The variability of the cyclotron line centroid energy in the spectra of XRPs is considered to be related to the geometry of accretion flow in close proximity to the surface of an NS. The geometry of the emitting region is related to the mass accretion rate. At low mass accretion rates, the radiation pressure is small, and one expects hot spots at the NS surface. At high mass accretion rates, the radiation pressure is high enough to stop accreting material above the NS surface. In this case, the accretion column supported by radiation pressure and confined by a strong magnetic field arises above the stellar surface (Basko & Sunyaev 1976; Wang & Frank 1981; Mushtukov et al. 2015a). The luminosity separating these two accretion regimes is called the “critical” luminosity Lcrit. The critical luminosity was shown to be dependent on the magnetic field strength (Mushtukov et al. 2015b). The dynamics of the cyclotron line was shown to be dependent on the luminosity state of XRPs (see Staubert et al. 2019 for review). In particular, the positive correlation between the cyclotron line centroid energy and accretion luminosity is considered to be typical for subcritical XRPs (Staubert et al. 2007; Klochkov et al. 2012; Fürst et al. 2014), while the negative correlation was detected in bright supercritical sources (Mihara et al. 2004; Tsygankov et al. 2006; Boldin et al. 2013). At the same time, there are some sources without any observed correlation between the cyclotron line energy and luminosity (Caballero et al. 2007). Several theoretical models are aiming to explain the variability of a cyclotron line. The positive correlation was explained by the Doppler effect in the accretion channel (Mushtukov et al. 2015c) and alternatively by the onset of collisionless shock above hot spots at low mass accretion rates (Shapiro & Salpeter 1975; Rothschild et al. 2017). The negative correlation was explained by the variations of accretion column height above the NS surface. Different models consider different locations of a line-forming region at supercritical mass accretion rates, which might be a radiation-dominated shock on top of an accretion column (Becker et al. 2012) or NS surface (see e.g., Poutanen et al. 2013; Lutovinov et al. 2015; Mushtukov et al. 2018), which reprocesses a large fraction of beamed radiation from the accretion column. Alternatively, Nishimura (2014) argued that some variations of cyclotron line centroid energy could be related to the changes of a beam pattern. By considering the structure of an accretion column in two dimensions, Nishimura (2019) suggested that the line-forming region is a region around an accretion mound in which the bulk velocity in the line-forming region can be considerably different from that in the continuum-forming region, which is assumed to be inside an accretion mound, so that the variation of Ecyc results from the motion of the accretion mound in the different luminosity ranges.


Let ui be the evolved quantity at the coordinate position xi. Then, THC_M1 approximates the derivative of the flux f(u) at the location xi as
(31)$$\begin{eqnarray}
\partial _x f (u) \simeq \frac{F_{i - 1 / 2} - F_{i + 1 / 2}}{\Delta x},
\end{eqnarray}$$where Fi − 1/2 and Fi + 1/2 are numerical fluxes defined at $x_i \mp \frac{\Delta x}{2}$, respectively. The fluxes are constructed as linear combination of a non-diffusive second order flux $F^{\operatorname{HO}}$ and a diffusive first order correction $F^{\operatorname{LO}}$:
(32)$$\begin{eqnarray}
F_{i + 1 / 2} = F_{i + 1 / 2}^{\operatorname{HO}} - A_{i + 1 / 2} \varphi _{i + 1 / 2} \left(F_{i + 1 / 2}^{\operatorname{HO}} - F_{i + 1 / 2}^{\operatorname{LO}}\right) .
\end{eqnarray}$$The term φi + 1/2 is the so-called flux limiter (LeVeque 1992), while Ai + 1/2 is a coefficient introduced to switch off the diffusive correction at high optical depth (more below). The role of the flux limiter is to introduce numerical dissipation in the presence of unresolved features in the solution u and ensure the non-linear stability of the scheme. In particular, if Ai + 1/2φi + 1/2 = 0 the second-order flux is used, while if Ai + 1/2φi + 1/2 = 1, then the low order flux is used. A standard second order non-diffusive flux is used for $F^{\operatorname{HO}}$, while the Lax–Friedrichs flux is used for $F^{\operatorname{LO}}$:
(33)$$\begin{eqnarray}
F^{\operatorname{HO}}_{i + 1 / 2} = \frac{f (u_i) + f (u_{i + 1})}{2},
\end{eqnarray}$$(34)$$\begin{eqnarray}
F^{\operatorname{LO}}_{i + 1 / 2} = \frac{1}{2} [f (u_i) + f (u_{i + 1})] - \frac{c_{i + 1 / 2}}{2} [u_{i + 1} - u_i] .
\end{eqnarray}$$The characteristic speed in the Lax–Friedrichs flux ci is taken to be the maximum value of the speed of light between the right and left cells
(35)$$\begin{eqnarray}
c_{i + 1 / 2} = \max _{a \in \lbrace i, i + 1 \rbrace } \left\lbrace \left| \alpha _a \sqrt{\gamma _a^{x x}} \pm \beta _a^x \right| \right\rbrace .
\end{eqnarray}$$We remark that it is known that the M1 system can, in some circumstances, lead to acausal (faster than light) propagation of neutrinos in GR (Shibata et al. 2011). For this reason, one might argue that a better choice of the characteristic velocity for the Lax–Friedrichs formula would have been given by the eigenvalue of the Jacobian of $\boldsymbol{F}$. These values are known analytically (Shibata et al. 2011), however in our preliminary tests we found that the use of the full eigenvalues resulted did not improve on the stability or accuracy of the M1 solver.


Spider pulsar systems are characterized by having a low-mass companion star in a compact orbit with an energetic millisecond pulsar (MSP) resulting in heavy irradiation of the companion by the pulsar’s wind. The spider pulsar population has been observed to have a clearly bimodal distribution of companion star masses (Roberts 2011; Strader et al. 2019) made up of two distinct subgroups: black widows (BW) with companion star masses ∼0.01–0.05 M⊙, and redbacks (RB) with companion star masses ∼0.1–1 M⊙. A large proportion of the spiders, whether BWs or RBs, have been observed to exhibit (quasi-)periodic eclipses of the pulsars’ radio emission (e.g. Fruchter, Stinebring & Taylor 1988; Lyne et al. 1990) that are generally attributed to excess material in the orbits – that has been driven from the companion stars by the pulsar irradiation (Podsiadlowski 1991; van den Heuvel & van Paradijs 1988; Phinney et al. 1988; Kluzniak et al. 1988) – interfering with the propagation of the radio emission. Studies of such eclipses are key for understanding mass loss from the irradiated companion stars, the properties of the medium causing the eclipses, interactions between the pulsar wind and the eclipse medium, and the mechanisms responsible for the apparent attenuation of pulsar radio emission during the eclipse. In the years after the initial BW discovery (Fruchter et al. 1988), there were a number of excellent early works (e.g. Ryba & Taylor 1991; Stappers et al. 2001a) investigating the observed radio eclipses. However, unfortunately, a lack of further in-depth eclipse analyses – largely as a result of difficult observing requirements and (until recently) a low number of known spider pulsars – has meant slow progress in reaching an understanding in any of these topics. However, the last few years have marked a revival of the field with detailed and novel studies beginning to give important insight into the nature of eclipsing pulsar systems (e.g. Broderick et al. 2016; Main et al. 2018; Li et al. 2019).

Background Background Background Motivation


A significant limitation of the model is that it ignores the well-established hydrodynamic evolution of the loop during the cooling process, involving the substantial transfer of mass between the chromosphere and the corona. For large downward heat fluxes, the transition region is unable to radiate the supplied energy, resulting in the deposition of thermal energy in the dense chromosphere. The resulting two to three orders-of-magnitude temperature enhancements create a large pressure gradient that drives an upward enthalpy flux of “evaporating” plasma. However, as the loop cools, the decreased heat flux becomes insufficient to sustain the radiation emitted in the now-dense transition region and hence an inverse process of downward enthalpy flux starts to occur. It has been suggested (Klimchuk et al. 2008) that the enthalpy fluxes associated with both evaporating and condensing plasma are at all times in approximate balance with the excess or deficit of the heat flux relative to the transition region radiation loss rate. This basic idea has allowed the development of global “Enthalpy-Based Thermal Evolution of Loops” (EBTEL) models that describe the evolution of the average temperature and density in the coronal part of the loops; these models are generally in good agreement with one-dimensional hydrodynamic simulations (Klimchuk et al. 2008; Cargill et al. 2012a, 2012b). It is, in principle, possible to include the effects of a turbulence-controlled heat flux in EBTEL (or 1D hydrodynamic) models. If this heat flux is reduced sufficiently relative to its collisional value, then, for the reasons explained above, there will be a significant impact on the thermal evolution of the loop. Doing so, however, would still require a numerical treatment, which is beyond the scope of the present work (but which it is our intention to carry out in a future work). Instead, we adopt a simpler approach that allows a systematic and fairly transparent quantitative analysis of the impact of turbulence on the thermodynamics of post-flare loops.


VHE gamma-rays from distant blazars suffer serious EBL absorption. For instance, the attenuation factor of the flux at ∼1 TeV emitted at z = 0.6 due to EBL absorption is ∼10−4 (Domínguez et al. 2011). If VHE gamma-rays from a blazar are produced in its jet, the intrinsic VHE spectrum after de-absorption would be very hard and not a simple power law (e.g. Archambault et al. 2014), which is hardly explained plausibly in leptonic models. In such a case, a leptohadronic jet model is used to explain VHE spectra of distant blazars (e.g. Böttcher, Reimer & Marscher 2009; Yan & Zhang 2015), in which VHE emission is attributed to synchrotron emissions of relativistic protons and pair cascades created in proton–photon (pγ) interaction. However, the jet model cannot explain the VHE emission from PKS 1424+240 if its redshift z > 0.7–0.8 (Yan & Zhang 2015). Alternatively, it is recently proposed that VHE gamma-rays from distant blazars may be the secondary gamma-rays produced in the rectilinear propagation of the UHECRs escaping from these blazars (e.g. Essey & Kusenko 2010; Essey et al. 2010). In the latter case, UHECRs interact with background photons, i.e. EBL photons and microwave background (CMB) photons, which creates UHE electrons and photons through Bethe–Heitler (BH) pair production and photo-meson production. These UHE electrons and photons would interact with background photons again, and then pair cascades are induced; the secondary photons are inverse-Compton-scattered (ICS) CMB photons by the pair cascades. Because of the large mean interaction path of UHECRs, these secondary photons are produced relatively close to the Earth (e.g. Lee 1998; Essey et al. 2011b; Murase et al. 2012). The UHECR induced cascade model is successful in explaining the observed VHE spectra of extreme high-synchrotron-peaked BL Lacertae objects (HBLs; e.g. Essey & Kusenko 2010, 2014; Murase et al. 2012; Aharonian et al. 2013; Takami et al. 2013). In particular, UHECR induced cascade model is able to explain the VHE emission from a blazar with redshift z > 1 (Aharonian et al. 2013; Essey & Kusenko 2014). However, in the previous works the primary emission produced in the jet is either simply assumed (e.g. Aharonian et al. 2013; Essey & Kusenko 2014) or neglected (e.g. Takami et al. 2013).


There are numerous models which lay down nucleosynthesis yields from CCSN explosions (Burbidge et al. 1957; Woosley & Weaver 1986, 1995; Thielemann et al. 1996; Heger & Woosley 2002, 2010; Rauscher et al. 2002; Kobayashi et al. 2006; Nomoto et al. 2006; Woosley & Heger 2007; Pignatari et al. 2015; Sukhbold et al. 2016). In this work, we utilize the zonal yield sets from Sukhbold et al. (2016; hereafter, S16), which used the modified 1D hydrodynamic code KEPLER4

4

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 along with P-HOTB. P-HOTB stands for Prometheus-Hot Bubble and was used to study core collapse (Janka & Mueller 1996; Kifonidis et al. 2003), whereas KEPLER was used to evolve the star along zero age main sequence and calculate nucleosynthesis yields and light curves (Weaver et al. 1978). For models that exploded, isotopic yields were generated post explosion. The zonal yields were obtained for three particular models, 15.2 M⊙, 20.1 M⊙, and 25.2 M⊙, ∼200 s after the explosion, before any mixing could take place.5

5
T. Sukhbold (2017, private communication).
 Although all models in S16 data set assume solar metallicity and do not take into account the effects of rotation, they can account for (1) detailed neutrino transport calculations using an improved explosion mechanism, as compared to Rauscher et al. (2002) and Woosley and Heger (2007); (2) a central engine that considers matter inside the collapsed core, unlike certain other models that investigated only the matter exterior to the central engines used; and (3) unlike previous nucleosynthesis models, all models6

6
Each model has a particular progenitor mass.
 used here are not exploded by injecting artificial energy because (a) models below 15 M⊙ almost always explode, (b) models in 20–30 M⊙ rarely explode, and (c) most models above 30 M⊙ implode and become black holes (see Figure 14 in S16 for the probability of explosion of different progenitor masses). In fact, the few models above 30 M⊙ in which explosion does take place is due to their core being ripped apart by winds to sizes comparable to ∼15 M⊙. The decimals in the progenitor masses of these models might seem bizarre; the reason is that the authors have tried to explode all possible progenitor masses in steps of 0.1 M⊙ between 12 and 30 M⊙; however, 15.0, 15.1, 20.0, 25.0, and 25.1 M⊙ imploded in their simulations. This apparently small change in progenitor mass, which leads to an altogether different end scenario, is due to small but significant variations in the progenitor compactness (O’Connor & Ott 2011) rather than the central engine characteristics (Pejcha & Thompson 2015). This effect is more pronounced near progenitor masses of ∼20 M⊙ because the carbon burning stage changes to the radiative pathway from a convective mechanism. In fact, it has been recently shown that two similar progenitors with identical masses but slightly different input physics can lead to totally different scenarios (Sukhbold et al. 2017). Thus it is not unusual for such stark differences to show up between two similar progenitor stars. Throughout this paper, we frequently approximate 15.2–15, 20.1–20, and 25.2–25 M⊙ models for the sake of simplicity.


In an interstellar cloud the spatial transition from atomic to molecular gas takes place in photon dominated regions (PDRs; see reviews by Hollenbach & Tielens 1997, 1999), which are also a source of a significant fraction of the far-infrared emission from the Milky Way and other galaxies. Exterior to the PDRs the gas makes the transition from neutral to ionized hydrogen. The ionized gas can take the form of a low density ionized boundary layer (IBL) in the case of weak UV fields, or a dense H II region in the proximity to a strong UV field arising from massive star formation. PDRs and H II regions are the boundary regions where the effects of star formation on molecular clouds manifest themselves. They have been the focus of a considerable modeling effort (see Tielens & Hollenbach 1985; Sternberg & Dalgarno 1989; Kaufman et al. 1999; Abel et al. 2005; Le Petit et al. 2006; Bron et al. 2018, and references therein). The observational analysis of PDRs, H II regions, and IBLs has improved considerably since the availability of far-infrared spectroscopic data from the Herschel Space Observatory (see Ossenkopf et al. 2013; Köhler et al. 2014; Stock et al. 2015; Joblin et al. 2018; Wu et al. 2018, and references therein) and the Stratospheric Observatory for Infrared Astronomy (SOFIA; e.g., Schneider et al. 2012; Pérez-Beaupuits et al. 2015; Pabst et al. 2017; Mookerjea et al. 2018). Most of these studies of the ionized and PDR layers have focused on very bright H II regions where high UV flux, density, and temperature produce strong far-infrared emission, making such regions easily observable in key gas tracers such as the fine-structure lines of C+, N+, and O. Less is known about the IBL–PDR conditions for typical molecular clouds where the UV field is smaller and, thus, the lines are weaker. The Herschel Space Observatory HIFI GOT C+ survey (Langer et al. 2010; Pineda et al. 2013) took a step in studying molecular cloud PDRs and IBLs in that it sampled [C II] along several hundred lines of sight (LOS) in the Galaxy producing an unbiased database of a few thousand clouds of various evolutionary stages with most LOS not containing H II regions as indicated by weak [C II] emission. However, because [C II] samples both weakly and highly ionized regions, there remains some uncertainty about the relative contributions of the ionized and PDR regions. Furthermore, because [C II] has only one fine-structure transition one cannot solve uniquely for the properties of the gas. For the GOT C+ survey Langer et al. (2014) derived the column density of material traced by [C II] by assuming a thermal pressure and its Galactic gradient.


The coronal rain plasma can be distinguished from the prominence material by looking at their trajectories and average speeds. The timescale on which the coronal rain forms following the heating onset is much shorter than for the quiescent scenario; in the studied event condensations appear 10 min after the reconnection event, whereas observations of quiescent rain suggests it recurs in the same loop of the order of hours (Antolin & Rouppe van der Voort 2012; Kohutova & Verwichte 2016). The period of the loop heating-condensation cycle in the quiescent scenario is equivalent to the time it takes for the sustained footpoint heating to refill the loop sufficiently with evaporated plasma to reach the thermally unstable regime, after the loop has been evacuated by the previous coronal rain event. This short timescale for coronal rain formation is likely a consequence of the heating input being much greater than in the quiescent case and of the short length of the studied loop. The 1D numerical simulations suggest that although the loop length is a contributing factor, the heating input is the main factor affecting the coronal rain formation timescale (Froment et al. 2018). The thermal instability in the case associated with magnetic reconnection is also more concentrated spatially and only a certain fraction of the loop with a cross section of around 5 Mm width becomes unstable. This implies that the heating that triggers the thermal instability is more localised and only affects a small number of field lines in the loop. As the thermal conduction acts predominantly along the magnetic field, most of the matter and energy transport occurs along the affected field lines. Comparing this to the quiescent scenario, the typical width of the loop bundles observed to undergo condensation formation is around 10−15 Mm (Antolin & Rouppe van der Voort 2012; Kohutova & Verwichte 2016), and in some cases, reaching 40 Mm (Auchère et al. 2018; Froment et al. 2019).


Understanding the redshift dependence of galaxy evolution requires disentangling the evolution of an individual galaxy within a population from the evolution in the average properties of the population as a whole (due to the continual addition of newly formed galaxies with different properties to the extant population). A key method of studying how both individual and populations of galaxies evolve is by analysing their growth in mass and size. An individual star-forming galaxy is expected to evolve along the relation of star-forming galaxies in the mass–size plane (Lilly et al. 1998; Ravindranath et al. 2004; Trujillo et al. 2006; Pezzulli et al. 2015; van Dokkum et al. 2015), while the average size of the population as a whole increases with decreasing redshift due to processes such as mergers (Hopkins et al. 2009; Naab, Johansson & Ostriker 2009) and ’puffing up’ due to strong active galactic nuclei (AGNs) feedback (Fan et al. 2008, 2010), but also new galaxies forming with larger radii (van Dokkum & Franx 2001; Carollo et al. 2013). In other words, star-forming galaxies at both low and intermediate redshifts follow parallel tracks in the mass–size plane (Speagle et al. 2014; van Dokkum et al. 2015), with their starting location in the plane depending on redshift. Therefore, when analysing star-forming galaxies in the mass–size plane at $z$ ∼ 0, we are observing the combined effect of both the evolution of individual galaxies throughout their lifetimes and the evolution of the population due to the addition of new members and loss of old members (as they quench). We want to disentangle these two effects to understand how the redshift range over which a galaxy formed and evolved influences its evolutionary path. We therefore need to understand how the processes influencing a galaxy may be regulated by the broader conditions of the Universe and how these conditions change with redshift. An important tool to measure the impact of various processes is the analysis of scaling relations, which quantify the link between different galaxy parameters to determine their dependence. Specifically, scaling relations between stellar population parameters and galaxy structure allow us to quantify how processes involved in star formation and stellar mass assembly interrelate with processes dominating structural and dynamical changes.

Background Background Background Background


One explanation for the low effective temperatures is that the TDE thermal emission does not originate from the accretion disc, but from an outflow supported by radiation pressure from the disc’s super-Eddington accretion rate (e.g. Loeb & Ulmer 1997; Strubbe & Quataert 2009; Lodato & Rossi 2011; Metzger & Stone 2016; Roth et al. 2016; Curd & Narayan 2019). The TDE’s high photosphere radius and low temperature can then be explained by the increased emitting area from the optically thick, expanding outflow, launched from the accretion disc or SMBH. These outflows can explain the nearly constant temperatures inferred from the spectrum of optically bright TDEs (Strubbe & Quataert 2009; Miller 2015), and may lead to observable emission or absorption line features in the TDE’s spectrum (Strubbe & Quataert 2011; Roth et al. 2016; Roth & Kasen 2018). Hydrodynamical simulations show that the outflow can be supported not only by radiation pressure from the compact disc (Dai et al. 2018; Curd & Narayan 2019) but also by shocks driven by stream–stream collisions during the circularization of stellar debris (Liptai et al. 2019; Lu & Bonnerot 2020). However, to power the outflow, a significant fraction of the tidally disrupted star’s rest-mass energy must be liberated ($0.05 \, {\rm M}_\odot \, c^2 \sim 10^{53} \, {\rm erg}$), much larger than the typical energy liberated by an optically bright TDE’s early emission ($\sim \! 10^{49}{\!-\!}10^{51} \, {\rm erg}$; e.g. Komossa 2015; van Velzen et al. 2020). This so-called missing energy problem has a number of proposed solutions. For instance, some argue most of the rest-mass energy is radiated in the unobservable far-UV wavelength bands (e.g. Lu & Kumar 2018; Jonker et al. 2020), while others propose this energy is carried away by a jet whose emission is unobservable for most TDE viewing angles (Dai et al. 2018). Some authors suggest this energy is never emitted in the first place, but rather becomes trapped due to the TDE disc and outflow’s high optical depth (photon trapping; e.g. Curd & Narayan 2019). The wind model has yet to conclusively address the missing energy problem.


The H α studies of Umeda et al. (2004) and Stroe et al. (2014a, 2015) are tracing instantaneous (averaged over 10 Myr) SF and little is known about SF on longer time-scales and the reservoir of gas that would enable future SF. An excellent test case for studying the gas content of galaxies within merging clusters with shocks is CIZA J2242.8+5301 (Kocevski et al. 2007). For this particular cluster unfortunately, its location in the Galactic plane, prohibits studies of the rest-frame UV or FIR tracing SF on longer time-scales, as the emission is dominated by Milky Way dust. However, the rich multiwavelength data available for the cluster give us an unprecedented detailed view on the interaction of their shock systems with the member galaxies. CIZA J2242.8+5301 is an extremely massive (M200 ∼ 2 × 1015 M⊙; Dawson et al. 2015; Jee et al. 2015) and X-ray disturbed cluster (Akamatsu & Kawahara 2013; Ogrean et al. 2013, 2014) which most likely resulted from a head-on collision of two, equal-mass systems (van Weeren et al. 2011; Dawson et al. 2015). The cluster merger induced relatively strong shocks, which travelled through the ICM, accelerated particles to produce relics towards the north and south of the cluster (van Weeren et al. 2010; Stroe et al. 2013). There is evidence for a few additional smaller shock fronts throughout the cluster volume (Stroe et al. 2013; Ogrean et al. 2014). Of particular interest is the northern relic, which earned the cluster the nickname ‘Sausage’. The relic, tracing a shock of Mach number M ∼ 3 (Stroe et al. 2014c), is detected over a spatial extent of ∼1.5 Mpc in length and up to ∼150 kpc in width and over a wide radio frequency range (150 MHz–16 GHz; Stroe et al. 2013, 2014b). There is evidence that the merger and the shocks shape the evolution of cluster galaxies. The radio jets are bent into a head–tail morphology aligned with the merger axis of the cluster. This is probably ram pressure caused by the relative motion of galaxies with respect to the ICM (Stroe et al. 2013). The cluster was also found to host a high fraction of H α emitting galaxies (Stroe et al. 2014a, 2015). The cluster galaxies not only exhibit increased SF and AGN activity compared to their field counterparts, but are also more massive, more metal rich and show evidence for outflows likely driven by SNe (Sobral et al. 2015). Stroe et al. (2015) and Sobral et al. (2015) suggest that these relative massive galaxies (stellar masses of up to ∼1010.0–10.7 M⊙) retained the metal-rich gas, which was triggered to collapse into dense star-forming clouds by the passage of the shocks, travelling at speeds up to ∼2500 km s−1 (Stroe et al. 2014c), in line with simulations by Roediger et al. (2014).


An extreme example of a rapid decay is V404 Cyg. The parameters of the long Porb system were chosen to be close to those of V404 Cyg, whose last outburst lasted only a couple of weeks and showed a pronounced disc outflow, conjectured to be a thermal wind (Muñoz-Darias et al. 2016). The mass in the disc at the onset of the outburst is ∼3 × 10−7 M⊙ in our models1. Blowing away most of this mass would require a sustained outflow rate of ≈ 10−5 M⊙ yr−1 ≈ 30 ṀEdd over 15 days for a 9 M⊙ black hole. Such very high mass outflow rates may be reached close to the Eddington luminosity as electron scattering contributes to the driving force of the wind. Figure 7 shows the mass outflow rate in the wind diverges near L ≈ 0.7 LEdd due to the estimated radiation driving correction (Eq. (4)). In principle, it might thus be possible to shorten the outburst of V404 Cyg to a couple of weeks by fine-tuning the model parameters to sample this high luminosity region. In support, observations of V404 Cyg do indicate the source likely reached LEdd (Kimura et al. 2016) and was enshrouded by rapidly varying Compton-thick outflowing material (Sánchez-Fernández et al. 2017) with an estimated Mw ≈ 4 × 10−6 M⊙ lost to the wind (Casares et al. 2019). The lower effective gravity due to the high radiation should enhance the wind (Proga & Kallman 2002) but the mass loss must saturate at some level as the outflow becomes optically thick. Higginbottom et al. (2019) do not find a significantly enhanced Ṁw near LEdd in their radiation-hydrodynamic simulations of thermal winds, but these neglect radiative driving by electron scattering. If winds are boosted near Eddington, a puzzle is why GRS 1915+105 has not been affected as much as V404 Cyg despite its luminosity also being close to Eddington and its disc size even greater. If the short duration of the V404 Cyg was due to a thermal wind, then this wind likely required very specific conditions. Instead, we speculate that the angular momentum transport was instead dominated by the jet. The system likely stayed in the (very) bright hard state during the outburst, where it has a strong jet which is almost certainly coupled to the accretion flow via the magnetic fields and could be responsible for angular momentum transport through the hot flow in this state (e.g. Ferreira et al. 2006).


Photometric Calibration: Photometric calibration was performed using the stellar locus regression technique (SLR: Ivezić et al. 2004; MacDonald et al. 2004; High et al. 2009; Gilbank et al. 2011; Coupon et al. 2012; Desai et al. 2012; Kelly et al. 2014). Our reference stellar locus was empirically derived from the globally calibrated DES Y1A1 stellar objects in the region of the Y1A1 footprint with the smallest E(B − V) value from the Schlegel et al. (SFD; 1998) interstellar extinction map. We performed a 1″ match on all Y1 and Y2 objects with S/N > 10 observed in r-band and at least one other band. We then applied a high-purity stellar selection based on the weighted average of the spread_model quantity for the matched objects (

; see below). The average zero point measured in Y1A1, ZPgrizY = {30.0, 30.3, 30.2, 29.9, 28.0}, was assigned to each star as an initial estimate. Starting from this coarse calibration, we began an iterative procedure to fix the color uniformity across the survey footprint. We segmented the sky into equal-area pixels using the HEALPix scheme (Górski et al. 2005). For each ∼0.2 deg2 (resolution nside = 128) HEALPix pixel, we chose the DES exposure in each band with the largest coverage and ran a modified version of the Big MACS SLR code (Kelly et al. 2014)41

41

https://code.google.com/p/big-macs-calibrate/

 to calibrate each star from the reference exposure with respect to the empirical stellar locus. These stars became our initial calibration standards. We then adjusted the zero points of other CCDs so that the magnitudes of the matched detections agreed with the calibration set from the reference exposure. CCDs with fewer than 10 matched stars or with a large dispersion in the magnitude offsets of matched stars (σZP > 0.1

) were flagged. For each calibration star, we computed the weighted-average magnitude in each band using these new CCD zero points; this weighted-average magnitude was used as the calibration standard for the next iteration of the SLR. In the first iteration, we assigned SLR zero points to the calibration stars based on the HEALPix pixel within which they reside. In subsequent iterations, we assigned SLR zero points to the calibration stars based on a bi-linear interpolation of their positions onto the HEALPix grid of SLR zero points. After the second iteration, the color zero points were stable at the 1–2

 level. The absolute calibration was set against the 2MASS J-band magnitude of matched stars (making use of the stellar locus in color-space), which were de-reddened using the SFD map with a reddening law of AJ = 0.709 × E(B − V)SFD from Schlafly & Finkbeiner (2011). The resulting calibrated DES magnitudes are thus already corrected for Galactic reddening by the SLR calibration.


Within the framework of the fireball shock model, Pe'er etÂ al. (2007) proposed a method to infer central engine parameters using observed data. With the measured temperature and flux of an identified thermal component in the spectrum, along with a flux ratio between thermal and non-thermal components, one may infer the size of the jet at the base of the outflow, r0, and the dimensionless entropy of the outflow,  (which is also the bulk Lorentz factor of the outflow, if the photosphere radius is greater than the fireball coasting radius). Some authors have applied this method to some Fermi GRBs (Iyyani etÂ al. 2013; Preece etÂ al. 2014; Ghirlanda etÂ al. 2013). The derived central engine parameters are sometimes ad hoc or inconsistent. For instance, the analyses for GRB 110721A (Iyyani etÂ al. 2013) and for GRB 130427A (Preece etÂ al. 2014) led to a curious conclusion that the bulk Lorentz factor of the outflow of different layers are decreasing with time. This would lead to no, or at most very inefficient, internal shock emission. Yet both bursts have dominant non-thermal emission. More curiously, the data of GRB 110721A (Iyyani etÂ al. 2013) require that r0 is rapidly varying with time by 2â3 orders of magnitudes. This is hard to imagine given the well-believed paradigm of the GRB central engine: If the engine is naked, the size of the engine (a hyper-accreting black hole or a millisecond magnetar) is around r0 â¼ 107âcm; if an extended envelope of a collapsar progenitor is considered, the fireball may be âre-born,â with r0 â¼ R*Î¸j â¼ 109R*, 10Î¸j, â1âcm (where R* is the size of the progenitor star, and Î¸j is the jet opening angle). If one considers the depletion of the envelope, r0 should decrease with time. However, Iyyani etÂ al. (2013) showed that r0 increases from 106âcm to 108âcm early on, and then decreases mildly after 2Â s. These absurd conclusions suggest that the starting point of the analysis, i.e., the assumption of a pure fireball model, might not be valid. It is interesting to see whether a hybrid ejecta photosphere model may solve the problem. Incidentally, Ghirlanda etÂ al. (2013) analyzed another burst, GRB 100507, using the fireball framework (Pe'er etÂ al. 2007), but found that the derived r0 remains constant and reasonable. The jet composition of that burst may be more close to a fireball. It would be interesting to see whether a general theoretical framework can be established, which may be reduced to the standard fireball framework when Ï0  1.


Over the last decade, the radial and vertical dependences of the metallicity-alpha-element distribution have been studied in more and more detail with increasingly larger samples (e.g., Bensby et al. 2011; Anders et al. 2014; Nidever et al. 2014; Hayden et al. 2015; Queiroz et al. 2020). Figure 6 is mostly consistent with similar plots shown in the above papers. In the inner 10 kpc, it displays two over-densities, a high alpha-element (here [Mg/Fe]), and a low one. Between Rg = 6 and 10 kpc, the two over-densities define two different sequences. In Appendix E, we note that when the sample is restricted to a ±500 pc layer around the Galactic plane, two close but separated sequences are observed in the Rg ∈ [4, 6] kpc interval. Because of their scale height (Bovy et al. 2012), kinematics (Bensby et al. 2003), and age properties (Haywood et al. 2013), these two sequences are associated with the thick disc (high-alpha) and thin disc (low-alpha), respectively. Moving inward of Rg = 4 − 6 kpc, Fig. 6 shows that the two over-densities connect through a zone of lower density to form a single sequence. This is in agreement with the observations of Hayden et al. (2015), Bensby et al. (2017), Zasowski et al. (2019), Bovy et al. (2019), and Lian et al. (2020a,b), who also report a single sequence in the inner disc and/or in the bulge/bar area. Conversely, Rojas-Arriagada et al. (2019) and Queiroz et al. (2020) observe two sequences in the inner regions. In Appendix F, we compare the distributions of different APOGEE DR16 alpha elements in the ([Fe/H], [α/Fe]) plane (restricting the sample to the stars contained in the Rg ∈ [0, 2] kpc interval). The different elements produce different patterns: the global alpha-element abundance5 and oxygen show a double sequence, while magnesium, silicon, and calcium present a single sequence. This could explain, at least partly, why Queiroz et al. (2020), who use a combined α-element abundance, observe a double sequence, while we see a single one with magnesium. However, this does not explain the discrepancy with Rojas-Arriagada et al. (2019), who also used magnesium. Beyond Rg = 10 kpc, the high-alpha sequence gradually vanishes. This is in agreement with the finding that the thick disc has a shorter scale length than the thin disc (Bensby et al. 2011; Cheng et al. 2012; Bovy et al. 2012). It should be emphasised that in this paragraph the term ‘sequence’ is used in the geometrical sense. It does not presuppose the number of chemical tracks that form the sequence or sequences. In particular, based on Fig. 6, it can not be excluded that the single geometrical sequence observed in the inner disc be made of two chemical tracks, with the low-alpha one restricted to a narrow metallicity range. We discuss and propose an interpretation of the inner disc sequence in Sect. 5.


In Figure 1, we show the archival 850 μm polarization data from the legacy program of SCUPOL with the JCMT (Matthews et al. 2009) (effective 20″ beam ∼0.76 pc). Matthews et al. (2009) collected the JCMT data toward the GC, where the B-field is sampled on a 10″ grid. The linearly polarized light from dust grains is frequently used to probe the integrated plane-of-sky B-field morphology. Interstellar dust grains are elongated, with their minor axes parallel to the B-field. The thermal emission from the aligned dust grains is then polarized with polarization segments perpendicular to the field lines (Cudlip et al. 1982; Hildebrand et al. 1984; Hildebrand 1988; Lazarian 2000; Andersson et al. 2015). The dust polarization can, therefore, reveal the plane-of-sky projected B-field orientations. In Figure 1, the SCUPOL B-field segments are overlaid on the JCMT 850 μm map (Di Francesco et al. 2008). The segments are plotted with p/dp ≥ 2, 3, 4. The 450 μm Submillimeter Polarimeter for Antarctic Remote Observing (SPARO) B-field (Novak et al. 2003) is also overlaid with a resolution of 6′ (corresponding to a linear scale of 13.7 pc). The low-resolution SPARO map traces the large-scale B-field which is parallel to the plane of our Galaxy. This alignment of field orientations with the Galactic plane is attributed to a large-scale toroidal B-field configuration (i.e., azimuthal field). The SCUPOL 850 μm B-field as well as the continuum are clearly detected and resolved along and across the sub-features in the GC at a 20″ resolution (=0.76 pc). The detected B-field orientations (ΦB) vary enormously over the entire map, ranging from −90° to 90° (0° is west, positive is counterclockwise). Nevertheless, the ΦB varies smoothly and systematically along certain substructures, revealing organized patches in, e.g., the CND, the giant molecular cloud 20/50 MC (e.g., Güsten et al. 1981), and the HVCC CO 0.02–0.02 (Oka et al. 1999). The azimuthal correlation seen in the CND is providing the link between the model proposed by Wardle & Königl (1990) and the SCUPOL 850 μm polarization data. We are exploring this link in the following sections. A comparison of polarization data is presented in Appendix A. We find that the morphology of the B-field is consistent with different threshold cuts in p/dp. Hence, in order to work with the maximum of independent data points, we present data with p/dp ≥ 2, and we will focus on the B-field structure of the CND.


Mainly four mechanisms are considered to be causing the observed solar radio wave emissions: two causing incoherent and two causing coherent emissions. Incoherent emissions can be due to bremsstrahlung and gyrosynchrotron radiation. In them, every electron radiates independent on the others. The total emission is simply the sum of the emissions of every single electron (Rybicki & Lightman 1979; Dulk 1985; Melrose 2017; Nindos 2020). Comparing with coherent emission mechanisms, the incoherent emission mechanisms are better understood. The two coherent emission mechanisms are plasma (Ginzburg & Zhelezniakov 1958; Melrose 1970a, 1970b; Zheleznyakov & Zaitsev 1970a, 1970b) and electron cyclotron maser (ECM) emissions (Twiss 1958; Gaponov 1959; Schneider 1959; Pritchett 1984a). In contrast to the incoherent emission mechanisms, coherent emission can explain (a) the high brightness temperatures, (b) the short eruption timescales, (c) the narrow frequency bands, and (d) the strong polarization of Type I, II, III solar radio bursts (SRBs) and solar radio spikes (Aschwanden 2005). And coherent radio emission mechanisms involve plasma instabilities. The plasma emission follows a beam or bump-on-tail instability, the source of free energy of which is related to an electron velocity distribution

u

∥

·
∂

f

u

∥

/

∂

u

∥

>
0

. On the other hand, the ECM instability requires a positive gradient in the electron velocity distribution along the direction perpendicular to the ambient magnetic field (

∂

f

u

⊥

/

∂

u

⊥

>
0

) (see Melrose 1986, 2017). Here, f is the electron distribution function and u∥, u⊥ are the electron velocities parallel and perpendicular to the ambient magnetic field, respectively. The expressions

u

∥

·
∂

f

u

∥

/

∂

u

∥

>
0

 and

∂

f

u

⊥

/

∂

u

⊥

>
0

 indicate an electron beam and a ring (or a loss cone or a horseshoe) distribution in the direction parallel and perpendicular to the ambient magnetic field, respectively. The existence of electron velocity distributions

u

∥

·
∂

f

u

∥

/

∂

u

∥

>
0

 has been conjectured based on observations of, e.g., SRBs and hard X-ray bursts as well as confirmed by observations of solar energetic particles (Chen et al. 2015, 2018; Cairns et al. 2018). Numerical simulations have also considered possible formation mechanisms of velocity distributions with

∂

f

u

⊥

/

∂

u

⊥

>
0

, e.g., by magnetic reconnection (e.g., Büchner & Kuska 1996; Bessho et al. 2014; Shuster et al. 2014; Zhou et al. 2016; Treumann & Baumjohann 2017; Muñoz & Büchner 2018a; Voitcu & Echim 2018; Yao et al. 2021a). It is, however, still under debate which coherent mechanism dominates and which role different features of the electron distribution play for the emission of coherent radio waves from the solar corona.


As mentioned in the previous section, if a magnetized WD rotates with a misalignment between its magnetic field and rotation axes (similar configuration to a pulsar), it can emit a continuous GW. We already provided a detailed discussion on GWs emitted from WDs with different magnetic field geometries and strengths in GR (Kalita & Mukhopadhyay 2019b; Kalita et al. 2020). Figure 7 shows an illustrative diagram of a magnetized WD where the magnetic field is along the z′-axis and rotation is along the z-axis, with χ being the angle between these two axes. We calculate the amplitude of GW using the set of Equations (34) assuming the difference in radii of the WD between those along x- and z-axes to be 0.01%, i.e.,

, due to the presence of a very weak magnetic field and slow rotation. The choice of weak fields and slow rotation assures that the underlying WD mass–radius solutions do not practically differ from the solutions based on the f(R) gravity without magnetic fields and rotation. In future, we plan to check rigorously by solving the set of equations, if indeed such ϵ is possible in the presence of weak magnetic fields and rotation keeping the mass and radius practically intact. As we will show below, however, the chosen ϵ appears to be the minimally required value to have any appreciable effect. Nevertheless, there are examples of weakly magnetized WD pulsars, which can be explained even in the GR framework, e.g., AE Aquarii (Bookbinder & Lamb 1987) and AR Scorpii (Marsh et al. 2016), where magnetic fields hardly affect their mass–radius relations. Figure 8 shows the PSD as a function of frequency for various detectors along with

 over 5 s integration time for various f(R)-gravity-induced WD pulsars with different i assuming χ = 90° and r = 100 pc. It is evident that while DECIGO and BBO can immediately detect such weakly magnetized super-Chandrasekhar WDs, the Einstein Telescope can detect them in

 minutes with S/N ≈ 5 (see Figure 5(b)). However, for ALIA and LISA, the corresponding integration time respectively turns out to be

 days and

 yr3

3
Note that even if the threshold S/N for detection increases slightly (say, from 5 to 20), many of these sources can still be detected in a few seconds to a few days of integration time depending on the type of the detectors.
. Hence, it is also possible to detect such weakly magnetized WDs using ALIA, whereas for LISA it is quite impossible. Figure 5(b) depicts

 for these WDs with different integration times to show that S/N increases if the integration time increases so that various detectors can detect them eventually. For such a system, the GW luminosity is given by (Zimmermann & Szedenits 1979)
43

It is expected that a source can emit electromagnetic radiation in the presence of a magnetic field, and it is the dipole radiation in the case of a WD pulsar. However, because of the presence of a weak magnetic field, the dipole radiation emitted from such a WD is minimal, and the corresponding dipole luminosity is negligible as compared to LGW. Hence, the spin-down timescale is mostly governed by LGW, given by (Kalita et al. 2020)
44

Figure 9 shows the variation of LGW and P with respect to M for various WDs with χ = 90°. The maximum LGW in the case of a WD is ∼1037 erg s−1. The empirical relations of LGW and P, in various branches, are same as in the previous case provided in Table 1. It is also clear from the figure that the massive WD pulsars are short-lived as compared to the lighter ones.


Various methods have been developed in the past to derive stellar atmospheric parameters from large number of medium-to-low resolution spectra (Recio-Blanco, Bijaoui & de Laverny 2006; Lee et al. 2008a, Wu et al. 2011). The approaches generally fall into two main categories of method (Wu et al. 2011): the minimum distance method (MDM) and non-linear regression method. Both categories of method have been applied to large stellar spectroscopic surveys, including the SEGUE (Yanny et al. 2009), RAVE (Steinmetz et al. 2006), APOGEE (Majewski et al. 2010) and LAMOST (Zhao et al. 2012). The MDM is usually based on spectral template matching, and searches for the template spectrum that has the shortest distance in parameter space from the target spectrum. The χ2 minimization, cross-correlation, weighted mean algorithm and the k-nearest neighbour are thought to be specific cases of MDM (Wu et al. 2011). Software and pipelines developed based on those algorithms include the tgmet (Katz et al. 1998), matisse (Recio-Blanco et al. 2006), sspp (Lee et al. 2008a), ulyss (Koleva et al. 2009), that of Allende Prieto et al. (2006) and of Zwitter et al. (2008). The non-linear regression method is sometimes also referred to as the artificial neural network. The method constructs a functional mapping between the spectra and stellar atmospheric parameters by training a library of template spectra with non-linear algorithms such as the principal component analysis, and then apply the mapping to target spectra. Related work can be found in Re Fiorentin et al. (2007) and Lee et al. (2008a). In addition to the above two categories of method, other approaches have been developed, for example, the line-index method based on the relations between the stellar atmospheric parameters and the equivalent widths (EWs) of spectral features and/or photometric colours (Beers et al. 1999; Wilhelm, Beers & Gray 1999; Cenarro et al. 2002). More recently, a Bayesian approach to determine stellar atmospheric parameters combing spectral and photometric measurements has been developed by Schönrich & Bergemann (2014).


We first discuss the relative bulk enrichment. From internal structure models one can derive the MZ necessary to reproduce the observed mass and radius, and—for the Solar System planets—the gravitational moments. Studies inferring in this way eZ,rel,int of transiting exoplanets have found that eZ,rel,int decreases with increasing mass (left panel of Figure 3). The planetary mass where

 defines the parity mass M1 (see Appendix B). It is shown in the right panel of Figure 3. It is extrapolated to be between ∼13 and 60 MJup (Miller & Fortney 2011; Thorngren et al. 2015). The planets analyzed in these studies have equilibrium temperatures of less than ∼1000 K (corresponding to an orbital distance of about 0.08 au for a solar-like star) so that they are not affected by the aforementioned bloating mechanisms. A similar decrease of eZ,rel,int with increasing mass is found for the bulk metal content of Solar System giants (Saumon & Guillot 2004; Helled et al. 2011), where the mass where eZ,rel,int reaches 1 is extrapolated to be at about 11 MJup. From theoretical planet population syntheses based on the core accretion theory one finally finds that the parity mass is at about 10 to 18 MJup (Mordasini et al. 2014). Considering that of the 255 extrasolar giant planets (M sin i > 0.1 M) inside of 0.1 au currently listed on www.exoplanets.org (Han et al. 2014) only 4 have a mass exceeding 10 M (which is not an observational bias). We thus deduce that at least based on their masses regarding the bulk composition of hot Jupiters, it appears that almost all of them should be dominated by planetesimal enrichment. We add the caveat that the bulk heavy element content cannot be inferred directly for typical hot Jupiters at equilibrium temperatures of Teq ≳ 1500 K because of bloating mechanisms. But the fact that both the planets analyzed by Miller & Fortney (2011), Thorngren et al. (2015) (a = 0.03–1 au, Teq ≲ 1000 K) and the solar system planets (a ≈ 5–30 au) follow the same trend, makes it appear unlikely—even though in principle not excluded—that the hot Jupiters at a ∼ 0.04 au do not follow the same enrichment pattern.


The theoretical models mentioned above have made assumptions. The validity of these assumptions needs to be examined. Besides, all these models involve parameters to be determined. These problems could be better understood through numerical simulations. Hurlburt et al. (1994) performed a two-dimensional numerical simulation of compressible flow on downward overshooting. They investigated the dependence of subadiabatic extent δ on stability parameter S. They revealed a scaling law of δ ∝ S−1 in the penetrative layer and δ ∝ S−1/4 in the overshooting layer, respectively. This result is in good agreement with Zahn’s analytic model (Zahn 1991). Freytag et al. (1996) performed two-dimensional numerical simulations with a realistic description of radiation and ionization on A-type stars and DA white dwarf stars. They described the overshooting as a diffusive process, and derived an exponential decay parameter for the diffusion. Early attempts of low-resolution three-dimensional numerical simulations on overshooting were made by Singh et al. (1994; upward overshooting), Singh et al. (1995, 1998), and Saikia et al. (2000) (downward overshooting). The scaling laws derived from the numerical simulations of downward overshooting (Singh et al. 1995, 1998; Saikia et al. 2000) agree well with Zahn’s analytical model. Only a qualitative result was given in the numerical simulations of upward overshooting (Singh et al. 1994). High-resolution numerical simulations of downward overshooting across a wide range of parameters were presented by Brummell et al. (2002). They confirmed the −1/4 scaling law of the thermal adjustment overshooting layer, while the −1 scaling law of the nearly adiabatic penetrative layer was absent in the simulations. Based on a semianalytic model, Rempel (2004) argued that the absence of the nearly adiabatic penetrative layer is caused by the large energy flux specified in the numerical simulations. Numerical experiments on Boussinesq flow were performed by Korre et al. (2019). They reported steeper scaling laws of δ ∝ S−1/3 or δ ∝ S−1/2, depending on the steepness of the background radiative temperature gradient. Simulations with realistic physical variables on stellar core convection were performed by Browning et al. (2004) and Brun et al. (2005). The effects of rotation and magnetic field are considered. They found that the penetrative convection yields a prolate shape of a nearly adiabatic region. Kitiashvili et al. (2016) performed 3D radiative hydrodynamic simulations of the outer layer of a moderate-mass star (1.47 solar mass). Their result discovered a nearly adiabatic layer and a deeper subadiabatic layer. The recent work of Brun et al. (2017) simulated the differential rotation and overshooting in solar-like stars. Their result indicated that slow rotators favor a wider overshooting region near the poles and fast rotators at mid-to-low latitude. Hotta (2017) performed numerical simulations on the solar overshooting with very low energy fluxes F. He found that the overshooting distance obeys a scaling law of δ ∝ F0.31. Käpylä (2019) conducted numerical experiments on downward overshooting by considering the effect of the smoothness of the heat conduction profiles. He discovered that the power-law index of the overshooting distance on the energy flux is smaller in the smooth heat conduction profile than in the step profile. Efforts on prediction of 321D turbulent theory were made by Arnett et al. (2015) and Arnett & Moravveji (2017). They separated the overshooting region into three layers: a fully mixed layer, a partially mixed wave layer, and an extra diffusive mixing layer. With the scale analysis of turbulent plumes and eddies, Viallet et al. (2015) discussed the three possible regimes of turbulent overshooting: a diffusion-dominated regime (only mix composition), a penetrative regime (transition within the boundary layer), and an entrainment regime (mix both entropy and composition). The selection criterion of different regimes during a stellar evolution calculation is not well defined yet.


(i) Our capability of reproducing the lensed images down to the noise level without fully correctly modelling the cored central mass density distribution of the lenses indicates some form of the source-position transformation (SPT; Schneider & Sluse 2014), in line with the previous findings by Unruh et al. (2017). As a consequence, our reconstructions lead to a systematic fractional error on the Hubble constant of $25_{-19}^{+37}$ per cent (in comparison to a statical error of $12_{-3}^{+6}$ per cent when the shape of the lensing potential is perfectly known). This result is in agreement with the latest analysis of Blum, Castorina & Simonović (2020) that shows that cored (dark matter) mass density distributions give rise to approximate MSDs, and an error on the inferred Hubble constant. The latest cosmographic analyses (see e.g. Wong et al. 2019) have attempted to break these degeneracies by including the information contained in the kinematic properties of the lens galaxies and the positions of the lensed quasar images. However, the validity of this approach has been recently debated by Kochanek (2020), who has demonstrated that departures from single power-law mass distributions are responsible for a fractional error on the Hubble constant of 30 per cent. While the cores in the simulations analysed in this paper are artefacts related to limited resolution, cored mass density distribution in real galaxies may be developed by the effect of baryonic processes (see e.g. Chan et al. 2015) or changes in the dark matter properties (Schive, Chiueh & Broadhurst 2014; Spergel & Steinhardt 2000). Moreover, similar additional complexities exist in real galaxies are related, for example, to the presence of faint discs (Hsueh et al. 2018), bars, or other (baryonic) structures (Gilman et al. 2018; Xu et al. 2013). More generally, there exist many plausible deviations from a smooth power-law distribution, such as broken power laws (see e.g. Du et al. 2020) or multiple component models (see e.g. Nightingale, Dye & Massey 2018), which can produce comparable degeneracies. Together with the findings of Blum et al. (2020), our results have important implications for the analysis of time delays and a potential solution to the H0 tension (Wong et al. 2019).


The equations of motions and hydrodynamics are solved by the ramses code (Teyssier 2002). The gas follows a piecewise polytropic equation of state (EoS) fitting the heating/cooling equilibrium (see Kraljic et al. 2014, and references therein). A Jeans polytrope sets a pressure floor in the most refined volumes, to prevent artificial fragmentation. We refer the reader specifically to sections 2.2, 2.3 and 4.2 of R+13 and Kraljic et al. (2014) for further discussions on this and specific EoS (see Robertson & Kravtsov 2008; Tasker & Bryan 2008; Dobbs, Burkert & Pringle 2011; Bonnell, Dobbs & Smith 2013, for a few other implementation schemes). The resulting gas density probability distribution function (PDF; McKee & Ostriker 2007, and references therein) in the present simulation follows a classic log-normal shape (Nordlund & Padoan 1999; Padoan & Nordlund 2002) with an additional few per cent of the mass in a power-law tail at high density (Choi, Shlosman & Begelman 2013; R+13), as expected from gravity and observed in real molecular clouds and galaxies (Lombardi, Alves & Lada 2010; Druard et al. 2014). While changes in the AMR grid refinement can locally bias the velocity dispersion, the density and velocity power spectra are thus clearly dominated by a single turbulence cascade with a well-identified injection scale at the average Jeans length (Bournaud et al. 2010; R+13). The very high resolution of the present simulation allows to resolve the turbulent cascade with a realistic power spectrum (Combes et al. 2013; R+13) and density distribution (Druard et al. 2014) down to the parsec scale. The simulation comprises conversion of gas into stellar particles (down to 160 M⊙) where the volume density ρ0 exceed 2000 cm−3, assuming that the local star formation rate depends on the free-fall time and with the star formation efficiency set at 3 per cent (R+13). This recipe does not take into account additional physics that may impact on the formation of molecules, and we thus rely on the EoS to follow the cloud collapse, the low temperatures and the high density which triggers the formation of new stars. These newly formed ‘stars’ are evolved on the AMR grid, i.e. with gravitational softening down to 0.05 pc, much smaller than the dark matter and primordial stellar components. The implementation of stellar feedback includes photoionization through heating, radiative pressure via injection of momentum and supernova explosions in the kinetic form (see R+13 for more details). A more thorough study of the impact of resolution or metallicity in such simulations was conducted by Kraljic et al. (2014), who have shown that the artificial density threshold ρ0 does not tune the efficiency of star formation, which mostly depends on the turbulence level (e.g. Mach number; see Klessen 2000; Li et al. 2004; Audit & Hennebelle 2010; Renaud, Kraljic & Bournaud 2012).


The collision velocity dependence of the coefficient of restitution between particles was observed in experiments (Bridges et al. 1996; Higa et al. 1996) and is discussed in the literature (e.g., Ramírez et al. 1999; Zhang & Vu-Quoc 2002). However, the experiments by Heißelmann et al. (2010), used in the present paper to support our assumption of a constant coefficient of restitution, do not see a variation of the coefficient of restitution between particles at low collision velocities (≤ 1 cm s−1). This discrepancy in results might originate in the nature of the collisions studied in these different experiments: Bridges et al. (1996) and Higa et al. (1996) performed collisions of a particle with a flat surface, while Heißelmann et al. (2010) observed particle-particle collisions in a free-floating environment. The latter is an experimental environment very similar to NanoRocks. In such inter-particle collisions in free-floating environments, other physical effects lead to a different behavior of the energy dissipation during collisions. In particular, the damping behavior of a large plate or surface is expected to differ from that of a same-sized particle, so that the velocity dependence of the coefficient of restitution might be an effect of the experimental setup in Bridges et al. (1996) and Higa et al. (1996). Colwell et al. (2016) and Brisset et al. (2018) studied collisions between a round cm-sized particle and a flat surface of fine grains. They also observed an increase of the coefficient of restitution with decreasing collision velocity. While the composition of the target surface was different than in Bridges et al. (1996) and Higa et al. (1996) (fine granular material vs. solid ice), the similar behavior of the coefficient of restitution supports the fact that particle-surface collisions are very different from particle-particle collisions, and coefficients of restitution are only velocity dependent for collisions with particles with very different sizes (a much larger particle can be approximated as a target surface).


Understanding how the diverse populations of protoplanetary discs in young stellar regions results in the range of exoplanet types and architectures found in the Galaxy is one of the major goals of planet-formation theory. This is an extremely challenging task due in part to the limited observational constraints available. The Atacama Large Millimetre/submillimetre Array (ALMA) is providing truly transformational images of protoplanetary discs with unprecedented sensitivity and resolution (Andrews 2020). However, millimetre wavelength images reveal the locations of small dust grains but provide little information on the presence of larger particles, beyond centimetre scales. Gas giant planets are mostly made of hydrogen and helium, which ALMA cannot directly observe; therefore, the information on the gas content relies on the observations of less abundant molecules, such as CO and its isotopologues, that are subjected to uncertain depletion processes in the gas-phase (e.g. Miotello et al. 2016). Planets might be detectable by ALMA, although indirectly, by the effects they have on the gas and/or dust in the disc. When planets become massive enough, they can carve gaps (e.g. Rice et al. 2006; Pinilla, Benisty & Birnstiel 2012; Zhu et al. 2012) and disturb the dynamics of the gas (Teague et al. 2018; Casassus & Pérez 2019; Pinte et al. 2019). The minimum gap-opening mass depends on the viscosity and scale-height of the disc (Crida, Morbidelli & Masset 2006; Duffell & MacFadyen 2013), but mini-Neptune-mass (Pérez et al. 2019) or even Earth-mass planets (Rosotti et al. 2016; Dong & Fung 2017) could produce detectable gaps. Gaps consistent with fully formed planets have been imaged by ALMA in discs with estimated ages ranging from  1 Myr (HL Tau and Elias 2–24; ALMA Partnership et al. 2015; Cieza et al. 2017) to ∼10 Myr (TW Hydra; Andrews et al. 2016). However, the origin of these gaps still remains to be established and several alternative explanations have been proposed, including the effect of snow-lines on the dust/gas evolution of different volatiles (Zhang, Blake & Bergin 2015), magneto-hydrodynamic effects (Flock et al. 2015), secular gravitational instability (e.g. Youdin 2011; Takahashi & Inutsuka 2014), and viscous ring-instabilities (Dullemond & Penzlin 2018). Each one of the proposed mechanisms has their merits and shortcomings, and it is possible that different mechanisms operate together or dominate in different objects or in the same object at at different times. For a recent review on disc (sub)structures, see Andrews (2020). Substructures are also expected to be ubiquitous in protoplanetary discs from a theoretical point of view. Without substructures to halt the migration of mm-size grains at large radii, dust particles should migrate towards the innermost part of the disc in time-scales shorter than 0.1 Myr (e.g. Brauer et al. 2007), which is inconsistent with the observations showing significant mm emission at large radii (≳10 au) at much older ages. Understanding the origin and evolution of substructures in protoplanetary discs and their implications for planet formation is currently one the main challenges in the field. To better understand the incidence and properties of disc substructures in any given molecular cloud, here we present 1.3 mm/230 GHz continuum ALMA long-baseline observations at 3–5 au resolution of the 10 brightest targets of the ‘Ophiuchus DIsc Survey Employing ALMA’ (ODISEA) project (Cieza et al. 2019) that were not included in ‘The disc Substructures at High Angular Resolution Project’ (DSHARP) ALMA Cycle-4 Large Program (Andrews et al. 2018). Our new observations result in the largest sample of disc images at ∼3–5 au resolution in any star-forming region observed so far at mm wavelengths (15 objects when combined with the brightest Ophiuchus objects in DSHARP). In Section 2, we discuss the sample selection, the long-baseline observations, and the data reduction. In Section 3, we characterize the observed substructures, including gaps, rings, inner discs, and cavities. In Section 4, we discuss individual objects and use the full sample of 15 bright Ophiuchus discs observed at high-resolution to construct a tentative evolutionary sequence in which the observed substructures are mostly driven by dust evolution and the formation of giant planets. We also discuss possible connections between the substructures observed in primordial discs and those seen in more evolved debris disc systems. A summary of our results and conclusions is presented in Section 5.


As stated earlier, during review of this manuscript Croviser et al. announced in a CBET a tentative water production rate approximately five times larger than our reported value. While the brief nature of the CBET precludes a detailed comparison, we discuss some possible reasons for this discrepancy. At the high airmass of these observations and the small dimensions of the ARCES slit, differential refraction can result in wavelength-dependent slit loss, which can skew flux measurements. However, this is not expected for [O i] 6300 Å emission because this feature is close to the guiding wavelength (∼5500 Å). We confirmed that this is indeed negligible for [O i] 6300 Å emission based on observations of comet C/2012 S1 (ISON) that were performed at a similarly high airmass with ARCES, and found that the production rates derived from the ISON [O i] 6300 Å measurements were consistent with values determined using other methods (McKay et al. 2018). Therefore, we do not consider this or other airmass-dependent phenomena as the reason for the discrepancy. At certain geocentric velocities the cometary [O i] 6300 Å emission sits on top of a strong telluric absorption, and at high airmass inaccurate removal of this feature can result in a decrease in the measured flux and therefore production rate. This was observed for C/2012 S1 (ISON) (McKay et al. 2018). However, the geocentric velocity of 2I/Borisov during our observations was ∼−35 km s−1, while the effect on observed [O i] 6300 Å line fluxes in comet ISON was only observed at geocentric velocities of ∼−50 km s−1. Therefore, this is also not a likely candidate to explain the discrepancy. It is also possible that the activity is highly variable, and we observed Borisov at a minimum in activity, while the Nançay observations, which were coadded over three weeks of observations, provide a long-term average production rate. However, no such variability is observed for CN, with the CN production rate being fairly constant over a several week period (Kareta et al. 2019; Opitom et al. 2019).


There have been rather strong claims of AGN QPOs in different bands of the electromagnetic spectrum, ranging from minutes through days through months and years (e.g. Gierliński et al. 2008; Lachowicz et al. 2009; Gupta, Srivastava & Wiita 2009; Gupta et al. 2018, 2019; King et al. 2013; Gupta 2014, 2018; Ackermann et al. 2015; Pan et al. 2016; Zhou et al. 2018; Bhatta 2019; and references therein). However, many of the claimed QPOs, particularly those made earlier, were marginal detections (Gupta 2014), lasting only a few cycles, and the originally quoted statistical significances are probably overestimates (Gupta 2014; Covino, Sandrinelli & Treves 2019). Among the better recent claims of QPOs in the gamma-ray band are of ∼34.5 d in the blazar PKS 2247–131 (Zhou et al. 2018) and of ∼71 d in the blazar B2 1520+31 (Gupta et al. 2019) found as part of a continuing analysis of blazar Fermi–LAT observations. A recent claim of a ∼44 d optical band QPO in the narrow-line Seyfert 1 galaxy KIC 9650712 from densely sampled Kepler data has been made by Smith et al. (2018); it was supported by an independent analysis of the same data, indicating a QPO contribution at 52 ± 2 d (Phillipson et al. 2020). Some possibly related QPOs of a few hundred days in two widely separated bands have been reported (Sandrinelli et al. 2016a; Sandrinelli, Covino & Treves 2016b; Sandrinelli et al. 2017). However, an analysis of the Fermi–LAT and aperture photometry light curves by Covino et al. (2019) argued that some multiwaveband QPOs, along with many earlier claims of gamma-ray QPOs, are not significant. Among the gamma-ray QPOs with month-like periods, none showed simultaneous oscillations in a different wavebands. Evidence for related QPOs in multiple wavebands was observed in PG 1553+113, where a QPO was detected in the 0.1–300 GeV and the optical waveband (Ackermann et al. 2015). The observed QPO had a dominant period of ∼754 d and the source showed strong inter-waveband cross-correlations.


Figures 8 and 9 show the effects of solar activity on the CR parallel λ∥ (blue line), perpendicular λ⊥ (red line), and radial mean free path λrr (gray line) for a proton with rigidity 445 MV (corresponding to a 100 MeV proton) for the inwardly and outwardly directed IMF, respectively. As described in Zank et al. (1998), the parallel mean free path (mfp) based on standard QLT and assuming magnetostatic turbulence is approximated by
13

where

,

, and

. RL is the particle Larmor radius, P is the particle rigidity, and B0 is the mean magnetic field strength. The analytic form of the perpendicular mfp based on NLGC theory is given by (Zank et al. 2004; Shalchi et al. 2010)
14

where a2 = 1/3 is a factor related to the gyrocenter velocity.

 is a constant such that ν = 5/6 yields a Kolmogorov (1941) spectrum. Note that Equation (14) was derived under the assumption of a specific form of 2D wave spectrum, which is a constant at large turbulence scales. It means that the 2D turbulence spectrum is independent of wavenumber in the energy range in Equation (14). Observation of magnetic fluctuations in the SW indicates that omnidirectional power spectra approach a k−1 wavenumber dependence and that low-frequency turbulence exhibits some sunspot cycle variability (Bieber et al. 1991). Based on this, Engelbrecht & Burger (2015) derived the perpendicular mfp by specifying the energy range spectral index of 2D turbulence power spectra as −1. A more general form of the 2D power spectrum with an energy range spectral index q was proposed by Shalchi et al. (2010). They show that the spectral index has a strong influence on the perpendicular diffusion coefficient. In their model, negative values of q correspond to a decreasing spectrum in the energy range, q = 0 corresponds to the constant spectrum we use here, and positive values of q correspond to an increasing spectrum. Matthaeus et al. (2007) presented a similar spectrum in different regimes: energy range, inertial range, and intermediate regime where the spectrum is proportional to k−1 to coincide with observations (Bieber et al. 1991; Goldstein & Roberts 1999). However, Shalchi (2013) argues that a spectrum that behaves like k−1 does not provide a different perpendicular diffusion coefficient (see also Shalchi et al. 2010), since the field lines for such length scales behave superdiffusively as in the inertial range (Shalchi & Kourakis 2007). In view of this uncertainty, we do not take into account a more elaborate spectrum in the present paper. The behavior of the 2D wave spectrum in the energy range, which may also be correlated with the sunspot cycle, is an important factor in deriving the CR perpendicular mfp. A general form (e.g., Shalchi et al. 2010; Shalchi 2013) should be employed in future studies of CR diffusion.


In light of this, the allowed structure of gamma-ray burst (GRB) jets and the efficiency at which it produces gamma-rays at large angles remains a topic of major importance, and it is useful to consider what types of jet structures are consistent with GRB observations (see also Beniamini et al. 2018b). Previous studies have considered the implications of structure models on the true energetics and rates of GRBs (Frail et al. 2001; Lipunov, Postnov & Prokhorov 2001; Rossi, Lazzati & Rees 2002; Zhang & Mészáros 2002; Eichler & Levinson 2004; van Eerten & MacFadyen 2012; Pescalli et al. 2015), on the shape of the afterglow light curve (Granot & Kumar 2003; Kumar & Granot 2003; Salmonson 2003) or on detectability of orphan afterglows (Lamb & Kobayashi 2017). Here, we propose a novel way to test the allowed structure of GRBs (in terms of both the energy and Lorentz factor angular distributions), by applying three independent techniques. We focus on long GRBs for which more detailed observations are available. First, we compare the predictions of these models regarding the EX/Eγ distribution (i.e. the isotropic equivalent early X-ray afterglow to prompt gamma-ray energy ratio) to the observations. We show that a variety of structure models predict large variations in this quantity, in contrast with results from GRB observations. Secondly, we reconsider the effect of the structure on the observed luminosity function and show that a large family of models can be ruled out as they lead to an overproduction of bursts with gamma-ray luminosities below the peak of the observed luminosity function. Both these considerations imply that while the energy angular profile may be steep, the Lorentz factor of GRBs must remain large at any region that produces gamma-rays efficiently. However, even such models typically lead to very peculiar light curves that can be ruled out by observations. The most likely implication is that efficient gamma-ray emission must be confined to a narrow opening angle around the jet’s core, where the isotropic equivalent energy is not much lower than that of the core. This will naturally resolve all the problems mentioned above.


The photoionization of the disc surface is characterized by two main parameters: irradiating X-ray continuum flux and disc density. Thus the measurement of the disc ionization parameter can address various aspects of the disc/corona interplay. According to Ballantyne, McDuffie & Rusin (2011), the ionization parameter of a radiation pressure supported disc illuminated by a geometrically thick corona in the SS73 model can be approximated as
(11)$$\begin{eqnarray*}
\xi &\approx & 4.33\times 10^{9}\left(\frac{\eta }{0.1}\right)^{-2}\left(\frac{\alpha }{0.1}\right)\left(\frac{r}{r_{\rm g}}\right)^{-\frac{7}{2}} \nonumber\\
&&\times \,f_{\rm c}(1-f_{\rm c})^{3}\Big (\frac{L_{{\rm bol}}}{L_{{\rm E}}}\Big)^{3}GR_{{\rm corr}},
\end{eqnarray*}$$where radiative efficiency η ≈ 0.1 (Davis & Laor 2011); viscosity parameter α = 0.1 (Shakura & Sunyaev 1973); coronal dissipation fraction fc ≈ 0.45 (Vasudevan & Fabian 2007); r is the disc radius in units of rg; $GR_{{\rm corr}}=R_{{\rm R}}^{3}R_{{\rm z}}^{-2}R_{{\rm T}}^{-1}$ is a general relativistic correction factor and is solely dependent on the dimensionless black hole spin a (e.g. Novikov & Thorne 1993; Krolik 1999). Therefore, the dependence of disc ionization log ξ on the Eddington ratio (Lbol/LE) is predominantly determined by two parameters: r and a:
(12)$$\begin{eqnarray*}
\log \xi \simeq \varPsi (r,a)\Big (\frac{L_{\rm bol}}{L_{\rm E}}\Big).
\end{eqnarray*}$$Fig. 4 shows the dependence of the disc ionization parameter on the Eddington ratio for the low-mass sample. We show the SS73 model predicted log ξ–Lbol/LE relationships for five different combinations of BH spin and inner disc radius: (a = 0.998, r = 2rg), (a = 0.998, r = 6rg), (a = 0.75, r = 6rg), (a = 0.5, r = 6rg), and (a = 0.25, r = 6rg). We noticed that the inferred ionization states of low-mass AGN discs are consistent with the SS73 model predicted solutions. The derived log ξ–Lbol/LE plane suggests that if the relativistic reflection originated from within 6rg of the inner accretion disc, then the measured ionization parameters of the low-mass sample require spins to be in the range of a ∈ [0.25, 0.998] with a median value of ∼0.75.


Set II: we now focus on the observational constraints on the model parameters after the inclusion of the local measurement of H0 by Riess et al. (2018) with the previous data sets (CMB, Pantheon, and CC) in order to see how the parameters could be improved with the inclusion of this data point. Since for this present UM, the estimation of H0 from CMB alone is compatible with the local estimation of H0 by Riess et al. (2018), thus, we can safely add both the data sets to see whether we could have something interesting. Following this, we perform another couple of tests after the inclusion of R18. The observational results on the model parameters are summarized in Table 4. However, comparing the observational constraints reported in Table 3 (without R18 data) and Table 4 (with R18), one can see that the inclusion of R18 data (Riess et al. 2018) does not seem to improve the constraints on the model parameters. In fact, the estimation of the Hubble constant H0 remains almost similar to what we found in Table 3. In order to be more elaborate in this issue, we have compared the observational constraints on the model parameters before and after the inclusion of R18 to other data sets. In Figs 7 (CMB versus CMB+R18), 8 (CMB+CC versus CMB+CC+R18), 9 (CMB+Pantheon versus CMB+Pantheon+R18), and 10 (CMB+Pantheon+CC versus CMB+Pantheon+CC+R18), we have shown the comparisons which prove our claim. One can further point out that the strong correlation between the parameters μ and H0 as observed in Fig. 5 still remains after the inclusion of R18 [see specifically the (μ, H0) planes in Figs 7–10]. The physical nature of μ does not alter at all. That means the correlation between H0 and μ is still existing after the inclusion of R18 to the previous data sets, such as CMB, Pantheon, and CC. In addition to that since μ ≲ 0.9 according to all the observational data sets, thus, the transition from past decelerating era to current accelerating era occurs to be around z ≲ 0.6, similar to what we have found with previous data sets (Table 3).


The gap and ring resemble those in the HL Tau system, recently found by the ALMA long baseline campaign (ALMA Partnership et al. 2015). Our result shows that gaps and rings in the (sub)millimeter dust continuum can exist, not only in relatively young disks (0.1–1 Myr) but also in relatively old disks (3–10 Myr). One possible mechanism for opening a gap is the gravitational interaction between a planet and the gas (e.g., Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Fung et al. 2014). Such an interaction may also produce the spiral density waves recently found in optical and near-infrared scattered light imaging of dust grains in protoplanetary disks (e.g., Muto et al. 2012). According to recent theoretical analyses of gap structure around a planet (Kanagawa et al. 2015a, 2015b, 2016), the depth and width of the gap are controlled by the planetary mass, the turbulent viscosity, and the gas temperature. The shape of the gap is strongly influenced by angular momentum transfer via turbulent viscosity and/or instability caused by a steep pressure gradient at the edges of a gap. The observed gap has an apparent width and depth of

 au and

, respectively. This is too shallow and too wide compared with that predicted by theory. However, the observations are limited to an angular resolution of ∼15 au, and the depth and width could be deeper and narrower in reality. For instance, if we assume that the gap depth times the gap width retains the value derived from the observations, it is possible for the gap to have a width and depth of

 6 au and

, which is similar to the GPI result (Rapson et al. 2015). Such a gap could be opened by a super-Neptune-mass planet, depending on the parameters of the disk, such as the turbulent viscosity (Kanagawa et al. 2015a, 2015b, 2016). If the gap in the larger dust grains is deeper than that in the gas, the planet could be lighter than super-Neptune mass. We note that a planet of even a few Earth masses, although it cannot open a gap in the gas, can open a gap in the dust distribution if a certain amount of pebble-sized particles, whose motions are not perfectly coupled to that of gas, are scattered by the planet and/or the spiral density waves excited by the planet (Paardekooper & Mellema 2006; Muto & Inutsuka 2009).


Several studies indicate that the solar wind produced by the WTD mechanism has higher AHe and vαp, while the solar wind may have lower AHe and vαp when the RLO mechanism is at work. While it is challenging to obtain direct observations of the AHe in the solar corona, first ionization potential (FIP) bias measurements are more readily available. It is found that generally the FIP bias is higher in AR and QS regions (mainly occupied by closed loops) than in CH regions (generally taken up by open magnetic field lines) (Widing & Feldman 2001; Feldman et al. 2005; Brooks & Warren 2011; Baker et al. 2013). It is believed that the reason for the enrichment of the low FIP ions in the corona and the solar wind is that they are ionized earlier in comparison to high FIP elements. The helium has the highest FIP and remains neutral longest. This results in the enrichment/depletion of low FIP elements/helium because only ions interact with waves (Laming 2012, 2015, 2017). It means that the helium abundance should be inversely proportional to the low FIP bias elements in the corona. Thus, the AHe is higher in open magnetic field structures and lower in closed loops if the above mechanism is valid. The helium abundance for the fast SW coming from large CHs is higher and remarkably stable (Schwenn 2006). In contrast, Rakowski & Laming (2012) suggested that the helium is depleted in closed loops and the depletion efficiency is higher in larger loops, and lower in smaller loops. Furthermore, from simulations, Laming (2017) showed that the AHe is higher in open magnetic field regions and it is lower in closed loops. Suess et al. (2009) found that the solar wind that comes from big streamers has lower helium abundance. The solar wind can be produced by interchange reconnection in the streamers (Huang et al. 2016). This gives the observational support to the notion that the AHe is lower in the closed loops as streamer structures are composed by very large closed loops.


The GW sources that LISA will observe at cosmological distances can be used as standard sirens. These include MBHBs, EMRIs, and SOBHBs. Unfortunately, only for the first of these types of sources are EM counterparts plausibly expected to be produced and observed by future EM facilities (Tamanini et al. 2016). MBHBs are in fact expected to emit a large amount of EM radiation in different bands at merger or during long-lasting (∼ weeks/months) afterglows (see, e.g. Palenzuela, Lehner & Liebling 2010; Dotti, Sesana & Decarli 2012; Giacomazzo et al. 2012; Moesta et al. 2012), and possibly even through pre-merger signals (Kocsis, Haiman & Menou 2008; Kaplan et al. 2011; O’Shaughnessy et al. 2011; Haiman 2017; Dal Canton et al. 2019). If sufficiently accurate sky localization can be attained from the GW parameter estimation analysis and if the EM counterpart is sufficiently powerful to be spotted by EM telescopes, then we expect to identify the host galaxy of up to a few LISA MBHB mergers per year (Tamanini et al. 2016; Tamanini 2017). These golden sources can then be used as high-redshift standard sirens to map the expansion of the Universe up to z ∼ 10. Although the low number of expected EM counterparts and the high redshift of MBHB mergers are not ideal to test standard cosmological models such as ΛCDM or to place constraints on late-time dark energy (DE) (Tamanini et al. 2016; Tamanini 2017; Belgacem et al. 2019b), they can efficiently be used to probe deviations from ΛCDM at earlier cosmological epochs, specifically in the interval 3 ≲ z ≲ 10 (Caprini & Tamanini 2016; Cai, Tamanini & Yang 2017; Belgacem et al. 2019b; Speri et al. 2020). Standard siren analyses with MBHBs would moreover definitely benefit from a network of space-based detectors, e.g. LISA and Taiji, which would greatly improve the sky location accuracy of each MBHBs and thus provide better chances to spot the EM counterpart (see, e.g. Shuman & Cornish 2021; Wang et al. 2021; Yang 2021).


Filippenko & Chornock (2001) first presented the dynamical estimate of mass of the source to be around 7.4 ± 1.1 M⊙. Recently, Radhika & Nandi (2014) claimed that the mass of XTE J1859+226 is perhaps in between 6.58 and 8.84 M⊙ which is similar to the prediction of Shaposhnikov & Titarchuk (2009), although the lower mass limit is estimated as 5.4 M⊙ by Corral-Santana et al. (2011). However, we consider the typical mass of the source as 7 M⊙. The distance of this source is around d ∼ 11 kpc (Filippenko & Chornock 2001). Steiner et al. (2013) measured the spin as ak ∼ 0.4; however, Motta et al. (2014b) recently reported that the spin of the source is ak ∼ 0.34. Since the spin predictions are quite close, we use ak ∼ 0.4 for this analysis. We estimate the fluxes Fx (see Table 1) of LHS and HIMS of the 1999 outburst of the source (Radhika & Nandi 2014). The corresponding disc luminosities are calculated as $L_{\rm disc}^{{\rm LHS}}=8.26 \times 10^{37}\ {\rm erg\ s^{-1}}$ and $L_{\rm disc}^{{\rm HIMS}}=1.85 \times 10^{38}\ {\rm erg\ s^{-1}}$, respectively. Now, it is reasonable to assume the accretion efficiency for rotating BH as η = 0.3 which corresponds to the accretion rate of the inflowing matter as ${\dot{M}}_{{\rm acc}}^{{\rm LHS}} = 0.304 {\dot{M}}_{{\rm Edd}}$ in LHS and ${\dot{M}}_{{\rm acc}}^{{\rm HIMS}} = 0.680 {\dot{M}}_{{\rm Edd}}$ in HIMS. For LHS, we use $R_{\dot{m}}=9.83$ per cent following our theoretical estimate where xs = 64.6rg for ak = 0.4, ${\mathcal {E}}=0.001\,98$ and λ = 3.18. Incorporating these inputs in equation (15), we obtain the jet kinetic power as $L^{{\rm LHS}}_{{\rm jet}} = 2.52\times 10^{37}\ {\rm erg\ s^{-1}}$. The maximum mass outflow rate for HIMS corresponding to ak = 0.4 is obtained from Fig. 9 as $R^{\rm max}_{\dot{m}}=17.5$ per cent for ${\mathcal {E}}=0.005\,47$ and λ = 3.1, where the shock transition occurs at 21.9rg. Using these values in equation (15), we obtain the maximum jet kinetic power as $L^{{\rm HIMS}}_{{\rm jet}} = 1.08\times 10^{38}\ {\rm erg\ s^{-1}}$ which we regard to be associated with the HIMS of this source.


Galaxies hosting active galactic nuclei (AGN) are not only promising candidates for providing direct evidence for such a co-evolution, they can also be used to study the scaling relations over cosmic times. Many studies suggest that BH growth precedes spheroid assembly (Treu, Malkan & Blandford 2004; McLure et al. 2006; Peng et al. 2006a,b; Shields et al. 2006; Woo et al. 2006, 2008; Salviander et al. 2007; Treu et al. 2007; Gu, Chen & Cao 2009; Jahnke et al. 2009; Bennert et al. 2010, 2011; Decarli et al. 2010; Merloni et al. 2010; Wang et al. 2010); however, other studies find no significant evolution with redshift (e.g. Shields et al. 2003; Shen et al. 2008a; Salviander & Shields 2013; Schramm & Silverman 2013; Salviander, Shields & Bonning 2015; Shen et al. 2015; Sun et al. 2015). The likely reason for the disagreement is that the studies are differently affected by intrinsic scatter in the relation, selection effects, and observational biases (see e.g. Lauer et al. 2007; Volonteri & Stark 2011; Schulze & Wisotzki 2014). Another issue is that the MBH–σ* relation is not well constrained even for local AGN host galaxies at the high-mass end. Especially for the AGN appearing as quasi-stellar objects (QSOs), the bright nuclear point source often outshines its host galaxy. Measurements of the spheroid properties are therefore difficult, in particular the stellar velocity dispersion for which high signal-to-noise (S/N) spectra are needed. Thus, many studies focus on Seyfert galaxies often using aperture spectra which integrates over the central few kpc of the host galaxy (e.g. Greene & Ho 2006; Woo et al. 2006, 2008; Treu et al. 2007; Shen et al. 2008a; Matsuoka et al. 2015). The side effect is that, if present, a significant disc contribution may be included, questioning the definition of the spheroid stellar velocity dispersion in these cases. Bennert et al. (2015) study the effect of different definitions of σ* in the literature for a sample of 66 local Seyfert-1 galaxies and find that it can vary by up to 40 per cent.


As regards the modeling of BGK modes, there are two main theoretical approaches: the integral solution or BGK methodology and the differential (or Schamel) technique. In the former method (BGK), one assumes that the initial particle distribution function and the electrostatic potential profiles are known, so these are substituted into the Poisson equation and the integral equation is solved to obtain the trapped particle distribution function (Bernstein et al. 1957; Aravindakshan et al. 2018a, 2018b, and the references therein). In Schamel’s approach, the form of the trapped particle distribution function and of the passing (i.e., free, nontrapped) particle distribution function is assumed and substituted in Poisson’s equation, leading to a differential equation that is then solved to obtain the form of the potential (Schamel 1986; Luque & Schamel 2005, and the references therein). A distinguishing factor in the former (BGK) approach is that it involves a condition in the form of an inequality to be satisfied by the potential parameters (width and amplitude) in order for a BGK mode to be sustained. The BGK approach will be adopted in this work. The above models tacitly assume a collisionless electron-ion plasma. These assumptions are acceptable in the Earth’s magnetosphere. However, as we move farther from near-Earth plasma environments, the presence of charged dust in the plasma cannot be neglected. In the case of Saturn, there are observations of streaming ions by the Cassini spacecraft (Badman et al. 2012a, 2012b). We know that these streaming ion flows can lead to the generation of ion holes. Electrostatic solitary waves have been observed in Saturn’s magnetosphere (Williams et al. 2006) and in the dusty environment near its moon Enceladus (Pickett et al. 2015). Williams et al. (2006) reported observations of solitary structures in the vicinity of Saturn’s magnetosphere. They detected a series of bipolar pulses and speculated that these could be either electron holes or ion holes (Williams et al. 2006). Later on, Pickett et al. (2015) observed solitary wave pulses within 10 Rs (Rs is the Saturn radius) and near Enceladus. Near the Enceladus plume, they discussed how dust impacts affected the observed solitary waves. In fact, Pickett et al. (2015) pointed out that some of the bipolar electric field pulses associated with the solitary waves observed had an inverse polarity (i.e., a positive pulse first, followed by a negative pulse in a short time period) and suggested that this might be due to either an inverse direction of propagation or to a true inverse potential pulse polarity (sign). Moreover, Farrell et al. (2017) examined the conditions that allow low-energy ions, such as those produced in the Enceladus plume, to be attracted and trapped within the sheath of negatively charged dust grains. Using particle-in-cell simulations, they showed that with dust in the system, the large electric field from the grain charge disrupts pickup and leads to ion trapping. Their simulation results also reveal that the bipolar pulses reported in the Enceladus plume by Williams et al. (2006) and Pickett et al. (2015) could most probably be ion holes. In the light of the above information, we may suggest that the formation of ion holes is highly likely in the dusty plasma of environments such as the one found in Saturn. Importantly, the instrument on board Cassini, the Radio Plasma Wave Science instrument, does not have the ability to determine the polarity of the electric fields associated with the ESWs observed with 100% certainty as it lacks the ability to perform interferometry (Williams et al. 2006).


In Figure 8, we also find small differences in the [Na, Al, O/Fe] abundance ratios between the stars in the bright and faint RC groups, although it is not as clear as in the case of [Na, Al, O/H] abundances. In particular, unlike Figure 7, stars in the bRC group are more enhanced in [Na/Fe] but appear to be more depleted in [Al/Fe] and [O/Fe] than those in the fRC group. The mean differences are 0.053 ± 0.021 dex, 0.032 ± 0.018 dex, and 0.071 ± 0.045 dex in [Na/Fe], [Al/Fe], and [O/Fe], respectively, which are marginally significant at p-values of 0.22, 0.18, and 0.23. When the relative fraction of RC stars is taken into account (27%; see Section 4), the difference in [Na/Fe] between the genuine RC stars would correspond to Δ[Na/Fe] ∼ 0.20 dex, which is comparable to that expected from our chemical evolution model for the bulge stars (Δ[Na/Fe] = 0.2 ∼ 0.3 dex; Kim & Lee 2018; Lee et al. 2019).10

10
The previous study by Lee et al. (2019) noted a clear separation of the two groups according to Na abundance among bright RGB stars in the outer bulge. The apparent lack of such a distinct difference between the two groups in this study may be due to a larger uncertainty on abundances of relatively faint sample stars.
 The overall chemical patterns, however, are not identical to those observed in typical GCs, where the later-generation stars are more enhanced in [Na, Al/Fe] and more depleted in [O, Mg/Fe] than the first-generation stars at a given metallicity, although the trend of [Na, Al O/Fe] between the two RCs is less clear. Figure 9 shows the comparison of stars in this study with stars in metal-rich GCs ([Fe/H] > −1.0) on the Na–O diagram. The stars used in this study have a different distribution from stars in GCs. Although the bRC group is slightly more enhanced in [Na/Fe] and more depleted in [O/Fe] than the fRC group, the [Na/Fe] variation of RC stars is smaller than that of GC stars. This discrepancy might imply the different chemical evolution between stars in the bulge and typical GCs. We note, however, that even though we employ only metal-rich GC stars for the comparison, the majority of stars are still far more metal-poor ([Fe/H]  −0.5) than stars in the bulge. Because the relatively small [Na/Fe] variation is expected from the chemical evolution model for metal-rich bulge stars and the O-depletion is indistinct in some metal-rich GCs, such as NGC 6121 and 47 Tuc (see Kim & Lee 2018; Lee et al. 2019), the direct comparison of bulge stars with similarly metal-rich GCs on the Na–O plane would require further spectroscopic observations for such GCs in the bulge.


Historically, the explanation of soft state spectra of NSs demanded the presence of a blackbody emission from the boundary layer of an NS (Mitsuda et al. 1984). For the harder states with a power-law tail in the energy spectrum, the need of Compton scattering became evident (White et al. 1986; Mitsuda et al. 1989). The difference between these two models was that while the former assumed a cooler boundary layer, the latter assumed a hotter one, compared to the accretion disc. Sunyaev and his collaborators (Inogamov & Sunyaev 1999, hereafter IS99; Popham & Sunyaev 2001, hereafter PS01; Gilfanov & Sunyaev 2014 , hereafter GS14) assume that the KD reaches all the way to the NS and is connected with the boundary layer where the thickness increases due to higher temperature. Most of these studies were done to address the soft state spectra of NSs. The state transition of NSs in Low Mass X-ray Binaries (LMXBs), presented another problem. The fact that disc accretion rate was not the single factor that controlled the size or temperature of the Compton cloud, used to model the hard state spectra, lead to the conclusion that some unknown parameter, related to the truncation radius of the disc, is responsible for the hard X-ray tail (Barret 2001; Barret & Olive 2002; Di Salvo & Stella 2002). Paizis et al. (2006) found a systematic positive correlation between the X-ray hard tail and the radio luminosity, inferring that the Compton cloud might serve as the base of radio jets (see Chakrabarti 2016, and references therein). Recent phenomenological works places a transition layer (TL) or Compton cloud between the KD and the boundary layer (Farinelli et al. 2008; Titarchuk, Seifina & Shrader 2014, hereafter TSS14). It has been argued in the past (Chakrabarti 1989; C96; Chakrabarti & Sahu 1997) that while in black hole accretion, passing of the flow through the inner sonic point ensures that the flow becomes sub-Keplerian just outside the horizon, in the case of NSs, the Keplerian flow velocity must slow down to match with the sub-Keplerian surface velocity. Numerical simulations clearly showed that jumping from a KD to a sub-Keplerian disc is mediated by a super-Keplerian region (Chakrabarti & Molteni 1995). In Titarchuk, Lapidus & Muslimov (1998, hereafter TLM98), a super-Keplerian TL was invoked to explain the kHz quasi-periodic oscillations (QPOs) and in TSS14 the TL was expanded several fold to explain the spectral properties. In reality, there are two such layers simultaneously present in an NS accretion: One is similar to the NBOL and the other is similar to the CENBOL in a black hole accretion (CT95). In a black hole accretion, only CENBOL is present. All these approaches clearly point to the existence of a CENBOL type hot electron reservoir, which naturally occurred in black hole accretion, confirming Chakrabarti & Sahu (1997) conclusions that the solutions of the transonic flows are modified only in the last few Schwarzschild radii as per the boundary condition of the gravitating object.