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).


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),

and later on with the Atacama Large Millimeter/submillimeter Array (ALMA,


Among the HMCs studied, G31 stands out as the only source that displays accelerating infall and rotational spin-up


At an angular resolution of 0.′′2 (~ 750 au), the dust continuum emission traces a well-resolved, monolithic, and featureless core, called Main by


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.


Within molecular clouds, several studies have found a power-law correlation between the surface densities of YSOs and gas


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.


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.


These amplitudes are excluded by K2 observations which do not show any significant variations at the level of 8–10 min, asdemonstrated by


We estimated the transit timing variations using the TTVFaster code


This is also in line with results showing that tightly packed multi-transiting planets have low eccentricities


Observations of transits of planet e using CHEOPS would be beneficial to constrain further the mass of planet f through a photodynamical analysis


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.


As a result, the calculation provides the mass fractions at t = 1 yr evaluated with the nuclear reaction network (network A in

), by using Y
e = 0.10–0.45 in steps of 0.05.


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}


Under this assumption, the ξ factor in equation (35) reads

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}


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.


One type of radio transient expected to be found in image plane transient searches is flares from stars and stellar systems (see e.g.

for a summary). Radio flare stars are usually M-type dwarf stars that emit coherent radio bursts on time-scales of minutes to hours.


detected 22 coherent radio bursts from M dwarfs using the VLA at 300 MHz and 1–6 GHz


detected several pulses from the M-dwarf UV Ceti with ASKAP.


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.


RS CVn emit in both the radio and X-ray while in quiescence (e.g.


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.


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


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


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.


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

and $\tilde{H}\ $ is the altitude up to which a homogeneous atmosphere would extend.


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.


and panels on the right use normalization to the latest αν = 2.46 neutrino spectrum from


Using the latest neutrino spectrum with αν = 2.46

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.


Here we point out that using the latest neutrino fluxes with spectrum  αν = 2.46

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.


Panels on the left show SFCR curves after normalization to the αν = 2.0 neutrino spectrum from


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


Normalization to the αν = 2.0 neutrino spectrum from

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


The polarized emission of these OH masers was studied several times in the past (e.g., Fish et al. 2005, 2006; Green et al. 2007; Fish & Reid 2007; Fish & Sjouwerman 2010). Fish et al. (2005) detected linearly polarized emission toward 24 OH maser spots at 1.665 GHz with the VLBA (⟨Pl⟩ = 36%). From these detections, they measured a mean polarization angle of + 59°. Under the assumption that the magnetic field is perpendicular to the linear polarization vectors, the magnetic field is orientated on the plane of the sky with an angle

$\Phi_{\rm{B}}^{\rm{1.6~GHz~OH}}={-}31$

Φ
B

1.6 GHz OH

=−31

°. From the 6.0 GHz OH masers, the linearly polarized emission was also detected, which led Green et al. (2007) and Fish & Sjouwerman (2010) to measure ⟨Pl⟩ = 13.6% and 19%, respectively. Green et al. (2007) measured a mean linear polarization angle of − 58° (

$\Phi_{\rm{B}}^{\rm{6.0~GHz~OH}}={+}32$

Φ
B

6.0 GHz OH

=+32

°) with the Multi-Element Radio Linked Interferometer Network (MERLIN), while Fish & Sjouwerman (2010) measured ⟨χ⟩ = −60° (

$\Phi_{\rm{B}}^{\rm{6.0~GHz~OH}}={+}30$

Φ
B

6.0 GHz OH

=+30

°) with a global VLBI network. In addition, from the right circular and left circular polarization spectra of the 1.6 and 6.0 GHz OH masers, it was possible to measure the magnetic field strength. From 11 and 10 Zeeman pairs of 1.7 GHz OH masers, a magnetic field on the plane of the sky B|| = −3.3 mG (Fish et al. 2005) and −4.5 mG (Fish & Reid 2007) was measured, respectively; while from 7, 6, and 11 Zeeman pairs of 6.0 GHz OH masers, B|| = −5.6 mG (Fish et al. 2006), − 3.5 mG (Green et al. 2007), and −4.8 mG (Fish & Sjouwerman 2010) was estimated, respectively. Green et al. (2007) reported a linearly polarized emission from two 6.7 GHz CH3OH maser features with the MERLIN (beam size 43 mas× 43 mas). In particular, for their features C (

$V_{\rm{lsr}}^{\rm{C}}={+}14.62$

V

lsr
C

=+14.62

 km s−1, IC = 53.56 Jy beam−1,

$\Delta v{_{\textrm{L}}}^{\rm{C}}=0.27$

Δv{_L}^C=0.27

 km s−1,

$P_{\rm{l}}^{\rm{C}}=0.2\%$

P
l
C

=0.2%

) and D (

$V_{\rm{lsr}}^{\rm{D}}={+}14.62$

V

lsr
D

=+14.62

 km s−1, ID = 20.03 Jy beam−1,

$\Delta v_{\textrm{L}}^{\rm{D}}=0.27$

Δ
v
L
D

=0.27

 km s−1,

$P_{\rm{l}}^{\rm{D}}=1.3\%$

P
l
D

=1.3%

), they measured a linear polarization angle of χC = +20.6° and χD = −76.7°, respectively.Green et al. (2007) also reported the very first Zeeman-splitting measurement for 6.7 GHz CH3OH maser emission using the cross-correlation method (Modjaz et al. 2005); for maser feature D, they measured

$P_{\rm{V}}^{\rm{D}}=0.6\%$

P
V
D

=0.6%

 and ΔVZD = 0.9 ± 0.3 m s−1.


From the 6.0 GHz OH masers, the linearly polarized emission was also detected, which led

and Fish & Sjouwerman (2010) to measure ⟨Pl⟩ = 13.6% and 19%, respectively.


measured a mean linear polarization angle of − 58° (

$\Phi_{\rm{B}}^{\rm{6.0~GHz~OH}}={+}32$

Φ
B

6.0 GHz OH

=+32

°) with the Multi-Element Radio Linked Interferometer Network (MERLIN), while Fish & Sjouwerman (2010) measured ⟨χ⟩ = −60° (

$\Phi_{\rm{B}}^{\rm{6.0~GHz~OH}}={+}30$

Φ
B

6.0 GHz OH

=+30

°) with a global VLBI network.


reported a linearly polarized emission from two 6.7 GHz CH3OH maser features with the MERLIN (beam size 43 mas× 43 mas). In particular, for their features C (

$V_{\rm{lsr}}^{\rm{C}}={+}14.62$

V

lsr
C

=+14.62

 km s−1, IC = 53.56 Jy beam−1,

$\Delta v{_{\textrm{L}}}^{\rm{C}}=0.27$

Δv{_L}^C=0.27

 km s−1,

$P_{\rm{l}}^{\rm{C}}=0.2\%$

P
l
C

=0.2%

) and D (

$V_{\rm{lsr}}^{\rm{D}}={+}14.62$

V

lsr
D

=+14.62

 km s−1, ID = 20.03 Jy beam−1,

$\Delta v_{\textrm{L}}^{\rm{D}}=0.27$

Δ
v
L
D

=0.27

 km s−1,

$P_{\rm{l}}^{\rm{D}}=1.3\%$

P
l
D

=1.3%

), they measured a linear polarization angle of χC = +20.6° and χD = −76.7°, respectively.


while from 7, 6, and 11 Zeeman pairs of 6.0 GHz OH masers, B||


also reported the very first Zeeman-splitting measurement for 6.7 GHz CH3OH maser emission using the cross-correlation method

for maser feature D, they measured

$P_{\rm{V}}^{\rm{D}}=0.6\%$

P
V
D

=0.6%

 and ΔVZD = 0.9 ± 0.3 m s−1.


We compute the integrated polarized emission of our radio relic following the formalism of Burn (1966):
(6)$$\begin{eqnarray*}
P_{\mathrm{pol}}(\lambda ^2) = \frac{\sum \limits _{\mathrm{los}} P_{\mathrm{tot}} \Pi \exp \left(2i \left(\epsilon _{\mathrm{int}} + \mathrm{RM}\lambda ^2 \right) \right) \mathrm{d}s}{\sum \limits _{\mathrm{los}} P_{\mathrm{tot}} \mathrm{d}s},\end{eqnarray*}$$using the emission per volume in each simulation cell Ptot, the intrinsic degree of polarization Π and the intrinsic angle of polarization ϵint. RMλ2 accounts for Faraday rotation. The intrinsic degree of polarization Π at observing frequency νobs is computed using the parallel and perpendicular components of the radio emission, equations (3) and (4) (e.g. Rybicki & Lightman 1986):
(7)$$\begin{eqnarray*}
\Pi = \frac{P_{\parallel }-P_{\perp }}{P_{\parallel }+P_{\perp }}.\end{eqnarray*}$$The intrinsic angle of polarization, ϵint, is computed with respect to the horizontal axis of the projected maps. Each simulation cell can be considered to be filled with a uniform magnetic field and, in this case, the intrinsic angle of polarization is perpendicular to the direction of the projected magnetic field. If the emission is going through a magnetized medium, the intrinsic angle of polarization is Faraday rotated. In equation (6), RMλ2 determines the amount of Faraday rotation. Here, λ is the wavelength corresponding to the observation frequency and RM is the rotation measure (RM) of the ambient medium. The RM at a distance x from the observer is computed as:
(8)$$\begin{eqnarray*}
\mathrm{RM}= 812 \int _{0}^x \frac{n_\mathrm{e}}{10^{-3} \ \mathrm{cm}^{-3}} \frac{B_{\mathrm{para}}}{\mu \mathrm{G}} \frac{\mathrm{d}l}{\mathrm{kpc}} \ \left[\mathrm{\frac{rad}{m^2}}\right].\end{eqnarray*}$$The integral is taken along the LoS. ne and Bpara are the thermal electron number density and parallel magnetic field component, respectively, along the LoS. Faraday rotation occurs either outside of the emitting region, external Faraday rotation, or inside the source, internal Faraday rotation.


Table 1 provides the details of the X-ray observations of Holmberg IX X-1 presented in this work. Our data reduction largely follows the procedures outlined in Walton et al. (2014). The XMM-Newton data for epochs 1 and 2 were re-reduced using SAS v14.0.0 in exactly the same manner as described in our previous work for both the EPIC-pn and EPIC-MOS detectors (Strüder et al. 2001; Turner et al. 2001), using more recent calibration files (up to date as of 2015 July). Our new Suzaku observations (epochs 3–6) were also reduced with  HEASOFT v6.18 in the same manner as the epoch 1 data; as before, we only used the XIS detectors (Koyama et al. 2007) given the high-energy NuSTAR coverage. The only major update to our data reduction procedure in comparison to Walton et al. (2014) is for the NuSTAR data. The standard science data (mode 1) were reduced with  NUSTARDAS v1.5.1 in the same manner as in that work, again with updated calibration files (NuSTAR CALDB 20150316). However, in order to maximize the signal-to-noise ratio (S/N), in this work we also make use of the spacecraft science data (mode 6) following the procedure outlined in Walton et al. (2016c; see also the NuSTAR Users Guide11

11

http://heasarc.gsfc.nasa.gov/docs/nustar/analysis/nustar_swguide.pdf

). In brief, spacecraft science mode refers to data recorded during periods of the observation in which the X-ray source is still visible, but the star tracker on the optics bench cannot return a good aspect solution. During such times, the star trackers on the spacecraft bus are used instead. For these observations, this provides an additional ∼20%–40% good exposure (depending on the observation). The inclusion of the mode 6 data allows us to fit the NuSTAR data for Holmberg IX X-1 from 3 to 40 keV for all epochs. As in Walton et al. (2014), we model the XMM-Newton spectra over the 0.3–10.0 keV range, and the Suzaku data over the 0.6–10.0 keV range for the front-illuminated XIS units (XIS0, XIS3) and the 0.7–9.0 keV range for the back-illuminated XIS1, excluding 1.7–2.1 keV owing to known calibration issues around the instrumental edges.


The shape of the fireball is an open question. Thompson & Duncan (2001) considered spherical or cylindrical structures, but the exact shape trapped in a realistic magnetar field structure, which may sustain local twists, is unclear. Feroci et al. (2001) showed that the time-dependence of the X-ray luminosity Lx(t) for the light curve of the giant flare from SGR 1900+14 on 1998 August 27 was well fit by the following function:

(1)

\begin{equation}
L_{\rm x}(t) = L_{\rm x}(0)\left(1 - \frac{t}{\tau _\mathrm{evap}}\right)^\chi ,
\end{equation}

where τevap is an evaporation time-scale. The parameter χ = a/(1 − a) can be related, in a simple model where the fireball has uniform energy density and surface flux, to the shape of the fireball (Thompson & Duncan 2001). For the 1998 giant flare from SGR 1900+14, the best-fitting value was a = 0.75 (Feroci et al. 2001), while for the 2004 December 27 giant flare of SGR 1806–20, Hurley et al. (2005) found a = 0.6. A homogeneous fireball cannot have a value higher than a = 2/3, which is also the value for a homogeneous sphere, so these results show that the SGR 1900+14 fireball may have a more complex structure (Thompson & Duncan 2001). In this paper, we choose to model the magnetic field as a pure dipole, which is sufficient to explore the generic character of outflows and fireballs, and their coupling. This choice leads to a torus-shaped fireball, as that is the only shape that can be trapped in the closed field-lines of a dipole field. We use the equations for a dipolar magnetic field in full general relativity given by Wasserman & Shapiro (1983). When considered in two dimensions (as our model geometry has rotational symmetry) the boundary of the fireball lies along a single closed field line, which is determined by a single parameter. We choose to use the diameter of the torus-shaped fireball as this parameter, defined as the equatorial radius of the outermost field line minus the radius of the star. We let the diameter vary between one and one hundred kilometres (∼0.1–10 stellar radii), to cover all possible fireball sizes.


For the 1998 giant flare from SGR 1900+14, the best-fitting value was a = 0.75

while for the 2004 December 27 giant flare of SGR 1806–20, Hurley et al. (2005) found a = 0.6.


Star-forming galaxies 'At all the (relatively low, z≲1\documentclass[12pt]{minimal}
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				\begin{document}$$z\lesssim 1$$\end{document}) redshifts probed by present facilities, radio-emitting star-forming galaxies appear in all respect identical to star-forming galaxies selected at any other wavelength, with spectral types ranging from spirals to irregulars and dwarfs. Although only ≲10-4\documentclass[12pt]{minimal}
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				\begin{document}$$\lesssim 10^{-4}$$\end{document} of the bolometric luminosity produced by a non-AGN powered galaxy is radiated at radio wavelengths (Condon 1992), the importance of radio selection for star-forming galaxies relies on the fact that it can provide estimates of the star-formation activity of such sources in a way which is unaffected by the presence of dust up to the earliest epochs. To do that, a solid calibration of the relation between radio luminosity and infrared luminosity (or alternatively SFR) is mandatory. This is why a lot of effort has been put in the recent years to provide calibrations as precise as possible at all redshifts and for all galaxy types. From an observational point of view, it has been found at least in the local universe that radio and infrared luminosities show an incredibly tight, approximately linear, correlation over more than five orders of magnitude in infrared luminosity (e.g., Helou et al. 1985; De Jong et al. 1985; Condon 1992; Yun et al. 2001), irrespective of galaxy type and level of star-forming activity, as long as the galaxy hosts episodes of star-formation. This relation, originally found at 1.4 GHz, has also been observed to hold at different frequencies (e.g., Delhaize et al. 2017; Gürkan et al. 2018). The theoretical reasons behind it are still not fully understood, as many factors involved in the process of shaping a galaxy should intervene to produce at least some modifications which are not (yet) clearly visible in the available data.

Background Background Background Background


Observations of the Lyman series forests in z ≳ 6 quasars indicate that the IGM is already highly ionized by z ∼ 6 (e.g., Fan et al. 2006; Bosman et al. 2018; Eilers et al. 2018, 2019; Yang et al. 2020), although the final completion of reionization might extend down to z ∼ 5.5 (e.g., Becker et al. 2015; Davies et al. 2018a; Kulkarni et al. 2019; Keating et al. 2020; Nasir & D’Aloisio 2020). However, the Lyman series forests are very sensitive to neutral hydrogen and saturate even at low IGM neutral fraction (i.e.,

). On the other hand, if the neutral fraction is of order unity, one would expect to see appreciable absorption redward of the wavelength of the Lyα emission line, resulting in a damping wing profile (e.g., Miralda-Escudé 1998) due to significant optical depth on the Lorentzian wing of the Lyα absorption. The first quasar with a damping wing detection is ULAS J1120+0641 (Mortlock et al. 2011) at z = 7.09, although different analyses yielded different constraints on

 (Bolton et al. 2011; Mortlock et al. 2011; Bosman & Becker 2015; Greig et al. 2017; Davies et al. 2018b), ranging from

 to

 at z ∼ 7.1. Recently, the spectrum of quasar ULAS J1342+0928 (Bañados et al. 2018) at z = 7.54 shows a robust detection of the damping wing signal (Bañados et al. 2018; Davies et al. 2018b; Greig et al. 2019; Ďurovčíková et al. 2020), yielding

 at z ∼ 7.5. Compared to other probes of reionization history, such as cosmic microwave background (CMB) polarization (Planck Collaboration et al. 2018) and Lyα emission line visibility in high-redshift galaxies (e.g., Ouchi et al. 2010; Mason et al. 2018), a main advantage of IGM damping wing measurement is that it can be applied to individual quasar sight lines, thereby constraining not only the average neutral fraction, but also its scatter in different regions of the IGM. However, the damping wing experiment is only feasible at very high redshifts where the IGM is relatively neutral, and current damping wing analyses have been limited to these two sight lines due to the lack of bright quasars at z ≳ 7. Thus, it is crucial to investigate the damping wing experiment along more z > 7 quasar sight lines.


The linearized equation of motion (see derivation of Equation (A12)) for the unmagnetized case is given by (also refer to Christensen-Dalsgaard 2003):
9

where cs(r), ρ0(r), and g(r) are the sound speed, density, and gravity (directed radially inward), respectively, and

 denotes the covariant spatial derivative operator. For all ensuing calculations and derivations, we write Equation (9) in the form

, where the magnetically unperturbed wave operator

 is self-adjoint (Goedbloed & Poedts 2004). To solve for the eigenmodes of the unperturbed model S, the boundary conditions employed are: (a)

 and the Eulerian pressure perturbation at r = 0 are finite, and (b) the Lagrangian pressure perturbation at r = R⊙ vanishes (see Section 17.6 in Cox 1980). As already mentioned earlier, the Cowling approximation is used, and hence, the gravitational Poisson equation is not needed while finding the eigenmodes. We suppress the subscript “0” in the unperturbed eigenfunctions

 and eigenfrequencies ωk,0 for the rest of this paper. Unless specified otherwise, any instance of ωk or

 should be assumed to imply eigenfrequencies and eigenfunctions of Equation (9), respectively. The Sun is treated as a fluid body with a vanishing shear modulus and, hence, is unable to sustain shear waves (although the presence of magnetic fields complicates this assumption). Thus, the eigenfunctions of the background model contain no toroidal components (see Chapter 8 of Dahlen & Tromp 1998), rendering them purely spheroidal. We write the displacement field

 in the basis of vector spherical harmonics (and thereafter GSH) as follows:
10

11

Here,

 denote spherical polar coordinates, with basis vectors

 and

 where n is the radial order, ℓ is the angular degree, and m is the azimuthal order. The dimensionless lateral covariant derivative operator is denoted by

. The basis vectors in spherical polar coordinates are related to those in the GSH basis via
12


A gravitational wave (GW) event originated by the merger of a binary neutron star (BNS) system was detected for the first time by aLIGO/Virgo (GW 170817; Abbott et al. 2017a), and was found to be associated to the weak short gamma-ray burst (GRB) GRB 170817A detected by the Fermi and INTEGRAL satellites (Goldstein et al. 2017; Savchenko et al. 2017), marking the dawn of multi-messenger astronomy (Abbott et al. 2017b). The proximity of the event ~41 Mpc; Hjorth et al. 2017, Cantiello et al. 2018) and the relative accuracy of the localisation (~30 deg2, thanks to the joint LIGO and Virgo operation) led to a rapid (Δt  11 h) identification of a relatively bright optical electromagnetic counterpart (EM), named AT2017gfo, in the galaxy NGC 4993 (Arcavi et al. 2017; Coulter et al. 2017; Lipunov et al. 2017; Melandri et al. 2017; Tanvir et al. 2017; Soares-Santos et al. 2017; Valenti et al. 2017). The analysis and modelling of the spectral characteristics of this source, together with their evolution with time, resulted in a good match with the expectations for a “kilonova” (i.e. the emission due to radioactive decay of heavy nuclei produced through rapid neutron capture; Li & Paczyński 1998), providing the first compelling observational evidence for the existence of such elusive transient sources (Cowperthwaite et al. 2017; Drout et al. 2017; Evans et al. 2017; Kasliwal et al. 2017; Nicholl et al. 2017; Pian et al. 2017; Smartt et al. 2017; Villar et al. 2017). While the bright kilonova associated to GW 170817 has been widely studied and its main properties relatively well determined, the observations of the short GRB 170817A are more challenging for the current theoretical frameworks. Indeed, the properties of this short GRB appear puzzling in the context of observations collected over the past decades (Berger 2014; D’Avanzo 2015; Ghirlanda et al. 2015). The prompt γ-ray luminosity was significantly fainter (by a factor ~2500) than typical short bursts (see, e.g. D’Avanzo et al. 2014). A faint afterglow was detected in the X-ray and radio bands only at relatively late times (starting from ~9 and 16 d after the GW/GRB trigger, respectively; Alexander et al. 2017; Haggard et al. 2017; Hallinan et al. 2017; Margutti et al. 2017; Troja et al. 2017a), while earlier observations provided only upper limits in these bands (Evans et al. 2017; Hallinan et al. 2017).


Gough’s (1977a) concept of averaging over convective eddies is based on Spiegel’s (1963) finding that the linearized equations of motions for determining the convective growth rate σ of a convective eddy, i.e. the rate with which the convective fluctuations grow with time, satisfy a variational equation for σ, if the locally defined superadiabatic temperature gradient β(r0) is replaced by the averaged, i.e. non-local, value (Gough 1977a,b)
(2)
\begin{eqnarray*}
{\cal B}(r)=\frac{2}{\ell }\int \beta (r_0)\cos ^2\left[\pi (r_0-r)/\ell \right]\, {\rm d}r_0\, .
\end{eqnarray*}
In expression (2), the averaging is obtained by taking account of contributions from convective eddies centred at r0 and the range of integration is the vertical extent ℓ, which scales with the local pressure scale height. Based on Spiegel’s finding on the convective growth rate, Gough (1977a) suggested similar expressions for the averaged, non-local, convective heat, ${\cal {F_{\mathrm{c}}}}$, and momentum, $\mathcal {P}_{\rm t}$, fluxes, with β(r0) in equation (2) being replaced by the locally computed convective fluxes Fc(r0) and pt(r0), respectively. This would, however, lead to a system of integro-differential equations for the stellar structure, which would be difficult to solve numerically. Gough (1977a) suggested therefore to approximate the kernel ${\cal K}:=2\cos ^2[\pi (r_0-r)/\ell ]$ in equation (2) by the expression ${\cal K}\simeq a\exp (-a|r_0-r|/\ell)/2$, and setting the integration limits formally to ±∞, which leads to an integral expression that represents the solution to the second-order differential equation
(3)
\begin{eqnarray*}
\frac{{\rm d}^2{\cal {F_{\mathrm{c}}}}}{{\rm d}\ln p^2}=\frac{a^2}{\alpha ^2}\left({\cal {F_{\mathrm{c}}}}-{F_{\mathrm{c}}}\right)\,
\end{eqnarray*}
(Gough 1977a; Balmforth 1992a; Houdek & Dupret 2015) for the non-local convective heat flux, ${\cal {F_{\mathrm{c}}}}$, for example, where ln p (p is total pressure) is now the new independent depth variable (as implemented in the calculations), α ≔ −ℓdln p/dr is the mixing-length parameter, and a is another dimensionless, non-local, parameter. Similar expressions are obtained for the averaged, non-local, superadiabatic temperature gradient, $\cal B$, and turbulent pressure, $\cal P_{\mathrm{t}}$, introducing the remaining non-local (dimensionless) parameters b and c, respectively. These non-local parameters control the spatial coherence of the ensemble of eddies contributing to the total heat (a) and momentum (c) fluxes, and the degree to which the turbulent fluxes are coupled to the local stratification (b). Roughly speaking, the parameters a, b, and c control the degree of ‘non-locality’ of convection; low values imply highly non-local solutions, and in the limit a, b, c → ∞ the system of equations formally reduces to the local formulation (except near the boundaries of convection zones, where the local equations are singular). Theoretical values for the dimensionless, non-local, parameters a, b, and c can be obtained by demanding that the terms in a Taylor expansion about r of the exact, 2cos 2[π(r0 − r)/ℓ], and approximate, aexp (− a|r0 − r|/ℓ)/2, kernels differ only at fourth order, resulting in a theoretical estimate for a ≃ 7.8 (Gough 1977a). The standard mixing-length assumption of assigning a unique scale to turbulent eddies at any given location can, however, cause too much smoothing, leading to larger values for a, b, or c (Houdek & Gough 2002). Therefore, values for a, b, and c, which typically differ, need to be determined from calibration in a similar way as it is common for the mixing-length parameter, α, in stellar evolutionary theory.1It should be mentioned that our adopted non-local convection formulation does not treat the overshoot regions correctly, where the non-local enthalpy flux, $\cal {F_{\mathrm{c}}}$, remains positive although it should be negative. The overshoot regions may therefore not be suitable for calibrating the non-local parameters to 3D simulation results (see also discussion in Section 5.2). The effect of such positive $\cal {F_{\mathrm{c}}}$ values in the overshoot regions on the damping rates and eigenfrequencies is, however, negligible.


To explore the link between angular momentum and local gravitational instability in nearby star-forming spirals, we need a reliable disc instability diagnostic. Contrary to what is commonly assumed, the gas Toomre parameter is not a reliable diagnostic: stars, and not molecular or atomic gas, are the primary driver of disc instabilities in spiral galaxies, at least at the spatial resolution of current extragalactic surveys (Romeo & Mogotsi 2017). This is confirmed by other investigations (Marchuk 2018; Marchuk & Sotnikova 2018; Mogotsi & Romeo 2018), and is true even for a powerful starburst+Seyfert galaxy like NGC 1068 (Romeo & Fathi 2016). The stellar Toomre parameter is a more reliable diagnostic, but it does not include the stabilizing effect of disc thickness, which is important and should be taken into account (Romeo & Falstad 2013). The simplest diagnostic that does this accurately is the Romeo–Falstad $\mathcal {Q}_{N}$ stability parameter for one-component (N = 1) stellar (⋆) discs, which we consider as a function of galactocentric distance R:
(1)
\begin{eqnarray*}
\mathcal {Q}_{\star }(R)= Q_{\star }(R)\, T_{\star }\, ,
\end{eqnarray*}
where Q⋆ = κσ⋆/πGΣ⋆ is the stellar Toomre parameter (σ denotes the radial velocity dispersion), and T⋆ is a factor that encapsulates the stabilizing effect of disc thickness for the whole range of velocity dispersion anisotropy (σ$z$/σR) observed in galactic discs:
(2)
\begin{eqnarray*}
T_{\star }= \left\lbrace \begin{array}{ll}\displaystyle 1+0.6\left(\frac{\sigma _{z}}{\sigma _{R}}\right)_{\star }^{2} &\quad \text{if }0\le (\sigma _{z}/\sigma _{R})_{\star }\le 0.5\, , \\
{\displaystyle 0.8+0.7\left(\frac{\sigma _{z}}{\sigma _{R}}\right)_{\star }} &\quad \text{if }0.5\le (\sigma _{z}/\sigma _{R})_{\star }\le 1\, . \end{array} \right.
\end{eqnarray*}
Observations do not yet constrain the radial variation of (σ$z$/σR)⋆, hence that of T⋆ (Gerssen & Shapiro Griffin 2012; Marchuk & Sotnikova 2017; Pinna et al. 2018).


One of the important findings of the COS-Halos survey is that

 is omnipresent in the halos of SF galaxies which harbor a major reservoir of galactic metals (i.e., Tumlinson et al. 2011a). Outflows from the SF galaxies are thought to be the origin of highly ionized oxygen in the CGM. However, the outflows need not be active at the time of observations. Active outflows that are detected primarily via low-ions (e.g.,

,

,

) absorption in the spectra of host-galaxies (i.e., the “down-the-barrel” outflows) are also found to be ubiquitous at both high and low redshifts (Shapley et al. 2003; Martin 2005; Rupke et al. 2005; Veilleux et al. 2005; Rubin et al. 2010, 2014). Despite the ubiquity of winds, several basic questions that are key to quantifying galaxy feedback remain unanswered. How far do they propagate? What is the baryon mass that are processed through them? What is the mass-flow rate through winds? What are the mass loading factors and kinetic power of the winds? To what degree are metals processed through them? This is essentially because the location of the outflow with respect to host-galaxy is an unknown in “down-the-barrel” outflows. Galaxy outflow probed by a background QSO has the advantage of having the minimum distance between host-galaxy and outflowing material. For example, by analyzing a post-starburst outflow from a galaxy at

 detected in the spectra of QSO PG 1206+459 at an impact parameter of ∼68 kpc, Tripp et al. (2011) have shown that the entrained gas mass could be as large as ∼ few

 Numerous strong and large velocity spread (

 km s−1) metal absorption lines along with solar to super-solar metallicities in different absorption components suggest that the absorber is tracing an active outflow. Such wide velocity spread

 absorbers at low-z are also reported by Tumlinson et al. (2011b) and Muzahid (2014). In both these cases metallicity of the

 bearing gas is low (e.g.,

) which suggest that

 is probably tracing an “ancient outflow” (Ford et al. 2014) rather than an active wind. Interestingly, strong

 is detected in PG 1206+459 which is not present in the latter two systems. We note that the location of the absorbing gas with respect to the host-galaxy’s projected major and minor axes are not known for any of these strong

 absorbers.


Accreting supermassive BHs can have a profound impact on the evolution of the host galaxies (see review by Alexander & Hickox 2012), as testified by the observed tight relationships between the relic BH masses and the physical properties of the hosts, most noticeably the stellar mass or velocity dispersion of the bulge component (e.g., Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Kormendy & Ho 2013; McConnell & Ma 2013; Reines & Volonteri 2015; Shankar et al. 2016, 2020a; Sahu et al. 2019; Zhu et al. 2021). These suggest that (apart from short-time stochastic fluctuations) the BH and the bulge stellar mass must have coevolved over comparable timescales, possibly determined by the energy feedback from the BH itself on the gas/dust content of the host (see Tinsley 1980; Silk & Rees 1998; Fabian 1999; King 2005; Lapi et al. 2006, 2014, 2018; for a review, see King & Pounds 2015). In fact, targeted X-ray observations in the high-redshift star-forming progenitors of local massive galaxies have started to reveal the early growth of a dust-enshrouded (super)massive BH in their nuclear regions (e.g., Mullaney et al. 2012; Page et al. 2012; Delvecchio et al. 2014; Rodighiero et al. 2015, 2019; Fiore et al. 2017; Stanley et al. 2015, 2017; Massardi et al. 2018; Combes et al. 2019; D’Amato et al. 2020) before it attains a high enough mass and power to manifest as a bright AGN and eventually reduce/quench star formation and partly evacuate gas and dust from the host (e.g., Granato et al. 2001, 2004; Lapi et al. 2011,2014, 2018). Another, albeit more indirect, indication of coevolution for the bulk of the BH and the host stellar mass comes from the similarity between the activity timescales of the central BH to the transition timescale of (green valley) galaxies from the blue cloud to the red sequence (see Wang et al. 2017; Lin et al. 2021, 2022; this is true apart from rejuvenations at late cosmic times; see Martin-Navarro et al. 2022).


Accreting supermassive BHs can have a profound impact on the evolution of the host galaxies (see review by


as testified by the observed tight relationships between the relic BH masses and the physical properties of the hosts, most noticeably the stellar mass or velocity dispersion of the bulge component (e.g.,

In fact, targeted X-ray observations in the high-redshift star-forming progenitors of local massive galaxies have started to reveal the early growth of a dust-enshrouded (super)massive BH in their nuclear regions (e.g.,


before it attains a high enough mass and power to manifest as a bright AGN and eventually reduce/quench star formation and partly evacuate gas and dust from the host (e.g.,


Another, albeit more indirect, indication of coevolution for the bulk of the BH and the host stellar mass comes from the similarity between the activity timescales of the central BH to the transition timescale of (green valley) galaxies from the blue cloud to the red sequence (see


Theoretical and observational work argue for a strong correlation between the NIR size and the AGN luminosity (Suganuma et al. 2006; Kishimoto et al. 2011; Koshida et al. 2014; GRAVITY Collaboration 2019). In the standard picture, the NIR emission originates from the dust sublimation region, which is determined by the dust sublimation temperature Tsub. The temperature at which dust sublimates depends on the grain size and composition and ranges from Tsub ≈ 1500 K for silicate (Si) grains to Tsub ≈ 2000 K for graphite (C) grains (Baskin & Laor 2018). By assuming a grain size distribution typical for the interstellar medium (ISM), the mean sublimation distance can be calculated for graphite and silicate dust as

R
≈

a
X

×

L

46

1
/
2

$ R\approx a_{\mathrm{X}}\times L_{46}^{1/2} $

 [pc] with aC = 0.5 and aSi = 1.3 (e.g. Barvainis 1987; Netzer 2015; for grain size dependence see Baskin & Laor 2018). However, there is evidence both from SED fitting (Mor & Netzer 2012) and from dust reverberation mapping (Kishimoto et al. 2007; Koshida et al. 2014) that, close to the AGN, one may find only larger graphite grains. Since, at the same distance from the AGN, large grains are cooler than small grains, one might naturally expect to find a distribution of large graphite grains at the smallest radii. In addition changes in the brightness of the AGN may lead to changes in the dust sublimation radius (Kishimoto et al. 2013) or instead, depending on the dust distribution, its temperature (Schnülle et al. 2015). As such, the sublimation radius is really rather a sublimation region according to grain size and species (e.g. Hönig & Kishimoto 2017; Baskin & Laor 2018). Empirical studies have found R2.2 μm ≈ 0.4 pc × (LAGN/1046 erg s−1)1/2 (assuming the V-band to Rsub relation from Koshida et al. 2014, and using the conversion LAGN ∼ 8L5500 Å from Netzer 2015), which corresponds to a good approximation to the sublimation radius of graphite. For NGC 1068 the observed relation from Koshida et al. (2014) predicts a dust sublimation radius in the range R2.2 μm ≈ 0.08–0.27 pc, for the luminosity range quoted in Table 1.


It is expected that the polarization properties produced by the thermal–nonthermal electrons have wide applications to many kinds of synchrotron sources. Here, we propose some examples. Covino et al. (2017) detected the low-level linear polarization of AT 2017gfo/GW170817 during the kilonova phase. Although the measured polarization seems to be due to the Galactic dust along the line of sight, we think that future synchrotron polarization detections for electromagnetic counterparts of gravitational wave events can be applied to further investigate compact object merger systems and constrain detailed physics of gravitational wave events. Fast radio burst (FRB) 140514 has the linear polarization limit of

 and the circular polarization degree of 21 ± 7% (Petroff et al. 2015). The development of the radio polarization measurements on FRBs are carried out to reveal the FRB origin. Covino & Götz (2016) summarized the polarization observation of GRB prompt and afterglow emissions, and the GRB jet geometry related to the polarization was also mentioned. Steele et al. (2017) presented the optical polarization results of some GRBs, and the linear polarization degrees are relatively low (see also the recent report of Wiersema & Covino 2018 on GRB 180205A). Moreover, the circular polarization was detected in GRB 121024A (Wiersema et al. 2014). Spectropolarimetric detection of either supernova Ia (SN Ia) or core-collapse supernova was performed (Tanaka et al. 2012; Porter et al. 2016). The polarization properties can be helpful to further explore SN explosion mechanisms. Thermal electrons cannot be ignored in the celestial objects mentioned above, and we should consider the thermal electron contribution to the synchrotron polarization. Moreover, some objects, such as kilonova, GRB, and SN as mentioned above, have dense environments (D’Elia et al. 2007; Kasen et al. 2013; Tanaka & Hotokezaka 2013; Buckey et al. 2018; Yang et al. 2018). Polarization radiative transfer should be taken into account when we consider the polarization measurements in the optical and radio bands. In this paper, we do not attempt to reproduce these different polarization properties with exact modeling parameters. However, these polarization detections of the different populations encourage us to further explore the possibility of applying our scenario, when we assume that all of these kinds of sources are dominated by the synchrotron radiation.


Recent work by Tseliakhovich & Hirata (2010) showed that, at the time of recombination, baryons were moving in a coherent manner (on scales of ≲3 Mpc comoving) with respect to the dark matter and had an rms velocity of ∼30 km s−1. Since, on small scales, the baryons effectively displayed a streaming motion with respect to the dark matter component, their relative velocity is usually referred to as a ‘stream velocity’ in the literature, and we adopt this convention throughout this work. Tseliakhovich & Hirata (2010) also argued that, since shortly after recombination the temperature of the baryons (and hence the sound speed) drops precipitously, the streaming motion of the baryons becomes supersonic and it can potentially have a dramatic impact on structure formation. Subsequent studies, both theoretical and numerical, showed that the large stream velocity of baryons has profound implications on the total number density of haloes (Tseliakhovich & Hirata 2010; Maio, Koopman & Ciardi 2011; Stacy, Bromm & Loeb 2011; Tseliakhovich, Barkana & Hirata 2011; Fialkov et al. 2012; Naoz, Yoshida & Gnedin 2012; O'Leary & McQuinn 2012; Bovy & Dvorkin 2013; Tanaka, Li & Haiman 2013; Tanaka & Li 2014; Asaba, Ichiki & Tashiro 2016), the size of the haloes that are able to retain gas at a given redshift (Naoz, Yoshida & Gnedin 2013), the overall gas fraction in haloes (Dalal, Pen & Seljak 2010; Greif et al. 2011; Maio et al. 2011; Tseliakhovich et al. 2011; Fialkov et al. 2012; Naoz et al. 2012, 2013; O'Leary & McQuinn 2012; Richardson, Scannapieco & Thacker 2013; Asaba et al. 2016), the gas density and temperature profiles (Greif et al. 2011; Liu & Wang 2011; Fialkov et al. 2012; O'Leary & McQuinn 2012; Richardson et al. 2013), the halo mass threshold in which star formation occurs (Liu & Wang 2011; Maio et al. 2011; Greif et al. 2011; Fialkov et al. 2012; O'Leary & McQuinn 2012; Bovy & Dvorkin 2013), as well as black hole evolution (Tanaka et al. 2013; Tanaka & Li 2014; Latif, Niemeyer & Schleicher 2014) and magnetic fields (Naoz & Narayan 2013). A thorough review of the implications of the stream velocity on structure formation is presented in Fialkov (2014).


the halo mass threshold in which star formation occurs


A thorough review of the implications of the stream velocity on structure formation is presented in


showed that, at the time of recombination, baryons were moving in a coherent manner (on scales of ≲3 Mpc comoving) with respect to the dark matter and had an rms velocity of ∼30 km s−1.


also argued that, since shortly after recombination the temperature of the baryons (and hence the sound speed) drops precipitously, the streaming motion of the baryons becomes supersonic and it can potentially have a dramatic impact on structure formation.


Gohil & Ballantyne (2017) showed that the AGN obscuration phenomenon depends on the gas fraction in NSDs. With a higher gas fraction, a greater amount of gas is available to accrete, which increases a column of gas in annulus Σmp and, in turn, controls expansion of the atmosphere. The overall gas fraction (f0) in galaxies is observed to be increasing with redshift. f0 is related to the depletion time and the specific star-formation rate (sSFR) through
(16)
\begin{eqnarray}
f_0(z)=\frac{1}{1+[t_{\rm dep}(z){\rm sSFR}(z)]^{-1}}.
\end{eqnarray}
The depletion time is estimated to be (Saintonge et al. 2013)
(17)
\begin{eqnarray}
t_{\rm dep}=1.5(1+z)^{\alpha } [{\rm Gyr}],
\end{eqnarray}
with α = −1.0 (Tacconi et al. 2013). Lilly et al. (2013) provide an analytical expression for the evolution of sSFR, given by
(18)
\begin{eqnarray}
{\rm sSFR}(M_*,z)= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}0.07\Big (\frac{M_*}{10^{10.5} \, \mathrm{M}_{\odot }}\Big )^{-0.1} (1+z)^3\ \ & z<2,\\
 0.30\Big (\frac{M_*}{10^{10.5} \, \mathrm{M}_{\odot }}\Big )^{-0.1} (1+z)^{5/3}\ \ & z\ge 2, \end{array}\right.
\end{eqnarray}
which was motivated by observational data (e.g. Daddi et al. 2007; Elbaz et al. 2007; Noeske et al. 2007; Pannella et al. 2009; Stark et al. 2013). Since there is a weak dependence on the galaxy mass M*, we choose M* = 1010.5 M⊙. Then the distribution of the gas fraction (Φf) is assumed to be Gaussian around f0 and is given by
(19)
\begin{eqnarray}
\Phi _{\rm f}(z,f)=\frac{1}{\sqrt{2 {\pi} }\sigma _{\rm f}}\exp \Big [-\frac{(f-f_0)^2}{2\sigma _{\rm f}^2}\Big ].
\end{eqnarray}
The dispersion σf is estimated to be 0.2 by roughly fitting a Gaussian to the results of Tacconi et al. (2013). Finally, we can compute the weights Wf of input parameter f ≡ fg, out from
(20)
\begin{eqnarray}
W_{\rm f}(z,f)=\frac{\Phi _{\rm f}(z,f)}{\sum \limits _{k}\Phi _{\rm f}(z,f_k)},
\end{eqnarray}
where
(21)
\begin{eqnarray}
f_k\in [0.2,0.4,0.6,0.8].
\end{eqnarray}
The right panel of Fig. 1 shows the evolution of 20, 40, 60 and 80 per cent gas fraction with redshift, represented by black, red, blue and green curves, respectively. At low redshift, NSDs with low gas fraction dominate the sample, while NSDs with a 40 per cent gas fraction dominate at 1  z  2. Beyond z = 2, NSDs with a 60 per cent gas fraction dominate. Wf of NSDs with low gas fraction decrease overall with redshift and the reverse is true for NSDs with high gas fraction (60 and 80 per cent). Similarly to the observational evidence, the figure illustrates that the dominant gas fraction increases with z and its evolution flattens out at higher redshift.


PHL 1811 itself has an estimated Eddington ratio of

 (Leighly et al. 2007b). PHL 1811, along with several PHL 1811 analogs and WLQs with rest-frame optical spectra (J1521+5202, 2QZ J2154–3056, and the two high-redshift WLQs in Shemmer et al. 2010), have weak or undetected [O iii]

 narrow emission lines, suggestive of high Eddington ratios (e.g., Boroson & Green 1992; Shen & Ho 2014). Moreover, several studies have found that as

 increases, the C iv REW generally decreases and the C iv blueshift also increases (e.g., Bachev et al. 2004; Baskin & Laor 2004; Richards et al. 2011; Shen & Ho 2014; Sulentic et al. 2014; Shemmer & Lieber 2015). In these correlations, there is only limited sampling in the super-Eddington or low (

 Å) C iv REW regime, but the overall trends suggest that the Eddington ratio grows as one moves from typical quasars toward WLQs in the C iv REW versus blueshift space (Figure 3(a)). By the nature of our selection of the PHL 1811 analogs and WLQs, demanding that the BELR does not produce normal emission lines, we may have recovered effectively a population of quasars with (extremely) high Eddington ratios. It is difficult to measure

 directly for our exceptional objects, as the SMBH masses estimated from the line-based virial method are likely highly uncertain and perhaps systematically in error (Section 5.3). However, based on the empirical Γ

 relations, our joint spectral analysis of the X-ray normal subsample in Sections 5.2 and 5.3 does indicate a high Eddington ratio (

) in general for our quasars.33

33
Narrow-line Seyfert 1 galaxies (NLS1s) also generally have steep X-ray spectra and high Eddington ratios. PHL 1811 itself is considered a NLS1 (Leighly et al. 2001), and NIR spectroscopy of a limited sample of WLQs shows that they have in general narrow Hβ and strong optical Fe ii emission (see Section 5.1 of Plotkin et al. 2015), similar to NLS1s. However, as the line widths and REWs have likely dependences on luminosity, our PHL 1811 analogs and WLQs are not directly comparable to local, less-luminous NLS1s.