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)$$\begin{eqnarray*}
j(x)=\frac{\int \int f_{\rm {e}}(p)\mathrm{ d}p P(t,p)\frac{x^{\prime 2}\mathrm{ d}x^{\prime }}{e^{x^{\prime }}-1}x}{\Delta x}.
\end{eqnarray*}$$


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).


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.


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.


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.


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)).


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.


To evaluate the turbulence heating term St, we need to describe the transport and dissipation of turbulence throughout the heliosphere, ensuring that the appropriate sources of turbulence are identified as well as that associated with pickup ion creation in the distant solar wind. The turbulence transport model we use is that of Zank et al. (2017). The current paradigm for fully developed turbulence in the solar wind is that it is a superposition of a majority quasi-2D component and a minority slab component, e.g., Matthaeus et al. (1995). The nearly incompressible (NI) reduction of MHD in the plasma beta

 regime, appropriate to most of the solar wind, shows that such a decomposition (quasi-2D plus slab) arises naturally in the presence of a sufficiently strong guide magnetic field (Zank & Matthaeus 1992, 1993; Hunana & Zank 2010). The stationary form of the NI MHD transport equations describing the evolution of the majority quasi-2D component in a spherically symmetric solar wind with an azimuthal magnetic field can be expressed as (Adhikari et al. 2017; Zank et al. 2017)
22

23

24

25

Here,

 are the forward and backward Elsässer variables, combining the quasi-2D fluctuating velocity field

 and the fluctuating magnetic field

 through the fluctuating (turbulent) Alfvén velocity

. The variance of the fluctuating Elsässer variables is denoted by

, where the operator

 averages over small scales and high frequencies. The quasi-2D residual energy (or energy difference) is defined by

. The total fluctuating quasi-2D energy is denoted by

, and the quasi-2D correlation lengths

 are related to the correlation functions

 through

. Finally,

 is a correlation function related to the residual energy. The energy for the forward and backward slab Elsässer variables

 and the slab correlation length

 are discussed further below. However, we note that the quasi-2D and slab components are coupled, capturing the 3-mode coupling (Shebalin et al. 1983; Zank et al. 2017) that governs spectral transfer in the inertial range. This is expressed by the source-like terms in Equations (22) and (23) that contain the slab variables. Equations (22) and (23) contain genuine source terms associated with stream shear (Adhikari et al. 2017; Zank et al. 2017), where

 denotes the strength of the stream-shear source of forward or backward quasi-2D fluctuations,

 the characteristic velocity jump across the stream, and VA the large-scale Alfvén speed. A lengthy derivation and discussion of these equations can be found in Zank et al. (2017) and specific solutions appropriate to a prescribed large-scale solar wind can be found in Adhikari et al. (2017).


To characterize the motions of the deeply embedded gas that could reveal signatures of rotation and infall and that would help us to understand the formation of massive stars, for many years we have been carrying out multiple high angular resolution studies of hot molecular cores (HMCs), first with the IRAM Plateau de Bure (PdB) and Submillimeter Array (SMA) interferometers (Beltrán et al. 2004, 2005, 2011a,b), and later on with the Atacama Large Millimeter/submillimeter Array (ALMA, Cesaroni et al. 2017; Beltrán et al. 2018, 2021; Moscadelli et al. 2018, 2021; Goddi et al. 2020) and the IRAM NOrthern Extended Millimeter Array (NOEMA, Gieser et al. 2019, 2021). Cesaroni et al. (2017) carried out an extensive and systematic search for circumstellar disks around O-type stars by observing a sample of HMCs at 1.4 mm and 0.′′2 resolution with ALMA. One of the HMCs of the sample is G31.41+0.31 (hereafter G31), which is located at 3.75 kpc Immer et al. (2019), has a luminosity of ~ 5 × 104 L⊙ Osorio et al. (2009), and displays a clear NE–SW velocity gradient suggestive of rotation (e.g., Beltrán et al. 2004). Among the HMCs studied, G31 stands out as the only source that displays accelerating infall and rotational spin-up (Beltrán et al. 2018). This characteristic suggests that the source is in an earlier evolutionary stage compared to all other targets in the Cesaroni et al. sample. At an angular resolution of 0.′′2 (~ 750 au), the dust continuum emission traces a well-resolved, monolithic, and featureless core, called Main by Beltrán et al. (2018), with no hint of fragmentation despite the high mass (~ 70 M⊙; Cesaroni et al. 2019) and the diameter of ~8000 au. However, new ALMA observations at 1.4 and 3.5 mm at a higher angular resolution of ~ 0.′′1 (~ 375 au), and Very Large Array (VLA) observations at 7 mm and ~ 0.′′05 (~ 190 au) resolution have resolved for the first time the HMC into a small protocluster composed of at least four massive sources, named A, B, C, and D and with masses ranging from ~15 to ~ 26 M⊙, within the central 1″ (~ 3750 au) region of the core (Beltrán et al. 2021). These observations have revealed that the homogeneous appearance previously observed at 1.4 mm and 0.′′2 is a consequence of both high dust opacity and insufficient angular resolution. Besides the four sources embedded in the Main core, there are six additional millimeter sources located very close to the core which appear to outline streams or filaments of matter pointing to the HMC (Beltrán et al. 2021).


The efficiency of pebble accretion is determined by the pebble scale height in the disc given by (5)\begin{equation} \label{eq:Hpebble}  H_{\rm peb} = H\sqrt{\alpha/\tau_{\rm f}} , \end{equation}Hpeb=Hα/τf,where τf is the Stokes number of the particles. If the planetary seed’s Hill radius is larger than the pebble scale height, the planetary seed can accrete efficiently in a 2D manner (6)\begin{equation} \label{eq:Mdotpebble}  \dot{M}_{\rm c, 2D} = 2 \left(\frac{\tau_{\rm f}}{0.1}\right)^{2/3} r_{\rm H} v_{\rm H} \Sigma_{\rm peb} , \end{equation}Ṁc,2D=2τf0.12/3rHvHΣpeb,where rH = r [Mc/ (3M⋆)] 1/3 is the Hill radius, vH = ΩKrH the Hill speed, and Σpeb the pebble surface density. For Stokes numbers larger than 0.1, the (τf/ 0.1)2/3 term vanishes in Eq. (6), because the planetary seed cannot accrete particles from outside its Hill radius (Lambrechts & Johansen 2012). However for small planetary seeds, the Hill sphere can be smaller than the pebble scale height, so that the seed accretes in a slow 3D fashion (7)\begin{equation} \dot{M}_{\rm c, 3D} = \dot{M}_{\rm c, 2D} \left( \frac{\pi (\tau_{\rm f}/0.1)^{1/3} r_{\rm H}}{2 \sqrt{2 \pi} H_{\rm peb}} \right) \cdot \end{equation}Ṁc,3D=Ṁc,2Dπ(τf/0.1)1/3rH22πHpeb·The transition from 3D to 2D pebble accretion is then reached (Morbidelli et al. 2015) when (8)\begin{equation} \label{eq:2D3D}  \frac{\pi (\tau_{\rm f}/0.1)^{1/3} r_{\rm H}}{2 \sqrt{2 \pi}} > H_{\rm peb} . \end{equation}π(τf/0.1)1/3rH22π>Hpeb.Clearly this transition depends on the scale height of the pebbles and thus on the scale height of the gas. Therefore, pebble accretion is slower in the initial planetary growth phase in the outer parts of the disc, because the aspect ratio is larger there. The core can then finally reach its pebble isolation mass (Lambrechts et al. 2014) (9)\begin{equation} \label{eq:Misolation}  M_{\rm iso} \sim 20 \left( \frac{H/r}{0.05}\right)^3~{M}_{\rm Earth} , \end{equation}Miso~20H/r0.053MEarth,where pebble accretion is terminated and the contraction of the gaseous envelope can start. We use for the contraction of the gaseous envelope the semi-analytical model described by Piso & Youdin (2014), where the contraction phase becomes shorter for more massive planetary cores. This contraction phase additionally depends on the opacity inside the of the planetary envelope, as this determines the cooling and with this the contraction rate of the envelope. Our nominal opacity in the envelope corresponds to similar values as in Movshovitz & Podolak (2008). We discuss the influence of a slower contraction rate in Appendix B. As soon as Menv = Mcore, rapid gas accretion can start.


Solar flares and coronal mass ejections (CMEs) are the most energetic phenomena in the solar coronae, which are considered as different manifestations of the same physical process (Zhang et al. 2001; Priest & Forbes 2002; Zhang et al. 2004; Shibata & Magara 2011). However, flares that are associated with no CMEs are known as “confined flares” or “failed eruptions” (e.g., Ji et al. 2003; Liu et al. 2014; Zhou et al. 2019; Mrozek et al. 2020). Whether an eruption is successful or not is heavily dependent on the strapping force exerted by the overlying magnetic field on the eruptive structure (e.g., Török & Kliem 2005; Guo et al. 2010; Cheng et al. 2011; Zuccarello et al. 2014; Myers et al. 2015; Sun et al. 2015; Amari et al. 2018), which is often a magnetic flux rope (MFR; Vourlidas et al. 2013; Liu 2020). A flux rope undergoes the torus instability (Bateman 1978; Kliem & Török 2006), when the magnitude of the overlying field decreases with height at a sufficiently steep rate, quantified by the decay index

n=−dlnB/dlnh

 (Bateman 1978), exceeding about 1.5; the corresponding height is referred to as the critical height. Nevertheless, one should bear in mind that the theoretical critical value of n
cr = 1.5 adopted in many studies, as well as in this study, is derived from a thin, toroidal current ring (Bateman 1978; Kliem & Török 2006). In practice, n
cr typically scatters around in a range as large as [0.8, 2] for different theoretical considerations (e.g., Démoulin & Aulanier 2010; Olmedo & Zhang 2010), observational studies (e.g., Jing et al. 2018; Duan et al. 2019), numerical simulations (e.g., Fan & Gibson 2007; Török & Kliem 2007), and laboratory experiments (e.g., Myers et al. 2015; Alt et al. 2021). Obviously, n
cr is a function of the flux-rope parameters including the geometry and internal current profile, yet the discrepancy in different studies is also affected by where the decay index is computed (Zuccarello et al. 2016; Alt et al. 2021).


A natural experiment for deciding between these possibilities is to search for a KS relation within individual molecular clouds using counts of the recently formed stars or protostars identified by their bright infrared emission from circumstellar dust. The rarity of massive stars and their disruptive effect on their host cloud means they are poor tracers on cloud scales. Protostars, by contrast, have the advantage that they sample a much shorter time interval and therefore provide a much better estimate of the “instantaneous” star formation rate (SFR), and they allow measurements of the SFR even in clouds that lack massive stars and have not been significantly affected by feedback. Studies based on this method generally do find a reasonable correlation between the number of young stellar objects (YSOs) in a cloud and its gas mass above a certain density, or its gas mass divided by its mean-density freefall time (Krumholz et al. 2012a, 2019; Lada et al. 2012; Heyer et al. 2016). Within molecular clouds, several studies have found a power-law correlation between the surface densities of YSOs and gas (Gutermuth et al. 2011; Lada et al. 2013; Willis et al. 2015). Most recently, Pokhrel et al. (2020) used high-accuracy YSO catalogs and high dynamic range gas column densities to show the presence of these laws in 12 nearby clouds. This correlation is consistent with a star formation surface density being proportional to the gas surface density squared. The scaling of this law, however, varies significantly between clouds. Moreover, their analysis technique examines the density of gas around stars on a star-by-star basis, and therefore cannot easily determine whether there is a KS relation based on the volume density of gas. Using the same data, we apply a different approach to determine the star formation law that includes a dependence on the volume density of the gas. We find that the star formation law can be recast as an effectively universal linear dependence of the surface densities between star formation rate and gas mass per freefall time, with much less scatter between clouds.


In the above equations, ρ is the mass density, v is the velocity, ψ(= −GM/r) is the Newtonian potential (where r is the distance from the central black hole, M is the black hole mass, and G is the gravitational constant), pgas is the gas pressure, σ is the viscous stress tensor, e is the gas internal energy, and Fc is the thermal conduction. Here, d/dt ≡ ∂/∂t + v · ∇ represents the Lagrangian or comoving derivative. The equation of state of the ideal gas is considered as

p

gas

=

γ
−
1

ρ
e

, with γ = 5/3 being the adiabatic index. In the purely HD limit, like our case, Fc is defined as
4

F

c

=
−
λ
∇
T
,

where λ is the thermal diffusivity and T is the gas temperature. We use spherical coordinates (r, θ, ϕ) to solve the above set of equations. In most previous semianalytical studies on hot accretion flow, they assumed that the system is in a steady state and axisymmetric, i.e., ∂/∂ϕ = ∂/∂t = 0. These assumptions imply that all physical quantities are independent of the azimuthal angle, ϕ, and the time, t. Following numerical simulations of hot accretion flow (e.g., Stone et al. 1999; Yuan et al. 2012a), we consider the following components of the viscous stress tensor:
5

σ

r
θ

=
ρ
ν

r

∂

∂
r

v

θ

r

+

1

r

∂

v

r

∂
θ

,

6

σ

r
ϕ

=
ρ
ν

r

∂

∂
r

v

ϕ

r

,

7

σ

θ
ϕ

=
ρ
ν

sin
θ

r

∂

∂
θ

v

ϕ

sin
θ

,

where ν is called the kinematic viscosity coefficient. By substituting all the above assumptions and definitions into Equations (1)–(3), we obtain following partial differential equations (PDEs) in spherical coordinates. Hence, the continuity equation is reduced to the following form
8

1

r

2

∂

∂
r

r

2

ρ

v

r

+

1

r
sin
θ

∂

∂
θ

ρ

v

θ

sin
θ

=
0
.

The three components of the momentum equation are as follows:
9

ρ

v

r

∂

v

r

∂
r

+

v

θ

r

∂

v

r

∂
θ

−

v

θ

−

v

ϕ

2

r

=

−

GM
ρ

r

2

−

∂

p

gas

∂
r

+

1

r
sin
θ

∂

∂
θ

sin
θ

σ

r
θ

,

10

ρ

v

r

∂

v

θ

∂
r

+

v

θ

r

∂

v

θ

∂
θ

+

v

r

−

v

ϕ

2

r

cot
θ

=

−

1

r

∂

p

gas

∂
θ

+

1

r

3

∂

∂
r

r

3

σ

r
θ

,

11

ρ

v

r

∂

v

ϕ

∂
r

+

v

θ

r

∂

v

ϕ

∂
θ

+

v

ϕ

r

v

r

+

v

θ

cot
θ

=

1

r

2

∂

∂
r

r

2

σ

r
ϕ

+

1

r
sin
θ

∂

∂
θ

sin
θ

σ

θ
ϕ

+

1

r

σ

r
ϕ

+

σ

θ
ϕ

cot
θ

.

The energy equation is written as
12

ρ

v

r

∂
e

∂
r

+

v

θ

r

∂
e

∂
θ

−

p

gas

ρ

v

r

∂
ρ

∂
r

+

v

θ

r

∂
ρ

∂
θ

=

∂

v

θ

∂
r

σ

r
θ

+

∂

v

ϕ

∂
r

σ

r
ϕ

+

1

r

∂

v

r

∂
θ

−

v

θ

σ

r
θ

+

1

r

∂

v

ϕ

∂
θ

σ

θ
ϕ

−

v

ϕ

r

σ

r
ϕ

+

σ

θ
ϕ

cot
θ

+

1

r

2

∂

∂
r

r

2

λ

∂
T

∂
r

+

1

r
sin
θ

∂

∂
θ

λ
sin
θ

r

∂
T

∂
θ

.


NGC 6397 is the second-nearest globular cluster to the Sun and has been thoroughly observed by the Hubble Space Telescope. Thus, we have high-quality deep images for it that have allowed us to study not only the lower main sequence, but also the white dwarf cooling sequence. NGC 6397 is old and metal-poor, its metallicity is [ Fe/H ]  =  −1.8 (Hansen et al. 2013), although there exists some discrepancy about its precise value (Richer et al. 2008) in the recent literature. The same is true for the age of the cluster. For instance, Hansen et al. (2007) analyzed the position of the cutoff of the white dwarf luminosity function and, comparing this with theoretical cooling models, derived an age Tc = 11.47 ± 0.47 Gyr. Similarly, Winget et al. (2009) simultaneously fitted the main-sequence, the pre-white dwarf, and the white dwarf regions of the color–magnitude diagram, and obtained an age \hbox{$T_{\rm  c}=12.0^{+0.5}_{-1.0}$}Tc=12.0-1.0+0.5 Gyr. These ages, which are based on white dwarf evolutionary models, need to be compared with those obtained by fitting the luminosity of the main-sequence turn-off (MSTO) – see Richer et al. (2008) for a discussion of different methods and age estimates of NGC 6397. In particular, it is worth highlighting that using this method, Gratton et al. (2003) derived an age of 13.9 ± 1.1 Gyr. This prompted Strickler et al. (2009) to claim that this age determination was compatible with the possible existence of a putative population of very old helium white dwarfs. On the other hand, Anthony-Twarog & Twarog (2000) obtained 12.0 ± 0.8 Gyr and Chaboyer et al. (2001) derived 13.4  ±  0.8 Gyr. Thus, the precise age of NGC 6397 remains unclear and needs to be independently evaluated using the most recent white dwarf evolutionary tracks of the appropriate metallicity. To this we add that NGC 6397 shares some other interesting characteristics with other Galactic globular clusters, such as the unusual lithium enhancement of some of their stars – see Pasquini et al. (2014) and references therein – or the possible existence of multiple populations (di Criscienzo et al. 2010; Milone et al. 2012). Thus, deriving an independent white dwarf cooling age for this globular cluster is of crucial importance.


As reported in Section 2.4, the uncertainties introduced by chromatic aberration in our observations prevented an unambiguous determination of the detailed characteristics of the spectral slope at wavelengths in the Balmer continuum range (Î»  3646âÃ). However, we do note the apparent lack of a jump in excess intensity or flare contrast at Î»â  â3646âÃ relative to the intensity at redder wavelengths (Figure 9). A dominant optically thin 10,000Â K spectrum would have produced a large Balmer jump ratio (i.e., the ratio of intensity at blue wavelengths to that at red wavelengths of 3646âÃ) that is â¼14 (Kunkel 1970; Neidig et al. 1993); we think that this should have been noticeable even with the data quality degraded due to chromatic aberration at Î»â  â3646âÃ. It is possible that in our spectra the blending of Stark-broadened high-order Balmer lines77The spectral region from 3654â3674âÃ contains the rapidly converging hydrogen Balmer lines H23 through at least H40; the center wavelengths of H23 and H24 are separated by â¼2.5âÃ whereas H39 and H40 are separated by only 0.5âÃ. combined with the Stark broadening of the Balmer recombination edge could smear the Balmer jump making it non-detectable. Still, we note that other WL flares reported in the literature either did not display a Balmer jump, or had other properties not readily explained by a Hydrogen recombination spectrum. In particular, a strong blue continuum emission at Î»  4000âÃ has been often reported (Hiei 1982; Neidig & Wiborg 1984), and shown by Donati-Falchi et al. (1985) as the result of the blending of Stark-broadened high-order Balmer lines in a dense chromosphere. In the model of Donati-Falchi et al. (1985), this âbumpâ in the blue continuum peaks at Î»â â¼ â3675âÃÂ  and becomes more prominent and shifts to redder wavelengths as electron density in the flare region increases. Existing models of Stark broadening imply that such blue âcontinuumâ emission originates from a location with electron density in excess of 1013 cmâ3 and electron temperature between 7000 and 10,000Â K. Interestingly, in our excess spectra we observe a relatively featureless, broad bump peaking at Î» â¼Â 3675âÃ (Figure 9). Taken at face value, the electron densities inferred from interpreting this feature via Stark broadening appear at odd with those derived in Section 5.2, unless more optically thin features such as the higher order Balmer lines probe different portions of the flaring atmosphere. However, additional spectra that are not affected by chromatic aberration will be needed to confirm and understand this feature.


K2-138 is a remarkable dynamical system with five planets in a chain close to the 3:2 resonance. In addition, these planets form a chain of three-body Laplace resonances (Christiansen et al. 2018). This makes K2-138 an ideal target to study transit timing variations. We estimated the transit timing variations using the TTVFaster code (Agol & Deck 2016). We first assumed zero eccentricities for the six planets and took the median masses from Table A.2. We found amplitudes of the order of 2.0, 4.1, 7.3,4.5, 6.4, and 0.02 min for planets from b to g, respectively. Then, assuming median eccentricities from Table A.2, we obtained amplitudes of the order 34, 42, 66, 37, and 41 min for planets b to f (Fig. 9). These amplitudes are excluded by K2 observations which do not show any significant variations at the level of 8–10 min, asdemonstrated by Christiansen et al. (2018). Therefore, the planets are likely close to circular orbits, which is compatible with the posterior distributions we obtained. This is also in line with results showing that tightly packed multi-transiting planets have low eccentricities (Van Eylen et al. 2019). Finally, using the upper limits derived for the masses of planets f and g, we computed the corresponding TTVs of planets e and f. Planet g does not impact TTVs of planet f in a significant (detectable) way as it is far from resonance, being located after a double gap in the chain of resonance. Therefore, it is not possible to use TTVs of planet f to constrain the mass of planet g even if there are undetected and non-transiting planets filling the gap in the chain. Still assuming circular orbits, TTVs of planet e reach an amplitude of the order eight minutes only for masses of planet f above 12 M⊕, which is above the upper limit we derived. However, as we cannot exclude an absorption of the signal of planet f by the GP, it is more conservative to include masses for planet f up to around 12 M⊕. Using the upper limit mass for planet f, the predicted TTVs of planet e are of the order of 6.4 min, and using the median mass they are of 4.5 min. Both are well within reach of space mission like CHEOPS which has a cadence of up to 60 s. Observations of transits of planet e using CHEOPS would be beneficial to constrain further the mass of planet f through a photodynamical analysis (Barros et al. 2015). Additionally, more transit observations, with a higher cadence, of planets b, c and d would allow to constrain the masses of planets c, d and e from TTVs which would make K2-138 an interesting benchmark system for comparing RV and TTV masses. This would allow to better calibrate the two mass measurement techniques. Nevertheless, at this stage, and without more precise photometric observationsof planets e and f, an analysis including TTVs would not allow to measure the masses of planets f and g.


The de-reddened photometry of TYC 8252-533-1 B and its host star are presented in Table 5. The listed Gaia photometry is directly obtained from the EDR3 catalog and the infrared photometry originates from our high-contrast imaging data. We applied the previously determined extinction of AV = 0.13 mag to correct the presented fluxes. As the SPHERE H band data from 2017-04-02 were collected in poor atmospheric conditions and with a very unstable AO performance, we disregarded the photometry that was extracted from these observations in our further analysis. We fitted the full optical and infraredSED of the companion utilizing the MCMC approach described in Bohn et al. (2020b). We used a linearly interpolated grid of BT-Settl models (Allard et al. 2012; Baraffe et al. 2015) with effective temperatures, Teff, between 1500 K and 4000 K, surface gravity in the range of

$0<\log\left(g\right)<6$0log(g)6
, and solar metallicity. We further allowed object radii R from 0.5 RJup to 5 RJup. The MCMC sampler was implemented in the emcee framework (Foreman-Mackey et al. 2013) and we used 100 walkers with 10 000 steps to sample the posterior distribution. In accordance with the computed autocorrelation time of approximately 200 steps, the first 1000 samples of each chain were discarded as burn-in phase and we further continued using each twentieth step of the remaining chains. This provided 45 000 samples for our posterior distribution in (Teff,

$\log\left(g\right)$log(g)
, R), which are visualized in Fig. D.1. From these distributions we derived an effective temperature of

$T_{\textrm{eff}}=3092^{+186}_{-91}\,K$Teff=3092−91+186 K
, a surface gravity of

$\log\left(g\right)=3.41^{+1.07}_{-0.31}$log(g)=3.41−0.31+1.07
 dex, and a radius of

$R=3.5^{+0.3}_{-0.4}\,R_{\textrm{Jup}}$R=3.5−0.4+0.3 RJup
 for TYC 8252-533-1 B. These values were obtained as the 95% confidence intervals around the medians of the posterior distributions. From these three parameters we derived the object luminosity as

$\log\left(L_*/L_{\odot}\right)=-1.99^{+0.01}_{-0.02}$log(L*/L⊙)=−1.99−0.02+0.01
. The results of this SED fit are presented in Fig. 9. Interestingly, the derived object radius is markedly larger than usual radii of field brown dwarfs of this mass, which are on the order of 1 RJup (e.g., Chabrier et al. 2009). This inflated photometric radius, however, is nothing unusual for sub-stellar companions at this young age, and similarly large values have been reported for various planetary-mass objects (e.g., Schmidt et al. 2008; Bohn et al. 2019; Stolker et al. 2020).


In our study, the central engine of GRB 100814A is assumed to be a newborn magnetar with spin evolution. The early-time optical shallow decay phase and X-ray plateau come from the initial spin-down process and the late-time optical rebrightening results from the enhanced energy injection generated by the spinning up of the magnetar. When calculating the dynamic process of the multi-band afterglow of GRB 100814A, we use the Equations for beamed GRB outflows developed by Huang et al. (1999, 2000) considering continuous energy injection from the central magnetar with spin evolution. The bulk Lorentz factor (γ) of the shocked interstellar medium is described by the following differential Equation (for details, see Huang et al. 1999):
6

where Mej is the initial ejecta mass, m is the swept-up interstellar medium mass, and  is the radiative efficiency. When the energy injection due to the magnetic dipole radiation from the central magnetar is taken into account, the above differential Equation can be modified to be (Kong & Huang 2010; Geng et al. 2013)
7

where

 can be calculated from Equation (1). In our calculations, several effects, such as lateral expansion, electron synchrotron cooling, and equal arrival time surfaces, have been incorporated. In the absence of electron synchrotron cooling, the comoving frame distribution of the shock-accelerated electrons is usually assumed to be a power-law function,

, where p is the power-law index. After considering the electron synchrotron cooling effect, the distribution function will be changed to

 for electrons above a critical Lorentz factor

 (Sari et al. 1998). For detailed description of electron distribution, see Huang et al. (2000) and Huang & Cheng (2003). We neglect the adiabatic pressure and energy losses due to adiabatic expansion, which might also have a minor effect on the dynamical evolution of the blast wave (van Eerten et al. 2010; Pe’er 2012; Nava et al. 2013). For radiative process, the multi-band afterglow emission mainly comes from synchrotron radiation of the shock-accelerated electrons due to their interaction with the magnetic field (Sari et al. 1998; Sari & Piran 1999). We present our numerical results below.


For the first step, we adopted the nucleosynthesis calculation for around 4070 nuclei performed by Fujimoto et al. (2007), which was cooled using the adiabatic expansion modeled from Freiburghaus et al. (1999) to provide the elemental composition ratios of nuclei for Y
e = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, and 0.45. This estimation assumes that the initial environment has a temperature of 9 × 109 K, a radius of 100 km, entropy per baryon of 10 k
B, where k
B is the Boltzmann constant, and a velocity of 2 × 109  cm s−1, along with the initial abundances of the 4070 nuclei in nuclear statistical equilibrium. As a result, the calculation provides the mass fractions at t = 1 yr evaluated with the nuclear reaction network (network A in Fujimoto et al. 2007), by using Y
e = 0.10–0.45 in steps of 0.05. To set up the mass distribution of nuclei for the NSMs at t = 1 yr, we blended the nuclei with the mass fraction using the Y
e provided in Wanajo et al. (2014). Specifically, the fractions are 4.54%, 4.85%, 14.6%, 29.7%, 10.3%, 25.1%, 10.5%, and 0.33% for Y
e = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, and 0.45, respectively. Note that this Y
e-fraction model by Wanajo et al. (2014) describes a slightly less neutron-rich environment than those by the recent dynamical-ejecta models after the kilonova observations of the gravitational event GW170817, such as four models in Kullmann et al. (2022) under two kinds of equations of state, density dependent 2 (DD2; Hempel & Schaffner-Bielich 2010; Typel et al. 2010), and SFHo (Steiner et al. 2013). In this paper, we adopted the first one by Wanajo et al. (2014) as a pessimistic case for the r-process site, but changing the Y
e-fraction models does not change the conclusions from the gamma-ray spectra as tested in Section 3. Figure 1 shows the mass fraction of multiple nuclei at t = 1 yr generated in an NSM case, information that is given in the table of nuclides (neutron number N versus atomic number Z). Using the same data set, Figure 2 summarizes the distribution of nuclei with mass number A at t = 1 yr, showing the contributions of Y
e. This plot demonstrates that the environment with lower Y
e contributes to the generation of heavier elements.


In this section, we study the feasibility of detecting non-thermal SZ with future CMB experiments. The distortion in intensity Iν of CMB at a frequency ν and towards a localized concentration of hot electrons can be written as,
(19)$$\begin{eqnarray*}
\Delta I_{\nu }=\Delta I_{\rm {y}}+\Delta I_{\rm {\mu }}+\Delta I_{\rm {kSZ}}+\Delta I_{\rm {rSZ}}+\Delta I_{\rm {NT}},
\end{eqnarray*}$$where ΔIy is the typically dominant non-relativistic y distortion, ΔIμ is the μ distortion, ΔIkSZ is the temperature shift of the CMB blackbody due to kSZ effect (Sunyaev & Zeldovich 1980), ΔIrSZ is the relativistic SZ signal from a massive galaxy cluster, and ΔINT is the non-thermal SZ distortion. The expression for ΔIy, ΔIμ, and ΔIkSZ are given by (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1970; Illarionov & Siuniaev 1975),
(20)$$\begin{eqnarray*}
\Delta I_{\rm {y}}=y\frac{2\rm {h}{\nu }^3}{\rm {c^2}}\frac{x\rm {e^x}}{(\rm {e^x}-1)^2}\left[x\frac{\rm {e^x}+1}{\rm {e^x}-1}-4\right]
\end{eqnarray*}$$(21)$$\begin{eqnarray*}
\Delta I_{\rm {\mu }}=\mu \frac{2\rm {h}{\nu }^3}{\rm {c^2}}\frac{\rm {e^x}}{(\rm {e^x}-1)^2}\left[\frac{x}{2.19}-1\right]
\end{eqnarray*}$$(22)$$\begin{eqnarray*}
\Delta I_{\rm {kSZ}}=\left(\frac{\Delta T}{T}\right)\frac{2\rm {h}{\nu }^3}{\rm {c^2}}\frac{\rm {e^x}}{(\rm {e^x}-1)^2}
.
\end{eqnarray*}$$The y-distortion can have contribution from both pre-recombination Universe as well as post-recombination Universe while μ-distortion can only be created at redshifts greater than 2 × 105. The SZ spectrum from electrons with energy ≳keV is obtained by including Klein–Nishina corrections for Compton scattering which can be calculated perturbatively as an expansion in $\frac{T_{\rm {e}}}{m_{\rm {e}}}$ (Challinor & Lasenby 1998; Itoh, Kohyama & Nozawa 1998; Sazonov & Sunyaev 1998; Dolgov et al. 2001). For example, the temperature of massive galaxy cluster with mass 1014 times solar mass can be ∼5 keV (Erler et al. 2018), and with a typical y ∼ 10−5.


Stars with masses 130 M
⊙ ≲ M
ZAMS ≲ 250 M
⊙ are subject to the pair instability, which disrupts them completely, and hence no BH forms. Stars with M
ZAMS ≳ 250 M
⊙ can collapse directly to IMBHs with a mass ≳135 M
⊙. Thus, in the standard picture, there should be an upper stellar-mass BH gap in the range [65, 135] M
⊙, and any BHs observed in this range (e.g., the primary BH of GW190521) have to form via other formation channels—for example, through hierarchical coalescence of smaller BHs or direct collapse of a stellar merger between an evolved star and a main-sequence companion (e.g., Quinlan & Shapiro 1989; Portegies Zwart & McMillan 2000; Ebisuzaki et al. 2001; Miller & Hamilton 2002; O’Leary et al. 2006; Gerosa & Berti 2017; Antonini et al. 2019; Di Carlo et al. 2019, 2020; Rodriguez et al. 2019; Gayathri et al. 2020a; Kimball et al. 2021; Mapelli et al. 2021). However, the exact mass boundaries of the gap depend on parameters that are uncertain. For example, the 12C(α, γ)16O nuclear reaction rate, which converts carbon to oxygen in the core, can affect the boundary significantly (Takahashi 2018; Farmer et al. 2020; Costa et al. 2021; Woosley & Heger 2021). Here, to better determine whether future GW observations will be able to observe BHs in the mass gap, we recompute the mass-gap boundaries with updated 12C(α, γ)16O reaction rates and increased mass and temporal resolution. The complexity of this system has made a reliable analysis of the reaction a decades-old challenge.
7

7
The reaction rate for 12C(α, γ)16O is determined by the quantum structure of the compound nucleus 16O as an α cluster system. It is characterized by the interfering ℓ = 1 waves of the J

π
 = 1− resonances and sub-threshold levels defining an E1 component for the reaction cross section as well as by the ℓ = 2 components and interference from broad J

π
 = 2+ resonances and the nonresonant E2 external capture to the ground state of 16O. In addition to these two main E1 and E2 ground-state components, transitions to higher lying excited states occur that also add to the total cross section (see, e.g., Buchmann & Barnes 2006; deBoer et al. 2017). The rapidly declining cross section at low energies has prohibited a direct measurement of the reaction at stellar temperatures, and the reaction rate is entirely based on the theoretical analysis and extrapolation of the experimental data toward lower energies. The rate newly derived by deBoer et al. (2017), using a multichannel analysis approach, derives for the first time a reliable prediction for the interference patterns within the reaction components by taking into account all available experimental data sets that cover the near-threshold energy range of the 12C(α, γ)16O process.


Another possible explanation for the peculiar solar composition is that some of the dust in the pre-solar nebula was radiatively cleansed by luminous hot stars in the solar neighbourhood before the formation of the Sun and its planets. This dust-cleansing scenario is supported by the finding that the open cluster M67 seems to have a chemical composition closer to the solar composition than most solar twins (Önehag et al. 2011, hereafter O11 and Önehag, Gustafsson & Korn 2014, hereafter O14). They suggested that the proto-solar nebula was dust-cleansed by massive stars, similar to what happened for the proto-cluster cloud of M67, while the majority of solar twins in the field would presumably have formed in less massive clusters where no nearby high-mass star (≥15 M⊙) was formed. M67 offers the possibility for studying the solar-type stars in a dense cluster environment. This cluster has about solar metallicity ([Fe/H] in the range −0.04 to +0.03, e.g. Hobbs & Thorburn 1991; Yong, Carney & Teixera de Almeida 2005; Randich et al. 2006; Pasquini et al. 2008). The age of M67 is also comparable with that of the Sun: 3.5–4.8 Gyr (Yadav et al. 2008). Pasquini et al. (2008) listed 10 solar-twin candidates in M67, including M67-1194 which hosts a hot Jupiter with a period of 6.9 d and a minimum mass of 0.34 MJup (Brucalassi et al. 2014). M67-1194 was studied by O11 and re-visited by O14. They found that unlike nearby solar twins, which were systematically analysed by Meléndez et al. (2009) and Ramírez, Meléndez & Asplund (2009), the chemical composition of M67-1194 is more solar-like and therefore suggested that the Sun may have been formed in a similar cluster environment, perhaps even in M67. While this scenario is plausible when considering chemical abundances and ages, the dynamics are problematic: the Sun has an orbit close to the Galactic plane, while M67 is presently some 450 pc above the plane. The probability that the Sun, if formed in M67 with an orbit similar to the present cluster orbit, was scattered or diffused out of the cluster into the present solar orbit, is found to be quite low, if the existence of the outer planetary system is taken into account (Pichardo et al. 2012). Gustafsson et al. (2016) suggested that the high-altitude metal-rich clusters (such as M67) were formed in orbits close to the Galactic plane and later scattered to higher orbits by interaction with giant molecular clouds and spiral arms. Thus, it is possible, though not very probable, that the Sun formed in such a cluster before scattering occurred. Currently, only one solar twin in M67 (M67-1194) was spectroscopically analysed with high precision (∼0.02 dex). It is thus crucial to analyse the chemical composition of additional solar-type stars in M67 of very high precision.


Let us consider a SMBH with mass MBH kicked with velocity vk from the galactic centre (r = 0). The SMBH will be ejected from the galactic halo if vk exceeds the escape velocity of the system:

(33)

\begin{eqnarray}
v_{\rm esc}= \sqrt{2G\left( \frac{M_{\rm BCG}}{r_{\rm H}} + c \,g_{\rm c} \frac{M_{\rm DM}}{r_{\rm v}}\right)} \,.
\end{eqnarray}

If vk  vesc, the SMBH will stop at a distance rmax from the centre. Gualandris & Merritt (2008) showed that the maximum displacement rmax can be estimated simply through energy conservation neglecting star friction (see their fig. 2):

(34)

\begin{eqnarray}
\frac{1}{2} v_{\rm k}^{2}+\phi (0)=\phi (r_{\rm max})\,.
\end{eqnarray}

The initial displacement is reached in a time which is typically 100 times smaller than the sinking time-scale (Gualandris & Merritt 2008) and will be therefore neglected. Here, we estimate the time needed to sink back to r = 0 integrating the DF equation on quasi-circular orbits. The frictional force exerted on to the black hole is given by (e.g. Binney & Tremaine 1987)

(35)

\begin{eqnarray}
F(r)=\frac{4 \pi G^2 M_{\rm BH}^2 \rho (r)\,\xi (r) \ln \Lambda }{v_{\rm c}^2(r)}\,,
\end{eqnarray}

where $v_{\rm c}(r)=\sqrt{r\, {\rm d}\phi /{\rm d}r}$ is the circular velocity, ln Λ is the Coulomb logarithm and the factor ξ(r) depends on the stellar velocity distribution. We take ln Λ = 2.5, as observed by Gualandris & Merritt (2008) in the very first phase of their simulated orbits (see also Escala et al. 2004). We assume the velocity distribution to be locally Maxwellian, with velocity dispersion σ(r). Although not exact, the Maxwellian distribution is approached as a consequence of collisionless relaxation processes (Lynden-Bell 1967). Under this assumption, the ξ factor in equation (35) reads (Binney & Tremaine 1987)

(36)

\begin{eqnarray}
\xi (r) = {\rm erf}\left[\frac{v_{\rm c}(r)}{\sqrt{2} \sigma (r)}\right] - \sqrt{\frac{2}{\pi }}\frac{v_{\rm c}(r)}{\sigma (r)}\exp \left[-\frac{v_{\rm c}^2(r)}{2\sigma ^2(r)}\right]\,.
\end{eqnarray}

The velocity dispersion σ(r) is computed from our galactic potential using the expression provided by Binney (1980) when isotropy is assumed. The frictional force F(r) is tangential and directed opposite to the SMBH velocity. The SMBH angular momentum L(r) = MBHrvc(r) is lost at the rate dL(r)/dt = −rF(r) by Newton's third law, causing the SMBH to slowly inspiral while remaining on a quasi-circular orbit. The DF time-scale, over which the SMBH sinks back to the galactic centre r = 0 from its initial position rmax , is thus given by6
(37)

\begin{eqnarray}
t_{\rm DF}=-\int _{r_{\rm max}}^0 \frac{\mathrm{d} L(r)}{\mathrm{d}r} \frac{1}{r F(r)} \mathrm{d}r\,.
\end{eqnarray}


We consider the MW satellites with parallax and proper-motion measurements from Gaia DR2 (Gaia Collaboration et al. 2018c) and the corresponding galactocentric coordinates and velocities computed and documented by Riley et al. (2019, their Table A.2; see also Li et al. 2020) and Vasiliev & Belokurov (2020). Of these, we only consider the satellites with known dynamical mass estimates (Simon & Geha 2007; Bekki & Stanimirović 2009; Łokas 2009; Erkal et al. 2019). Adopting the initial conditions for galactocentric positions (R, z, ϕ) and velocities (v

R
, v

z
, v

ϕ
) as the median values quoted by Riley et al. (2019) and Vasiliev & Belokurov (2020), we simulate the orbits of the galaxies in the combined gravitational potential of the MW halo, disk, and bulge, which are respectively modeled by a spherical NFW (Navarro et al. 1997) profile (virial mass M

h
 = 9.78 × 1011
M
⊙, scale radius r

h
 = 16 kpc, and concentration c = 15.3), a Miyamoto−Nagai (Miyamoto & Nagai 1975) profile (mass M

d
 = 9.5 × 1010
M
⊙, scale radius a = 4 kpc, and scale height b = 0.3 kpc), and a spherical Hernquist (1990) profile (mass M

b
 = 6.5 × 109
M
⊙ and scale radius r

b
 = 0.6 kpc).
5

5
Our MW potential is similar to GALPY MWPOTENTIAL2014 (Bovy 2015) except for the power-law bulge, which has been replaced by an equivalent Hernquist bulge. The total mass of our fiducial MW model is thus 1.08 × 1012
M
⊙. We evolve the positions and velocities of the satellites both forward and backward in time from the present day, using a second-order leap-frog integrator. For simplicity, we ignore the effect of dynamical friction.
6

6
Dynamical friction might play an important role in the orbital evolution of massive satellites like the Large Magellanic Cloud (LMC) and Sgr, pushing their orbital radius farther out in the past. From each individual orbit, we note the time, t
cross, when the satellite crosses the disk (i.e., crosses z = 0) and record the corresponding distance, x
P, from the Sun, which we integrate backward/forward in time using a purely circular orbit up to t
cross. We also record the velocity,

vP=vR2+vz2+vϕ2

, and the angle of impact with respect to the disk normal,

θP=cos−1(vz/vP)

. Finally, we compute the disk response to the satellite encounter using Equation (49). Results are summarized in Table 1 and Figures 6 and 7.


One type of radio transient expected to be found in image plane transient searches is flares from stars and stellar systems (see e.g. Osten 2008, for a summary). Radio flare stars are usually M-type dwarf stars that emit coherent radio bursts on time-scales of minutes to hours. Recently, Villadsen & Hallinan (2019) detected 22 coherent radio bursts from M dwarfs using the VLA at 300 MHz and 1–6 GHz, and Zic et al. (2019) detected several pulses from the M-dwarf UV Ceti with ASKAP. As well as flare stars, binary systems such as RS Canum Venaticorum (RS CVn), cataclysmic variables (CVs), and symbiotic binaries are known to flare in the radio. RS CVn are binary systems consisting of a late-type giant or sub-giant star with a late-type main-sequence star companion (e.g. Hall 1976; Craig et al. 1997; García-Sánchez, Paredes & Ribó 2003). RS CVn are chromospherically active and the giant or sub-giant rotates quasi-synchronously with the orbital period. Periods for RS CVn are typically 1–20 d, and radio flares on RS CVn systems have been observed to last up to a few days (e.g. Walter et al. 1987; Trigilio, Umana & Migenes 1993). RS CVn emit in both the radio and X-ray while in quiescence (e.g. Gunn 1996). CVs are binary systems with a white dwarf primary accreting matter from Roche lobe overflow of the secondary star, usually a main-sequence star (e.g. Coppejans et al. 2016). Dwarf novae from magnetic CVs have been observed to exhibit radio outbursts that can last for weeks, with rapid radio flaring on time-scales less than an hour (Mooley et al. 2017). Symbiotic binaries also have a white dwarf primary, but the companion is a red giant star and the orbit is wide. In these systems, mass is accreted on to the white dwarf via stellar winds (O’Brien et al. 2015). Radio variability over time-scales of a few hours have been detected on symbiotic binaries, for example RX Puppis (Seaquist 1977). AR Scorpii is another stellar binary that is observed in the radio, it consists of a white dwarf primary and an M-dwarf companion. AR Scorpii has an orbital period of 3.56 h, is observed to pulsate in the optical, radio and X-ray with a period of 1.97 min (Marsh et al. 2016; Takata et al. 2018), and is highly polarized (e.g. Buckley et al. 2017). Radio flaring stellar systems vary on a variety of time-scales, making it difficult to detect when these systems are in outburst (e.g. Osten & Bastian 2008). This means that wide-field monitoring is an important method for discovering and investigating these sources in the radio.


It has been shown by Kocifaj et al. (2019) that night-sky luminance due to a ground-based light source (city or town) can be approximated for intermediate and/or long distances $D$ by the formula
(1)$$\begin{eqnarray*}
J \left( z \right) = {I_0}\ \frac{9}{{32\pi }}\frac{{\tilde{\tau }{\mathrm{ e}^{ - D\tilde{\tau }/\left( {3\tilde{H}} \right)}}}}{{\mu \tilde{H}D}}\left\{ {Ci\left( {aD} \right)\sin\left( {aD} \right) + \left[ {\frac{\pi }{2} - Si\left( {aD} \right)} \right]\cos\left( {aD} \right)} \right\},
\end{eqnarray*}$$where $z$ is the observational zenith angle, $\mu = \cos( z )$, ${I_0}$ the luminance of a city normalized to its area, i.e. the city interpreted as a point light source (${I_0}$ is measured in cd or lm sr−1). The functions $Si( x )$ and $Ci( x )$ introduced above are sine and cosine integrals, respectively (Press et al. 2007).The argument $x$ is the product of distance $\ D$ and parameter $a$, with
(2)$$\begin{eqnarray*}
a\ = \frac{{\tilde{\tau }}}{{\tilde{H}}}\ \left( {1 + \frac{{{{\tilde{\mu }}_0}}}{\mu }} \right),
\end{eqnarray*}$$where ${\mu _0}$ is the mean effective cosine of the emission angle (see section 2 of Kocifaj et al. 2019) and $\tilde{H}\ $ is the altitude up to which a homogeneous atmosphere would extend. Here we assume that $\tilde{H}\ $ is composed of the weighed contributions of molecular and aerosol constituents, i.e.
(3)$$\begin{eqnarray*}
\tilde{H} = \frac{{{H_\mathrm{ R}}{{\tilde{\tau }}_\mathrm{ R}} + {H_\mathrm{ a}}{{\tilde{\tau }}_\mathrm{ a}}}}{{{{\tilde{\tau }}_\mathrm{ R}} + {{\tilde{\tau }}_\mathrm{ a}}}},
\end{eqnarray*}$$where ${\tilde{\tau }_\mathrm{ R}}$ and ${\tilde{\tau }_\mathrm{ a}}$ are the Rayleigh and aerosol components of the total optical depth of a cloudless atmosphere except for ozone or water-vapour absorption bands, i.e. $\tilde{\tau } = {\tilde{\tau }_\mathrm{ R}} + {\tilde{\tau }_\mathrm{ a}}$ (Utrillas et al. 2000). Here, ${H_\mathrm{ R}}$ and ${H_\mathrm{ a}}$ are the Rayleigh and aerosol components of the atmospheric scaleheights. Aerosol scaleheight typically ranges from 1–3 km (Wu et al. 2011), while ${H_\mathrm{ R}}$ = 8 km is a commonly accepted value in many studies (see e.g. Waquet et al. 2009). The emission function for a city, i.e. the luminous intensity emitted by that city, is generally unknown, so an isotropic-like radiator appears to be a useful zero-order approximation, resulting in the following formula:
(4)$$\begin{eqnarray*}
J \left( z \right) = {W_0}\ \frac{9}{{64{\pi ^2}}}\frac{{\tilde{\tau }{\mathrm{ e}^{ - D\tilde{\tau }/\left( {3\tilde{H}} \right)}}}}{{\mu \tilde{H}D}}\left\{ {Ci\left( {aD} \right)\sin\left( {aD} \right) + \left[ {\frac{\pi }{2} - Si\left( {aD} \right)} \right]\cos\left( {aD} \right)} \right\},
\end{eqnarray*}$$where ${W_0}$ (lm) is the total amount of lumens directed above the horizontal; consequently, it is obtained as a superposition of both direct emission upwards and isotropic reflection from the ground. If the entirety of light approaching the sky is due only to diffuse reflection from the ground, then ${W_0} = G \cdot W_0^{\rm installed}$, where $G$ is the diffuse reflectance and $W_0^{\rm installed}$ is the total lumen output from all light sources in the city.


Assuming that some or the entire detected high-energy neutrino flux originates from cosmic rays accelerated in structure-formation accretion shocks, we find the corresponding gamma-ray flux and determine its contribution to the EGRB for different assumed cosmic-ray spectra and plot these results in Figure 1. Panels on the left show SFCR curves after normalization to the αν = 2.0 neutrino spectrum from Aartsen et al. (2014b), and panels on the right use normalization to the latest αν = 2.46 neutrino spectrum from Aartsen et al. (2015). Top panels represent SFCR curves (dot–dashed line) derived for the case of strong shocks, i.e., for cosmic-ray spectral index α = 2.0, which matches that of neutrinos from Aartsen et al. (2014b). Normalization to the αν = 2.0 neutrino spectrum from Aartsen et al. (2014b) yields an upper limit to the SFCR contribution of ≈46% of the EGRB observed by Fermi, which is around the contribution of some of the major components, such as unresolved star-forming galaxies contributing ≲50% (Fields et al. 2010; dashed line), or blazars contributing ≈16% (Abdo et al. 2010a; dotted line). If, for this same SFCR model, the attenuation of highest-energy gamma-ray photons by extragalactic background light (EBL; Gilmore et al. 2012) is included (thin red solid line), we find that this reduces the SFCR contribution to ≲18% of the observed EGRB. The sum of all components is shown with a thick blue solid line. Using the latest neutrino spectrum with αν = 2.46 (Aartsen et al. 2015) as the upper limit constraint, we find that for SFCRs accelerated in strong shocks, i.e., with spectral index α = 2.0, their contribution to the EGRB can be ≈29%, or ≲12% after the EBL attenuation was included. However, SFCRs with such a spectrum cannot explain the entire detected high-energy neutrino flux, and additional sources would be needed. If, on the other hand, SFCRs are assumed to have softer spectra with indices α > 2.0, their resulting fluxes would quickly violate the Fermi EGRB observations. The middle panels of Figure 1 show SFCR gamma-ray fluxes for source spectra α = 2.3, while the bottom panels correspond to α = 2.6. Here we point out that using the latest neutrino fluxes with spectrum  αν = 2.46 (Aartsen et al. 2015) is more constraining in the sense that it does not allow for this entire neutrino flux to be of the SFCR origin because the accompanying gamma-ray flux quickly violates the observed EGRB, as we can see in the bottom two panels on the right side of Figure 1. Such a soft neutrino spectrum, together with gamma-ray observations from Fermi, allows for only a small fraction of its flux to be made by SFCRs with hard spectra α ≲ 2.2.