The main purpose of this paper is to calibrate the mass–observable relation from a joint measurement of the abundance (number counts) and the stacked cluster weak lensing profiles. We develop and apply our method to the Sloan Digital Sky Survey (SDSS) red-sequence Matched-filter Probabilistic Percolation (redMaPPer) cluster catalog that is constructed by identifying overdensities of red sequence galaxies with similar colors to galaxy clusters from the SDSS ugriz photometries (Rozo & Rykoff 2014; Rykoff et al. 2014; Rozo et al. 2015a, 2015b; see most recently Rykoff et al. 2016, for the details of the method). Since the cluster finder gives an estimation of the optical richness, λ, for each cluster, we will constrain the scaling relation between the optical richness and mass for the clusters. In this paper we develop a forward modeling approach, where we constrain the probability distribution of richness for a given halo mass,

. This is in contrast with previous studies (Baxter et al. 2016; Jimeno et al. 2017; Melchior et al. 2017; Simet et al. 2017), where the backward modeling approach is employed to constrain the probability of mass for a given richness,

. The forward modeling approach has several advantages. First, we can use the abundance measurements more easily to constrain the mass–observable relation, as Saro et al. (2015) constrained

 from the abundance measurements of SZ-selected clusters after matching to redMaPPer clusters. Second,

 can be inferred from

 once the halo mass function

 is given, based on the Bayes theorem, while the opposite transformation, i.e., inferences of

 from

, is not straightforward, because this requires knowledge of the richness function

 over the whole range of λ, which is not generally available, or is at least very noisy (and possibly affected by contamination), for richness below a threshold richness in cluster catalogs. Third, the forward modeling is convenient to generate mock catalogs of clusters by populating halos in N-body simulations with galaxies, e.g., to test systematics in a cluster-finding algorithm.


In the last few decades, the dynamic properties of the quiet Sun have been thoroughly investigated using a range of substantially different techniques, allowing us to elaborate a consistent picture of the photospheric dynamics by approaching the problem from different points of view. Particularly interesting and promising are the studies involving the tracking of small-scale magnetic fields in the quiet photosphere. Such investigations reveal features that still cannot be captured by theoretical models and/or simulations because of the complexity of the system and the simultaneous coupling of a wide range of spatial and temporal scales. These studies are based on the hypothesis that magnetic fields are passively transported by the plasma flow and provide a characterisation of advection and diffusion processes in the quiet photosphere from granular to supergranular scales (see, e.g. Wang 1988; Berger et al. 1998; Cadavid et al. 1998, 1999; Hagenaar et al. 1999; Lawrence et al. 2001; Sánchez Almeida et al. 2010; Abramenko et al. 2011; Manso Sainz et al. 2011; Lepreti et al. 2012; Giannattasio et al. 2013, 2014a,b; Giannattasio et al. 2019; Keys et al. 2014; Caroli et al. 2015; Del Moro et al. 2015; Yang et al. 2015a,b; Roudier et al. 2016; Abramenko 2017; Kutsenko et al. 2018; Agrawal et al. 2018; Giannattasio & Consolini 2021). These studies reveal an anomalous scaling of magnetic field transport with a superdiffusive character consistent with a Levy walk inside supergranules and a more Brownian-like motion in their boundary. This has provided constraints on the magnetic flux emergence and evolution that models have to consider in order to fully explain the dynamics governing these environments. At the same time, the scaling laws affecting magnetic fields in the quiet Sun are crucial to understanding how such fields vary with scale size, despite the small scales at which dissipation occurs still being inaccessible with the currently available observations (see, e.g. Lawrence et al. 1994; Stenflo 2012, and references therein). For example, in the milestone work by Stenflo (2012), the spectrum of magnetic flux density in the quiet Sun was found to be consistent with a Kolmogorov power-law scaling. The scale at which scale invariance is broken lies below the current resolution limit. This latter author argued that the collapse of magnetic fields in Kilogauss flux tubes injects energy that is expected to cascade down because of the flux decay occurring via interchange instability, and to fragment into weaker ‘hidden’ fields at smaller scales (down to ∼10 km). As far as we know, no other studies have focused on the scaling properties characterising the magnetic fields in a quiet Sun region within a range of spatial and temporal scales from (sub)granular to supergranular in the time domain. In this work, for the first time we apply the structure function analysis typical of complex systems (Frisch 1995) to fill this gap. This approach complements the studies based on feature tracking mentioned above. The main difference is that, while in those works statistical properties of the photospheric plasma flows are investigated via the transport of small-scale magnetic fields in a frozen-in condition, here we directly study the magnetic field variations emerging from magnetogram time-series. The paper is organised as follows. Section 2 describes the data set used and the analysis techniques applied. Section 3 describes the obtained results, while Sect. 4 is devoted to their discussion in the light of current literature. Finally, in Sect. 5, we present our conclusions and present future perspectives.


Some studies have used the 2.5D mean-field dynamo approach to do so, extending solar mean-field dynamo models to other stellar spectral types (Chabrier & Küker 2006; Jouve et al. 2010; Küker et al. 2011; Kitchatinov et al. 2018, and references therein). While these studies are very helpful, most of them lack the full nonlinearity and genuine parametric dependence of 3D magnetohydrodynamic (MHD) simulations. Recent developments by Pipin (2021) are starting to overcome these limits and have extended the work of Rempel (2006) on the Sun to solar-type stars with various rotation rates. Nevertheless, with the arrival of more powerful supercomputers, other authors have used instead global 3D MHD simulations to model DR and stellar magnetism in the convection zone of solar-like stars (Glatzmaier & Gilman 1982; Miesch et al. 2000, 2006; Brun et al. 2004, 2011; Brown et al. 2008, 2010; Ghizaru et al. 2010; Käpylä et al. 2011, 2014; Gastine et al. 2014; Augustson et al. 2015; Karak et al. 2015). These studies pointed out the large magnetic temporal variability and the critical effect of stellar rotation and mass on magnetic field generation through dynamo mechanism, leading in some parameter regimes to configurations with cyclic activity (Gilman & Miller 1981; Gilman 1983; Glatzmaier 1985a; Brown et al. 2011; Racine et al. 2011; Augustson et al. 2013, 2015; Käpylä et al. 2013; Nelson et al. 2013; Beaudoin et al. 2016; Guerrero et al. 2016, 2019; Strugarek et al. 2017, 2018; Viviani et al. 2018, 2019; Warnecke 2018; Matilsky & Toomre 2020). Several studies pointed out the positive effect of a stable region underneath the convection zone (Parker 1993) on the efficient storage of intense toroidal field and the lengthening of the stellar dynamo cycle period (Glatzmaier 1985b; Browning et al. 2006; Lawson et al. 2015; Beaudoin et al. 2016; Guerrero et al. 2016, 2019; Käpylä et al. 2019; Bice & Toomre 2020). Over the last decade, significant progress has been made in successfully simulating large-scale mean flows and stellar activity cycle using different numerical codes and methods (Jones et al. 2011). This is quite reassuring that a global consensus is growing on the nature of solar-like star dynamos. It is common knowledge that there are still key transitions in Rossby number (at low and high values of this parameter) that need to be understood further, as well as what is the exact type of convective dynamos realized in solar-like stars as their global parameters are varied. This study continues this effort by doing an even broader systematic parametric study of solar-like star dynamos coupled to a stably stratified layer below than what have been published so far. It extends the work published in Varela et al. (2016) and Brun et al. (2017) with the MHD anelastic spherical harmonic code (ASH) (Brun et al. 2004). In particular, we wish to better characterize energy transfers and how much of a star’s energy (luminosity) is converted into magnetic energy by nonlinear global convective dynamos over a wide range of Rossby numbers, generalizing to solar-like stars the work by Starr & Gilman (1966) and Rempel (2006).


In order to understand this discrepancy, we consider the BL Lac object nature of NGC 1275 (Veron 1978). The BL Lac object properties of 3C 84 have been noted in connection with the synchrotron optical emission that hides the accretion generated continuum. The BL Lac object aspect can be very pronounced with optical polarization that changes dramatically in amplitude and position angle, with the largest optical polarization reaching 6% (Angel & Stockman 1980). Such extreme blazar-like properties suggest a small LOS to the jet axis (Lind & Blandford 1985). We qualify this statement by noting that the evidence does not support extreme BL Lac object behavior. The range of polarization is 1%–6% in Angel & Stockman (1980) and more extensive polarization data9

9
From http://www.bu.edu/blazars/VLBAproject.html.
 covering the time period 2011–2018 indicates an optical polarization that is usually below 2% and extremely rare (2) instances of polarization >3% were reported. This suggests a slightly off angle BL Lac object. The Hβ BEL is considered to be rotating gas in a flat “pancake-like” region in which the normal to the BEL disk is parallel to the jet axis (Wills & Browne 1986). Thus, the FWHM that appears in Equation (4) depends on the LOS. For polar lines of sight, Equation (4) will under estimate Mbh. In the virial mass estimation,
5

where G is the gravitational constant, RBLR is the orbital radius of the BLR (broad line region) and vBLR is the velocity of the BEL gas. In order to relate vBLR to an observed quantity, one defines the de-projection factor, f,
6

Equations (5) and (6) indicate that

. The de-projection factor has been estimated for various classes of objects that are believed to be differentiated by an LOS (Antonucci 1993; DeCarli et al. 2011). The method of DeCarli et al. (2011) was to estimate Mbh from the bulge luminosity of the host galaxy. Using this estimate to set the value of Mbh in the virial formula, they were able to estimate f for various classes of objects,
7

Equation (4) is derived based on assuming an isotropic distribution of BEL gas velocity. Thus, we adopt a correction factor for the estimate in Equation (4) of

 for BL Lac object orientations such as the one that exists in 3C 84. Taking the nominal value of DeCarli et al. (2011) in Equation (7), we expect a correction factor of 64. The orientation corrected central black hole mass estimate based on the data in Table 1 yields

. Alternatively, if we use the blazar correction of 42 associated with the nominal value of fblazars in Equation (7) instead, we get

.


Strong gravitational lensing of high-z star-forming galaxies offers a unique way to examine highly magnified molecular gas. The method for selecting strongly lensed dusty galaxy candidates, at z > 1, is primarily based on unusually bright (sub)millimeter fluxes compared to the expected steep drop-off in (sub)millimeter number counts (e.g., Negrello et al. 2007, 2010). This method has since identified a large number across the extragalactic sky, i.e., more than 100 lensed candidates at z > 1 (Ivison et al. 2010; Vieira et al. 2010, 2013; Bussmann et al. 2013, 2015; Wardlow et al. 2013; Weiß et al 2013; Cañameras et al. 2015; Harrington et al. 2016; Strandet et al. 2016; Díaz-Sánchez et al. 2017; Negrello et al. 2017; Bakx et al. 2018). The lensed population of dusty star-forming galaxies selected by the South Pole Telescope (SPT), Herschel Space Observatory, and Planck has now been detected in more than two CO transitions (e.g., Spilker et al. 2016; Strandet et al. 2017; Yang et al. 2017; Bakx et al. 2020; this work). The Herschel-selected, strongly lensed galaxy sample (Bussmann et al. 2013) offered the first systematic approach to producing a statistically significant sample of CO/[C i] lines (Yang et al. 2017), followed by a compilation in 11 Planck- and Herschel-selected lensed galaxies (Cañameras et al. 2018b), including four galaxies with both [C i] lines detected (Nesvadba et al. 2019). The IR-to-CO luminosity relations of local starbursts and high-z star-forming galaxies explored by Greve et al. (2014) indicate that the ISM radiation field is an important component to consider when understanding CO line excitation, yet this investigation was limited to 23 unlensed and 21 lensed dusty star-forming systems—all with more than three frequency measurements of the dust continuum and usually a single CO line detection. Most previous studies used only single- and/or double-component gas-emitting regions to reproduce the observed CO emission, excluding the simultaneous modeling of the available [C i] emission, but also ignoring the role of the dust continuum emission as a heating source of the gas.


Studies of molecules play a prominent role in explaining the physical, chemical, and kinematic properties of the interstellar medium (ISM) in galaxies (Omont 2007; Tielens 2013). One such molecule is H2O, the third most abundant molecule in the warm dense ISM after H2 and CO (Neufeld et al. 1995). As an asymmetric rotor with a large electric dipole moment, H2O has a rich and complex spectrum giving rise to emission and absorption lines mainly in the submillimeter and far-infrared (FIR) regimes of the electromagnetic spectrum. Observations from local galaxies (van der Werf et al. 2010; Weiß et al. 2010; Rangwala et al. 2011; Yang et al. 2013), high-redshift ultra-luminous infrared galaxies (ULIRGs; Omont et al. 2013; Yang et al. 2016), and active galactic nuclei (AGNs; van der Werf et al. 2011) have shown H2O emission to be ubiquitous with intensities as bright as CO lines. Modeling has shown that, in addition to infrared pumping where H2O is excited by FIR photons, collisions also contribute to the intensities of low-excitation transitions (e.g., González-Alfonso et al. 2010, 2012). This is best represented in Figure 3 from Liu et al. (2017), which shows prominent H2O lines in different ISM components. The low-excitation lines become weaker or completely disappear in the warm and hot regions (>40 K) where infrared pumping dominates over collisions. The higher excitation transitions that require strong far-infrared radiation density are mainly found in the hotter regions (100–200 K) of the galaxy. The cascading emission lines,

 (Eup = 100.8 K,νrest = 987.927 GHz),

 (Eup = 137 K, νrest = 752.033 GHz), and p-H2O (22,0 − 21,1) (Eup = 196 K, νrest = 1228.789 GHz) are pumped by 101 μm photons from the base 11,1 level and are primarily excited in the warm regions of the galaxy. The collisional excitation of the low-lying levels (11,1 and 20,2) in optically thin or high-density hot regions might also contribute to the emission of the

 line. Hence, H2O transitions probe the infrared radiation field density and physical properties of the ISM such as gas density and kinetic temperature (e.g., Weiß et al. 2010; González-Alfonso et al. 2014; Liu et al. 2017).


Recent theoretical work has suggested that the presence, or lack thereof, of long-period giant planets could affect the formation of such systems. Batygin & Laughlin (2015) argued that the migration of Jupiter within our own solar system might have disrupted a massive primordial inner protoplanetary disk that could have formed multiple short-period super-Earths; they predicted that systems like the Kepler short-period multiple systems should typically lack long-period giant planets. A related question is, how common are planetary systems broadly similar in architecture to our solar system, with small close-in planets and more distant giant planets? We can begin to answer these questions in the near future through the combination of searches for short-period super-Earths and data from the long-term RV programs that have been monitoring many bright FGK stars for well over a decade. Super-Earths can be found with either high-precision RV observations or space-based transit searches. Such high-precision RV surveys include those being undertaken currently with HARPS (e.g., Díaz et al. 2016), HARPS-N (M15), APF (Vogt et al. 2014), and CHIRON (Tokovinin et al. 2013), and in the near future with MINERVA (Swift et al. 2015), CARMENES (Quirrenbach et al. 2014), ESPRESSO (Mégevand et al. 2014), and SPIRou (Artigau et al. 2014). The major upcoming space-based transit survey is that of TESS (Ricker et al. 2015). Long-term RV programs include the McDonald Observatory Planet Search (e.g., Endl et al. 2016), the Anglo-Australian Planet Search (e.g., Jones et al. 2010), the Lick-Carnegie Exoplanet Survey (e.g., Rowan et al. 2016), the CORALIE planet search (Marmier et al. 2013), and the planet search at ESO (e.g., Zechmeister et al. 2013). Long-period giant planets will also be found by Gaia, which will produce a huge sample of astrometrically detected planets (Perryman et al. 2014). While most of the Kepler sample is too faint to have been observed previously by long-term RV surveys (e.g., Coughlin et al. 2015), Gaia will be able to astrometrically detect long-period planets around many of these stars. Our own McDonald Observatory Planet Search program now has a baseline of 12–15 years for ∼200 FGKM stars, and a handful of stars also have lower precision observations dating back more than 25 years. HD 219134 is one of these stars, and here we present an analysis of our RV observations of this star, as well as our data on the stellar activity.


“Wiggling” outflows are often observed among young stellar jets and protostellar molecular outflows (Eisloffel et al. 1996; Terquem et al. 1999). Terquem et al. (1999) investigated such binary systems where the accretion disk, from which the jet originates, is inclined to the binary orbital plane. They concluded that the observed jet “wiggling” is a consequence of the jet precession caused by tidal interactions in such non-coplanar binary systems. Nichols & Slavin (2009) as well as Hollis & Michalitsianos (1993) suggested that the precession of the accretion disk around the WD may be responsible for the bending of the wide-angle outflow found in the previous studies. In analogy to these observations of young stellar jets, we suggest that the “wiggling” that we also find here for the R Aqr jet may result from disk precession as well. We estimated the precession period using the theory from Terquem et al. (1999) and the binary parameters taken from Gromadzki & Mikołajewska (2009) – Mh = 0.8M⊙ (the mass of the hot WD companion), Mp∕Mh = 1.65 (where Mp is the mass of the primary companion), D = 15.5 AU (the mean semi-major axis of the system), and e = 0.25 (eccentricity). The value of Rd is unknown in our case and we used Rd = 5 AU giving D∕Rd ≈ 3 which corresponds to the average value of 2 ≤ D∕Rd ≤ 4 (the range taken from Terquem et al. 1999). For this calculation, we assumed that the angle δ between the disk plane and that of the binary orbit is small enough (10°) and we adopted cosδ = 1. Using Eq. (1) from Gromadzki & Mikołajewska (2009), we derived the precession time of T ≈ 530 yr. This value is quite large for the wiggling waves that we see. We estimated the projected spatial wavelength λproj of the “wiggling” wave according to λ = λproj∕sin i, where i is the angle between the jet symmetry axis and the line of sight, and T = λ∕υ, where υ is the jet velocity, from Gromadzki & Mikołajewska (2009). Using i = 72° and υ ~ 100 km/s, we derive λproj ≈ 10 500 AU which is more than 20 times larger than the projected length of the observed wiggling outflow (2′′ ≈ 440 AU). However, we should note that the precessing time strongly depends on the D∕Rd; the T decreases significantly with increasing R. It may also be the case that the “wiggling” model developed for YSO jets is not fitting for R Aqr which consists of evolved objects, and both the WD and the disk where the jet probably forms are much hotter than YSO systems. Furthermore, we cannot exclude that the steady wiggling might be a sequence of dynamical interactions of the two collimated flows tilted to each other.


For instance, due to its dissipative nature, gas can be very efficient in absorbing and transporting outwards the angular momentum of the pair, likely leading to a rapid evolution and eventual coalescence. From observations, as well as numerical simulations, it has been established that in gas-rich galaxy mergers there is a large inflow of gaseous material to the central kiloparsec of the galactic remnant, often resulting in a massive circumnuclear disc (Barnes & Hernquist 1992; Sanders & Mirabel 1996; Mayer et al. 2007). Driven by dynamical friction and global torques from this disc, the pair of MBHs decays very efficiently down to separations of the order of ∼1–0.1 pc, where it forms a gravitationally bound binary (Escala et al. 2004, 2005; Mayer et al. 2007; Fiacconi et al. 2013; del Valle et al. 2015; Roškar et al. 2015). At these sub-parsec scales, most theoretical and numerical studies have focused on the evolution of binaries surrounded by a gaseous circumbinary disc, often either co-rotating (see e.g. Ivanov, Papaloizou & Polnarev 1999; Armitage & Natarajan 2005; Cuadra et al. 2009; Haiman, Kocsis & Menou 2009; Lodato et al. 2009; Nixon et al. 2011; Roedig et al. 2011, 2012; Kocsis, Haiman & Loeb 2012; Amaro-Seoane, Brem & Cuadra 2013; D’Orazio, Haiman & MacFadyen 2013; Muñoz & Lai 2016; Miranda, Muñoz & Lai 2017; Tang, MacFadyen & Haiman 2017) or counter-rotating (see e.g. Roedig & Sesana 2014; Nixon & Lubow 2015) with respect to the binary’s orbital motion. These discs are generally assumed to be well-defined, smooth, and relaxed, with no attempt to link their presence to the gaseous environment around the binary, nor to the fuelling mechanisms that bring gas to the nucleus. Furthermore, all these idealized scenarios are subject to the disc consumption problem, namely, if the disc dissolves through some process (e.g. star formation, active galactic nucleus (AGN)/supernovae feedback, Lupi et al. 2015), the evolution of the binary orbit stops. In fact, the evolution of MBHBs in gas-rich environments is intimately related to the unsolved problem of gas supply to the centre of galactic nuclei.


The first multi-messenger GW event was discovered on 17 August, 2017. About 1.7 s after the GW170817 signal detected by LIGO and Virgo (Abbott et al. 2017a), the Fermi Gamma-ray Burst Monitor was successfully triggered by GRB 170817A (Abbott et al. 2017b; Goldstein et al. 2017; Zhang et al. 2018) and subsequently a large number of follow-up observations monitored the afterglow emission in different electromagnetic bands from the radio to X-rays (Alexander et al. 2017; Hallinan et al. 2017; Margutti et al. 2017; Troja et al. 2017; D’Avanzo et al. 2018; Ghirlanda et al. 2019; Lazzati et al. 2018; Lyman et al. 2018) and the kilonova AT 2017gfo in the ultraviolet–optical–infrared band (Abbott et al. 2017c; Andreoni et al. 2017; Arcavi et al. 2017; Chornock et al. 2017; Coulter et al. 2017; Covino et al. 2017; Cowperthwaite et al. 2017; Evans et al. 2017; Hu et al. 2017; Kilpatrick et al. 2017; Lipunov et al. 2017; Nicholl et al. 2017; Smartt et al. 2017; Soares-Santos et al. 2017; Tanvir et al. 2017). The observations of GRB 170817A and its afterglows robustly confirmed the long-standing hypothesis that SGRBs can originate from compact binary mergers. Moreover, it became possible to explore the angular structure of the SGRB jet from an off-axis view (Lamb & Kobayashi 2017; Granot et al. 2018; Lazzati et al. 2018; Mooley et al. 2018a,b; Li et al. 2019). Meanwhile, the observations of AT 2017gfo indicated the existence of the merger ejecta, which suggests that the progenitor binary should at least contain one NS. In more detail, the existence of a “blue” and possibly also a “purple” component in the AT 2017gfo emission further indicated that the merger product of the GW170817 event is very likely to be a hypermassive NS, which lasted for at least a few hundred milliseconds, as an immediately formed black hole can only be associated with a “red” kilonova1 (Cowperthwaite et al. 2017; Perego et al. 2017; Tanaka et al. 2017; Tanvir et al. 2017; Villar et al. 2017; Kawaguchi et al. 2018). Therefore, in summary, the progenitor of the GW170817 event can be identified as a DNS system, which is consistent with the result of the GW analysis.


The stellar photospheric parameters we collected from literature for the benchmark stars are summarized in Table A.1. Although most benchmark stars have v sini  2 km s−1 (Reiners et al. 2018), there are two stars with larger values: J07558+833 (12.1 km s−1) and J13005+056 (16.4 km s−1). These stars are useful to investigate the performance of the algorithms when dealing with higher rotational velocities. The literature values were derived with different methods. These methods include: interferometry to estimate the stellar radius and Teff (Boyajian et al. 2012; Ségransan et al. 2003; von Braun et al. 2014; Berger et al. 2006; Newton et al. 2015), synthetic model fitting using BT-Settl models to determine Teff (Gaidos et al. 2014; Lépine et al. 2013; Gaidos & Mann 2014; Mann et al. 2015) and log g (Lépine et al. 2013), empirical relations to derive stellar mass in the form of mass-luminosity relations (Mann et al. 2015; Khata et al. 2020; Boyajian et al. 2012; Berger et al. 2006; Ségransan et al. 2003), along with the mass-magnitude relations (Maldonado et al. 2015), mass-radius relations (von Braun et al. 2014), mass–Teff relations (Gaidos & Mann 2014; Gaidos et al. 2014), empirical relations to derive the stellar radius in the form of mass-radius relations (Maldonado et al. 2015) and Teff–radius relations (Gaidos & Mann 2014; Gaidos et al. 2014; Houdebine et al. 2019), pEW measurements to determine Teff (Maldonado et al. 2015; Neves et al. 2014; Newton et al. 2015) and [Fe/H] (Maldonado et al. 2015; Neves et al. 2014; Gaidos et al. 2014; Mann et al. 2015), the definition of spectral indices such as the H2O-K2 index to estimate Teff (Rojas-Ayala et al. 2012), as well as the combination of the H2O-K2 index with pEWs to derive [Fe/H] (Rojas-Ayala et al. 2012; Khata et al. 2020), the stellar radius and Teff (Khata et al. 2020), and spectral curvature indices for the determination of Teff (Gaidos & Mann 2014). Additionally, [Fe/H] was derived by using color-magnitude metallicity relations (Dittmann et al. 2016), atomic line strength relations (Gaidos & Mann 2014), and spectral feature relations (Terrien et al. 2015). Terrien et al. (2015) used K-band magnitudes and the Dartmouth Stellar Evolution Program (Dotter et al. 2008) to derive the stellar radius, whereas Mann et al. (2015) employed the Boltzmann equation with Teff determined from synthetic model fits. Last, but not least, Houdebine et al. (2019) derived Teff from photometric colors. For more details on the individual methods, we refer to the descriptions in the corresponding works.


Discovery of exoplanetary systems has presented a rather more complex picture of planetary architectures. Transiting exoplanets, those that cross the visible disk of their host stars from our vantage point, permit the measurement of the spin-orbit misalignment between the planetary orbital plane and the stellar equatorial plane that is projected onto the sky, as well as other crucial characteristics (Triaud 2018). This is what is referred to as the sky-projected obliquity angle (λ hereafter). Its measurement is performed through the observations of the Rossiter-McLaughlin effect (Rossiter 1924; McLaughlin 1924). For exoplanets, it entails observations of the stellar radial velocity (RV) during the planetary transit. The anomaly in the measured RV values arises from the deformation of absorption lines from which they are determined, which is caused by the transiting planet occulting either the blue- or red-shifted portion of the spinning stellar disk. The measurement of this effect is possible thanks to precision, high dispersion spectrographs at large telescopes that allow for one to obtain high resolution and large signal-to-noise ratio (S/N) spectra at relatively high temporal sampling. The measurement of λ has been performed for a large number of transiting exoplanets, full details of which can be found in TEPCAT1 (Southworth 2011). These have revealed a surprising diversity in the orbital alignments (for example Queloz et al. 2000; Winn et al. 2005, 2006, 2009; Triaud et al. 2009; Gandolfi et al. 2010; Mancini et al. 2018; Yu et al. 2018; Lendl et al. 2020; Sedaghati et al. 2021), which is in contrast to the Laplacian ideals of planets forming inside a flat disk, coplanar with the stellar equator and staying there (de Laplace 1796). A surprising picture that has emerged is that a significant fraction of those close-in hot-Jupiter regime planets are on misaligned orbits, as is evident in panel b of Fig. 1 (Albrecht et al. 2012; Dawson 2014). Furthermore, the spectral type of the host also appears to play a role, whereby giant planets around hot stars seem to exist on more oblique orbits (panel a of Fig. 1), perhaps pointing to a different, more chaotic formation history, as compared to their cooler counterparts. The relation between the obliquity and the host star temperature was observed by Winn et al. (2010), who placed the boundary between the two regimes at T⋆ = 6250 K (namely theKraft break; Kraft 1967). Hébrard et al. (2011) also point out a lack of planets with masses > 3 MJup on retrograde orbits, the distribution for which is shown in panel c of Fig. 1. Tidal interactions over time with the host star are also expected to realign orbits of close-in, massive planets (Zahn 1977). Attempts have been made to study the impact of stellar age on the obliquity of planetary orbits (for example Safsten et al. 2020; Anderson et al. 2021), with Triaud (2011) finding that hot-Jupiters around younger A stars are more misaligned, setting the age barrier at 2.5 Gyr. However, a lack of precision in the measured stellar ages and the absence of uniform and homogeneous studies estimating those ages have hindered any concrete conclusions being drawn with regard to the impact of stellar ages on planetary orbital alignments. This fact is evident in panel d of Fig. 1.


The EWs were measured with the methods described in Vásquez et al. (2015). As in Paper I, we used the sum of the EWs of the two strongest CaT lines (λ8542, λ8662) as a metallicity estimator, following the Ca II triplet method of Armandroff & Da Costa (1991). Different functions have been tested in the literature to measure the EWs of the CaT lines, depending on the metallicity regime. For metal-poor stars ([Fe/H] ≲ −0.7 dex) a Gaussian function was used with excellent results (Armandroff & Da Costa 1991), while a more general function (such as a Moffat function or the sum of a Gaussian and Lorentzian, G + L) is needed to fit the strong wings of the CaT lines observed in metal-rich stars (Rutledge et al. 1997; Cole et al. 2004). Following our previous work (Gullieuszik et al. 2009; Saviane et al. 2012) we have adopted here a G+L profile fit. To measure the EWs, each spectrum was normalised with a low-order polynomial using the IRAF continuum task, and set to the rest frame by correcting for the observed radial velocity. The two strongest CaT lines were fitted by a G+L profile using a non-linear least squares fitting routine part of the R programming language. Five clusters from the sample of Saviane et al. (2012) covering a wide metallicity range were re-reduced and analysed with our code to ensure that our EWs measurements are on the same scale as the template clusters used to define the metallicity calibration. Figure 5 shows the comparison between our EWs measurements and the line strengths measured by Saviane et al. (2012) (in both cases the sum of the two strongest lines) for the five calibration clusters. The observed scatter is consistent with the internal errors of the EW measurements, computed as in Vásquez et al. (2015). The measurements show a small deviation from the unity relation, which is more evident at low metallicity. A linear fit to this trend gives the relation: ΣEW(S12) = 0.97 ΣEW(this work) + 0.21, with an rms about the fit of 0.13 Å. This fit is shown in Fig. 5 as a dashed black line. For internal consistency, all EWs in this work have been adjusted to the measurement scale of Saviane et al. (2012) by using this relation. In Table 3 we provide the coordinates, radial velocities, and the sum of the equivalent widths for the cluster member stars, both measured (“m”) and corrected (“c”) to the system of Saviane et al. 2012.

Following our previous work
we have adopted here a G+L profile fit.


The Orion Bar is an ideal source for probing the excitation and chemistry of molecules in PDRs, thanks to its close distance of 414 pc (Menten et al. 2007) and its well-known structure and geometry. The Orion Bar is located between the Orion molecular cloud and an Hii region illuminated by the Trapezium cluster. The FUV radiation field of the Trapezium cluster at the location of the Orion Bar is equivalent to (1−4) × 104χ0 in Draine (1978) units. Its orientation changes from face-on to nearly edge-on where the molecular emissions peak. The observations presented in this paper also correspond to the nearly edge-on orientation part of the Orion Bar. The geometrical enhancement of the column densities toward the nearly edge-on part of the Orion Bar was derived by multiple studies and is in the range between 4 and 20. The tilt angle compared to a completely edge-on orientation was suggested to be 3° in the model of Hogerheijde et al. (1995) and Jansen et al. (1995). A tilt angle of 3° is equivalent to an enhancement factor of 20 for the measured column densities. Based on Oi 1.317 μm emission, Walmsley et al. (2000) find a model that requires a geometrical enhancement factor of 5 to convert the observed column densities into face-on values. Neufeld et al. (2006) find the geometrical enhancement factor to be 4 based on measured C+ column densities. Using a clumpy 3D PDR model, Andree-Labsch et al. (2014) successfully reproduced the Orion Bar stratification using a clumpy edge-on cavity wall, and they claim that a model of a convex filament fails to describe the structure of the Orion Bar. The average kinetic temperature was estimated to be 85 K (Hogerheijde et al. 1995). Closer to the ionization front, higher temperatures are also measured; for example, OH transitions observed with Herschel/PACS are consistent with 160−220 K gas (Goicoechea et al. 2011) and CH+ observations with temperatures around 500 K (Nagy et al. 2013). Part of the molecular line emission measured toward the Orion Bar corresponds to an “interclump medium” with densities between a few 104 and 2 × 105 cm-3 (Simon et al. 1997). It has been suggested that other molecular lines originate in clumps with densities in the range between 1.5 × 106 and 6 × 106 cm-3 (Lis & Schilke 2003).


Cluster radio relics are usually found in the outskirts of merging galaxy clusters. They exhibit elongated morphologies and high degrees of polarisation above 1 GHz (up to 70%, Ensslin et al. 1998; Bonafede et al. 2014; Loi et al. 2019; de Gasperin et al. 2022). The resolved spectral index in radio relics shows a gradient: it steepens towards the cluster centre and flattens towards the outskirts. Their size can reach up to ∼2 Mpc, and high-resolution observations have revealed filamentary structures within relics themselves (Di Gennaro et al. 2018; Rajpurohit et al. 2020, 2022a,b; de Gasperin et al. 2022). The Largest Linear Sizes (LLS) and radio powers of relics are correlated, as well as the integrated spectral index and the radio power (van Weeren et al. 2009b; Bonafede et al. 2012; de Gasperin et al. 2014). Relics trace ICM shock waves with relatively low (M   3) Mach numbers (Finoguenov et al. 2010; Akamatsu et al. 2013; Shimwell et al. 2015; Botteon et al. 2016). The acceleration of electrons is believed to proceed via diffusive shock acceleration (DSA) in the ICM (Ensslin et al. 1998; Roettiger et al. 1999), in which particles scatter back and forth across the shock front gaining energy at every crossing. Nevertheless, this mechanism has been shown to be rather inefficient in accelerating electrons from the thermal pool (Vazza & Brüggen 2014; Vazza et al. 2016; Botteon et al. 2020a; Brüggen & Vazza 2020; see Brunetti & Jones 2014 for a review). Recently, it has been suggested that seed electrons could originate from the tails and lobes (driven by AGN outflows) of cluster radio galaxies (Bonafede et al. 2014; van Weeren et al. 2017; Stuardi et al. 2019), which alleviates the requirements of high acceleration efficiencies at cluster shocks (e.g., Markevitch et al. 2005; Kang et al. 2012, 2017; Botteon et al. 2016; Eckert et al. 2016). In some cases, double relics have been detected on opposite sides of the cluster centre (e.g., Rottgering et al. 1997; van Weeren et al. 2010, 2012b; Bonafede et al. 2012; de Gasperin et al. 2015a). In these clusters it is possible to constrain the merger history, providing important information about the formation processes of relics.


Remnant BHs of Population III stars with ${\sim }10^{2}\, \mathrm{M}_{\odot }$ are one of the candidates for such seeds (e.g. Madau & Rees 2001). They can grow SMBHs in the available time if either with continuous accretion at the Eddington limit or with short episode of super-Eddington growth. In reality, however, the BH’s growth can easily be hindered by its own radiative feedback (Milosavljević, Couch & Bromm 2009; Park & Ricotti 2011; Orofino, Ferrara & Gallerani 2018; Sugimura et al. 2018). An alternative and attractive pathway for seed BH formation is via the so-called direct collapse (e.g. Bromm & Loeb 2003), where a supermassive star (SMS) with ${\sim } 10^{5}\, \mathrm{M}_{\odot }$ collapses into a BH with a similar mass by general relativistic instability (Umeda et al. 2016; Woods et al. 2017; Haemmerlé et al. 2018). Here, SMSs are supposed to form from a primordial gas in some peculiar sites. Unlike in ordinary first star formation, which is driven by H2 cooling (see e.g. Bromm & Larson 2004; see also Glover 2013 for a review), H2 is dissociated by strong external far-ultraviolet (UV) radiation from nearby galaxies and the contraction of clouds is solely caused by H atomic cooling in the most intensively studied channel for SMS formation (Omukai 2001; Omukai, Schneider & Haiman 2008; Shang, Bryan & Haiman 2010; Regan, Johansson & Wise 2014; Sugimura, Omukai & Inoue 2014). Such clouds contract almost isothermally at ∼104 K without experiencing vigorous fragmentation. The protostar formed at the centre subsequently accretes the gas at a high rate of $10^{-1}\, \mathrm{M}_{\odot }\, \mathrm{yr}^{-1}$ due to high temperature (Latif et al. 2013; Inayoshi, Omukai & Tasker 2014; Becerra et al. 2015; Chon, Hosokawa & Yoshida 2018). Such rapidly accreting protostar inflates greatly in radius with effective temperature of several 1000 K and grows supermassive with ${\gtrsim } 10^{5}\, \mathrm{M}_{\odot }$ avoiding ionization radiation feedback on the accretion flow (Hosokawa, Omukai & Yorke 2012; Hosokawa et al. 2013), before collapsing by general relativistic instability.


SN2006jc, PS15dpn and other narrow-line SNe. Two out of the three SNe we considered above are super-luminous, however the final collapse of a PPI+CC progenitor or PISNe does not need to be superluminous (Woosley 2017). The PPI is just one possible mechanism to create CSM, which can produce extreme luminosities by generating radiation from the kinetic energy of the ejecta and/or narrow emission lines (even if the luminosity does not reach extreme values). The detection of narrow H lines determines the classification of a SN as a type IIn, while the detection of narrow He emission lines determines the classification as type Ibn. Both kinds of event are too common to be entirely explained with PPI+CC progenitors, and it is likely that both classes contain events with a diversity of physical mechanisms (e.g., Pastorello et al. 2008 but see also Hosseinzadeh et al. 2017). Nevertheless, it is possible that at least some of these events might correspond to the observational counterpart of the death of PPI+CC progenitors. In particular, our simulations can produce several solar masses of H-free CSM moving at a few thousand km s−1, which correspond to the width of the He lines detected in some SN Ibn without any fine-tuning required. Even if the detection of narrow lines alone is not sufficient to associate a specific SN to a PPI event, combining evidences from previous coincident transients, large ejecta masses or long lightcurve durations, large 56Ni yields, an extremely young surrounding stellar population, and/or nucleosynthetic signatures might strengthen the case for associating specific event with this scenario. Possible examples of SN Ibn that might correspond to PPI+CC are SN2006jc and PS15dpn. The former showed relatively narrow He lines possibly hinting to asphericity of the CSM (Foley et al. 2007) and was spatially coincident with an unexplained outburst two years earlier (e.g., Pastorello et al. 2007; Foley et al. 2007). For the latter, Wang & Li (2019) proposed to fit the light curve by combining CSM interaction and radioactive decay, and inferred CSM and 56Ni masses of ∼0.8 M⊙ and ∼0.1 M⊙, respectively, in good agreement with our models.


The task of estimating stellar ages has been addressed by several authors and lots of different methods are found in the literature. For instance, there are (i) empirical methods, which use a deterministic relation between a given parameter and the age of a star (first proposed by Skumanich 1972). This is the case of gyrochronology (e.g. Barnes 2003, 2007; Mamajek & Hillenbrand 2008; Collier Cameron et al. 2009), decay of chromospheric activity (e.g. Soderblom, Duncan & Johnson 1991; Rocha-Pinto & Maciel 1998; Rocha-Pinto et al. 2000; Pace & Pasquini 2004; Lyra & Porto de Mello 2005; Pace et al. 2009; Zhao et al. 2011; Pace 2013), lithium depletion (Sestito & Randich 2005; Jackson & Jeffries 2014; Carlos, Nissen & Melendez 2016), and ‘magnetochronology’ (proposed by Vidotto et al. 2014). (ii) Model-dependent methods, which are based on the comparison between measurable physical quantities and the ones expected from stellar structure models that use age as one of its parameters. Isochrone fitting (Edvardsson et al. 1993; Nordström et al. 2004; Pont & Eyer 2004; Jørgensen & Lindegren 2005; Silaj & Landstreet 2014; Maxted, Serenelli & Southworth 2015) and asteroseismology (Cunha et al. 2007; Vauclair 2009; Metcalfe et al. 2010; Silva Aguirre et al. 2017) are classified in this category. (iii) Semifundamental methods, those that are based on well-known fundamental physics and employ only few assumptions. These are the cases for the method of cluster expansion (e.g. Makarov 2007) and nucleocosmocronology, known to predict unreliable ages (Ludwig et al. 2010). (iv) Statistical methods, which use statistical relations, like the age–metallicity relation (AMR) and the age–velocity dispersion relation (AVR), between a given property and the age. These relations have not been much explored in the literature as a direct tool to estimate stellar ages. Some few examples of its usage are found in Lachaume et al. (1999) and Maciel, Rodrigues & Costa (2011) (for the AVR) and Spina et al. (2016) (for the AMR, especially [Y/Mg] × age and [Y/Al] × age).


The kinetic energy stored in the CO shell can be estimated as $E_{\rm kin} = 0.5\, M_{\rm shell}\, V^2_{\rm exp}$, where Vexp is the expansion velocity of the shell and Mshell is the total (molecular, atomic, and ionized) shell mass. Adopting an expansion velocity equal to half the velocity interval where the structure is observed, Vexp = 7.0 ± 1.3 km s− 1 , the molecular mass given in Table 1 and the atomic and ionized masses estimated by Cichowolski et al. (2001), 1450 and 3000 M⊙, respectively, we obtain Ekin = (2.5 ± 1.0) × 1049 erg, assuming a 40 per cent error for the masses.. Although Cichowolski et al. (2001) concluded that WR 130 could have alone created the observed structure, it is important to note that they did not take into account the molecular mass present in the shell, which considerably increases the kinetic shell energy. Thus, we can compare now the new value obtained for Ekin with the mechanical energy deposited in the ISM by the wind of the WR star, Ew = (0.7–2.2) × 1050 erg (Cichowolski et al. 2001). We obtain ϵ = Ekin/Ew = 0.007–0.5. The ratio ϵ measures the energy conversion efficiency in the shell, and according to evolutionary models ϵ ≤ 0.2 (Koo & McKee 1992). Thus, not all the possible values of ϵ are compatible with the scenario where the energy injected during the WR phase is enough to create the structure. In this case, the contribution of the energy injected during the O-star phase and/or other massive stars, should be considered. As mentioned in the Introduction, WR 130 is a WNH star, and according to Smith & Conti (2008) its age would be of about 2–3 Myr and its initial mass of at least 60 M⊙. A rough estimation of the energy injected by such a star during its main sequence yields Ew = (2.5–3.5) × 1050 erg (de Jager, Nieuwenhuijzen & van der Hucht 1988), which would be enough to create the observed structure. We have nevertheless looked for the presence of other massive stars in the region. We queried the available catalogues such as the Galactic O-Star Catalog (Maíz Apellániz et al. 2013), the Early-Type Emission-Line Stars Catalogue (Wackerling 1970), the Catalogue of Be stars (Jaschek & Egret 1982), the H-alpha Stars in the Northern Milky Way Catalogue (Kohoutek & Wehmeyer 1997), and the Catalog of Galactic OB Stars (Reed 2003), for early-type and emission stars. No stars were found in any catalogue. The only massive star located nearby is, as mentioned by Cichowolski et al. (2001), an OB star, which has an uncertain spectral type and no distance estimate (Stock, Nassau & Stephenson 1960). It is located in projection not in the centre of the structure but on to the shell (there is a second OB star mentioned by Cichowolski et al. 2001 but its location is actually outside the structure, see fig. 1 of Cichowolski et al. 2001). Although we cannot completely rule out the possibility that the OB star may be playing a role in creating the shell structure, we think that the action of WR 130 is sufficient and most likely dominant in the region.


The use of N-body simulations that include fragmentation allows us to perform a more detailed study of the final composition of the planets formed. In particular, we can study the water loss and/or accretion of the final planets more realistically than in the classic models of accretion. Marcus et al. (2010) presented two empirical models for the mantle stripping in differentiated planetary embryos after a collision. The authors set a simple planet structure of two layers, assuming differentiation in core and mantle, where the mantle could be composed by silicate or ice. In this work, the authors concluded that the more energetic the collision, the more mass from the mantle is lost. Therefore, for violent collisions, water could be more easily removed. Dvorak et al. (2015) performed SPH (smoothed particle hydrodynamics) simulations and studied water loss in planetary embryos and water retained in significant fragments after a collision. They concluded that the impact velocity and the impact angle play a key role in the water loss of a planetary embryo after a collision. The investigations developed by Marcus et al. (2010) and Dvorak et al. (2015) suggest that incorporating a realistic model of volatile transport and removal in an N-body code, may lead to reduced water contents on the resulting terrestrial-like planets, in comparison with those derived from classical models that assume perfect mergers. Burger et al. (2018) studied the volatile loss and transfer. The authors focused on hit-and-run encounters using SPH simulations. They concluded that the cumulative effect of several hit-and-run collisions could efficiently strip off volatile layers of protoplanets. Driven by this, Dugaro et al. (2019) studied the water delivery in planets formed in the habitable zone (HZ), using the mantle stripping models derived by Marcus et al. (2010) in their N-body simulationswith fragmentation. The authors showed that fragmentation is not a barrier for the surviving of water worlds in the HZ, and fragments may be important in the final water content of the potentially habitable terrestrial planets formed in situ.


In Fig. 5, we consider how the velocity dispersion profile scales with radius. Specifically, we plot the power-law index (η) versus the central velocity dispersion (σ0). The vast majority of the galaxies with σ0 ≲ 2.45 are BGGs and these have negative η values. This includes most of the Romulus galaxies (red filled and open circles), the L18 BGGs (blue crosses) and the early-type galaxies that comprise the SAURON sample (Cappellari et al. 2006; grey line and shaded area). In contrast, nearly all of simulated BGGs with σ0 ≲ 2.45) from the DIANOGA Hydro-10x simulations Marini et al. (2021) have positive η values. For σ0 ≳ 2.45, the spread of η for the observed galaxies (e.g. L18 and Newman et al. 2013 BCGs) broadens and spans both positive and negative η values. In fact, majority of the galaxies tend to have positive ηs. This change in behaviour is well known. A number of studies have noted that on the group-scale and lower, the stellar velocity dispersion profile of the central galaxies tend to decrease with increasing radius. On the cluster-scale, the BCGs typically have rising velocity dispersion profiles with increasing radius (Von Der Linden et al. 2007; Bender et al. 2015; Veale et al. 2017). The origin of this flip is still not well understood. We leave a more detailed investigation of this change to future work. Here, we simply mention two possible explanations: The change in slope may be a reflection of the differences in the dynamical state (e.g. mass-to-light ratio; M/L) at the outskirts of BCGs (Dressler 1979; Fisher, Illingworth & Franx 1995; Sembach & Tonry 1996; Carter et al. 1999; Kelson et al. 2002; Loubser et al. 2008; Newman et al. 2013; Schaller et al. 2015; Marini et al. 2021), or it could be due to increased contribution from the intragroup/intracluster light along the line-of-sight and the increased leverage of tangential orbits (Loubser et al. 2020). All of these effects are linked to the increased frequency of galaxy–galaxy interactions and more specifically, central-satellite interactions, implicated in the build-up of extended diffuse stellar component. And, as discussed by Schaye et al. (2015), Oppenheimer et al. (2021), and the EAGLE simulations clearly show that the extended stellar halo becomes increasingly more important, and hosts a non-trivial fraction of the total stellar mass towards the cluster scale.


Similarly to the [α/Fe] ratio, the ratio of the slow (s-) neutron capture process elements to iron can be regarded as a cosmic clock. Ba, Sr, La, and Y are mainly s-process elements produced on long timescales by low mass AGB stars (Matteucci 2012). Since a low mass star must evolve to the AGB phase before the s-process can occur, the s-process elements are characterized by a delay in the production, much like the delay of iron production by SNe Ia relative to the α elements production by core collapse SNe. Among the four s-process elements mentioned above, GES provides the abundances of Y II (the first s-process peak element) and Ba II (the second s-process peak element) for all our sample stars. Their abundances behave differently in the Galactic thick and thin discs (Bensby et al. 2005, 2014; Israelian et al. 2014; Bisterzo et al. 2017; Delgado Mena et al. 2017). Unlike the Galactic thick disc stars, which show an almost constant [Ba/Fe] abundance close to the solar value, the Galactic thin disc stars have their [Ba/Fe] abundances increasing with [Fe/H] and reaching their maximum values around solar metallicity, after which a clear decline is seen (see also Cristallo et al. 2015a,b, for the most recent s-process calculation in AGB yields). The same trend is observed in our sample. In Fig. 13 we display the Li-[Ba/Fe], [Ba/Fe] as a function of [Fe/H], and the evolution of absolute Ba abundance A(Ba), as derived from Ba II lines. Similar figures are also plotted for yttrium (Y II). [Ba/Fe] and [Y/Fe] values here are derived from MCMC simulations, taking into account the measurement uncertainties of A(Ba II)/A(Y II) and [Fe/H]. By applying the same MCMC setups used for [α/Fe] (see Sect. 3.1), we calculate the mean values of [Ba/Fe] and [Y/Fe] for each star. These values, together with their corresponding 1σ uncertainties, are listed in Table 1. In the literature there are several theoretical works on the evolution of [Ba/Fe] and [Y/Fe] in the Galactic thin disc (e.g. Pagel & Tautvaisiene 1997; Travaglio et al. 1999, 2004; Cescutti et al. 2006; Maiorca et al. 2012; Bisterzo et al. 2017). For comparison, we show in Fig. 13 the predictions of the most recent one (Bisterzo et al. 2017) where the updated nuclear reaction network was used.


Diffusive shock acceleration (DSA) operating at expanding shock waves of supernova remnants (SNRs) is widely believed to be the mechanism converting the kinetic energy released by supernova explosions into the energy of cosmic rays (CRs) (e.g., Malkov & Drury 2001). In the DSA theory, CRs being accelerated at shocks must be scattered by self-generated magnetic turbulence. Since the highest-energy CRs in the shock precursor are prone to lack self-generated turbulence, they are expected to escape the shock. The DSA theory generally predicts that a substantial fraction of the shock energy is carried away by escaping CRs. In the presence of molecular clouds surrounding the SNR, escaping CRs can illuminate the clouds through pp interactions, producing gamma-ray emission with a flux depending on the amount of nuclear CRs released by an SNR and the diffusion coefficient in the interstellar medium (ISM; Aharonian & Atoyan 1996; Aharonian et al. 2004; Rodriguez Marrero et al. 2008; Gabici et al. 2009). High Energy Stereoscopic System (H.E.S.S.) observations reveal a complex of sources (HESS J1800-240A, B, and C) ∼0.5° south of the SNR W28, coincident with molecular clouds in the field, and the Large Area Telescope (LAT) on board the Fermi satellite reveals a similar structure in gigaelectronvolt energies (Abdo et al. 2010; Hanabata et al. 2014). The gigaelectronvolt–teraelectronvolt gamma-ray emission around W28 can be regarded as a realization of this scenario (Aharonian et al. 2008). Another example is the detection of two extended gamma-ray structures located at two opposite edges of SNR W44 by Fermi-LAT (Uchiyama et al. 2012; Peron et al. 2020). The gamma-ray emission coincides with the molecular cloud complex that surrounds SNR W44. The gamma-ray emission that appears to come from the surrounding molecular cloud complex can be ascribed to the CRs that have escaped from W44. The total kinetic energy channeled into the escaping CRs is estimated to be larger than a few 1049 erg in both W28 and W44, although the exact number depends on the value of the diffusion coefficient of escaping CRs.


The standard single-frequency detection methods for point sources in the CMB and far IR are based on wavelet techniques (Vielva et al. 2003; Barnard et al. 2004; González-Nuevo et al. 2006) or on the matched filter (or MF hereafter, Tegmark & de Oliveira-Costa 1998; Herranz et al. 2002; Barreiro et al. 2003; López-Caniego et al. 2006, see also Herranz & Vielva 2010 for a review.). Wavelets are well suited for the detection of compact sources due to their good position-scale determination properties, whereas the MF is the optimal linear detector-estimator because it provides the maximum signal-to-noise (S/N) amplification for a source with a known shape (usually the point-spread function, or PSF hereafter, of the telescope) embedded in statistically homogeneous and spatially correlated noise. By default, these techniques are applicable only to single-frequency sky images: even for multiwavelength observatories such as the Herschel Space Observatory (Pilbratt et al. 2010) or Planck (Tauber et al. 2010), the standard detection pipelines have produced individual source catalogs for each frequency band (see e.g., Planck Collaboration VII 2011; Planck Collaboration XXVIII 2014; Planck Collaboration XXVI 2016; Maddox et al. 2018). The next logical step is to boost the signal of faint sources by combining the different bands into a single detection, that is, “multifrequency detection”. Most of the blind component separation algorithms that are used for diffuse components in microwave and far IR astronomy can not deal with the high diversity of spectral behaviors associated to the different populations of extragalactic compact sources (see for example Leach et al. 2008). However, over the last few years a number of multifrequency compact source detection techniques have been proposed in the literature (Herranz & Sanz 2008; Herranz et al. 2009; Lanz et al. 2010, 2013; Planck Collaboration Int. LIV 2018). A review on the topic can be found in Herranz et al. (2012). In particular, if the spatial profile and the spectral energy distribution (SED) of the sources are known, and if the cross-power spectrum is known, or can be estimated from the data, the optimal linear detection method is the matched multifilter (or MMF hereafter, Herranz et al. 2002). Lanz et al. (2010) also showed that the MMF can be generalized for the case where the SED of the sources is not known. This generalization outperforms the single-frequency MF in terms of S/N and can be used to infer the spectral index of synchrotron-dominated radio sources, as shown in Lanz et al. (2013). However, in this paper we will incorporate a specific SED to the MMF in order to derive a photometric redshift estimation of dusty galaxies and high-redshift star forming galaxies detected in the IR part of the spectrum1. We will do so by applying the multifrequency MMF filter to the first and second data releases of the Herschel Astrophysical Terahertz Large Area Survey (the Herschel-ATLAS or H-ATLAS, Eales et al. 2010), the largest single key project carried out in open time with the Herschel Space Observatory. We restrict our multifrequency analysis to the three wavelength bands covered by the SPIRE instrument aboard Herschel (Griffin et al. 2010), centered around 250, 350 and 500 μm. As discussed in Hopwood et al. (2010), Lapi et al. (2011), González-Nuevo et al. (2012), Pearson et al. (2013) and Donevski et al. (2018), the SPIRE bands are ideal for capturing the peak in the SED corresponding to dust emission of star-forming galaxies at z ∼ 2, that is redshifted from its rest-frame wavelength around 70–100 μm to the SPIRE wavelengths: This is the redshift range where galaxies have formed most of their stars. At higher redshifts, dusty star-forming galaxies (DSFGs) occupy the most massive halos and are among the most luminous objects found at z ≳ 4 (Michałowski et al. 2014; Oteo et al. 2016; Ikarashi et al. 2017). These high-redshift DSFGs have markedly red colors as seen by SPIRE, with rising flux densities from 250 to 500 μm (the so-called “500 μm-risers”), and have received a great deal of attention in the recent years (see for example Ivison et al. 2016; Negrello et al. 2017; Strandet et al. 2017). The DSFGs, and particularly the 500 μm risers uncovered by Herschel, are providing much insight into the early star forming history of the universe. However, sensitivity and limited angular resolution severely constrain the power of this type of objects as astrophysical probes. The sensitivity of SPIRE allows for the direct detection of only the brightest, and thus rarest objects, at the bright end of the luminosity function. By means of our multifrequency MMF technique, we intend to enhance the detectability and statistical significance of very faint red objects in the H-ATLAS source catalog and so expand the list of reliable 500 μm-riser candidates.

Background Background Background Motivation


Our spectroscopic coverage includes the damped Lyα absorption (DLA) feature from neutral hydrogen. Analysis of this feature is complicated by blending between the host and the companion, as well as uncertainties in the continuum modeling, and it is difficult to constrain the individual contributions of the two component systems to the overall line. Modeling the line as two separate, blended Voigt profiles, we can robustly constrain the total column density of the two systems together to 5 × 1019 cm−2 ≤ N(H i) ≤ 9 × 1019 cm−2; our best-fit estimate is (6.7 ± 1.2) × 1019 cm−2, making the system technically a sub-DLA (Péroux et al. 2003). This spread in N(H i) is due to uncertainties in the relative contributions of the two systems and continuum placement. The derived N(H i) is an order of magnitude less than a typical value through a disk of a spiral galaxy with stellar mass >1010 M⊙, measured by The H i Nearby Galaxy Survey (THINGS; Walter et al. 2008; Leroy et al. 2008). This result has two implications. One is that SN2017egm may have exploded on the near side of the galaxy, where there is less neutral H i material along the line of sight. One may argue for the second possibility that UV fluxes from the SN explosion could photoionize a large fraction of neutral H i in the disk of NGC 3191, and SN2017egm could be anywhere in the disk. To validate this hypothesis, we calculate the time required, tphot, to photoionize N(H i) ∼ 1021 cm−2 over a scale height R of an H i disk. H i-ionizing flux, JUV, at the time of SN explosion is poorly constrained by observation. Let us take JUV as a fraction of fUV of the peak bolometric luminosity (Lbol); thus we have

. The Milky Way thin disk (stellar) has a scale height of 100 pc.7

7
The H i gas can be more extended (Marasco & Fraternali 2011).
 As the most conservative assumption, let us take R = 50 pc if the SN is at the mid-plane; thus we have

 cm−2 s−1. In this equation, without any knowledge of the early-time UV fluxes from SN 2017egm, we assume it is only 10% of the estimated bolometric luminosity (fUV = 0.01). Here, the maximum Lbol is 2 × 1044 erg s−1. To photoionize a column of 1021 cm−2 H i atoms, we need the same number of H i-ionizing photons, thus, over a timescale of tphot, we have JUVtphot ≃ 1021 cm−2, leading to tphot = 3 × 109 s. This is two orders of magnitude longer than the time lag between the explosion and the HST spectroscopy date for SN2017egm. Since

, a larger fUV value will shorten the photoionization timescale tphot by a factor of a few, but not enough to change our conclusion. This result suggests that photoionization due to the SN explosion is probably localized within a 5 pc region.


Among the several features, the distribution of chemical elements across the Galactic disc historically constitutes the most important constraint to chemo-dynamical models of our Milky Way. A number of studies (e.g. Tosi 1988; Hayden et al. 2014, 2015; Anders et al. 2017) have shown the spatial distributions of chemical abundances and their ratios across the Galactic disc. However, these studies are mainly based on field stars, which also include very old populations that had time to migrate significantly and redistribute the chemical elements across the Galaxy (e.g. Sellwood & Binney 2002; Roškar et al. 2012; Martinez-Medina et al. 2016). Open clusters are a valuable alternative, being on average younger (Magrini et al. 2017), and therefore a better tracer of the gradients in the disc out of which the most recent stars formed. Since the work of Janes (1979), much observational evidence has established that the metallicity distribution (often abbreviated by the iron-to-hydrogen ratio [Fe/H]) traced by clusters throughout the Milky Way disc shows a significant decrease with increasing distance from the Galactic Centre. This ‘radial metallicity gradient’ – in its apparent simplicity – reflects a complex interplay between several processes that are driving the evolution of our Galaxy, including star formation, stellar evolution, stellar migration, gas flows, and cluster disruption (Cunha & Lambert 1992, 1994; Friel 1995; Stahler & Palla 2004; Carraro et al. 2006; Boesgaard, Jensen & Deliyannis 2009; Magrini et al. 2009; Frinchaboy et al. 2013; Netopil et al. 2016; Anders et al. 2017; Spina et al. 2017; Bertelli Motta et al. 2018; Quillen et al. 2018). Complementary to the study of the overall metallicity distribution, the abundance ratios of several other elements, such as α-elements, iron peak, odd-z, and neutron capture, can provide deep insight into the variety of nucleosynthesis processes, with their production sites and time-scales (e.g. Carrera & Pancino 2011; Ting et al. 2012; Reddy, Lambert & Giridhar 2016; Duffau et al. 2017; Magrini et al. 2017, 2018; Donor et al. 2020; Casamiquela et al. 2020). Therefore, understanding the distribution of metals traced by clusters across the Galactic disc is fundamental for explaining the birth, life, and death of both stars and clusters, the recent evolution of our own Milky Way, and the evolution of other spiral galaxies (Boissier & Prantzos 2000; Bresolin 2019).


We analyze the 10 BBH mergers reported by LIGO and Virgo in their O1 and O2 observing runs (Abbott et al. 2019a; LIGO Scientific Collaboration & Virgo Collaboration 2019). Before discussing results, it is useful to review expectations from the literature for the spin distributions resulting from different formation scenarios. Isolated binary evolution is predicted to yield black holes with spins preferentially aligned with their orbit. Although spin misalignments may be introduced by natal supernova kicks, episodes of mass transfer and tidal torques serve to realign component spins before the formation of the final black hole binary (Rodriguez et al. 2016; Zevin et al. 2017; Gerosa et al. 2018; Qin et al. 2018; Zaldarriaga et al. 2018; Bavera et al. 2020). The black holes’ spin magnitudes in this scenario are much more uncertain. Recent work indicates that angular momentum is efficiently transported away from stellar cores, leaving black holes with natal spins as low as a ∼ 10−2 (Qin et al. 2018; Fuller & Ma 2019). While tides on the progenitor of the second-born black hole can spin up the progenitor star (Zaldarriaga et al. 2018), this effect can be counteracted by mass loss in stellar winds, and more detailed simulations find only low or moderate spin increases due to tides (Qin et al. 2018; Bavera et al. 2020). Meanwhile, dynamically formed systems in dense stellar clusters have no a priori preferred axis, and so are likely to have random spin configurations (Rodriguez et al. 2016, 2018, 2019; Doctor et al. 2020). Once again, however, the expected spin magnitudes are largely unknown, subject to the same uncertainties mentioned above regarding natal black hole spins. One firm prediction of the dynamical scenario concerns the spins of second-generation binaries, whose components were themselves formed from previous mergers. Regardless of their component spins, black hole mergers generally yield remnants with a ∼ 0.7; thus the effective spin of two such second-generation binaries may be large (Fishbach et al. 2017; Gerosa & Berti 2017; Rodriguez et al. 2018, 2019; Doctor et al. 2020).


Many studies often have environmental classes simply divided into (relaxed) ‘clusters’ or ‘fields’. However, in a Λ cold dark matter (ΛCDM) Universe, most clusters are expected to be the result of group/smaller cluster mergers – some of which can be extremely violent. Little is known about the role of cluster and group mergers in galaxy formation and evolution, and whether they could be important in setting the environmental trends which have now been robustly measured and described. It is particularly important to understand if cluster mergers trigger star formation (e.g. Miller & Owen 2003; Ferrari et al. 2005; Owen et al. 2005; Hwang & Lee 2009; Wegner, Chu & Hwang 2015), if they quench it (e.g. Poggianti et al. 2004), or, alternatively, if they have no direct effect (e.g. Chung et al. 2010). Results from Umeda et al. (2004), studying a merging cluster at z ∼ 0.2 (Abell 521) found tentative evidence that merging clusters could perhaps trigger star formation. More recently, Stroe et al. (2014a) conducted a wide field Hα narrow-band survey over two merging clusters with a simple geometry, with the merger happening in the plane of the sky. Stroe et al. (2014a) find a strong boost in the normalization of the Hα luminosity function of the CIZA J2242.8+5301 (‘Sausage’) cluster, several times above the field and other clusters. The authors suggest that they may be witnessing star formation enhancement or triggered due to the passage of the shock wave seen in the radio and X-rays. Interestingly, Stroe et al. (2014a) do not find this effect on the other similar merging cluster studied (‘Toothbrush’), likely because it is a significantly older merger (about 1 Gyr older; cf. Stroe et al. 2014a, 2015), and thus displays only the final result (an excess of post-starburst galaxies instead of Hα emitters). The results are in very good agreement with simulations by Roediger et al. (2014) and recent observational results by Pranger et al. (2014).


The possible presence of dust in the habitable zones of nearby main-sequence stars is considered a major threat for the direct imaging and characterization of Earth-like extrasolar planets (exo-Earths) with future dedicated space-based telescopes. Several independent studies have addressed this issue and concluded that visible to mid-infrared direct detection of exo-Earths would be seriously hampered in the presence of dust disks 10 to 20Â times brighter than the solar zodiacal cloud assuming a smooth brightness distribution (e.g., Beichman et al. 2006; DefrÃ¨re et al. 2010; Roberge et al. 2012). The prevalence of exozodiacal dust at such a level in the terrestrial planet region of nearby planetary systems is currently poorly constrained and must be determined to design these future space-based instruments. So far, only the bright end of the exozodi luminosity function has been measured on a statistically meaningful sample of stars (Lawler et al. 2009; Kennedy & Wyatt 2013). Based on WISE observations and extrapolating over many orders of magnitude, Kennedy & Wyatt (2013) suggest that at least 10% of gigayear-old main-sequence stars may have sufficient exozodiacal dust to cause problems for future exo-Earth imaging missions. To determine the prevalence of exozodiacal dust at the faint end of the luminosity function, NASA has funded the Keck Interferometer Nuller (KIN) and the Large Binocular Telescope Interferometer (LBTI) to carry out surveys of nearby main-sequence stars. Science observations with the KIN started in 2008 and the results were reported recently (Millan-Gabet et al. 2011; Mennesson et al. 2014). One of their analyses focused on a sample of 20 solar-type stars with no far infrared excess previously detected (i.e., no outer dust reservoir). Assuming a log-normal luminosity distribution, they derived the median level of exozodiacal dust around such stars to be below 60Â times the solar value with high confidence (95%). Yet, the state-of-the-art exozodi sensitivity achieved per object by the KIN is approximately one order of magnitude larger than that required to prepare future exo-Earth imaging instruments.


Here, we invoke an extend accretion disc corona (ADC) model to explain the long-term time lags detected in NS-LMXBs, including the long-term time lags of GX 349+2 that we derive in this work. Analysing the dip and non-dip spectra of NS-LMXBs, Church & Bałucińska-Church (1993, 1995) proposed a Birmingham model which consists of a blackbody component interpreted as the emission from a point source, i.e. the NS, and a power-law component that might be resulted from the Comptonization of thermal emission in an ADC above the accretion disc. Through dip ingress time technique, Church & Bałucińska-Church (2004) measured the radius of the ADC and developed the Birmingham model into an extended ADC model. The measured radius of a thin, hot corona above the accretion disc varies in the range of ∼(2–70)× 104 km. Therefore, the corona is very extended and the disc is substantially covered by the corona. In the extended ADC model, almost all soft X-ray photons from the accretion disc are inversely Comptonized by the energetic electrons from the extended ADC, which produces the observed hard X-rays, while the observed soft X-rays are interpreted as the emission from the NS; the accretion disc is illuminated by the emission of the NS, leading to the production of the extended ADC above the disc. This model was successfully applied to Cyg-like Z sources (Church, Halai & Bałucińska-Church 2006; Jackson, Church & Bałucińska-Church 2009; Bałucińska-Church et al. 2010) and Sco-like Z sources (including GX 349+2) (Church et al. 2012), as well as atoll sources (Church et al. 2014), so it could be a universal model for NS-LMXBs. Since the extended ADC model is a unified model for NS-LMXBs, we try to interpret the long-term time lags in NS-LMXBs with the help of this model. The extended ADC and the NS are two independent emitting regions, which satisfies the request that the hard and soft X-rays for long-term time lags are emitted from two distant regions, as discussed above. In order to explain the long-term time lags detected in NS-LMXBs in terms of the extended ADC model, we introduce two time-scales. One is the Comptonization time-scale during which the disc seed photons are inversely Comptonized by the high-energy electrons in the extended ADC, and another is the viscous time-scale in the order of hundreds of seconds (Lei et al. 2008), during which the accreting matter flows from the disc on to the NS. The hard X-ray time lags will be produced if the Comptonization time-scale is less than the viscous time-scale, and, contrarily, the soft X-ray time lags will be observed if the Comptonization time-scale is larger than the viscous time-scale. It is noted that a minority of positively correlated short-term time lags (1 s) are listed in Tables 1 and 3, which are derived with the CCF method in our work. These short-term time lags cannot be explained by the models reviewed in section 4.1, because those models are used to interpret the short-term time lags produced in two adjacent energy intervals, while these short-term time lags obtained in this work are derived from two distant energy intervals, i.e. 2–5 kev and 16–30 keV energy intervals. In the frame of the extended ADC model, we propose that these short-term time lags will be observed under the circumstance that the Comptonization time-scale is comparable with the viscous time-scale.


The binary characteristics of early-type magnetic stars may provide crucial clues, allowing one to test alternative fossil field hypotheses. The non-magnetic chemically peculiar stars of Am (A-type stars with enhanced lines of Fe-peak elements) and HgMn (late-B stars identified by strong lines of Hg and/or Mn) types are frequently found in close binaries (Gerbaldi, Floquet & Hauck 1985; Ryabchikova 1998; Carquillat & Prieur 2007), including eclipsing systems (Nordstrom & Johansen 1994; Strassmeier et al. 2017; Takeda et al. 2019). In contrast, only about ten close (Porb  20 d) spectroscopic binaries containing at least one magnetic ApBp star are known (Landstreet et al. 2017). The overall incidence rate of magnetic upper main sequence stars in close binaries is less than 2 per cent (Alecian et al. 2015), although this fraction is significantly higher if one includes wide long-period systems (Mathys 2017). This low incidence of magnetic ApBp stars in close binaries is frequently considered as an argument in favour of the stellar merger origin of fossil fields (de Mink et al. 2014; Schneider et al. 2016). In this context, confirmation of magnetic ApBp stars in short-period binary systems gives support to alternative theories or, at least, demonstrates that early-type stars may acquire magnetic fields through different channels. In addition, detached close binary stars, particularly those showing eclipses, are valuable astrophysical laboratories that provide model-independent stellar parameters and allow one to study pairs of co-evolving stars formed in the same environment. Until recently, no early-type magnetic stars in eclipsing binaries were known. The first such system, HD 66051, was identified by Kochukhov et al. (2018). The second system, HD 62658 containing twin components of which only one is magnetic, was found by Shultz et al. (2019). Several other eclipsing binaries containing candidate ApBp stars were proposed (Hensberge et al. 2007; González, Hubrig & Castelli 2010; Skarka et al. 2019), but the magnetic nature of these stars has not been verified by direct detections of their fields using the Zeeman effect. In this paper, we put a spotlight on another candidate eclipsing magnetic Bp star, which received little attention prior to our work despite being significantly brighter than the confirmed magnetic eclipsing systems HD 62658 and HD 66051.


Below, we describe a simple emission model that can explain most of the key emission features of the pulsars presented in this paper. It draws significantly from the model proposed in Timokhin (2010) that explains nulling and mode changing in pulsars by shrinking and expanding the magnetosphere of the neutron star. The model assumes a dipole magnetic field line structure and a schematic diagram is shown in Fig. 10. The emission beam is assumed to be a ‘fan beam’ with multiple flux tubes along the open field lines (Karastergiou & Johnston 2007; Oswald, Karastergiou & Johnston 2019). We assume the fan beam to be “patchy” and partially filled meaning that only a portion of the tube is emitting radiation. The patchy beam scenario is one of the common beam models that has been used to explain the emission properties of pulsars (see Lyne & Manchester 1988; Manchester et al. 2010). As shown in Fig. 10, it is assumed that the emission patch is fixed within the polar cap region, i.e. the region consists of open magnetic field lines with respect to the boundary of the comoving magnetosphere (known as the light cylinder). When the magnetosphere shrinks the polar cap region expands accordingly with the dipole field lines and changes the orientation of the emission flux tube produced from the patchy region (see Fig. 10b). Due to these changes, our line of sight (LoS) encounters a different part of the flux tube, resulting in variation in the observed pulse profile shape, switching the pulsar from the normal to swoosh emission. It has been postulated that low-frequency radio emission is produced at a higher altitude compared to the high-frequency emission in the magnetosphere (i.e. – RFM, Cordes 1978; Oswald et al. 2019). Therefore, due to the curvature of the dipole magnetic field lines, the flux tube of the patchy beam at a lower frequency can be oriented in a direction that is out of our LoS (see Fig. 10b). This results in apparent nulls at low frequencies, which is consistent with the observations of PSR B0919+06 (see Fig. 1, and also Shaifullah et al. 2018).


In Fig. 11, we compared the WL-derived masses (MWL) with those derived from the dynamical analysis (Mdyn) and listed in Table 1. The dynamical masses were computed under the assumption of the singular isothermal model and velocity dispersions (Haines et al. 2018). We obtained the WL masses for the four massive clusters (A 3556, A 3558, A 3560, and A 3562) with the MCMC method. On the other hand, we only gave the upper limit of the masses for six out of seven low-mass clusters due to the large shape noise in the WL analysis. In the figure, we labelled the low-mass clusters as black crosses, AS 0726 as black box and the four massive clusters as cyan circles. For AS 0726, we only indicatively adopted the average mass for the low-mass clusters derived using the stacked shear profile. We notice that (i) the masses obtained from the dynamical and WL analysis are consistent within 1σ for all clusters except A 3554, A 3558, and AS 0726; (ii) the dynamical masses turn out to be systematically higher than the WL-derived ones. Previous studies showed that WL masses obtained from the tangential shear fitting were biased low up to 10 per cent with a scatter of ∼25 per cent (Oguri et al. 2005; Sereno & Umetsu 2011; Sereno & Ettori 2015). The main source of the bias are due to substructures and triaxiality. When a cluster whose major axis is perpendicular to the line of sight, i.e elongated in the sky-plane, a mass obtained with the spherical NFW profile is typically underestimated. The presence of substructures around clusters and uncorrelated large-scale structures along the line of sight also generate the biases for the estimation of the WL masses (Meneghetti et al. 2010; Becker & Kravtsov 2011; Giocoli et al. 2012, 2014). In addition, we point out that the dynamical mass of AS 0726 is actually an upper limit since the velocity distribution of member galaxies is strongly bimodal, and certainly not Gaussian (see fig. 14 in Haines et al. 2018). This system probably consists of two groups with velocity dispersions ∼300 km s−1 rather than one single system with σ ∼600 km s−1. This would reduce the mass estimate by a factor of 4. The complex structure of A 3558 and, in general, the dynamical activity in the SSC core (see Bardelli et al. 1998; Ettori et al. 2000; Finoguenov et al. 2004; Rossetti et al. 2007a) may explain the systematic differences between the two mass determinations quoted above, since the virial mass tends to overestimate the mass of unrelaxed clusters. Moreover, Rossetti et al. (2007b) studied A 3558 with an X-ray observation and showed the possibility of the presence of substructures along the line of sight, which interact with the cluster. Since such structures along the line of sight broaden velocity distribution, masses obtained from the dynamical analysis can be overestimated up to 100 per cent (Takizawa, Nagino & Matsushita 2010; Pratt et al. 2019). Fig. 14 in Haines et al. (2018) showed the broad velocity distribution of galaxies for the cluster. This indicates the presence of the substructures along the line of sight and the possibility for overestimating the dynamical mass. A 3554 also shows strong substructures in the velocity distribution diagram.


Silicon monoxide (SiO) is predicted to be the most abundant Si-bearing molecule in the entire 1–10 R* range in the atmospheres of M stars. In S-type atmospheres, the calculated abundance of SiO decreases by two orders of magnitude in the 1–5 R* but retains a very high abundance beyond, and the same occurs in C-rich atmospheres, although in this case, the abundance drop in the 1–5 R* is even more pronounced (see Fig. 2; see also Agúndez & Cernicharo 2006). Observations indicate that the abundance of SiO does not differ significantly between envelopes around M-, S-, and C-type stars, although in all them the SiO abundance decreases with increasing mass-loss rate (González Delgado et al. 2003; Schöier et al. 2006; Ramstedt et al. 2009; Massalkhi et al. 2019, 2020). This decline in the SiO abundance with increasing envelope density is not a consequence of chemical equilibrium (Massalkhi et al. 2019), but has been interpreted as evidence that SiO disappears from the gas phase at high densities to be incorporated into dust grains (González Delgado et al. 2003; Schöier et al. 2006; Ramstedt et al. 2009; Massalkhi et al. 2019, 2020). It therefore appears that the gradual abundance decline calculated for SiO in the 1–5 R* region from stellar type in the sense M → S → C does not have a direct consequence in the SiO abundance that is injected into the expanding wind. However, this behavior predicted by chemical equilibrium probably explains why SiO masers are observed in M-type stars but not toward carbon stars (e.g., Pardo et al. 2004). Except for these details, chemical equilibrium and observations agree in the fact that SiO is one of the most abundant carriers of silicon in the atmospheres of M-, S-, and C-type stars. Calculations and observations also agree for SiS in that it is an abundant molecule regardless of the C/O. However, observations indicate a differentiation between C- and O-rich envelopes, with SiS being on average one order of magnitude more abundant in carbon-rich sources (Schöier et al. 2007; Danilovich et al. 2018; Massalkhi et al. 2019, 2020). Moreover, in some oxygen-rich envelopes, the fractional abundance of SiS relative to H2 is as low as ~10−8, which is well below the predictions of chemical equilibrium (Danilovich et al. 2019; Massalkhi et al. 2020).


After the gravitational collapse and if the total mass of the individual cloud is approximately the mass of the Sun (2 × 1030 Kg; (van Dishoeck 2014), a new astrophysical system forms that is dominated gravitationally by a low-mass protostar known as a young stellar object (YSO). The protostar is in the center of the system and is surrounded by a Keplerian-rotating envelope of dust and gas, that is gravitationally connected and in which the angular momentum is conserved (Cassen & Moosman 1981). The distance to the star will define the surrounding energy, which is dominantly thermal and capable of heating the more distant dustgrain mantles. The processes progressively inject chemical constituents into the gas-phase (Ceccarelli et al. 2001)and determine what types of physical-chemical processes govern in every region of the disk. Circumstellar envelopes oflow-mass protostars (CELMP) are environments that are extraordinarily rich in organic molecules as H2CO or HNCO and in iCOMs such as CH3CN, CH3CHO, and C2 H5OH (Schöier et al. 2002), in addition to other O- and N-bearing complexes (Jørgensen et al. 2012). IRAS 16293-2422 (hereafter IRAS 16293) was chosen for this work as the quintessential example of such an object. IRAS 16293 is adequate for this role because formaldehyde has previously been detected with notable variations in abundances in three differentiated regions of its protodisk (Ceccarelli et al. 2001; Jørgensen et al. 2012; Jaber et al. 2014; van der Wiel et al. 2019). Additionally, glycolaldehyde (C2 H4O2), the simplest form of sugar and one of the first intermediates in the formose reaction (Larralde et al. 1995), has also been detected in this object for the first time in space in 2012 (Jørgensen et al. 2012). The more distant regions of the disk (>4000 AU) contain species such as CO, H2CO, CH3OH, or H2O (van Dishoeck et al. 1995) at low temperatures (10–20 K) where the gas is slightly warmer than the dust grains as they are tightly coupled to them. Energy from collisions among dust and gas is therefore considered to be the main heating mechanism in this region. Nevertheless, the thermal energy generated by those collisions is not enough to activate the formation of molecules with barriers at or below 20 K (0.0397 kcal mol−1). The gas column density of molecular hydrogen has been estimated to be N(H2) = 1.3 × 1023 cm−2 (Ward-Thompson et al. 1999), with a fractional abundance of formaldehyde – N(H2CO)/N(H2) – ~ 4 × 10−10 cm−3 in the gas-phase (Ceccarelli et al. 2001). When the temperature rises above ~ 20 K, CO starts to desorb from the ice grains and enters the gas-phase with an increase of ~ 103 cm−3 in detected densities with respectto H2 (Cassen & Moosman 1981; Aikawa et al. 2015). Formaldehyde starts to deplete from frozen grains at around ~ 40 K and it is fully desorbed at ~ 60 K (Ceccarelli et al. 2001). The additional H2CO mixes with the existing circumstellar mass of gas, which may justify why at ~ 700 AU from the core and at gas temperatures between 80 and 100 K ~ 50 kcal mol−1), the detected fractional abundances of formaldehyde reach N(H2CO)/N(H2) = ~ 4.0 × 10−9 cm−3 (Ceccarelli et al. 2001). This implies an H2 column density that is estimated to be N(H2) = ~ 5.0 × 1021 cm−2 (Bottinelli et al. 2014). The inner part of the envelope at ~ 150 AU in a region with temperatures of 100–150 K has a higher density of formaldehyde with an N(H2CO)/N(H2) of ~ 10 × 10−7 cm−3, as well as an increase in fractional abundances for H2O (which desorbs from iced mantles at ~80 K) (Ceccarelli et al. 2001). This region also produces new molecules principally due to the thermal energy emitted from the YSO (99.99 kcal mol−1). One example is trans-HONO. This chemical compound has recently been detected for first time in space and in this part of the disk (Coutens et al. 2019). Its proposed formation has inspired some reactions proposed in this work that may lead to H2CO. In this region, a column density for molecular hydrogen is considered like that in region II (N(H2) = ~ 5.0 × 1021 cm−2). The three regions of IRAS 16293 dictate the physical parameters for the presentation of our computations which are defined as follows:

Region I/d ~ 4000 AU, Tgas = 20 K, Pgas = 2.29 × 107 K cm−3;
Region II/d~ 700 AU, Tgas = 80 K, Pgas = 7.06 × 107 K cm−3;
Region III/d~150 AU, Tgas = 150 K, Pgas = 6.08 × 108 K cm−3.


RR Lyrae are old low-mass stars that, during the central helium-burning phase, show mainly radial pulsation while crossing the classical instability strip in the colour–magnitude diagram. From the observational point of view, they represent the most numerous class of pulsating stars in the Milky Way and, being associated with old stellar populations, are typically found in globular cluster and abundant in the Galactic halo and bulge. The investigation of RR Lyrae properties is motivated by their important role both as distance indicators and tracers of old stellar populations. In particular, evolving through the central helium-burning phase, they represent the low-mass, Population II counterparts of Classical Cepheids, as powerful standard candles and calibrators of secondary distance indicators. In particular, they can be safely adopted to infer distances to Galactic globular clusters (see e.g. Coppola et al. 2011; Braga et al. 2016, 2018, and references therein), the Galactic centre (see e.g. Contreras Ramos et al. 2018; Marconi & Minniti 2018; Griv, Gedalin & Jiang 2019), and Milky Way satellite galaxies (see e.g. Coppola et al. 2015; Martínez-Vázquez et al. 2019; Vivas et al. 2019, and references therein). Being associated with old stellar populations, they represent the basis of an alternative Population II distance scale (see e.g. Beaton et al. 2016, to the traditionally adopted Classical Cepheids), more suitable to calibrate secondary distance indicators that are not specifically associated with spiral galaxies (e.g. the globular cluster luminosity function, see Di Criscienzo et al. 2006, and references therein). The properties that make RR Lyrae standard candles are (i) the well-known relation connecting the absolute visual magnitude MV to the metal abundance [Fe/H] (see e.g. Sandage 1993; Caputo et al. 2000; Cacciari & Clementini 2003; Catelan, Pritzl & Smith 2004; Di Criscienzo, Marconi & Caputo 2004; Federici et al. 2012; Marconi 2012; Marconi et al. 2015, 2018; Muraveva et al. 2018, and references therein); (ii) the period–luminosity relation in the near-infrared (NIR) filters and in particular in the K 2.2 μm band (see e.g. Longmore et al. 1990; Bono et al. 2003; Dall’Ora et al. 2006; Coppola et al. 2011; Ripepi et al. 2012; Coppola et al. 2015; Marconi et al. 2015; Muraveva et al. 2015; Braga et al. 2018; Marconi et al. 2018, and references therein). In spite of the well-known advantage of using NIR filters (see e.g. Marconi 2012; Coppola et al. 2015, and references therein), in the last decades there has been a debate on the coefficient of the metallicity term of the KB and PL relation (see e.g. Bono et al. 2003; Sollima, Cacciari & Valenti 2006; Marconi et al. 2015, and references therein). On the other hand, it is interesting to note that many recent determinations (see e.g. Sesar et al. 2017; Muraveva et al. 2018) seem to converge towards the predicted coefficient by Marconi et al. (2015), with values in the range 0.16-0.18 mag dex−1. As for the optical bands, our recently developed theoretical scenario (Marconi et al. 2015) showed that, apart from the MV−[Fe/H] relation that is affected by a number of uncertainties (e.g. a possible non-linearity, the metallicity scale with the associated α elements enhancement and helium abundance variations, as well as evolutionary effects, see Caputo et al. 2000; Marconi et al. 2018, for a discussion), the metal-dependent Period–Wesenheit (PW) relations are predicted to be sound tools to infer individual distances. In particular, for the B-, V- band combination, Marconi et al. (2015) demonstrated that the inferred PW relation is independent of metallicity. In order to test this theoretical tool, we need to compare the predicted individual distances with independent reliable distance estimates, for example, the astrometric ones recently obtained by the Gaia satellite (Gaia Collaboration 2016). To this purpose, in this paper we transform the predicted light curves derived for RR Lyrae models with a wide range of chemical compositions (Marconi et al. 2015, 2018) into the Gaia bands, derive the first theoretical PW relations in these filters and apply them to Gaia Data Release 2 Data base (hereinafter Gaia DR2; Gaia Collaboration 2018; Clementini et al. 2019; Ripepi et al. 2019). The organization of the paper is detailed in the following. In Section 2, we summarize the adopted theoretical scenario, while in Section 3 we present the first theoretical light curves in the Gaia filters. From the inferred mean magnitudes and colours, the new theoretical PW relations are derived in Section 4 that also includes a discussion of the effects of variations in the input chemical abundances. In Section 5, the obtained relations are applied to Gaia Galactic RR Lyrae with available periods, parallaxes, and mean magnitudes to infer independent predictions on their individual parallaxes, to be compared with Gaia DR2 results. The conclusions close the paper.


In Sifón et al. (2013), we used the σ–M200 scaling relation of Evrard et al. (2008) to estimate dynamical masses. As discussed in Section 1, the scaling relation of Evrard et al. (2008) was calibrated from a suite of N-body simulations using DM particles to estimate velocity dispersions. However, the galaxies, from which velocity measurements are made in reality do not sample the same velocity distribution as the DM particles. They feel dynamical friction and some are tidally disrupted, which distorts their velocity distribution and biases their dispersion (e.g. Carlberg 1994; Colín et al. 2000). Recent high-resolution hydrodynamical simulations of ‘zoomed’ cosmological haloes have shown that there is a significant difference between the velocity distributions of DM particles and galaxies themselves; whether galaxies (i.e. overdensities of stars in hydrodynamical simulations) or DM subhaloes are used makes comparatively little difference (Munari et al. 2013). Results from state-of-the art numerical simulations depend on the exact definition of a galaxy and the member selection applied, but the current consensus is that galaxies are biased high (i.e. at a given mass the velocity dispersion of galaxies or subhaloes is larger than that of DM particles) by 5–10 per cent with respect to DM particles (Lau et al. 2010; Munari et al. 2013; Wu et al. 2013), translating into a positive 15–20 per cent bias in dynamical masses when using DM particles. This is illustrated in Fig. 5: DM particles are not significantly impacted by either dynamical friction or baryonic physics; therefore, the scaling relations for DM particles are essentially the same for all simulations. In contrast, DM subhaloes are affected by baryons in such a way that including baryonic feedback (most importantly feedback from active galactic nuclei – AGN, but also from cooling and star formation) makes their velocity dispersions much more similar to those of simulated galaxies. This means we can rely on our analysis of the previous section, based on DM subhaloes, to correct the velocity dispersions measured for ACT clusters, and then estimate dynamical masses using predictions obtained either from galaxies or subhaloes. The difference between the Saro et al. (2013) and Munari et al. (2013) galaxy scaling relations depends on the details of the semi-analytic and hydrodynamical implementations used in Saro et al. (2013) and Munari et al. (2013), respectively. The different cosmologies used in the Millenium simulation (in particular, σ8 = 0.9; Springel et al. 2005) by Saro et al. (2013) and the simulations by (Munari et al. 2013, σ8 = 0.8) may also play a role.

Munari et al. (2013)
Munari et al. (2013)