According to chemical models, the molecules HCS+ and CS are among the most closely related species present in the ISM, as they participate in a direct exchange in both the formation and destruction of CS (Drdla et al. 1989; Lucas & Liszt 2002). In the reaction network considered by Drdla et al. (1989) and Lucas & Liszt (2002), CS is believed to be formed primarily from the dissociative recombination reaction of (6)\begin{equation} \label{eq:HCSdisRecomb} \text{HCS}^+ + {\rm e}^- \rightarrow \text{CS} + \text{H}, \end{equation}HCS++e−→CS+H,where HCS+ is first formed by reactions beginning with S+. However, recent experiments indicate that this product channel occurs in only 19 percent of collisions, with 81 percent of collisions forming CH + S (Montaigne et al. 2005). Three mechanisms dominate the destruction of CS in these networks. In the first two, photoionization and ion-exchange reactions destroy CS to form CS+, by (7)\begin{equation} \label{eq:CSphotoIo} \text{CS} + \gamma \rightarrow \text{CS}^+ + {\rm e}^- , \end{equation}CS+γ→CS++e−,and (8)\begin{equation} \label{eq:CSionExchange} \text{CS} + X^+ \rightarrow \text{CS}^+ + X, \end{equation}CS+X+→CS++X,where X+ is a cationic species and X is the corresponding neutral species. CS+ then quickly reacts to form HCS+: (9)\begin{equation} \label{eq:CSio_recomb} \text{CS}^+ + \text{H}_2 \rightarrow \text{HCS}^+ + \text{H} . \end{equation}CS++H2→HCS++H.In the third destruction route, CS reacts with H\hbox{$_3^+$}+3 to form HCS+ directly: (10)\begin{equation} \label{eq:CSandH3plus} \text{CS} + \text{H}_3^+ \rightarrow \text{HCS}^+ + \text{H}_2 . \end{equation}CS+H3+→HCS++H2.As CS is formed by a reaction of HCS+ and the dominant destruction pathways for CS form HCS+, we would expect the abundances of CS and HCS+ to be delicately balanced, and for the two abundances to track each other. While chemical models of sulfur-bearing chemistry in diffuse clouds underpredict the abundance of HCS+ by multiple orders of magnitude, Lucas & Liszt (2002) determined that if the observed abundance of HCS+ is injected into a diffuse cloud, it is possible to account for the CS abundances observed in their sample, further emphasizing the theoretical importance of NCS/NHCS+. In clouds in the Galactic center, where H\hbox{$_3^+$}+3 is ≳10 times more abundant compared to diffuse clouds in the disk (Oka et al. 2005), we might expect an offset in the ratio, tending toward a higher relative abundance of HCS+.


Simultaneously solving the MHD equations and the global angle- and energy-dependent radiation transport equation, in general relativity, is both computationally expensive (typically prohibitively so) and technically challenging. Even so, significant progress has been made in the last decade, though the problem is usually made tractable by introducing at least one of the following simplifying assumptions: abandoning general relativity in favor of a pseudo-Newtonian description of the gravitational potential while performing realistic, multi-angle group radiation transport (Jiang et al. 2014a, 2014b, 2019a, 2019b); limiting the possible angular dependence of the radiation field by invoking either flux-limited diffusion (Zanotti et al. 2011; Roedig et al. 2012) or, more recently, the “M1 closure” relation in either axisymmetric (2D; Sadowski et al. 2014) or 3D simulations (Fragile et al. 2012, 2014; McKinney et al. 2014; Sadowski et al. 2016); or using Monte Carlo (Ryan et al. 2015) / hybrid Monte Carlo techniques (Ryan & Dolence 2020). Most attempts have eschewed energy-dependent transfer in favor of a “gray” atmosphere—the radiation field is treated as monochromatic, coupled to the fluid only through the Rosseland mean opacity (Rybicki & Lightman 1986). The first of these approximations, the pseudo-Newtonian potential, is especially problematic in regions close to the black hole, where general relativistic effects play a critical role in determining both the structure of the accretion flow and the photon trajectories. The others are essentially variants of a diffusion approximation and are best suited to the cooler, denser, and optically thick body of the accretion disk, where the environment is similar to those found in stellar atmospheres—the field from which these methods, and gray transfer, originate (Chandrasekhar 1960). With the exception of Monte Carlo methods (Ryan et al. 2015; Ryan & Dolence 2020), these are all especially poorly suited to the diffuse, hot, optically thin corona, especially at small radii near the black hole.


At each heliocentric distance rh, the activity model (Fulle et al. 2020b) is defined by five analytical equations fixing (i) the gas pressure P(s) depending on the depth s from the nucleus surface (Fig. 1 for the CO2 case), (ii) the gas flux Q from the nucleus surface, (iii) the temperature gradient ∇T at depths of a few cm, (iv) the heat conductivity λs at depths of a few cm below the nucleus surface, and (v) the temperature Ts of the nucleus surface
(3)$$\begin{eqnarray*}
P(s) = P_0 ~f(s) ~\exp \left[{- {T_0 \over {T_s - s ~\nabla T}}}\right]
\end{eqnarray*}$$(4)$$\begin{eqnarray*}
Q = {{14 ~r ~P(R)} \over {3 ~R}} \sqrt{{2 ~m} \over {\pi k_B ~(T_s - R ~\nabla T)}}
\end{eqnarray*}$$(5)$$\begin{eqnarray*}
\nabla T = {\sqrt{\Lambda ~Q ~/ ~\sigma _B} \over {8 ~(T_s - R ~\nabla T) ~R}}
\end{eqnarray*}$$(6)$$\begin{eqnarray*}
\lambda _s = {32 \over 3} ~(T_s - R ~\nabla T)^3 ~\sigma _B ~R
\end{eqnarray*}$$(7)$$\begin{eqnarray*}
(1 - A) ~I_\odot ~\cos \theta ~r_h^{-2} = \epsilon \sigma _B T_s^4 + \lambda _s ~\nabla T + \Lambda ~Q
,
\end{eqnarray*}$$where P0, T0, and Λ values are listed in Table 1, s is the depth from the nucleus surface, $f(s) = 1 - (1 - {s \over R})^4$ for s ≤ R, f(s) = 1 elsewhere, r ≈ 50 nm and R ≈ 5 mm are the radii of the grains of which cometary dust consists (Levasseur-Regourd et al. 2018; Güttler et al. 2019; Mannel et al. 2019) and of the pebbles of which cometary nuclei consist (Blum et al. 2017; Fulle et al. 2020b), m is the mass of the gas molecule, kB is the Boltzmann constant, σB is the Stefan–Boltzmann constant, A is the nucleus Bond albedo (e.g. A = 1.2 per cent measured at 67P; Fornasier et al. 2015), I⊙ is the solar flux at the heliocentric distance of Earth, θ is the solar zenithal angle, and ϵ ≈ 0.9 is the nucleus emissivity. Since the gas originates from the superficial pebbles and is assumed to share the temperature Ts − s ∇T of refractories and ices, the thermal diffusion due to gas convection is negligible with respect to the sublimation sink Λ Q. A nucleus is active if the gas pressure overcomes the tensile strength S bonding dust particles to the nucleus surface (Skorov & Blum 2012), thus defining the activity onset for each ice (Table 2), occurring (i) at rh = 85 au for carbon monoxide (Fulle et al. 2020a); (ii) at rh = 60 au for molecular oxygen; (iii) at rh = 52 au for methane; (iv) at rh = 18 au for ethane; (v) at rh = 13 au for carbon dioxide (dotted line in Fig. 1); and (vi) at rh = 3.8 au for water (Fulle et al. 2020b; Ciarniello et al. 2021). The value R ≈ 5 mm has been constrained by several data collected at comet 67P, by laboratory experiments of dust accretion in conditions expected to occur in the solar protoplanetary disc and by observations of other protoplanetary discs (Blum et al. 2017). Other R-values would not provide the best fit of the 67P water-loss time-evolution (Ciarniello et al. 2021).


As we said in the previous section, the evolution of the magnetic fields in the two domains (the disk and the magnetosphere) are related. It is important to note here that there is an important complication that enters into the matching between the two domains and this has to do with the abrupt transition between the dense turbulent accreting flow and the almost empty ionized (probably also evaporating and outflowing) magnetosphere. This is very similar to the well known transition region in the Sun where, within a very thin layer of about 100 km, the temperature rises by almost two orders of magnitude to about one million degrees, and the matter density drops by about one order of magnitude. The solar transition region is the subject of ongoing intense theoretical and numerical investigations. The astrophysical disk transition region is at least equally complicated since it is not only the base of the disk corona, but also the origin of purported disk winds and outflows (e.g., Ferreira & Pelletier 1995; Li 1995). It is not the aim of the present study to solve in detail for the structure of the flow and magnetic field in this region. In practice, we define a transition region of two θ-grid zones above

. In that region we set
14

This matching allows for magnetic field loops that reach the disk surface zones to escape to the magnetosphere. We also set the field lines in that region in rotation by introducing a magnetospheric poloidal electric field
15

Finally, we slightly modify the induction equation that we solve for the disk material in that region by introducing an extra phenomenological term that mimics various complex physical effects taking place in the surface layers of astrophysical accretion disks such as surface convection, buoyancy of magnetic field loops, and the fact that the surface layers are highly ionized (thus also fully conducting) due to cosmic ray irradiation, namely
16

where,

 is the disk magnetic field in the transition layer. The introduction of this transition layer effectively shields the disk interior: (a) it prevents any generated magnetospheric field from “entering” the disk from above (especially in the case of turbulent high η flow conditions), and (b) it “pushes outward” (in the θ direction) at a very small fraction of the speed of light any magnetic field loop that enters it from below. We implemented different fractions (

,

, etc.) and the results were qualitatively similar. As we said above, we are not claiming that we study in detail the disk-magnetosphere transition region. This is the reason we opted for a minimal two θ-zone layer that only serves the purpose of shielding the disk interior from the disk magnetosphere. Note that Parfrey et al. (2015) ignored the important physical significance of a transition layer by directly coupling their magnetosphere to a prescribed distribution of the magnetic field on the surface of the accretion disk without solving for the magnetic field distribution in the disk interior.


We note that it is unlikely that poloidal magnetic fields are responsible for the observed RMs of M87 because in that case one expects ρ ∝ r0 from B ∝ r−2, which is impossible to explain with the accretion models currently available (Yuan & Narayan 2014). However, there is indication of non-negligible poloidal fields as well as toroidal fields—resulting in helical magnetic fields—in the jet environment of other distant AGNs, which results in transverse RM gradients with no sign changes (e.g., Asada et al. 2002; Zamaninasab et al. 2013; Gómez et al. 2016; Gabuzda et al. 2018, see also Section 3.5.2). The existence of non-negligible poloidal fields was indicated even for the M87 jet at HST-1 from the observed moving knots with both fast and slow velocities, which could be explained by quad relativistic MHD shocks in a helical magnetic field permeating the jet (Nakamura et al. 2010; Nakamura & Meier 2014). In Sections 3.5.2 and 4.1, we explained that poloidal magnetic fields might be very weak at distances ≳5000 rs probed in this study and we concluded that hot accretion flows and winds are more probable to be the Faraday screen than the jet sheath. However, if the jet experiences recollimation, which may lead to formation of standing shocks (e.g., Daly & Marscher 1988; Gómez et al. 1995; Agudo et al. 2001; Mizuno et al. 2015; Martí et al. 2016; Fuentes et al. 2018), then the strength of poloidal fields could be substantially enhanced. Indeed, the width of HST-1 is significantly smaller than expected from the parabolic (conical) width profile inside (outside) the Bondi radius (Asada & Nakamura 2012), which has been explained with a hydrodynamic recollimation shock (e.g., Stawarz et al. 2006; Bromberg & Levinson 2009; Asada & Nakamura 2012). Also, the core of blazars is often identified with a recollimation shock (e.g., Daly & Marscher 1988; Marscher 2008; Cawthorne et al. 2013). This may explain the presence of non-negligible poloidal fields in the sheath of blazar jets and in HST-1, but not in the M87 jet inside the Bondi radius.


However, a significant outstanding problem of using red giants is that modeling their individual frequencies is too time consuming for the analysis of tens of thousands of stars. We therefore rely on using asteroseismic scaling relations,

ν

max

∝

gT

eff

−
1

/

2

 and

Δ
ν
∝
ρ

 (Brown et al. 1991; Kjeldsen & Bedding 1995), to estimate their radius and mass (and hence age). Here,

ν

max

 is the frequency of maximum amplitude and

Δ
ν

 the average large frequency separation. These relations assume that the structure of a red giant star is homologous with respect to the Sun. In reality this assumption is not strictly correct and verification of the relations is ongoing. However, the independent high-precision estimates of mass and radius required for this verification are difficult to obtain. For subgiants and dwarfs, the

ν

max

 scaling relation has been shown to work well (Bedding 2014) and recently, Coelho et al. (2015) found the proportionality

ν

max

∝

gT

eff

−
1

/

2

 to be accurate to within 1.5%. Using Hipparcos parallaxes and/or interferometry, the asteroseismic radii calculated from scaling relations have been found to be accurate to within 5% (Bruntt et al. 2010; Huber et al. 2012; Silva Aguirre et al. 2012). For giants we generally do not have accurate parallaxes, so such studies are awaiting results from Gaia (Perryman 2002). Open clusters have been used to test the scaling relations for giants (Brogaard et al. 2012; Miglio et al. 2012; Sandquist et al. 2013). Miglio et al. (2012) found agreement to within 5% for scaling relation-based radii. Testing of masses is more challenging. For a few cases where such verification have been performed, the scaling relation-based masses seem to be overestimated for giants (Miglio et al. 2012; Frandsen et al. 2013; Epstein et al. 2014). For two lower red giant branch stars (Epstein et al. 2014) find evidence that the mass estimated by using only

Δ
ν

 (but with additional

ν

max

 independent quantities) is lower compared to using both

Δ
ν

 and

ν

max

. Based on this they suggested that a modification to the

ν

max

 scaling relation might be required. Theoretical modeling has suggested corrections to the

Δ
ν

 scaling relation (Stello et al. 2009; White et al. 2011; Miglio et al. 2013a), but there has been no comprehensive study to verify the corrections. In relation to

ν

max

, Houdek et al. (1999) and Chaplin et al. (2008) had suggested theoretically that

ν

max

 coincides with the plateau of the damping rate with frequency. Belkacem et al. (2011) confirmed this for the Sun using SoHO GOLF observations. Balmforth (1992) suggested that this is caused by a resonance between the thermal adjustment time of the superadiabatic boundary layer and the mode frequency, which was also confirmed by the theoretical study of Belkacem et al. (2011; see also Belkacem 2012; Belkacem et al. 2013). However, there is currently no way to accurately predict

ν

max

 from theory.


This paper presents a new 3D RT code, RAdiation SCattering in Astrophysical Simulations (RASCAS), which was designed to construct accurate multi-wavelength mock observations (spectra, images, or datacubes) from high-resolution simulations. RASCAS deploys a general two-step methodology (e.g. Hummels et al. 2017; Barrow et al. 2017). The first step consists in extracting all relevant information from the simulation outputs: (1) the information concerning the medium through which light will propagate, and (2) the information concerning the sources of radiation. Point (1), for example, determines the number density of H I atoms, their thermal velocity dispersion and bulk velocity, and the dust density, everywhere in a chosen volume. Fully coupled radiation-hydrodynamic simulations would naturally provide the information about ionised states, but retrieving this information in pure hydrodynamic simulations may be tricky. In such case, it is necessary to process the simulation outputs with additional software and models, typically with CLOUDY (Ferland et al. 2013), as in Hummels et al. (2017) and Barrow et al. (2017), or with independent codes which solve for H and He ionisation by propagating ionising radiation in post-processing (e.g. Li et al. 2008; Yajima et al. 2012). Point (2) is also involved to a greater or lesser extent, depending on the sources. Computing the continuum emission from star particles is relatively straightforward, using spectro-photometric models of stellar populations (e.g. Bruzual & Charlot 2003; Eldridge et al. 2017). However, computing the emission lines from the gas (e.g. in the Lyman-α line or in other nebular lines) again requires a detailed knowledge of the ionisation and thermal state of the emitting species. This may be provided by the simulation code, as is the case for H and He with RAMSES-RT (Rosdahl et al. 2013), or it may be necessary to post-process the simulation to estimate the emissivities of the gas. This first step is very simulation- and model-dependent, and RASCAS chooses to encapsulate it in a simulation-plugin module and to implement two stand-alone pre-processing codes which generate an adaptive mesh with all the needed physical information about the gaseous medium, and the initial conditions for light emission in the form of lists of photon packets. These datasets, which could easily be generated from other simulations with any post-processing code, serve as inputs to RASCAS to perform the radiative transfer computation.


We note that Pavesi et al. (2019) derived a lower κs parameter log(κs) ≈ −1 for HZ10, by observing the CO(2-1) line, which implies a low star formation efficiency for this source. The conflict between the two results can be explained by the fact that Pavesi et al. (2019) estimated the gas mass by adopting a large CO-to-Mgas conversion factor αCO = 4.5 M⊙ (K km s−1 pc2)−1, a value that is close to the Galactic conversion factor αCO = 4.36 M⊙ (K km s−1 pc2)−1 (Bolatto et al. 2013). Although the Galactic conversion factor is a derived value for Milky Way and normal, star-forming galaxies in the local Universe, it may not be applicable for more extreme environments of starburst galaxies at high-z (see Carilli & Walter 2013 for a review). The conversion factor depends on the physical conditions of the gas in the ISM (temperature, surface density, dynamics, and metallicity), as well as the star formation and associated feedback (Narayanan et al. 2011, 2012; Genzel et al. 2012; Feldmann et al. 2012; Renaud et al. 2019; see, e.g., Bolatto et al. 2013 for a review). It is typically in the range between 0.8 and 4.36 M⊙ (K km s−1 pc2)−1 (see, e.g., Bolatto et al. 2013; Carilli & Walter 2013, and Combes 2018 for reviews). Low metallicities (Z = 0.6 Z⊙ for HZ10) will drive αCO towards values higher than the Galactic value (Narayanan et al. 2012; Genzel et al. 2012; Popping et al. 2014), although αCO spans a broad range of values of αCO ∼ 0.4 − 11 M⊙ (K km s−1 pc2)−1, due to large uncertainties (see, e.g., Fig. 9 of Bolatto et al. 2013). On the other hand, high values of temperature, surface density, and velocity dispersion in a turbulent ISM of starbursts and merging systems will shift αCO towards lower values (Narayanan et al. 2011, 2012; Vallini et al. 2018). HZ10 has an extremely high value of the burstiness parameter log(κs) ∼ 1.4 and high total density of the [C II] emitting gas log(n) ∼ 3.35 cm−3, and it is also a multi-component system (Jones et al. 2017, Carniani et al. 2018a). Thus, for this source we assumed αCO = 0.8 M⊙ (K km s−1 pc2)−1, usually adopted for starburst galaxies (e.g., Downes & Solomon 1998; Bolatto et al. 2013). We obtained log(κs) = 0.53 ± 0.34, which is within the 2σ uncertainties of the

log

(

κ
s

)

=
1
.

43

−
0.53

+
0.38

$ \log{(\kappa_s)} = 1.43_{-0.53}^{+0.38} $

, estimated exploiting the C III] emission and the Vallini et al. (2020) model.


We validated our results against different selection effects due to redshift, luminosity, incompleteness of the FIRST survey, and incomplete radio counter-part identifications. However, we have not explicitly checked for any bias due to incomplete target selection in the DR16Q catalogue. As quasars are chosen for spectroscopic observations in SDSS primarily based on their photometric properties, our analysis can be biased due to the differential target selection completeness for BALQSOs and non-BALQSOs. As BALQSOs are significantly dust-reddened compared to non-BALQSOs, the completeness of BALQSOs is expected to be less. Allen et al. (2011) computed this difference in the selection probabilities using simulated BAL and non-BALQSO magnitudes and found that the completeness is very high for both BALQSOs and non-BALQSOs with z  2.1 and i  19.1, drops sharply for a narrow region close to z ∼ 2.6 and rises again for higher redshifts. This completeness difference can introduce some bias for analysis involving optically selected samples. To our rescue, eBOSS also targets all SDSS point sources that are within 1 arcsec of a radio detection in the FIRST point source catalogue (Myers et al. 2015). As our final sample is radio-selected BAL quasars and non-BAL quasars, we do not expect the behaviour of the BAL fraction to be severely influenced by any incompleteness effects. As with the case of dust-reddening, strong absorption troughs can make the broadband magnitude appear fainter for a BALQSO than an equivalent non-BALQSO. C iv BAL troughs fall in the SDSS i band for the redshift range 3.5  z  4.4. Our BAL quasar sample contains 82 quasars (4 per cent) in the redshift range where the i band luminosities are underestimated due to the presence of C iv BAL absorption. Additionally, a small fraction of quasars (1 per cent) should be LoBALs and FeLoBALs where the i-band luminosities are affected by Mg ii, Al iii, and Fe ii broad absorption. The orientation indicator, radio-to-optical core luminosity has the i-band optical luminosity in the denominator. Although the fraction of BALQSOs containing strong BAL features within the i-band wavelengths is less, any correction of the optical luminosities will further push these sources to lower log(RI) bins, thereby increasing the BAL fraction at higher orientation angles. Likewise, Hewett & Foltz (2003) suggest that optically bright BAL quasars are half as likely as non-BAL quasars to be detectable as S$_{1.4 \,\mathrm{ GHz}}\, \ge$ 1 mJy sources. To probe this, we divided the sample into three optical luminosity bins, (i) Log(Li band) ≤ 24 W/Hz, (ii) 24 W/Hz  Log(Li band) ≤ 25 W/Hz, and (iii) Log(Liband) > 25 W/Hz, and studied the distribution of radio luminosity of BAL and non-BAL quasars. We do not see any difference in the distribution of radio luminosity between BAL and non-BAL quasars for the lower optical luminosity bins. But, for the highest luminosity bin, we see that the BAL quasars have statistically lower radio luminosities as compared to the non-BAL quasars (p-value of KS test  0.05). Thus, a flux-limited survey like FIRST would have missed a fraction of optically bright BAL quasars that have radio fluxes below the FIRST sensitivity. As these missed BAL quasars have low radio luminosities and high optical luminosities, a correction for the missed quasars would only populate the lower log(RI) bins. This again will amplify the already seen trend of BAL fraction increase at high orientation angles. To ensure that the optically bright sources are not biasing the BAL fraction trend, we excluded the optically bright sources from our sample and then studied the variation of BAL fraction as a function of orientation. The pruned sample also follows the trend of high BAL fraction for high orientation angles.


Flattened distributions of very likely co-orbiting satellite galaxies around the MW (Lynden-Bell 1976, 1982; Kroupa, Theis & Boily 2005; Pawlowski, Pflamm-Altenburg & Kroupa 2012) and M31 (Metz, Kroupa & Jerjen 2007; Ibata et al. 2013) have long posed a challenge to our understanding of galaxy formation in the ΛCDM context. Recent proper motion data confirm that most of the classical MW satellites do indeed have a common orbital plane (Pawlowski, Kroupa & Jerjen 2013; Pawlowski & Kroupa 2020) aligned with the plane normal defined by the satellite positions alone (Santos-Santos, Domínguez-Tenreiro & Pawlowski 2020). Their velocities show a very significant bias towards the tangential direction, as occurs for a rotating disc (Cautun & Frenk 2017). Proper motions of two M31 satellite plane members indicate that this structure is likely also coherently rotating (Sohn et al. 2020), as suggested by the RVs of satellites in this nearly edge-on structure (Ibata et al. 2013). After careful consideration of several proposed scenarios for how primordial CDM-rich satellites might end up in a thin plane, Pawlowski et al. (2014) concluded that none of them agree with observations for either the MW or M31. Structures as extreme as those observed are exceedingly rare in cosmological simulations (Ibata et al. 2014; Pawlowski & McGaugh 2014b), including hydrodynamical simulations (Ahmed, Brooks & Christensen 2017; Shao, Cautun & Frenk 2019; Pawlowski & Kroupa 2020) and simulations which model the effects of a central disc galaxy (Pawlowski et al. 2019). The arguments raised by Metz et al. (2009) and Pawlowski et al. (2014) against the group infall and filamentary accretion scenarios were later independently confirmed by Shao et al. (2018) using the EAGLE hydrodynamical cosmological simulation (Crain et al. 2015; Schaye et al. 2015). For a recent review of the satellite plane problem, we refer the reader to Pawlowski (2018), who considered both LG satellite planes and the recently discovered one around Centaurus A (Müller et al. 2018, 2021).


The cosmic web, consisting of haloes, voids, filaments, and walls in large-scale structure is predicted by the cold dark matter model (Bond, Kofman & Pogosyan 1996; Pogosyan et al. 1998) and confirmed by large galaxy surveys (e.g. de Lapparent, Geller & Huchra 1986; Colless et al. 2003; Alam et al. 2015). Among these large-scale structures, the underdensities of the universe, i.e. cosmic voids, have been shown to have great potential for constraining dark energy and testing theories of gravity via several measurements. These measurements include: distance measurement via the Alcock–Paczyński test (Ryden 1995; Lavaux & Wandelt 2012; Sutter et al. 2014), weak gravitational lensing of voids (Krause et al. 2013; Clampitt & Jain 2015; Melchior et al. 2014; Gruen et al. 2016; Sánchez et al. 2016), the signal of the integrated Sachs–Wolfe effect associated with voids (Sachs & Wolfe 1967; Granett, Neyrinck & Szapudi 2008; Nadathur, Hotchkiss & Sarkar 2012; Flender, Hotchkiss & Nadathur 2013; Ilić, Langer & Douspis 2013; Cai et al. 2014; Planck Collaboration XIX 2014; Aiola, Kosowsky & Wang 2015; Kovács & Granett 2015; Planck Collaboration XXI 2015), void ellipticity as a probe for the dark energy equation of state (Lee & Park 2009; Lavaux & Wandelt 2010; Bos et al. 2012; Pisani et al. 2015; Sutter et al. 2015), void abundances and profiles for testing theories of gravity and cosmology (Li, Zhao & Koyama 2012; Clampitt, Cai & Li 2013; Barreira et al. 2015; Cai, Padilla & Li 2015; Lam et al. 2015; Massara et al. 2015; Zivick et al. 2015), coupled dark energy (Pollina et al. 2016), the nature of dark matter (Yang et al. 2015), baryon acoustic oscillations in void clustering (Kitaura et al. 2016; Liang et al. 2016), and redshift-space distortions in voids (Hamaus et al. 2015, 2016; Cai et al. 2016). Despite their popularity and great potential as a cosmological tool, a gap of knowledge between the evolution of individual voids through simulations and observations versus theory still persists. How voids evolve from the initial conditions and how dark energy or alternative theories of gravity shape this process still lacks a complete analytical understanding. As with the formation history of haloes, the initial conditions and evolution history of voids sets the base for their two fundamental properties: profile and abundance. As these are crucial for constraining cosmological parameters, it is therefore important to bridge the gap between theory and observations. This is the main goal of our study.


Scheeres (2007) and Nesvornỳ & Vokrouhlickỳ (2007) established averaging methods for YORP effects experienced by small bodies (mostly asteroids) taking real-time insolation and shape into account. However, unlike thermal radiation, as Fig. 2 shows, the CO (or H2O) production rate variation versus solar distance is too complex to be written into a linear or polynomial formula (over the entire range we want to investigate). Sidorenko, Scheeres & Byram (2008) applied the averaging method to obtain evolutionary equations in order to study the long-term variations in the nucleus spin state induced by outgassing torques, but the illumination variation is not considered in their simplified model. There are further analytical approaches described in the literature (Samarasinha & Mueller 2013; Steckloff & Jacobson 2016; Steckloff & Samarasinha 2018), although these models predicted a change in the spin state of the nucleus without taking into consideration the exact 3D shape model and its variation induced by mass loss. Therefore, to our knowledge, there exists no approximate framework that is able to combine spin and mass loss together with orbital evolution for the 3D shapes (accounting for shadowing and self-heating) we are exploring in this paper. Currently, only fully numerical schemes are appropriate to investigate sublimation torques in our context. At the same time, it is not feasible to include such numerically expensive considerations when exploring the parameter space of orbits, mass-loss function and axis orientations. Therefore, in this work, we consider the rotation of the nucleus only for the stable case, without taking into consideration the wobbling rotation excitation and the change in spin states as a result of sublimation torque. However, we plan to perform these in the future, relying on the knowledge gained from physically and scientifically interesting cases in this work. As a general remark, the spin (and/or orientation) changes induced by sublimation torque are not expected to play a role only if the orientation variation time-scale is significantly larger than the time-scale of shape modification. However, if the time-scale is significantly shorter, new results are expected as the variation period of θ and φ might bring new periodical effects. A tumbling spin state with large procession angle or even a chaotic rotation may average the mass loss on the surface to yield an ‘averaged’ shape as a result. Therefore, if a nucleus exhibits an asymmetry shape believed to be driven by sublimation activity, it would indicate that the nucleus rotates in a stable stage for a long-term period in its dynamical history. In other words, this implies that the time-scale of wobbling excitation of such objects might be much longer than the one for shape modification.


The temperature of the IGM can be measured using the effect of Doppler broadening on the forest of Ly α absorption line seen in the spectra of high-redshift quasars and galaxies (Rauch 1998; Savaglio, Panagia & Padovani 2002). Traditionally, the absorbing clouds were treated as being distinct and fitting Voigt profiles (e.g. van de Hulst & Reesinck 1947; Carswell & Webb 2014; Webb, Carswell & Lee 2021) to features seen in spectra leads to two parameters, column densities and temperatures (e.g. Hu et al. 1995). However, since the identification of the Ly α forest in hydrodynamic simulations (Cen et al. 1994; Zhang, Anninos & Norman 1995; Hernquist et al. 1996), it was realized that in the context of cosmological models for structure formation, the Ly α forest is due to a fluctuating, continuous IGM (e.g. Bi 1993; Weinberg et al. 2003). Because of this, Hubble flow is the major source of broadening, and fitting the width of features with a thermally broadened profile of a discrete object will not yield the temperature directly. Many other techniques have therefore been developed to constrain the IGM temperature from the Ly α forest. One is the relatively straightforward use of Voigt profile fit parameters to try to find the envelope that corresponds to a minimum IGM temperature (e.g. Garzilli, Theuns & Schaye 2020). Others include wavelet transforms (Meiksin 2000; Garzilli et al. 2012), or measuring statistical properties of the Ly α forest flux, such as the power spectrum (Croft et al. 1998; Lai et al. 2006), which respond to the small-scale smoothing that results from increased IGM temperatures. More recent ideas include the reconstruction techniques of Müller, Behrens & Marsh (2021) that use 3D tomographic data. In the current work, we will use a neural network (NN, see below) trained on simulated Ly α spectra to infer the temperature from the smoothness of absorption features. We will be looking in particular for discontinuous T jumps that could result from discrete sources of heating such as quasars reionizing He ii. We note that the temperature of gas will also influence the pressure smoothing scale (Gnedin & Hui 1998), which physically makes gaseous structures smoother than their underlying dark matter counterparts (Peeples et al. 2010). In our current preliminary work, we will not model this effect (see also the approach of McQuinn et al. 2011, who used a similar approximation), so that the smoothing of absorption features will be entirely attributed to Doppler broadening.


The origin of giant pulses has been remaining a mystery since the discovery of giant pulses from Crab pulsar (Staelin & Reifenstein 1968). The generation of giant pulse activity was pointed to be an intrinsic phenomenon within the pulsar (Hankins 1971). The giant pulses are supposed to be the product of induced Compton scattering of the radio radiation off the plasma in the pulsar magnetosphere (Petrova 2006). The extremely high intensity is as well caused by an enhanced number of charges partaking in the nonthermal, coherent radiation processes (Hankins et al. 2003). Alternatively, the origination of giant pulses is proposed from the coherent instability of plasma near the magnetic equator of light cylinder (Wang et al. 2019). Singal & Vats (2012) suggested that the giant pulse emission and nulling may be opposite manifestations of the same physical process. The giant pulses are suggested to occur in pulsars with extremely high magnetic fields at the light cylinder of BLC > 105 G (Cognard et al. 1996). Therefore, the giant pulses are proposed to originate near the light cylinder (Istomin 2004). However, the giant pulses are also detected in the pulsars with ordinary magnetic fields at the light cylinder of BLC  100 G, such as PSRs B0031−07 (Kuzmin & Ershov 2004), B1112+50 (Ershov & Kuzmin 2003), J1752+2359 (Ershov & Kuzmin 2005), B0950+08 (Smirnova 2012), B0656+14 (Kuzmin & Ershov 2006), B1237+25 (Kazantsev & Potapov 2017), and B0301+19 (Kazantsev et al. 2019), and it does not seem to support the high BLC hypothesis. The Vela giant micropulse emission physics maybe independent on the high magnetic field at the light cylinder. Although Vela’s BLC is about 20 times smaller than that of PSR B1937+21 and the Crab pulsar, it is still in the top 5% of pulsars with BLC estimate. The giant pulses from PSR J1824−2452A occur in narrow phase windows that correlate in phase with X-ray emission, and the two emission phenomena likely originate from the similar magnetospheric regions but not the same physical mechanism (Knight et al. 2006). In order to reveal the nature of the giant micropulses, simultaneous radio and X-ray observations on the Vela pulsar will be required.


We constructed a sample of artificial stars representing the TT and HAeBe regime by using synthetic models of stellar atmospheres (Kurucz 1993). The properties of each object are provided in Table 1. Columns two and three show the stellar luminosity and effective temperature. From these, the stellar radii was derived, spanning between 0.7 and 4 R⊙ (column 4). The stellar masses (column 5) were derived assuming log g = 4, and cover the 0.2–6 M⊙ range. Magnetospheric accretion (MA) shock modelling was carried out for each star by adding (blackbody) accretion contributions to the photospheric (Kurucz) spectra (see e.g. the reviews in Calvet, Hartmann & Strom 2000; Mendigutía 2013). Two representative examples are presented in Fig. 2 (left-hand panel). The shock model was applied following the usual recipes for both the TTs and HAeBes, and we refer the reader to Calvet & Gullbring (1998); Mendigutía et al. (2011) and Fairlamb et al. (2015) for further details. Three different values for the UV excess in the Balmer region of the spectra (from ∼3500 to 4000 Å, as defined in Mendigutía et al. 2013) were modelled for each object assuming typical values for the inward flux of energy carried by the accretion columns (1012 erg cm2 s−1) and the disc truncation radius (5R*): a ‘maximum’ excess (0.70 mag), whose corresponding accretion contribution is Lacc ∼ L* for L* ≥ L⊙; a ‘minimum’ excess (0.01 mag) representative of the observational limit, and whose corresponding accretion contribution is Lacc ∼ 0.01L* for L* ≥ L⊙; and finally, a ‘typical’ excess in-between the two previous (0.12 mag). The resulting accretion luminosities are shown in the last three columns of Table 1. These are plotted versus the corresponding L* values (blue diagonal dashed lines in Fig. 1), matching the overall distribution of data. We note that excesses larger than 0.70 mag could still be measured for the less luminous sources (L* ≤ L⊙) without reaching the upper bound (Lacc ∼ L*).


Coronal loops are curvilinear structures in the outer layer of the solar atmosphere, and they are formed by thermal plasmas confined by magnetic fields and well reflect the coronal magnetic field configurations. Because a number of active phenomena in the solar atmosphere, such as flares, the solar wind, coronal mass ejection (CME) and coronal oscillation, are associated with the coronal magnetic field, instead of direct observation of the coronal magnetic field, solar physicists are able to indirectly obtain the structure and evolution of the coronal magnetic field by observing and studying the kinetic characteristics of coronal loops. On the other hand, when flares and CME occur, the coronal magnetic field structure will undergo drastic changes, which will result in the oscillation of coronal loops (Li, Liu & Tam 2017). Therefore, on the basis of the oscillation of coronal loops, especially their oscillation characteristics in frequency, phase and attenuation coefficient, solar physicists can further understand the relationships between the coronal magnetic field, flares and CME (Aschwanden et al. 1999; Jain, Maurya & Hindman 2015) and obtain corresponding physical parameters associated with oscillating coronal loops (Aschwanden & Schrijver 2011). Meanwhile, the study on coronal loops is also useful for us to well understand the properties of coronal plasmas (Parnell & De Moortel 2012), plasma motion inside coronal loops (Winebarger, DeLuca & Golub 2001; Uritsky et al. 2013; Zhou & Liang 2016), the propagation of MHD waves in coronal loops (Wang, Ofman & Davila 2013), the evolution of active regions (Yan et al. 2012) and coronal heating (Aschwanden et al. 2007a; Aschwanden & Peter 2017). In addition, in modelling the coronal magnetic field, the observed coronal loops can be taken as a constraint condition so as to obtain more realistic extrapolated results of the non-linear force-free (NLFF) magnetic field model (Su et al. 2009; Aschwanden 2013). In particular, along with the enrichment of observational data in multiple wavebands, the classification and study of different temperatures and densities of the coronal loops will be further expanded, and the meaningful physical mechanism of these phenomena will also be obtained (Aschwanden & Boerner 2011; Huang et al. 2012; Aschwanden et al. 2013a). However, it must be emphasized that the current observational data of coronal loops are just a two-dimensional (2D) projective result of the real shape of coronal loops on to the plane of sky. Therefore, to obtain the genuine shape of coronal loops, a three-dimensional (3D) reconstruction of coronal loops is needed (Aschwanden et al. 2008b). It is desirable to reconstruct the real shape of coronal loops, resulting from the fact that identical coronal loops from the 2D observation data of multiple viewing angles (the observation data of the Solar Terrestrial Relations Observatory (STEREO), Uritsky et al. 2013) can be detected and extracted.


EE Cet (ADS 2163 B) is the southern (slightly fainter) component of the visual binary WDS 02499+0856 (Mason et al. 2001). It was discovered by the HIPPARCOS mission (Perryman et al. 1997), by noticing the variability of the combined light of both visual components. Lampens et al. (2001) performed photometric measurements of the visual pair and gave (but only for one epoch) the following values V(A) = 9.47 mag and V(B) = 9.83 mag. Pribulla & Rucinski (2006) lists the orbital parameters (orientation and separation) of WDS 02499+0856 and gave θ = 194°, ρ = 5.66 arcsec and magnitude difference ΔV = 0.07 mag (the magnitude difference can be as large as ΔV = 0.36 mag, due to photometric variability of the eclipsing binary). WDS 02499+0856 turned out to be a quadruple system, when the northern component was found to be a double-lined (SB2) binary from the DDO spectroscopic observations (Pribulla & Rucinski 2006). D’Angelo et al. (2006) re-confirmed the multiplicity of the system, and listed it among the contact binaries with additional components. Radial velocity observations from Rucinski et al. (2002) resulted in a well-defined circular orbit of the contact binary, with K1 = 84.05 km s−1, K2 = 266.92 km s−1 (q = 0.315), and an F8V spectral type. Karami & Mohebi (2007) using their own velocity curve analysis method, arrived at almost identical results for the mass ratio. Djurašević et al. (2006) presented the first model, resulting in orbital inclination of i = 78.5° and a fill-out factor of f = 32.69 per cent, T2 = 6314 K, and T1 = 6095 K, when spots were added. Their no-spot model resulted in very close value for the fill-out factor but slightly different geometrical and orbital parameters. The physical parameters derived in this study were: M1 = 1.37 M⊙, M2 = 0.43 M⊙, and mean radii R1 = 1.35 R⊙, R2 = 0.82 R⊙. It is worth noting here that the light curves analysed by these authors included the visual component in the photometric aperture with a contamination of about 54 per cent.


Among the several features, the distribution of chemical elements across the Galactic disc historically constitutes the most important constraint to chemo-dynamical models of our Milky Way. A number of studies (e.g. Tosi 1988; Hayden et al. 2014, 2015; Anders et al. 2017) have shown the spatial distributions of chemical abundances and their ratios across the Galactic disc. However, these studies are mainly based on field stars, which also include very old populations that had time to migrate significantly and redistribute the chemical elements across the Galaxy (e.g. Sellwood & Binney 2002; Roškar et al. 2012; Martinez-Medina et al. 2016). Open clusters are a valuable alternative, being on average younger (Magrini et al. 2017), and therefore a better tracer of the gradients in the disc out of which the most recent stars formed. Since the work of Janes (1979), much observational evidence has established that the metallicity distribution (often abbreviated by the iron-to-hydrogen ratio [Fe/H]) traced by clusters throughout the Milky Way disc shows a significant decrease with increasing distance from the Galactic Centre. This ‘radial metallicity gradient’ – in its apparent simplicity – reflects a complex interplay between several processes that are driving the evolution of our Galaxy, including star formation, stellar evolution, stellar migration, gas flows, and cluster disruption (Cunha & Lambert 1992, 1994; Friel 1995; Stahler & Palla 2004; Carraro et al. 2006; Boesgaard, Jensen & Deliyannis 2009; Magrini et al. 2009; Frinchaboy et al. 2013; Netopil et al. 2016; Anders et al. 2017; Spina et al. 2017; Bertelli Motta et al. 2018; Quillen et al. 2018). Complementary to the study of the overall metallicity distribution, the abundance ratios of several other elements, such as α-elements, iron peak, odd-z, and neutron capture, can provide deep insight into the variety of nucleosynthesis processes, with their production sites and time-scales (e.g. Carrera & Pancino 2011; Ting et al. 2012; Reddy, Lambert & Giridhar 2016; Duffau et al. 2017; Magrini et al. 2017, 2018; Donor et al. 2020; Casamiquela et al. 2020). Therefore, understanding the distribution of metals traced by clusters across the Galactic disc is fundamental for explaining the birth, life, and death of both stars and clusters, the recent evolution of our own Milky Way, and the evolution of other spiral galaxies (Boissier & Prantzos 2000; Bresolin 2019).


In the turbulent gaseous halos of clusters, groups, and galaxies (particularly massive ones), extended filaments and clouds condense out of the hot plasma in a top-down nonlinear23

23
This nonlinear condensation process has properties significantly different from those of classic linear thermal instability (TI); the latter is mainly concerned with small overdensities overcoming buoyancy oscillations (e.g., Field 1965; Balbus & Soker 1989; Burkert & Lin 2000; Pizzolato & Soker 2005; McCourt et al. 2012—more in Section 5).
 condensation cascade, forming a chaotic multiphase rain. The thermal state and kinematics of the progenitor hot plasma halo drive the formation and evolution of all the condensed structures, which inherit some of the parent properties. Part of the inner condensed gas eventually accretes onto the central supermassive black hole (SMBH), igniting the feedback response and efficiently self-regulating the entire atmosphere over several gigayears (e.g., Gaspari et al. 2011a, 2011b, 2012a, 2012b; Li & Bryan 2014; Barai et al. 2016; Soker 2016; Yang & Reynolds 2016; Meece et al. 2017; Voit et al. 2017). This feeding process is known as chaotic cold accretion (CCA; Gaspari et al. 2013b) and can intermittently boost the accretion rates up to 100× the hot (Bondi) rate. If turbulence is subdominant, the halo tends instead to condense in a disk structure (due to the preservation of angular momentum), reducing feeding and feedback—this regime is more important for low-mass, spiral galaxies.24

24
The top-down rain differs from the bottom-up condensation in the disk of spiral galaxies, where the hot/warm phase is created in situ by supernovae, which drive compressive, non-solenoidal turbulence (e.g., McKee & Ostriker 1977; Kim et al. 2013). Nevertheless, the two complement each other, producing multiphase gas in the more extended halo and in the disk, respectively. Massive galaxies, groups, and clusters, lacking an extended disk (e.g., Werner et al. 2014), typically reside in the top-down condensation regime.
 Finally, if the entropy of the halo (or cooling time) becomes too high, the whole atmosphere may simply prevent condensation and remain hot for an extended period of time, dramatically stifling the feedback response. Overall, assessing the dynamical state of the multiphase halos is crucial to understand the past and to predict the future evolution of cosmic structures.


Jovian electrons were used as test particles to model the charged-particle transport computationally (see e.g. Chenette et al. 1977; Conlon 1978; Fichtner et al. 2000; Zhang et al. 2007, and references therein) to ascertain the diffusion coefficients parallel and perpendicular to the Heliospheric Magnetic Field (HMF). This was usually done by comparing computed with measured electron intensities at Earth during periods of good or bad magnetic connection. Furthermore, given the demonstrated sensitivity of computed low-energy galactic electron intensities to various turbulence quantities (see Engelbrecht & Burger 2010, 2013; Engelbrecht 2019), it may be possible to draw conclusions from Jovian electrons to better understand the behaviour of those quantities in regions of the heliosphere where spacecraft observations of this character do not exist (see, e.g., Engelbrecht 2017). Since these transport parameters and the diffusion coefficients they depend on are spatially dependent, the time that particles reside in a certain part of the heliosphere may yield significant insights to the modulation of GCRs as well. Florinski & Pogorelov (2009) showed this dependency for GCR protons, investigating the time they spend in the heliotail, in the heliosheath, and in the solar wind within the termination shock, respectively. Utilising both galactic electrons and protons, Strauss et al. (2011a) focussed on the connection between thetotal propagation time and energy losses. They find a significant non-linear dependency on the total propagation times, which is strong enough to influence also the observations of Jovian electrons. These energy losses are entirely caused by adiabatic effects as other possible influences such as particle-particle interactions are negligible in the TPE due to a lack of significance in the interplanetary medium. As the adiabatic energy changes d E∕d t ~−2∕3 ⋅ EuSW∕r are connected to the radial position, the corresponding energy loss rate per step only depends on the temporal step size Δs and the radial position after the step. The radial direction of the step is thereby irrelevant. This leads to particles spending more simulation time at small radii losing more energy and implicitly to a statistical connection between the average energy losses and the particle’s mean free paths.


Turbulence in plasmas is a complex phenomenon that is characterized by different regimes in different ranges of spatial and temporal scales. Turbulence in the solar wind has been extensively studied, both by detailed analyses of in situ measurements and from a theoretical point of view; see Bruno & Carbone (2005) for a review. Such studies often adopt complementary views that the turbulence may be described either as a collection of waves that interact nonlinearly, so-called wave turbulence, or else as a collection of broadband, essentially zero-frequency eddies or flux tubes that form a hierarchy of coherent structures. These approaches have been extensively reviewed (Barnes 1979; Matthaeus et al. 2015), and we do not attempt a critical comparison in the present work. Instead, we adopt mainly a wave taxonomy of the fluctuations based on linear theory in order to address a specific set of questions. As motivation, we note that a variety of observations in the solar wind (Bale et al. 2005; Sahraoui et al. 2012) have suggested that fluctuations near the end of the magnetohydrodynamic (MHD) inertial cascade range, and approaching the kinetic plasma range, may consist primarily of kinetic Alfvén waves (KAWs). Here we address in particular the nature of fluctuations produced due to nonlinear interactions near the proton inertial length dp and investigate in some detail the basis for identifying them as KAWs. We show how phase mixing of large-scale parallel-propagating Alfvén waves is an efficient mechanism for the production of KAWs at wavelengths close to dp and at a large propagation angle with respect to the magnetic field. To support the interpretation as KAWs, we perform and analyze MHD, Hall magnetohydrodynamic (HMHD), and hybrid Vlasov–Maxwell (HVM) simulations that model the propagation of Alfvén waves and their fully nonlinear interaction with a nonuniform plasma background. We will be able to characterize fluctuations produced by this phase-mixing-like interaction as highly oblique KAWs.


In addition to secular evolutionary processes, a disc galaxy like ours is expected to have experienced several accretion events in its recent and early past (Bullock & Johnston 2005; De Lucia & Helmi 2008; Stewart et al. 2008; Cooper et al. 2010; Font et al. 2011; Brook et al. 2012; Martig et al. 2012; Pillepich et al. 2015; Deason et al. 2016; Rodriguez-Gomez et al. 2016). While some of these accretions are currently being caught in the act, like for the Sagittarius dwarf galaxy (Ibata et al. 1994) and the Magellanic Clouds (Mathewson et al. 1974; Nidever et al. 2010; D’Onghia & Fox 2016), we need to find the vestiges of ancient accretion events to understand the evolution of our Galaxy and how its mass growth has proceeded over time. Events that took place in the far past are expected to have induced a thickening of the early Galactic disc, first by increasing the in-plane and vertical velocity dispersion of stars (Quinn et al. 1993; Walker et al. 1996; Villalobos & Helmi 2008, 2009; Zolotov et al. 2009; Purcell et al. 2010; Di Matteo et al. 2011; Qu et al. 2011; Font et al. 2011; McCarthy et al. 2012; Cooper et al. 2015; Welker et al. 2017), and second by agitating the gaseous disc from which new stars are born, generating early stellar populations with higher initial velocity dispersions than those currently being formed (Brook et al. 2004, 2007; Forbes et al. 2012; Bird et al. 2013; Stinson et al. 2013). These complementary modes of formation of the Galactic disc can be imprinted on kinematics-age and kinematics-abundance relations (Strömberg 1946; Spitzer & Schwarzschild 1951; Nordström et al. 2004; Seabroke & Gilmore 2007; Holmberg et al. 2007, 2009; Bovy et al. 2012a, 2016; Haywood et al. 2013; Sharma et al. 2014; Martig et al. 2016; Ness et al. 2016; Mackereth et al. 2017; Robin et al. 2017), and distinguishing between them requires full 3D kinematic information for several million stars, in order to be able to separate the contribution of accreted from in-situ populations, and to constrain impulsive signatures that are typical of accretions (Minchev et al. 2014) versus a more quiescent cooling of the Galactic disc over time. Accretion events that took place in the more recent past of our Galaxy can also generate ripples and rings in a galactic disc (Gómez et al. 2012b), as well as in the inner stellar halo (Jean-Baptiste et al. 2017). Such vertical perturbations of the disc are further complicated by the effect of spiral arms (D’Onghia et al. 2016; Monari et al. 2016b), which together with the effect of accretion events might explain vertical wave modes as observed in SEGUE andRAVE (Widrow et al. 2012; Williams et al. 2013; Carrillo et al. 2018), as well as in-plane velocity anisotropy (Siebert et al. 2012; Monari et al. 2016b). Mapping the kinematics out to several kiloparsec from the Sun is crucial for understanding whether signs of these recent and ongoing accretion events are visible in the Galactic disc, to ultimately understand to what extent the Galaxy can be represented as a system in dynamical equilibrium (Häfner et al. 2000; Dehnen & Binney 1998), at least in its inner regions, or to recover the nature of the perturber and the time of its accretion instead from the characteristics and strength of these ringing modes (Gómez et al. 2012b).


The spectral energy distribution of the flare determines the altitude range in the planet’s atmosphere that is affected by the flare. Intense flares on AU Mic display a strong continuum emission enhancement in the XUV (see Fig. 2c) which results in increased ionisation over a broad altitude range in the thermosphere. Large continuum enhancement during flares have also been seen in other wavebands. Strong increases in the FUV have been observed on AU Mic (Robinson et al. 2001), and also on another active M dwarf, AD Leo (Hawley & Pettersen 1991). More recently, Kowalski et al. (2010) showed that continuum emission may be the dominant luminosity source in the near ultraviolet (NUV) during flares. There have not been many simultaneous multi-wavelength studies of flaring stars (Hawley et al. 2003; Osten et al. 2005). There is a particular dearth of simultaneous data in the EUV which is critical for studies of exoplanetary upper atmospheres, but where measurements are difficult due to ISM absorption. More observations are needed, but if young, magnetically active stars display broadband continuum enhancements during flares, similar to AU Mic, this would mean that large altitude ranges in planetary atmospheres are affected. Programmes such as Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary Systems (MUSCLES) have begun to provide more information on the high-energy spectral shape of K and M dwarf stars, including in the EUV (France et al. 2016; Youngblood et al. 2016; Loyd et al. 2016). However, in these studies, the EUV flux is computed using reconstructed Lyman alpha flux assuming a spectral energy distributio similar to the Sun. This approach could lead to incorrect results for active stars. Further development of coronal models driven by multi-wavelength observations would provide the most accurate high energy spectra for active low-mass stars. Such observations should be provided by the Multiwavelength Observations of an eVaporating Exoplanet and its Star (MOVES) program which has begun to undertake a long-term study of the HD 189733 system (Fares et al. 2017).


Having intermediate colours between the blue cloud and the red sequence, galaxies populating the so-called green-valley (e.g. Salim 2014; Schawinski et al. 2014) are generally considered as the transiting objects par excellence (Martin et al. 2007; Mendel et al. 2013; Salim 2014; Schawinski et al. 2014). Among these, the most interesting population certainly consists of those galaxies which have just entered the quenching phase (within a few Myr). Although hampered by the short duration of the quenching process, the search for galaxies in this critical phase of evolution has been carried on by several authors in the past decades. Galaxies characterized by both a tidally disturbed morphology and intermediate colours (e.g. Schweizer & Seitzer 1992; Tal et al. 2009) or low level of recent SF (Kaviraj 2010), young elliptical galaxies (Sanders et al. 1988; Genzel et al. 2001; Dasyra et al. 2006) and very recent post-merger remnants with strong morphological disturbances (Hibbard & van Gorkom 1996; Rothberg & Joseph 2004; Carpineti et al. 2012) have been considered as valid 'recent time' quenching candidates. Moreover, many attempts aimed at spectroscopically identifying quenching galaxies come from the investigations of the post-starburst (E+A or K+A) galaxies' UV and optical spectra, whose strong Balmer absorption lines and missing [O ii] λ3727 (hereafter [O ii]) and Hα emission lines (Couch & Sharples 1987; Poggianti et al. 2004; Quintero et al. 2004; Balogh et al. 2011; Muzzin et al. 2012; Mok et al. 2013; Wu et al. 2014) have been interpreted as signs of a recent halt of the SF (Dressler & Gunn 1983; Zabludoff et al. 1996; Quintero et al. 2004; Poggianti et al. 2008; Wild et al. 2009). The scarcity of galaxies that are in the transition phase suggests that, whatever mechanism may be responsible for the SF shut-off, it has to happen on short time-scales (Tinker, Wechsler & Zheng 2010; Salim 2014). The relatively short duration of the quenching process is also suggested by the surprising identification of a significant number of galaxies that look already quiescent at z ∼ 4–5, when the Universe was only ∼1–1.5 Gyr old (e.g. Mobasher et al. 2005; Wiklind et al. 2008; Juarez et al. 2009; Brammer et al. 2011; Marsan et al. 2015; Citro et al. 2016). However, a general and coherent picture concerning the SF quenching is still lacking. Very recently, evidence that the quenching of the SF could be a separated process with respect to the morphological transformation has come from the photometric and spectroscopic investigations of passive spiral galaxies (Fraser-McKelvie et al. 2016).


The 21 cm global spectrum experiments aim to measure the sky-averaged spectrum with high precision so as to probe the early epochs of the universe. There are a number of such ground-based experiments, including the Experiment to Detect the Global Epoch-of-Reionization Signature (EDGES; Bowman et al. 2008; Bowman & Rogers 2010; Monsalve et al. 2017), the Sonda Cosmológica de las Islas para la Detecciónde Hidrógeno Neutro (SCI-HI; Voytek et al. 2014), the Probing Radio Intensity at high-z from Marion (PRIzM; Philip et al. 2019), the Shaped Antenna measurement of the background RAdio Spectrum (SARAS; Patra et al. 2013; Singh et al. 2017, 2018), the Cosmic Twilight Polarimeter (CTP; Nhan et al. 2019), the Broadband Instrument for Global Hydrogen Reionization Signal (BIGHORNS; Sokolowski et al. 2015), the Large-Aperture Experiment to Detect the Dark Age (LEDA; Bernardi et al. 2016; Bernardi 2018; Price et al. 2018), and the Radio Experiment for the Analysis of Cosmic Hydrogen (REACH; de Lera Acedo 2019). Compared to the 21 cm tomography experiments, measurement of the global 21 cm emission has a higher raw sensitivity and requires smaller collecting area, so that it could be conducted even with a single antenna. The EDGES (Bowman et al. 2018) reported the detection of a strong absorption feature at ∼78 MHz, which has a cosmic dawn 21 cm spectrum interpretation, though it has an unexpectedly large amplitude (0.5 K) and an unusual flattened profile. If it originated from the cosmic 21 cm spectrum, this would suggest possibly new physics or astrophysics (Chen & Miralda-Escudé 2004, 2008; Creasey et al. 2011; Nelson et al. 2013; Barkana 2018; Fialkov et al. 2018; Fraser et al. 2018; Barkana et al. 2018; Slatyer & Wu 2018; Li et al. 2021; Houston et al. 2018; Hirano & Bromm 2018; Muñoz et al. 2018; Yang 2021), though this result was not confirmed by a recent measurement of the SARAS experiment (Singh et al. 2022). It is imperative to check this result and improve upon it with further and more precise observations.


The ORCA software (version 4.0.2) (Neese 2012) was used for all geometry minimizations, potential energy surface (PES), and vibrational frequency analyses using the global hybrid functional PW6B95 (Zhao & Truhlar 2005) coupled to the split valence triple-ζ def2-TZVPP basis set with two sets of polarization functions (Weigend & Ahlrichs 2005) and the atom-pairwise dispersion correction energy with Becke–Johnson damping (D3BJ) (Grimme et al. 2010; Grimme, Ehrlich & Goerigk 2011). The selected level of theory (PW6B95-D3BJ/def2-TZVPP) is a reliable and accurate theoretical tool in the estimation of general main group thermochemistry, kinetics, and non-covalent interactions after the double hybrid functionals (Goerigk et al. 2017). The reliability of the used method is also underlined by the good qualitative agreement between the Density Functional Theory (DFT) and MP2-F12 calculations as found in previous works (Fioroni & DeYonker 2016; Fioroni et al. 2018; Fioroni, Savage & DeYonker 2019). To speed up calculations, the RI (Resolution of the Identity) (Neese 2003) and RIJCOSX (Neese et al. 2009b) algorithms were used coupling the Coulomb-fitting basis sets def2/J (Weigend 2006). Because all the considered TSs involve a simple bond breaking/formation or a dihedral rotation, after a PES search performed on the desired reaction coordinate, the eigenvector following method (Schlegel 1982; Horn et al. 1991; Eckert, Pulay & Werner 1997), as implemented in ORCA, was used. Furthermore, the computed structures were verified to represent a minimum or a transition state by analysis of harmonic vibrational frequency calculations. Finally, the visualization of the normal mode associated to the TS was analysed. The obtained enthalpies: (HTot = [EEl. + EZPE + EVib. + ERot. + ETrans.] + kBT) and S values (STot = SEl. + SVib. + SRot. + STrans.) were used to estimate the free energies (G) at T = 200 K. The selected T = 200 K is lower bound to the HCN polymerization to progress efficiently. When referring to astronomical bodies, such temperatures can be experienced, for example, by comets where temperature rises periodically by surface heating to release HCN and H molecules (Hoang et al. 2019) or by dust particles or greater bodies in the turbulent phase of a proto-planetary disc and planetary system evolution.


The early history of galaxy clusters is a poorly constrained aspect of galaxy and large-scale structure formation. Hierarchical clustering models predict that massive elliptical galaxies will form in the cores of what will become the most massive galaxy clusters today, but the epoch of the bulk of star formation for these galaxies is unclear. Observations of high redshift clusters (z = 1–1.5) by the ISCS project (IRAC Shallow Cluster Survey; Eisenhardt et al. 2008) suggest that this is at z > 3, and the presence of well-defined red sequences of galaxies in clusters out to z ∼ 2 (Andreon & Huertas-Company 2011; Gobat et al. 2011; Santos et al. 2011; Pierini et al. 2012) supports this conclusion. Theoretical models by Granato et al. (2004) suggest that forming clusters will go through a phase in which multiple members will undergo near-simultaneous massive bursts of star formation. The spectral energy distribution of these objects would be dominated by the far-IR, as is the case for local massive starbursts (e.g. Clements, Dunne & Eales 2010). A galaxy cluster or protocluster (we use the term protocluster to indicate a structure that has yet to become virialized) going through such a formative phase would appear as a clump of dusty protospheroidal galaxies, and might be detected by observations in the far-IR and submm bands. Hints of such objects may already have been found by Spitzer (Magliocchetti et al. 2007) and SCUBA (Ivison et al. 2000; Priddey, Ivison & Isaak 2008; Stevens et al. 2010 and references therein). A recent study by Ivison et al. (2013) has uncovered a group of HLIRGs and ULIRGs at z = 2.41 thought to be the progenitor of a 1014.6 M⊙ cluster. At still higher redshifts, the highest redshift protocluster currently known, at z ∼ 5.3 (Riechers et al. 2010; Capak et al. 2011), and a group of objects associated with a z = 5 quasar (Husband et al. 2013) both contain at least one submm-luminous object, while the best studied group of high z submm-luminous sources to date is probably the four objects associated with a source in the GOODS-North field designated GN20, all lying at z = 4 (Daddi et al. 2009; Carilli et al. 2011), though this group is extended over a broad range of redshifts Δz ∼ 0.1.


For the TRAPPIST-1 system, data obtained by HST provide initial constraints on the extent and composition of the planet’s atmospheres, suggesting that the four innermost planets do not have a cloud/haze-free H2-dominated atmosphere (de Wit et al. 2016, 2018). However, follow-up work by Moran et al. (2018) have shown that HST data can also be fit to a cloudy/hazy H2-dominated atmosphere. Complementary to HST, NASA’s Spitzer Space Telescope—which played a major role in the discovery and orbital determination of TRAPPIST-1d, e, f, and g (Gillon et al. 2017)—has also allowed us to put additional constraints on the atmospheric composition of TRAPPIST-1b. Transit observations with Spitzer (Delrez et al. 2018) have found a +208 ± 110 ppm difference between the 3.6 and 4.2 μm bands, suggesting CO2 absorption. Spitzer also showed that transit depth measurements do not show any hint of significant stellar contamination in the 4.5 μm spectral range. Morris et al. (2018) reached the same conclusion using a “self-contamination” approach based on the Spitzer data set. Spitzer's “Red Worlds” Program encompassed over 1000 hours of observations of the TRAPPIST-1 system, whose global results have been presented (Ducrot et al. 2020). HST and Spitzer measurements have also been combined with transit light curves obtained from space with K2 (Luger et al. 2017) and from the ground with the SPECULOOS-South Observatory (Burdanov et al. 2018; Gillon 2018) and Liverpool Telescope (Steele et al. 2004) where Ducrot et al. (2018) produced featureless transmission spectra for the planets in the 0.8–4.5 μm wavelength range, showing an absence of significant temporal variations of the transit depths in the visible. Additional ground-based observations with the United Kingdom Infra-Red Telescope, Anglo-Australian Telescope, and Very Large Telescope also show no substantial temporal variations of transit depths for TRAPPIST-1 b, c, e, and g (Burdanov et al. 2019). While the K2 optical data set detected a 3.3 day periodic 1% photometric modulation, it is not present in the Spitzer observations (Delrez et al. 2018). Further constraints on the molecular weight and presence/absence of atmospheres on the TRAPPIST-1 planets will require additional observations with future facilities.


As a class, the fastest stars in our Galaxy are expected to be hypervelocity stars (HVSs). These were first theoretically predicted by Hills (1988) as the result of a three-body interaction between a binary star and the massive black hole in the Galactic Centre (GC), Sagittarius A*. Following this close encounter, a star can be ejected with a velocity ∼1000 km s−1, sufficiently high to escape from the gravitational field of the MW (Kenyon et al. 2008; Brown 2015). The first HVS candidate was discovered by Brown et al. (2005); a B-type star with a velocity more than twice the Galactic escape speed at its position. Currently about ∼20 unbound HVSs with velocities ∼300–700 km s−1 have been discovered by targeting young stars in the outer halo of the MW (Brown, Geller & Kenyon 2014). In addition, tens of mostly bound candidates have been found at smaller distances but uncertainties prevent the precise identification of the GC as their ejection location (e.g. Hawkins et al. 2015; Vickers, Smith & Grebel 2015; Zhang, Smith & Carlin 2016; Marchetti et al. 2017; Ziegerer et al. 2017). HVSs are predicted to be ejected from the GC with an uncertain rate around 10−4 yr−1 (Yu & Tremaine 2003; Zhang, Lu & Yu 2013), two orders of magnitude larger than the rate of ejection of RSs with comparable velocities from the stellar disc (Brown 2015). Because of their extremely high velocities, HVS trajectories span a large range of distances, from the GC to the outer halo. Thus, HVSs have been proposed as tools to study the matter distribution in our Galaxy (e.g. Gnedin et al. 2005; Sesana, Haardt & Madau 2007; Kenyon et al. 2014; Fragione & Loeb 2017; Rossi et al. 2017; Contigiani, Rossi & Marchetti 2018) and the GC environment (e.g. Zhang et al. 2013; Madigan et al. 2014), but a larger and less observationally biased sample is needed in order to break degeneracies between the GC binary content and the Galactic potential parameters (Rossi et al. 2017). Using the fact that their angular momentum should be very close to zero, HVSs have also been proposed as tools to constrain the solar position and velocity (Hattori, Valluri & Castro 2018a). Other possible alternative mechanisms leading to the acceleration of HVSs are the encounter between a single star and a massive black hole binary in the GC (e.g. Yu & Tremaine 2003; Sesana, Haardt & Madau 2006, 2008), the interaction between a globular cluster with a single or a binary massive black hole in the GC (Capuzzo-Dolcetta & Fragione 2015; Fragione & Capuzzo-Dolcetta 2016), and the tidal interaction of a dwarf galaxy near the centre of the Galaxy (Abadi et al. 2009). Another possible ejection origin for HVSs and high-velocity stars in our Galaxy is the Large Magellanic Cloud (LMC; Boubert & Evans 2016; Boubert et al. 2017; Erkal et al. 2018), orbiting the MW with a velocity ∼380 km s−1 (van der Marel & Kallivayalil 2014).


A color–color diagram in V − R versus B − V of the individual SSCs projected against the central spiral is shown in Figure 2 (left panel). The manner in which photometry of the individual stellar clusters was performed, including background subtraction, is described at length in Lim et al. (2020). In brief, we used the StarFinder algorithm (Diolaiti et al. 2000) to perform PSF-fitting photometry that includes subtraction of the local background. Only objects that exceed 4σ
local, where σ
local is the local root-mean-square noise level, are accepted as detections. Conversions to magnitude are based on the standard zero-points in the Vega system as described in the ACS Data Handbook (Lucas et al. 2018), and corrected for Galactic extinction. Two age loci, each of which is based on a model single stellar population (SSP; i.e., population of stars all having the same age and metallicity) as taken from Zackrisson et al. (2011), are plotted in Figure 2 for different metallicities of Z = 0.4 Z
⊙ (black) and Z = Z
⊙ (yellow). The model SSPs are based on those utilized in the Starburst99 software (Leitherer et al. 1999; Vázquez & Leitherer 2005), and their colors include both continuum and line emission from H ii regions corresponding to an adopted unity filling factor for the gas left over from star formation. The resulting line emission, in particular Hα+[N ii], that is contained in the R band causes the steep rise in V − R starting below B − V ≈ 0.0 when the star clusters are younger than ∼10 Myr. The two loci having different metallicities closely overlap, such that the inferred age at a given color differs little between the two loci. The  SSCs selected for study as indicated in Figure 1 (lower row) are concentrated around B − V ≈ 0.35, corresponding to an age around ∼500 Myr (more details below). The spread in their colors is larger than can be accounted for by photometric uncertainties alone, although there may well be a degree of contamination by stellar clusters associated with the HVS, as well as the chance projection of SSCs lying beyond 5 kpc but positioned along the sightline toward the central spiral. By contrast, as shown in the color–color diagram of Figure 2 (right panel), the SSCs located beyond the central spiral (omitting the region occupied by the HVS and the outer regions of NGC 1275 dominated by GCs, as described by Lim et al. 2020) span a continuous range of ages from a few Myr to at least ∼1 Gyr, beyond which they cannot be distinguished from the even more numerous GCs around NGC 1275.


We also detect a Compton hump at around 15–30 keV during the 2016 NuSTAR observation. The inclusion of the 10–50 keV spectral data does not affect the spectral model parameters (disc inclination angle, spin, emissivity indices, and break radius) obtained from fitting of the 0.3–10 keV spectrum only. The best-fitting value of the disc electron density as derived from the modelling of the broad-band (0.3–50 keV) spectral data is $n_{\rm e}=5.2^{+5.2}_{-4.2}\times 10^{16}$ cm-3. Mrk 1044 is known to be a highly accreting AGN with the dimensionless mass accretion rate of $\dot{m}=\frac{\dot{M}c^{2}}{L_{\rm E}}=16.6^{+25.1}_{-10.1}$ (Du et al. 2015). At high accretion rate, the inner region of a standard α-disc (Shakura & Sunyaev 1973) is radiation pressure dominated and the electron density of the disc can be written as (Svensson & Zdziarski 1994)
(11)
\begin{eqnarray*}
n_\mathrm{ e}=\frac{1}{\sigma _{\rm T}R_{\rm S}} \frac{256\sqrt{2}}{27}\alpha ^{-1}r^{3/2}\dot{m}^{-2} [1-(3/r)]^{-1} (1-f)^{-3}.
\end{eqnarray*}
where σT = 6.64 × 10−25 cm2 is the Thomson scattering cross section, RS = 2GMBH/c2 is the Schwarzschild radius, MBH is the black hole mass, α = 0.1 is the disc viscosity parameter, r = R/RS, R is the characteristic disc radius, $\dot{m}=\frac{\dot{M}c^{2}}{L_{\rm E}}$ is the dimensionless mass accretion rate, and f is the fraction of the total power released by the disc into the corona. The variation of the disc electron density (ne) with the dimensionless mass accretion rate ($\dot{m}$) for α = 0.1, $M_{\rm BH}=3\times 10^{6}\,\mathrm{M}_{\odot }$, f = 0.9, and r = 10 is shown as the solid curve in Fig. 17. As evident from Fig. 17, the assumption of constant disc density (ne = 1015 cm-3) is not physically realistic for low-mass AGNs even when the mass accretion rate is very high. The observed best-fitting value for the disc electron density of the source and its 90 per cent confidence limits are shown as the dotted and dashed lines in Fig. 17, respectively. The corresponding dimensionless mass accretion rate of Mrk 1044 estimated using equation (11) is $\dot{m}\approx 10-32$, which is in agreement with that found by Du et al. (2015). We further verified the SMBH mass with the use of X-ray variability techniques as pioneered by Ponti et al. (2012). The relation between the SMBH mass (MBH, 7) in units of $10^{7}\,\mathrm{M}_{\odot }$ and normalized excess variance ($\sigma _{\rm NXS}^{2}$) in the 2–10 keV light curves of 10 ks segments and the bin size of 250 s, can be written as
(12)
\begin{eqnarray*}
\log (\sigma _{\rm NXS}^{2})=(-1.83\pm 0.1)+(-1.04\pm 0.09)\log (M_{\rm BH,7}).
\end{eqnarray*}
The SMBH mass of Mrk 1044 measured using equation (12) is $M_{\rm BH}=(4-5)\times 10^{6}\,\mathrm{M}_{\odot }$ which is close to that measured by Du et al. (2015).


An et al. (1988) and Wu et al. (1990) investigated the effects of plasma injection on the formation of the Kippenhahn-Schlüter model of prominence in optimum conditions. These authors found that for high values of the plasma-β parameter (the ratio of plasma pressure to magnetic pressure) the magnetic arcade develops a magnetic dip at the centre of the structure that supports the prominence plasma. However, comparing with our study, in the low plasma-β regime (or under others injection conditions) they found that the dip is less deep and the system develops two additional plasma enhancements located at the lateral edges of the magnetic arcade. Recent works suggest that the deformation of the magnetic field lines is determined by the parameter δ (the ratio of the gravity to the magnetic pressure) (Zhou et al. 2018; Zhang et al. 2019). An et al. (1988) suggested that the steady lateral plasma accumulates because of both the injection process and because the field lines without dips do not geometrically contain the injected plasma, but Wu et al. (1990) proposed that the prominence mass is also supported by an increase in the pressure gradient. Since in this study we consider that magnetic field lines do not change owing to the presence of the dense thread we investigate in detail the results of An et al. (1988) and Wu et al. (1990) in part II of this paper. On the contrary, studies of the formation of 1D filament threads by chromospheric heating in the presence of non-adiabatic effects, such as radiative losses and thermal conduction, show that for magnetic loops without a dip, the plasma condenses but it streams along the magnetic field and disappears after falling to the footpoints (Antiochos et al. 2000; Karpen et al. 2006). Moreover, when the thread is initially in a thermal and force-balance equilibrium state but it is disturbed by a strong velocity perturbation, the prominence mass drains down to the chromosphere. Dense blobs of falling plasma have been habitually observed (Schrijver 2001; de Groof et al. 2005), therefore it seems that threads cannot be held static along vertical magnetic flux tubes in the corona. The coronal part of the tube can only slow down the falling blobs. Müller et al. (2004) proposed that the acceleration reduces because the pressure of the cooling plasma underneath the radiating blobs slows down the descent, and Oliver et al. (2014, 2016) and Martínez-Gómez et al. (2020) argue that pressure gradient is the main force that opposes the action of gravity. Our study proposes that the pressure gradient can cause the equilibrium of threads in quasi-vertical flux tubes without dips even though it has not been corroborated by observations. Besides, our model is relatively simple to study the stability of filament threads in the magnetic field without dips. Other processes such as radiation and heat conduction must be considered, which might change the stability results.


The other criteria that we can use to categorize the reconstruction methods is whether the reconstruction is performed using forward-modelling or uses a direct inversion from the data. Inverting non-linear problems from partial, noisy, observations is an ill-posed inverse problem, which makes forward-modelled Bayesian methods particularly suitable for the task of reconstruction of high-dimensional fields. Bayesian reconstruction methods have become increasingly popular in cosmology and have been applied in a range of different applications such as initial conditions reconstruction (Jasche & Wandelt 2013; Modi, Feng & Seljak 2018; Jasche & Lavaux 2019), weak lensing (Fiedorowicz et al. 2022; Porqueres et al. 2021, 2022; Boruah, Rozo & Fiedorowicz 2022), and CMB lensing (Millea et al. 2021; Millea, Anderes & Wandelt 2020). Such methods have also been used for the local velocity field reconstruction. The simplest of such methods uses a Wiener filtering technique (Zaroubi, Hoffman & Dekel 1999). This approach assumes that the density/velocity field is described as a Gaussian random field and the Wiener filtered reconstruction is the maximum-a-posteriori (MAP) solution for the problem. The Wiener filtering approach has been extended to account for uncertainties and biases in the reconstruction using a constrained realization approach (Hoffman & Ribak 1991; Hoffman, Courtois & Tully 2015; Hoffman et al. 2018; Lilow & Nusser 2021) An alternative way to account for the biases in the reconstruction in Wiener filtering is using the unbiased minimal variance approach (Zaroubi 2002). Another similar approach is the Bayesian hierarchical method, virbius (Lavaux 2016), which is based on the constrained realization approach but accounts for many different systematic effects in its analysis. This approach has been been applied to the Cosmicflows-3 (Tully, Courtois & Sorce 2016) data set by Graziani et al. (2019). A similar reconstruction code, hamlet, was introduced in Valade et al. (2022). However, these methods fail to account for the inhomogeneous Malmquist (IHM) bias which is an important source of systematic error in peculiar velocity analysis. The IHM bias arises from an incorrect assumption on the distribution of peculiar velocity tracers due to neglecting the line-of-sight inhomogeneities.


Before 1500 A.D., when the magnetic latitude of China was higher than that at present, most of the observed aurorae were caused by CIRs and moderate CMEs. After 1500 A.D., the GNP moved to Canada. This caused the geomagnetic field latitude of China to decrease considerably. Meanwhile, the field intensity decreased to its minimum value in the last 2000 yr. Under these conditions, it would have been very rare to observe the aurora in most places in China after the 16th century, just like the present case (Wu et al. 2016a, 2016b; Hayakawa et al. 2020b, 2020c). However, this provides the best opportunity to investigate the existence of great-storm CMEs, which generate a wider auroral belt and equatorward shift of the EBAO and the EBAV (Boudouridis et al. 2003; Shue et al. 2009; Sigernes et al. 2011). If auroras were observed in China at lower magnetic latitudes than the EBAV related to the great storms, this indicates that the strong CMEs occurred. To quantify our results, we applied the classification of magnetic storms at the great level having Dst  −350 nT (Loewe & Prölss 1997). According to the EBAV of the great level shown as the red dashed line in Figure 2(b), the auroras caused by great-storm CMEs could be identified, which were below the red dashed line. Surprisingly, there were many great-storm CMEs between 1500 and 1900, when Spörer minimum (1390–1550), Maunder minimum (1645–1715), and Dalton minimum (1797–1827) were inside (Lean et al. 1995; Solanki & Fligge 2000; Usoskin et al. 2015; Cliver & Herbst 2018; Hayakawa et al. 2020e). There were auroral sighting records in the same year, such as 1620 and 1646, from different places in China. As previously reported by Hayakawa et al. (2018b) there were simultaneous auroral records on the same day from different places in China in 1730. There are quite a few studies about the aurorae in September 1770 (Willis et al. 1996; Ebihara et al. 2017; Hayakawa et al. 2017a). Apart from the case analyses for extreme space weather events (Hattori et al. 2019; Isobe et al. 2019), our results indicate that there are more great-storm CMEs in the past 400 yr. Some of the great-storm CMEs happened in the solar maximum years, and some happened in the solar minimum years, as shown in Figure 7.


There are several different methods used in the literature for creating mock catalogues and light-cones. In purely empirical methods such as the JAdes extraGalactic Ultradeep Artificial Realizations (jaguar) models used to create mock catalogues in support of the JADES survey (Williams et al. 2018), observed galaxy properties are interpolated or extrapolated, and there is no underlying physics model nor setting within a Λ cold dark matter (ΛCDM) context. In what are sometimes called ‘semi-empirical’ methods [also called subhalo abundance matching (SHAM) or halo occupation distribution (HOD) models; see Wechsler & Tinker 2018], galaxy properties are mapped on to the properties of dark matter haloes such that a set of observational quantities is reproduced (Behroozi, Conroy & Wechsler 2010; Moster, Naab & White 2013, 2018; Behroozi et al. 2019; Wechsler et al. 2022). Both of these methods have the advantage that they are computationally efficient, are not dependent on a specific model for galaxy formation, and are guaranteed to match the observations that were used to calibrate them. However, they have the disadvantage that using them for forecasts for new observations is highly uncertain, and they are of limited use for interpretation. Semi-empirical models are typically calibrated using derived physical properties such as stellar masses and star formation rates (SFRs), which are highly uncertain at high redshifts, leading to models that are nominally calibrated on the same observations, but which have very different predictions for the link between galaxy and dark matter halo properties (see e.g. Yung et al. 2019b). We note that this is only a general overview for the semi-empirical modelling approach. These models are designed with different purposes in mind and adopt different calibration criteria. For example, UniverseMachine used high-redshift ultraviolet (UV) luminosity functions (LFs) to calibrate galaxy growth at the highest redshifts (Behroozi et al. 2019, 2020) and the semi-empirical model presented in Behroozi & Silk (2015) allows forecasts under the assumption that galaxy–halo growth relationships are given by power laws.


Observing with instruments such as the Atacama Large Millimeter/submillimeter Array (ALMA) is crucial to our understanding of planet-formation mechanisms, as we can observe at wavelengths that trace continuum emission from the cold midplane (e.g., Testi et al. 2014), where we expect planets to be forming or have already formed. In the case of midplane spiral structures, their origin may be linked to the presence of a companion—stellar, fly-by, or planetary (Pohl et al. 2015; Bae & Zhu 2018a; Dong et al. 2018; Forgan et al. 2018b; Cuello et al. 2019; Keppler et al. 2020). Spirals may also be excited if the system is gravitationally unstable. Gravitationally instability is expected in cool and massive disks, where the disk-to-star mass ratio is larger than 0.1 (Bell et al. 1997; Gammie 2001; Lodato & Rice 2004; Cossins et al. 2009; Hall et al. 2016; Kratter & Lodato 2016; Rice 2016; Hall et al. 2019; Zhang & Zhu 2020). To date, not many spirals in dust continuum emission have a clear origin, except for those in multiple systems where the presence of spirals has been linked to stellar interactions (Kurtovic et al. 2018; Rosotti et al. 2020). On the other hand, there are disks where spirals have been reported at millimeter wavelengths and where no companion to which the spiral origin may be linked has been detected yet (to date these are Elias 27, IM Lup, WaOph 6, and MWC 758; Pérez et al. 2016; Dong et al. 2018; Huang et al. 2018c). If no companion is detected and the disk is massive compared to the host star mass, the gravitational instability (GI) scenario arises as a possible explanation for the origin of the observed spirals. Studying disks undergoing GI is important, as population synthesis models show that GI primarily ends up forming brown dwarf mass objects (Hall et al. 2017; Forgan et al. 2018a). It seems that giant-planet formation through GI is rare (Rice et al. 2015), but it may still be the formation mechanism for important systems like HR 8799 (Vigan et al. 2017).


In a quasi-isothermal jet, Uj is (17)\begin{equation} \label{eq:U_j_quasi} U_{\rm j} = \zeta n_0 m_{\rm p} c^2\left(\frac{\gamma_{\rm j}\beta_{\rm j}}{\gamma_0\beta_0}\right)^{-\Gamma}\left(\frac{z}{z_0}\right)^{-2} \cdot \end{equation}Uj=ζn0mpc2γjβjγ0β0−Γzz0-2·Substituting Eqs. (17) and (13) into Eq. (10), and assuming the jet is launched with an initial γ0β0 equal to the sound speed (Eq. (16)), the 1D Euler equation that results is \begin{eqnarray} \label{eq:AGNJET_Corrected} &&\left\{\gamma_{\rm j}\beta_{\rm j}\frac{\Gamma+\xi}{\Gamma-1}-\Gamma\gamma_{\rm j}\beta_{\rm j}-\frac{\Gamma}{\gamma_{\rm j}\beta_{\rm j}}\right\}\frac{\partial \gamma_{\rm j}\beta_{\rm j}}{\partial z} = \frac{2}{z}; \\ &&\xi = \frac{1}{\zeta}\left(\frac{\gamma_{\rm j}\beta_{\rm j}}{\gamma_0\beta_0}\right)^{\Gamma-1}; \qquad \gamma_0\beta_0=\sqrt{\frac{\zeta\Gamma(\Gamma-1)}{1+2\zeta\Gamma-\zeta\Gamma^2}} \cdot \end{eqnarray}γjβjΓ+ξΓ−1−Γγjβj−Γγjβj∂γjβj∂z=2z;ξ=1ζγjβjγ0β0Γ−1; γ0β0=ζΓ(Γ−1)1+2ζΓ−ζΓ2·The above equation should reduce to the jet Lorentz factor profile used in Falcke (1996), Markoff et al. (2005) when ζ = 1. However, it differs from Eq. (2) in Falcke (1996): (20)\begin{equation} \label{eq:Heino96} \left\{\gamma_{\rm j}\beta_{\rm j}\frac{\Gamma+\xi}{\Gamma-1}-\frac{\Gamma}{\gamma_{\rm j}\beta_{\rm j}}\right\}\frac{\partial \gamma_{\rm j}\beta_{\rm j}}{\partial z} = \frac{2}{z}; \end{equation}γjβjΓ+ξΓ−1−Γγjβj∂γjβj∂z=2z;(21)\begin{equation} \xi = \left(\gamma_{\rm j}\beta_{\rm j}\frac{\Gamma+1}{\Gamma(\Gamma-1)}\right)^{1-\Gamma} \cdot \end{equation}ξ=γjβjΓ+1Γ(Γ−1)1−Γ·The difference between our equation and the equation in Falcke (1996) can be accounted for as follows: the − Γγjβj term in Eq. (18) results from a neglected \hbox{$\frac{\partial}{\partial z}(U_{\rm j}/n)$}∂∂z(Uj/n) term, the difference in the exponent in ξ results from an arithmetic error, and finally the difference in the inside of the parenthesis of ξ terms is from setting \hbox{$\gamma_0\beta_0 = \beta_{\rm s0}^{2}$}γ0β0=βs02 instead of using the proper value given in Eq. (16). The difference between the solutions of Eqs. (18) and (20) are small and shown in Fig. 1. In Fig. 1, we also include solutions to the 1D Euler equations when the jet is isothermal (Tj = const., i.e., Eq. 20 with ξ = 1) and adiabatic (Tj ∝ (γjβj)1 − Γz2 − 2Γ, see Eq. (25)).


The orbital decay of BSBHs may slow down or stall at ∼pc scales (e.g. Begelman et al. 1980; Milosavljević & Merritt 2001; Zier & Biermann 2001; Yu 2002; Vasiliev, Antonini & Merritt 2014; Dvorkin & Barausse 2017; Tamburello et al. 2017), or the barrier may be overcome in gaseous environments (e.g. Gould & Rix 2000; Escala et al. 2004; Hayasaki, Mineshige & Sudou 2007; Hayasaki 2009; Cuadra et al. 2009; Lodato et al. 2009; Chapon, Mayer & Teyssier 2013; Rafikov 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g. Yu 2002; Berczik et al. 2006; Preto et al. 2011; Khan et al. 2013, 2016; Vasiliev, Antonini & Merritt 2015; Gualandris et al. 2017; Kelley, Blecha & Hernquist 2017a), and/or by interacting with a third SMBH in hierarchical mergers (e.g. Valtonen 1996; Blaes, Lee & Socrates 2002; Hoffman & Loeb 2007; Kulkarni & Loeb 2012; Tanikawa & Umemura 2014; Bonetti et al. 2018). The accretion of gas and the dynamical evolution of BSBHs are likely to be coupled (Ivanov, Papaloizou & Polnarev 1999; Armitage & Natarajan 2002; Haiman, Kocsis & Menou 2009; Bode et al. 2010, 2012; Farris, Liu & Shapiro 2010, 2011; Kocsis, Haiman & Loeb 2012; Shi et al. 2012; D’Orazio, Haiman & MacFadyen 2013; Shapiro 2013; Farris et al. 2014, 2015) such that the occurrence rate of BSBHs depends on the initial conditions and gaseous environments at earlier phases (e.g. thermodynamics of the host galaxy interstellar medium; Dotti et al. 2007, 2009; Dotti, Sesana & Decarli 2012; Fiacconi et al. 2013; Mayer 2013; Tremmel et al. 2018). Quantifying the occurrence rate of BSBHs at various merger phases is therefore important for understanding the associated gas and stellar dynamical processes. This is a challenging problem for three main reasons. First, BSBHs are expected to be rare (e.g. Foreman, Volonteri & Dotti 2009; Volonteri, Miller & Dotti 2009), and only a fraction of them accrete enough gas to be ‘seen’. Secondly, the physical separations of BSBHs that are gravitationally bound to each other (≲a few pc) are too small for direct imaging. Even VLBI cannot resolve BSBHs except for in the local universe (Burke-Spolaor 2011). CSO 0402+379 (discovered by VLBI as a double flat-spectrum radio source separated by 7 pc) remains the only secure case known (Rodriguez et al. 2006; Bansal et al. 2017, see Kharb, Lal & Merritt 2017; however, for a possible 0.35-pc BSBH candidate in NGC 7674). Thirdly, various astrophysical processes complicate their identification such as bright hot spots in radio jets (e.g. Wrobel, Walker & Fu 2014b). Until recently, only a handful cases of dual active galactic nuclei (AGNs) – galactic-scale progenitors of BSBHs – were known (Owen et al. 1985; Junkkarinen et al. 2001; Komossa et al. 2003; Ballo et al. 2004; Hudson et al. 2006; Max, Canalizo & de Vries 2007; Bianchi et al. 2008; Guidetti et al. 2008). While great strides have been made in identifying dual AGNs at kpc scales (e.g. Gerke et al. 2007; Comerford et al. 2009, 2012, 2015; Green et al. 2010; Liu et al. 2010, 2013, 2018; Fabbiano et al. 2011; Fu et al. 2011, 2012, 2015a,b; Koss et al. 2011, 2012, 2016; Rosario et al. 2011; Teng et al. 2012; Woo et al. 2014; Wrobel, Comerford & Middelberg 2014a; McGurk et al. 2015; Müller-Sánchez et al. 2015; Shangguan et al. 2016; Ellison et al. 2017; Satyapal et al. 2017), there is no confirmed BSBH at sub-pc scales (for recent reviews, see e.g. Popović 2012; Burke-Spolaor 2013; Bogdanović 2015; Komossa & Zensus 2016).


The uncertainties in number counts were derived from Poisson statistics, which apply when event rates are calculated from small numbers of observed events (Gehrels 1986). Our results are given in Table 6 and plotted in Figure 7. For comparison, we also show the integral number counts from the lensing cluster surveys of Knudsen et al. (2008) and Johansson et al. (2011), from the SCUBA Half-Degree Extragalactic Survey (SHADES; Coppin et al. 2006), from LESS (Weiß et al. 2009), from SCUBA-2 (Hsu et al. 2016), and from the high-resolution ALMA follow-up of LESS (Karim et al. 2013). We find that, for the intrinsic flux density range covered by our survey, results are consistent within uncertainties with previous single-dish surveys conducted in blank fields (e.g., Coppin et al. 2006), toward lensing clusters (Knudsen et al. 2008; Johansson et al. 2011), and combining both cluster and blank fields (Hsu et al. 2016). The exception is the data point at Sint = 6.3 mJy, which is fainter than the 4σ detection threshold for all clusters and therefore comprises sources that are necessarily magnified. The discrepancy is possibly due to the uncertainties in our analytical lens models, which may generally underestimate magnification factors relative to those derived from strong-lensing models, as suggested by the comparison presented in Figure 5. If magnification factors are minimally increased across the field, intrinsic flux densities and binned number counts do not vary significantly, but the detectable area Asource where μ > 4σ/Sint is increased, thus affecting the number counts per unit area. To test this hypothesis, we repeated our calculations but slightly scaled our analytical magnification maps by a factor of ∼1. We find that a satisfactory match between our resulting number counts at Sint = 6.3 mJy and previous surveys can be reached if all magnification estimates are varied by only ∼5%, which is within the uncertainties obtained for μ and listed in Table 5.


Our assumption of a constant f
dg does not account for vertical settling and depletion of dust in the disk atmosphere, which could in principle reduce the height of the different τ = 1 surfaces shown in Figure 6. Even though the Stokes number remains smaller than 1 for all considered particle sizes well above the τ = 1 surfaces for irradiation, the actual dust distribution in the atmosphere depends on the balance between turbulent stirring and vertical settling. The dust distribution away from the midplane is better captured by hydrodynamical simulations than by simple prescriptions relying on a parameterization of the turbulence strength (see, e.g., Dubrulle et al. 1995; Dullemond & Dominik 2004b), since it depends on the particular source of turbulence. An example of this is the anisotropic turbulence produced by the VSI, whose vertically elongated modes significantly increase the vertical stirring of dust with respect to models of homogeneous isotropic turbulence (Stoll & Kley 2016; Stoll et al. 2017). We note that these authors obtain a vertical Gaussian dust distribution of micron-sized particles with the same scale height as the gas in the entirety of a domain reaching z = 5H at 5 au, which supports the validity of our prescription. On the other hand, dust sedimentation of small grains in outer regions can even be affected by nonideal magnetohydrodynamical effects and magnetized winds (Riols & Lesur 2018; Booth & Clarke 2021; Hutchison & Clarke 2021). In principle, we do not expect dust settling effects to be relevant for our work as long as they occur in optically thin regions where radiative transport prevents any shadowing from occurring. Future disk models could test this hypothesis by studying the stability of disks in which the opacity is computed as a function of a dynamically evolving small dust distribution. We have also assumed instantaneous thermal equilibrium of gas and dust particles, which is not verified at high altitudes, as collisions between dust and gas particles become less frequent for smaller densities (see, e.g., Malygin et al. 2017; Pfeil & Klahr 2021). We do not expect this effect to alter our results, since at most it could increase the time it takes to form temperature perturbations in such regions.