The third and higher harmonic plasma emissions, at the frequency of multiples of the local plasma frequency, were also observed in SRBs (Takakura & Yousef 1974; Cairns 1986; Reiner & MacDowall 2019). Various schemes have been proposed to explain the formation of higher harmonic plasma emission. Zlotnik (1978) put forward that the third harmonic emission H
3 is generated by the coalescence process of the Langmuir waves and the harmonic emission H, i.e., L + H → H
3. Cairns (1988) generalized this idea to explain higher harmonics of transverse wave emissions, claiming that nth harmonic waves H

n
 could be generated by a coalescence of beam-generated Langmuir waves L and adjacent transverse waves H

n−1, i.e., L + H

n−1 → H

n
. An alternative way for the generation of harmonic transverse waves would be an interaction of Langmuir waves L and harmonic electrostatic waves L

n−1, i.e., L + L

n−1 → H

n
 (Gaelzer et al. 2003; Yi et al. 2007). Ziebell et al. (2015) numerically solved the weak-turbulence equations and theoretically demonstrate the process of multiple-harmonic plasma emission, including the nonlinear conversion from Langmuir turbulence to electromagnetic radiation. Rhee et al. (2009), Ziebell et al. (2015), Yao et al. (2021) investigated both schemes of multiple-harmonic plasma emissions by means of Particle-In-Cell (PIC) code simulations. They found that Cairns (1988)'s scheme can qualitatively explain the third and fourth harmonic emissions while Yi et al. (2007)'s theory is appropriate to explain the simultaneously generated harmonics of electrostatic waves. PIC code simulations of electron beams demonstrated that fundamental transverse waves at the plasma frequency can be generated because of the wave–wave interaction processes involving ion-acoustic waves (Thurgood & Tsiklauri 2015; Henri et al. 2019). The PIC simulations of ring-beam EVDFs by Zhou et al. (2020) also provided evidence for the generation of the second and third harmonic emission via wave–wave interactions. Annenkov et al. (2019) used PIC code simulations to describe localized beams and found that the emissions at harmonics of the plasma frequency can be generated due to an antenna mechanism.


Measurements of the scale length (h

R
) and scale height (h

Z
) are important to trace the structure, size, mass distribution, and radial luminosity profile of the Galactic disk components (e.g., Dehnen & Binney 1998). In order to calculate h

R
 and h

Z
 of our Atari disk sample, we solve the fundamental collisionless Boltzmann equation of axisymmetric systems, which is expressed as follows (see Equation (4.12) in Binney & Tremaine 2008):
10

∂f∂t+vR∂f∂R+vϕR2∂f∂ϕ+vz∂f∂z−∂Φ∂R−vϕ2R3∂f∂vR−∂Φ∂ϕ∂f∂vϕ−∂Φ∂z∂f∂vz=0,

where f is the number of objects in a small volume and Φ is the gravitational potential. It is then convenient to derive the Jeans equation from the Boltzmann equation in the radial and Z-component directions as follows (see Equation (9) in Gilmore et al. 1989):
11

ρKR=1R∂(RρσVR2)∂R+∂(ρσVR,Z2)∂Z−ρσVϕ2R−ρRVϕ¯2

12

ρKZ=∂(ρσVZ2)∂Z+1R∂(RρσVR,Z2)∂R,

where ρ(R, Z) is the space density of the stars in the thick disk and K

R
 =

∂ϕ∂R

 and K

Z
 =

∂ϕ∂Z

 are the derivatives of the potential. Assuming an exponential density profile, the radial Jeans equation can be rewritten as follows (Li et al. 2018):
13

σVϕ2σVR2−2+2RhR−Vc2−Vϕ¯2σVR2+σVZ2σVR2=0,

where h

R
 is the scale length. By substituting our calculated velocity dispersions from the Atari disk sample, within ≈3 kpc of the Sun in the cylindrical R coordinate and ≈2 kpc above or below the Galactic plane (6347 stars), into Equation (13), we obtain a radial scale length of h

R
 = 2.48 kpc. Calculating the scale length using different metallicity bins shows a small increase from 2.38 kpc among the higher-metallicity stars up to 2.91 kpc for the low-metallicity stars. The results are detailed in Table 3. In general, these results point to the Atari disk being comparable in size in the radial direction to the thick and thin disk. For reference, the scale length of the canonical thick disk has been measured as 2.0 kpc (Bensby et al. 2011), 2.2 kpc (Carollo et al. 2010), and 2.31 kpc (Sanders & Binney 2015), although larger values have also been reported previously (Chiba & Beers 2000; de Jong et al. 2010). Thin disk values refer to an overall similar spatial distribution, although it is likely somewhat more extended (h

R
 > 3.0 kpc;  e.g., Bensby et al. 2011; Sanders & Binney 2015). See Table 2 for further details.


Our fitting is performed based on the Markov Chain Monte Carlo (MCMC) method to produce posterior predictions for the model parameters. The MCMC method is widely used in finding the best set of parameters for a specified model, e.g., GRB 080413B (Geng et al. 2016); GRBs 100418A, 100901A, 120326A, and 120404A (Laskar et al. 2015). In our work, fitting the afterglows of the four bursts with the MCMC method is to test whether or not the late simultaneous bumps followed by a steep decay can be explained with an external-forward shock in a free-to-shocked wind environment. The posterior probability density functions for the physical parameters, i.e., Ek,iso,on, Γ0, θc, θjet/θc, θv/θc, p, ϵe, ϵB, A*, and Rtr, are presented in Figure 6, where only the fitting result of GRB 120326A is shown as an example. The optimal result from MCMC fitting is shown in Figure 5 with the blue line (XRT) and red line (optical), and the obtained parameters at the 1σ confidence level are reported in Table 1, where the values of the transition radius

R

tr

 (i.e., 1.05 × 1017, 1.91 × 1017, 3.98 ×1017, and 6.31 × 1016 cm for GRBs 120326A, 100901A, 100814A, and 120404A, respectively) are consistent with those found in other bursts (e.g., Ramirez-Ruiz et al. 2001; Kong et al. 2010; Feng & Dai 2011; Li et al. 2020). It can be found that both the X-ray afterglow and the optical afterglow of these bumps can be well modeled with an external-forward shock in a free-to-shocked wind environment.

1.
The theoretical light curves do not well fit the transition behavior from the free-wind phase to the shocked-wind phase in the afterglows of GRB 120326A. This may imply that the density jump factor ξ from the free-wind medium to the shocked-wind medium may be less than 4, i.e., ξ  4. Figure 1 reveals that the external-forward shock propagating in a shocked-wind environment can yield a plateau before the late simultaneous bumps in the case of θv = 4θc. Then, we try to fit the afterglows of GRB 120326A with the emission of the external-forward shock propagating in a homogeneous medium, where the priors of

log
10

(

E

k
,
iso
,
on

erg

-
1

)

,

log

10

Γ

0

, θc, θjet/θc, θv/θc, p,

log

10

ϵ

e

,

log

10

ϵ

B

, and

log

10

n

0

 are set as a uniform distribution in the range of (52, 55), (1.5, 3.0), (0.3, 8.0), (2.5, 4.5), (0.0, 8.0), (2.1, 2.9), (−3.2, −0.5), (−6.5, −2.0), and (0.0, 1.7) in our MCMC fittings, respectively. However, no well-fitting result is found. The fitting result with minimum reduced χ2 is shown with dashed lines in the top left panel of Figure 5. It can be found that the early plateau of GRB 120326A could not be well fitted based on the emission of the external-forward shock in a homogeneous medium.

2.
There seems to be a plateau at tobs ∼ 600 s in the optical afterglow of GRB 100814A, which may indicate an energy injection into the external shock. If so, our model could not well describe the afterglows of GRB 100814A in its early phase. This may be the reason for the deviation in our theoretical results relative to the observations in the early phase. In addition, the X-ray and optical afterglows of GRB 100814A show achromatic behavior, which could not be explained only by our scenario (i.e., an off-core observed structured jet propagating into a free-to-shocked wind environment) since the appearance of the late simultaneous bump/plateau in our scenario is achromatic. Other emission rather than that of the external-forward shock may contribute to the long X-ray shallow decay. In addition, our model could not well describe the X-ray emission at tobs ≳ 3 × 105 s in GRB 100901A, which seems to be another component in this phase.

3.
The fitting result of GRB 120404A shows that this burst is observed in the core. Based on the fitting result of this burst, the deceleration radius of the (0, 0) jet can be estimated and is 2.43 × 1016 cm, which is at around the transition radius

R

tr

=
6.31
×

10

16

 cm of this burst.


For the cosmic variance term, any two samples are correlated so the off-diagonal
            element of Gij
              (i ≠
              j) is non-zero. But for the measurement noise term,
            small scale, and intrinsic dispersion, the two samples are not correlated, so they only
            contribute to the diagonal term in the Gij matrix. The first
            term is the real space velocity correlation function, which is related to the matter
            power spectrum in Fourier space (Watkins et al.
              2009; Ma & Scott 2013),
              (10)\begin{eqnarray} \bigl\langle \bigl(\hat{{\vec r}}_{i} \cdot {\vec v}({\vec r}_{i})\bigr)  \bigl(\hat{{\vec r}}_{j} \cdot {\vec v}({\vec r}_{j})\bigr) \bigr\rangle  =\frac{H^{2}_{0}f^{2}(z=0)}{2 \pi^{2}}\int \mathrm{\ } \der k  \mathrm{\ } P(k) \mathrm{\ } F_{ij}(k), \label{vel_rnrm1} \end{eqnarray}

r

ˆ

i

·

v

(

r

i

)

r

ˆ

j

·

v

(

r

j

)

=

H

0

2

f

2

(
z
=
0
)

2

π

2

∫

d
k

P
(
k
)

F

ij

(
k
)
,

where P(k) is
            the matter power spectrum that we output from public code camb (Lewis et al. 2000). The f(z) =
                dlnD/ dlna is
            the growth rate function that characterizes how fast the structures grow at different
            epochs of the Universe. Since the Cosmicflows-2  samples peak at the redshift
              z ≃
              0.0167 (lower panel of Fig. 2),
            we use the zero-redshift growth function \hbox{$f(z=0)=\Omega_{\rm m}^{0.55}$}
f
(
z
=
0
)
=

Ω
m
0.55

 (Watkins
              et al. 2009; Ma & Scott 2013) in Eq.
              (10). In the future, if the survey
            probes deeper region of the space, one should use the corresponding growth function
              f(z) in Eq. (10), so that the joint constraints on
              f(z)σ8
            can be obtained, which constitutes a sensitive test of modify gravity models (Hudson & Turnbull 2012; Planck Collaboration XIII 2015).


we use the zero-redshift growth function \hbox{$f(z=0)=\Omega_{\rm m}^{0.55}$}
f
(
z
=
0
)
=

Ω
m
0.55


Solar flares are one of the most spectacular activities in the solar system. Up to 1029–1032 erg magnetic free energy is impulsively released, accompanied by rapid increases of emissions in various wavelengths (Shibata & Magara 2011). It is generally believed that magnetic reconnection plays a key role in the reconfiguration of magnetic field lines and the conversion of magnetic energy into kinetic and thermal energies of plasma (Forbes & Acton 1996; Priest & Forbes 2000). In the context of the standard solar flare model (Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976), accelerated nonthermal electrons propagate downward and heat the chromosphere, forming bright flare ribbons in the Hα, Ca ii H, ultraviolet (UV), and extreme-ultraviolet (EUV) wavelengths. The flare ribbons have diverse shapes, with two parallel flare ribbons (PFRs) being the most commonplace (Li & Zhang 2015). Sometimes, circular flare ribbons (CFRs) appear accompanied by remote brightenings, which are generally believed to be associated with a magnetic null point (

) and a dome-like spine–fan topology (Masson et al. 2009; Zhang et al. 2012; Pontin et al. 2016). More complex flare ribbons have also been observed (Joshi et al. 2015). Apart from the localized heating, hard X-ray (HXR) emissions are generated via Coulomb collisions (Brown 1971). If open magnetic field lines are involved, it is likely that the nonthermal electrons at speeds of 0.06–0.25c (c is the speed of light) escape from the Sun into interplanetary space and generate type III radio bursts with frequencies ranging from 0.2 to hundreds of MHz (Dulk et al. 1987; Aschwanden et al. 1995; Zhang et al. 2015). Sometimes, the HXR and radio emissions of a flare show quasi-periodic pulsations (QPPs), with their periods ranging from milliseconds (Tan et al. 2010) through a few seconds (Kliem et al. 2000; Asai et al. 2001; Ning et al. 2005; Nakariakov et al. 2010; Hayes et al. 2016) to several minutes (Ofman & Sui 2006; Nakariakov & Melnikov 2009; Sych et al. 2009; Ning 2014). So far, QPPs have been extensively explored using both imaging (Su et al. 2012a, 2012b) and spectral observations (Mariska 2005). Li et al. (2015b) studied the X1.6 flare on 2014 September 10. Four-minute QPPs are evident not only in the HXR, EUV, UV, and radio light curves, but also in the temporal evolutions of the Doppler velocities and line widths of the C i, O iv, Si iv, and Fe xxi lines. Brosius & Daw (2015) studied the M7.3 flare on 2014 April 18. The chromospheric and transition region line emissions show quasi-periodic intensity and velocity fluctuations with periods of ∼3 minutes during the first four peaks and ∼1.5 minutes during the last four peaks.


Figures 2 and 3 demonstrate that the GW alert system is capable of providing GW alerts before merger, but they do not consider the prospects for detection from an astrophysical source population. We generate a population of simulated BNS signals, henceforth referred to as injections, using the TaylorF2 (Sathyaprakash & Dhurandhar 1991; Blanchet et al. 1995, 2005; Buonanno et al. 2009) waveform model. Both source-frame component masses are drawn from a Gaussian distribution between 1.0 M⊙ m1, m2  2.0 M⊙ with mean mass of 1.33 M⊙ and standard deviation of 0.09 M⊙, modeled after observations of galactic BNSs (Özel & Freire 2016).40

40
Note that if GW190425 is a BNS, then galactic measurements are not representative of neutron star masses.
 The neutron stars in the population are nonspinning, motivated by the low spins of BNSs expected to merge within a Hubble time (Burgay et al. 2003; Zhu et al. 2018). The signals are distributed uniformly in comoving volume up to a redshift of z = 0.2. We consider a network of four GW detectors: LIGO-Hanford, LIGO-Livingston, Virgo, and KAGRA at their projected O4 sensitivities.41

41

https://dcc.ligo.org/LIGO-T2000012/public

 We simulate the results of an early warning matched-filtering pipeline by considering six different discrete frequency cutoffs: 29, 32, 38, 49, 56, and 1024 Hz to analyze signal recovery at (approximately) 58, 44, 28, 14, 10, and 0 s before merger, motivated by Sachdev et al. (2020). We calculate the network S/N of each injection at each frequency cutoff and consider the events that pass an S/N cutoff of 12.0 as “detected.” We then calculate the sky posteriors for each of the detected signals by using BAYESTAR (Singer & Price 2016). We use the most recent BNS local merger rate from Abbott et al. (2020b) of

 to estimate the number of events detected per year in the detector network. In Figure 4(a) we see that our optimistic scenario predicts

 GCN will be received 1 s before merger per year, while our pessimistic scenario predicts

 GCN will be received 1 s before merger per year considering the higher end of the BNS rate. Figure 4(b) predicts that ∼9 events will be detected per year, out of which ∼20% (∼1.3%) will be detected 10 s (60 s) before merger. Further, ∼3% of the detectable events (∼1 BNS every 3–4 yr) will be detected 10 s prior to merger and have a localization less than 100 deg2 at O4 sensitivities. This highlights the need for continued latency improvements in advance of O4 to maximize the potential of capturing prompt emission.


Solar system planetary bodies are generally volatile depleted compared to chondritic meteorites (e.g., O’Neill & Palme 2008; Braukmüller et al. 2019). For example, Rb–Sr and K–U are elemental pairs with similar geochemical behaviors, but Rb and K are much more volatile than Sr and U, and Earth has Rb/Sr and K/U ratios that are ∼10 times lower than that of Ivuna-like carbonaceous (CI) chondrites (Allègre et al. 2001; Halliday & Porcelli 2001). However, the mechanisms and conditions that led to this relative volatile-element depletion remain ambiguous. The stable isotopes of elements can be fractionated by volatilization processes and the shifts in their isotopic compositions have been applied to understand volatile-element depletion processes (Humayun & Clayton 1995; Luck et al. 2005; Hin et al. 2017; Kato & Moynier 2017b; Pringle & Moynier 2017; Young et al. 2019; Bloom et al. 2020; Hellmann et al. 2021). Cr is a moderately volatile element (Lodders 2003; Wood et al. 2019) that may show a special behavior during volatilization at higher temperatures. Under moderately reducing to oxidizing conditions (fO2 > IW+1), relevant during evaporation processes from magma oceans on planets (Visscher & Fegley 2013), Cr may be present in the gas as oxide species CrO(g), CrO2(g) and CrO3(g) (Chase 1998). Equilibrium isotopic fractionation between these gas phase species and melt will cause enrichment of the lighter Cr isotopes in the condensed phases, in which Cr is either di- or trivalent, relative to the vapor (Sossi et al. 2018). This contrasts with its speciation under more reduced nebular conditions in which Cr may be present as Cr2+ (major; in silicates) or Cr0 (minor; in Fe–Ni metal) in condensed phases and Cr0 in the gas (Grossman et al. 2008), in which case, through either equilibrium or kinetic fractionation, the condensed phase will be enriched in the heavy Cr isotopes. Solids or magmas that evaporated significant quantities of moderately volatile elements such as Zn, K, Rb, or Ga normally show enrichments of heavy isotopes of these elements, both for kinetic and equilibrium isotope fractionation processes (Paniello et al. 2012; Wang & Jacobsen 2016; Kato & Moynier 2017a; Pringle & Moynier 2017). The behavior of Cr during evaporation has been a motivation to apply this element to study the origin of volatile depletion in the Moon and of the eucrite parent body (likely asteroid 4-Vesta) (Sossi et al. 2018; Zhu et al. 2019b). Both lunar samples and HED meteorites from Vesta are enriched in the lighter isotopes of Cr compared to Earth and chondrites. These isotopic enrichments have been interpreted to reflect equilibrium Cr isotope fractionation between melt (reduced, and enriched in CrO) and gas (oxidized, and enriched in CrO2) during evaporation from a magma ocean. Given that at equilibrium, isotopic fractionation scales with 1/T
2, the Cr evaporation from the Moon and Vesta was estimated to have occurred under relatively low temperatures (∼2000 K), which in the case of the Moon is comparable to the surface temperature of a lunar magma ocean (Sossi et al. 2018).


The behavior of Cr during evaporation has been a motivation to apply this element to study the origin of volatile depletion in the Moon and of the eucrite parent body (likely asteroid 4-Vesta)


Given that at equilibrium, isotopic fractionation scales with 1/T
2, the Cr evaporation from the Moon and Vesta was estimated to have occurred under relatively low temperatures (∼2000 K), which in the case of the Moon is comparable to the surface temperature of a lunar magma ocean


The semiforbidden [C iii] λ1907 + C iii] λ1909 doublet (hereafter C iii] λ1909 or C iii]) is one of the strongest nebular emission line features observed in the rest-frame ultraviolet (rest-UV) spectrum of low-metallicity, 12+log(O/H) ≤ 8.4 (Z  0.5 Z⊙) star-forming galaxies (SFGs) both locally (Garnett et al. 1995; Rigby et al. 2015; Berg et al. 2016, 2019; Senchyna et al. 2017, 2019) and at high redshifts (Fosbury et al. 2003; Erb et al. 2010; Bayliss et al. 2014; Stark et al. 2014, 2015a, 2017; de Barros et al. 2016; Vanzella et al. 2016; Amorín et al. 2017; Laporte et al. 2017; Maseda et al. 2017; Berg et al. 2018; Hutchison et al. 2019; Le Fèvre et al. 2019). The C iii] emission line is frequently observed in gravitationally lensed SFGs at z > 2–6 as the strongest line after Lyα λ1216 emission. Since the Lyα emission at z > 6 is expected to be severely quenched by the increasingly neutral intergalactic medium (IGM), C iii] is emerging as an alternative redshift indicator for galaxies in the reionization era (Stark et al. 2015a, 2017; Ding et al. 2017). At z > 6, the rest-UV spectrum of galaxies is redshifted to the near-infrared (NIR) wavelengths and is accessible to the spectrographs on upcoming facilities (such as the James Webb Space Telescope (JWST) and the >20 m class Extremely Large Telescopes), which will address the key goal of identifying the sources of reionization. In recent years, there has been considerable effort to develop spectral diagnostics involving the C iii] emission line and other UV spectral features (e.g., C iv λλ1548, 1550, He ii λ1640, and O iii] λλ1661, 1666 (hereafter O iii] λ1663 or O iii])) that can be used to understand the nature of the reionizers, their hard ionizing continua, and the physical conditions in their ISM (Feltre et al. 2016; Gutkin et al. 2016; Jaskot & Ravindranath 2016, hereafter JR16; Byler et al. 2018; Nakajima et al. 2018b). The C iii] emission observed at high redshifts is mostly from strongly lensed low-metallicity (0.5 Z⊙), low-mass (1010 M⊙) galaxies with high specific star formation rates (SSFRs ≳ 2 Gyr−1), and their C iii] equivalent widths (EWs) show a broad range: EW(C iii]) ∼ 3–25 Å (Fosbury et al. 2003; Erb et al. 2010; Bayliss et al. 2014; Stark et al. 2014; de Barros et al. 2016; Vanzella et al. 2016; Berg et al. 2018). The composite rest-UV spectra of Lyman-break galaxies at z ∼ 3 (Shapley et al. 2003) and SFGs at z = 1–2.5 (Steidel et al. 2016; Amorín et al. 2017) also reveal the C iii] λ1909 and O iii] λ1663 nebular lines at subsolar metallicities.


While the properties of the SLSN hosts themselves are of interest, they are most diagnostic when compared to other classes of extragalactic transient whose progenitors are better understood. To this end, we employ a comparison sample of LGRB and CCSN3 host galaxies. In principle, CCSNe should trace all core-collapse events, although the mass function means they will be dominated by stars at the lower mass end (∼8 M⊙ to ∼25 M⊙). There also remains a possibility that some very massive stars can undergo core collapse without yielding a LSNe (e.g. Smartt 2009; Ugliano et al. 2012; Kochanek 2014) such that CCSNe samples might only provide a census of lower mass core collapsing stars (e.g. 8  M*  20 M⊙). Indeed, constraints from explosion parameters have shown the majority of CCSNe to be consistent with lower mass progenitors, as opposed to more massive Wolf–Rayet stars (Cano 2013; Lyman et al. 2016) GRBs likely represent a population with rather larger initial masses (Larsson et al. 2007; Raskin et al. 2008). LGRBs are now known to be associated with the core collapse of massive stars, and broad line SN Ic are near ubiquitously associated with low-z events (where such signatures can be seen; Hjorth et al. 2012). When compared to the hosts of CCSNe they are generally smaller and of lower luminosity, consistent with an origin in galaxies of lower metallicity (Fruchter et al. 2006; Svensson et al. 2010). In relatively local examples, where spatially resolved gas phase metallicities can be obtained, these indeed appear to be lower for GRBs than for CCSNe, even in cases where the luminosity of the galaxy is relatively high (i.e. the GRB host galaxies lie off the mass–metallicity relation; Modjaz et al. 2008; Graham & Fruchter 2013). Hence, comparing the hosts of SLSNe to these events allows us to test the large-scale environments of SLSNe against those of the bulk core-collapse population and a subset which appears to derive largely from massive stars at lower metallicity, although we note that agreement on this matter is not complete (e.g. Podsiadlowski, Joss & Hsu 1992; Eldridge, Izzard & Tout 2008; Smartt 2009; Drout et al. 2011). By exploiting both LGRB and CCSN host samples we can ascertain if there is a strong metallicity dependence in SLSN production, and if this is more or less extreme than that observed in GRB hosts.


In relatively local examples, where spatially resolved gas phase metallicities can be obtained, these indeed appear to be lower for GRBs than for CCSNe, even in cases where the luminosity of the galaxy is relatively high (i.e. the GRB host galaxies lie off the mass–metallicity relation;


The distribution of optical depth values at 10 au and 1.3 mm (Figure 7, top) has its maximum at

log

τ

1.3

mm
,
10

au

=
−
0.25

 (corresponding to

τ

1.3

mm
,
10

au

∼
0.5

), and a secondary peak at

log

τ

1.3

mm
,
10

au

=
−
1

. We note that the shape of this distribution is determined mostly by Taurus, given the lack of sufficient long-wavelength data for most objects in Chamaeleon I and Ophiuchus, and the distributions in these regions appear to be broader than the one in Taurus (again, this interpretation is limited by the small number statistics in these regions). For comparison, reasonable assumptions about the dust opacity and surface density based on observations of the solar system bodies yield

τ

1

mm

=
1

 at ∼10 au for the Minimum Mass Solar Nebula (Davis 2005), suggesting that several of the modeled protoplanetary disks may have optical depth profiles (and hence possibly surface densities) similar to that of the parental disk of the solar system. In the case of β, values smaller than the one measured for the ISM (∼1.6–2; see Draine 2006 and references therein) imply some degree of grain growth. Almost the entirety of the Taurus and Ophiuchus distributions (and part of Chamaeleon I) are constrained within that value, in correspondence with the observational result discussed in Section 3.3. As with

α

mm

, Chamaeleon I shows a different behavior (an excess of high β values) that will be discussed further on. We note that the distributions of β should be considered with caution, not only due to the aforementioned caveats, but also because degenerate cases have been removed from the analysis. Because these occur when

α
=
2

, this procedure inevitably discards objects with

β
∼
0

. There is a tentative bimodality in both distributions, especially in the case of β, with a tentative secondary peak occurring at ∼1.4–1.5. Given the limited size of the modeled sample and simplicity of the models used, we do not investigate this issue in detail here. However, we speculate that, if real, it may hint at a quick transition from micron-sized grains (large β values) to mm/cm-sized dust (smaller β).


Our SBs and MSs are located in a dense environment. To better understand the environmental effect on gas turbulence, we compare the four cluster members with field galaxies from the literature. The data included in Figs. 5 and 6 contain molecular and ionized gas observations. The ionized gas observations of the star-forming galaxies, sorted by redshifts from the lowest to the highest, are from surveys GHASP (log M*[M⊙]=9.4–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.6; Epinat et al. 2010), DYNAMO (log M*[M⊙]=9.0–11.8; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.3; Green et al. 2014), MUSE and KMOS (log M*[M⊙]=8.0-11.1; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.4, 9.8 at z ∼ 0.7, 1.3; Swinbank et al. 2017), KROSS (log M*[M⊙] = 8.7–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.9; Johnson et al. 2018), KMOS3D (log M*[M⊙]=9.0–11.7; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.5, 10.6, 10.7 at z ∼ 0.9, 1.5, 2.3; Wisnioski et al. 2015; Übler et al. 2019), MASSIV (log M*[M⊙]=9.4–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.2; Epinat et al. 2012), SIGMA (log M*[M⊙]=9.2–11.8; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.0; Simons et al. 2016), SINS (log M*[M⊙]=9.8–11.5; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.8; Förster Schreiber et al. 2009; Cresci et al. 2009), LAW09 (log M*[M⊙]=9.0–10.9; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.3; Law et al. 2009), AMAZE (log M*[M⊙]=9.2–10.6; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.0; Gnerucci et al. 2011), and KDS (log M*[M⊙]=9.0–10.5; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.8; Turner et al. 2017). The molecular gas observations of the star-forming galaxies, sorted by redshifts, are from the HERACLES survey (log M*[M⊙]=7.1–10.9; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.5; Leroy et al. 2008, 2009) and the PHIBSS survey (log M*[M⊙]=10.6-11.2; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=11.0; Tacconi et al. 2013). We note that although the velocity dispersion measured from the molecular gas is ∼10−15 km s−1 lower than from the ionized gas (with extra contributions from thermal broadening and expansion of the HII regions) in the local Universe, this difference becomes smaller with increasing redshift (Übler et al. 2019). Some field SBs with individual CO observations (Calistro Rivera et al. 2018; Barro et al. 2017; Swinbank et al. 2011; Tadaki et al. 2017, 2019) are indicated with blue triangles in Figs. 5 and 6. When possible, we also identified the field starbursts (orange symbols in Figs. 5 and 6) within the above star-forming samples, requiring their SFR to be at least 0.5 dex higher than the MS. In general, the gas in these field starbursts is slightly more turbulent than in main-sequence galaxies, showing a higher σ0, especially at 2   z   4. A similar trend of increasing σ0 as galaxies move above the MS at a fixed stellar mass is shown for star-forming galaxies at z = 0−3 (e.g., Perna et al. 2022; Wisnioski et al. 2015), suggesting that mergers or interactions would increase gas σ0 as they enhance star formation.


Analytical expressions for the IP and OP spectra obtained by IR–MAIRS are theorized by Itoh et al. (2009) and Shioya et al. (2019). In short, Shioya et al. showed both experimentally and theoretically that the IP spectrum (A
IP) quantitatively corresponds to the normal-incidence transmission spectrum,

Athinθ=0

 (see the Appendix for details; Shioya et al. 2019).
5

AIP=Athinθ=0=8πdaλln102n2k2xy.

A
OP is expressed similarly to A
IP, while the TO energy-loss function is replaced with a longitudinal optic (LO) energy-loss function: i.e., (2n
2
k
2z
)/(

n22

 + k
2z

2)2 (Shioya et al. 2019; Hama et al. 2020).
6

AOP=8πdaHλln10(2n2k2z)n22+k2z22,

where H is a substrate-specific correction factor (0.33 for Si) that accounts for the intensity ratio of the electric fields at the optical interface along the surface-parallel and surface-perpendicular directions (Shioya et al. 2019). Because amorphous H2O vapor-deposited at 80–100 K has values of k
2 that are approximately 2 orders of magnitude smaller than n
2 (k
2 ≪n
2) at 3750–3650 cm−1 in the bulk (Mastrapa et al. 2009), Equation (6) simplifies to
7

AOP≈8πdaHλln10(2n2k2z)n24.

From Equations (5) and (7), k
2xy
 and k
2z
 are derived as
8

k2xy=AIPλln1016πdan2

9

k2z=n24HAOPλln1016πdan2.

Therefore, optically isotropic k
2 and σ can be obtained from A
IP and A
OP even for a thin sample (Shioya et al. 2020):
10

k2=2k2xy+k2z3=2AIP+n24HAOP3λln1016πdan2

11

σ=4πdk2λN=4πdλN2AIP+n24HAOP3λln1016πdan2=2AIP+n24HAOP3ln104an2N.

The band strength β (cm molecule−1) can be also obtained by integrating

σ(ν˜)

 at

ν˜=

 3710–3680 cm−1.
12

β=∫σν˜dν˜=23∫AIPν˜dν˜+n24H3∫AOPν˜dν˜ln104an2N,

where

AIPν˜

 and

AOPν˜

 is the IP and OP absorbance at a given wavenumber (

ν˜

), respectively. This study uses n
1 = 1 for vacuum, n
2 = 1.26 for amorphous water at 90 K (Kofman et al. 2019), and n
3=3.41 for Si (Tasumi 2014; Shioya et al. 2019). Hence, a = 0.290 and

n24

H = 0.832. Seven independent measurements for amorphous water formed after 32 minutes of exposure at 90 K gave A
IP  = 0 [

∫AIPν˜dν˜=0

 (cm−1)] and

n24HAOP

 = 6.2 ± 0.4 × 10−5 at 3696 cm−1 [

n24H∫AOPν˜dν˜

 = 8.4 ± 0.4 × 10−4 (cm−1)], considering amorphous water formed on both sides of the Si substrate. Therefore, σ is derived as 1.0 ± 0.2 × 10−18 cm2 at 3696 cm−1 from Equation (11) by adopting N = 3.3 ± 0.6 × 1013 molecules cm−2. β is also derived as 1.4 ± 0.3 × 10−17 cm molecule−1 at

ν̃

 = 3710–3680 cm−1 from Equation (12).


Radio interferometric observations of the redshifted 21-cm signal directly measures the complex visibilities that are the Fourier components of the intensity distribution on the sky. The radio telescope typically has a finite beam that allows us to use the ‘flat-sky’ approximation. Ideally, the fields κ and δT are expanded in the basis of spherical harmonics. For convenience, we use a simplified expression for the angular power spectrum by considering the flat sky approximation whereby we can use the Fourier basis. Using this simplifying assumption, we may approximately write the cross-correlation angular power spectrum as (Dash & Guha Sarkar 2021)
$$\begin{eqnarray}
C^{ T \kappa }_\ell &=& \frac{1 }{\pi (\chi _2- \chi _1)} \sum _{\chi _1}^{\chi _2} \frac{\Delta \chi }{\chi ^2} ~ \mathcal {A}_T (\chi) \mathcal {A}_\kappa (\chi) D_{+}^2 (\chi) \int _0^{\infty } \mathrm{ d}k_{\parallel }\\
&&\times \,\left[ 1 + \beta _T(\chi) \frac{k_{\parallel }^2}{k^2} \right] P (k),
\end{eqnarray}$$where $k = \sqrt{k_{\parallel }^2 + \left(\frac{\ell }{\chi } \right)^2 }$, D+ is the growing mode of density fluctuations, and βT = f/bT is the redshift distortion factor – the ratio of the logarithmic growth rate f and the bias function and bT(k, z). The redshift-dependent function $\mathcal {A}_{T}$ is given by (Bharadwaj & Ali 2005; Datta, Choudhury & Bharadwaj 2007; Guha Sarkar et al. 2012)
(6)$$\begin{eqnarray}
\mathcal {A}_{T} = 4.0 \, {\rm {mK}} \, b_{T} \, {\bar{x}_{\rm H\,{\small I}}}(1 + z)^2\left(\frac{\Omega _{b0} h^2}{0.02} \right) \left(\frac{0.7}{h} \right) \left(\frac{H_0}{H(z)} \right) .
\end{eqnarray}$$The quantity bT(k, z) is the bias function defined as ratio of H i-21 cm power spectrum to dark matter power spectrum $b_T^2 = P_{\mathrm{ H}\,{\small I}}(z)/P(z)$. In the post-reionization epoch z  6, the neutral hydrogen fraction remains with a value ${\bar{x}_{\rm H\,{\small I}}} = 2.45 \times 10^{-2}$ (adopted from Noterdaeme et al. 2009; Zafar et al. 2013). The clustering of the post-reionization H i is quantified using bT. On sub-Jean’s length, the bias is scale dependent (Fang et al. 1993). However, on large-scale the bias is known to be scale independent. The scale above which the bias is linear, is however sensitive to the redshift. Post-reionization H i bias is studied extensively using N-body simulations (Bagla et al. 2010; Guha Sarkar et al. 2012; Sarkar et al. 2016; Carucci, Villaescusa-Navarro & Viel 2017). These simulations demonstrate that the large-scale linear bias increases with redshift for 1  z  4 (Marín et al. 2010). We have adopted the fitting formula for the bias bT(k, z) as a function of both redshift z and scale k (Guha Sarkar et al. 2012; Sarkar et al. 2016) of the post-reionization signal as
(7)$$\begin{eqnarray}
b_{T}(k,z) = \sum _{m=0}^{4} \sum _{n=0}^{2} c(m,n) k^{m}z^{n} .
\end{eqnarray}$$The coefficients c(m, n) in the fit function are adopted from Sarkar et al. (2016).


Post-reionization H i bias is studied extensively using N-body simulations


We have adopted the fitting formula for the bias bT(k, z) as a function of both redshift z and scale k

of the post-reionization signal as
(7)$$\begin{eqnarray}
b_{T}(k,z) = \sum _{m=0}^{4} \sum _{n=0}^{2} c(m,n) k^{m}z^{n} .
\end{eqnarray}$$


To investigate this further, we performed an additional test where we froze the slopes for both the fossils and groups to 3.0 and only fit the normalisation for both samples. The value of 3.0 for the slope was chosen as generally most groups and clusters can be described well by this particular slope, after factoring in selection effects. The fossils now have a normalisation of 0.38 ± 0.11 and the groups sample have a normalisation of 0.22 ± 0.13. We now proceeded to remove the influence of selection effects on the normalisation of the scaling relation for both samples by generating mock samples of objects as in Bharadwaj et al. (2015) by varying the input normalisations. For each mock sample, the slope was always fixed to 3 and the intrinsic scatter was fixed to the observed values. Flux and luminosity cuts were applied to both the 400d sample and the groups sample to match the true sample of objects. For the 400d fossil sample, the selection criteria were taken from Voevodkin et al. (2010), and can be described as follows: a lower flux cut of 1.4 ×  10-13 erg / s / cm2, an upper redshift cut of 0.2, and a lower luminosity cut of 1043 erg/s. Here, fluxes and luminosities are in the ROSAT (0.5–2.0 keV) band. After performing the bias corrections, the fossils scaling relation has a normalisation of 0.30 ± 0.10 vs. 0.0078 ± 0.13 for the groups, indicating that the large normalisations seem to persist even after accounting for selection effects. Despite a nearly 2.3σ higher normalisation for fossils with respect to non-fossils, we still only treat this finding as an indication because on top of the statistical uncertainties in both quantities (which is large currently for the 400d fossils), there could also be an additional systematic uncertainty introduced due to differences in the luminosity determination in the different parent catalogues. Secondly, the archival nature of this subsample of 400d objects is biased towards systems lacking a SCC (Table 2) and, assuming that the remaining systems are SCC, then adding these objects could potentially increase the normalisation for the scaling relation, and the difference between the fossils and the groups sample would be higher than what we demonstrate here. We plan to explore this in greater detail in a future study of fossils scaling relations.


In this work, we focus on constraining the variability of the fine-structure constant, α ≡ e2/ℏc, which represents the coupling strength of the electromagnetic force. In the last 15 years, there have been many attempts to measure this constant in absorption systems along the lines of sight to distant quasars. The approach called the ‘Many Multiplet’ (MM) method, pioneered by Dzuba, Flambaum & Webb (1999) and Webb et al. (1999), compares the relative velocity spacing between different metal ion transitions and relates it to possible variation in α. For example, considering just a single transition, variation in α is related to the velocity shift Δvi of a transition

(1)

\begin{equation}
\Delta \alpha /\alpha \equiv \frac{\alpha _{\rm obs}-\alpha _{\rm lab}}{\alpha _{\rm lab}}\approx \frac{-\Delta v_i}{2c}\frac{\omega _i}{q_i},
\end{equation}

where c is the speed of light, qi is sensitivity of the transition to α-variation, calculated from many body-relativistic corrections to the energy levels of ions and ωi is its wavenumber measured in the laboratory. There have been two MM method studies of large absorber samples: the Keck High Resolution Echelle Spectrometer (HIRES/Keck) sample of 143 absorption systems (Webb et al. 1999, 2001; Murphy et al. 2001a, 2004; Murphy, Webb & Flambaum 2003b) and the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope (UVES/VLT) sample of 154 absorption systems (Webb et al. 2011; King et al. 2012). These studies reported weighted mean values of Δα/α = −5.7 ± 1.1 parts per million (ppm) and 2.29 ± 0.95 ppm, respectively. Webb et al. (2011) and King et al. (2012) also combined both samples and found a 4.1σ statistical preference for a dipole-like variation in α across the sky. Contrary to these studies, several other attempts to measure α from individual absorbers (Quast, Reimers & Levshakov 2004; Levshakov et al. 2005, 2006, 2007; Chand et al. 2006) or smaller samples of absorption systems (Chand et al. 2004; Srianand et al. 2004) reported Δα/α consistent with zero. However, these analyses contained shortcomings that produced biased values and/or considerably underestimated error bars (Murphy, Webb & Flambaum 2007a, 2008b; cf. Srianand et al., 2007). For example, a reanalysis, including remodelling of the Chand et al. (2004) spectra by Wilczynska et al. (2015), yielded a weighted mean of Δα/α = 2.2 ± 2.3 ppm which, given the distribution of the 18 quasars across the sky, was not inconsistent with the evidence for dipole-like α-variation.


There have been two MM method studies of large absorber samples: the Keck High Resolution Echelle Spectrometer (HIRES/Keck) sample of 143 absorption systems

These studies reported weighted mean values of Δα/α = −5.7 ± 1.1 parts per million (ppm) and 2.29 ± 0.95 ppm, respectively.


Even when imaged into 40 km s−1 channels, CO(2−1) absorption is apparent in NGC 4374 and IC 4296 against their strong nuclear continuum sources. While extragalactic CO absorption is not frequently observed due to low equivalent widths, ALMA’s high angular resolution and sensitivity allow its detection against some nearby active nuclei (e.g., Davis et al. 2014; Rangwala et al. 2015). To better characterize the central absorption features, we also imaged their visibility data (without continuum subtraction) into cubes with 1.28 km s−1 channels. The primary absorption in IC 4296 is consistent with the galaxy’s assumed systemic velocity (Table 1), although there may be a small secondary absorption feature ∼15 km s−1 redward of

. The CO(1−2) transition seen in the nucleus of NGC 4374, however, is redshifted by ∼100 km s−1 from the

 value derived using stellar kinematics (Emsellem et al. 2004) and by ∼50 km s−1 with respect to the best-fitting

 from ionized gas dynamical modeling (e.g., Walsh et al. 2010). The absorption lines in these two galaxies have narrow widths (≲7 km s−1 FWHM) that are only slightly smaller than the minimum line widths in our CO-bright subsample. Maximum absorption depths are (−3.26 ± 0.74) and (−5.25 ± 0.35) K with integrated absorption intensities

 of (19.4 ± 2.5) and (44.3 ± 1.4) K km s−1 for NGC 4374 and IC 4296, respectively. From these

 measurements, we estimate corresponding H2 column densities

 of

 and

 cm−2 after assuming an absorption depth ratio CO(1−2)/CO(0−1) ≈ 0.6 (based on 13CO absorption line ratios; e.g., see Eckart et al. 1990) and a CO-to-H2 conversion factor

 corresponding to

 M pc−2 (K km s−1)−1 (see Section 3.1.2; Sandstrom et al. 2013). These

 estimates are much more uncertain than indicated by the formal associated uncertainties, since the

 and CO(1−2)/CO(1−0) values carry large systematic uncertainties, and faint nuclear CO emission may somewhat dilute these equivalent widths.


In order to examine how our qcr(M1) would affect the brightness distribution of the violent merger scenario, we adopt the same assumptions as in Ruiter et al. (2013) except for qcr(M1). For the primary at the time of merging, we use the same WD mass distribution as Ruiter et al. (2013; see also Ruiter et al. 2011), although it is highly uncertain whether the WD can increase its mass by avoiding the formation of a common envelope during the very rapid accretion from the He star companion. We also assume the same

m

WD

–

M

bol

 relation of SNe Ia as Ruiter et al. (2013; see Figure 4 in Ruiter et al. 2013; Sim et al. 2010) and a flat mass ratio distribution of DD systems for simplicity. Using either our critical mass ratio or theirs, we calculated the brightness distribution and compared them with each other (Figure 11). We find that there is no significant difference between them qualitatively. This result implies that the qcr of the violent merger-induced explosion is not so crucial for the brightness distribution of SNe Ia, as mentioned in Ruiter et al. (2013). Our results are closest to the η = 1.5 case among Ruiter et al.'s three cases. In the cases of our qcr and Ruiter et al.'s η = 1.5, the brightness distribution concentrates around −19.0 mag. In Figure 11, we add the observational volume-limited brightness distributions of SNe Ia. One in panel (a) is derived by the Lick Observatory Supernova Search (LOSS; Li et al. 2011), and the other in panel (b) is obtained by ROTSE-IIIb (see Table 1 in Quimby et al. 2012). The fraction of faint events in our models is lower than the observation of LOSS. This discrepancy might decrease if we consider the viewing angle effects (see also the discussion in Ruiter et al. 2013), which might be increasingly important especially for a large value of qcr. On the other hand, the results of ROTSE-IIIb are consistent with ours, except most luminous (≲−19.5 mag) events. It should be noted, however, that there are large uncertainties for the observational brightness distribution (e.g., Quimby et al. 2012) and our models are too simple to be compared with the observational results. More detailed studies are required to reach a definitive conclusion.


For the primary at the time of merging, we use the same WD mass distribution as

although it is highly uncertain whether the WD can increase its mass by avoiding the formation of a common envelope during the very rapid accretion from the He star companion.


We also assume the same

m

WD

–

M

bol

 relation of SNe Ia as


This result implies that the qcr of the violent merger-induced explosion is not so crucial for the brightness distribution of SNe Ia, as mentioned in

Using either our critical mass ratio or theirs, we calculated the brightness distribution and compared them with each other (Figure 11).

We find that there is no significant difference between them qualitatively.


This discrepancy might decrease if we consider the viewing angle effects (see also the discussion in

which might be increasingly important especially for a large value of qcr.


Since its discovery in 1983, Phaethon has remained a well-studied yet poorly understood object, known to exhibit both cometary and asteroidal characteristics, and widely presumed to be the parent to the Earth-crossing Geminid meteor shower (Green & Kowal 1983; Gustafson 1989; Williams & Wu 1993). Significant efforts have been made to study the physical properties of the asteroid such as size (6 km in diameter, Taylor et al. 2019), spectral shape (B type, e.g., Licandro et al. 2007), and albedo (0.11, Green et al. 1985), as well as the Geminids (e.g., Hughes & McBride 1989; Jenniskens 1994; Arlt & Rendtel 2006; Blaauw 2017), their relationship to one another (e.g., Gustafson 1989; de León et al. 2010), and the proposed relationship to asteroids 2005 UD and 1999 YC (e.g., Ohtsuka et al. 2006, 2008; Kareta et al. 2021). Several Earth-based studies of Phaethon have sought to investigate the perihelion activity of the asteroid, attempting—but failing—to detect small fragments (Jewitt et al. 2018; Ye et al. 2018) or nearby dust grains (Jewitt et al. 2019) released during its close (0.14 au) passage by the Sun. Despite this, spacecraft observations of Phaethon’s activity at perihelion obtained by the Sun-Earth Connection Coronal and Heliospheric Investigation (Howard et al. 2008) on NASA’s Solar Terrestrial Relations Observatory (Kaiser et al. 2008) during the asteroid’s perihelion passages in 2009, 2012, 2016, and 2022, have shown that the asteroid is clearly active at perihelion, displaying a short tail and reported to be releasing mass on the order ∼2.5 × 108
a
1 kg (where a
1 is the grain radius in millimeters, and noting that the size of dust released at perihelion remains unknown; Jewitt & Li 2010; Jewitt et al. 2013; Hui & Li 2017). This is insufficient by orders of magnitude to support the best estimates for the mass of the Geminids. Shortly before submission of this manuscript, Phaethon was also observed by the Solar and Heliospheric Observatory Large Angle Spectrometric Coronagraph (Brueckner et al. 1995) C2 coronagraph during its 2022 May 15 perihelion passage, constituting the first known detection of Phaethon by this instrument.


Since its discovery in 1983, Phaethon has remained a well-studied yet poorly understood object, known to exhibit both cometary and asteroidal characteristics, and widely presumed to be the parent to the Earth-crossing Geminid meteor shower


Significant efforts have been made to study

and the proposed relationship to asteroids 2005 UD and 1999 YC (e.g.,


We consider a radio galaxy of SMBH mass M = 109
M
9
M
⊙ with a mass accretion rate of

Ṁ=ṁLEdd/c2≃1.4×1022M9ṁ−4gcm−2

, where c is the speed of light and L
Edd is the Eddington luminosity. The gravitational radius of the BH is r

g
 = GM/c
2 ≃ 1.5 × 1014
M
9 cm. We consider that the accretion flow is in the MAD state, and then the magnetic field strength around the SMBH is estimated to be

Bmad=ṀcΦmad2/(4π2rg2)≃1.1×103M9−1/2ṁ−41/2Φmad,1.7G

 (e.g., Yuan & Narayan 2014), where Φmad ≈ 50Φmad,1.7 is the saturated magnetic flux (Tchekhovskoy et al. 2011; Narayan et al. 2012; McKinney et al. 2012; White et al. 2019). The high-resolution GRMHD simulation with a BH spin parameter a = 0.9375 suggests that magnetic reconnection occurs at a distance of r
rec ∼ 2r

g
 (Ripperda et al. 2022). The value of r
rec could depend on a or other parameters, but we fix r
rec = 2r

g
 throughout this paper for simplicity. We estimate the reconnecting magnetic field strength to be (see Appendix A)
1

Brec≈2Bmadrrecrg−2≃3.9×102M9−1/2ṁ−41/2Φrec,1.2G,

where

Φrec=2Φmad(rrec/rg)−2

 is the effective magnetic flux at the reconnection region. The magnetosphere will be formed around the SMBH. The minimum number density of the magnetosphere that can maintain the electric current for the BZ process is (Goldreich & Julian 1969; Levinson & Cerutti 2018)
2

nGJ=BrecΩF2πec≈Brec8πerg≃2.2×10−4M9−3/2ṁ−41/2Φrec,1.2cm−3,

where Ω
F
 ≈ ac/(4r

g
) is the field line angular velocity (Tchekhovskoy et al. 2010; Nathanail & Contopoulos 2014; Ogihara et al. 2021; Camilloni et al. 2022) and we assume the BH spin parameter as a ∼ 1. For the magnetosphere, which consists of e
+
e
− pair plasma with the density n
GJ, the magnetization parameter is
3

σB,GJ=Brec24πnGJmec2≈6.8×1013M91/2ṁ−41/2Φrec,1.2.

This value should be regarded as an upper limit, because the number density of the magnetosphere can be higher than n
GJ. Various mechanisms of particle injection into the BH magnetosphere have been proposed (see Appendix B), which can lead to multiplicity of κ
± ≡ n/n
GJ ∼ 1 − 103. This results in the magnetization parameter of σ

B
 ≳ 1010.


It is well known that energetic electrons can boost the CMB photons to higher energy through inverse Compton scattering creating a distortion in the CMB blackbody spectrum. If the energy distribution of the electrons is non-relativistic and thermal, then the distortion has y-distortion shape (Zeldovich & Sunyaev 1969) (see Section 6 for further discussions on corrections to y-distortion). For relativistic electrons, with a Lorentz factor γ, a photon with energy ϵ gets boosted to γ2ϵ. In this case, the spectral distortion shape will be a function of the electron energy distribution. The intensity of the CMB spectrum per frequency is given by,
(9)$$\begin{eqnarray*}
I_{\nu }(x)=2\frac{(k_{\rm {B}}T_{\rm {CMB}})^3}{({hc})^2}\frac{x^3}{e^x-1}=I_0I(x),
\end{eqnarray*}$$where $I(x)=\frac{x^3}{e^x-1}$, I(x) is the dimensionless intensity, x is the dimensional frequency which is given by $x=\frac{E_{\gamma }}{k_{\rm {B}}T_{\rm {CMB}}}$, where Eγ is the energy of photon, kB is the Boltzmann constant, TCMB is the CMB temperature, and other symbols have usual meanings. The intensity of distorted CMB spectrum is independent of redshift for a given population of electrons. The CMB distortion in the optically thin limit can be written as (Zeldovich & Sunyaev 1969; Birkinshaw 1999),
(10)$$\begin{eqnarray*}
\Delta I(x)=(j(x)-i(x))\tau ,
\end{eqnarray*}$$where j(x) is the spectral intensity of photons at frequency x after being upscattered while i(x) is the intensity of photons at frequency x before upscattering, τ = σT∫nedl where ne is the electron number density and dl is the line-of-sight width of this electron population. i(x) is non-zero over the range of x where the CMB photons are present. Equation (10) can be recast to include a y-parameter as ΔI(x) = yg(x), where $y=\frac{\sigma _{\rm {T}}}{m_{\rm {e}}\rm {c}^2}\int n_{\rm {e}}k_{\rm {B}}\overset{\sim }{T_{\rm {e}}}\mathrm{ d}l$ with $k_{\rm {B}}\overset{\sim }{T_{\rm {e}}}=\frac{P_{\rm {e}}}{n_{\rm {e}}}$. Here $\overset{\sim }{T_{\rm {e}}}$ refers to a fictitious temperature scale for a non-thermal distribution to make the equations look suggestively similar to thermal SZ (more details can be found in Enßlin & Kaiser 2000). In order to distinguish non-thermal spectral distortion shape from y-type distortion (the well-known thermal $\rm {SZ}$ effect), we will refer to non-thermal distortion amplitude as yNT, such that,
(11)$$\begin{eqnarray*}
\Delta I_{\rm {NT}}(x)=y_{\rm {NT}}g_{\rm {NT}}(x),
\end{eqnarray*}$$where gNT(x) is the spectral distortion function resulting from scattering of the CMB in a non-thermal population of clusters. We use the formalism of Enßlin & Kaiser (2000) to compute gNT(x) and we refer the reader to the paper for details. The pressure for a distribution of relativistic electrons is given by,
(12)$$\begin{eqnarray*}
P_{\rm {e}}=n_{\rm {e}}\int \mathrm{ d}p f_{\rm {e}}(p)\frac{1}{3}p{v}(p)m_{\rm {e}}c,
\end{eqnarray*}$$where fe(p) is the normalized electron spectrum i.e. ∫fe(p)dp = 1 with electron dimension-less momentum $p=\sqrt{(\gamma ^2-1)}$, v = βc, where γ is the Lorentz factor and β is the boost factor of energetic electrons. The number of CMB photons which get upscattered from energy x′ to x is given by,
(13)$$\begin{eqnarray*}
N(x^{\prime }-\gt x)= P(t,p)\times 2\frac{(k_{\rm {B}}T_{\rm {CMB}})^2}{({hc})^2}\frac{x^{\prime 2}\mathrm{ d}x^{\prime }}{e^{x^{\prime }}-1},
\end{eqnarray*}$$where P(t, p) is the kernel of the inverse Compton scattering which captures the kinematics of photon scattering with the electrons with electron energy p, $t=\frac{x}{x^{\prime }}$. The number of CMB photons within energy x′ and x′ + dx′ is $2\frac{(k_{\rm {B}}T_{\rm {CMB}})^2}{({hc})^2}\frac{x^{\prime 2}dx^{\prime }}{e^{x^{\prime }}-1}$ and ∫dtP(t, p) = 1, which conserves the number of photons. The formula for P(t, p) is given by (Enßlin & Kaiser 2000),
(14)$$\begin{eqnarray*}
P(t;p)&=&\frac{-3\left|(1-t)\right|}{32p^6t}[1+(10+8p^2+4p^4)t+t^2]+\frac{3(1+t)}{8p^5}\nonumber\\
&&\times \left[\frac{3+3p^2+p^4}{\sqrt{1+p^2}}-\frac{3+2p^2}{2p}(2\operatorname{arcsinh}{p}-\left| \ln {t}\right|)\right],\nonumber\\
\end{eqnarray*}$$The spectral intensity of upscattered photons per frequency, in frequency bin x and x + Δx, is given by,
(15)$$\begin{eqnarray*}
j(x)=\frac{\int \int f_{\rm {e}}(p)\mathrm{ d}p P(t,p)\frac{x^{\prime 2}\mathrm{ d}x^{\prime }}{e^{x^{\prime }}-1}x}{\Delta x}.
\end{eqnarray*}$$


to compute gNT(x) and we refer the reader to the paper for details.


Hosler et al. (1957) measured pull-off forces of spherical water-ice particles with radius r0 = 7.366 mm in either air or nitrogen vapour. Prior to their measurements, the two spheres suspended by cotton threads were brought into contact at the position of just touching, resembling the experimental setup of Nakaya & Matsumoto (1954). Therefore, we assume zero indentation indicating that equation (21) applies to their pull-off force measurements, although Hosler et al. (1957) did not report whether or not the particles rotate prior to separation. Since their experiments were carried out in a wide range of temperatures from T = 193 to 273 K and the saturated vapour pressure exponentially decreases with the inverse of the temperature, we may need to consider a variation of relative humidity with the temperature. The saturated vapour pressure psat of water molecules is given by (Kimura, Ishimoto & Mukai 1997)
(22)$$\begin{eqnarray*}
p_\mathrm{sat}(T) &=& p_\infty \exp \left({-\frac{\Delta H_\mathrm{s}}{k_\mathrm{B} T}}\right) ,
\end{eqnarray*}$$where $p_\infty = \displaystyle \lim _{T \rightarrow \infty } p_\mathrm{sat}(T)$ and ΔHs denotes the enthalpy of sublimation. According to Prialnik (1992), we assume the following form:
(23)$$\begin{eqnarray*}
p_\mathrm{sat}(T) &=& 3.56 \times {10}^{12}~\mathrm{Pa} \, \exp \left[{-\left({\frac{T}{6141.667~\mathrm{K}}}\right)^{-1}}\right] .
\end{eqnarray*}$$Fig. 10 depicts the experimental data of Hosler et al. (1957) with open circles for their measurements in air and open squares for those in nitrogen vapour. Also plotted are the temperature variations of equation (21) with the vapour pressure pv = 63.2 in air and 468 Pa in nitrogen vapour. Although the assumption of a constant pv value may be too crude to reproduce the experimental data, equation (21) approximates the experimental data of Hosler et al. (1957) in air and nitrogen vapour within a factor of 2.


NR simulations of BH–NS mergers are computationally intensive and typically conclude a few ms after the merger. To follow the post-merger dynamics, the unbound ejecta must be transferred to a hydrodynamics simulation that can accommodate the rapid expansion of the distance scale. Few studies have explored the long-term evolution of the unbound component while incorporating the effect of r-process radioactive heating, which can affect the evolution; as an estimate, the total radioactive energy deposited over the first ∼10−4 d is Erad ∼ 1049 erg (Metzger et al. 2010), which is nontrivial compared to the total kinetic energy

E

kin

∼

M

d

v

d

2

/

2
∼
few
×

10

50

 erg. Fernández et al. (2015) studied the long-term behavior using Newtonian hydrodynamics with a prescription for r-process heating; they found that heating enlarges the ejecta and smooths small-scale irregularities. This corroborated earlier work in the NS–NS merger context by Rosswog et al. (2014), who used a Newtonian smoothed particle hydrodynamics (SPH) code with heating and a nuclear network and additionally found that the ejecta reaches homology at the ≲1% level by ∼102 s and the abundance of nucleosynthesis products remains roughly unaffected by heating. These two studies used Newtonian merger simulations to initialize the unbound ejecta. Roberts et al. (2017) simulated the hydrodynamic evolution of the BH–NS merger unbound component to generate thermodynamic trajectories as inputs for a nuclear reaction network; since their main aim was to study r-process abundances, they did not study the back-reaction of the nuclear heating on the ejecta structure. Kawaguchi et al. (2021) recently examined the NS–NS post-merger evolution using 2D axisymmetric hydrodynamics with radioactive heating. They found that heating only modestly affects the ejecta structure and hydrodynamics minimally impacts the nucleosynthesis, results that are dependent on the details of the NR handoff. The overall ejecta remains mildly prolate with a lanthanide-present torus and some matter falls back to the BH-disk system.


From our MUSE

 and quasar spectra, we can estimate how many of our nebulae would have been missed by NB filter observations because of filter losses. For this thought experiment, we will assume that quasars would have been selected for NB observations if their estimated systemic redshift from corrected broad-line emission (C iv in our case, but typically Mg ii at

) would have placed the expected

 emission at the peak of the NB filter transmission. In reality, the selection is made on a range of systemic redshifts (typically within 500 km s−1 from the filter center), but we will assume that this range is absorbed by the intrinsic scatter in the C iv-corrected redshift estimation (

 km s−1 at 1σ, Shen et al. 2016). Assuming a typical NB filter width of 3000 km s−1 (FWHM), we can notice from Figure 2 that two nebulae would have been totally missed by NB observations (#2, #6) while for the other five (#1, #10, #11, #12, #16) the displacement between the

 emission and the filter transmission peak would have resulted in a substantial flux loss that would likely have compromised the detectability of the nebulae, especially in the faintest but most extended regions. However, at least nine nebulae (#3, #4, #5, #7, #8, #9, #13, #14, #15, #17), which is at least 50% of the sample, have

 emission so close to the estimated systemic redshift that they would not have been missed by NB observations, including also the systemic redshift errors from C iv or Mg ii. This factor is much larger than the 10% detection rate estimated from NB imaging. Therefore, we conclude that filter losses cannot completely explain the discrepancy between MUSE detection rates and those of NB surveys, though they can certainly reduce the NB detection rate significantly. We note that radio-loud quasars typically have much broader

 emission and therefore filter losses would be even less important for these systems. This may partially explain the higher detection rate of radio-loud nebulae versus radio-quiet ones.


The fireball model remains the most popular scenario for the gamma-ray burst (GRB) phenomenon (Cavallo & Rees 1978; Goodman 1986; Paczynski 1986; Shemi & Piran 1990; Rees & Mészáros 1992, 1994; Mészáros & Rees 1993). In this model, the GRB central engine is a stellar-mass black hole or a rapidly spinning and highly magnetized neutron star formed by either the collapse of a supermassive star (collapsar; Woosley 1993; MacFadyen & Woosley 1999; Woosley & Heger 2006) or the merger of two compact objects (Paczynski 1986; Fryer et al. 1999; Rosswog 2003). In both cases, the original explosion creates a bipolar collimated jet composed mainly of photons, electrons, positrons, and a small fraction of baryons. The relativistic explosion ejecta within the jet are not homogeneous—they form multiple high density layers, which propagate at various velocities. When the fastest layers catch up with the slowest, the charged particles contained in the layers are accelerated through mildly relativistic collisionless shocks (internal shocks; Rees & Mészáros 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998). The particles subsequently cool via emission processes such as synchrotron, Synchrotron Self Compton (SSC), and Inverse Compton (IC). The internal shock phase is usually associated with the so-called GRB prompt emission,17

17
See Pe’er (2015) for a recent review of GRB prompt emission.
 mainly observed in the keV−MeV energy range (see, e.g., the spectral catalogs by Gruber et al. 2014; von Kienlin et al. 2014) and usually lasting from a few ms up to several tens to hundreds of seconds. As the ejecta interact with the interstellar medium they slow down via relativistic collisionless shocks (external shocks; Rees & Mészáros 1992; Mészáros & Rees 1993), accelerating charged particles, which then emit non-thermal synchrotron photons. This external shock phase is usually associated with the so-called GRB afterglow emission observed at radio wavelengths up to X-rays and in some cases even up to the GeV regime hours after the prompt phase, and days and even years for the lowest frequencies. The detailed origin of the gamma-ray emission, however, is not fully understood and many theoretical difficulties remain, such as the composition of the jet, the energy dissipation mechanisms, as well as the radiation mechanisms (e.g., Zhang 2011).


The fireball model remains the most popular scenario for the gamma-ray burst (GRB) phenomenon


In this model, the GRB central engine is a stellar-mass black hole or a rapidly spinning and highly magnetized neutron star formed by either the collapse of a supermassive star


for a recent review of GRB prompt emission.


The detailed origin of the gamma-ray emission, however, is not fully understood and many theoretical difficulties remain, such as the composition of the jet, the energy dissipation mechanisms, as well as the radiation mechanisms (e.g.,


Once relatively high-resolution cross sections have been computed, it is reasonably simple to produce k-tables using what is known as the correlated k-distribution method; this method is well described in the literature (see, e.g. Lacis & Oinas 1991; Pierrehumbert 2010), and is extensively used for radiative transfer calculations in the context of planetary and substellar atmospheres (see, e.g. Irwin et al. 2008; Showman et al. 2009; Freedman et al. 2008, 2014; Lee et al. 2019; Mollière et al. 2015; Amundsen et al. 2014; Sharp & Burrows 2007; Malik et al. 2017; Drummond et al. 2016; Phillips et al. 2020). K-tables are generally considered faster (and more accurate for the same

R

=

λ

Δ
λ

$ R~=~\frac{\lambda}{\Delta \lambda} $

) than cross sections, but they also come with their own assumptions and therefore limitations (see, e.g. Rocchetto 2017). However, these can be assumed to be negligible when compared to other unknowns. The general principle is to order spectral lines within a given spectral bin, producing a smooth cumulative distribution function to represent opacity, which can be more efficiently sampled. The number of points used for sampling within a given spectral bin is determined by a set of Gaussian quadrature points, which are assigned corresponding weights. These are often chosen so as to sample the extremes of the bin more finely so as not to miss the weakest and strongest lines (i.e. the distance between sampling points within a bin is not constant). One of the methods used in this work, for example, is based on the use of Gauss Legendre polynomials (see Sect. 4 for details of the opacities produced for individual retrieval codes). K-tables are produced using a method of opacity sampling which enables low-resolution computations while still taking strong opacity fluctuations at high resolution into account; see, for example, Min (2017). The assumption made for the k-distribution method that the k-coefficients at each Guassian quadrature point are correlated, breaks down for inhomogeneous atmospheres (Lee et al. 2019)


Our starting point of modelling the different two-point functions in the 3×2pt data vector is the three-dimensional (3D) non-linear matter power spectrum P(k, z) at a given wavenumber k and redshift z. We obtain it by using either of the Boltzmann solvers CLASS11 or CAMB12 to calculate the linear power spectrum and the HALOFIT fitting formula (Smith et al. 2003) in its updated version (Takahashi et al. 2012) to turn this into the late-time non-linear power spectrum. From this 3D power spectrum, the angular power spectra required for our three two-point functions [cosmic shear (κκ), galaxy–galaxy lensing (δgκ), and galaxy–galaxy clustering (δgδg)] in the Limber approximation are given by (e.g. Limber 1953; Krause et al. 2017):
(4)$$\begin{eqnarray*}
C^{ij}_{\kappa \kappa }(\ell) = \int \mathrm{ d}\chi \frac{q^i_\kappa (\chi) q^j_\kappa (\chi)}{\chi ^2} P\left(\frac{\ell +\frac{1}{2}}{\chi }, z(\chi)\right),
\end{eqnarray*}$$(5)$$\begin{eqnarray*}
C^{ij}_{\delta _\mathrm{ g} \kappa }(\ell) = \int \mathrm{ d}\chi \frac{q^i_\delta \left(\frac{\ell +\frac{1}{2}}{\chi },\chi \right) q^j_\kappa (\chi)}{\chi ^2} P\left(\frac{\ell +\frac{1}{2}}{\chi }, z(\chi)\right),
\end{eqnarray*}$$(6)$$\begin{eqnarray*}
C^{ij}_{\delta _\mathrm{ g} \delta _\mathrm{ g}}(\ell) = \int \mathrm{ d}\chi \frac{q^i_\delta \left(\frac{\ell +\frac{1}{2}}{\chi },\chi \right) q^j_\delta \left(\frac{\ell +\frac{1}{2}}{\chi },\chi \right)}{\chi ^2} P\left(\frac{\ell +\frac{1}{2}}{\chi }, z(\chi)\right)\ ,
\end{eqnarray*}$$where χ is the comoving radial distance, i and j denote different combinations of pairs of redshift bins, and the lensing efficiency $q^i_\kappa$ and the radial weight function for clustering $q^i_\delta$ are, respectively, given by
(7)$$\begin{eqnarray*}
q^i_\kappa (\chi) &=& \frac{3H_0^2 \Omega _\mathrm{ m}\chi }{2a(\chi)} \int _{\chi }^{\chi _\mathrm{ h}} \mathrm{ d}\chi ^\prime \left(\frac{\chi ^\prime - \chi }{\chi }\right) n^i_\kappa (z(\chi ^\prime)) \frac{\mathrm{ d} z}{\mathrm{ d}\chi ^\prime } , \nonumber \\
q^i_\delta (k,\chi) &=& b^i(k, z(\chi))\ n^i_\delta (z(\chi)) \frac{\mathrm{ d} z}{\mathrm{ d}\chi }\ .
\end{eqnarray*}$$Here, H0 is the Hubble parameter today, Ωm is the ratio of today’s matter density to today’s critical density of the Universe, a(χ) is the Universe’s scale factor at comoving distance χ, and bi(k, z) is a scale- and redshift-dependent galaxy bias. Furthermore, $n^i_{\kappa , \mathrm{ g}}(z)$ denote the redshift distributions of the different DES-Y3 redshift bins of source and lens galaxies, respectively, normalized such that
(8)$$\begin{eqnarray*}
\int \mathrm{ d}z \,\, n^i_{\kappa , \mathrm{ g}}(z) = 1\ .
\end{eqnarray*}$$Note that on large angular scales the DES-Y3 analysis does not make use of the Limber approximation for galaxy clustering but instead employs the method derived in Fang, Eifler & Krause (2020a).


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

.


For the five He ii sightlines recorded at sufficiently high spectral resolution (

), coeval spectra of the optically thin H i Lyα forest provide estimates of the number density ratio

. In a fully reionized IGM

 probes the spectral shape of the UV background and its source population (e.g., Miralda-Escudé & Ostriker 1990; Madau & Meiksin 1994; Fardal et al. 1998; Haardt & Madau 2012), so observational constraints are of great astrophysical interest. The patches of strong quasi-continuous He ii absorption at

 typically require

 (Reimers et al. 1997, 2005; Heap et al. 2000; Smette et al. 2002; Shull et al. 2010; Syphers & Shull 2014), whereas the

 He ii Lyα forest revealed order-of-magnitude fluctuations around

 on scales down to ∼1 Mpc (Kriss et al. 2001; Shull et al. 2004; Zheng et al. 2004b; Fechner et al. 2006; Fechner & Reimers 2007). However, these studies were affected by various systematics, most importantly by uncertainty in the zero level of the FUSE data (Fechner & Reimers 2007) and H i continuum uncertainty (e.g., Heap et al. 2000). We recently showed that by accounting for H i continuum systematics with realistic mock spectra from numerical simulations, the higher-quality COS spectra from the two sightlines sampling

 are consistent with

 without the need for fluctuations exceeding a factor of two (McQuinn & Worseck 2014). This implies that in the post-reionization IGM, radiative transfer effects (Maselli & Ferrara 2005; Tittley & Meiksin 2007; Meiksin & Tittley 2012) or Poisson fluctuations in the number density of quasars (Bolton et al. 2006) generate only modest UV background fluctuations. However, all existing inferences on the epoch of He ii reionization have been tempered by the very small sample of sightlines available to study He ii Lyα absorption. Indeed, expanding the data set has been the most pressing need to advance our understanding of He ii reionization.


However, these studies were affected by various systematics, most importantly by uncertainty in the zero level of the FUSE data


In order to improve the census of X-ray sources in old open clusters, we are undertaking a survey with Chandra of open clusters with ages between 3.5 and 10 Gyr. The observations are designed to reach a limiting luminosity of

 (0.3–7 keV), or better, at the distance of the clusters. As part of this survey, we have carried out the first X-ray study of Collinder 261 (Cr 261), and we present the results of our efforts in this paper. With an estimated age of 6–7 Gyr (Bragaglia & Tosi 2006), Cr 261 is one of the oldest open clusters in the Galaxy, being superseded in age by NGC 6791 (8–9 Gyr) and Berkeley 17 (8.5–10 Gyr) only. The cluster metallicity is close to solar (Drazdauskas et al. 2016), and reported values for the distance and reddening lie between 2.2–2.7 kpc and

, respectively (see, e.g., Gozzoli et al. 1996; Carraro et al. 1999; Bragaglia & Tosi 2006), with a higher value of the reddening considered more plausible (Friel et al. 2003). In this paper, we adopt a distance of 2.5 kpc and

, unless stated otherwise. The latter corresponds to a V-band extinction AV = 1.05 for the canonical ratio

, and a neutral hydrogen column density

 (Predehl & Schmitt 1995). The Galactic coordinates of Cr 261 are

,

 due to its low Galactic latitude and location toward the bulge, the number of foreground and background stars projected onto the cluster is high. Cluster membership is poorly constrained for the majority of stars in the field. Cr 261 is included in the star cluster catalog of Kharchenko et al. (2013), which lists structural parameters such as the overall size of the cluster and the radius of its central region. In this work, we present an estimate for the half-mass radius rh and the approximate mass of Cr 261, which, to our knowledge, have not been reported in the literature before. These parameters facilitate a uniform comparison with the X-ray properties of other old Galactic clusters.


reported values for the distance and reddening lie between 2.2–2.7 kpc and

, respectively (see, e.g.,


We determined the luminosity of the L and T components using J-, H-, and K-band bolometric corrections (Golimowski et al. 2004b; Nakajima et al. 2004; Vrba et al. 2004) valid for field dwarfs and the spectral intervals L6–L8 (Luhman 16A) and L9–T1 (Luhman 16B), which accounts for the spectral types of the pair members and their associated uncertainties. Note that the H-band bolometric correction (BC) shows the smallest dispersion among the three bands. We also used a solar bolometric luminosity of 4.73 mag and the trigonometric distance of 2.02  ±  0.02 pc determined for the system by Boffin et al. (2014). The near-infrared magnitudes of each pair member provided by Burgasser et al. (2013) and Kniazev et al. (2013) were averaged. We derived the following values: log L/L⊙ =  −4.68 ± 0.08 (L component) and −4.66  ±  0.08 dex (T component). The luminosity error bars consider the uncertainties in the photometry (typically ±0.03 mag), the distance modulus (±0.02 mag), and the corresponding BCs (typically ±0.15 mag for J and K, and ±0.08 for H). Our determination for the L component broadly agrees with Faherty et al. (2014), who found log L/L⊙ =  −4.67 ± 0.04 dex after the integration over their optical (0.6−0.9 μm) and near-infrared (0.8−2.5 μm) spectra supplemented with synthetic spectra for wavelengths longer than 2.5 μm. This supports the reliability of the BCs for these spectral types. Our luminosity determination for the T component is consistent with that of Faherty et al. (2014, −4.71 ± 0.10 at the 1σ level, although our measurement suggests a slightly brighter luminosity. In a similar manner as described in Faherty et al. (2014), we also integrated our X-Shooter spectra and found that the L and T dwarfs have nearly identical luminosity, with the T object being 0.01 dex more luminous than the L dwarf, in agreement with the luminosities inferred from the BCs. The T component has been reported to show stronger variability than the L source at optical and near-infrared wavelengths with peak-to-peak amplitudes of 11−13.5% in continuous observations and strong night-to-night evolution according to Gillon et al. (2013), Biller et al. (2013), and Burgasser et al. (2014). This might affect the luminosity determination for this particular dwarf. To account for the observed variability amplitude, an uncertainty of ±0.05 dex should be added to the error bar in the luminosity derivation of the T pair member, thus yielding log L/L⊙ =  −4.66 ± 0.13 dex. One striking result is that, despite the differing spectral classifications, both objects, Luhman 16A and B, have consistent L/L⊙ values at the 1σ level.


Our luminosity determination for the T component is consistent with that of

although our measurement suggests a slightly brighter luminosity.


we also integrated our X-Shooter spectra


Despite the differences between the methods used to determine GC ages, they paint a similar picture: GC populations are typically older than field stars, as they mostly formed before the peak of the cosmic star formation history (z ≃ 2, Madau & Dickinson 2014), and metal-poor GCs seem to have formed coevally to or earlier than the metal-rich ones. In the Milky Way, the population of massive (M > 105 M⊙) GCs with metallicities −2.5  [Fe/H]  −0.5 is ≃12.2 Gyr old (z ∼ 4, Kruijssen et al. 2019a, based on measurements from Forbes & Bridges 2010; Dotter et al. 2010; Dotter, Sarajedini & Anderson 2011; VandenBerg et al. 2013), which is older than the inferred mean star formation time based on the star formation history of the Milky Way, τf = 10.5 ± 1.5 Gyr (Snaith et al. 2014). Despite the relatively large uncertainties (∼1 Gyr), several studies find an age–metallicity relation among the GCs in the Milky Way, with metal-poor GCs being the oldest and younger objects having higher metallicities. The exact age offset between both subpopulations depends on the catalogue considered and the metallicity range, but overall metal-poor GCs are found to be coeval to or older (by up to ∼1.25 Gyr) than the metal-rich subpopulation within the uncertainties (considering $\rm [Fe/H]\gtrless -1.2$ between the metal-poor and metal-rich subpopulations; Dotter et al. 2010; Forbes & Bridges 2010; Dotter et al. 2011; VandenBerg et al. 2013). Subsamples of GCs in different metallicity intervals are also observed to have radial age gradients, as seen in M31 (Beasley et al. 2005) and in 11 early-type galaxies from the SLUGGS survey (Forbes et al. 2015). The implied differences in formation epoch have been proposed to explain the observed differences in spatial distributions and kinematics between these metallicity GC subpopulations (Brodie & Strader 2006), and some authors also suggest they indicate different formation mechanisms (Griffen et al. 2010).


The first step of the main process aims at selecting as few modes as possible to model the magnetic topology of the star, using sparse approximation. We begin by combining the sequence of observed Stokes V profiles in sliding subsets of n consecutive profiles (with n ≪ q, the total number of observed profiles) and we associate each group of profiles to the mean date over the corresponding subset. This date is then converted into a rotation cycle using the stellar rotation period, assumed to be known or derived from the GPR modelling of the longitudinal field (see Section 2.3.2). We assume that the magnetic field we want to reconstruct is not too strong, to ensure that the relation between the Stokes V data and the magnetic map remains linear. We come back in Section 5 on this limitation. Our problem thus amounts to look for the simplest linear combination of data base profiles that can reproduce the selected subset of observations to a given precision σ. In practice, this is achieved through sparse approximation (e.g. Mallat & Zhang 1993; Tibshirani 1996; Donoho & Elad 2003; Donoho, Elad & Temlyakov 2006), this problem being known as Basis Pursuit denoizing (Chen, Donoho & Saunders 1998) and formalized as follows:
(9)$$\begin{eqnarray}
\rm minimize \, \, \, \Vert \boldsymbol {X}\Vert _1 \, \, \, \rm s.t. \, \, \, \Vert \boldsymbol{ \mathsf {A}}\boldsymbol{X}-\boldsymbol{ \boldsymbol {B}}\Vert _2 \le \sigma ,
\end{eqnarray}$$where $\boldsymbol{ B}$ is a vector containing the N data points in the group of n profiles, $\boldsymbol{ \mathsf {A}}$ the dictionary, an N × p matrix for which each column contains the spectral signatures of each mode (included in the data base) taken at the observed phases, and σ the level at which we wish the data to be fitted. The components Xi of the vector $\boldsymbol {X}$ are directly associated with the modes to be reconstructed, i.e. proportional to the real or imaginary part of one of the αℓ,m, βℓ,m, and γℓ,m coefficients describing the field (up to a degree ℓmax ). Taking into account the noise in the data, the previous problem can be generalized by including an N × N diagonal matrix $\boldsymbol{ \mathsf {W}}$ containing the inverse of the errors on the spectral points. Problem (9) then becomes:
(10)$$\begin{eqnarray}
\rm minimize \, \, \, \Vert \boldsymbol {X}\Vert _1 \, \, \, \rm s.t. \, \, \, \Vert\boldsymbol{ \mathsf {W}}(\boldsymbol{ \mathsf {A}}\boldsymbol{X}- \boldsymbol {B})\Vert _2 \le \tau
\end{eqnarray}$$with τ2 being the chi-square (χ2) level at which we wish the data to be fitted.


CO emission is derived for each object in the catalog by converting infrared luminosity to CO line strength with the well-established LIR–

 correlation (e.g., Carilli & Walter 2013, hereafter CW13; Greve et al. 2014, hereafter G14; Dessauges-Zavadsky et al. 2015). We can then use the measured stellar mass functions of galaxies at 0  z  2 to calculate the power of CO line foregrounds, with the ability of monitoring different subsets (e.g., different stellar mass bins, quiescent versus star-forming galaxies, etc.) Now the total mean intensity of CO contamination can be expressed as
7

where

, 3 ≤ Jupp ≤ 6, representing all CO transitions acting as foregrounds and

 being the stellar mass function measured by the COSMOS/UltraVISTA Survey (Muzzin et al. 2013b).

 represents an evolving mass cut that measures the depth of foreground masking (see Section 4.3 for a detailed discussion) and is set to

 when no masking is applied. The factor

 accounts for the mapping of frequency into distance along the line of sight (Visbal & Loeb 2010). The comoving radial distance χ, the comoving angular diameter distance DA, and the luminosity distance DL are related by

. In the presence of scatter (as is always the case), the expectation value of a function

 of CO luminosity at a best-fit

 given by simstack can be written as
8

and
9

where
10

is derived from the best-fit

 correlation, with rJ being some scaling factor for different J. Consequently, in the presence of scatter, Equation (7) becomes

, which describes the expectation value of the total CO mean intensity, averaged over the probability distribution of

 as specified by μ and

. In our calculation, we consider two prescriptions: (i) CW13, who give α = 1.37 ± 0.04, β = −1.74 ± 0.40, and scaling relations appropriate for sub-millimeter galaxies which are used to convert to transitions higher than

, and (ii) G14, who provide α and β coefficients for each individual J transition (i.e., rJ = 1) based on samples of low-z ultra-luminous infrared galaxies and high-z dusty star-forming galaxies comparable to CW13. Since the total CO foreground consists of multiple J transitions, we deem the G14 prescription more appropriate for our purposes, because it treats both the slope and intercept as free parameters when fitting to galaxies observed in different J. Henceforth, we present our results based on the G14 model unless otherwise stated.


We applied the convex inversion method of Kaasalainen et al. (2001) and Kaasalainen & Torppa (2001) to the optical dataset of 70 light curves (described in Appendix A and listed in Table A.1) following exactly the procedure of Hanuš et al. (2016). Specifically, we scanned rotation periods in the proximity of the expected value while testing ten initial pole solutions for each sampled period. Four poles were selected on the equator with 90° difference in longitude, and three poles in each hemisphere with the latitude ±60° and with 120° difference in longitude. We assumed that all solutions within a 3σ uncertainty interval had

χ
2

(
1
+
3

2
/
ν

)

χ
min
2

$ \chi^2 < (1+3\sqrt{2/\nu})\,\chi^{2}_{\mathrm{min}} $

, where

χ

min

2

$ \chi _{{\rm{min}}}^2 $

 is the χ2 of the best-fitting solution and ν is the number of degrees of freedom. This threshold to consider the solution acceptable was used before in Vokrouhlický et al. (2017) or Ďurech et al. (2018)
3 and corresponded to a ∼7% increase in

χ

min

2

$ \chi _{{\rm{min}}}^2 $

 value. Only the best-fitting solution fulfilled the 3σ condition on the χ2. To further verify that the best-fitting solution was the only one acceptable, we also visually inspected the light-curve fit with the second-best-fitting period, similarly as in Hanuš et al. (2016), see their Figs. 3 and 4 for illustration. This solution was already inconsistent with several individual light-curves. Therefore, we considered the difference in χ2 as significant and rejected all periods except for the best-fitting one. Next, we ran the convex inversion with the unique period and multiple pole orientations (isotropically distributed on a sphere with a 30° difference) as starting points of the optimization procedure and derived a single solution within the 3σ uncertainty interval defined above. Again, we visually inspected the light-curve fit with the second-best-fitting pole orientation and rejected this solution and also considered all other solutions as non-acceptable. The final solution is given in Table 1 together with the previous determinations. It is notable that our analysis and the recent study of Kim et al. (2018) provide for the first time a unique shape and spin solution that is consistent with the preferred solution of Hanuš et al. (2016). There are two differences between our old and revised models: (i) the relative dimension along the rotation axis (or the c/a ratio) is now smaller by ∼10%, which is expected because this dimension is generally the least constrained by the optical data, and (ii) the pole directions are about 8° apart.


We note that Mrk 590 does not fit this description. The soft excesses in obs1 and obs2 were described by a nonrotating black hole (r
in fixed at 6r

g
) and a slightly ionized accretion disk. This result contradicts recent studies of other type 1 AGNs, where the reflection model favors a rotating black hole and the inner part of the disk approaches the innermost stable circular orbit (1.25r

g
; García et al. 2019; Ghosh & Laha2020, 2021). We note a significant increase in disk ionization between observations (from

logξ=0.52−0.30+0.77

 in obs2 to

logξ∼3

 in obs3). We also found a significant decrease in Γ value from obs1 to obs8, indicating a spectral hardening. For obs7 and obs8, we found a well-constrained, high-energy cutoff of

92−25+55keV

 and

60−8+10keV

, respectively, better described by the relxill model. Although these values are relatively lower compared to other Seyfert 1 galaxies (200–300  keV; Ghosh et al. 2016; Ricci et al. 2017; Fabian et al. 2017; Akylas & Georgantopoulos 2021), similar low values of E
cut have been found in recent sample studies of Swift/BAT-selected AGNs (Kamraj et al. 2022). This low-energy cutoff may indicate a decrease in the plasma temperature of the corona only if it was higher during obs1 and obs2. But we cannot test this scenario owing to the nonavailability of data beyond 10 keV. Now hard X-ray photons illuminate the disk, so less energetic photons mean low illumination. However, we note that the disk is still moderately ionized and is capable of Ross & Fabian (2005) producing fluorescent emission lines. More importantly, we did not find any broad Fe emission line or Compton hump in any of the source spectra. The reflected flux and reflection fraction do not show statistically significant variation and remain within the 3σ value. When we plotted the soft excess flux versus the power-law flux (2–10  keV), we did not find any significant correlation (see Figure 8). This result is consistent with Boissay et al. (2016), where the shape of reflection at hard X-rays stays constant when the soft excess varies, showing an absence of a link between reflection and soft excess. In Mrk 590, the power law, the Fe line emission, and UV monochromatic flux follow the same temporal pattern (Figure 3). This result suggests that the disk and corona are most likely evolving together. However, the soft excess is not responding to this change in disk–corona properties. Hence, the soft excess emission observed in Mrk 590 is not due to ionized reflection from the disk.


We also note that we do not have reason to think that we may be underestimating the stellar mass of Was 49b due to dust obscuration. We examined this possibility by measuring the K-band light ratio of Was 49b to Was 49a using an image from the UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2007), taken on 2010 February 27. At 2.2 μm the AGN is manifest as a dominant, unresolved source in Was 49b (Figure 6, left). Because of the dominance of the AGN and the shallowness of the UKIDSS image, it was difficult to model any extended emission in Was 49b using galfit, as was done for the DCT images. We instead subtracted the AGN and measured the remaining emission by determining the expected K-band AGN luminosity given its intrinsic luminosity found in Section 3.1. Using five bright stars within a few arcminutes of Was 49, we made an empirical template of the PSF. We found that we could accurately model the stars in the image with a PSF with three Gaussian components. With our PSF template, we fit the unresolved source in Was 49b, allowing only the amplitude and position to vary (Figure 6, middle). We found a PSF K-band magnitude of 13.52, which corresponds to

 erg s−1. Before subtracting this unresolved source from the K-band image, we performed a check to ensure that it is consistent with the expected AGN emission. Using the value of AV ∼ 12 for the direct AGN continuum calculated above and the extinction curve of Cardelli et al. (1989), we found AK ∼ 1.3, which implies an intrinsic AGN luminosity of LK ∼ 1044 erg s−1. If we assume that the intrinsic spectral energy distribution is approximately flat in the near-IR to the mid-IR for luminous AGNs, then we can use LK ∼ L6 μm ∼

 (e.g., Mateos et al. 2015) to calculate

 erg s−1, consistent with the intrinsic value, given the scatter on EB–V/

 and L6 μm/

. As the observed unresolved source in Was 49b is consistent with expectations for the AGN, we subtracted it from the background-subtracted K-band image and photometered the remaining emission in Was 49b for comparison with Was 49a. We did not attempt to model any contribution to the K-band emission from the AGN in Was 49a, as the faint, soft X-ray source in Was 49a (Section 3.1) suggests that the 2–10 keV luminosity and therefore the expected K-band luminosity are negligible. We used the position and inclination angle for the disk of Was 49a (Table 2) to build an elliptical isophotal profile of Was 49a, which we interpolated across the position of Was 49b using a low-order polynomial and by taking the median isophotal value. Subtracting this profile and estimating the variance from a source-free region in the UKIDSS image, we found that the K-band light ratio is consistent with the Was 49 system being a

 merger, where the confidence intervals are ∼90%. We emphasize that these confidence intervals assume that the AGN has been accurately subtracted from Was 49b. This may not be the case, as there is scatter in the relationship between the intrinsic X-ray luminosity and intrinsic IR luminosity of AGNs. If our model of the AGN overestimated the observed K-band luminosity of the AGN, then the K-band light ratio would be somewhat lower. It is less likely, however, that our model underestimated the observed K-band luminosity of the AGN, as increasing the modeled luminosity would quickly lead to negative flux values when subtracting it from the K-band data. With these considerations, we consider Was 49 to be a minor merger, with a mass ratio between ∼1:7 and ∼1:15.


How does excitation affect 6.2 μm/7.7 μm? A harder illuminating spectrum contains more UV photons at higher energy and hence directly enhances PAH emission at shorter wavelengths relative to that at longer wavelengths (Draine et al. 2021; Rigopoulou et al. 2021). If this were true, the 11.3 μm/7.7 μm band ratio should decrease strictly with increasing excitation, because a harder radiation field preferentially enhances PAH emission at shorter wavelengths and would boost the fraction of ionized PAH molecules. However, as explained above, the variation of 11.3 μm/7.7 μm with [Ne iii]/[Ne ii] is not monotonic. Alternatively, a harder radiation field can modify PAH size distribution through preferential photo erosion of smaller PAH grains (Allain et al. 1996a; Micelotta et al. 2010a, 2010b; Murga et al. 2016). The normally narrow peak size distribution of 150 C atoms for astronomical PAHs (Weingartner & Draine 2001a; Draine & Li 2007) would shift with photo erosion toward sizes of fewer than 100 C atoms (with the total amount of smaller PAHs reduced as well). This effect also can potentially account for the increase of 6.2 μm/7.7 μm, but it cannot explain why 11.3 μm/7.7 μm does not simultaneously increase in lockstep. The 11.3 μm/7.7 μm band ratio first decreases and then increases with [Ne iii]/[Ne ii]. To account for this peculiar behavior, we propose that, during the photo erosion process, a harder radiation field first elevates the ionization fraction of the PAH molecules up to a certain threshold, which results in the decrease of 11.3 μm/7.7 μm, beyond which photo erosion proceeds to dominate by modifying the PAH size distribution, which results in the increase of 6.2 μm/7.7 μm and 11.3 μm/7.7 μm. In addition, preferential photodestruction of the ionized PAH grains, which are more vulnerable relative to the neutral ones (Allain et al. 1996b; Holm et al. 2011), also may contribute to the increase of 11.3 μm/7.7 μm. This scenario further provides a plausible explanation for the severe depletion of the overall PAH emission among the dwarf galaxies with the hardest radiation field (Figure 6(a)).


A harder illuminating spectrum contains more UV photons at higher energy and hence directly enhances PAH emission at shorter wavelengths relative to that at longer wavelengths


In addition, preferential photodestruction of the ionized PAH grains, which are more vulnerable relative to the neutral ones


The normally narrow peak size distribution of 150 C atoms for astronomical PAHs

would shift with photo erosion toward sizes of fewer than 100 C atoms (with the total amount of smaller PAHs reduced as well).


It has been known for more than 30 years that quasar emission lines are often shifted from the systemic velocity. Most notable is the average blueshift of the high-ionization broad C iv line (e.g., Gaskell 1982; Wilkes & Carswell 1982; Tytler & Fan 1992; Richards et al. 2002) by hundreds of

 from systemic, with a strong dependence on luminosity (e.g., Richards et al. 2011). Low-ionization broad lines (such as Mg ii), however, are observed to have an average velocity closer to the systemic velocity (e.g., Hewett & Wild 2010). While high-ionization narrow emission lines such as [O iii] are not as blueshifted as high-ionization broad lines, they can have line peak shifts18

18
[O iii] often shows a blue asymmetry (e.g., the “blue wing”) that depends on the properties of the quasar (e.g., Heckman et al. 1981; Peterson et al. 1981; Whittle 1985; Veilleux 1991; Zhang et al. 2011; Shen & Ho 2014). The velocity offset of this wing [O iii] component appears to increase with luminosity and Eddington ratio (e.g., Zhang et al. 2011; Shen & Ho 2014), and can reach extreme values of ≳1000

 at the highest quasar luminosities (e.g., Shen 2016; Zakamska et al. 2016). However, the velocity offset of the core [O iii] component or the [O iii] peak is generally more stable and lies within ∼50

 of the systemic velocity, with only mild dependences on quasar luminosity and Eddington ratio (e.g., Figure 2 and Figure E2 in Shen & Ho 2014, also see, Hewett & Wild 2010).
 of tens of

, with some extreme cases exceeding a hundred

  (e.g., the so-called “blue outliers,” Zamanov et al. 2002; Boroson 2005; Komossa et al. 2008; Marziani et al. 2015). These average trends have been generally confirmed with recent large spectroscopic quasar samples (e.g., Vanden Berk et al. 2001; Richards et al. 2002, 2011; Shen et al. 2007, 2008, 2011; Hewett & Wild 2010; Zhang et al. 2011; Shen & Ho 2014), although differences in measuring the line center and the reference line used for systemic velocity will affect the exact value of the inferred velocity shifts.


The velocity offset of this wing [O iii] component appears to increase with luminosity and Eddington ratio (e.g.,


However, the velocity offset of the core [O iii] component or the [O iii] peak is generally more stable and lies within ∼50

 of the systemic velocity, with only mild dependences on quasar luminosity and Eddington ratio (e.g.,


These average trends have been generally confirmed with recent large spectroscopic quasar samples (e.g.,