platais, I., Monet, D. G., & Lopez, C. E. 2004, AJ, 127, 3060 * Johnston et al. (2005) Johnston, K. V., Law, D. R., & Majewski, S. R. 2005, ApJ, 619, 800 (J05) * Kalirai et al. (2004) Kalirai, J. S., et al. 2004, ApJ, 601, 277 * Kuijken & Tremaine (1994) Kuijken, K., & Tremaine, S. 1994, ApJ, 421, 178 * Law et al. (2005) Law, D. R., Johnston, K. V., & Majewski, S. R. 2005, ApJ, 619, 807 (L05) * Majewski (2004) Majewski, S. R. 2004, Pub.Astr.Soc. Australia, 21, 197 * Majewski et al. (2004) Majewski, S. R., et al. 2004, AJ, 128, 245 * Majewski et al. (2003) Majewski, S. R., Skrutskie, M. F., Weinberg, M. D., & Ostheimer, J. C. 2003, ApJ, 599, 1082 (MSWO) * van der Marel et al. (2002) van der Marel, R. P., Alves, D. R., Hardy, E., & Suntzeff, N. B. 2002, AJ, 124, 2639 * Newberg et al. (2002) Newberg, H. J., et al. 2002, ApJ, 569, 245 * Olling & Merrifield (1998) Olling, R. P. & Merrifield, M. R. 1998, MNRAS, 297, 943 * Reid & Brunthaler (2004) Reid, M. J., & Brunthaler, A. 2004, ApJ, 616, 872 * Richter & Sancisi (1994) Richter, O.-G., & Sancisi, R. 1994, A&A, 290, L9 * Rix & Zaritsky (1995) Rix, H., & Zaritsky, D. 1995, ApJ, 447, 82 * Shapley (1918) Shapley, H. 1918, ApJ, 48, 154 * Vivas et al. (2001) Vivas, A. K., et al. 2001, ApJ, 554, L33 * Walker et al. (1996) Walker, I. R., Mihos, C., & Hernquist, L. 1996, ApJ, 460, 121 * Zacharias et al. (2000) Zacharias, N., Rafferty, T. J., & Zacharias, M. I. 2000, ASP Conf. Ser., 216, 427	[]
	f two in the flux values and a similar shape can be recognized for all experiments with a knee at energies of about 4 PeV. Typical values for the systematic uncertainties of the absolute energy scale for air shower experiments are about 15 to 20%. Renormalizing the energy scales of the individual experiments to match the all-particle spectrum obtained by direct measurements in the energy region up to almost a PeV requires correction factors in the order of \(\pm 10\)% [60]. A remarkable result, indicating that behind an absorber of 11 hadronic interaction lengths or 30 radiation lengths the energy of the primary particle is determined with an absolute error in the order of \(\pm 10\)%. One should keep in mind that the experiments investigate different air shower components, are situated at different atmospheric depths, and use different interactions models to interpret the observed data. Nevertheless, the systematic differences are relatively small and the all-particle spectrum seems to be well known. Average experimental fluxes are tabulated in [60]. The lines in Fig. 1 indicate sum spectra obtained by extrapolations of the energy spectra for individual elements from direct measurements assuming power laws with a cut-off proportional to the charge of the respective element according to the poly-gonato model [60]. Sum spectra for elements from hydrogen to nickel and for all elements are shown. The extrapolated all-particle flux is compatible with the flux derived from air shower experiments in the knee region. Above \(10^{8}\) GeV the flux of galactic cosmic rays is not sufficient to account for the observed all particle spectrum, and an additional, presumably extragalactic component is required. ## 3 Average mass of cosmic rays At energies below a PeV energy spectra for individual elements have been observed above the atmosphere [69]. At higher energies this is presently not possible due to the low flux values and the large fluctuations in the development of extensive air showers. Thus, mostly the mean mass is investigated. An often-used quantity to characterize the composition above 1 PeV is the mean logarithmic mass, defined as \(\langle\ln A\rangle=\sum_{i}r_{i}\ln A_{i}\), \(r_{i}\) being the relative fr Figure 2: Mean logarithmic mass of cosmic rays derived from the measurements of electrons, muons, and hadrons at ground level. Results are shown from CASA-MIA [61], Chacaltaya [62], EAS-TOP electrons and GeV muons [63], EAS-TOP/MACRO (TeV muons) [64], HEGRA CRT [65], KASCADE electrons and muons interpreted with two hadronic interaction models [50], hadrons and muons [66], as well as an analysis combining different observables with a neural network [52], and SPASE/AMANDA [67]. The lines indicate expectations according to the poly-gonato model. For comparison, results from direct measurements are shown as well from the JACEE [68] and RUNJOB [32] experiments.	[ { "caption": "Figure 2. Mean logarithmic mass of cosmic rays derived from the measurements of electrons, muons, and hadrons at ground level. Results are shown from CASA-MIA [61], Chacaltaya [62], EAS-TOP electrons and GeV muons [63], EAS-TOP/MACRO (TeV muons) [64], HEGRA CRT [65], KASCADE electrons and muons interpreted with two hadronic interaction models [50], hadrons and muons [66], as well as an analysis combining different observables with a neural network [52], and SPASE/AMANDA [67]. The lines indicate expectations according to the poly-gonato model. For comparison, results from direct measurements are shown as well from the JACEE [68] and RUNJOB [32] experiments.", "captionBoundary": { "x1": 95, "x2": 699, "y1": 346, "y2": 474 }, "figType": "Figure", "imageText": [ "JACEE", "direct", ":", "RUNJOB", "❄", "KASCADE", "(nn)", "KASCADE", "(h/m", ")", "⊗", "KASCADE", "(e/m)", "QGSJET", "⊕", "KASCADE", "(e/m)", "SIBYLL", "✧", "CASA-MI", "A", "Chacaltaya", "✣", "EAS-TOP", "+", "MACRO", "✕", "EAS-TOP", "(e/m", ")", "∅", "HEGRA", "(CRT)", "∇", "SPASE/AMANDA", "∇", "∇", "∇", "∇", "∇", "∇", "✕", "✕", "✕", "✕", "✕", "✕", "✣", "✣", "✣", "✣", "✣", "✣", "✣", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "❄", "❄", "❄", "❄", "❄", "❄", "❄", "❄", "∅", "∅", "∅", "∅", "∅", "∅", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "F", "e", "M", "g", "N", "B", "e", "H", "e", "H", ">", "ln", "A", "s", "<", "m", "as", "hm", "ic", "ga", "rit", "n", "lo", "M", "ea", "Energy", "E", "0", "[GeV]", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "4", "3.", "5", "3", "2.", "5", "2", "1.", "5", "1", "0.", "5", "0" ], "name": "2", "regionBoundary": { "x1": 115, "x2": 676, "y1": 85, "y2": 332 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure2-1.png", "source": "fig" } ]
	action of nuclei of mass \(A_{i}\). In the superposition model of air showers, the shower development of nuclei with mass \(A\) and energy \(E_{0}\) is described by the sum of \(A\) proton showers of energy \(E=E_{0}/A\). The shower maximum \(t\) penetrates into the atmosphere as \(t\propto\ln E\), hence, most air shower observables at ground level scale proportional to \(\ln A\). Frequently, the ratio of the number of electrons and muons is used to determine the mass composition. The experiments CASA-MIA [61], EAS-TOP [63], or KASCADE use muons with an energy of several 100 MeV to 1 GeV. To study systematic effects two hadronic interaction models are used to interpret the data measured with KASCADE [50]. High energy muons detected deep below rock or antarctic ice are utilized by the EAS-TOP/MACRO [64] and SPASE/AMANDA [67] experiments. Also the correlation between the hadronic and muonic shower components has been investigated by KASCADE [66]. The production height of muons has been reconstructed by HEGRA/CRT [65] and KASCADE [70]. Results from various experiments measuring electrons, muons, and hadrons at ground level are compiled in Fig. 2. At low energies the values for the mean logarithmic mass are compared to results from direct measurements. A clear increase as function of energy can be recognized. The experimental values follow a trend predicted by the poly-gonato model as indicated by the lines in the figure. However, individual experiments exhibit systematic deviations of about \(\pm 1\) unit in \(\langle\ln A\rangle\) from the line. Of particular interest are also the investigations of the KASCADE experiment, interpreting the same measured data with two different models for the interactions in the atmosphere results in a systematic difference of about 0.7 to 1 in \(\langle\ln A\rangle\). Another technique to determine the mass of cosmic rays are measurements of the average depth of the shower maximum using non-imaging and imaging Cerenkov detectors as well as fluorescence telescopes at the highest energies. The results of several experiments are presented in Fig. 3 as function of energy. The observed values are compared to predictions of air shower simulations for primary protons and iron nuclei using the program CORSIKA [78] with the hadronic interaction model QGSJET 01 [79] and a modified version w Figure 3: Average depth of the shower maximum \(X_{max}\) as function of primary energy as obtained by BLANCA [39], CACTI [71], DICE [41], Fly’s Eye [43], Haverah Park [72], HEGRA [46], HiRes/MIA [73], Mt. Lian Wang [74], SPASE/VULCAN [75], Tunka-25 [58], and Yakutsk [76]. The lines indicate simulations for proton and iron induced showers using the CORSIKA code with the hadronic interaction model QGSJET 01 (solid line) and a version with lower cross sections and slightly increased elasticities (dashed line, model 3 in [77]).	[ { "caption": "Figure 3. Average depth of the shower maximum Xmax as function of primary energy as obtained by BLANCA [39], CACTI [71], DICE [41], Fly’s Eye [43], Haverah Park [72], HEGRA [46], HiRes/MIA [73], Mt. Lian Wang [74], SPASE/VULCAN [75], Tunka-25 [58], and Yakutsk [76]. The lines indicate simulations for proton and iron induced showers using the CORSIKA code with the hadronic interaction model QGSJET 01 (solid line) and a version with lower cross sections and slightly increased elasticities (dashed line, model 3 in [77]).", "captionBoundary": { "x1": 95, "x2": 700, "y1": 379, "y2": 472 }, "figType": "Figure", "imageText": [ "p", "Yakutsk", "Fe", "⊗", "SPASE/VULCAN", "∇", "Tunka-25", "Mt.", "Lian", "Wang", "⊕", "HiRes/MIA", "HEGRA", "(Airobicc)", "Haverah", "Park", "✡", "Fly's", "Eye", "DICE", "✲", "CACTI", "BLANCA", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✲", "✲", "✲", "✲", "✲", "2", "]", "/c", "m", "ax", "[g", "X", "m", "Energy", "E0", "[GeV]", "10", "10", "10", "9", "10", "8", "10", "7", "10", "6", "10", "5", "800", "750", "700", "650", "600", "550", "500", "450", "400" ], "name": "3", "regionBoundary": { "x1": 114, "x2": 663, "y1": 92, "y2": 362 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure3-1.png", "source": "fig" } ]
	ith lower cross sections and larger values for the elasticity of the hadronic interactions (model 3a in [77]). The latter is compatible with measurements at colliders, for details see [77]. The lower values for the total inelastic proton-antiproton cross sections yield also lower inelastic proton-air cross sections, which are in good agreement with recent measurements from the HiRes experiment [80, 81]. In principle, the difference between the solid and the dashed lines in the figure represents an estimate of the projection of the experimental errors from collider experiments on the average depth of the shower maximum in air showers. At \(10^{9}\) GeV the difference between the two model versions for primary protons is about half the difference between proton and iron induced showers. This illustrates the significance of the uncertainties of the collider measurements for air shower observables. Knowing the average depth of the shower maximum for protons \(X_{max}^{p}\) and iron nuclei \(X_{max}^{Fe}\) from simulations, the mean logarithmic mass is derived in the superposition model of air showers from the measured \(X_{max}^{meas}\) using \(\langle\ln A\rangle=(X_{max}^{meas}-X_{max}^{p})/(X_{max}^{Fe}-X_{max}^{p}) \cdot\ln A_{Fe}\). The corresponding \(\langle\ln A\rangle\) values, obtained from the results shown in Fig. 3, are plotted in Fig. 4 versus the primary energy using both interaction models to interpret the observed data. Up to Figure 4: Mean logarithmic mass of cosmic rays derived from the average depth of the shower maximum, see Fig. 3. Two hadronic interaction models are used to interpret the measurements: QGSJET 01 (top) and a modified version with lower cross sections and a slightly increased elasticity (model 3a [77], bottom). For references, see caption of Fig. 3. For comparison, results from direct measurements are shown as well from the JACEE [68] and RUNJOB [32] experiments. The lines indicate expectations according to the poly-gonato model.	[ { "caption": "Figure 4. Mean logarithmic mass of cosmic rays derived from the average depth of the shower maximum, see Fig. 3. Two hadronic interaction models are used to interpret the measurements: QGSJET 01 (top) and a modified version with lower cross sections and a slightly increased elasticity (model 3a [77], bottom). For references, see caption of Fig. 3. For comparison, results from direct measurements are shown as well from the JACEE [68] and RUNJOB [32] experiments. The lines indicate expectations according to the poly-gonato model.", "captionBoundary": { "x1": 95, "x2": 699, "y1": 559, "y2": 652 }, "figType": "Figure", "imageText": [ "JACEE", "direct", ":", "RUNJOB", "Haverah", "Park", "⊕", "HiRes", "DICE", "✡", "Fly's", "Eye", "BLANCA", "✲", "CACTI", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "⊗", "⊗", "∇", "∇", "∇", "∇", "∇", "∇", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊗", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "⊕", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✲", "✲", "✲", "✲", "✲", "F", "e", "M", "g", "N", "B", "e", "H", "e", "H", ">", "ln", "A", "s", "<", "m", "as", "hm", "ic", "ga", "rit", "n", "lo", "M", "ea", "4", "3.", "5", "3", "2.", "5", "2", "1.", "5", "1", "0.", "5", "0", "Yakutsk", "⊗", "SPASE/VULCAN", "∇", "Tunka-25", "HEGRA", "(Airobicc)", "Mt.", "Lian", "Wang", "∇", "∇", "∇", "∇", "⊗", "⊗", "⊗", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "⊗", "⊗", "⊗", "⊗", "⊗", "⊕", "⊕", "⊗", "⊕", "⊕", "⊕", "✡", "⊕", "⊕", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✲", "✲", "✲", "✲", "✲", "F", "e", "M", "g", "N", "B", "e", "H", "e", "H", ">", "ln", "A", "s", "<", "m", "as", "hm", "ic", "ga", "rit", "n", "lo", "M", "ea", "Energy", "E", "0", "[GeV]", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "4", "3.", "5", "3", "2.", "5", "2", "1.", "5", "1", "0.", "5", "0" ], "name": "4", "regionBoundary": { "x1": 115, "x2": 676, "y1": 85, "y2": 545 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure4-1.png", "source": "fig" } ]
	about a PeV there are only marginal differences between the two interpretations. On the other hand, at large energies a significantly heavier composition is obtained for the modified version (model 3a). At \(10^{9}\) GeV the differences amount to about 1 in \(\langle\ln A\rangle\). These examples illustrate how strong the interpretation of air shower measurements depends on model parameters such as the inelastic cross sections or elasticities used. At Tevatron energies the cross sections vary within the error range given by the experiments and at \(10^{8}\) GeV the proton air cross sections of QGSJET and model 3a differ only by about 10%, but the general trend of the emerging \(\langle\ln A\rangle\) distributions proves to be significantly different. This underlines the importance to test and improve the understanding of hadronic interactions in the atmosphere with air shower experiments [82]. It seems that the interpretation with model 3a is better compatible with the mean logarithmic masses derived from electrons, muons, and hadrons already shown in Fig. 2 and also with the predictions of the poly-gonato model as indicated by the lines. ## 4 Spectra for elemental groups A significant step forward in the understanding of the origin of cosmic rays are measurements of energy spectra for individual elements or at least groups of elements. Up to about a PeV direct measurements have been performed with instruments above the atmosphere. As examples, results for primary protons, helium, and iron nuclei are compiled in Figs. 5 to 7. Recently, also indirect measurements of elemental groups became possible. A special class of events, the unaccompanied hadrons were investigated by the EAS-TOP and KASCADE experiments [99, 97]. Simulations reveal that these events, where only one hadron is registered in a large calorimeter, are sensitive to the flux of primary protons. The derived proton fluxes agree with the results of direct measurements as can be inferred Figure 5: Energy spectrum for protons. Results from direct measurements above the atmosphere by AMS [83], ATIC [84], BESS [85], CAPRICE [86], HEAT [87], Ichimura _et al.[88]_, IMAX [89], JACEE [90], MASS [91], Papini _et al.[92]_, RUNJOB [32], RICH-II [93], Ryan _et al.[94]_, Smith _et al.[95]_, SOKOL [33], Zatsepin _et al.[96]_, and fluxes obtained from indirect measurements by KASCADE electrons and muons for two hadronic interaction models [50] and single hadrons [97], EAS-TOP (electrons and muons) [98] and single hadrons [99], HEGRA [100], Mt. Chacaltaya [101], Mts. Fuji and Kanbala [102], Tibet burst detector (HD) [103] and AS\(\gamma\) (HD) [104]. The line indicates the spectrum according to the poly-gonato model.	[ { "caption": "Figure 5. Energy spectrum for protons. Results from direct measurements above the atmosphere by AMS [83], ATIC [84], BESS [85], CAPRICE [86], HEAT [87], Ichimura et al [88], IMAX [89], JACEE [90], MASS [91], Papini et al [92], RUNJOB [32], RICH-II [93], Ryan et al [94], Smith et al [95], SOKOL [33], Zatsepin et al [96], and fluxes obtained from indirect measurements by KASCADE electrons and muons for two hadronic interaction models [50] and single hadrons [97], EAS-TOP (electrons and muons) [98] and single hadrons [99], HEGRA [100], Mt. Chacaltaya [101], Mts. Fuji and Kanbala [102], Tibet burst detector (HD) [103] and ASγ (HD) [104]. The line indicates the spectrum according to the poly-gonato model.", "captionBoundary": { "x1": 95, "x2": 699, "y1": 379, "y2": 507 }, "figType": "Figure", "imageText": [ "∅", "Zatsepin", "Ryan", "Smith", "SOKOL", "RUNJOB", "⊕", "RICH-II", "Papini", "JACEE", "✢", "MASS", "Ichimura", "∇", "IMAX", "✴", "✴⊗", "AMS", "✴", "ATIC", "✡", "BESS", "✧", "CAPRICE", "98", "¤", "HEAT", "✴", "✴", "✴", "✴", "✴", "✧", "✧", "✧", "✧", "✢✢✢✢✢✢✢✢✢✢✢", "✢", "✢✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "∅", "∅", "∅", "∅", "∅", "∅", "∅", "∅", "⊕", "⊕", "⊗", "⊗", "⊗", "⊗⊗", "⊗", "⊗⊗", "⊗⊗", "⊗", "✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡", "EAS-TOP", "¤", "¤", "¤", "¤", "¤", "¤", "¤", "⊕⊕⊕", "⊕", "Mt.", "Fuji", "◆", "HEGRA", "✥", "Mt.", "Chacaltaya", "EAS-TOP", "SH", "✦", "Tibet-ASγ", "(HD)", "KASCADE", "SH", "Tibet-BD", "(HD)", "KASCADE", "SIBYLL", "Proton", "KASCADE", "QGSJET", "✥", "✥", "✥", "✥", "✥", "✥", "✦", "✦", "✦", "✦", "✦", "✦", "✦", "◆", "◆◆", "◆", "◆", "◆", "◆", "◆", "◆", "◆", "◆", "5", "]", "eV", "1.", "-1", "G", "r-1", "s", "-2", "s", "5", "[m", "0", "2.", "0", "⋅", "E", "/d", "E", "x", "dΦ", "F", "lu", "Energy", "E0", "[GeV]", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "10", "3", "10", "2", "10", "4", "10", "3", "10", "2" ], "name": "5", "regionBoundary": { "x1": 118, "x2": 676, "y1": 91, "y2": 362 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure5-1.png", "source": "fig" } ]
	from Fig. 5, indicating a reasonably good understanding of the hadronic interactions in the atmosphere for energies below 1 PeV. At higher energies a breakthrough has been achieved by the KASCADE experiment. Measuring simultaneously the electromagnetic and muonic component of air showers and unfolding the two dimensional shower size distributions, the energy spectra of five elemental groups have been derived [50]. In order to estimate the influence of the hadronic interaction models used in the simulations, two models, namely QGSJET 01 and SIBYLL [106], have been applied to interpret the measured data. It turns out that the all-particle spectra obtained agree satisfactory well within the statistical errors. For both interpretations the flux of light elements exhibits individual knees. The absolute flux values differ by about a factor of two or three between the different interpretations. However, it is evident that the knee in the all-particle spectrum is caused by a depression of the flux of light elements. The KASCADE results are illustrated in Figs. 5 to 7. In the figures also results from other air shower experiments are shown. EAS-TOP derived spectra from the simultaneous observation of the electromagnetic and muonic components. HEGRA used an imaging Cerenkov telescope system to derive the primary proton flux [100]. Spectra for protons and helium nuclei are obtained from emulsion chambers exposed at Mts. Fuji and Kanbala [102]. The Tibet group performs measurements with a burst detector as well as with emulsion chambers and an air shower array [103, 57]. Over the wide energy range depicted, the measurements seem to follow power laws with a cut-off at high energies. The spectra according to the poly-gonato model are indicated in the figures as lines. It can be recognized that the measured values are compatible with cut-offs at energies proportional to the nuclear charge \(\hat{E}_{Z}=Z\cdot 4.5\) PeV. Figure 6: Energy spectrum for helium nuclei. Results from direct measurements above the atmosphere by ATIC [84], BESS [85], CAPRICE [86], HEAT [87], Ichimura _et al.[88]_, IMAX [89], JACEE [90], MASS [91], Papini _et al.[92]_, RICH-II [93], RUNJOB [32], Smith _et al.[95]_, SOKOL [33], Webber [105], and fluxes obtained from indirect measurements by KASCADE electrons and muons for two hadronic interaction models [50], Mts. Fuji and Kanbala [102], and Tibet burst detector (HD) [103]. The line indicates the spectrum according to the poly-gonato model.	[ { "caption": "Figure 6. Energy spectrum for helium nuclei. Results from direct measurements above the atmosphere by ATIC [84], BESS [85], CAPRICE [86], HEAT [87], Ichimura et al [88], IMAX [89], JACEE [90], MASS [91], Papini et al [92], RICH-II [93], RUNJOB [32], Smith et al [95], SOKOL [33], Webber [105], and fluxes obtained from indirect measurements by KASCADE electrons and muons for two hadronic interaction models [50], Mts. Fuji and Kanbala [102], and Tibet burst detector (HD) [103]. The line indicates the spectrum according to the poly-gonato model.", "captionBoundary": { "x1": 95, "x2": 699, "y1": 379, "y2": 490 }, "figType": "Figure", "imageText": [ "⊕", "RICH-II", "RUNJOB", "Smith", "SOKOL", "❄", "Webber", "Papini", "JACEE", "✢", "MASS", "Ichimura", "∇", "IMAX", "✴", "ATIC", "✡", "BESS", "✧", "CAPRICE", "98", "¤", "HEAT", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "✴", "❄", "❄", "❄", "❄❄❄❄❄", "❄❄❄❄", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "∇", "∇", "✧✧", "✧✧✧✧", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "∇", "✢✢✢", "✢✢✢✢✢", "✢✢✢✢✢✢", "✢∇", "∇", "∇", "∇", "✡✡✡", "✡✡✡✡✡✡✡✡✡✡✡✡", "⊕", "✡", "✡✡✡", "¤", "¤", "¤", "¤", "¤", "¤", "¤⊕", "⊕⊕⊕", "⊕", "eV", "1.", "5", "]", "Helium", "KASCADE", "QGSJET", "KASCADE", "SIBYLL", "Tibet-BD", "(HD)Mt.", "Fuji", "-1", "G", "r-1", "s", "-2", "s", "5", "[m", "0", "2.", "0", "⋅", "E", "/d", "E", "x", "dΦ", "F", "lu", "Energy", "E0", "[GeV]", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "10", "3", "10", "2", "10", "4", "10", "3", "10", "2" ], "name": "6", "regionBoundary": { "x1": 118, "x2": 676, "y1": 91, "y2": 362 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure6-1.png", "source": "fig" } ]
	## 5 Conclusion and outlook In the last decade substantial progress has been achieved and the knowledge about galactic cosmic rays has been significantly increased. The all-particle energy spectrum is reasonably well known. For the first time, spectra for groups of elements could be derived from air shower measurements. The observed spectra seem to exhibit cut-offs proportional to the nuclear charge. The increase of the mean logarithmic mass derived from air shower observations seems to be compatible with subsequential cut-offs for individual elements. However, the interpretation of the measurements is still limited by the uncertainties of the description of hadronic interactions in the atmosphere. The implications of recent measurements on the contemporary understanding of the origin of the knee have been discussed elsewhere [113]. With the recent measurements of TeV \(\gamma\)-rays from supernova remnants [25] a new window has been opened, and the hypothesis of cosmic-ray acceleration in supernova remnants is tested directly. With the KASCADE-Grande experiment [114], investigations of the energy spectra for groups of elements will be extended into the region of the expected iron knee (\(\sim 100\) PeV) and up to the second knee (\(\sim 400\) PeV). In particular, the inclusion of the hadronic component will help to improve the interaction models at these energies. A new technique to measure air showers is about to be established: The LOPES experiment has registered first air showers observing geosynchrotron emission in the radio frequency range from 40 to 80 MHz [115]. The presently largest balloon borne detector, the TRACER experiment [111], almost reaches the energy region of indirect measurements. Successor experiments will provide twofold progress for the understanding of the origin of galactic cosmic rays: an extension of the measurements of the ratio of secondary to primary nuclei to energies approaching the knee will improve the knowledge about the propagation of cosmic rays and the extension of energy spectra with individual element resolution towards the air shower regime will provide useful information Figure 7: Energy spectrum for iron nuclei. Results from direct measurements above the atmosphere by CRN [107], HEAO-3 [108], Juliusson _et al.[109]_, Minagawa _et al.[110]_, TRACER [111] (single element resolution) and Hareyama _et al.[112]_, Ichimura _et al.[88]_, JACEE [31], RUNJOB [32], SOKOL [33] (iron group), as well as fluxes from indirect measurements (iron group) by KASCADE electrons and muons for two hadronic interaction models [50] and EAS-TOP [98]. The line indicates the spectrum according to the poly-gonato model.	[ { "caption": "Figure 7. Energy spectrum for iron nuclei. Results from direct measurements above the atmosphere by CRN [107], HEAO-3 [108], Juliusson et al [109], Minagawa et al [110], TRACER [111] (single element resolution) and Hareyama et al [112], Ichimura et al [88], JACEE [31], RUNJOB [32], SOKOL [33] (iron group), as well as fluxes from indirect measurements (iron group) by KASCADE electrons and muons for two hadronic interaction models [50] and EASTOP [98]. The line indicates the spectrum according to the poly-gonato model.", "captionBoundary": { "x1": 95, "x2": 699, "y1": 369, "y2": 462 }, "figType": "Figure", "imageText": [ "❄", "Minagawa", "RUNJOB", "SOKOL", "✧", "TRACER", "✡", "Hareyama", "Ichimur", "a", "JACEE", "∇", "Juliusson", "CRN", "HEAO-", "3", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "✧", "❄", "❄", "❄", "❄", "❄", "❄", "❄", "❄", "∇", "∇", "∇", "∇", "∇", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "✡", "∇", "KASCADE", "SIBYLL", "EAS-TOP", "Iro", "n", "KASCADE", "QGSJET", "]", "eV", "1.", "5", "-1", "G", "r", "-1", "s", "-2", "s", "2.", "5", "[", "m", "⋅", "E", "0", "E", "0", "Φ", "/d", "x", "d", "F", "lu", "Energy", "E", "0", "[", "GeV", "]", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "10", "3", "10", "2", "10", "4", "10", "3", "10", "2" ], "name": "7", "regionBoundary": { "x1": 114, "x2": 676, "y1": 81, "y2": 354 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508014-Figure7-1.png", "source": "fig" } ]
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	with mass number \(A\), and \(M\) the solar modulation parameter. A value \(M=750\) MeV is used for the parametrization. Above \(Z\cdot 10\) GeV, the modulation due to the magnetic field of the heliosphere is negligible and the energy spectra of cosmic-ray nuclei are assumed to be described by power laws. However, extrapolating the elemental spectra with power laws, the all-particle spectrum obtained will overshoot the measured all-particle spectrum above several PeV. Hence, a cut-off in the spectra for the individual elements is introduced. Such a cut-off is motivated by various theories for the origin of the knee[3]. Inspired by such theories the following ansatz is adopted to describe the energy dependence of the flux for particles with charge \(Z\) \[\frac{d\Phi_{Z}}{dE_{0}}(E_{0})=\Phi_{Z}^{0}E_{0}^{\gamma_{Z}}\left[1+\left( \frac{E_{0}}{\hat{E}_{Z}}\right)^{\epsilon_{c}}\right]^{\frac{ -\Delta\gamma}{\epsilon_{c}}}.\] (2) The absolute flux \(\Phi_{Z}^{0}\) and the spectral index \(\gamma_{Z}\) quantify the power law. The flux above the cut-off energy is modeled by a second and steeper power law. \(\Delta\gamma\) and \(\epsilon_{c}\) characterize the change in the spectrum at the cut-off energy \(\hat{E}_{Z}\). Both parameters are assumed to be identical for all elements, \(\Delta\gamma\) being the difference in the spectral indices below and above the respective knees and \(\epsilon_{c}\) describes the smoothness of the transition from the first to the second power law. To study systematic effects also a common spectral index for all elements above their respective knee has been tried, see [2]. Different relations for the cut-off energy \(\hat{E}_{Z}\) have been checked. The investigations showed that a rigidity dependent cut-off with \(\hat{E}_{Z}=\hat{E}_{p}\cdot Z\), with the proton cut-off energy \(\hat{E}_{p}\), fits the data best [2]. The all-particle spectrum is obtained by summation of the flux \(d\Phi_{Z}/dE_{0}(E_{0})\) for all cosmic-ray elements. The absolute flux and the spectral indices for the elements from protons to nickel are derived from direct measurements above the atmosphere. For the heavier elements only abundances are known at energies around 1 GeV/n. In Fig. 2 the relative abundance normalized to Fe \(\equiv 1\) is presented as function of the nuclear charge number. One recognizes that all elements up to the end of the periodic table are present. At energies of 1 GeV/n the measured abundances are	[ { "caption": "Figure 3. Cosmic ray energy spectra according to the poly-gonatomodel. The data points represent the normalized all-particle flux obtained by several experiments [2].", "captionBoundary": { "x1": 95, "x2": 382, "y1": 359, "y2": 417 }, "figType": "Figure", "imageText": [ "eV", "1.", "5", "]", "1", "2", "3-9", "10-24", "25-27", "28-92", "-1", "G", "r-1", "s", "-2", "s", "5", "[m", "0", "2.", "0", "⋅", "E", "/d", "E", "x", "dΦ", "F", "lu", "Energy", "E0", "[GeV]", "10", "9", "10", "8", "10", "7", "10", "6", "10", "5", "10", "4", "10", "3", "10", "2" ], "name": "3", "regionBoundary": { "x1": 100, "x2": 389, "y1": 82, "y2": 320 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508015-Figure3-1.png", "source": "fig" } ]
	heavily suppressed by the heliospheric modulation of cosmic rays -- this is taken into account in the poly-gonato model using eq. (1). In addition, the measured abundances are extrapolated to high energies using an empirical relation for the spectral indices \(-\gamma_{Z}=A+B\cdot Z^{C}\) with the values \(A=2.70\pm 0.19\), \(B=(-8.34\pm 4.67)\cdot 10^{-4}\), and \(C=1.51\pm 0.13\). The average all-particle flux obtained by several air shower experiments is given in Fig. 4 by the data points. A fit to these points yields the parameters for the poly-gonato model \(\hat{E}_{p}=4.49\pm 0.51\) PeV, \(\Delta\gamma=2.10\pm 0.24\), and \(\epsilon_{c}=1.90\pm 0.19\). The expected spectra for elemental groups corresponding to these parameters are shown in Fig. 4. The all-particle spectrum obtained by summation of all elements fits the experimental values well. It can be recognized that within this approach the knee is a consequence of the subsequential cut-offs for the individual elemental spectra, starting with protons at 4.5 PeV. The shape of the all-particle spectrum above this energy is determined by the overlay of the individual cut-offs for all elements. The second knee is related to the cut-offs for the heaviest elements at the end of the periodic table or the end of the galactic component. The relative contributions of elemental groups to the all-particle spectrum are presented in Fig. 4. The most dominant group is the iron group (\(Z=25-27\)), at energies around 70 PeV more than 50% of the all-particle flux consists of these elements. Ultra heavy elements (\(Z\geq 28\)) are expected to contribute at 300 PeV with slightly less than 40% to the all-particle flux. Around 300 PeV measurements of the average depth of the shower maximum (\(X_{max}\)) are available from the Fly's Eye and HiRes experiments, see Fig. 5. These data are used to check the predictions of the poly-gonato model. With the air shower simulation program CORSIKA [10] and a modified version of the hadronic interaction model QGSJET with lower cross sections [9] the expected \(X_{max}\) distribution has been calculated. The composition of galactic cosmic rays has been assumed according to the poly-gonato model. The resulting distribution is given in Fig. 5 as solid histogram. It should be emphasized that this is not a fit to the measured \(X_{max}\) values, instead, the fluxes are taken as predicted by the model. As can be inferred from the figure, the left-hand side of the \(X_{max}\) distribution is represented quite reasonably. An ad-hoc component has been introduced to describe also the right-hand part of the distribution, for details see [9]. It should be pointed out that the three parameters of the model, namely \(\hat{E}_{p}\), \(\epsilon_{c}\), and \(\Delta\gamma\) in eq. (2) have been determined by a fit to the all particle spectrum derived from indirect measurements. That means the flux values for individual elements in the region of their cut-off are real predictions of the model. A comparison of the predicted fluxes with the energy spectra for groups of elements as recently derived from air shower measurements shows a quite Figure 5: Distribution of the depth of the shower maximum \(X_{max}\) measured by the Fly’s Eye [7] and HiRes [8] experiments. The measured values are compared with simulated results using QGSJET with lower cross sections [9]. Galactic component according to the poly-gonato model (solid line) and galactic plus ad-hoc component (dashed line), see also [9].	[ { "caption": "Figure 5. Distribution of the depth of the shower maximumXmax measured by the Fly’s Eye [7] and HiRes [8] experiments. The measured values are compared with simulated results using QGSJET with lower cross sections [9]. Galactic component according to the poly-gonato model (solid line) and galactic plus ad-hoc component (dashed line), see also [9].", "captionBoundary": { "x1": 413, "x2": 700, "y1": 173, "y2": 317 }, "figType": "Figure", "imageText": [ "3-5", "108", "GeV", "galactic", "gal.+ad", "hoc", "HiRes", "ax", "Fly's", "Eye", "X", "m", "dN", "/d", "Xmax", "[g/cm", "2]", "400", "500", "600", "700", "800", "900", "100011001200", "0.01", "0.008", "0.006", "0.004", "0", "0.002" ], "name": "5", "regionBoundary": { "x1": 96, "x2": 381, "y1": 90, "y2": 313 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508015-Figure5-1.png", "source": "fig" } ]
	satisfactory compatibility between the poly-gonato model and the measurements [3]. Two questions may rise when inspecting Fig. 4: Why are the spectra of heavy elements flatter than the spectra for light elements? And: can there be a sufficient contribution of ultra heavy elements at energies around \(10^{8}\) GeV? To obtain a quantitative estimate on this items in the following section the propagation of cosmic rays in the Galaxy is discussed. ## 3 Propagation of Cosmic Rays in the Galaxy To study cosmic-ray propagation in the Galaxy, a detailed knowledge of the structure of the magnetic fields is essential. Unfortunately, the question about the configuration of the galactic magnetic field remains open -- different models exist based on experimental data [11, 12, 13, 14]. How cosmic rays are accelerated to extremely high energies is another unanswered question. Although the popular model of cosmic-ray acceleration by shock waves in the expanding shells of supernovae (see e.g. [15, 16, 17]) is almost recognized as "standard theory," there are still a number of unresolved problems. Furthermore, the question about other acceleration mechanisms is not quite clear, and could lead to different cosmic-ray energy spectra at the sources [11]. Different concepts are verified by the calculation of the primary cosmic-ray energy spectrum, making assumptions on the density of cosmic-ray sources, the energy spectrum at the sources, and the configuration of the galactic magnetic fields. The diffusion model can be used in the energy range \(E<10^{17}\) eV, where the energy spectrum is calculated using the diffusion equation for the density of cosmic rays in the Galaxy. At higher energies this model ceases to be valid, and it becomes necessary to carry out numerical calculations of particle trajectories for the propagation in the magnetic fields of the Galaxy. This method works best for the highest energy particles, since the time for the calculations required is inversely proportional to the particle energy. Therefore, the calculation of the cosmic-ray spectrum in the energy range \(10^{14}-10^{19}\) eV has been performed within the framework of a combined approach, the use of a diffusion model and the numerical integration of particle trajectories. ### Assumptions High isotropy and a comparatively long retention of cosmic rays in the Galaxy (\(\sim 10^{7}\) years for the disk model) reveal the diffusion nature of particle motion in the interstellar magnetic fields. This process is described by a corresponding diffusion tensor [11, 13, 18]. The steady-state diffusion equation for the cosmic-ray density \(N(r)\) is (neglecting nuclear interactions and energy losses) \[-\nabla_{i}D_{ij}(r)\nabla_{j}N(r)=Q(r).\] (3) \(Q(r)\) is the cosmic-ray source term, \(D_{ij}(r)\) is the diffusion tensor. Under the assumption of azimuthal symmetry and taking into account the predominance of the toroidal component of the magnetic field, eq. (3) is presented in cylindrical coordinates as \[\left[-\frac{1}{r}\frac{\partial}{\partial r}rD_{\perp}\frac{\partial}{ \partial r}-\frac{\partial}{\partial z}D_{\perp}\frac{\partial}{\partial z}- \frac{\partial}{\partial z}D_{A}\frac{\partial}{\partial r}+\frac{1}{r}\frac{ \partial}{\partial r}rD_{A}\frac{\partial}{\partial z}\right]N(r,z)=Q(r,z),\] (4) where \(N(r,z)\) is the cosmic-ray density averaged over the large-scale fluctuations with a characteristic scale \(L\sim 100\) pc [13]. \(D_{\perp}\propto E^{m}\) is the diffusion coefficient, where \(m\) is much less than one (\(m\approx 0.2\)), and \(D_{A}\propto E\) the Hall diffusion coefficient. The influence of Hall diffusion becomes predominant at sufficiently high energies (\(>10^{15}\) eV). The sharp enhancement of the diffusion coefficient leads to the excessive cosmic-ray leakage from the Galaxy at energies \(E>10^{17}\) eV. For investigating the cosmic-ray propagation at such energies it is necessary to carry out calculations of the trajectories for individual particles. The numerical calculation of trajectories is based on the solution of the equation of motion for a charged particle in the magnetic field of the Galaxy. The calculation was carried out using	[]
	a fourth order Runge-Kutta method. Trajectories of cosmic rays were calculated until they left the Galaxy. Testing the differential scheme used, it was found that the accuracy of the obtained trajectories for protons with an energy \(E=10^{15}\) eV after passing a distance of 1 pc amounts to \(5\cdot 10^{-8}\) pc. The retention time of a proton with such an energy averages to about 10 million years, hence, a total error for the trajectory approximation by the differential scheme is about 0.5 pc. The magnetic field of the Galaxy consists of a large-scale regular and a chaotic component \(\vec{B}=\vec{B}_{reg}+\vec{B}_{irr}\). A purely azimuthal magnetic field was assumed for the regular field \[B_{z}=0,\quad B_{r}=0,\quad B_{\phi}=1~{}\mu\mbox{G}\cdot\exp\left(-\frac{z^{2 }}{z_{0}^{2}}-\frac{r^{2}}{r_{0}^{2}}\right),\] (5) where \(z_{0}=5\) kpc and \(r_{0}=10\) kpc are constants [13]. The irregular field was constructed according to an algorithm used in [19], that takes into account the correlation of the magnetic field intensities in adjacent cells. The radius of the Galaxy is assumed to be 15 kpc and the galactic disk has a half-thickness of 200 pc. The position of the Solar system was defined at \(r=8.5\) kpc, \(\phi=0^{\circ}\), and \(z=0\) kpc. A radial distribution of supernovae remnants along the galactic disk was considered as sources [20]. ### Results The results for the calculations of the cosmic-ray proton spectrum are presented in Fig. 7. These results were obtained using the diffusion model and numerical calculations of trajectories. It is evident from the graph that both methods give identical results up to about \(3\cdot 10^{7}\) GeV. At higher energies there is a continuous decrease of the intensity in the diffusion spectrum, which corresponds to the excessive increase in the diffusion coefficient that leads to a large leakage of particles from the Galaxy. An energy of \(10^{17}\) eV can be defined as the conventional boundary to apply the diffusion model. At this energy the results obtained with the two methods differ by a factor of two and for higher energies the diffusion approximation becomes invalid. The predicted spectra are compared to direct and indirect measurements of the primary proton flux in Fig. 7. In the depicted range there is almost no difference between the two	[]
	approaches. The relatively steep decrease of the flux at energies exceeding 4 PeV is not reflected. On the other hand, the data are described reasonably well by the poly-gonato model, as also shown in the figure. The observed change in the spectral index \(\Delta\gamma\approx 2.1\) according to the poly-gonato model has to be compared to the value predicted by the diffusion model. In the latter the change should be \(1-m\approx 0.8\)[13]. The observed value is obviously larger, which implies that the remaining change of the spectral shape should be caused by a change of the spectrum at the source, e.g. due to the maximum energy attained in the acceleration process. The maximum energy and, therefore, the energy at which the spectrum steepens depends on the intensity of the magnetic fields in the acceleration zone and on a number of assumptions for the feedback of cosmic rays to the shock front. The uncertainty of the parameters yields variations in the maximum energy predicted by different models up to a factor of 100 [3, 16]. Thus, there is no consensus about what the "standard model" is considered to predict. For the time being, it is difficult to make definite conclusions from the comparison between the experimental spectra for different elemental groups and the "standard model" of cosmic ray acceleration at ultra high energies. The obtained path length (\(\lambda_{dif}\)) in the Galaxy for protons as function of energy is presented in Fig. 9. The interstellar matter density was taken as \(n_{d}=1\) cm\({}^{-3}\) for the galactic disk and \(n_{h}=0.01\) cm\({}^{-3}\) for the halo. For heavier nuclei with charge \(Z\) the path length scales with the rigidity, i.e. is related to the values for protons \(\lambda(E)\) as \(\lambda(E,Z)=\lambda(E/Z)\). At the corresponding knees, the amount of traversed material is less than 1 g/cm\({}^{2}\). The dashed dotted line indicates a trend at lower energies according to \(\lambda\propto E^{-\delta}\). To reach values around 10 g/cm\({}^{2}\) as obtained around 1 GeV, see below, one needs a relatively small slope \(\delta\approx 0.2\) -- much lower than the value usually assumed \(\delta\approx 0.6\).	[ { "caption": "Figure 9. Rayleigh amplitudes as function of energy for various experiments, for references see [22]. Additionally, model predictions for leaky-box models [23] and a diffusion model [24] are shown. The lines indicate the expected anisotropy for primary protons, iron nuclei, and all particles.", "captionBoundary": { "x1": 413, "x2": 700, "y1": 359, "y2": 469 }, "figType": "Figure", "imageText": [ "Mt.", "Norikura", "1990", "EAS-TOP", "1996", "EAS-TOP", "2003", "Akeno", "1986", "Adelaide", "1983", "Energy", "log10", "E", "[GeV]", "5", "5.5", "6", "6.5", "7", "7.5", "l.b.", "with", "reacc.", "eaky", "box", "stand", "ard", "l", "Diffusio", "n", "(Fe)", "(H)", "e)", "Diffu", "sion", "(all", "p", "articl", "Diffu", "sion", "-1", "10", "KASCADE", "95%", "confidence", "level", "-2", "10", "-3", "10", "-4", "10", "de", "pl", "itu", "A", "m" ], "name": "9", "regionBoundary": { "x1": 405, "x2": 699, "y1": 141, "y2": 323 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508015-Figure9-1.png", "source": "fig" } ]
	Measurements of the ratio of secondary to primary cosmic-ray nuclei at GeV energies are successfully described using a leaky-box model. For example, assuming the escape path length as \[\lambda_{lb}=\frac{26.7\beta~{}\mbox{g/cm}^{2}}{(\beta R/1.0~{}\mbox{GV})^{0.5 8}+(\beta R/1.4~{}\mbox{GV})^{-1.4}}\] (6) various secondary to primary ratios obtained by the ACE/CRIS and HEAO-3 experiments can be described consistently in the energy range from \(\sim 70\) MeV to \(\sim 30\) GeV [25]. A similar approach is the residual path length model [26], assuming the relation \[\lambda_{rp}=\left[6.0\cdot\left(\frac{R}{10~{}\mbox{GV}}\right)^{-0.6}+0.013 \right]~{}\mbox{g/cm}^{2}\] (7) for the escape path length. Both examples are compared to the predictions of the diffusion model in Fig. 9. Extrapolating these relations to higher energies, the strong dependence of the path length on energy (\(\propto E^{-0.6}\)) leads to extremely small values at PeV energies. Above \(10^{5}\) GeV the traversed matter would be less than the matter accumulated along a straight line from the galactic center to the solar system \(\lambda_{gc}=8~{}\mbox{kpc}\cdot 1~{}\mbox{proton/cm}^{3}\approx 0.04~{}\mbox{ g/cm}^{2}\). This value is indicated in Fig. 9 as dotted line. A similar conclusion can be derived from anisotropy measurements. Rayleigh amplitudes observed by different experiments are compiled in Fig. 9[22]. Leaky-box models, with their extremely steep decrease of the path length \(\lambda\propto E^{-0.6}\), yield relative large anisotropies even at modest energies, which seem to be ruled out by the measurements. Two versions of a leaky-box model [23], with and without reacceleration are represented in the figure. On the other hand, a diffusion model [24], which is based on the same basic idea [13] as the present work, seems to be compatible with the measured data. For this model the expected Rayleigh amplitudes are given for primary protons and iron nuclei, as well as for a mixture of all elements. Using nuclear cross sections according to the hadronic interaction model QGSJET [27] and assuming a number density \(n_{p}=1\) proton/cm\({}^{3}\) in the galactic disk, the interaction length of	[]
	nuclei has been calculated. The results for four different elements are presented in Fig. 11. The values decrease slightly as function of energy. Values for protons are in the range \(20-55\) g/cm\({}^{2}\), the values decrease as function of nuclear charge and reach values \(<1\) g/cm\({}^{2}\) for the heaviest elements. The interaction probability for different nuclei has been calculated based on the obtained path length and interaction parameters according to the QGSJET model. Nuclear fragmentation is taken into account in an approximate approach [28]. It should be pointed out that a nuclear fragment conserves the trajectory direction of its parent if \(Z/A\) in question is the same as for the primary nucleus and for most stable nuclei the ratio \(Z/A\) is close to \(1/2\). The resulting fraction of nuclei which survive without an interaction is presented in Fig. 11 for selected elements. It turns out that at the respective knees (\(Z\cdot 4.5\) PeV) more than about 50% of the nuclei survive without interactions, even for the heaviest elements. This is an important result, since the poly-gonato model relates the contribution of ultra-heavy cosmic rays to the second knee in the all-particle spectrum around 400 PeV. It should be noted that the fraction of surviving nuclei would be even larger for a leaky-box model, with its low path length at such energies. ## 4 Discussion The energy spectrum of cosmic rays at their source \(Q(E)\) is related to the observed values at Earth \(N(E)\) as \[N(E)=Q(E)\left(\frac{1}{\lambda_{esc}(E)}+\frac{1}{\lambda_{int}(E)}\right)^{-1}\] (8) with the propagation path length \(\lambda_{esc}\) and the interaction length \(\lambda_{int}\). The corresponding values are presented in Fig. 9 and Fig. 11, respectively. It is frequently assumed that the propagation path length decreases as function of energy \(\lambda_{lb}\propto E^{-0.6}\), as discussed above. Since the interaction length is almost independent of the primary energy this necessitates a spectrum at the sources \(Q(E)\propto E^{-2.1}\) to explain the observed spectrum at Earth \(N(E)\propto E^{-2.7}\). However, the model by Berezhko _et al.[16]_ predicts even flatter spectra at the sources before the knee and an even stronger dependence of \(\lambda_{esc}\) on energy is needed. Recent measurements of the TeV \(\gamma\) ray flux from a shell type supernova remnant yield a spectral index \(\gamma=-2.19\pm 0.09\pm 0.15\)[29] in agreement with the "standard model". However, e.g. for the Crab Nebula a steeper spectrum with \(\gamma=-2.57\pm 0.05\) has been obtained [30], indicating that not all sources exhibit the same behaviour. As has been discussed above, the dependence of the propagation path length \(\lambda_{esc}\propto E^{-0.6}\) can not be extrapolated to knee energies. Taking a value \(\lambda_{esc}\propto E^{-0.2}\) as discussed in relation with Fig. 9 necessitates additional assumptions on the spectral shape \(Q(E)\) at the source in order to explain the observed spectra with spectral indices in the range \(-\gamma\approx 2.55-2.75\)[2]. Direct measurements seem to indicate that the spectra of light elements are flatter as compared to heavy elements [2]. The values of \(\lambda_{int}\) for protons are at all energies larger than the escape path length \(\lambda_{dif}\). Hence, for protons the escape from the Galaxy is the dominant process influencing the shape of the observed energy spectrum. For iron nuclei at low energies hadronic interactions are dominating (\(\lambda_{int}\approx 2~{}\mbox{g/cm}^{2}<\lambda_{dif}\)) and leakage from the Galaxy becomes important at energies approaching the iron knee (\(26\times\hat{E}_{p}\)). However, for elements heavier than iron the interaction path length is smaller than the escape path length for all energies, except above the respective knees. At low energies the propagation path length exceeds the interaction path length by about an order of magnitude. For these elements nuclear interaction processes are dominant for the shape of the observed spectrum. This may explain why energy spectra for heavy elements should be flatter as compared to light nuclei. In particular, the ultra-heavy elements suffer significantly from interactions at low energies. In summary, the results obtained show the effectiveness of the combined method to calculate the cosmic-ray spectrum using a numerical calculation of trajectories and a diffusion	[]
	approximation. The calculated dependence of the propagation path length on energy suggests that the difference between the predicted spectral index at the source (\(\gamma\approx-2.1\)) in the "standard model" and the experimental value (\(\gamma\approx-2.7\)) can not be explained by the energy dependence of the escape path length solely. The compatibility of the observed cosmic-ray energy spectrum with the "standard model" requires additional assumptions. Most likely, it can be concluded that the knee in the energy spectrum of cosmic rays has its origin in both, acceleration and propagation processes. In the poly-gonato model the knee in the energy spectrum at 4.5 PeV is caused by a cut-off of the light elements and the spectrum above the knee is determined by the subsequent cut-offs of all heavier elements at energies proportional to their nuclear charge number. The second knee around \(400~{}\mbox{PeV}\approx 92\cdot\hat{E}_{p}\) is due to the cut-off of the heaviest elements in galactic cosmic rays. Considering the calculated escape path length and nuclear interaction length within the diffusion model, it seems to be reasonable that the spectra for heavy elements are flatter as compared to light elements. The calculations show also that even for the heaviest elements at the respective knee energies more than 50% of the nuclei survive the propagation process without interactions. This may explain why ultra-heavy elements are expected to contribute significantly (\(\sim 40\%\)) to the all-particle flux at energies around 400 PeV. ## Acknowledgements The authors are grateful to J. Engler, A.I. Pavlov, and V.N. Zirakashvili for useful discussions. N.N.K. and A.V.T. acknowledge the support of the RFBR (grant 05-02-16401). ## References * [1] M. Nagano & A.A. Watson, _Rev. Mod. Phys._72, 689 (2000). * [2] J.R. Horandel, _Astropart. Phys._19, 193 (2003). * [3] J.R. Horandel, _Astropart. Phys._21, 241 (2004). * [4] J.T. Link _et al._, _Proc. 28th Int. Cosmic Ray Conf., Tsukuba_4, 1781 (2003). * [5] A.G.W. Cameron, _Space Sci. Rev._15, 121 (1973). * [6] G. Bonino _et al._, _Proc. 27th Int. Cosmic Ray Conf., Hamburg_9, 3769 (2001). * [7] T.K. Gaisser _et al._, _Phys. Rev. D_47, 1919 (1993). * [8] T. Abu-Zayyad _et al._, _Astrophys. J._557, 686 (2000). * [9] J.R. Horandel, _J. Phys. G: Nucl. Part. Phys._29, 2439 (2002). * [10] D. Heck _et al._, Report FZKA 6019, Forschungszentrum Karlsruhe (1998). * [11] V.S. Berezinsky _et al._, _Astrophysics of Cosmic Rays_, North-Holland (1990). * [12] A.A. Ruzmaikin _et al._, _Magnetic Fields of Galaxies_, Kluwer, Dordrecht (1988). * [13] S.V. Ptuskin _et al._, _Astron. & Astroph._268, 726 (1993). * [14] E.V. Gorchakov & I.V. Kharchenko, _Izv. RAN ser. phys._64, 1457 (2000). * [15] D.C. Ellison _et al._, _Astrophys. J._488, 197 (1997). * [16] E.G. Berezhko & L.T. Ksenofontov, _JETP_89, 391 (1999). * [17] L.G. Sveshnikova _et al._, _Astron. & Astroph._409, 799 (2003). * [18] N.N. Kalmykov & A.I. Pavlov, _Proc. 26th Int. Cosmic Ray Conf., Salt Lake City_4, 263 (1999). * [19] V.N. Zirakashvili _et al._, _Izv. RAN ser. phys._59, 153 (1995). * [20] K. Kodaira, _Publ. Astron. Soc. Japan_26, 255 (1974). * [21] J.R. Horandel, _astro-ph/0407554_ (2004). * [22] T. Antoni _et al._, _Astrophys. J._604, 687 (2004). * [23] V.S. Ptuskin, _Adv. Space Res._19, 697 (1997). * [24] J. Candia _et al._, _J. Cosmol. Astropart. Phys._5, 3 (2003). * [25] N.E. Yanasak _et al._, _Astrophys. J._563, 768 (2001). * [26] S.P. Swordy, _Proc. 24th Int. Cosmic Ray Conf., Rome_2, 697 (1995). * [27] N.N. Kalmykov _et al._, _Nucl. Phys. B (Proc. Suppl.)_52B, 17 (1997). * [28] N.N. Kalmykov & S.S. Ostapchenko, _Yad. Fiz._56, 105 (1993). * [29] F. Aharonian _et al._, _Nature_432, 75 (2004). * [30] C. Masterson _et al._, _2nd Int. Symp. on High Energy Gamma Ray Astronomy, Heidelberg, 2004, APS Conf. Proc. 745_ p. 617 (2004).	[]
	# The Dependence of Galaxy Colors on Luminosity and Environment at \(z\sim 0.4\)1 Footnote 1: affiliationtext: Based on observations from the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, le Centre Nationale de la Recherche Scientifique and the University of Hawaii. H.K.C. Yee2 , B.C. Hsieh3 4 , H. Lin5 , M.D. Gladders6 Footnote 2: affiliation: Department of Astronomy & Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada; email: hyee@astro.utoronto.ca Footnote 3: affiliation: Institute of Astronomy, National Central University, No. 300, Jhongda Rd. Jhongli City, Taoyuan County 320, Taiwan, R.O.C. Footnote 4: affiliation: Institute of Astrophysics & Astronomy, Academia Sinica, P.O. Box 23-141, Taipei 106, Taiwan, R.O.C; email: paul@cluster.asiaa.sinica.edu.tw Footnote 5: affiliation: Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510; email: hlin@fnal.gov Footnote 6: affiliation: Carnegie Observatories, Pasadena, CA 91101, USA; email: gladders@ociw.edu ###### Abstract We analyse the \(B-R_{c}\) colors of galaxies as functions of luminosity and local galaxy density using a large photometric redshift catalog based on the Red-Sequence Cluster Survey. We select two samples of galaxies with a magnitude limit of \(M_{R_{c}}<-18.5\) and redshift ranges of \(0.2\leq z<0.4\) and \(0.4\leq z<0.6\) containing \(\sim 10^{5}\) galaxies each. We model the color distributions of subsamples of galaxies and derive the red galaxy fraction and peak colors of red and blue galaxies as functions of galaxy luminosity and environment. The evolution of these relationships over the redshift range of \(z\)\(\sim\)0.5 to \(z\)\(\sim\)0.05 is analysed in combination with published results from the Sloan Digital Sky Survey. We find that there is a strong evolution in the restframe peak color of bright blue galaxies in that they become redder with decreasing redshift, while the colors of faint blue galaxies remain approximately constant. This effect supports the "downsizing" scenario of star formation in galaxies. While the general dependence of the galaxy color distributions on the environment is small, we find that the change of red galaxy fraction with epoch is a function of the local galaxy density, suggesting that the downsizing effect may operate with different timescales in regions of different galaxy densities. Subject headings: Galaxies: evolution -- galaxies: fundamental parameters + Footnote †: journal: To appear in ApJL, V. 629 1 Footnote 1: affiliationtext: Based on observations from the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, le Centre Nationale de la Recherche Scientifique and the University of Hawaii. ## 1. Introduction Many investigations in the past 30 years have shown that galaxy populations, manifested by galaxy colors, spectra, and morphological distributions, have a strong dependence on the environment and luminosity of the galaxies (e.g., Melnick & Sargent 1970; Dressler 1980; de Vaucouleurs 1961 and many other subsequent studies). More recently, much more detailed analyzes of galaxy color distributions in the local universe have become available using large galaxy samples of tens of thousands from the Sloan Digital Sky Survey (SDSS). Baldry et al. (2004) and others show that the colors of galaxies can be neatly separated into components of red and blue galaxies. Balogh et al. (2004, hereafter B04) were able to delineate the dependence of galaxy colors on galaxy luminosity and environment, concluding that there is only a weak dependence on the latter. In this Letter we present a similar study of galaxy samples at redshift between 0.2 and 0.6 using the photometric redshift galaxy catalogs from the Red-Sequence Cluster Survey (RCS) from Hsieh et al. (2005, hereafter H05). Combined with results from a \(z\)\(\sim\)0.05 sample from the SDSS (B04), we examine the trend of the dependence of galaxy colors on luminosity and environment at different epochs. In SS2 we briefly describe the photometric redshift galaxy sample and the measurement techniques. We present the results in SS3; their implications are discussed and summarized in SS4. We adopt a flat cosmology with \(\Omega_{m}=0.3\), \(\Omega_{\Lambda}=0.7\) and \(H_{0}=70\) km s\({}^{-1}\)Mpc\({}^{-1}\). ## 2. The Photometric Redshift Galaxy Data The RCS is a 92 square degree imaging survey in the \(z^{\prime}\) and \(R_{c}\) bands conducted with the CFHT 3.6m and the CTIO 4m to search for galaxy clusters in the redshift range of \(z<1.4\) (see Gladders & Yee 2005). Additional imaging in the \(V\) and \(B\) bands was also obtained for 33.6 square degrees in the northern patches using the CFH12K camera. The observation and data reduction techniques are discussed in detail in Gladders & Yee (2005) for the \(z^{\prime}\) and \(R_{c}\) data, and in H05 for the \(V\) and \(B\) data. Photometric redshifts for 1.2 million galaxies using the four-color photometry are derived using an empirical training set method with 4,924 spectroscopic redshifts. Detailed descriptions of the photometric redshift method and the precision and completeness of the sample are presented in H05. We select samples of galaxies covering two moderate redshift intervals using conservative criteria to minimize redshift and color errors. We choose galaxies with absolute magnitudes \(M_{R_{c}}<-18.5\) in the photometric redshift bins: \(0.2\leq z<0.4\) (hereafter, the \(z\)\(\sim\)0.3 sample) and \(0.4\leq z<0.6\) (hereafter, the \(z\)\(\sim\)0.5 sample) with redshift uncertainty \(\sigma_{z}/(1+z)<0.1\), where \(\sigma_{z}\) is the computed photometric redshift uncertainty (see H05). The absolute magnitudes are corrected for K-correction and estimated evolution. For each galaxy, we use model colors computed from the spectral energy distributions of galaxies from Coleman et al. (1980) to estimate a galaxy spectral type based on the observed \(R_{c}-z^{\prime}\) color, and derive K-corrections for the \(B\) and \(R_{c}\) magnitudes. We approximate the \(R_{c}\) band luminosity evolution by \(M(z)=M(0)-Qz\) (see Lin et al. 1999), where we adopted \(Q=1.5\) for early-type galaxies (\(B-R_{c}>1.8\)	[]
	) and \(Q=1.0\) for late-type galaxies (\(B-R_{c}<1.8\)). We further limit our galaxy sample to be at least 100\({}^{\prime\prime}\) from the edges of the fields. These criteria produce a sample of 106,095 and 124,004 galaxies for the \(z\)\(\sim\)0.3 and \(z\)\(\sim\)0.5 sample, respectively. We estimate the local galaxy density using a projected surface density as a proxy (see, e.g., Dressler 1980). For each galaxy in the primary sample, we measure \(R_{5}\), the distance to the fifth nearest neighbor with \(M_{R_{c}}\leq-19.5\), from which a local surface density parameter \(\Sigma_{5}\) in units of Mpc\({}^{-2}\) (proportional to \(R_{5}^{-2}\)) is computed. For counting the five objects we select only those with photometric redshifts within 2\(\sigma_{z}\) of the primary galaxy. A completeness correction for each neighbor is applied, as not all galaxies within the sampling magnitude range satisfy the redshift uncertainty criterion. The completeness factor is estimated using the ratio of the total number of galaxies within a 0.1 mag bin at the magnitude of the neighbor galaxy to the number of galaxies in that bin with photometric redshifts satisfying the redshift uncertainty criterion. Since red galaxies on average have lower photometric redshift uncertainties, we refine our completeness correction by computing the factor separately for red and blue galaxies, separated at \(B-R_{c}=1.8\). The average completeness correction factor is \(\sim\)1.10. We count the nearest neighbors to the primary galaxy by summing the corrected weight for each galaxy until the total is \(\geq 5.0\). To be able to put the \(\Sigma_{5}\) parameter for galaxies at different redshift on a comparative scale, a background correction must be applied to \(\Sigma_{5}\). This is estimated for each primary galaxy by computing the average number of galaxies in a circle of radius \(R_{5}\) using galaxies in the same RCS patch (of size \(\sim 2.5\) deg \({}^{2}\), see Gladders & Yee 2005) that satisfy the redshift and magnitude criteria used for the counting of the neighbors. Completeness corrections are also applied in the counting of the background galax	[ { "caption": "Fig. 1.— Upper large panel: Distributions of rest (B−R) colors for galaxies in subsamples of different surface density (Σ5) and luminosity (MRc) for 0.2 ≤ z < 0.4. Subpanels for different densities are plotted horizontally, while subpanels for different absolute magnitudes are plotted vertically. Lower large panel: same for 0.4 ≤ z < 0.6. Solid lines are two-Gaussian fits to the distributions of blue and red galaxies.", "captionBoundary": { "x1": 68, "x2": 749, "y1": 754, "y2": 782 }, "figType": "Figure", "imageText": [ "-19.5", "-18.5", "-19.5", "~", "-20.5", "-21.5", "~", "-20.5", "~", "<-21.5", "~", "-20.5", "~", "~", "-19.5", "-18.5", "-19.5", "-20.5", "-21.5", "<-21.5", "RcM", "MRc", "(B", "-", "Rc)0", "b", "in", "m", "ag", "0.", "1", "pe", "r", "ie", "s", "al", "ax", "of", "g", "be", "r", "N", "um", "0.4", "<", "z", "<", "0.6", "0.2", "<", "z", "<", "0.4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "Σ", "<−2.03", "−2.03<Σ", "<0.53", "0.53<Σ", "<4.11", "4.11<Σ", "<14.76", "Σ", ">14.76", "Σ", "<−2.01", "−2.01<Σ", "<0.49", "0.49<Σ", "<3.95", "3.95<Σ", "<13.88", "Σ", ">13.88", "1", "2", "1", "2", "1", "2", "1", "2", "500", "1000", "1", "2", "500", "1000", "1500", "500", "1000", "100", "200", "300", "1", "2", "1", "2", "1", "2", "1", "2", "500", "1000", "1500", "1", "2", "500", "1000", "1500", "300", "600", "100", "200", "300" ], "name": "1", "regionBoundary": { "x1": 127, "x2": 680, "y1": 207, "y2": 729 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508016-Figure1-1.png", "source": "fig" } ]
	ies. The average background correction applied is 5.59 galaxies, with a scatter of 4.43. The stochastic nature of the background produces negative local densities; for convenience, we add 5 to \(\Sigma_{5}\) for the purpose of displaying some of the results. ## 3. Results The two panels of Figure 1 show the rest \((B-R_{c})_{0}\) distributions in bins of absolute magnitude and \(\Sigma_{5}\) for the \(z\)\(\sim\)0.3 and \(z\)\(\sim\)0.5 bins. We divide the galaxies into subsamples of local density such that the first bin contains the least dense 12.5% of the galaxies; the next three bins, each of the successive 25%; and the last bin, the most dense 12.5%. The solid lines are the two-Gaussian fits for the red and blue populations. Qualitatively, the two-Gaussian model that has been used for the SDSS \(z\sim 0.05\) galaxy samples by Baldry et al. (2004) and B04 is also a reasonable descriptor of the data at these higher redshifts. For comparison purposes, we note that the approximate transformations between the SDSS system and our data are: \(r\sim R_{c}+0.5\) and \(u-r\sim 2(B-R_{c})-0.4\) (based on galaxy color data from Fukugita et al. 1995). We also repeat the calculation using a sample with a more liberal error criterion, \(\sigma_{z}/(1+z)<0.4\); this produces qualitatively identical results, showing that the incompleteness correction works well. Figure 2 shows the fraction of red galaxies, \(f_{r}\), derived from integrating the Gaussian fits, as a function of local density for each magnitude bin for the two redshift samples. The uncertainty for each \(f_{r}\) value is estimated by generating 100 Monte-Carlo realizations based on the errors on the parameters of the corresponding two-Gaussian fit. We take the 68% confidence limits on \(f_{r}\) for the 100 results to be the \(\pm 1\sigma\) uncertainty. For both redshift bins, there is a strong dependence of \(f_{r}\) on luminosity, covering a factor of 5-6, with the luminous galaxy samples having the higher values. Within each magnitude bin, the dependence of \(f_{r}\) on local density, however, is relatively moderate, especially for the fainter galaxies. Figure 3 illustrates the dependence of the peak \((B-R_{c})_{0}\) of the Gaussian fits on local density for galaxies of different luminosities. Similar to the SDSS result at \(z\)\(\sim\)0.05 (B04; Hogg et al. 2003), the peak colors of the color distributions of both the red and the blue galaxies have only a weak dependence on the local galaxy density. We note that because of the scatter of the background corrections, there will be some blurring of the local density parameter; however, this smoothing should have minimal effect at the high density bins, where the density is much higher than the scatter in the background correction. ## 4. Discussion and Conclusions Using the three redshift epochs, we find a number of interesting evolutionary effects in the color distribution of galaxies. We find the peak colors of the red galaxy distributions to be remarkably similar over the three redshift bins. The color peaks for both the \(z\)\(\sim\)0.3 and \(z\)\(\sim\)0.5 samples for the different magnitude subsamples range from \((B-R_{c})_{0}\) of \(\sim\) 1.3 to \(\sim 1.6\), with the brighter red galaxies being redder. This is essentially identical to the \(z\)\(\sim\)0.05 SDSS samples of B04 over a similar luminosity range, which have \(u-r\sim 2.2\) to 2.5 (or \(B-R_{c}\sim 1.3\) to 1.5). Hence, there has been only minimal changes in the colors of red galaxies in all environments from \(z\)\(\sim\)0.5 to \(z\)\(\sim\)0.05. The small color changes suggest that red galaxies of all luminosities and in different environments are already well-evolved by redshift \(\sim 0.6\) (e.g., see models in Bruzual & Charlot 1993). The difference in \((B-R_{c})_{0}\) for red galaxies of different absolute magnitudes is likely due to the slope of the color-magnitude relation (CMR) for early-type galaxies. The range of \((B-R_{c})_{0}\) of \(\sim\) 0.3 mag over \(\sim 4.5\) mag in \(M_{R_{c}}\) is equivalent to a CMR slope of \(\sim\)0.06, very similar to the slope of 0.03 to 0.08 found in the CMR of galaxy clusters (e.g., Lopez-Cruz et al. 2004). The blue galaxy color distributions, however, show a strong evolutionary effect from \(z\)\(\sim\)0.5 to \(z\)\(\sim\)0. For the \(z\)\(\sim\)0.5 sample, the peak colors of the blue galaxies are essentially independent of the luminosity of the galaxies, with values bunching up around \((B-R_{c})_{0}\)\(\sim 0.85\) Figure 2.— Galaxy color (\((B-R_{c})_{0}\)) peak of the blue and red galaxy distributions as a function of local galaxy density for samples of different luminosities for the two redshift bins. Figure 1.— Red galaxy fraction, \(f_{r}\), as a function of local galaxy density for samples of different luminosities for the two redshift bins.	[ { "caption": "Fig. 3.— Galaxy color ((B − Rc)0) peak of the blue and red galaxy distributions as a function of local galaxy density for samples of different luminosities for the two redshift bins.", "captionBoundary": { "x1": 68, "x2": 393, "y1": 771, "y2": 798 }, "figType": "Figure", "imageText": [ "0.2", "<", "z", "<", "0.4", "0.4", "<", "z", "<", "0.6", "Blue", "Red", "Blue", "Red", "Rc", "Rc", "Rc", "Σ", "+55", "Rc", "1", "10", "1", "10", "1", "10", "M", "<-21.5", "-21.5<M", "<-20.5", "-20.5<M", "<-19.5", "-19.5<M", "<-18.5", "tio", "n", "ri", "bu", "di", "st", "re", "d", "lu", "e/", "of", "b", "lo", "r", "n", "co", "M", "ea", "2.0", "1.8", "1.6", "1.4", "1.2", "1.0", "0.8", "0.6", "1", "10" ], "name": "3", "regionBoundary": { "x1": 70, "x2": 383, "y1": 512, "y2": 756 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508016-Figure3-1.png", "source": "fig" }, { "caption": "Fig. 2.— Red galaxy fraction, fr, as a function of local galaxy density for samples of different luminosities for the two redshift bins.", "captionBoundary": { "x1": 68, "x2": 394, "y1": 451, "y2": 479 }, "figType": "Figure", "imageText": [ "Rc", "Rc", "Rc", "Rc", "0.2", "<", "z", "<", "0.4", "0.4", "<", "z", "<", "0.6", "1", "10", "Σ", "+55", "M", "<-21.5", "-21.5<M", "<-20.5", "-20.5<M", "<-19.5", "-19.5<M", "<-18.5", "io", "n", "ib", "ut", "is", "tr", "ed", "d", "in", "r", "ie", "s", "al", "ax", "of", "g", "tio", "n", "Fr", "ac", "1.0", "0.8", "0.6", "0.4", "0.2", "1", "10", "0.0" ], "name": "2", "regionBoundary": { "x1": 73, "x2": 383, "y1": 192, "y2": 436 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508016-Figure2-1.png", "source": "fig" } ]
	# Heteroclinic orbit and tracking attractor in cosmological model with a double exponential potential Xin-zhou Li kychz@shtu.edu.cn Yi-bin Zhao Chang-bo Sun Shanghai United Center for Astrophysics(SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234,China (October 12, 2023) ###### Abstract In this paper, the dynamical heteroclinic orbit and attractor have been employed to make the late-time behaviors of the model insensitive to the initial condition and thus alleviates the fine tuning problem in cosmological dynamical system of barotropic fluid and quintessence with a double exponential potential. The late-time asymptotic behavior of the double exponential model does not always correspond to the single case. The heteroclinic orbits are the non-stationary solutions and in general they will interpolate between the critical points. Although they can not be shown analytically, yet a numerical calculation can reveal most of their properties. Varied heteroclinic orbits and attractors including tracking attractor and de Sitter attractor have been found. pacs: 04.40.-b, 98.80.Cq, 98.80.Es ## I Introduction Astronomical observation on the cosmic microwave background(CMB) anisotropybennett , supernova type Ia(SNIa)tonry and SLOAN Digital Sky Survey(SDSS)tegmark depicted that our Universe is spatially flat, with about seventy percent of the total density resulting from dark energy that has an equation of state \(w<-1/3\) and accelerates the expansion of the Universe. Several candidates to represent dark energy have been suggested and confronted with observation: cosmological constant, quintessence with a single fieldPeebles or with N coupled fieldli , phantom field with canonicalcaldwell or Born-Infield type Lagrangianhao1 , k-essencemuk and generalized Chaplygin gas(GCG)sen . One of the most important issues for dark energy models is the fine tuning problem, and a good model should limit the fine tuning as much as possible. The dynamical attractor of the cosmological system has been employed to make the late time behaviors of the model insensitive to the initial condition of the field and thus alleviates the fine tuning problem. In quintessencewang and phantomhao2 models, the dynamical systems have tracking attractors that make the quintessence and phantom evolve by tracking the equation of state of the background cosmological fluid so as to alleviating the fine tuning problem. In addition, there are also two late time attractors in the phantom system corresponding to the big rip phasehao3 and de Sitter phasehao4 . On the other hand, exponential potentials can arise from string/\(M\)-theory, e.g.via compactification on product spaces possibly with fluxes. In this case, the equations of motion can be written as an autonomous system, and the power-law and de Sitter solutions can be determined by an algebraic method. The properties of the attractor solutions of exponential potentials can lead to models of quintessence exp . And general exact solution for double exponential potential with one exponent is the negative of the other for quintessence was presented in Ref.Rubano . The aim of this paper is to study the cosmological dynamics of barotropic fluid and scalar field with a double exponential potential and point out that the late-time asymptotic behavior does not always corresponding to the single-exponential casecopeland . We show that the existence of tracking attractor and de Sitter attractor. We also find some heteroclinic orbits which mean solutions interpolate between different critical points. Emphasis must be placed on that the tracking orbits have the similar but not exactly equal dynamical behavior and the initial possibilities consist in a wide range. ## II Phase space and critical points We consider 4-dimensional gravity with barotropic fluid and a scalar \(\phi\) which only depend on cosmic time \(t\). The scalar has a double exponential potential \[V(\phi)=\lambda_{1}e^{-\alpha_{1}\phi}+\lambda_{2}e^{-\alpha_{2}\phi}.\] (1)	[]
	Since current observations favor flat Universe, we will work in the spatially flat Robertson-Walker metric. The corresponding equations of motion and Einstein equations could be written as \[\dot{H}=-\frac{\kappa^{2}}{2}(\rho_{\gamma}+p_{\gamma}+\dot{\phi} ^{2}),\] \[\dot{\rho_{\gamma}}=-3H(\rho_{\gamma}+p_{\gamma}),\] \[\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,\] \[H^{2}=\frac{\kappa^{2}}{3}(\rho_{\gamma}+\rho_{\phi}),\] (2) where \(\kappa^{2}=8\pi G\), \(\rho_{\gamma}\) is the density of fluid with a barotropic equation of state \(p_{\gamma}=(\gamma-1)\rho_{\gamma}\), where \(0\leq\gamma\leq 2\) is a constant that relates to the equation of state by \(w=\gamma-1\). The overdot represents a derivative with respect to \(t\), the prime denotes a derivative with respect to \(\phi\). \(\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)\) and \(p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)\) are the energy density and pressure of the \(\phi\) field respectively, and \(H\) is the Hubble parameter. Phase space methods are particularly useful when the equations of motion are hard to solve analytically for the presence of barotropic density. In fact, the numerical solutions with random initial conditions are not a satisfying alternative because of these may not reveal all the important properties. Therefore, combining the information from the critical points analysis with numerical solutions, one is able to give the complete classification of solutions according to their late-time behavior. Similarly as in Ref.wang ; copeland , we introduce the following dimensionless variables \(x=\frac{\kappa}{\sqrt{6}H}\dot{\phi}\), \(y=\frac{\kappa\sqrt{\lambda_{1}e^{-\alpha_{1}\phi}}}{\sqrt{3}H}\), \(z=\frac{\kappa\sqrt{\lambda_{2}e^{-\alpha_{2}\phi}}}{\sqrt{3}H}\), \(\Gamma=\frac{V(\phi)V^{\prime\prime}(\phi)}{V^{\prime 2}(\phi)}\) and \(N=\log a\). Then, the Eqs.(II) could be reexpressed as the following system of equations: \[\frac{dx}{dN} = \frac{3}{2}x[\gamma(1-x^{2}-y^{2}-z^{2})+2x^{2}]-[3x-\frac{1}{ \kappa}\sqrt{\frac{3}{2}}(\alpha_{1}y^{2}+\alpha_{2}z^{2})],\] \[\frac{dy}{dN} = \frac{3}{2}y[\gamma(1-x^{2}-y^{2}-z^{2})+2x^{2}]-\frac{1}{\kappa} \sqrt{\frac{3}{2}}\alpha_{1}xy,\] \[\frac{dz}{dN} = \frac{3}{2}z[\gamma(1-x^{2}-y^{2}-z^{2})+2x^{2}]-\frac{1}{\kappa} \sqrt{\frac{3}{2}}\alpha_{2}xz.\] (3) Also, we have a constraint equation \[\Omega_{\phi}+\frac{\kappa^{2}\rho_{\gamma}}{3H^{2}}=1,\] (4) where \[\Omega_{\phi}={\kappa^{2}\rho_{\phi}\over 3H^{2}}=x^{2}+y^{2}+z^{2}.\] (5) Different from the case in the single exponential potential, the parameter \(\Gamma\) is dependent on \(\phi\): \[\Gamma=\frac{(\alpha_{1}^{2}y^{2}+\alpha_{2}^{2}z^{2})(y^{2}+z^{2})}{(\alpha_{ 1}y^{2}+\alpha_{2}z^{2})^{2}}\] (6) The equation of state for the scalar field could be expressed in terms of the new variables as \[w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{x^{2}-y^{2}-z^{2}}{x^{2}+y^{2}+z^{ 2}}.\] (7) According to the Eqs.(II), one can obtain the critical points and study the stability of these points. Substituting linear perturbation \(x=x+\delta x\), \(y=y+\delta y\), \(z=z+\delta z\) near the critical points into the three independent equations, to first order in the perturbations, gives the evolution equations of the linear perturbations, from which we could yield three eigenvalues. Stability requires the real part of all eigenvalues to be negative. The results are contained in Table	[]
	I. Cases (i) and (ii): These are tracking attractors. The linearization of system (II) about these critical points yields three eigenvalues: [3(a1-a2)g2a1,3{a12(g-2)-a12(g-2)[a12(9g-2)-24g2k2]}4a12, 3{a12(g-2)+a12(g-2)[a12(9g-2)-24g2k2]}4a12]; (8) [3(a2-a1)g2a2,3{a22(g-2)-a22(g-2)[a22(9g-2)-24g2k2]}4a22, 3{a22(g-2)+a22(g-2)[a22(9g-2)-24g2k2]}4a22]. (9) According to the eigenvalue (8), the stability of the attractor requires the condition \(\sqrt{3\gamma\kappa}<\alpha_{1}<\alpha_{2}\) or \(\alpha_{2}<\alpha_{1}<-\sqrt{3\gamma\kappa}\). For the eigenvalue (9), it requires the condition \(\sqrt{3\gamma\kappa}<\alpha_{2}<\alpha_{1}\) or \(\alpha_{1}<\alpha_{2}<-\sqrt{3\gamma\kappa}\). Cases (iii) and (iv): These are quintessence attractors. The linearization of system (II) about these critical points yields three eigenvalues: \[\Bigl{[}\frac{\alpha_{1}(\alpha_{1}-\alpha_{2})}{2\kappa^{2}}, \frac{\alpha_{1}^{2}}{2\kappa^{2}}-3,\frac{\alpha_{1}^{2}}{\kappa^{2}}-3\gamma \Bigr{]};\] (10) \[\Bigl{[}\frac{\alpha_{2}(\alpha_{2}-\alpha_{1})}{2\kappa^{2}}, \frac{\alpha_{2}^{2}}{2\kappa^{2}}-3,\frac{\alpha_{2}^{2}}{\kappa^{2}}-3\gamma \Bigr{]}.\] (11) According to the eigenvalue (10), the stability of the attractor requires the condition \(0<\alpha_{1}<\sqrt{3\gamma\kappa}\) and \(\alpha_{1}<\alpha_{2}\) or \(-\sqrt{3\gamma\kappa}<\alpha_{1}<0\) and \(\alpha_{1}>\alpha_{2}\). For the eigenvalue (11), it requires the condition \(0<\alpha_{2}<\sqrt{3\gamma\kappa}\) and \(\alpha_{2}<\alpha_{1}\) or \(-\sqrt{3\gamma\kappa}<\alpha_{2}<0\) and \(\alpha_{2}>\alpha_{1}\). Cases (v): The critical point is a attractor corresponding to equation of state \(w=-1\) and cosmic energy density parameter \(\Omega_{\phi}=1\), which is a de Sitter attractor. The condition of such a de Sitter attractor is \(\alpha_{1}\alpha_{2}<0\). The linearization of system (II) about these critical points yields three eigenvalues: \[\Bigl{[}-3\gamma,\frac{-3\kappa^{2}-\sqrt{12\alpha_{1}\alpha_{2} \kappa^{2}+9\kappa^{4}}}{2\kappa^{2}},\frac{-3\kappa^{2}+\sqrt{12\alpha_{1} \alpha_{2}\kappa^{2}+9\kappa^{4}}}{2\kappa^{2}}\Bigr{]}.\] (12) Cases (vi): These critical points corresponding to kinetic-dominated solutions in the asymptotic regime. The linearization of system (II) about these critical points yields three eigenvalues: \[\Bigl{[}6-3\gamma,3-\sqrt{\frac{3}{2}}\frac{\alpha_{1}}{\kappa},3 -\sqrt{\frac{3}{2}}\frac{\alpha_{2}}{\kappa}\Bigr{]};\] (13) \[\Bigl{[}6-3\gamma,3+\sqrt{\frac{3}{2}}\frac{\alpha_{1}}{\kappa},3 +\sqrt{\frac{3}{2}}\frac{\alpha_{2}}{\kappa}\Bigr{]}.\] (14) Although the kinetic-dominated critical points are always unstable, we will find some new properties in numerical analysis. \begin{table} \begin{tabular}{c c c c c} \hline Case & Critical points (x,y,z) & \(\Omega_{\phi}\) & \(w_{\phi}\) & Stability \\ \hline (i) & \(\sqrt{\frac{3}{2}}\frac{\gamma\kappa}{\alpha_{1}}\),\(\sqrt{\frac{3}{2}}\sqrt{\frac{(2\gamma-\gamma^{2})\kappa^{2}}{\alpha_{1}^{2}}}\),0 & \(\frac{3\gamma\kappa^{2}}{\alpha_{1}^{2}}\) & \(\gamma-1\) & stable \\ (ii) & \(\sqrt{\frac{3}{2}}\frac{\gamma\kappa}{\alpha_{2}}\),0,\(\sqrt{\frac{3}{2}}\sqrt{\frac{(2\gamma-\gamma^{2})\kappa^{2}}{\alpha_{2}^{2}}}\) & \(\frac{3\gamma\kappa^{2}}{\alpha_{2}^{2}}\) & \(\gamma-1\) & stable \\ (iii) & \(\frac{\alpha_{1}}{\sqrt{6}\kappa}\),\(\sqrt{1-\frac{\alpha_{1}^{2}}{6\kappa^{2}}}\),0 & 1 & \(-1+\frac{\alpha_{1}^{2}}{3\kappa^{2}}\) & stable \\ (iv) & \(\frac{\alpha_{2}}{\sqrt{6}\kappa}\),0,\(\sqrt{1-\frac{\alpha_{2}^{2}}{6\kappa^{2}}}\) & 1 & \(-1+\frac{\alpha_{2}^{2}}{3\kappa^{2}}\) & stable \\ (v) & 0,\((1-{\alpha_{1}\over\alpha_{2}})^{-1/2}\),\((1-{\alpha_{2}\over\alpha_{1}})^{-1/2}\) & 1 & -1 & stable \\ (vi) & \(\pm 1\),0,0 & 1 & 1 & unstable \\ \hline \end{tabular} \end{table} Table 1: The properties of the critical points.	[ { "caption": "TABLE I: The properties of the critical points.", "captionBoundary": { "x1": 284, "x2": 537, "y1": 232, "y2": 238 }, "figType": "Table", "imageText": [ "(v)", "0,(1−", "α1", "α2", ")−1/2,(1−", "α2", "α1", ")−1/2", "1", "-1", "stable", "(vi)", "±1,0,0", "1", "1", "unstable", "1−", "α", "2", "2", "6κ2", "1", "−1", "+", "α", "2", "2", "3κ2", "stable", "√", "(iv)", "α2√", "6κ", ",0,", "1−", "α", "2", "1", "6κ2", ",0", "1", "−1", "+", "α", "2", "1", "3κ2", "stable", "√", "(iii)", "α1√", "6κ", ",", "γ", "−", "1", "stable", "α2", "2", "3γκ2", "α2", "2", "(2γ−γ2)κ2", "√", "3", "2", ",0,", "√", "3", "2", "γκ", "α2", "(ii)", "√", "γ", "−", "1", "stable", "α2", "1", ",0", "3γκ", "2", "α2", "1", "(2γ−γ2)κ2", "√", "3", "2", ",", "√", "3", "2", "γκ", "α1", "(i)", "√", "3", "Case", "Critical", "points", "(x,y,z)", "Ωφ", "wφ", "Stability" ], "name": "I", "regionBoundary": { "x1": 249, "x2": 750, "y1": 39, "y2": 216 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508019-TableI-1.png", "source": "fig" } ]
	## III Heteroclinic orbits Critical points are always exact constant solution in the context of autonomous dynamical systems. These points are often the extreme points of the orbits and therefore describe the asymptotic behavior. If the solutions interpolate between critical points they can be divided into heteroclinic orbit and homoclinic orbit(closed loop). The heteroclinic orbit connects two different critical points and the homoclinic orbit is an orbit connecting a critical point to itself. If the numerical calculation is associated with the critical point analysis, then we will find all kinds of heteroclinic orbit, as show in Figure.1 to Figure.3. The initial point is \((\pm 1,0,0)\) for all orbits since it is a repeller. Especially, the heteroclinic orbit is shown in Figure 2, which connects the tracking attractor to \((1,0,0)\). In Figure 3, the heteroclinic orbit connects the de Sitter attractor to \((1,0,0)\). Tracking behavior consists in the possibility to avoid the fine tuning of initial conditions, obtaining the similar behavior for a while range of initial possibilities. If we take cosmic time \(t=t_{0}\), the phase space contain a two-dimensional submanifold corresponding to set of initial possibilities. This initial submanifold consists with the intersection set of heteroclinic orbit and \(t=t_{0}\) surface. In Figure 4, we plot the dynamical evolution of matter, radiation and dark energy for the model with double exponential potential, in which tracker mechanism was used to provide a non-negligible energy density at early epochhao5 . The dynamical evolution of equation of state \(w\) vs \(N\) are given in Figure 5. Figure 1: The attractor property of the quintessence at the presence of dust matter. We choose \(\alpha_{1}=2\), \(\alpha_{2}=\sqrt{5}\), \(\gamma=1\), \(\kappa=1\). The heteroclinic orbit connects the critical point which corresponds to the case (i) to the kinetic-dominated critical point \((1,0,0)\). Figure 2: The attractor property of the quintessence at the presence of dust matter. We choose \(\alpha_{1}=1\), \(\alpha_{2}=2\), \(\gamma=1\), \(\kappa=1\). The heteroclinic orbit connects the critical point which corresponds to the case (iii) to the kinetic-dominated critical point \((1,0,0)\).	[ { "caption": "FIG. 1: The attractor property of the quintessence at the presence of dust matter. We choose α1 = 2, α2 = √ 5, γ = 1, κ = 1. The heteroclinic orbit connects the critical point which corresponds to the case (i) to the kinetic-dominated critical point (1, 0, 0).", "captionBoundary": { "x1": 72, "x2": 749, "y1": 302, "y2": 336 }, "figType": "Figure", "imageText": [ "z", "0.4", "0.2", "0", "-0.2", "y", "0.6", "0", "0.2", "0.4", "1", "x", "0.5", "0.75", "4", "0", "0.25" ], "name": "1", "regionBoundary": { "x1": 282, "x2": 750, "y1": 39, "y2": 278 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508019-Figure1-1.png", "source": "fig" }, { "caption": "FIG. 2: The attractor property of the quintessence at the presence of dust matter. We choose α1 = 1, α2 = 2, γ = 1, κ = 1. The heteroclinic orbit connects the critical point which corresponds to the case (iii) to the kinetic-dominated critical point (1, 0, 0).", "captionBoundary": { "x1": 72, "x2": 749, "y1": 590, "y2": 624 }, "figType": "Figure", "imageText": [ "z", "0.4", "0.2", "0", "0.4", "0.60.8", "y", "0", "0.2", "0.4", "0.6", "0.8", "1", "x" ], "name": "2", "regionBoundary": { "x1": 278, "x2": 546, "y1": 359, "y2": 565 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508019-Figure2-1.png", "source": "fig" } ]
	# SAPAC: a SBF analysis pipeline for the astronomical community Laura P. Dunn Helmut Jerjen RSAA, Mt Stromlo Observatory, ANU, Cotter Road, Weston ACT 2611 Australia email: laurad@mso.anu.edu.au, jerjen@mso.anu.edu.au (2005; ?? and in revised form ??) ###### Abstract Large volumes of CCD imaging data that will become available from wide-field cameras at telescopes such as the CFHT, SUBARU, VST, or VISTA in the near future are highly suitable for systematic _distance surveys of early-type galaxies_ using the Surface Brightness Fluctuation (SBF) method. For the efficient processing of such large data sets, we are developing the first semi-automatic SBF analysis pipeline named SAPAC. After a brief description of the SBF method we discuss the image quality needed for a successful distance measurement and give some background information on SAPAC. keywords: Methods: data analysis, galaxies: distances and redshifts + Footnote †: editors: H. Jerjen & B. Binggeli, eds. + Footnote †: editors: H. Jerjen & B. Binggeli, eds. + Footnote †: editors: H. Jerjen & B. Binggeli, eds. ## 1 The SBF method in a nutshell Employing the surface brightness fluctuation signal of unresolved stars in distant galaxies is an effective and inexpensive new way to measure accurate distances to early-type (dwarf) galaxies. Unlike other extragalactic distance indicators (e.g. TRGB, RR Lyrae stars), this method does _not_ require resolved stars therefore allowing distance measurements for early-type galaxies far beyond the practical limits of any of the classical distance indicators (\(\sim\)5 Mpc). With Fourier analysis techniques, the SBF method quantifies the mean stellar flux per CCD pixel and rms variation due to Poisson noise across a designated area in a dwarf galaxy. Initially the SBF method was almost exclusively applied on nearby giant ellipticals and MW globular clusters (e.g. Tonry et al. 1989, 1994) but was found to work equally well with dwarf elliptical (dE) galaxies (e.g. Jerjen et al. 1998, 2000, 2001, 2004, and Rekola et al. 2005). As dE galaxies are by far the most numerous galaxy type at the current cosmological epoch, the SBF method in combination with wide-field CCD imaging offers the opportunity for the first time to spatially locate dEs in vast numbers and thereby to map in 3D the densest environments of the local Universe (for first results see contributions by Cote et al., Jerjen, Jordan et al., and Rekola et al. in this volume). First SBF distances are published for dEs as distant as 15Mpc (using 2m ground-based telescopes) and 25Mpc (using 8m VLT+FORS and HST & ACS). ## 2 Analysis prerequisites The _minimal requirements_ for the SBF analysis of an early-type galaxy are: * Galaxy morphology: the light distribution of the stellar system must be radially symmetric and have minimal structure. An overall elliptical shape of the galaxy is crucial as this is modelled and subtracted as part of the SBF analysis. *Photometry: calibrated CCD images are required in two photometric bands, e.g. (\(B,R\)) or (\(g_{475}\), \(z_{850}\)), as the fluctuation magnitude shows a colour dependency.	[]
	* Image quality: FWHM <=reff['']/20, where \(r_{\rm eff}\) is the half-light radius of the galaxy. *Integration time: \(t=\)S/N\(\cdot 10^{0.4\cdot(\mu_{\rm gal}-\mu_{\rm sky}+DM+\overline{M}-m_{1})}\), where \(\mu_{\rm gal}\) is the mean surface brightness of the galaxy, \(\mu_{\rm sky}\) the surface brightness of the sky background, \(DM\) the estimates distance modulus of the galaxy, \(\overline{M}\) the fluctuation luminosity of the underlying stellar population, and \(m_{1}\) the magnitude of a star providing 1 count/sec on the CCD detector at the telescope. To give a general idea of these constraints, Fig. 1 illustrates the depth required for an image of a dE at the distance of the Fornax cluster observed with VLT+FORS1. The SBF amplitude above the shot noise level (signal-to-noise) in the power spectrum is shown as a function of integration time and mean effective surface brightness of the galaxy. A SBF distance can be determined when the S/N is approximately 0.5, (see Fig. 8 in Rekola et al. 2005), but that depends largely on the image quality i.e. seeing. For example, to achieve a S/N\(\sim\)2 in the galaxy power spectrum, the minimum exposure time required for a dE with a mean surface brightness of 25 mag arcsec\({}^{-2}\) is 1600s. It is interesting to note that this exposure time is by a factor of 20 shorter than the 32,000s of HST time spent by Harris et al. (1998) to measure the TRGB distance of a dwarf elliptical at a similar distance. ## 3 SBF Reduction Pipeline Previous SBF work has entailed individuals hand selecting regions in galaxy images for the analysis. To make the results as impartial as possible and data reduction more Figure 1: An illustration how the signal-to-noise in the SBF power spectrum increases with length of exposure time and galaxy surface brightness at the distance of the Fornax Cluster.	[ { "caption": "Figure 1. An illustration how the signal-to-noise in the SBF power spectrum increases with length of exposure time and galaxy surface brightness at the distance of the Fornax Cluster.", "captionBoundary": { "x1": 134, "x2": 632, "y1": 543, "y2": 562 }, "figType": "Figure", "imageText": [], "name": "1", "regionBoundary": { "x1": 243, "x2": 524, "y1": 153, "y2": 530 }, "renderDpi": 150, "renderURL": "/data/data1/arivx_dataset/output/0508/astro-ph0508020-Figure1-1.png", "source": "fig" } ]
	efficient we are developing a rapid, semi-automatic SBF analysis package named SAPAC that can process large numbers of galaxies. SAPAC is a software package that carries out a semi-automatic SBF analysis of any early-type galaxy for which CCD data meets the requirements as discussed above. For a detailed description of the fluctuation magnitude calibration and the individual reduction steps such as the modelling of the galaxy, foreground star removal, selection of SBF fields etc. we refer the reader to Jerjen (2003). SAPAC consists of Perl scripts using and IRAF module and uses a sophisticated graphical user interface, also written in Perl. The average processing time for 10 SBF fields in a galaxy and measuring a distance is approximately 20 minutes. Initially we have concentrated the pipeline on \(B\), \(R\) images, but the implementation of calibration information for a wider range of commonly used filter sets for SBF work like \(J,H,K\) of the SDSS \(g,z\) filters is in process. Potential users of SAPAC who are interested in testing this package for calculating accurate distances of early-type dwarfs are welcome to contact Laura Dunn. This software package will be made available to the astronomical community soon. ###### Acknowledgements. L.P.D would like to acknowledge partial financial support from the Astronomical Society of Australia, the International Astronomical Union, and Alex Rodgers Travel Scholarship. ## References * [Harris et al. (1998)] Harris, W. E., Durrell, P. R., Pierce, M. J., Secker, J. 1998, _Nature_, 395, 45 * [Jerjen (2003)] Jerjen, H. 2003, _A&A_, 398, 63 * [Jerjen, Binggeli & Barazza (2004)] Jerjen, H., Bingeli, B., Barazza, F. 2004, _AJ_, 127, 771 * [Jerjen, Freeman & Binggeli (1998)] Jerjen, H., Freeman, K.C., Binggeli, B. 1998, _AJ_, 116, 2873 * [Jerjen, Freeman, & Binggeli (2000)] Jerjen, H., Freeman, K.C., Binggeli, B. 2000, _AJ_, 119, 166 * [Jerjen, Rekola, et al. (2001)] Jerjen, H., Rekola, R., Takalo, L., Coleman, M., Valtonen, M. 2001, _A&A_, 380,90 * [Rekola, Jerjen & Flynn (2005)] Rekola, R., Jerjen, H., Flynn, C. 2005, _A&A_, in press * [Tonry, Ajhar & Luppino (1989)] Tonry, J.L., AJhar, E.A., Luppino, G.A. 1989, _ApJ_, 346, L57 * [Tonry & Schneider (1988)] Tonry, J.L. & Schneider, D.P. 1988, _AJ_, 96, 807	[]
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	# Observational Constraints on the Role of the Crust in the Post-glitch Relaxation M. Jahan-Miri Department of Physics, Shiraz University, Shiraz 71454, Iran jahan@physics.susc.ac.ir ###### Abstract The observed large rates of spinning down after glitches in some radio pulsars have been previously explained in terms of a long-term spin-up behavior of a superfluid part of the crust of neutron stars. We argue that the suggested mechanism is not viable; being inconsistent with the basic requirements for a superfluid spin-up, in addition to its quantitative disagreement with the data. Hence, the observed post-glitch relaxations may not be interpreted due only to the effects of the stellar crust. stars: neutron - hydrodynamics - pulsars ## 1 Introduction Glitches, and post-glitch relaxations, are widely believed to be effects caused by (the superfluid component in) the crust of neutron stars (see, eg., Lyne & Graham-Smith 1998; Krawczyk et. al. 2003). There exist, however, observational data on glitches that could not be possibly due to the role of the stellar crust. The data have been previously explained (Alpar, Pines & Cheng 1990; hereafter APC) in terms of a suggested spin-up of (a part of) the crustal superfluid by the spinning down crust ("the container") over a time much larger than the associated relaxation timescale. However, a closer look at the relative rotation of the superfluid and its vortices reveals that the suggested mechanism fails quantitatively by, at least, more than one order of magnitude. In addition, the suggested spin-up process is also argued to be in contradiction with the well-known requirements for a superfluid spin-up. Hence, the (pinned) superfluid in the crust is not the primary cause of the post-glitch relaxation. On the other hand, the pinning of the superfluid vortices in the crust, and also in the core, of a neutron star has recently been objected by some authors on the account of the possible observations of long period precession in isolated pulsars (Jones & Anderson 2001; Link 2003; Buckley, Metlitski & Zhitnitsky 2004). Thus the post-glitch relaxation must be driven by mechanism(s) other than that due only to the crust superfluidity, whether the pinning is realized or not. In section 2, the general role of an assumed superfluid component in (the crust or in the core of) a neutron star on the observable post-glitch behavior of the star is briefly described. In section 3, the relevant observational data and the problem raised by these observations against any model of post-glitch relaxation based on the effects of the crust alone are highlighted. The earlier suggested resolution (APC) of the problem is then stated. In section 4, a quantitative evaluation of the suggested mechanism is given, indicating a large disagreement with the data. The subsection 4.1 presents a more detailed discussion of the rotation of the different components in the crust, paying particular attention to the vortex lines, and arrives at the same conclusion as already deduced, in the section 4. In section 5, the feasibility of the suggested process, of spinning up of a pinned superfluid component in the crust during the spinning down of the crust itself, is questioned, altogether. The possibility of such a process is argued to be ruled out, on a general dynamical ground, and also according to the vortex creep formulation. We conclude in section 6, with a speculative suggestion for the possible	[]