X-ray radiation observed from clusters of galaxies is emitted by hot, low-density and metal-enriched plasmas filling the space between galaxies (for a review, see Sarazin 1986). We focus attention on relaxed galaxy clusters that are not disturbed by ongoing mergers with other groups (or clusters) of galaxies. The measured X-ray surface brightness and derived plasma density profiles of such clusters are sharply centrally peaked and decrease with radius, while the derived plasma temperature profiles have minima at the centres (for a review, see e.g. Peterson  Fabian 2006, and references therein). These central low-temperature dense regions are coined as cool-cores (Molendi  Pizzolato 2001, formerly known as cooling flows) and galaxy clusters having cool cores are called cool-core clusters. If no external energy sources are present in cool cores, the radiative losses in clusters centres would lead to gas cooling well below X-ray temperatures in a fraction of the Hubble time. The absence of the low temperature emission lines in the spectra of cool-core clusters (e.g. Tamura etal. 2001) and the lack of intense star formation suggest that external energy sources for plasma heating do exist in the cool cores. A likely explanation is that the energy produced by the central active galactic nuclei (AGNs) balances radiative losses of the X-ray emitting plasma (e.g. Churazov etal. 2000; McNamara etal. 2000). The activity of the central AGNs manifests itself via the presence of X-ray cavities (or bubbles) inflated by relativistic outflows from the AGNs that are present in majority of cool-core clusters (for a review, see Fabian 2012; Bykov etal. 2015). These bubbles are typically bright in the radio band (e.g. Boehringer etal. 1993), implying that relativistic electrons (and, perhaps, positrons) are present inside them. The bubbles rise buoyantly in cluster atmospheres and pure mechanical interaction can provide efficient energy transfer from the bubble enthalpy to the gas (e.g. Churazov etal. 2000, 2001, 2002). These buoyancy arguments suggest that the energy supplied by the AGNs matches approximately the gas cooling losses. This conclusion has been confirmed by the analysis of many systems that differ in X-ray luminosity by several orders of magnitude (e.g. Brzan etal. 2004; Hlavacek-Larrondo etal. 2012).


Similar to Balachandran & Flanagan (2007), who also studied the detectability of mode resonances (see also Flanagan & Racine 2007), we assume that the frequency-domain GW signal has the form
(30)
\begin{equation}
\tilde{h}(f) = \mathcal {A} f^{-7/6} \text{e}^{\text{i}\Psi (f)},
\end{equation}
where the amplitude (Maggiore 2007)
(31)
\begin{equation}
\mathcal {A}=\left(\frac{5}{24\pi ^{4/3}}\right)^{1/2}\left(\frac{G\mathcal {M}}{c^3}\right)^{-5/6}\frac{\mathcal {K}}{d}.
\end{equation}
Here $\mathcal {M}=[(M M^{\prime })^{3}/(M+M^{\prime })]^{1/5}$ is the chirp mass, d is the distance to the source and $\mathcal {K}$ is the antenna response (for an optimally oriented source $\mathcal {K}=1$). Our model of the phase evolution Ψ(f) accounts for the zeroth-order post-Newtonian point-particle contribution and for the resonant tidal excitation of individual g-modes. We ignore higher order post-Newtonian terms, spin, the equilibrium tide and non-linear tidal effects (such as those considered in Essick, Vitale & Weinberg 2016), and assume that the signal shuts off at the GW frequency corresponding to the innermost stable circular orbit f = 2fISCO ≃ 1.6 × 103 Hz [2.8 M⊙/(M + M΄)] (Cutler & Flanagan 1994; Poisson & Will 1995). In Appendix B, we show that under these assumptions,
(32)
\begin{equation}
\Psi (f) = \Psi _{\rm pp}(f) - \sum _a\left(1-\frac{f}{f_a}\right) \delta \phi _a \Theta \left(f-f_a\right),
\end{equation}
where, by the stationary-phase approximation, the point-particle phase is (Cutler & Flanagan 1994)
(33)
\begin{equation}
\Psi _{\rm pp}(f) = 2\pi f t_{\rm c} -\phi _{\rm c}-\frac{\pi }{4} +\frac{3}{4}\left(\frac{8\pi G\mathcal {M}f}{c^3}\right)^{-5/3},
\end{equation}
δϕa is the phase shift due to the tidal resonance with a mode a with eigenfrequency fa, and Θ(f − fa) is the Heaviside step function. Here tc and ϕc are constants of integration that set a reference time and phase. The duration of each resonance is, in general, much shorter than the orbital decay time-scale due to radiation reaction (their ratio is ${\simeq } 0.1\times [(\mathcal {M}/1.2\,\mathrm{M}_{\odot })(f/500 \text{ Hz})]^{5/6}$; see Lai 1994; Balachandran & Flanagan 2007; Flanagan & Racine 2007). We therefore model the resonance as an instantaneous process, which should be a good approximation since nearly all the g-modes we consider have $f_a^{\rm (GR)} < 500\,{\rm Hz}$ (the one exception is the na = 1 hyperonic g-mode of the 1.6 M⊙ HS model which has $f_a^{\rm (GR)}=740\,{\rm Hz}$).


An LyC candidate at z = 3.212 in the GOODS-Southern field, named Ion2 (GDS-ID 033203.24-274518.8), was first identified by Vanzella et al. (2015). This candidate is a

 galaxy with SFR

 (sSFR

), EW(Lyα)

 Å, and EW(

)

 Å (de Barros et al. 2016). The compact star-forming region, showing strong [O iii]λλ4959,5007 emission lines (with a rest-frame equivalent width of 1500 Å) and a large Oxygen ratio

/

 (

 at 1σ, de Barros et al. 2016) measured with Keck/MOSFIRE, makes Ion2 the highest redshift “Green Pea” galaxy the currently known and, accordingly to the photoionization models (Jaskot & Oey 2013; Nakajima & Ouchi 2014), an ideal candidate LyC emitter. A plausible spectroscopic LyC detection was subsequently discussed in de Barros et al. (2016). However, the presence of a close companion not resolved with ground-based spectroscopy and imaging (02, see Figure 1) cast some doubts on the association of the observed flux with Ion2,and thus on the reliability of the LyC leakage. Thanks to the Lyβ and Lyγ detection in the Very Large Telescope (VLT)/VIMOS Ion2 spectrum (see Figure 6 panel (A), or de Barros et al. 2016, Figure 2), the spectroscopic redshift of component A is unambiguously z = 3.2. The 4 hr exposure is not enough to detect such absorption features with a signal-to-noise ratio (S/N) of

 (as observed here) for component B, which at these wavelengths is mag = 27.2 in the continuum.11

11
From ESO/Exposure Time Calculator, an S/N

 is expected at this magnitude, wavelength, and integration time. This would imply an S/N

 if Lyβ and Lyγ absorption lines arise from component B.
 Therefore these features arise from component A at z = 3.2. The [O iii] emission is also mainly associated with component A because the K-band magnitude is dominated by the strong [O iii] lines and the K-band spatial emission is more prominent from A. This is shown in Figure 2 where the VLT/HAWKI K-band image has been extracted from the HUGS survey (Fontana et al. 2014).


The observation of FRB 200428 suggests the FRBs are accompanied by XRBs. The energy ratio between radio burst and XRB is about

. If this value is valid for FRB 121102, the X-ray energy release in one period for FRB 121102 is about
17

The typical active timescale of magnetars is about 100 yr (Beloborodov & Li 2016). The total energy release of XRBs in the active phase is
18

where τ is the active timescale of the magnetar. For FRB 121102,

 is assumed. Compared to the rotational energy, the magnetic energy is the main reservoir responsible for powering FRBs. The magnetic energy of magnetar is
19

where

 G is the interior magnetic field strength, and R = 12 km is the magnetar radius. This value is smaller than the required XRB energy. The XRB energy is shared by the ultrarelativistic ejecta and the baryon shell with a subrelativistic velocity vw. It has been found that the kinetic energy of the baryon shell is comparable to the flare energy (Metzger et al. 2019; Margalit et al. 2020b). We assume that the baryon shell is subrelativistic with a typical velocity

. Based on the above assumptions, the baryonic mass ejected by magnetar in the active time can be derived as
20

This value is much larger than the typical mass (

) of a magnetar outer crust (Gudmundsson et al. 1983; Glendenning & Weber 1992). According to the structure of magnetars, the core is mainly composed of superfluid. The ejected baryon matter of the magnetar is mainly provided by crust. The baryonic mass estimated in this model is larger than the typical mass of a magnetar outer crust, indicating that the outer crust of the magnetar cannot eject enough baryonic matter required by the model. In Figure 2, we show the constraints on MB using Equation (20). If the radiation efficiency

 from FRB 200428 is used, the required baryonic mass MB is always larger than

 for the whole parameter ranges as shown in the left panel of Figure 2. If τ = 100 yr is fixed, the same result is shown in the right panel of Figure 2. The outer crust of magnetars is not sufficient to provide such a large baryonic mass, which challenges the synchrotron maser shock model. Interestingly, if the inner crust is included, the total mass of crust is about 0.01 M⊙ (Chamel & Haensel 2008), which is larger than the required baryonic mass. Theoretically, the physical mechanism and the rate of the baryonic mass ejection are both uncertain. More investigations are required. We also use the bursts of FRB 121102 observed by FAST to estimate the required baryonic mass (Li et al. 2019). The observation of FAST provides the information on the lowest-energy bursts of FRB 121102 so far. In a 56.5 hr observation, 1121 bursts were observed. The total energy of these bursts is 3.14

 erg.3

3
D. Li & P. Wang 2020, private communications.
 Therefore, the average energy release in one period is about

 erg. Using the same formulae as above, the baryon mass can be approximated as
21

which is similar to that derived from GBT observation.


The intrinsic (edge-on) ellipticity (ϵint) has been estimated using the observed V/σ and ϵ based on the theoretical prediction of intrinsic shape for varying inclination under the assumption of rotating oblate system (Binney 2005). Cappellari et al. (2007) has shown that intrinsic ellipticity correlates with the anisotropy parameter using early-type galaxies from the SAURON IFS survey (de Zeeuw et al. 2002). Cappellari et al. (2016) presented the relationship between the intrinsic and observed ellipticities:
(14)$$\begin{eqnarray*}
\epsilon = 1-\sqrt{1+\epsilon _{\rm int} (\epsilon _{\rm int}-2) \sin ^2 i}.
\end{eqnarray*}$$The observed V/σ with an inclination i can be corrected to the intrinsic (V/σ)int for an edge-on (i = 90°) view (Binney & Tremaine 1987):
(15)$$\begin{eqnarray*}
\left(\frac{V}{\sigma }\right) = \left(\frac{V}{\sigma }\right)_{\rm int} \frac{\sin i}{\sqrt{1-\delta \cos ^2 i}},
\end{eqnarray*}$$where δ is an approximation to the anisotropy parameter β such as δ ≈ β = 0.7ϵint. For each component, we generated a grid with varying ϵint and i. Then, we calculated corresponding ϵ and V/σ for each ϵint and i using equations (14) and (15). The ϵint has been derived from the grid which has the best matched ϵ and V/σ to the observation. Note that V/σ for the bulge and disk components are measured using Re, PA, and ϵ of each component in this section, whereas all kinematic measurements in the other sections except for this section and Fig. 13 used the structural parameters from galaxies. Measuring V/σ at the effective radius of each component gives more galaxies with kinematics for both components: 664 and 633 are available for measuring bulge and disk components (respectively), and 468 galaxies have kinematics available for both components. On the other hand, the above method is not valid for slowly rotating bulges (e.g. below the green line in Fig. 13) because they are not oblate rotators. In that instance, we adopt ϵint = ϵ as our best estimate of the intrinsic ellipticity (although it is strictly a lower limit).


This section discusses the results of planet–planet scattering in systems with two giant planets in four simulation sets (Table 1). The statistics use only simulations with one surviving planet after the other planet is ejected via planet–planet scattering (23.26% of all simulations; see Table 1), which is around 5500 simulations in each set. In two-planet systems, the instability boundary is very narrow. If both planets survive, its very likely they have none or very little interactions, so it is not very interesting to study them for our purpose. Simulations with planet–star (2.71% of all simulations) or planet–planet collisions (73.61% of all simulations; see Table 1) are excluded from the statistics, as obliquity evolution for the collision of fluid bodies is beyond the scope of this work (see Li & Lai 2020; Li et al. 2020). Simulations with energy error

∣

δ
E

E

∣

 greater than

(

10

−
4

)

 are also excluded (0.36% of all simulations), which is an adequate threshold for multiplanet systems (Barnes & Quinn 2004; Raymond et al. 2010). A negligible fraction of systems (0.03% of all simulations) that do not experience planet destruction in 108 yr are also excluded. Figures 5 and 6 show the distribution of the surviving planet’s final spin orientation and obliquity for simulation sets 1–3. The spin inclination is measured relative to the z-axis, which is aligned with the inner planet’s initial spin and orbit normal. The final spin and obliquity distribution is quite broad compared with the orbital inclination distribution (Figure 7). Overall, the final spin and obliquity distributions (Table 3) are broader than the inclination distributions, with 19.51% planets’ obliquity beyond 40°, and 8.77% of the surviving planets are on retrograde obliquity (i.e., the rotation is in reversed direction relative to the orbit). The inclination distribution only has 0.12% planets beyond 40°. The rate of highly oblique planets (>40°) are 17.86%, 31.98%, and 10.43% in sets 1, 2, and 3. There are 5.07% surviving planets on retrograde obliquity in the base set, 21.46% in set 2, and 3.21% in set 3. The distribution of obliquity for set 3 is more clustered at lower regions compared to the other sets. From the rate of high and retrograde obliquity in set 2 (the inner planet ends up at 0.19 au after the outer planet is ejected), we may expect a high rate of highly oblique planets close to the central star, and some oblique planets farther out (Figures 5 and 6). A moderate peak in Figure 6 for close-in planets in sets 1 and 2 between obliquities 170°–180° implies that Venus-like obliquity is possible. The simulation results imply that planets closer to the star receive more torque from it and become more evolved in obliquity.


Recently, Esteban et al. (2022) examined the abundances in the Galaxy as measured from a number of sources, including H ii  regions from Arellano-Córdova et al. (2020, 2021), with the goal of assessing the large abundance dispersion that De Cia et al. (2021) measure in Galactic neutral clouds. The direct abundances in the H ii  regions are calculated using T

e
[N ii] in the low-ionization zone and T

e
[O iii] in the high-ionization zone, only applying the Esteban et al. (2009) T

e
–T

e
 relation when one of these temperatures is missing. From the 42 Galactic regions, the dispersion about the O/H gradient is ∼0.07 dex, which is similar to the dispersion in metallicity measured from B-type stars, classical Cepheids, and young clusters in the Galaxy (Esteban et al. 2022), and significantly less than the dispersion measured in the neutral clouds. Méndez-Delgado et al. (2022a) corroborate the low dispersion in the Arellano-Córdova et al. (2020, 2021) H ii  regions when considering nonzero temperature fluctuations and updated H ii  region positions. In the spiral galaxy M101, Kennicutt et al. (2003) determine the abundance gradient from 20 H ii  regions, the majority of which have direct T

e
[O iii] and T

e
[S iii] measurements, and nine of which have direct T

e
[N ii]. Citing the tight correlation they observe between T

e
[O iii] and T

e
[S iii], Kennicutt et al. (2003) use an average of the direct T

e
[O iii] and the inferred T

e
[O iii] from T

e
[S iii] to obtain the high-ionization zone temperature, and they use the direct T

e
[N ii] in the low-ionization zone, when available. Although this sample consists of only 20 regions, there is a clear abundance gradient in M101 with very little dispersion about the gradient. The dispersion reported by CHAOS (0.097 dex; Rogers et al. 2021) is slightly larger than the gradient from Kennicutt et al. (2003) would indicate, but this could be due to the inclusion of H ii  regions with a single direct T

e
.


The newly discovered arcs and new spectroscopic redshifts have been incorporated into updated HFF+ versions of the Abell 2744 and MACSJ0416.1-2403 lensing models (Table 5); many of these have

 (see Figure 8 for a comparison of the arc redshift distributions adopted by the pre-HFF and new HFF+ lensing models; Cypriano et al. 2004; Okabe & Umetsu 2008; Zitrin et al. 2009; 2013; Okabe et al. 2010a, 2010b; Merten et al. 2011; Christensen et al. 2012; Mann & Ebeling 2012; Jauzac et al. 2014; Lam et al. 2014; Richard et al. 2014; Balestra et al. 2015; Diego et al. 2015; Grillo et al. 2015; Jauzac et al. 2015; Rodney et al. 2015; Wang et al. 2015; Kawamata et al. 2016). The incorporation of these new multiple image systems often results in a reduction in the statistical uncertainty in the galaxy magnifications for a given model. All of the public HFF lensing models provide a range of possible realizations from which the statistical uncertainty of a given model set may be calculated (typically 100 but no fewer than 30). We plot the cumulative distribution of the galaxy magnification uncertainties σ(model)/

, for the galaxies and photometric redshifts provided by the ASTRODEEP catalogs (Merlin et al. 2016; Castellano et al. 2016a) for Abell 2744 (Figure 9) and MACSJ0416.1-2403 (Figure 10). Generally, the statistical uncertainties are reduced for the models computed with the new HFF data sets, with more dramatic reductions for the methods that rely strongly upon the strong-lensing constraints. The parametric methods (CATS, Sharon, Zitrin, GLAFIC) report median statistical magnification errors of 0.2%–5%, while the non-parametric methods (Bradac Williams, Diego) report median statistical magnification errors of 2%–11% for the post-HFF calculations (green curves), versus 2%–22% and 2%–17% respectively for pre-HFF models (blue curves). (We note that the statistical errors for the MACSJ0416.1-2403 Bradac post-HFF models (Hoag et al. 2016) included additional uncertainties due to the photometric redshift uncertainties of the multiple images. These were not included in the pre-HFF Bradac model, and thus may explain why the post-HFF statistical errors are larger for this model.)


As an illustrative example we apply the general results obtained in the last section to the IGM shortly after the reionization onset at a redshift of about z = 10. To this end we calculate the effective growth rate of a Maxwell–tensor distributed electron–proton plasma with a kinetic temperature of

 for both particle species and an electron density of

 being traversed by a highly relativistic beam with a Lorentz factor of

. The corresponding fluctuations of the background plasma are dominated by a recently discovered damped, aperiodic mode

 with

 (Felten et al. 2013). Schlickeiser & Felten (2013) found that its quantitative description can be simplified by introducing the dimensionless wavenumber κ and the dimensionless frequency y defined by
79

80

where kc is the inverse of the electron skin depth
81

modified in such a way that it absorbs all remaining prefactors that occur in the dispersion relation:
82

Here,

 denotes the thermal velocity of the electrons. In terms of these variables the dispersion relation takes a simple form and is given by (Schlickeiser & Felten 2013)
83

with
84

For our present purposes it is more convenient to consider the frequency y as the independent variable rather than the wavenumber κ. This approach can easily be reversed by the approximation for the inverse dispersion relation derived by Schlickeiser & Felten (2013),
85

(This procedure was also carried out during the creation of Figures 1 and 3 shown below.) Our task is to compute the effective growth rate (78), which for aperiodic modes reads
86

Making use of definitions (79)–(80) and the dispersion relation (83) we can rewrite this as
87

where we introduced the abbreviations
88

89

90

Next, we will turn to the transversal dispersion function of the unperturbed thermal background plasma (Schlickeiser & Felten 2013)
91

Making use of (79)–(80) and (83) once more we can write the derivative as a function of y alone:
92

Plugging this expression into Equation (87) we arrive at the effective growth rate
93

where
94

Note that to zeroth order in ε this result reproduces the dispersion relation (83) of the unperturbed growth rate, and that it contains an additional first order correction term that accounts for the presence of the beam.


Except for WR 134, the detected frequencies are high, from 3 d−1 up to 14 d−1. This cannot be easily reconciled with an orbital period (the companion would travel inside the WR star), nor a rotation rate (it would be above break-up velocity). Therefore, the most probable culprits are pulsations. WR 66 very probably possesses a close neighbour (see Section 2.1), which could cast doubt on the identification of the pulsating star, but that is not the case for the other stars (including WR 123: see Lefèvre et al. 2005). Pulsations here are thus considered to originate from the WR star itself, and such a possibility has actually been considered before. Pushed by the observational considerations of Vreux (1985), the first pulsation models for WRs were elaborated more than 30 years ago, with predicted periods of about one hour (Maeder 1985; Scuflaire & Noels 1986). Subsequent work by Glatzel et al. (1999) focused on strange modes in small and hot helium stars (often taken as behaving similarly to WRs) and the predicted pulsations had periods of several minutes with amplitudes of several mmag. After the discovery of a 2.45 d−1 signal in WR 123 (Lefèvre et al. 2005), i.e. at a much lower frequency than expected, the models were revisited. Strange modes were extended to stars with larger radii (Dorfi, Gautschy & Saio 2006; Glatzel 2008), while predictions of g-modes excited by the κ mechanism were made for WRs by Townsend & MacDonald (2006). In both cases, the predicted periods could be made compatible with the 10-h signal observed in WR 123. The signals we observe have frequencies in the same range, even for the 2-min cadence data, which allows us to probe very high frequencies, but no coherent signal is detected above 14 d−1: WRs thus seem to pulsate only at moderately high frequencies. Furthermore, Townsend & MacDonald (2006) expected pulsations with frequencies that are higher in early WN (2–8 d−1) than in late WN (1–2 d−1). For our pulsators, only WR 7 has an early WN type and its frequency values are not particularly higher: its main signals are in the same frequency range as those of the late-type WR 66. Moreover, the highest frequencies are all found in late WN stars. However, Townsend & MacDonald (2006) models were made for ‘general/typical’ stars, not specific ones. In addition, since the WR light mostly comes from inside the wind, the filtering effect of the wind on a signal from the underlying hydrostatic surface needs to be assessed in detail. New, dedicated models will be needed for a more in-depth comparison between observations and predictions.


To obtain reaction rates in a wide temperature range

T

9

=
0.001

–10 (where T9 is in units of 109 K) for BBN calculation inputs, continuous excitation functions over

10

−
8

 to a few MeV are needed for numerical integration. Therefore we performed an R-matrix analysis to compile both the present and the previous data of these three reaction channels by using AZURE2 (Azuma et al. 2010) including all the relevant partial widths

Γ

n

,

Γ

p
0

,

Γ

p
1

, and

Γ

α

. The 7Be

(
n
,

p

0

)

 data set used for the R-matrix analysis is based on the same selection as used in Damone et al. (2018), Damone (2018; part of Dam18 + Sek76 as displayed in Figure 2), and the present THM data, Bor63, and Pop76 are added. For the 7Be

(
n
,

p

1

)

 and 7Be

(
n
,
α
)

 channels, all the labeled data shown in Figure 2 (except for p1 Tom19 and

γ
α

 Bar16) were used without any data cutoff. Appendix B shows the adopted data sets more explicitly. We included nine known levels below the 8Be excitation energy of

E

x

=
24

MeV

 (Tilley et al. 2004) and an additional nonresonant background pole, with a common channel radius of 5 fm for each channel in the same manner as Adahchour & Descouvemont (2003). The most significant channel 7Be

(
n
,

p

0

)

7

Li is dominated by four of these levels as described in detail by a former R-matrix study (Adahchour & Descouvemont 2003; Descouvemont et al. 2004); the 2− resonance near the neutron threshold (labeled as level I) principally dominates the cross section up to the BBN energies, the two 3+ states at

E

x

=
19.24

MeV

 (level II) and

E

x

=
21.5

MeV

 well characterize the corresponding single peaks, and a 2+ background pole accounts for the enhancement at high energies. The most important resonances to expand analysis to the 7Be

(
n
,

p

1

)

7

Li* and 7Be

(
n
,
α
)

4

He channels are the 1− state at

E

x

=
19.4

MeV

 (level III) and the 2+ state at

E

x

=
20.1

MeV

 (level IV), respectively; the former expresses the 7Be

(
n
,

p

1

)

 resonance behavior peaked at

E

c
.
m
.

∼
0.5

MeV

 tailing down to the thermal neutron energy by the

1

/

v

 law, and the latter forms the first peak in the 7Be

(
n
,
α
)

 spectrum around

E

c
.
m
.

∼
1

MeV

. Despite its importance in the

(
n
,
α
)

 channel, level IV is much less significant in the total cross section especially at lower energies due to its p-wave nature. Therefore we imposed some restrictions on level IV, which made the analysis much simpler; fixing

Γ

p
0

,

Γ

p
1

, and

Γ

α

 at a known ratio

Γ

α

/

Γ

p

∼
4.5

 (Tilley et al. 2004), and freeing

Γ

n

 and resonance energy (refitted to be 19.87 MeV) not to significantly exceed the known total width

Γ
=
880

keV

 (Tilley et al. 2004). The other four higher-lying levels (0+ at

E

x

=
20.2

MeV

, 2+ at

E

x

=
22.24

MeV

, 1− at

E

x

=
22

 MeV, and 2− at

E

x

=
24

 MeV) play rather supplementary roles mainly for the higher-energy behavior. The 4+ and 4− higher-spin states (at

E

x

=
19.86

MeV

 and 20.9 MeV, respectively) were not included due to their limited influences. We do not introduce the γ-emission channels to fit the Bar16 plots (representing the 7Be

n
,
γ
α

4

He reaction channel Barbagallo et al. 2016), which appears significant only below the BBN energies. See Appendix C for more details on the present R-matrix analysis.


We first consider the case in which the star is surrounded by a gaseous atmosphere. The star surface is divided in six angular patches in magnetic colatitude, centred at the values θ = {0°, 10°, 30°, 50°, 70°, 89°}. By using the magnetic and temperature profiles previously described we compute, for each θ, the local magnetic field strength, B, the angle θB between the magnetic field and the normal to the surface, and hence the temperature, T. We then compute a set of atmospheric models corresponding to the six θ angles. Since the models are computed using different integration grids in the photon phase space (because the choice of the photon trajectories along which the radiative transfer is solved needs to be optimized to ensure fast convergence at the different values of magnetic field strength and inclination, see Lloyd 2003), we reinterpolate all model outputs over a common grid. This results in three 4D arrays for the emergent intensity $I^i_\nu (\boldsymbol k,\theta ) \equiv I^i(E,\mu _k,\phi _k,\theta )$ (i = 1, …, 3) which contain the total intensity and the intensities for the ordinary and extraordinary modes, respectively. In order to take into account the emission from the southern magnetic hemisphere of the star, we use the previous 4D arrays with the substitutions θB → π − θB and ϕk → π − ϕk, which is justified by the symmetry properties of the opacities. By using the ray tracing method described in Section 2, we can then compute light curves, phase resolved spectra and polarization fractions for each choice of the angles χ and ξ. As an example, Fig. 3 shows the X-ray light curve (0.12–0.39 keV) and the phase-dependent polarization degree in the X-ray and optical7 (B-filter) bands, for χ = 90° and ξ = 15°. For this particular viewing geometry, the X-ray pulsed fraction is 1 per cent, in agreement with the observed data and, as illustrated in the figure, the polarization degree is expected to be substantial and constant in phase.


However, this approximation holds only for luminosities â² 0.3LEdd. At higher accretion rates, the hardening factor Îº increases with luminosity and can be as high as â2.5â3 for some Galactic BHs (e.g., GRO J1655â40 at outburst peak, and GRS 1915+105) and for ULXs in the slim-disk state (Watarai & Mineshige 2003; Kawaguchi 2003; Shrader & Titarchuk 2003; Isobe et al. 2012). This is why at near-Eddington luminosities, the fitted color temperature Tin increases faster than the standard  high/soft state relation (âanomalous regimeâ; Kubota & Makishima 2004; Abe et al. 2005). Given the luminosities of M83 ULX-1 in the three XMM-Newton epochs, and the fact that a disk model with p â 0.6 provides the best fit to the data, we argue that the source was likely to be in the anomalous regime (slightly below Eddington) or in the slim disk regime (slightly above Eddington), and, therefore, we take a hardening factor Îº â 3 for BH mass estimates. Following Vierdayanti et al. (2008), we also take Î¾ â 0.353, which takes into account the transonic flow in the pseudo-Newtonian potential. Taking an average normalization constant Ndisk = (2.5 Â± 0.5) Ã 10â3 from our diskpbb fit (Table 2), we obtain a âtrueâ inner disk radius Rin â (73 Â± 15)(cosâÎ¸)â1/2âkm. This corresponds to an âapparentâ BH mass
2Finally, we need to take into account that the inner radius of a slim disk extends slightly inside the innermost stable circular orbit, so that the true mass is MBH â 1.2MX (Vierdayanti et al. 2008). This gives our final best estimate of the BH mass as
3In principle, the estimated mass can be as high as â60âM in the (implausible) extreme Kerr scenario; however, the fact that the system appears to be in the anomalous regime (upper end of the high/soft state) or in the slim disk state for a moderate bolometric luminosity â2 Ã 1039âergâsâ1 (and possibly even lower, considering the 2014 January spectrum) suggests that the BH mass is closer to â¼10â20âM. The same argument holds if we interpret the X-ray spectra as Comptonized emission from a warm, optically thick corona, which is typical of sources at LX â 1â3LEdd (Gladstone et al. 2009).


The ultraviolet (UV) upturn or excess is an unexpected rise in flux in the spectral energy distributions (SEDs) of early-type galaxies shortwards of 2500 Å. It appears to be a nearly ubiquitous property of spheroids and bulge-dominated galaxies (see e.g. Yi 2008, 2010 for a recent review), although their generally old, metal-rich, and quiescent stellar populations (e.g. Thomas et al. 2005, 2010) should contain no sources capable of providing significant flux below the 4000 Å break. While many candidates have been proposed, the source population is generally agreed to consist of hot horizontal branch (HB) stars (Greggio & Renzini 1990; Brown et al. 1998a) and it now appears most likely that this population is helium-rich and formed in situ at high redshift (Ali et al. 2018a,b,c). While such stars are now known to exist in globular clusters in our Galaxy (e.g. see Norris 2004; Piotto et al. 2005, 2007) and likely elsewhere (Kaviraj et al. 2007b; Mieske et al. 2008; Peacock et al. 2017), the origin of such high helium abundances is unclear.1 It has been suggested that the effect may depend on the environment: stratification of helium in the centres of clusters might create populations of galaxies with high helium abundances (Peng & Nagai 2009). On the other hand, it is also possible that the helium enrichment depends on the details of early star formation, as in globular clusters. Differences in the originating mechanisms or time-scales may be reflected in different evolutionary histories for galaxies in clusters and the field. Atlee, Assef & Kochanek (2009) find a slow decrease in the strength of the upturn for a sample of bright field elliptical galaxies at 0  z  0.65 compared to the nearly constant colour for very bright ellipticals in the works of Brown et al. (1998b, 2000, 2003) and brightest cluster galaxies (Loubser & Sánchez-Blázquez 2011; Boissier et al. 2018). In our work on much larger samples of cluster early types with luminosities down to L*, we also see no evolution at z  0.55 (Ali et al. 2018a,b,c) but then detect a rapid reddening in the UV colour at z = 0.7; this may be consistent with the last data point in Atlee et al. (2009), despite the large errors and small number statistics. Le Cras et al. (2016) also find evidence for evolution in the UV upturn at z > 0.6, albeit from stacked spectra and using spectroscopic indices sensitive to the HB population, for a sample of very luminous BOSS galaxies (as opposed to the UV photometry used by other studies).


It may be appropriate to single out at this point the recent and interesting works by Lin & Ishak (2017a,b) in which the authors run a so-called (dis)cordance test based on using a proposed index of inconsistency (IOI) tailored at finding possible inconsistencies/tensions between two or more data sets in a systematic and efficient way. For instance, it is well known that there is a persistent discrepancy between the Planck CMB measurements of H0 and the local measurements based on distance ladder (Riess et al. 2016, 2018b). At the same time, if one compares what is inferred from Planck 2015 best-fitting values, the LSS/RSD measurements generally assign smaller power to the LSS data parametrized in terms of the weighted linear growth rate f(z)σ8(z). This feature is of course nothing but the σ8-tension we have been addressing in this paper. It is therefore natural to run the IOI test for the different kinds of H0 measurements and also to study the consistency between the H0 and the growth data. For example, upon comparing the constraints on H0 from different methods, Lin & Ishak (2017b) observe a decrease of the IOI when the local H0 measurement is removed. From this fact they conclude that the local measurement of H0 is an outlier compared to the others, what would favour a systematics-based explanation. This situation is compatible with the observed improvement in the statistical quality of the fitting analysis by Solà, Gómez-Valent & de Cruz Pérez (2017b,c) when the local H0 measurement is removed from the overall fit of the data using the RVM and the ΛCDM. In this respect, let us mention that a recent model-independent analysis of data on cosmic chronometers and an updated compilation of SNIa seem to favour the lower range of H0 (Gómez-Valent & Amendola 2018), what would be more along the line of the results found here, which favour a theoretical interpretation of the observed σ8 and H0 tensions in terms of vacuum dynamics and in general of DDE (cf. Fig. 10).


These results show that the shape of the mass function and the position of the mass peak of massive clusters have little evolution over the course of the galaxy collision of more than 1 Gyr. We note that the evolution of SCs is subjected to several destructive processes (e.g., Gnedin et al. 1999a; Fall & Zhang 2001). For low-mass clusters (

), the destruction is mainly dominated by two-body relaxation processes, in which the mass of a cluster linearly decreases with time until it is destroyed. For more massive clusters, the evolution is primarily influenced by stellar evolution at early times (

 Myr) and by gravitational shocks at later times. These effects are included in our hydrodynamic simulations, but the mass resolution is not high enough to resolve the processes realistically, since the two-body relaxation and stellar evolution depend on individual stars. However, the cluster disruption timescale due to two-body relaxation is proportional to the cluster mass, trlx ∼ 1.7 Gyr × (Mclus/104

)0.62 × (T/104 Gyr−2)−0.5, where Mclus is the cluster mass and T is the tidal strength around the clusters (Kruijssen et al. 2012). Using the minimum cluster mass of

 in our simulation and a typical range of tidal strength in the nuclear region (since most of these clusters are concentrated around galaxy nuclei) of

 (Renaud 2010), we find that the range of the disruption timescale is

 Gyr. For more massive SCs, the timescale is even longer, beyond our run time of 1 Gyr. Therefore, the two-body relaxation may not have a major disruptive effect on these SCs. Furthermore, the mass loss timescale due to gravitational shocks (tsh) depends strongly on the cluster density,

 (Kruijssen et al. 2012). The majority of our clusters have a density range of

, as shown in Figure 9, which suggests

 Gyr, much longer than the Hubble time. So gravitational shocks may not have a significant impact on the clusters in our simulation. In addition, as demonstrated by Renaud & Gieles (2013), SCs formed in galaxy mergers are also affected by the intense tidal field of the galaxies, more so for clusters in the merger remnant than for the ejected ones. We note, however, the clusters in their simulations have masses

, and mass loss decreases as the cluster mass increases. For example, for a cluster to increase its mass by a factor of 2, from

 to

, its survival rate (fraction of initial mass survived) after 1 Gyr increases from 0.6 to 0.7. Extrapolating this trend to our clusters, which are ∼10 times more massive than those in Renaud & Gieles (2013), we argue that destruction from tidal fields likely has negligible effects on the clusters we consider here.


The second significant source of energy dissipation is stream–stream collisions near apocenter from GR apsidal precssion. As in Dai, McKinney & Miller (2015) and Bonnerot et al. (2017), we calculate the ingoing collision stream velocities by finding the intersection point of two elliptical orbits, shifted by an instantaneous pericenter shift δϖ from GR apsidal precession. Since an eccentric orbit undergoes apsidal precession at a rate (assuming af ≫ Rg)
(24)$$\begin{eqnarray*}
\left. \frac{\partial \varpi }{\partial t} \right|_{\rm GR} = \frac{3 R_{\rm g}n}{a (1-e^2)},
\end{eqnarray*}$$we have after one orbital period
(25)$$\begin{eqnarray*}
\delta \varpi \simeq \frac{3 \pi R_{\rm g}}{r_{\rm p}} = 11.5^\circ \, \beta \frac{\bar{M}_\bullet ^{2/3} \bar{M}_\star ^{1/3}}{\bar{R}_\star }.
\end{eqnarray*}$$With δϖ, one can calculate the relative stream velocities during the collision, leading to a loss of energy of (Dai et al. 2015; Bonnerot et al. 2017)
(26)$$\begin{eqnarray*}
\delta E_{\rm ss} &=& \frac{e_{\rm f}^2 G M_\bullet }{2a_{\rm f}(1-e_{\rm f}^2)} \sin ^2 \left(\frac{\delta \varpi }{2} \right) \simeq \frac{G M_\bullet }{4 r_{\rm p}} \sin ^2 \left(\frac{\delta \varpi }{2} \right),
\end{eqnarray*}$$hence
(27)$$\begin{eqnarray*}
\mathcal {V}_{\rm ss} = \frac{\delta E_{\rm ss}}{E_{\rm f}} = \frac{a_{\rm f}}{2 r_{\rm p}} \sin ^2 \left(\frac{\delta \varpi }{2} \right).
\end{eqnarray*}$$When δϖ ≪ 1, equation (27) reduces to
(28)$$\begin{eqnarray*}
\mathcal {V}_{\rm ss} \simeq \frac{9 \pi ^2}{16} \frac{R_{\rm g}^2 a_{\rm f}}{r_{\rm p}^3} = 0.125 \, \beta ^3 \frac{\bar{M}_\bullet ^{5/3} \bar{M}_\star ^{1/3}}{\bar{R}_\star ^2} \left(\frac{t}{t_{\rm f0}} \right)^{2/3}.
\end{eqnarray*}$$Different sources of energy dissipation need to be compared with the total loss of energy required to completely circularize the disc. If the debris circularizes with constant orbital angular momentum, the final semimajor axis is given by $a_{\rm circ} = (1 - e_{\rm f}^2)a_{\rm f}\simeq 2 r_{\rm p}$. Hence the orbital energy of a completely circularized debris stream is given by
(29)$$\begin{eqnarray*}
E_{\rm circ} = - \frac{G M_\bullet }{4 r_{\rm p}},
\end{eqnarray*}$$so the fractional binding energy which must be lost to circularize the stream is
(30)$$\begin{eqnarray*}
\mathcal {V}_{\rm circ} = \frac{E_{\rm circ}}{E_{\rm f}} = \frac{R_{\rm t}^2}{4 r_{\rm p}R_\star } \left(\frac{t}{t_{\rm f0}} \right)^{2/3} = \frac{25}{\beta } \frac{\bar{M}_\bullet ^{1/3}}{\bar{M}_\star ^{1/3}} \left(\frac{t}{t_{\rm f0}} \right)^{2/3}.
\end{eqnarray*}$$We see that $\mathcal {V}_{\rm circ} \gg 1$ for typical parameters.


While a non-negligible fraction of stars have disc-like kinematics, the majority of the UIP stars, as well as the VMP and EMP stars, have halo-like kinematics. This does not necessarily mean that they are all accreted, since a fraction of the halo can be made of stars formerly in the disc, but later kinematically heated to halo kinematics by one or several satellite accretions. This has indeed proven to be the dominant in-situ mode of formation of the Galactic halo for stars at higher metallicities, and at few kpc from the Sun (see Di Matteo et al. 2019), as we discuss more extensively in the next section. It is, however, interesting to note that, compared to stars at higher metallicities, such as stars in the Nissen & Schuster sample and stars in the Gaia DR2-APOGEE sample (middle and right columns in Fig. 5), in the Toomre diagram, the UIP and LP samples seem to lack stars with null angular momentum, meaning along the Lz = 0 line. At values of

(

V
R

2

+

V
Z

2

)

≳
200

$ \sqrt{({V_R}^2+{V_Z}^2)} \gtrsim 200 $

 km s−1, stars with [Fe/H]   −4 have rather prograde or retrograde motions, but none seem to lie along the sequence of accreted halo stars discovered by Nissen & Schuster (2010), and later confirmed in Gaia DR1 and DR2 data by Belokurov et al. (2018) (Gaia Sausage), Haywood et al. (2018), Helmi et al. (2018) (Gaia Enceladus). The Gaia Sausage, which is this group of halo stars with very radial orbits, and null VΦ, seems to indeed disappear at [Fe/H]   −2 in this plane. We emphasise that this apparent difference between the kinematics of VMP and EMP stars, on the one side, and stars with [Fe/H] >  −2, on the other side, is simply the consequence of these stars probing different regions and distances from the Galactic centre. Indeed, when one compares the kinematics of these different samples of stars in the quasi-integral-of-motion space Lz − Lperp plane (see Fig. 5, second row), rather than in the Toomre diagram, the kinematic properties of these samples are the same, over the whole [Fe/H] interval. Before moving further, we need, however, to emphasise two points of the comparison with samples at higher metallicity. Firstly, when compared to the Nissen & Schuster (2010) sample, which, we remind the reader, is a kinematically selected sample of thick disc and halo stars, the VMP and EMP stars in the LP First stars sample show an excess of stars at retrograde motions (positive Lz, i.e. Lz >  5) and high values of Lperp (Lperp >  15). None of the Nissen & Schuster (2010) stars occupy this region of the Lz − Lperp diagram, and we suggest this is a consequence of the “local” character of stars in the Nissen & Schuster (2010) study, which are all limited to a few hundred parsecs from the Sun. Indeed, when VMP and EMP stars are compared to stars in the Gaia DR2-APOGEE sample, one can see that stars with Lz and Lperp as extreme as Lz >  5 and Lperp >  15 are found also in the latter. Secondly, because in the comparison shown in this figure we used all stars in the Gaia DR2-APOGEE sample, not restricting ourself to stars from the kinematically defined thick disc and halo, the reader will not be surprised to find that the majority of stars in Gaia DR2-APOGEE sample are stars with cold (i.e. thin) disc-like kinematics, their distribution peaking at Lz ∼ 20 and Lperp ≤ 5. At this stage, what is important to retain is that the region occupied by all these samples, independently of their [Fe/H] ratio, is the same: the relative fraction of stars in one or in another region of the space under analysis can change from one sample to another, but not their overall distribution.


In this Letter we propose to address these apparent difficulties and account for many of the observed differences between the properties of planets in cool and hot stars by postulating that, in addition to the tidal interaction with existing close-in planets, a large fraction of the stellar hosts—both cool and hot—ingest a hot Jupiter early on in their evolution. This proposal is motivated by the expectation that a large fraction (up to 80% according to Trilling et al. 2002) of solar-type stars possess giant planets during their pre-MS phase, and that a large fraction of the giant planets that form in a protoplanetary disk on scales

 migrate close to their host star before the disk is dispersed (e.g., Ida & Lin 2004). Numerical simulations incorporating an N-body code and a 1D α-viscosity disk model (Thommes et al. 2008) demonstrated that this behavior can be expected for disks with

 that are sufficiently massive. The inward planet migration is likely stopped by the strong (

 kG) protostellar magnetic field that truncates the disk at a radius (rin) of a few stellar radii (e.g., Lin et al. 1996). Gravitational interaction with the disk causes a planet reaching rin to penetrate into the magnetospheric cavity and, if it is massive enough, to undergo eccentricity excitation that can rapidly lead to a collision with the star (e.g., Rice et al. 2008). It was, however, inferred that if Mp is sufficiently small (

), the planet would remain stranded at a distance where its orbital period is ∼0.5 of that at rin until well after the gas disk disappears (on a timescale of ∼ 106–107 years). In our proposed scenario, the primordial disk orientations span a broad angular range that is reflected in the orbital orientations of the stranded planets. When the latter are ingested by tidal interaction with the host star (on a timescale

 Gyr), the absorbed angular momentum is sufficient to align a solar-mass star in that general direction, but not an MS star with

 K. This is because cool stars have significantly lower angular momenta at the time of ingestion than hot stars as a result of a more efficient magnetic braking process and of a lower moment of inertia. Given the proximity of the stranded HJs (SHJs) to their host stars (

days), they can be expected to have been ingested by the time their parent planetary systems are observed; however, giant planets farther out can continue to interact with their host stars and potentially affect their measured obliquities. In our simplified formulation, we model the SHJs using as parameters their characteristic mass MSHJ and the fraction p of systems that initially harbored an SHJ. By comparing the predictions of this model with the observational data, we infer

 and

.


Low-frequency observation of radio sky is also essential for the study of the astrophysics at play in various evolutionary stages of different Galactic and extragalactic sources. Radio emission at low frequencies, together with their redshift information, can be used to infer several astrophysical properties associated with the sources. In general, the source distribution is assumed Poissonian (with the possibility of clustering following single power law) (Ali et al. 2008; Jelić et al. 2010; Trott et al. 2016). Source counts are also modelled via a single power-law distribution (Hurley-Walker et al. 2016; Intema et al. 2017; Franzen et al. 2019). However, several recent studies have shown deviation from the single power-law model (Williams et al. 2016; Prandoni et al. 2018; Hale et al. 2019). Thus more detailed studies both in wide field as well as deep fields are required for generating a fiducial model of sources at low frequencies. The differential source counts at these are also useful for constraining the nature of sources. At frequencies in the GHz range and upwards, the source properties are well characterized. However, at lower frequencies, there is a lack of consensus for the same. It has been seen in previous studies that at these frequencies, AGNs dominate in the flux density scales down to few 100 μJy while SFGs and radio quite AGNs become dominant below ∼100 μJy (Simpson et al. 2006; Mignano et al. 2008; Seymour et al. 2008; Smolčić et al. 2008; Padovani et al. 2009, 2011, 2015; Prandoni et al. 2018; Hale et al. 2019). This is inferred from the flattening of the source counts below 1 mJy. However, such studies are very few on account of the limitations in reaching the required SNR. Thus empirical constraints at frequencies ≲1.4 GHz are limited. Therefore the study of the low-frequency radio sky is vital for fiducial modelling of foregrounds for 21-cm cosmology as well as constraining the physics and the astrophysics of the sources.


Applying the P&T profiles defined in Sect. 2 for the three regions of IRAS 16293 our computations imply that 56 mechanisms possess valid reaction thermodynamic function values to proceed with the formation of H2CO. Twenty-six of these can produce H2CO in region I, 32 in region II, and 33 in region III (please see Table 7). These are the amended reactions with detected reagents in every region of the disk. The presence of methane (CH4) and neutral atomic carbon (CI) in this astronomical object, in addition to trans-HONO in region III (detected at ~ 150 AU from the core – Coutens et al. 2019), confirms the hypothetical feasibility of these mechanisms. Moreover, because nitrosyl hydride (HNO) has not been detected in IRAS 16293, R84 cannot be included in any circumstellar envelope region, as it was in DCDMCs. In Fig. 3 we show the abundance of reactants and formaldehyde itself already detected in the three regions of the disk. They have been calculated for the methylene (⋅ 3CH2) (Vasyunin & Herbst 2013), methyl radical (⋅CH3) (Sakai et al. 2012; Woon 2002), and methane (CH4) (Sakai et al. 2012) ground electronic state abundances rather than detected. The reason is that they are planar (or linear) symmetric-top molecules and therefore have a small permanent electric dipole moment, which makes them undetectable through terrestrial radio telescopes by rotational spectroscopy. On the other side of the spectrum, neither high-resolution infrared absorption nor ultraviolet-visible spectroscopy telescopes can identify them in the gas-phase because they lack of a powerful enough source of electromagnetic radiation. The abundance values for the triplet ground-state of methylene (⋅CH2(X3B1)) have therefore been estimated by using Monte Carlo algorithms (Vasyunin & Herbst 2013) which in turn are based on known abundances of derived species. The presence of the methyl radical (⋅CH3) in region I just before the CO depletion zone from icy grains should be negligible because ⋅CH3 is thought tobe mainly produced in this region by CH4 photodissociation (Sakai et al. 2012) and CH3OH (Woon 2002) desorbs from the grains in region II. Consequently, reactions based on ⋅CH3 were considered only for regions II and III, assuming for both an abundance of 6.40 × 10−9 cm−3 (Sakai et al. 2012).


Fukui et al. (2014, 2015) compared the Planck dust data and the gas emission data such as

 and

 for the solar neighborhood. The

 emission is generally assumed to be optically thin, as written in textbooks. If the dust properties are uniform and DGR is a constant, it is expected that the velocity-integrated intensity of the

 spectrum (

) is proportional to

 for the data points where the

 emission is not detected. Fukui et al. (2014, 2015), however, found that the correlation between them is not so good. By introducing

 into the

 correlation plot, these authors discovered that the poor correlation in the

 plot is mainly due to the data points where the density is high and

 is low. There are two main possibilities to explain this bad correlation between

 and

 one is the presence of optically thick

 gas, and the other is the presence of “

-dark

 gas,” which is

 gas without the

 emission (e.g., Wolfire et al. 2010; Planck Collaboration et al. 2011c; Langer et al. 2014). Fukui et al. (2014, 2015) investigated the

 fractions in the hydrogen gas by referring to the UV measurements (Gillmon et al. 2006), and found that the fractions are typically

 toward the lines of sight whose column densities are up to at least

.1

1
Note that in Figure 16 of Fukui et al. (2015), the

 fractions are somewhat larger than 10% toward two Galactic B-type stars, HD 210121 and HD 102065 (Rachford et al. 2002). These two stars may be contaminated by their own localized gas, and therefore the

 fractions for the local ISM are possibly not reliable toward them. For this reason, Fukui et al. (2017; see Section 4.3) did not use the results obtained in Rachford et al. (2002).
 That is,

 dominates

, and the “

-dark

 gas” would not be a dominant component in the local ISM. Therefore, these authors concluded that there exists a large amount of optically thick

 gas in the local ISM, whose typical optical depth is

, and the amount of the

 gas is underestimated by

 if a correction for the opacity effect is not applied. In Fukui et al. (2014, 2015), these analyses were made for the high Galactic latitude at

 where the gas density is low. An open issue discussed in Fukui et al. (2014, 2015) is if

 obeys a simple linear relationship with the

 or not. A study in the Orion A molecular cloud (Roy et al. 2013) indicates that the dust optical depth is proportional to the 1.28th power of

, rather than a simple linear relation. If correct, this nonlinearity may be ascribed to the dust evolution at high column density. This suggests that it is possible to apply a minor modification of the method of Fukui et al. (2014, 2015) in order to improve the accuracy in

.


Half a century ago, a series of theorems laid the ground for the Kerr hypothesis (Israel 1967; Carter 1971; Robinson 1975); according to these no-hair theorems, the only stationary, axisymmetric, asymptotically flat, regular outside of the horizon solution to four-dimensional GR when the matter fields feature the same isometries as the spacetime is the Kerr BH. Notwithstanding their significance, there are many ways with which to circumvent them and discover different solutions. Still, in four dimensions, hairy BHs have been described in different theories of gravity, such as Einstein–Yang–Mills (Bizon 1990; Künzle & Masood-ul-Alam 1990; Volkov & Galtsov 1990; Breitenlohner et al. 1992; Kleihaus & Kunz 1998, 2001; Kleihaus et al. 2004), scalar-tensor (Bocharova et al. 1970; Bekenstein 1974; Kleihaus et al. 2015; Collodel et al. 2020a), and Gauss–Bonnet theories (Kanti et al. 1996; Kleihaus et al. 2011, 2016; Antoniou et al. 2018; Doneva & Yazadjiev 2018; Silva et al. 2018; Cunha et al. 2019; Collodel et al. 2020b; Herdeiro et al. 2021; Berti et al. 2021). Remarkably, by dropping the assumption that the matter fields must be stationary and axisymmetric, Herdeiro and Radu found solutions in the context of GR where BHs have hair (Herdeiro & Radu 2014b, 2015), by minimally coupling to gravity a complex scalar field that depends on time and on the axial coordinate while its energy-momentum tensor still possesses the respective isometries; see Herdeiro et al. (2015, 2016a, 2016b), Brihaye et al. (2016), and Delgado et al. (2016) for generalizations. These are known as scalarized Kerr black holes (KBHsSH) and they are the object of study of this paper. In their domain of existence, they connect Kerr BHs (that is, with no hair) with pure solitonic solutions, also known as boson stars (BS), which are regular everywhere and feature no horizons. In this sense, one can think of the KBHsSH indeed as a combined system of a BS with a horizon at its center, and therefore it shares traits of both objects.


Column 11.—Adopted distance DH and its uncertainty σD, both in Mpc. For objects with cz⊙ > 6000

, the distance is simply estimated as czcmb/H, where czcmb is the recessional velocity measured in the cosmic microwave background reference frame (Lineweaver et al. 1996) and H is the Hubble constant, adopted to be 70

 Mpc−1. For objects with czcmb  6000

, we use the local peculiar velocity model of Masters (2005), which is based in large part on the SFI++ catalog of galaxies (Springob et al. 2007) and results from analysis of the peculiar motions of galaxies, groups, and clusters using a combination of primary distances from the literature and secondary distances from the Tully–Fisher relation. The resulting model includes two attractors with infall onto the Virgo cluster and the Hydra-Centaurus supercluster, as well as a quadrupole and a dipole component. The transition from one distance estimation method to the other is selected to be at cz⊙ = 6000

  because the uncertainties in each method become comparable at that distance. Where available, primary distances in the published literature are adopted; we also use secondary distances, mainly from Tully et al. (2013), for galaxies with czcmb  6000

. When the galaxy is a known member of a group (Springob et al. 2007), the group systemic recessional velocity czcmb is used to determine the distance estimate according to the general prescription just described. Where primary distances are not available, objects in the Virgo region are assigned to likely Virgo substructures (Mei et al. 2007), and then the distances to those subclusters are adopted following Hallenbeck et al. (2012). Errors on the distance are generated by running 1000 Monte Carlo iterations where peculiar velocities are drawn from a normal distribution each time, as described in Section 4.1 of Jones et al. (2018). Such errors are likely underestimates in the vicinity of major attractors such as Virgo. It should be noted that the values quoted here are Hubble distances, not comoving or luminosity distances.


The above formulation in principle applies to any planet with a Dungey cycle-type stellar wind–magnetosphere interaction, and we thus consider here the appropriate parameters for exoplanets orbiting at arbitrary distances, with an emphasis on close-orbiting hot Jupiters. As discussed above, whereas for the RBL the radio powers are computed as functions of incident kinetic or Poynting flux, in our model, the powers are principally functions of the motional electric field of the solar wind, the dynamic pressure of the stellar wind and the Pedersen conductance of the ionosphere, all of which are dependent on further stellar and planetary parameters as described below. We examine results for both a solar-like stellar wind, and that representative of a young Sun-like star with high-mass-loss rate and magnetic field strength relative to the Sun. Considering first the Sun-like stellar wind, the relevant parameters are shown in Fig. 4 versus radial distance d normalized by the solar radius Rs [we truncate the inner radial distance of the plot at 2 $\mathrm{{\rm {\it R}}_s}$, being the canonical location of the heliospheric magnetic field ‘source surface’ (Owens & Forsyth 2013)]. Absolute distances in au are shown on the top axis for information, although we recognize that in reality the conversion from $\mathrm{{\rm {\it R}}_s}$ to au depends on the individual star. Specifically, Fig. 4(a) shows with the solid line the incident velocity of the solar wind on the magnetosphere vm, which is a function both of the stellar wind speed and the planet's orbital speed. For simplicity, we employ Parker's isothermal solution for the stellar wind speed vsw (Parker 1958), which is fully parametrized by the sound speed cs, and which, as shown by Cranmer (2004) has the closed-form solution

(18)

\begin{equation}
{v_{{\rm sw}}}^2= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}-v_{\rm c}^2W_0[-D(d)] & {\rm if } d \le d_{\rm c} ,\\
-v_{\rm c}^2W_{-1}[-D(d)] & {\rm if } d \ge d_{\rm c} ,\\
\end{array}\right.
\end{equation}

where W0 and W−1 are branches of the Lambert W function, dc is the critical distance at which vsw passes through the sound speed cs, given by

(19)

\begin{equation}
d_{\rm c}=\frac{GM_{\rm s}}{2c_{\rm s}^2},
\end{equation}

where Ms = 1.9891 × 1030 kg is the solar mass and D(d) is given by

(20)

\begin{equation}
D(d)=\left(\frac{d}{d_{\rm c}}\right)^{-4}\exp \left[4 \left(1-\frac{d_{\rm c}}{d}\right)-1\right].
\end{equation}

For the Sun-like wind, we employ a sound speed cs = 130 km s−1 (which, for a Sun-like average particle mass of 1.92 × 10−27 kg corresponds to a temperature of ∼1.18 MK, though note for the present Sun calculation we actually make no assumptions in this regard), yielding a velocity at 1 au of ∼480 km s−1, consistent with observations, and ∼50–200 km s−1 in the hot Jupiter region of 3–10 Rs (indicated by the grey region). The dotted line indicates the Keplerian speed of a planet in a circular orbit vorb, and the solid line is the sum in quadrature of the two, giving the resultant incident stellar wind speed vm. Note that the two speeds are comparable in the inner region associated with hot Jupiters, and although this will modify the orientation of the magnetosphere with respect to the radial vector, it will not significantly alter the magnetospheric dynamics. We further show with the loosely dotted and dot–dashed lines the (constant) sound speed cs and the Alfvén speed vA given by equation (5). It is evident that the interaction is everywhere supersonic (modestly so in the hot Jupiter region, with a Mach number of ∼2) but becomes sub-Alfvénic inside of ∼15 $\mathrm{{\rm {\it R}}_S}$, such that Alfvén wings will form along the IMF field lines, as discussed by Saur et al. (2013), effectively shielding the stellar wind motional electric field and is related to KR saturation of the convection potential.


A simpler model with sinusoids only will be used when analyzing time series of activity indices. That is,
5

6

7

where

 is some measurement of an activity index (e.g., emission in the

 line, S-index, etc.) obtained with instrument INS. The parentheses in

 mean that fits with and without long-term trends will also be investigated. The significance assessments here are performed using frequentist estimations of the false alarm probability (FAP) following the recipes by Baluev (2009). Detecting a periodic signal in a time series will consist of  comparing a null model (e.g., no

 term) with a model with one more periodic signal (e.g., one sin/cos term). Since the period is a highly nonlinear parameter, many periods (about 20,000 equally spaced in 1/P or frequency domain) will be tested between 2 and 50,000 days. A plot of the improvement of the logarithm of the maximum likelihood statistic (

) against the period of the test signal produces a plot that is often called a periodogram. Periodograms for unevenly sampled data were developed in the late 1970s (e.g., Lomb 1976; Scargle 1982). In those cases, the merit statistic plotted against the test period was the improvement in

 or the F-ratio statistic. These early methods only accounted for a model with a single sin/cos component. A further refinement was later implemented by also including an offset (as our γ) as a free parameter to acknowledge the fact that the mean of the series was not known a priori. These periodograms were called floating mean periodograms (Ferraz-Mello 1981) or generalized least-squares (GLS, Zechmeister & Kürster 2009) periodograms. RRM15 relies on the use of GLS periodograms. GLS is adequate to identify and quantify the significance of periodicities if the data contain a single sinusoidal signal and uncertainties in the time series are realistic. Otherwise, analytic significance estimates of the favored solutions using GLS can only be considered to be approximate. In this sense, GLS is a very efficient method to guess the most likely signals left in residual data, but it cannot be used to provide a reliable assessment of its significance. This is discussed in the context of the basic model comparison in Anglada-Escudé & Tuomi (2015). Further refinements of the model (e.g., simultaneous fit to a trend, combining different instruments with different offsets, include basic models for unaccounted noise) require the use of more sophisticated tools beyond GLS, like the in-likelihood periodograms presented here. Note that the

 statistic used by GLS and previous periodograms is a particular case of the ln-likelihood statistic with

,

, and only one instrument INS. Finding the maximum likelihood statistic requires adjusting all of the free parameters of the model at each test period. For these reasons, it produces much more robust significance estimates than its predecessors, especially when the description of the data requires models more complex than a single sinusoid. Reviewing the properties of the likelihood function in the context of periodic signal detection is not the topic of this article. We refer the interested reader to Baluev (2009, 2013) for more detailed discussions and formal proofs.


We stress the uncertainty of our present interpretation, given that the density and temperature structures of our target sources are not yet constrained in a sufficiently detailed way on our resolved spatial scales. In addition, our 44 GHz observationsare still subject to missing short-spacing, which nevertheless should not impact regions with high intensity. To provide a quantitative sense of how missing short-spacing biased the observed Stokes I intensity fractionally, we assume that exterior to the disks (or disk-like structures) around IRAS 16293–2422 A and B there is ~2 M⊙ of gas on ≳10′′ scales (Mundy et al. 1990), and assume that the gas to dust mass ratio is 100. The averaged dust mass column density is Σdust ~ 0.1 g cm−2. Assuming the dust opacity at 230 GHz is κ230 GHz = 0.5

$^{0.5}_{-0.4}$−0.4+0.5
 cm2 g−1 (c.f., Draine 2006), and the dust opacity spectral index is β ~1−1.75 (Li et al. 2017), the derived averaged dust optical depth at 44 GHz is

$\tau_{\mbox{\tiny 44 GHz}}=\kappa_{\mbox{\tiny 230 GHz}}(44/230)^{\beta}\mathrm{\Sigma}_{\mbox{\tiny dust}} = $τ 44 GHz=κ 230 GHz(44/230)βΣ dust=
2.8

$^{16}_{-2.2}\,\times$−2.2+16 ×
 10−3. Assuming a nominal 20–30 K dust temperature on 10′′ scales (van Dishoeck et al. 1995; Ceccarelli et al. 2000; Chandler et al. 2005), the brightness temperature of the extended dust emission is ~ (25 ± 5) × (1 − e−τ44 GHz) = 0.069

$^{0.57}_{-0.058}$−0.058+0.57
 K. This is approximately how much missing short-spacing can bias the Stokes I intensity. If we take regions where the observed brightness temperature in the Robust = 0 weighted 44 GHz image is above 1.4 K (i.e., 6σ; the second contour from bottom in the right panel of Fig. 1) for example, missing flux can bias Stokes I intensity by at most ~(0.64 K)/(1.4 K + 0.64 K) ~ 31%. For structures detected at the 6σ significance, clean errors and thermal noise (e.g., 3σ) may dominate the errors in the measured polarization percentages. The brightness temperature of the missing flux can be comparable with the 6σ detection limit of our Robust = 2 weighted 44 GHz image (~0.6 K), and therefore the observed polarization percentages can be biased by over ~50% in the low intensity (3–6σ) area in that image (Table B.1). Our assessment is that for our present 44 GHz images, missing short-spacing biased the observed polarization percentages of the detected, spatially-extended, dust emission component. Since the brightness temperature of the extended component is low (Fig. 1) such that dust is likely to be optically thin (i.e., Tbg = TCMB), missing short-spacing does not dramatically flip the observed polarization position angles by 90° (Sect. 4.1; Figs. 2–4).


In this work we first used the data points of low-redshift Hubble diagrams for Pantheons, quasars, and GRBs to put constraints on the present value of cosmographic parameters in an independent cosmography approach. To do this, we used different combinations of data samples including Pantheon, Pantheon + quasars, Pantheon + GRB, finally Pantheon + quasars + GRB. In the context of a cosmography approach, we obtained the best-fit values of cosmographic parameters as well as their confidence regions up to 3 − σ uncertainties for different combinations of data samples. Our results showed that the best-fit value of the deceleration parameter q0 varies in the range of −0.844 to −0.702 and the best fit of jerk parameter j0 varies in the range of 1.60 to 2.61 for different combinations of data samples. Note that here we used the Hubble diagrams of quasars and GRBs, respectively, derived in Lusso & Risaliti (2016a) and Demianski et al. (2017a). In the calibration procedure to form the Hubble diagrams of both quasars and GRBs, they have used the SNIa data at low redshifts. Their results for quasars and GRBs samples are consistent with that of the SNIa samples at the low-redshift universe. Hence we adopted their calibrations and used their Hubble diagrams for quasars and GRBs. In the case of concordance ΛCDM cosmology, our results are also compatible with recent work in Lusso et al. (2019). They confirmed the presence of a tension between Λ cosmology and the best-fit cosmographic parameters ∼4σ with SnIa+quasars, at ∼2σ with SnIa+GRBs, and at  4σ with the whole SnIa+quasars+GRB data set (Lusso et al. 2019). Furthermore, we studied some relevant DE parameterizations as well as the concordance ΛCDM cosmology using the Hubble diagrams of Pantheons, quasars, and GRB observations in the context of a cosmography approach. The DE parameterizations studied in our analysis are wCDM, CPL, and Pade parameterizations. First, using the different combinations of data samples and in the context of the MCMC algorithm, we calculate the χ2 function of the distance modulus to find the best-fit values and also the 1 − σ uncertainty of cosmological parameters for each DE parameterization. Using the chain of data obtained for cosmological parameters, we found the best-fit values and the 1 − σ confidence region of the cosmographic parameters of DE parameterizations. Comparing the results for DE models with those obtained for the model-independent approach leads us to conclude that the model is in better (worse) agreement with Hubble diagrams of Pantheons, quasars, and GRBs. In the first stage, using the solely Pantheon sample, we found that the wCDM model is the most compatible model with the result of model-independent constraints and on the other hand the concordance ΛCDM model is the worst model. In the second step, by combining the GRB data to the Pantheon sample, we obtained disappointing results for the ΛCDM model. In this case the q0 parameter of the ΛCDM has a 3.8 − σ tension with that of the model-independent cosmography approach. Moreover, the j0 parameter of ΛCDM cosmology has roughly 5 − σ tension with that of the model-independent approach. These results will be more complicated when we see the results of other DE models and parameterizations that we studied in this work. We observed that wCDM, CPL, and Pade parameterizations are in better agreement with the results of the model-independent cosmography approach rather than the concordance model. In the third and fourth steps, using the combinations Pantheon+quasars and Pantheon+GRB+quasars data points, we obtained the same results again, supporting our results in previous steps. So we conclude that the concordance ΛCDM cosmology has much tension with observations of quasars and GRBs at higher redshift. Note that the DE parameterizations studied in this work are in better agreement with Hubble diagrams of high-redshift quasars and GRB observations. Finally, we reconstructed the Hubble parameter using the best-fit value of cosmographic parameters for both model-independent approaches, the ΛCDM model, and DE parameterizations. We observed that for different data sample combinations, the evolution of reconstructed H(z) in a concordance ΛCDM model has a maximum deviation from the confidence region compared to different DE parameterizations. Upon this result, we can conclude that among the different cosmological models studied in this work, the ΛCDM has the minimum compatibility with the predictions of a model-independent approach and thus it is falsified by a cosmography approach. The large value of tensions (between 3σ to 6σ for different data combinations) that we observed between the cosmographic parameters of ΛCDM and those we obtained in the model-independent approach support this claim again that we should explore other alternatives for standard ΛCDM cosmology. We observed that other DE parameterizations in this study cannot be refuted in the context of a cosmography approach. Our results for ΛCDM cosmology are in agreement with the results of recent work (Yang et al. 2019; Khadka & Ratra 2020), which were obtained using a different approach and different data sets. Although in the literature it has been thoroughly affirmed that the ΛCDM describes the evolution of the universe until recent times, our conclusion confirms the result of Benetti & Capozziello (2019) that show large tensions emerge at higher redshifts for ΛCDM. Our analysis can be extended by considering other cosmic observations in the context of a cosmography approach.


To definitively assess whether or not the growth of the vertical kinetic energy observed in our simulations is associated with the disc eccentricity, we performed three additional simulations using the parameters of the fiducial model, except that: (i) In the first simulation, the disc orbits a single star and the cooling time-scale is set to tc = 0.1Ω−1; (ii) In a second run, the disc orbits a single star and the cooling time-scale is set to tc = 0.001Ω−1; and (iii) In the third run, we restarted the fiducial model at time t0 = 955 but with the system slowly transitioning from a central binary system into a single star located at the centre of mass. This is done by treating the gravitational potential as the sum of two terms, one corresponding to the binary plus one corresponding to a single star, with weighting factors that change with time. More precisely, the gravitational potential for this simulation is given by (Mutter et al. 2017)
(23)$$\begin{eqnarray}
\Phi _{\rm trans}=(1-W (t))\Phi _{\rm bin}-\frac{GM_\star }{r} W(t)
\end{eqnarray}$$with
(24)$$\begin{eqnarray}
W(t) &= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1 & \text{if } \,\, t-t_0 \ge 100T_{\rm bin}\\
\frac{1}{2}\left[1-\cos \left(\frac{\pi (t-t_0)}{100T_{\rm bin}}\right) \right] & \text{if} \,\, t-t_0 \lt 100T_{\rm bin}\end{array}.\right.
\end{eqnarray}$$The aim of simulations (i) and (ii) is to check that our fiducial disc is stable to the VSI whereas run (iii) is useful to examine whether the turbulent flow is maintained under unforced conditions. For these three calculations, the time evolution of the vertical kinetic energy is shown in Fig. 5. Considering simulations (i) and (ii), exponential growth of the vertical kinetic energy due to growth of the VSI only occurs for the case with tc = 0.001 Ω−1. This is consistent with the results of Richard, Nelson & Umurhan (2016), who found that for h = 0.05 the disc remains stable to the VSI provided tc > 0.05 Ω−1. It also demonstrates that our fiducial model, which has tc = Ω−1, is stable to the VSI. For simulation (iii), the saturation level of the vertical kinetic energy remains essentially unchanged after the switch (t = 1055). Contours of the normalized density perturbation one scale height above the mid-plane, and of the meridional velocity in a [R, Z] plane located at azimuth φ = π, are presented in Fig. 6 at t = 1500. They clearly reveal that both the eccentric mode and the turbulence can be maintained under unforced conditions. The continued existence of the eccentric mode at that time is not surprising since m = 1 free eccentric modes can be long lived (Papaloizou 2005b). The persistence of the turbulent flow is a strong indication that the instability originates from the presence of an eccentric mode in the disc.


Another, probably more plausible, explanation for the systematic trend in Figure 5 might be that, in deriving SN explosion energy from the global parameters, it is difficult to take into account the complex environments of SNRs, and this is compounded by the parameter sensitivity of Equation (10) (and to a lesser extent, Equation (8)). As we have already seen in many observed SNRs (e.g., Figure 1) and simulated SNRs (e.g., Figure 3), the real ISM provides a very complex environment for SNRs. In this case, the background medium density parameter

 for describing SNR evolution is not well-defined. Fortunately, as demonstrated in 3D simulations considering inhomogeneous background medium (e.g., Figure 4; see also Cho et al. 2015; Iffrig & Hennebelle 2015; Kim & Ostriker 2015; Li et al. 2015; Martizzi et al. 2015; Walch & Naab 2015; Zhang & Chevalier 2019), the evolutionary tracks of integrated quantities such as total momentum and kinetic energy in the radiative SNRs (i.e., after the shell formation) are not sensitive to the complexity of the background medium. In particular, the total momentum does not evolve in time after reaching a terminal value. The model uncertainty due to the ISM inhomogeneity is less than a factor of two. However, a unitary background medium density

 does not seem be applicable to both

 and

. In our TS models, where the ambient medium has a large-scale nonuniformity, the thermal energy is considerably smaller than that of the uniform medium case with the same mean density (Figure 4(e)). This also happens for some SNRs in TI models (see Section 4.2). On the other hand, the observed thermal energies of the SNRs appear to be generally larger than those of the uniform medium cases with the same mean densities (Figure 4(f)). For the two prototypical SNRs interacting with MCs, W44 and IC 443, for example, Table 5 shows that

/

 = 6–8 when we adopt the mean ambient density

(H i+H2)(=50–70 cm−3). It is, however, clear that this density is much higher than the density of the material filling most of the volume of the ISM where the SN blast wave propagates. For example, the ambient density derived from an analysis of the X-ray surface brightness profile of W44 is ∼3 cm−3 (Harrus et al. 1997). An overestimated ambient density would predict a smaller

 or a larger

 to match the observed

 (see Equation (10)). On the other hand, Equation (10) is based on 1D simulations. In real SNRs in inhomogeneous/nonuniform media, the cooling could be significantly enhanced due to the mixing and diffusion between the engulfed dense material and the hot plasma, in which case the thermal energy of real SNRs might be smaller than the

 predicted from 1D simulations. Therefore, it is not obvious whether Equation (10) with

(H i+H2) would overpredict or underpredict

 for SNRs in complex environments (see also Section 4.2). High-resolution simulations of SNRs in realistic environments, including the complex physics at the hot/cool interface, are needed in oder to understand the environmental dependency of the evolution of thermal energy.


It has been clear since many years now (Matthews et al. 1964) that the majority of powerful radio-AGN is hosted within the most massive galaxies (stellar masses M∗∼1011M⊙\documentclass[12pt]{minimal}
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				\begin{document}$$M_* \sim 10^{11}\, M_\odot$$\end{document} and above) known in the universe. This result has been confirmed at least up to z∼1\documentclass[12pt]{minimal}
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				\begin{document}$$z\sim 1$$\end{document} (and in some cases beyond) both by direct estimates (e.g., Heeschen 1970; Ekers and Ekers 1973; Auriemma et al. 1977; Jenkins 1982; Best et al. 2005a; Mauch and Sadler 2007; Smolčić et al. 2009; Magliocchetti et al. 2016; Sabater et al. 2019; Capetti et al. 2022) and also by the extremely tight correlation in the MK-z\documentclass[12pt]{minimal}
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				\begin{document}$$M_K-z$$\end{document} plane exhibited by the hosts of radio-AGN (e.g., Lilly and Longair 1984; Eales et al. 1997; Jarvis et al. 2001; De Breuck et al. 2002; Willott et al. 2003) that can be explained by assuming that the observed K-band light is dominated by emission from the old stellar population and that radio-AGN have very similar (large) host masses all across half of the age of the universe. More recent results based on rest-frame 1.6 and 4.5 μ\documentclass[12pt]{minimal}
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				\begin{document}$$z\sim 3$$\end{document} if not beyond (Seymour et al. 2007; De Breuck et al. 2010; Gürkan et al. 2014; Drouart et al. 2016).


On a ∼30000 au scale, a bipolar outflow associated with this source was detected in the east-west direction by FCRAO single-dish observations in 12CO

J
=
1
−
0

 molecular line emission (Narayanan et al. 2012) and by James Clerk Maxwell Telescope (JCMT) single-dish observations in 12CO

J
=
3
−
2

 molecular line emission (Hogerheijde et al. 1998). Their results show that the blue and red robes of the outflows are on the eastern and western sides, respectively. On the other hand, inner parts (∼8000 au scale) of the outflow mapped with the Nobeyama Millimeter Array (NMA) in 12CO

J
=
1
−
0

 show the opposite distribution, i.e., stronger blueshifted emission on the western side and stronger redshifted emission on the eastern side (Tamura et al. 1996). Mid-infrared observations toward L1527 IRS with Spitzer Space Telescope shows bright bipolar scattered light nebulae along the outflow axis in the ∼20000 au scale (Tobin et al. 2008). They fitted a protostellar envelope model to near- and mid-infrared scattered light images and spectral energy distribution (SED). As a result, the inclination angle of the envelope around L1527 IRS was estimated to be

i
=
85
°

, where

i
=
90
°

 is the edge-on configuration. In addition, Oya et al. (2015) found that the western side is closer to observers. This inclination angle is consistent with a disk-like structure of dust highly elongated along the north-south direction, which was spatially resolved for the first time in 7-mm continuum emission by the Very Large Array (VLA; Loinard et al. 2002). By expanding the studies by Tobin et al. (2008, 2013), we fitted a model composed of an envelope and a disk to (sub)millimeter continuum emissions and visibilities observed with the Submillimeter Array (SMA) and the Combined Array for Research in Millimeter Astronomy (CARMA), as well as infrared images and SED. Their best-fitting model suggests a highly flared disk structure (

H
∝

R

1.3

, H = 48 au at R = 100 au) with a radius of 125 au. This study has geometrically distinguished the protostellar disk and envelope around L1527 IRS, although they were not kinematically distinguished from one another.


The error budget associated with Δtij was estimated by considering three sources of uncertainties. Firstly, we propagated the statistical error inferred from the MCMC sampling. This was simply done by constructing histograms for Δtij h−1 from the independent samples of SIEg model parameters reported in Figs. 3 and 4. An example of these histograms is illustrated in Fig. 7 for the case of zl = 0.7. Secondly, we considered the impact of hypothetical massive objects lying on the line-of-sight by scaling the theoretical time delay distance with an external convergence term κext such that

D

Δ
t

=

D

Δ
t

theory

/

(
1
−

κ

ext

)

$ D_{\Delta t} = D_{\Delta t}^{\text{ theory}} / (1 - \kappa_{\text{ ext}}) $

 (see, e.g., Keeton 2003). We applied a different scaling for each set of model parameters used to construct the histograms (see Fig. 7). The κext values were randomly drawn from a zero mean normal distribution and characterized by a conservative standard deviation σκ = 0.03 (see, e.g., Wong et al. 2017). When combined, the typical errors are ∼7.7% for ΔtCB, ∼7.8% for ΔtCA, and ∼7.6% for ΔtCD, compared to the median values. Thirdly, we considered the impact of the Source Position Transformation (SPT), a degeneracy existing between different lens density profiles that reproduce equally well the lensing observables, except for the product Δt H0 (Schneider & Sluse 2013, 2014). A short summary is given in Appendix A. We SPT-transformed the SIEg model using the modified deflection law

α

̂

=

α

SIEg

+

β

−

β

̂

(

β

)

$ {\hat{\boldsymbol{\alpha}}} = \boldsymbol{\alpha}_{\text{ SIEg}} + \boldsymbol{\beta} - {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) $

 along with a radial stretching of the source plane defined by

β

̂

(

β

)

=

[
1
+
f
(
|

β

|
)
]

β

$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = [1 + f(|\boldsymbol{\beta}|)]\ \boldsymbol{\beta} $

. In particular, we considered the special case where the deformation function f(|β|) is the lowest-order expansion of more general functions, f(|β|) = f0 + f2|β|2/(2θE), where the constants f0 = f(0) and

f
2

=
θ
E
2

f

′′

(0)

$ f_2 = \theta_{{\rm E}}^2 f^{\prime\prime}(0) $

 quantify the magnitude of the deformation. This choice explores a large variety of degeneracies characterized by an isotropic transformation of the source plane. In the end, this at least defines a lower limit on the impact of the SPT on the time delays. In the particular case of f2 = 0, the SPT reduces to the well-known mass-sheet degeneracy (MSD) characterized by

β

̂

(

β

)

=
λ

β

≡

(
1
+

f
0

)

β

$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = \lambda \boldsymbol{\beta} \equiv (1 + f_0) \boldsymbol{\beta} $

. In our analysis, we only explored degeneracies that cannot be explained by a MSD, hence f0 = 0.


For completeness and ease of our discussions, here we summarize briefly the main features of the NS EOS-metamodel we use in solving the NS inverse-structure problem. More details can be found in our previous publications (Zhang et al. 2018; Xie & Li 2019, 2020; Zhang & Li 2019a, 2019b, 2019c, 2020). We assume the cores of NSs are made of totally charged neutral neutrons, protons, electrons, and muons (the npeμ model) at β-equilibrium. Unlike the widely used composition-degenerate spectral functions and/or piecewise polytropes that directly parameterize the pressure as a function of energy or baryon density, to probe the high-density symmetry energy, we have to keep the isospin dependence of the EOS and retain explicitly the composition information at all densities. For this reason, we metamodel NS EOS by starting at the single-nucleon energy level with explicitly isospin dependence. Specifically, we parameterize the E0(ρ) and Esym(ρ) according to
2

E

0

(
ρ
)
=

E

0

(

ρ

0

)
+

K

0

2

ρ
−

ρ

0

3

ρ

0

2

+

J

0

6

ρ
−

ρ

0

3

ρ

0

3

,

3

E

sym

(
ρ
)

=

E

sym

ρ

0

+
L

ρ
−

ρ

0

3

ρ

0

+

K

sym

2

ρ
−

ρ

0

3

ρ

0

2

+

J

sym

6

ρ
−

ρ

0

3

ρ

0

3

where E0(ρ0) = −15.9 ± 0.4 MeV is the binding energy and K0 ≈ 230 ± 20 MeV (Shlomo et al. 2006; Piekarewicz 2010; Garg & Colò 2018) is the incompressibility at the saturation density ρ0 of SNM, while Esym(ρ0) = 31.7 ± 3.2 MeV is the magnitude and L ≈ 58.7 ± 28.1 MeV is the slope of symmetry energy at ρ0 (Li & Han 2013; Oertel et al. 2017) based on earlier surveys of over 50 analyses of both terrestrial experiments and astrophysical observations, respectively. A very recent survey of 24 new analyses of NS observations since GW179817 indicates that L ≈ 57.7 ± 19 MeV and Ksym ≈ −107 ± 88 MeV at 68% confidence level (Li et al. 2021). In this study, we use L = 58.7 as its most probable value and vary it within ±20 MeV. We keep the E0(ρ0), K0, and Esym(ρ0) at their most probable values given above as they have been relatively well determined and their variations within their remaining uncertain ranges have been shown to have little effects on the masses, radii, and tidal deformabilities of NSs Fattoyev et al. (2013, 2014), Xie & Li (2019, 2020). As a reference, we notice that very recent calculations based on a set of commonly used Hamiltonian operators including two- and three-nucleon forces derived from chiral effective field theory predicted that L = 47 ± 3 MeV, Ksym = −146 ± 43 MeV, Jsym = 90 ± 334 MeV (Somasundaram et al. 2021). While in the same calculations, with a prior J0 = −300 ± 400 MeV, the posterior ranges between J0 =−573 ± 133 MeV and J0 = −172 ± 243 MeV depending on the scaling schemes used. The EOS parameter ranges we used are consistent with these calculations.


Next, we estimated the distance and reddening. As in Brown et al. (2014), we compared the Eri II main sequence to a synthetic Victoria-Regina theoretical isochrone from VandenBerg et al. (2014) and the horizontal branch of Eri II to the horizontal branch of M92, which was observed with the same ACS filters by Brown et al. (2005). The synthetic main-sequence isochrone was constructed assuming an age of 13 Gyr, the metallicity distribution displayed in Figure 3, and a binary fraction of 0.48 (Geha et al. 2013). For M92, we assumed a distance modulus of 14.62 mag (the mean of literature measurements by Del Principe et al. 2005, Sollima et al. 2006, and Paust et al. 2007) and reddening of E(B − V) = 0.023 mag (Schlegel et al. 1998). We included main-sequence stars more than 0.5 mag below the main-sequence turnoff (m814 > 27.67) in order to avoid portions that are age-sensitive, and we used horizontal branch stars that are bluer than m606 − m814 = −0.68 in order to avoid the RR Lyrae instability strip. Fitting the main sequence and the horizontal branch simultaneously, we found m − M = 22.65 mag (d = 339 kpc) and E(B − V) = 0.03 mag, as shown in Figure 4. This distance modulus is consistent, within the uncertainties, with the distance derived from the Eri II RR Lyrae stars by A. Garofalo et al. (2021, in preparation). The statistical uncertainties on the distance fit are very small (0.015 mag in distance modulus, 0.003 mag in reddening), and they are certainly dominated by systematics in the choice of comparison cluster/isochrone and the magnitude and color range of stars to include. By analogy to Brown et al. (2014), we assume overall uncertainties of 0.07 mag in m − M and 0.01 mag in E(B − V). Similar to a few of the ultra-faint dwarfs studied by Brown et al. (2012, 2014), the distance modulus determined with this approach is in modest disagreement with some literature results (Crnojević et al. 2016 measured m − M = 22.8 ± 0.1 mag and Koposov et al. 2015 estimated m − M = 22.9 mag, although Bechtol et al. 2015 found a comparable value of m − M = 22.6 mag) and the derived reddening is larger than the E(B − V) = 0.009 mag value obtained by Schlafly & Finkbeiner (2011) (also see Mutlu-Pakdil et al. 2019). However, for consistency with previous star formation history analyses, we adopt the derived values for the remainder of this study.


A. Garofalo et al. (2021, in preparation)


To model the inner coma plasma, we use the simplified model by Eriksson et al. (2017). Neutral gas flows outward from the comet with a velocity of about 0.7 km s−1 (Gulkis et al. 2015), with the density decreasing as r−2(1)
\begin{equation}
n_{{\rm n}} (r) \approx n_{0} \left( \frac{R}{r} \right)^{2},
\end{equation}
where n0 is the neutral gas density at the surface of a comet with radius R, and nn is the density at the cometocentric distance r. Assuming that steady ionization by solar EUV radiation dominates, this gives a peak in plasma density at some cometocentric distance, and a decrease of this density as r−1 at large distances (since the density of neutrals is typically much higher than the plasma density):
(2)
\begin{equation}
n_{{\rm e}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where ne is the electron density at some distance r, ν is an ionization frequency depending on the distance to the sun and u is the outflow velocity of the plasma (Haser 1957; Mandt et al. 2016; Eriksson et al. 2017; Vigren & Eriksson 2017; Vigren 2017). Assuming that the plasma is dominated by water ions from the comet, ion–electron pairs are created with particle energies of less than 0.1 and about 12 eV, respectively (due to conservation of momentum). When the density is high enough close to the nucleus, neutral-electron collisions will cool the electrons to temperatures of 0.1 eV or less. Beyond some distance rce the electrons are not efficiently cooled, and a mixture of cool (0.1 eV) and warm (5–10 eV) electrons are observed (Edberg et al. 2015; Mandt et al. 2016; Eriksson et al. 2017). Collisions between neutrals and ions are important out to larger distances. We investigate the region where electron-neutral collisions can be neglected, and warm electrons are present and magnetized, while the ions are unmagnetized due to collisions and their large gyroradius. This is a region favourable for lower hybrid waves and is where Rosetta spend much of its time during the 2 yr investigation of 67P (Mandt et al. 2016). We use three populations to model the plasma, cold and warm electrons, and H2O + ions
(3)
\begin{equation}
n_{{\rm ew}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{r_{{\rm ce}}}{r}\right) \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(4)
\begin{equation}
n_{{\rm ec}} (r) \approx n_{{\rm e}} (r_{{\rm ce}}) \frac{r^2_{{\rm ce}}}{r^2} = n_0 \frac{\nu R}{u} \frac{Rr_{{\rm ce}}}{r^2} \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(5)
\begin{equation}
n_{{\rm i}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where new, nec and ni are the densities of warm and cold electrons, and ions, respectively (Eriksson et al. 2017). For distances smaller than rce, the electron motion is disrupted by collisions and lower hybrid waves are likely to be heavily damped.


Multiwavelength searches for the donor stars have been pursued for more than a decade. Some have observed ULXs in the optical range (e.g. Gutiérrez & López-Corredoira 2006; Ptak et al. 2006; Roberts et al. 2011; Gladstone et al. 2013; Fabrika et al. 2015). As several of the ULXs are located in or near young star clusters (e.g. Fabbiano, Zezas & Murray 2001; Roberts et al. 2002; Poutanen et al. 2013), some donor stars of ULXs might as well be RSGs (Copperwheat et al. 2005, 2007; Patruno & Zampieri 2008), which are bright in the near-infrared (NIR) band. In light of this, we started an observing campaign of ULXs in the NIR (Heida et al. 2014; López et al. 2017), tailored to detect RSG donor stars (−8  H, Ks  −11; Elias, Frogel & Humphreys 1985; Drilling & Landolt 2000). We targeted sources at distances of up to 10 Mpc, within uncertainties. Of the 97 ULXs observed by Heida et al. (2014) and López et al. (2017), we detected a NIR candidate counterpart for 33 ULXs. Of these 33 NIR sources, 19 had the absolute magnitude consistent with an RSG. So far, we performed spectroscopic follow-up observations for 7 out of these 19 candidate RSGs and confirmed the RSG nature for 3 counterparts: ULX RX J004722.4−252051 (in NGC 253; Heida et al. 2015a), ULX J022721+333500 (in NGC 925), and ULX J120922+295559 (in NGC 4136; Heida et al. 2016). Additionally, we discovered that four counterparts, initially classified as RSG based on photometry, are actually nebulae partially powered by the X-ray emission of the ULX: ULX J022727+333443 (in NGC 925), ULX J120922+295551 (in NGC 4136), ULX Ho II X-1 (in Holmberg II; Heida et al. 2016), and [SST2011] J110545.62+000016.2 (in NGC 3521; López et al. 2019). This means that, at the time of writing this manuscript, the photometric and spectroscopic observations provided 3 confirmed RSGs, 12 candidate RSGs, 4 confirmed nebulae, and 14 candidate stellar clusters/active galactic nuclei (AGNs). Recently, Lau et al. (2019) observed 96 ULXs within 10 Mpc in the mid-IR with the Spitzer Space Telescope. They detected and identified 12 ULXs with candidate counterparts whose absolute mid-IR magnitude is consistent with being RSGs.


We first focus on the properties of the PDLA obtained through the VPFit code. Together with the fitted absorption profile, both the column density and the redshift of the PDLA are derived: (8)\begin{equation} \log(N) = 20.9290 \pm 0.0024;~~~ z_{\rm abs} = 3.082097 \pm 0.000053. \label{eq:col_dens_zabs} \end{equation}log(N)=20.9290±0.0024;zabs=3.082097±0.000053.The redshift value obtained here differs by no more than 0.000163 from the value obtained based on metallic lines and is thus entirely consistent with it (see Sect. 3.2 above). We took the refractive index n of air into account that causes a shift in the observed wavelengths: λair = λvac/n. We used the Ciddor equation (Ciddor 1996), taking the atmospheric conditions at the time of the measurement into account (air temperature 12°C, pressure 744.9 hPa, and relative humidity 3%, as reported in the header of the original FITS file). The errors are the formal errors provided by VPFit and are certainly underestimated, as they do not include uncertainties linked to the continuum determination. In their pioneer study, Leibundgut & Robertson (1999) obtained a column density and redshift \begin{eqnarray*} \log(N) = 20.85 \pm 0.03;~~~ z_{\rm abs} = 3.0825. \end{eqnarray*}log(N)=20.85±0.03;zabs=3.0825.Our estimate of the column density is higher than theirs by 2.6σ (referring to their error estimate), and we find a slightly lower redshift (the difference amounts to 7.6σ according to our own error estimate). The latter discrepancy might be due to our lower spectral resolution (R ~ 1840 instead of 3300), which might cause the metallic absorption lines in the blue wing of the Lyα absorption trough to be less easily recognized and be partly included in the Voigt profile fit. This would result in a slightly overestimated column densitiy and in a slight blue shift. However, our VPFit estimate of the redshift, based on the Lyα line alone, agrees within only 0.5σ with the PDLA redshift estimate based on metallic lines.