We envision a situation where intense current sheets are developed within a magnetically dominated plasma. Earlier work in nonrelativistic low-β plasmas has shown that the gradual evolution of the magnetic field can lead to formation of intense, nearly force-free current layers where magnetic reconnection may be triggered (Galsgaard et al. 2003; Titov et al. 2003). In the present study, the critical parameter is the magnetization parameter defined as

, which roughly corresponds to the available magnetic energy per particle. The numerical simulations presented in this paper are initialized from a force-free current layer with

 (Che et al. 2011; Liu et al. 2013, 2014), which corresponds to a magnetic field with magnitude B0 rotating by 180° across the central layer with a half-thickness of λ. No external guide field is included in this study but there is an intrinsic guide field By associated with the central sheet. The plasma consists of electron–positron pairs with mass ratio

. The initial distributions are Maxwellian with a spatially uniform density n0 and a thermal temperature (

). Particles in the central sheet have a net drift

 to represent a current density

 that is consistent with

. Since the force-free current sheet does not require a hot plasma component to balance the Lorentz force, this initial setup is more suitable to study reconnection in low-β and/or high-σ plasmas. The full particle simulations are performed using the VPIC code (Bowers et al. 2009) and NPIC code (Daughton et al. 2006; Daughton & Karimabadi 2007), both of which solve Maxwell equations and push particles using relativistic approaches. The VPIC code directly evolves electric and magnetic fields, whereas in the NPIC code the fields are advanced using the scalar and vector potentials. Although the two codes have very different algorithms, all of the key results are in good agreement for this study, thus providing additional confidence in our conclusions. In addition, we have developed a particle-tracking module to analyze the detailed physics of the particle energization process. In the simulations, we define and adjust σ by changing the ratio of the electron gyrofrequency

 to the electron plasma frequency

,

. For 2D simulations, we have performed simulations with

 and box sizes

,

, and

, where di is the inertial length

. For 3D simulations, the largest case is

 with σ = 100. For high-σ cases (

), we choose cell sizes

 and

, so the particle gyromotion scale

 is resolved. The time step is chosen to correspond to a Courant number

, where

. The half-thickness of the current sheet is

 for

,

 for σ = 400, and

 for σ = 1600 in order to satisfy the drift velocity

. For both 2D and 3D simulations, we have more than 100 electron–positron pairs in each cell. The boundary conditions for 2D simulations are periodic for both fields and particles in the x direction, while in the z direction the boundaries are conducting for the field and reflecting for the particles. In the 3D simulations, the boundary conditions are periodic for both fields and particles in the y direction, while the boundary conditions in the x and z directions are the same as in the 2D cases. A weak long-wavelength perturbation (Birn et al. 2001) with

 is included to initiate reconnection. The parameters for different runs are summarized in Table 1, which also lists key results such as maximum energy of particles, spectral index, the fraction of kinetic energy converted from the magnetic energy, and the portion of energy gain arising from the perpendicular electric fields.


Here, for simplicity, computational convenience, and continuity with past works (Kopparapu et al. 2017; Haqq-Misra et al. 2018), we have assumed a slab ocean with zero ocean heat transport. Ocean heat transport may have a significant effect on climate and also the location of clouds (Way et al. 2018). Dynamic ocean heat transport on completely ocean-covered worlds (i.e., no continents) leads to warmer global mean temperatures and a reduction in day–night temperature differences on synchronously rotating planets (Hu & Yang 2014; Del Genio et al. 2019). Yang et al. (2019) argue that ocean heat transport is critical for the treatment of cold, partially ice-covered planets around M dwarf stars; however, it does not have a meaningful effect on the climate and thermal emission phase curves for warm terrestrial planets (Ts ∼ 300 K) residing close to the inner edge of the habitable zone. The presence of continents also presents a significant uncertainty in the net effect of ocean heat transport on climate, as continents can reroute or eliminate day-to-night ocean heat transport entirely, resulting in climate states that are similar to those found in simulations with zero ocean heat transport (Del Genio et al. 2019; Yang et al. 2019). In this work, we have argued that clouds are likely more important in modulating phase curves for habitable planets, compared to surface energy transports. Still, ocean heat transport patterns, combined with variations in continental configurations and topographic prominences, could have significant complex feedback on the surface temperature, sea ice and snow cover, and cloud distributions, respectively, all of which affect thermal and reflected light phase curves. In future works, our group plans to couple our GCM with a dynamic ocean model to study the interactions between oceans, continents, topography, and observable features. However, the long equilibration times for a coupled dynamic atmosphere-ocean model still present a significant computational hinderance for conducting 3D simulations across wider parameter spaces, as we have done here.


One possibility could be that the disk, since viewed at such high inclination, might create a “shadow” zone where part of the reflection incoming between Rin, 1 and Rin, 2 is obscured (see Fig. 10, panel a). As highlighted by the Figure, a jump in the scale height of the disk, located sufficiently far away from its inner edge, could in principle generate such self-shielding effect. A disk flared in the outer region was also put forward by Zdziarski et al. (2021) and Axelsson & Veledina (2021) in order to explain the presence of the outer reflection component and the variability associated to the iron line. However, it is not obvious whether a sudden puffing up of the disk could happen or not in a classical Shakura-Sunyaev disk, like the SAD. A jump could arise from a transition in the mechanism contributing mainly to the opacity in the disk. In the outer regions of a Shakura-Sunyaev disk, the opacity is expected to be mainly due to free-free absorption, while Thomson scattering dominates the opacity in the inner regions of the disk (see Eqs. (2.16) and (2.19) of Shakura & Sunyaev 1973). The geometrical thickness of the disk scales with the radius R following a (slightly) different relation; it is proportional to R−21/20 in the inner Thomson scattering-dominated, regions and to R−9/8 in the outer regions. At the boundary between these two zones (i.e., at a radius Rjump) a jump is expected. We calculated Rjump according to our results for Ṁin and assuming the same value for α in both equations. We found Rjump between 3 × 104 RG and 7 × 104 RG in all the epochs considered. In our fits the location of the outer reflector Rin, 2 was kept frozen to 300 RG. When the parameter is thawed, it is completely unconstrained, as shown in Fig. 11. Notwithstanding the uncertainty on Rin, 2, a boundary at beyond 104 RG is likely located too far away to produce an extra reflection component. However, a crucial role might be played by irradiation from the inner regions of the disk to the outer regions, an ingredient neglected in Shakura-Sunyaev disks. Beyond some boundary radius, we expect that the upper layers of the disk should be heated up by the impinging photons coming from the inner regions, leading to evaporation and to a larger effective scale height.


High-precision photometric measurements have shown that the present treatment of limb darkening is not sufficiently accurate and leads to systematic errors in the derived parameters of the exoplanets (e.g., Espinoza & Jordán 2016; Morello et al. 2017a; Maxted 2018). Typically, limb darkening is represented by rather simple laws, such as a linear law (Schwarzschild 1906), a quadratic law (Kopal 1950), a square-root law (Diaz-Cordoves & Gimenez 1992), a power-2 law (Hestroffer 1997), or a four-coefficients law (Claret 2000), so that when modeling the transit light curves, limb darkening is parameterized by some set of coefficients. Ideally, these coefficients should be constrained by the stellar modeling. Consequently, many libraries of limb-darkening coefficients covering a wide range of effective temperatures (Teff), surface gravity (log g), and metal-licities (M/H) have been produced (e.g., Claret 2000; Sing 2010; Claret & Bloemen 2011; Magic et al. 2015) using various radiative transfer codes, such as ATLAS (Kurucz 1993), NextGen (Hauschildt et al. 1999), PHOENIX (Husser et al. 2013), MARCS (Gustafsson et al. 2008), and STAGGER (Magic et al. 2013). However, the limb-darkening parameters diverge between different libraries and often lead to an inadequate quality of fits to the observed transit profiles (Csizmadia et al. 2013). First, this can be due to errors in limb-darkening coefficients introduced by interpolation from the grid of stellar parameters used in these libraries to the actual stellar fundamental parameters. Second, available stellar calculations may not treat mechanisms that affect limb darkening with sufficient accuracy, for instance, convection (Pereira et al. 2013; Chiavassa et al. 2017) or magnetic activity (Csizmadia et al. 2013). Therefore, limb-darkening coefficients are often left as free parameters in a least-squares fit to observed light curves (e.g., Southworth 2008; Claret 2009; Cabrera et al. 2010; Gillon et al. 2010; Csizmadia et al. 2013; Maxted 2018). While this method usually leads to a good quality fit to observed transit profiles, it introduces additional free parameters, resulting in possible biases and degeneracies in the returned planetary radii (Espinoza & Jordán 2015; Morello et al. 2017b). The way to reduce these biases and reliably determine planetary radii is to improve theoretical computations of stellar limb darkening.


One of our key results is that accounting for a complete stellar mass function can explain the post-starburst preference of TDE hosts. The alternative, currently most widespread explanation is that post-starburst galaxies are initially born with an extremely steep density profile (Stone et al. 2018). Fig. 6 compares the initial conditions adopted in our runs with the ultrasteep initial conditions adopted by Stone et al. (2018) to explain the post-starburst preference, rescaled to our MBH mass according to their footnote 12 (see also the caption of our Fig. 6). The central density in our and their profile differs by several orders of magnitude, and in general it is not straightforward to imagine how star formation could generate such an extremely steep cusp. In fact, Sanders (1998) points out that, too close to the MBH, molecular clouds should get disrupted by its gravitational field preventing star formation; even if the Milky Way nucleus features a population of young stars very close to the MBH (Morris 1993; Ghez et al. 2003; Lu et al. 2013), supporting instead the idea of possible central star formation, it seems unlikely it can develop such steep profiles. For instance, in these systems the time-scale for stellar collisions and mergers can be shorter than the relaxation time-scale at small radii (10−2 pc, Stone et al. 2018). Note that if the density profile is initially very steep, relaxation processes render it milder in time, as the equilibrium configuration is the Bahcall & Wolf (1976) solution (ρ∝r−1.75): this means that the ultrasteep configuration should be in place since the starburst stage, i.e. stars should be basically born with ρ∝r−γ, γ = 2.5–2.75. In order to compare our results with the ones by Stone et al. (2018), we evolve their (monochromatic) profiles shown in Fig. 6 with ρ∝r−2.5, r−2.75 and we compare the obtained rates with our complete mass function models in Fig. 7. Even if their rates are overall larger, the early to late event rate is compatible with what we find for much milder profiles featuring a complete mass function; in fact, the ratio between the average TDE rate in the range 250–750 Myr (i.e. roughly the age of post-starburst systems) and 10–12 Gyr is equal to 8–14 for the Stone et al. (2018) cases, i.e. they are compatible with or smaller than the same ratios computed in our runs (see numbers in parenthesis in Table 1).


The driving mechanisms and chronology of the buildup of bulges in late-type galaxies (LTGs) is an issue of key relevance to our understanding of galaxy evolution. According to our current knowledge on bulge demographics in the local universe, a large fraction of LTGs host pseudo-bulges (PBs; e.g., Gadotti 2009; Fisher & Drory 2011; Fernández Lorenzo et al. 2014) that substantially differ from classical bulges (CBs) in their spectrophotometric and kinematical characteristics. The latter resemble in many respects “old and dead” elliptical galaxies, lacking ongoing star-formation (SF), exhibit a spheroidal shape with inwardly steeply increasing surface brightness profiles (SBPs) being well approximated by the Sérsic (1963) fitting law with a high (≳3) exponent η, show stellar kinematics dominated by velocity dispersion (σ⋆) and obey the Kormendy (1977) scaling relations for normal elliptical galaxies (Fisher & Drory 2010). It is observationally established that CBs contain a super-massive black hole (SMBH) with a mass M∙ tightly correlating with their stellar mass

${\cal M}_{\star,\textrm{B}}, \sigma_{*}$M⋆,B,σ*
 and optical luminosity (Ho 2008; Kormendy & Ho 2013; see also Ferrarese & Merritt 2000). Traditionally, bulges were thought to invariably form early-on via violent quasi-monolithic gas collapse (Larson 1974) or mergers (Bender et al. 1992; Aguerri et al. 2001; Keselman & Nusser 2012) associated with vigorous nuclear starbursts (Okamoto 2012), with the disk gradually building up around them. Whereas this inside-out galaxy formation scenario appears consistent with important integral characteristics of CBs (e.g., their red colors), it does not offer a plausible explanation for the presence of PBs in present-day LTGs. These generally show ongoing SF, a significant degree of rotational support (Kormendy & Kennicutt 2004, for a review) and flatter/ellipsoidal shapes with nearly exponential SBPs (η≲2; e.g., Drory & Fisher 2007; Fisher & Drory 2010). Even though there is observational evidence that PBs also contain a SMBH (Kormendy et al. 2011; Kormendy & Ho 2013), in some cases revealing itself as an active galactic nucleus (AGN; e.g., Kotilainen et al. 2016; see Kormendy & Ho 2013 for a review), these do not follow the M∙ –σ* correlation for CBs, which appears to be consistent with a different formation route. Indeed, the prevailing concept on PB formation is that these entities emerge gradually out of galactic disks through gentle gas inflow spawning quasi-continuous SF and the emergence of a central bulge-like luminosity excess at their centers (e.g., Courteau et al. 1996; Carollo et al. 2001; Kormendy & Kennicutt 2004). Besides bar-driven gas inflow (e.g., Springel & Hernquist 2005), various other mechanisms, such as inward stellar migration, minor mergers with low-mass satellites, or a purely dynamical re-arrangement of the disk (Scannapieco et al. 2010; Guedes et al. 2013; Bird et al. 2012; Roskar et al. 2012; Grand et al. 2014; Halle et al. 2015) have been proposed as further contributors to PB growth along the Gyr-long secular evolution of LTGs.

Background Background Background Compare/Contrast


The second issue concerns the fact that inside a given region, coreshine is not detected in all the dense clouds observed by Paladini (2014) and Lefèvre et al. (2014) and that the proportion of clouds exhibiting coreshine varies from one region to another. For instance, 75% of the dense clouds detected in Taurus exhibit coreshine, whereas in most other regions the proportion is closer to 50% (such as Cepheus, Chamaeleon, and Musca)5. On the contrary, there are for instance very few detections in the Orion region. In THEMIS, most of the scattering efficiency originates in the accretion of an a-C:H mantle. This leads to three possible explanations for the absence of detectable coreshine. The first explanation is related to the amount of carbon available in the gas phase. The abundance used by Köhler et al. (2015) relies on the highest C depletion measurements made by Parvathi et al. (2012) towards regions with \hbox{$N_{\rm H} \geqslant 2 \times 10^{21}$}NH⩾ 2 × 1021 H/cm2. Parvathi et al. (2012) highlighted the variability in the carbon depletion in dust depending on the line of sight. Thus, there may be clouds were the amount of carbon available for a-C:H mantle formation is smaller or even close to zero: such regions would be populated with aggregates with a thinner H-rich carbon mantle or no second mantle at all and thus exhibit very little or no coreshine emission. A second explanation is related to the stability of H-rich carbon in the ISM, which depends strongly on the radiation field intensity to local density ratio (Godard et al. 2011; Jones et al. 2014). In low-density regions (according to Jones et al. 2014, \hbox{$A_{V} \leqslant 0.7$}AV⩽ 0.7 for the standard ISRF), UV photons are responsible for causing the photo-dissociation of CH bonds, a-C:H → a-C. In transition regions from diffuse ISM to dense clouds (Jones et al. 2014, \hbox{$0.7 \leqslant A_{V} \leqslant 1.2$}0.7 ⩽ AV⩽ 1.2 for the standard ISRF), better shielded from UV photons and where the amount of hydrogen is significantly higher, H-poor carbon can be transformed into H-rich carbon through H atom incorporation, a-C → a-C:H. Similarly, carbon accreted from the gas phase in these transition regions is likely to be and stay H-rich. Then, in the dense molecular clouds, most of the hydrogen is in molecular form and thus not available to produce a-C:H mantles on the grains. However, this approximately matches the density at which ice mantles start to accrete on the grains, which would partly protect a-C:H layers that had formed earlier (Godard et al. 2011, and references therein). The stability and hydrogenation degree of a-C:H, as well as the exact values of AV thresholds, are both dependent on the timescale and UV field intensity. The resulting a-C ↔ a-C:H delicate balance could explain why in a quiet region such as Taurus most of the clouds exhibit coreshine, whereas in Orion, where on average the radiation field intensity and hardness are much higher, most clouds do not. A third explanation is related to the age and/or density of the clouds. In a young cloud, where dust growth is not advanced, or in an intermediate density cloud (ρC ~ a few 103 H/cm3), the dust population may be dominated by CMM grains instead of AMM(I) dust. Such clouds would be as bright in the IRAC 8 μm band as in the two IRAC bands at 3.6 and 4.5 μm, thus not matching the selection criteria defined by Pagani et al. (2010) and Lefèvre et al. (2014) and would be classified as “no coreshine" clouds.


Galaxy clusters are the traces of the formation of the largest structures in the Universe and so reliable tools to investigate structures formation and evolution. In principle, this is possible only if and when we have full knowledge of the properties of these objects. The total mass (i.e. the total amount of the dark matter (DM), the intracluster medium (ICM), and the stellar components) is an invaluable quantity when exploring the abundances of clusters along the redshift: a standard way to infer cosmological parameters such as the mean matter density Ωm and the amplitude of matter perturbations σ8(Planck Collaboration XIII 2016). Furthermore, under the assumption of a simple self-similar model (Kaiser 1986; Voit 2005), we could derive the total mass of the clusters from a few observables in optical, X-ray, or millimetre band (Giodini et al. 2013). This approach results in a few scaling relations valuable when we are interested to obtain averaged results based on some statistics. However, it is prone to the assumed simplified approximations: hydrostatic equilibrium and isothermal and spherical distribution for DM and ICM (Bryan & Norman 1998). It is well known that the hydrostatic equilibrium in haloes is not always satisfied, due to non-thermal pressure contributions from internal motions and turbulence (see e.g. Fang, Humphrey & Buote 2009; Lau, Kravtsov & Nagai 2009; Laganá, de Souza & Keller 2010; Rasia et al. 2012; Nelson, Lau & Nagai 2014; Yu, Nelson & Nagai 2015; Biffi et al. 2016; Eckert et al. 2019; Angelinelli et al. 2020; Ansarifard et al. 2020; Gianfagna et al. 2020; Green et al. 2020), pointing out the impact that the dynamical state of those large gravitational bounded objects should have. Several attempts have been made to infer clusters dynamical state, using both observational data and simulations, by analysing the images of the emission in optical (see e.g. Ribeiro, Lopes & Rembold 2013; Wen & Han 2013) and in the X-ray band (see e.g. Rasia, Meneghetti & Ettori 2013; Lovisari et al. 2017; Nurgaliev et al. 2017; Bartalucci et al. 2019; Cao, Barnes & Vogelsberger 2020; Yuan & Han 2020) or of the diffusion of the cosmic microwave background (CMB) photons by thermal Sunyaev–Zel’dovich (tSZ) effect in the millimetre band (Cialone et al. 2018; De Luca et al. 2020, hereafter DL20), or a combination of some of them (see e.g. Mann & Ebeling 2012; Molnar, Ueda & Umetsu 2020; Ricci et al. 2020; The CHEX-MATE Collaboration 2020; Zenteno et al. 2020). Among the possibilities, we have to mention the studies of the clusters morphology in X-ray and tSZ maps. Several indicators are commonly used, such as asymmetry parameter (Schade et al. 1995), light concentration (Santos et al. 2008), third-order power ratio (Buote & Tsai 1995; Weißmann et al. 2013), centroid shift (Mohr, Fabricant & Geller 1993; O’Hara et al. 2006), strip parameter, Gaussian fit parameter (Cialone et al. 2018), and so on. They exploit the maps with different apertures and efficiencies and are applied individually or combined together, even with different weights (see e.g. Böhringer et al. 2010; Nurgaliev et al. 2013; Rasia et al. 2013; Weißmann et al. 2013; Mantz et al. 2015; Cui et al. 2016; Lovisari et al. 2017; Cialone et al. 2018; Cao et al. 2020; DL20; Yuan & Han 2020). A complementary approach is by applying thresholds on specific thermodynamic variables. Among the others, the central electron gas density and the core entropy are fairly reliable (Hudson et al. 2010). The azimuthal scatter in radial profiles of gas density, temperature, entropy, or surface brightness (Vazza et al. 2011) is also used as a proxy of the ICM inhomogeneities and correlated to the clusters dynamical state (see e.g. Roncarelli et al. 2013; Ansarifard et al. 2020). Alternatively, the projected sky separations between key positions in the images are resulting in reliable estimators of the dynamical state. Interestingly, the offsets between the bright central galaxy (BCG) and the peaks and/or the centroids of X-ray or tSZ maps are an indication of how much the relaxation condition is satisfied, with different efficiency (see e.g. Jones & Forman 1984; Katayama et al. 2003; Lin & Mohr 2004; Sanderson, Edge & Smith 2009; Mann & Ebeling 2012; Rossetti et al. 2016; Lopes et al. 2018; DL20; Ricci et al. 2020; Zenteno et al. 2020). To be mentioned also other approaches based on wavelets analysis (Pierre & Starck 1998), on the Minkowski functionals (Beisbart, Valdarnini & Buchert 2001), or on machine learning (see e.g. Cohn & Battaglia 2019; Green et al. 2019; Gupta & Reichardt 2020).


Stellar population evolution studies beyond z ≈ 1 have primarily focused on ages through studies of luminosity changes. Beifiori et al. (2017) used new data for 19 galaxies in z = 1.3–1.6 clusters obtained with the Very Large Telescope/KMOS to extend the redshift coverage of the results regarding the evolution of the mass-to-light (M/L) ratios of bulge-dominated passive galaxies. The authors used their new results together with the available literature results covering up to z = 1.3 (van Dokkum & Franx 1996; Jørgensen et al. 1999, 2006, 2014; Kelson et al. 2000; Wuyts et al. 2004; Holden et al. 2005, 2010; Barr et al. 2006; van Dokkum & van der Marel 2007; Saglia et al. 2010; Jørgensen & Chiboucas 2013) and low-redshift reference data for the Coma cluster (Jørgensen 1999; Jørgensen et al. 2006) to further solidify the evidence supporting passive evolution and a formation redshift zform ≈ 2. The formation redshift should be understood as the epoch of the last major star formation episode. At z ≈ 1 the massive (Mass > 1011 M) bulge-dominated galaxies in clusters appear to be in place and mostly passively evolving. Lower mass galaxies may still be added to the red sequence and from then on passively evolve (e.g., Sánchez-Blázquez et al. 2009; Choi et al. 2014), but see also Cerulo et al. (2016) for results supporting that the red sequence well below L⋆ is fully populated in rich clusters already at

. Ultimately, the properties of galaxies mapped over a large fraction of the age of the universe, may constrain the models for building the galaxies. It is difficult to understand within the prevailing hierarchical model favored by the ΛCDM (cold dark matter) cosmology, the existence of such massive passive galaxies with relatively old stellar populations at z ≈ 1, while less massive galaxies appear to harbor younger stellar populations, e.g., Jørgensen et al. (2017, and references therein), see Kauffmann et al. (2003) for a discussion of this tension between the observational results and the hierarchical models of galaxy formation. However, more recent cosmological simulations like Illustris (Genel et al. 2014; Vogelsberger et al. 2014; Wellons et al. 2015) and UniverseMachine (Behroozi et al. 2019) find that massive quiescent galaxies can be in place by z ≳ 2.


Understanding of the low-mass star formation process has been intensively studied from decades ago, mainly by theoretical work at the beginning (e.g., Larson 1969; Shu 1977; Shu et al. 1987; Inutsuka 2012). They explained that the fragmentation and condensation of molecular clouds result in forming dense cores, which undergo gravitational collapse to form stars. According to the theoretical studies, dense cores eventually harbor the first protostellar cores, the first quasi-hydrostatic object during the star formation process (hereafter the first core; e.g., Larson 1969; Masunaga et al. 1998; Tomida et al. 2013), which provide the initial condition of star formation. Recent magnetohydrodynamic (MHD) simulations demonstrated that the first core with a size of 1–100 au is formed when the central density exceeds ∼1010 cm−3 via the gravitational collapse. The first core is suggested to have a low-velocity (1–10 km s−1) molecular outflow with a wide opening angle (Machida et al. 2008), which is qualitatively different from the collimated jet driven by a mature protostar. However, it is difficult to identify such an object observationally because the first core has a short lifetime, 103–104 yr, depending on the physical condition of the parental core (Tomida et al. 2010), and does not show bright infrared emission. Although some candidates of the first core were already reported in the past decade (e.g., Chen et al. 2010, 2012; Enoch et al. 2010; Pineda et al. 2011; Pezzuto et al. 2012; Hirano & Liu 2014), the first-core phase is not fully explored observationally and it is supposed to still be the missing link between the isothermal and adiabatic collapse (i.e., prestellar and protostellar core). To search for candidates of the first core, it is essential to perform a survey-type observation toward a large number of starless cores. According to the early dense core survey with an average density of ≳105 cm−3 (Onishi et al. 2002), the lifetime of the starless phase is ∼4 × 105 yr (see also Ward-Thompson et al. 2007). The simple calculation tells us that only one out of a few × 10–100 cores harbors the first core(s).


The scenario presented above has been formulated in various flavors to individually explain several of the properties reported here. The main addition of this work, namely the excitation of CO in distant main-sequence and starburst galaxies, fits in the general picture that we sketched. The ensemble of properties and correlations that we reported here can be also used to revisit the definition of what a starburst is. A standard operational classification is based on the distance from the observed empirical M⋆-SFR correlation, the main sequence. This proved to be a useful distinction and an excellent predictor of several trends (e.g., Sargent et al. 2014), but recent results, including our present and previous analysis (Puglisi et al. 2019), show that the demarcation between starburst and main-sequence galaxies is more blurred that we previously considered. We do detect starburst-like behaviors in galaxies on the main sequence (Elbaz et al. 2018), likely linked to the existence of transitional objects (Popping et al. 2017; Barro et al. 2017b; Gómez-Guijarro et al. 2019; Puglisi et al. 2019, and in prep. to limit the references to recent works based on submillimeter observations). Such transition might well imply an imminent increase of the SFR, driving the object in the realm of starbursts (e.g., Barro et al. 2017b), or its cessation, bringing the system back onto or even below the main sequence (Gómez-Guijarro et al. 2019; Puglisi et al. 2019), with the CO properties potentially able to distinguish between these two scenarios. Regardless of these transitional objects, a definition of starburst based on ΣSFR, rather than ΔMS, would naturally better account for the observed molecular gas excitation properties, dust temperatures and opacities, or SFE (see also Elbaz et al. 2011; Rujopakarn et al. 2011; Jiménez-Andrade et al. 2018; Tacconi et al. 2020). As an example, in Fig. 8 we show the mean SLED of the subsample of galaxies with both CO (2 − 1) and CO (5 − 4) coverage, split at its median ΣSFR. While only tentative at this stage, this suggests a trend of increasing CO excitation with ΣSFR, consistently with Fig. 7 and what mentioned above.

Compare/Contrast
Compare/Contrast
Future Work


We ignore the density stratification effect in Case I, II, and IIa, because the width of the horizontal current sheet in our simulations is much shorter than the length. The simulation domain extends from x = 0 to x = L0 in the x-direction, and from

y
=
−
0.5

L

0

 to

y
=
0.5

L

0

 in the y-direction, in the three cases, with

L

0

=

10

6

 m. Outflow boundary conditions are used in the x-direction and inflow boundary conditions in the y-direction. For the inflow boundary conditions, the fluid is allowed to flow into the domain but not to flow out; the gradient of the plasma density vanishes; the total energy is set such that the gradient in the thermal energy density vanishes; a vanishing gradient of parallel components plus divergence-free extrapolation of the magnetic field. For the outflow boundary conditions, the fluid is allowed to flow out of the domain but not to flow in, and the other variables are set by using the same method as the inflow boundary conditions. The horizontal force-free Harris current sheet is used as the initial equilibrium configuration of magnetic fields in Case I,
13

B

x
0

=
−

b

0

tanh
[
y

/

(
0.05

L

0

)
]

14

B

y
0

=
0

15

B

z
0

=

b

0

/

cosh
[
y

/

(
0.05

L

0

)
]
.

The magnetic fields in the low solar atmosphere could be very strong (Jin et al. 2009, 2012; Khomenko et al. 2014; Peter et al. 2014; Vissers et al. 2015) and the magnetic field can exceed 0.15 T in both the intranetwork and the network quiet region (e.g., Orozco Suárez et al. 2007; Martínez González et al. 2008; Jin et al. 2009, 2012). In the work by Jin et al. 2012, the maximum of the field strength was found to be 0.15 T. The magnetic field could be even stronger in the active region near the sunspot. Therefore, we set b0 = 0.05 T in Case I and Case II, and b0 = 0.15 T in Case IIa. Due to the force-freeness and neglect of gravity, the initial equilibrium thermal pressure is uniform. The initial temperature and plasma density are set as T0 = 4200 K and ρ0 = 1.66057 × 10−6 kg m−3 in Case I, and T0 = 4800 K and ρ0 = 3.32114 × 10−5 kg m−3 in Case II and Case IIa. Therefore, the initial plasma β is calculated as β ≃ 0.0583 in Case I, β ≃ 1.332 in Case II, and β ≃ 0.148 in Case IIa. The initial ionization degree is assumed as Yi = 10−3 in Case I, and Yi = 1. 2 × 10−4 in Case II and IIa. The magnetic diffusion in this work matches the form computed from the solar atmosphere model in Khomenko & Collados (2012), and we set

η
=

[

5
×

10

4

(
4200

/

T
)

1.5

+
1.76
×

10

−
3

T

0.5

Y

i

−
1

]

 m2 s−1 in Case I, and

η
=
[
5
×

10

4

(
4800

/

T
)

1.5

+
1.76

×

10

−
3

T

0.5

Y

i

−
1

]

 m2 s−1 in Case II and IIa. The first part ∼ T−1.5 is contributed by collisions between ions and electrons, the second part

∼

T

0.5

Y

i

−
1

 is contributed by collisions between electrons and neutral particles. Small perturbations for both magnetic fields and velocities at t = 0 make the current sheet to evolve and secondary instabilities start to appear later in the three cases. The forms of perturbations are listed below:
16

b

x
1

=
−
pert
·

b

0

·
sin

2
π

y
+
0.5

L

0

L

0

·
cos

2
π

x
+
0.5

L

0

L

0

17

b

y
1

=
pert
·

b

0

·
cos

2
π

y
+
0.5

L

0

L

0

·
sin

2
π

x
+
0.5

L

0

L

0

18

v

y
1

=
−
pert
·

v

A
0

·
sin

π

y

L

0

·

random

n

Max

(

∣

random

n

∣

)

,

where pert = 0.08, vA0 is the initial Alfvén velocity, randomn is the random noise function in our code, and

Max
(
∣

random

n

∣
)

 is the maximum of the absolute value of the random noise function. This random noise function makes the initial perturbations for the velocity in the y-direction to be asymmetric, and such an asymmetry makes the current sheet gradually become more tilted, especially after secondary islands appear. The reconnection process is not really symmetrical in nature (Murphy et al. 2012), this is one of the reasons that we use such a noise function. Another reason is that the asymmetric noise function makes the secondary instabilities develop faster. Figure 1(a) shows the distributions of the current density and magnetic fields at t = 0 in case I.


Early theoretical work on the formation of the first stars, mostly through high-resolution simulations, found that Pop III stars with mass MIII,* ≳ 100 M⊙ are born in isolation inside minihalos, and thus the “one massive Pop III star per minihalo” paradigm was established (Abel et al. 2002; Bromm et al. 2002; Yoshida et al. 2006). Later, however, several higher-resolution simulations began to observe the formation of binary protostar systems with a smaller mass range, MIII,* ≃ [10–40] M⊙ (Turk et al. 2009; Stacy et al. 2010). While the universality of the latter finding is in doubt (Greif et al. 2012; Stacy  Bromm 2013; Susa 2013; Hirano et al. 2014; see also Becerra et al. 2015 for the formation of Pop III stars inside more massive halos), it certainly introduces very important subtleties to the old paradigm. One important aspect is that X-ray binary systems may remain after some of these stars die, and can emit X-rays very efficiently (we will quantify their X-ray emissivity in Section 3) through gas accretion rate comparable to the Eddington limit (Mirabel et al. 2011). This then could make the X-ray heating epoch occur earlier than previously thought, or even allow a model where the reionization is dominated by X-ray photons instead of UV photons. Reionization dominated by X-ray photons, due to their long mean free path, will occur much more smoothly in space than reionization by UV photons (e.g., Haiman 2011; Mesinger et al. 2013, and references therein). In addition, X-rays can heat the IGM (see, e.g., recent observational signature reported by Parsons et al. 2014), which impacts the dynamics of the IGM (e.g., Tanaka et al. 2012; Jeon et al. 2014) and δTb of IGM (e.g., Mesinger et al. 2013; Fialkov et al. 2014; Jeon et al. 2014). High-redshift X-ray binaries seem to dominate the X-ray background over the active galactic nuclei at the late stage of EoR (z ∼ 6–8), if their emissivity and spectral energy distribution (SED) is calibrated by the observed, low-z (0 ≤ z ≲ 4) X-ray binaries (Fragos et al. 2013).


The first investigations of the rotational spectra of ethyl isocyanide were carried out in 1966 by Bolton et al. (1966). The spectra of the first vibrational and torsional excited states were measured in the centimeter wave domain (Anderson & Gwinn 1968). In this initial study, the dipole moment was determined to be μa = 3.79 D and μb = 1.31 D; this value is usually large for a molecule that includes a CN group. This causes dense and intense rotational spectra in the millimeter wave range and also in the submillimeter wave range up to 900 GHz (bQ lines). Anderson & Gwinn (1968) also observed some A–E splittings due to the internal rotation motion of the methyl group. The most recent spectroscopic study is from Krüger & Dreizler (1992) who reinvestigated the internal rotation measurements and also determined hyperfine coupling parameters due to the nitrogen quadrupole. As in our previous studies of ethyl cyanide isotopologs, it was not possible to observe internal rotation and hyperfine splittings due to our Doppler limited resolution. Our analysis was rather easy, starting from a prediction based on Krüger & Dreizler (1992) parameters. First, we analyzed and fit the most intense transitions, the aRh transitions, up to 330 GHz. These transitions were shifted only a few MHz from the initial predictions. Then bR and bQ lines were searched and included in the fit up to 330 GHz. Next, all the spectra were analyzed up to 990 GHz without difficulty. For the fitting, we employed ASFIT (Kisiel 2001) and predictions were made with SPCAT (Pickett 1991). The global fits included 6 transitions from Anderson & Gwinn (1968), 29 lines from Krüger & Dreizler (1992), and 2906 from this work. The maximal quantum numbers are J = 103 and Ka = 30. Both reductions A and S were tested. A reduction permits us to check theagreement of our new parameters set with those from Krüger & Dreizler (1992) (Table 1). Using S reduction slightly decreases root mean square from 30.3 to28.7 kHz. The condition numbers are nearly the same: 295 and 310 for the A and S reductions, respectively.The A reduction requires 23 parameters, but 5 additional parameters are required for the S reduction (Table 2). For this reason we used the A reduction even if this molecule is close to the prolate limit with kappa = −0.9521. Part of the new measurements are in Table 3. Owing to its large size, the complete version of the global fit Table S1 is supplied at the CDS. The fitting files .lin (S2), .par (S3), and the prediction .cat (S4) are also available at CDS.


Several methods are commonly used to estimate gas masses. The first method is using CO emission line fluxes (e.g., Daddi et al. 2010; Genzel et al. 2010; Tacconi et al. 2010, 2013). This method has uncertainties on the CO-to-H2 conversion factor, which changes depending on metallicity (Genzel et al. 2012), and on the CO excitation states when higher-J CO lines are used (e.g., Daddi et al. 2015; Riechers et al. 2020). Furthermore, observations of CO lines for galaxies at high redshifts are observationally expensive. The second method is converting a dust mass into a gas mass with an assumed gas-to-dust mass ratio (e.g., Santini et al. 2014; Béthermin et al. 2015). Because the gas-to-dust mass ratio depends on the metallicity (Leroy et al. 2011; Rémy-Ruyer et al. 2014), metallicity measurements are crucial for estimating the gas mass accurately. Gas masses can also be estimated with an empirically calibrated relation between single-band dust continuum flux at the Rayleigh–Jeans (R-J) tail and gas mass (e.g., Scoville et al. 2014, 2016; Groves et al. 2015). These scaling relations are calibrated with local galaxies or with local galaxies and SMGs up to z ∼ 2. In this method, the gas-to-dust mass ratio is included in the conversion factor, and therefore does not need to be considered. It remains unclear whether the empirical calibration methods are applicable to galaxies at z > 3 or how much scatter there is in the relationships. Given that dust continuum observations take much less time compared to the CO observations, using dust continuum as a tracer of gas has the advantage of increasing the number of galaxies at higher redshifts with individual gas estimates, but these will only be reliable when precise metallicities are available as well. Metallicity measurements based on rest-frame optical emission lines for dustier star-forming galaxies are thought to have larger uncertainties due to strong dust obscuration (e.g., Santini et al. 2010). Herrera-Camus et al. (2018) reported a discrepancy between metallicities derived with rest-frame optical emission lines and far-infrared (FIR) fine-structure lines for local (ultra) luminous infrared galaxies ((U)LIRGs).


The band-merged Hi-GAL product catalogue (Elia et al. 2016) is built as in Elia et al. (2013) and provides spectral energy distribution (SED) fit parameters to the individual clumps. The average angular size of the clumps is 25′′ at 250 μm. Using the heliocentric distances provided in the Hi-GAL product catalogue (described in Sect. 3.2.) and the SED fit parameters, the authors of the catalogue are able to provide linear diameters and masses of the clumps. The catalogue explores a wide range of linear diameters and masses, from sub-parsec (≤ 0.1 pc) to parsec scale (1−5 pc) with masses from 1 M⊙ to 105M⊙. These wide ranges mean that we probably mix several types of objects, from single star-forming cores to clumps containing multiple cores, even to entire clouds, depending on the distance of object. Most of these sources, however, fulfil the definition of clump, according to the definition of Bergin & Tafalla (2007). Dust temperatures for these clumps have been estimated through a grey-body fit, and searched in the range T = 5−40 K. Most of them are found between 10 and 20 K. The appearance of the SED and the parameters obtained through the grey-body fit allow for classifying the evolutionary stage of these objects. Three stages are identified: starless unbound and bound (pre-stellar) objects, and proto-stellar objects. The pre- and proto-stellar stages are distinguished from each other by the presence of a 70 μm source in a proto-stellar clump (e.g.  Dunham et al. 2008; Ragan et al. 2012; Veneziani et al. 2013). The bound versus unbound identification is obtained by using the mass-radius relation, well known as Larson’s third law, originally formulated as M(r) > 460 M⊙(r/ pc)1.9, with r the radius of the source (Larson 1981). Beyond 4−5 kpc two effects could lead to misclassifying the pre- and proto-stellar stages. First, different sensitivities of PACS and SPIRE could lead to missing a possible 70 μm counterpart of a source detected with SPIRE. Second, at large heliocentric distances, two or more pre- and proto-stellar sources could be detected as a single object as a result of lacking resolution, globally and simply labelled as proto-stellar. The first effect was partially mitigated by searching for a possible 70 μm counterpart that was not originally listed in the single-band catalogue through performing additional source detection at this band using a threshold less demanding than the initial one. Elia et al. (2016) provide statistics and a discussion about the ratio between pre- and proto-stellar clumps. The distribution of the three evolutionary stages is shown in a portion of the Galactic Plane in Fig. 1. Each panel represents an evolutionary stage in the longitude range 26 ≤l≤ 31 deg. The pre-stellar clumps are more extended in Galactic latitude than the proto-stellar clumps. The unbound clump distribution is hard to characterise because it is obscured by the proto-stellar clumps in the mid-plane, therefore we only consider clustered over-densities composed of pre- and proto-stellar clumps. These distributions are observed across the entire longitude range of this study.


The connection between the accretion flow and compact jets is routinely studied in the X-ray–radio luminosity (LX–LR) diagram. Here, the former traces the accretion luminosity and is a proxy for accretion rate, while the latter traces the jet and its brightness. Compact BH jets show an LX–LR correlation across ∼eight orders of magnitude in X-ray luminosity (Hannikainen et al. 1998; Corbel et al. 2000, 2003; Gallo et al. 2006); a subset of sources follows a radio-bright correlation with a power-law slope of β ≈ 0.6 (where $L_R \propto L_X^{\beta }$, while others follow a steeper correlation with β ≳ 1 at high X-ray luminosities, before re-joining the other track around LX ≈ 1035 erg s−1 (e.g. Coriat et al. 2011; Soleri & Fender 2011; Carotenuto et al. 2021, although discussion exists regarding the statistical robustness of this separation; e.g. Gallo, Degenaar & van den Eijnden 2018). The situation for NS LMXBs is even more complex. While, as a sample, NS LMXBs are ∼22 times radio-fainter than BH systems, the individual NS systems do not appear to follow a single correlation; significant scatter exists both between sources and between outbursts of the same source. Furthermore, due to their radio faintness, few sources have been monitored over a large range of X-ray luminosity – particularly below LX ≈ 1035 erg s−1, few NS LMXBs have been detected in radio (e.g. Tudor et al. 2017; Gusinskaia et al. 2020). For the sample of NS LMXBs, however, a power-law slope of β ≈ 0.4– −0.5 has been measured (Gallo et al. 2018), which is similar to BHs. Different radiative efficiencies may be expected for the two types of LMXBs–BHs can advect a fraction of the liberated gravitational energy across the event horizon, while the presence of an NS surface implies that all this liberated energy should, eventually, be radiated. Therefore, a similarity in the LX–LR coupling of the two source classes, which depends on this radiative efficiency, is surprising.


In principle, Fig. 5, central panel G, shows that (469219) 2016 HO3 may have been locked in a Kozai–Lidov resonance with ω librating about 270° for nearly 100 kyr and probably more. Because of the Kozai–Lidov resonance, both e (central panel E) and i (central panel F) oscillate with the same frequency but out of phase (for a more detailed view, see Fig. 4, panels E and F); when the value of e reaches its maximum the value of i is the lowest and vice versa ($\sqrt{1 - e^2} \cos i \sim$ constant, see Fig. 4, panel B). During the simulated time and for the nominal orbit, 469219 reaches perihelion and aphelion the farthest possible from the ecliptic. Fig. 5, G-panels, show that for other incarnations of the orbit of 469219, different from the nominal one, ω may librate about 90° as well during the simulated time interval. However, is this a true Kozai–Lidov resonance? Namouni (1999) has shown that the secular evolution of co-orbital objects is viewed more naturally in the erωr-plane, where er and ωr are the relative eccentricity and argument of perihelion computed as defined in Namouni's work (see equations 3 in Namouni 1999); these are based on the vector eccentricity and the vector inclination. Fig. 6 shows the multi-planet erωr-portrait for the nominal orbit of this object. It clearly resembles figs 13 and 19 in Namouni (1999). Asteroid 469219 librates around $\omega _{\rm r}=-90{^\circ }$ for Venus, the Earth, and Jupiter. This behaviour corresponds to domain III in Namouni (1999), horseshoe-retrograde satellite orbit transitions and librations (around $\omega _{\rm r}=-90{^\circ }$ or 90°). For a given cycle, the lower part corresponds to the horseshoe phase and the upper part to the quasi-satellite or retrograde satellite phase. This is not the Kozai–Lidov resonance; in this case, the Kozai–Lidov domain (domain II in Namouni 1999) is characterized by libration around $\omega _{\rm r}=0{^\circ }$ (or 180°) which is only briefly observed at the end of the backwards integrations (see Fig. 6). The Kozai–Lidov resonance is however in action at some stage in the orbits displayed in Figs 5 and 8. Our calculations show that the orbital evolution followed by 469219 is the result of the dominant secular perturbation of Jupiter as the periodic switching between co-orbital states ceases after about 8 kyr if Jupiter is removed from the calculations. Fig. 7 shows that, without Jupiter, 469219 switches between the Kozai–Lidov domain and that of horseshoe-quasi-satellite orbit transitions and librations (including both −90°and 90°). Jupiter plays a stabilizing role in the dynamics of objects following orbits similar to that of 469219. It is not surprising that Jupiter instead of the Earth or Venus is acting as main secular perturber of 469219. Ito & Tanikawa (1999) have shown that the inner planets share the effect of the secular perturbation from Jupiter; in fact, Venus and our planet exchange angular momentum (Ito & Tanikawa 2002). In their work, these authors argue that the inner planets maintain their stability by sharing and weakening the secular perturbation from Jupiter. Tanikawa & Ito (2007) have extended this analysis to conclude that, regarding the secular perturbation from Jupiter, the terrestrial planets form a collection of loosely connected mutually dynamically dependent massive objects. The existence of such planetary grouping has direct implications on the dynamical situation studied here; if Jupiter is removed from the calculations, the overlapping secular resonances and the recurrent dynamics disappear as well.


Our models show that high signal to noise ratio (S/N) spiral structures in the gas dynamics, together with a spiral seen in the dust continuum surface density, can put constraints on grain size comparing the observed amplitude of the spiral in the dust with that in the gas. There are multiple candidate sources that are known to have spirals in the continuum or the gas. However, to give an accurate estimate of the surface density and to derive a velocity residual map, we need data with high spatial and velocity resolution. Out of the recent high-resolution Disk Substructures at High Angular Resolution Project (DSHARP) survey of 20 disks, 3 were found to harbor spirals: Elias 27, WaOph 6, and IM Lup (Huang et al. 2018a). Elias 27 harbors the highest contrast spiral arms of the survey, but gravitational instability is a convincing explanation for this source (Pohl et al. 2015; Meru et al. 2017; Forgan et al. 2018, although see also Hall et al. 2018), which makes it unusable for this purpose, as further discussed in Sect. 5.3. Furthermore, Elias 27 and WaOph 6 suffer from cloud and outflow contamination in the CO gas (Andrews et al. 2018), which makes it impossible to use the kinematic data published so far. The IM Lup disk has two high contrast spiral arms in the millimeter continuum, and no cloud contamination (Andrews et al. 2018), so is an ideal candidate to analyze. Unfortunately, no spiral signal is detected in the gas for the IM Lup disk, but in Sect. 4.3 we will show that we can make a prediction on what the spiral looks like in the two velocity components. Other sources are considered by surveying the literature. MWC 758 is a promising source (Dong et al. 2018; Boehler et al. 2018), but the observations are currently too limited to be used. HD 100453 has a spiral detected in scattered light, continuum, and CO gas (Benisty et al. 2017; Rosotti et al. 2020), but the spiral seen in the gas is beyond the size of the continuum disk and the inner region of the disk has a noisy velocity map, potentially due to a warp in the inner disk, which makes the source not useful in our analysis. TW Hya harbors the only spirals in gas dynamics, potentially in the azimuthal velocity (Teague et al. 2019). The spirals are detected in the CO gas emission as well, but no spiral signal is present inthe dust continuum images. In the next subsection, we will analyze the TW Hya disk further and show that the observed spiral signal in the dynamics is interesting for further study, but not useful to test our model.


Two major limitations, often neglected, affect this type of analysis: the adoption of oversimplified models to describe the wide variety of observed galaxy SEDs and the presence of ‘systematic’ model uncertainties. This second limitation has been addressed in several studies already (e.g. Charlot, Worthey & Bressan 1996; Cerviño, Luridiana & Castander 2000; Conroy, Gunn & White 2009; Percival & Salaris 2009; Conroy & Gunn 2010; Conroy, White & Gunn 2010a). The difficulty of precisely quantifying systematic model uncertainties has led to mainly qualitative conclusions, leaving the problem unsolved. The first limitation is easier to tackle, for example, by using more physically realistic models of galaxy SEDs and combining these with advanced statistical techniques to extract physical constraints from data. This appears as the most promising route to fully exploit the information gathered by modern photometric and spectroscopic galaxy surveys. Yet, the several tools proposed so far to interpret galaxy SEDs in terms of physical parameters do not allow one to fully exploit the high quality of modern data. For example, most existing approaches rely on the adoption of a rigid physical model (e.g. analytic, two-parameter star formation histories combined with a standard dust attenuation curve and the assumption that all stars in a galaxy have the same metallicity) to describe galaxy SEDs (e.g. Bolzonella et al. 2010; Wuyts et al. 2011; Hernán-Caballero et al. 2013; Ilbert et al. 2013; Bauer et al. 2013; Muzzin et al. 2013; Lundgren et al. 2014; Kochiashvili et al. 2015; Mortlock et al. 2015; Kawinwanichakij et al. 2016). Even with the inclusion of superimposed bursts of star formation (e.g. Kauffmann et al. 2003; Gallazzi et al. 2005; Pozzetti et al. 2007; Gallazzi & Bell 2009; da Cunha et al. 2010), this does not allow a physically consistent description of the contributions by stars, gas and dust to the integrated emission from a galaxy, nor the inclusion of a potential AGN component (a notable exception is the approach of Pacifici et al. 2012, who incorporate star formation and chemical enrichment histories from numerical simulations of galaxy formation and emission from photoionized gas). Also, current spectral analysis tools are generally optimized to interpret either photometric or spectroscopic observations of galaxies, but not arbitrary combinations thereof. Finally, most existing tools suffer from additional limitations: many focus on the selection of ‘best-fitting’ parameters rather than on the uncertainties associated with these parameters (e.g. chi-square minimization techniques; Arnouts et al. 1999; Bolzonella, Miralles & Pelló 2000; Kriek et al. 2009); when this is not the case, the number of free parameters that can be explored is generally limited (e.g. with grid-based Bayesian techniques; da Cunha, Charlot & Elbaz 2008; Noll et al. 2009; Pacifici et al. 2012); and when more sophisticated (e.g. Markov Chain Monte Carlo, hereafter MCMC) techniques allow the exploration of more parameters, instrumental effects are generally not incorporated in the analysis (e.g. Acquaviva, Gawiser & Guaita 2011; Serra et al. 2011; Han & Han 2014).


Over time, the data releases have treated the Balmer line regions in different ways. The presence of the artificial curvature was first reported by Busca et al. (2013) in the context of the DR9 data release. To minimize this effect, a different scheme was used in DR12 (Alam et al. 2015, see their table 2) by using a linear function (instead of an iterative b-spline procedure) to interpolate the flux over the masked regions. Surprisingly, we observe that this data reduction change was only applied to the Balmer β, γ, and δ lines but not applied to the Balmer α line. For the latter one, the original iterative spline interpolation scheme has been used all along. As a result, an absorption-like feature at the location of Balmer α is found in SDSS data releases 9 up to now, i.e. the latest data release 14 (Abolfathi et al. 2018). To illustrate this, we show examples of calibration vectors for SDSS BOSS DR9 (Ahn et al. 2012; Dawson et al. 2013), DR12 (Alam et al. 2015), eBOSS DR14 (Dawson et al. 2016; Abolfathi et al. 2018) data release as well as calibration vectors for the MaNGA survey (Bundy et al. 2015) DR14 data release (Abolfathi et al. 2018) in Fig. 6. In the upper panel, we show small-scale features in calibration vectors from the same plate (number 3647) processed by the DR9 (blue), DR12 (green), and DR14 (red) pipelines. The spectra are obtained by normalizing the large-scale features in calibration vectors with a cubic b-spline with break points separated by 50 Å. The three spectra are mostly identical while the smooth curving features at the wavelengths of Balmer series in DR9 disappear in DR12 and DR14. From DR12, the pipeline corrects the features with linear interpolation across wavelength regions with Balmer β, γ, and δ lines (Alam et al. 2015). One can also observe the change in the median residual spectra of DR9 and DR12 in Fig. 1. However, the Hα feature remains uncorrected from DR9 to DR14. We also show a DR7 calibration vector (black) in which most of the wiggles are absent. As pointed previously, this is due to the fact that the DR7 pipeline interpolates the calibration vectors using an effective scale larger than that used in subsequent data releases.


The best-fit parameters are shown in Table 3. We find that the uncertainties in M* and

 are considerably large due to a degeneracy between the two parameters when all parameters are variable. Plotted in Figure 1 is the faint-end slope α as a function of redshift. Our results indicate that the best-fit values of α are about −2 at

 to 10, which are steeper than those at lower redshift (e.g.,

 at

 in Bouwens et al. 2015). We show the fitting results at

, 8, 9, and 10 in Figures 2, 3, 4, and 5, respectively. The top and bottom panels present the observed number densities and the best-fit luminosity functions in the image plane and the source plane, respectively. We also plot the results of previous blank-field surveys (Ouchi et al. 2009; Bradley et al. 2012; McLure et al. 2013; Oesch et al. 2013; Schenker et al. 2013; Bowler et al. 2014; Bouwens et al. 2015; Finkelstein et al. 2015; Calvi et al. 2016) and recent HFF results in other studies (Atek et al. 2015a; Laporte et al. 2016; McLeod et al. 2016). The best-fit parameters are consistent with those in previous studies. In the top panel of Figure 2, there may be an excess in the observed surface number density at

. The reason for this excess is not clear, although using a size–luminosity relation that gives smaller sizes at faint magnitudes may reduce this excess. At

, the observed number densities at the bright end are slightly larger than the number densities in the simulation. This is probably due to the existence of an overdense region of

 dropouts in the Abell 2744 cluster field. We discussed the properties of the overdensity in Ishigaki et al. (2016; see also Zheng et al. 2014 and Atek et al. 2015b). At

, although we detect no galaxies, we can place a constraint on the luminosity function from the non-detection. Based on the best-fit parameters where only

 is variable, ∼1.4 galaxies are expected to be detected in the HFF fields. The middle panels of Figures 2–5 show histograms of the number of the dropouts. It is seen that our samples push the magnitude limits of the luminosity functions significantly by up to ∼3 magnitude. The correlations between M* and α at z ∼6–7 and 8 are presented in Figure 6.


While this large-scale structure of the Universe (LSS) is also composed of dark matter and gas, it was through the galaxy distribution that it has started to be detected. Galaxy clusters were the first cosmic web features to be identified and studied because they are easily detectable through various techniques. Only with the advent of wide-area spectroscopic redshift surveys have other structures such as filaments begun to be systematically identified. Surveys such as the Two-Degree Field Galaxy Redshift Survey (2dFGRS, Colless et al. 2001), the Sloan Digital Sky Survey (SDSS, York et al. 2000), the Galaxy And Mass Assembly survey (GAMA, Driver et al. 2009), the Vimos Public Extragalactic Redshift Survey (VIPERS, Scodeggio et al. 2018), or the COSMOS survey (Scoville et al. 2007) have allowed us to obtain statistical samples of filaments and other LSS features. For example, Chen et al. (2016) and Tempel et al. (2014a) have produced filament catalogues in the SDSS (but see also the works by Aragón Calvo 2007; Sousbie et al. 2011; Rost et al. 2020; Shuntov et al. 2020; Kraljic et al. 2020, some of which also used the same algorithm as we used here). Other works such as Kraljic et al. (2018) and Alpaslan et al. (2014) detected filaments in GAMA, while Malavasi et al. (2017) detected filaments in VIPERS. Additionally, Gott et al. (2005), Iovino et al. (2016), and Kraljic et al. (2018) also identified walls in the SDSS, COSMOS, and GAMA surveys, respectively, while several projects are devoted to the analysis of voids (see e.g. Colberg et al. 2008, for a summary). Recently, not only spectroscopic surveys, but also the increased precision of photometric redshifts (e.g. in COSMOS and in the Canada-France-Hawaii Telescope Legacy Survey, CFHTLS, Laigle et al. 2016; Coupon et al. 2009) allowed for the detection of filaments in volumes of the Universe up to z ∼ 1 (Laigle et al. 2018; Sarron et al. 2019, but also Darvish et al. 2014).


In spite of these similarities, the angular profiles of these properties in two-dimensional simulations reveal a totally different accretion pattern. Within the stagnation radius, winds are mainly accreted in the polar region as a result of the low angular momentum there, whereas material with high angular momentum in the mid-plane outflow is subject to outflow, because the angular momentum of the stellar winds at a given radius decreases towards the pole. This ‘funnel’ accretion scenario is also found in other simulations with a spherical-like distribution of rotationally accreting material (Proga & Begelman 2003; Ressler et al. 2018). The density and temperature at the equatorial plane are higher than those in the polar region, and thus the highest mass accretion rate is found at the boundary of the polar and disc regions. Therefore the accretion pattern of the two-dimensional simulations is totally different from the inflow–outflow structure in the one-dimensional calculations. Another difference between our results is that the parameters of our best-fitting model, fq and vw, sn, are generally smaller than those in one-dimensional calculations. The smaller fq can be explained in three ways. First, our BH mass is double that in the one-dimensional case, which results in a deeper gravitational potential that allows for a more massive gas reservoir. Thus fq should be smaller to achieve the observed gas density. Second, our outer boundary is set at 17 arcsec, which is larger than the radius of the one-dimensional outer boundary of 12 arcsec. The larger outer boundary causes the mass injection rate by stellar winds to increase by 50 per cent. Because stellar winds are the only mass source in our models, a smaller fq is required to cancel out this extra mass injection. Third, a smaller vw, sn could also lead to a smaller fq owing to less efficient heating by supernova explosions. The smaller vw, sn is constrained by the temperature at large radii, especially the temperature between 10 and 20 arcsec, which is not fitted in the one-dimensional calculations.


The heating due to the reprocessing of non-thermal photons produced in the nebula can be efficient even at late times. Here, we treat these processes in a simple way as follows. At early times, electromagnetic cascades proceed in the saturation regime, leading to a flat energy spectrum up to ∼1 MeV (Metzger et al. 2014). At later times, the spectrum depends on the seed photon spectra, but it can roughly be estimated to be a flat spectrum from ∼1 eV to ∼0.1 TeV, while the supernova emission continues, which is expected based on more detailed calculations (e.g., Murase et al. 2015; K. Murase et al. 2017, in preparation). High-energy γ-rays (≳1 MeV) heat up the ejecta through the Compton scattering and the pair production process. X-ray and UV photons are absorbed and heat up the ejecta through the photoelectric (bound-free) absorption unless the ejecta are fully ionized. Here, we describe the heating rate as
12

Q

˙

rad

(
t
)
≈

(

f

γ

+

f

X
−
UV

,
bf

)

L

sd

,

where fγ and

f

X
−
UV

,
bf

 are the heating efficiencies of γ-rays and X-ray and UV photons to the spin-down luminosity, respectively. We calculate the frequency dependent heating efficiency of γ-rays at each time:
13

f

γ

(
t
)
=

∫

ν

min

ν

max

d
ν

ν

min
(

K

γ
,
ν

τ

γ
,
ν

,
1
)

∫

1

eV

1

TeV

d
ν

ν

,

where the frequency range of γ-rays is

(
h

ν

min

,
h

ν

max

)

=
(
10

keV
,
1

TeV
)

, and h is the Planck constant. Here, τγ, ν is the optical depth of the ejecta to γ-rays and Kγ, ν is the photon inelasticity at a given frequency, where the Klein–Nishina cross section and the cross section for the Bethe–Heitler pair production in the field of a carbon nucleus are taken into account (Chodorowski et al. 1992; Murase et al. 2015). Note that the coefficient of the γ-ray optical depth depends on the density profile of the ejecta. Here, we simply assume a density profile to be constant with the radius. Adopting a realistic density profile may result in different ejecta mass and velocity estimates by a factor of a few.


Observations have shown that the shape of the Lyα line is diverse. It includes broad damped absorption profiles, P-Cygni profiles, double-peak profiles, pure symmetric emission profiles, and combinations thereof (Kunth et al. 1998; Mas-Hesse et al. 2003; Shapley et al. 2003; Møller et al. 2004; Noll et al. 2004; Tapken et al. 2004; Venemans et al. 2005; Wilman et al. 2005). This variety can be understood through a detailed radiative transfer calculation, which is analytically solvable only for simple cases (e.g., a static, plane-parallel slab by Harrington 1973 and Neufeld 1990, and a static uniform sphere by Dijkstra et al. 2006). Later, numerical algorithms based on Monte Carlo techniques were developed to solve radiative transfer for more general cases. Now theoretical studies mostly rely on them (e.g., Spaans 1996; Loeb & Rybicki 1999; Ahn et al. 2000, 2002; Zheng & Miralda-Escudé 2002; Richling 2003; Cantalupo et al. 2005; Dijkstra et al. 2006; Hansen & Oh 2006; Tasitsiomi 2006; Verhamme et al. 2006, 2015; Laursen et al. 2013; Behrens et al. 2014; Duval et al. 2014; Gronke et al. 2015; Smith et al. 2019; Lao & Smith 2020; Michel-Dansac et al. 2020). Meanwhile, a galaxy model needs to be constructed to perform such a radiative transfer calculation. One can adopt a realistic galaxy model from hydrodynamical simulations. Galaxies from such simulations can be useful for performing a statistical study of Lyα properties, but they cannot be directly used to model individual galaxies in observations. Therefore it would be better to adopt a simple but manageable toy model for the purpose of reproducing observations. One example for such models is a shell model, in which a central Lyα source is surrounded by a constantly expanding, homogeneous, spherical shell of H i medium with dust. Although this shell model has surprisingly well reproduced many observed Lyα line profiles (e.g., Ahn 2004; Schaerer & Verhamme 2008; Verhamme et al. 2008; Schaerer et al. 2011; Gronke et al. 2015; Yang et al. 2016; Gronke 2017; Karman et al. 2017), because of its extreme simplicity and contrivance, there is still room for improvement (e.g., see Section 7.2 in Yang et al. 2016; Orlitová et al. 2018).


The radial distribution of S/H ratios and the corresponding gradient are shown in panel (b) of Fig. 12. We obtain a slope of −0.035 ± 0.006 dex kpc−1 (very similar to the one we obtain with the ICF of ADIS20, as can be seen in Table 8), which is consistent with the one of −0.041 ± 0.014 dex kpc−1 estimated by Rudolph et al. (2006) using FIR lines and also very similar to the slope of our O/H gradient. We report a dispersion around the S/H gradient of 0.10 dex, somewhat larger than the individual observational uncertainties. Recently, Fernández-Martín et al. (2017) reported a slope of −0.108 ± 0.006 dex kpc−1 using optical spectra for H  ii regions located at RG between 5 and 17 kpc. That value of the slope is considerably much steeper than our determination and other previous estimates from the literature (e.g. Shaver et al. 1983; Simpson et al. 1995; Afflerbach, Churchwell & Werner 1997; Rudolph et al. 2006). Esteban & García-Rojas (2018) observed some H ii regions with a very low ionization degree – O2 +/O  0.03 – but measurable [S iii] 6312 or 9069 Å lines. For such nebulae is possible to assume that S/H ≈ S+/H+ + S2 +/H+, since the contribution of S3 +/H+ in those objects is expected to be negligible. The objects with such properties are IC 5146, Sh 2-235, Sh 2-257, and Sh 2-271, whose total abundances of S/H range from 6.70 to 7.01 but, unfortunately, cover a rather narrow range of RG – between 9.3 and 11.7 kpc – and no confident gradient can be estimated with so small baseline. Inspecting Tables 3 and 4 we can see that there is a sizable group of low-ionization degree H ii regions that lack of determination of their S2 +/H+ ratios. This is because the rather faint auroral [S iii] 6312 Å line could not be detected in those objects. We plan to obtain additional optical spectra covering the bright nebular [S iii] 9069, 9532 Å lines of that group of nebulae for trying to increase the number of objects with S/H ratios determined without ICF and estimate a more precise gradient for this element.


Our simulation results show that the median detectable distances of targeted GW events from BH–NS mergers for a single 2nd generation GW detector and a network of such detectors are ∼300 Mpc and ∼700 Mpc, respectively (see Table 4). For comparison, Figure 12 shows that the detection rate and detectable distance for HLV (O3) are approximately the same as those for the case when only bKAGRA is running. This is basically consistent with the detection rate and the distance distribution of BH–NS merger candidates detected during LVC O3 (e.g., Anand et al. 2020; Antier et al. 2020b; Coughlin et al. 2020b; Gompertz et al. 2020; Kasliwal et al. 2020). In Section 3, we have shown that the kilonova absolute magnitude at 0.5 days after a BH–NS merger is mainly distributed in the range ∼–10 to ∼–15.5. In view of the fact that the limiting magnitude of the follow-up wide-field survey projects is almost ≲21 mag (e.g., Antier et al. 2020b; Gompertz et al. 2020; Coughlin et al. 2020b; Kasliwal et al. 2020; Wyatt et al. 2020), the maximum detectable distance for BH–NS kilonovae would be ≲200 Mpc, which can hardly cover the horizon of GW-triggered BH–NS merger events that O3 found (as shown in Figure 10). However, although BH–NS merger kilonovae can hardly be detected for the present search depths, Figures 9 and 11 reveal that there are great opportunities to discover on-axis afterglows associated with sGRBs or orphan afterglows if the BH components have a high-spin distribution. In order to cover the distance range for searching for BH–NS kilonovae for the network of 2nd generation GW detectors as completely as possible, a search limiting magnitude mlimit ≳ 23–24 is required as shown in Figure 10. Present survey projects could reach this search limiting magnitude by increasing exposure times and the number of simultaneous exposures. However, the GW candidates during O3 had very large localization areas with an average of thousands of square degrees (Antier et al. 2020b). Increasing exposure times makes it hard for the present survey projects to cover such large localization areas. Therefore, during the HLVK era, we recommend that survey projects may search for jet afterglows after GW triggers with a relatively shallow search limiting magnitude. If BH–NS mergers have a high location precision, a limiting magnitude of mlimit ≳ 23–24 can be reached, which gives a higher probability of discovering associated kilonovae.


In this work, we follow the method for measuring the IGM thermal state based on Voigt profile decomposition of the Ly α forest (Schaye et al. 1999; Ricotti et al. 2000; McDonald et al. 2001). In this approach, a transmission spectrum is treated as a superposition of multiple discrete Voigt profiles, with each line described by three parameters: redshift zabs, Doppler broadening b, and neutral hydrogen column density $N_{\rm H\, {\small I}}$. By studying the statistical properties of these parameters, i.e. the $b-N_{{{{\rm H\, {\small I}}}}{}}$ distribution, one can recover the thermal information encoded in the absorption profiles. The majority of past applications of this method constrained the IGM thermal state by fitting the low-b-$N_{\rm H\, {\small I}}$ cutoff of the $b-N_{{{{\rm H\, {\small I}}}}{}}$ distribution (Schaye et al. 1999, 2000; Ricotti et al. 2000; McDonald et al. 2001; Rudie, Steidel & Pettini 2012; Boera et al. 2014; Bolton et al. 2014; Garzilli, Theuns & Schaye 2015, 2020; Hiss et al. 2018; Rorai et al. 2018). The motivation for this approach is that the Ly α lines are broadened by both thermal motion and non-thermal broadening resulting from combinations of Hubble flow, peculiar velocities, and turbulence. By isolating the narrow lines in the Ly α forest that constitutes the lower cutoff in $b-N_{{{{\rm H\, {\small I}}}}{}}$ distributions, of which the line-of-sight component of non-thermal broadening is expected to be zero, the broadening should be purely thermal, thus allowing one to constrain the IGM thermal state. However, this method has three crucial drawbacks. First, the IGM thermal state actually impacts all the lines besides just the narrowest lines. Therefore, by restricting attention to data in the distribution outskirts, this approach throws away information and reduces the sensitivity to the IGM thermal state significantly (Rorai et al. 2018; Hiss et al. 2019). Secondly, in practice, determining the location of the cutoff is vulnerable to systematic effects, such as contamination from the narrow metal lines (Hiss et al. 2018; Rorai et al. 2018). Lastly, the results from this approach critically depend on the choice of low-b cutoff fitting techniques, where different techniques might result in inconsistent T0 and γ measurements (Hiss et al. 2018; Rorai et al. 2018).


To fully describe a population of streams, we need a realistic model of the Galaxy. The latest generation of hydrodynamical simulations have produced models that well reproduce a multitude of features observed in galaxies, both as individual objects (e.g., Wetzel et al. 2016) and as a population (e.g., Nelson et al. 2018; Pillepich et al. 2018). Part of their success is in the achieved high resolution, so at the present a Milky-Way-like galaxy is modeled by up to 140 million particles (Wetzel et al. 2016). Ultimately, we would like to have a description of a galaxy that is representative of these models, but at a fraction of the numerical cost. In the classic study, Hernquist & Ostriker (1992) developed a set of basis functions for density and gravitational potential that can reproduce complex morphology of galaxies with a small number of terms (e.g., Lowing et al. 2011; Lilley et al. 2018a, 2018b). These expansions reproduce the force field of an N-body simulation with a precision of a few percent, so representing the gravitational potential in our model with an expansion of basis functions is tempting. However, a truly realistic solution needs to accurately capture not only the current structure of the Galaxy but also its evolution in time. Even though the Milky Way has had a relatively quiet recent merger history, it is currently undergoing a major merger with the Large Magellanic Cloud (e.g., Besla et al. 2007; Peñarrubia et al. 2016), which has been a source of gravitational perturbation for at least a billion years—sufficiently long to affect stellar streams. Specifically, Law & Majewski (2010) showed that the only static, ellipsoidal halo that reproduces both the positions and radial velocities along the Sagittarius stream is triaxial. This model correctly predicted proper motions along the stream (Sohn et al. 2015), so it appears to be describing the effective potential well, even though it is cosmologically improbable (Debattista et al. 2013). On the other hand, modeling Sagittarius in a combined system of the Milky Way and the LMC relaxes the requirement for the dark matter halo to be triaxial (Vera-Ciro & Helmi 2013), signaling that having a model of the potential that is correct on average is no guarantee of recovering the true halo shape. To ensure that the complexities added to the model are realistic, the basis function expansion should thus be time dependent, simultaneously describing the interaction between the Milky Way and the LMC, while maintaining a thin and old disk. We relegate the development of such a model and its implementation for mapping the dark matter in the Galaxy to a future study.


Another way of obtaining information about the nucleosynthesis processes involved in producing carbon is to compare it with other elements that are characterised by a well-known source of production, as in the case of oxygen. In Fig. 5, we show the variation of [C/O] as a function of [Fe/H], which serves as a first-order approximation to the evolution with time. To calculate the [C/O] ratios, two oxygen abundance indicators are used independently. At subsolar metallicities, the abundance ratios with both oxygen indicators are mostly negative and show an increasing trend towards higher metallicity. This is explained by the fact that oxygen is entirely produced by SNe Type II from massive progenitors, which started to release theiryields at earlier ages in the Galaxy and, hence, at lower metallicities (e.g. Woosley & Weaver 1995). The massive stars producing carbon at low metallicities might be less massive than those producing oxygen (i.e. having a longer life), explaining a delayed contribution of carbon, hence, the negative [C/O] ratios. Alternatively, this could be explained by increasing O/C yields for more massive progenitors of SNeII. Once metallicity starts to increase, low- and intermediate-mass stars release carbon and massive stars start to eject more carbon than oxygen (Carigi et al. 2005). The [C/O] ratio seems to have a constant rise towards higher metallicities when using the forbidden oxygen line. However, in the case when the O I 6158 Å line is employed, we do observe that the maximum in [C/O] takes places close to solar metallicity to then become flat or decrease. This suggests that low-mass stars mostly contribute to carbon around solar metallicity, whereas at super-solar metallicities, massive stars produce carbon together with oxygen, thereby flattening or even decreasing the [C/O] ratio. This trend is in agreement with the metallicity dependent yields from Carigi et al. (2005), which provide higher carbon as [Fe/H] increases from massive stars (i.e. also increasing the O production) but lower carbon from low and intermediate mass stars as [Fe/H] increases (i.e. less production of C). The turning point of increased relative production of carbon from massive stars takes place at A(O) ~ 8.7 dex (see Fig. 2 of Carigi et al. 2005) which equals to [O/H] ~ 0.0 dex. This observed behaviour of [C/O] is in contrast to the steady increase of [C/O] up to [Fe/H] ~ 0.3 dex found, for example, by Franchini et al. (2021). Nevertheless, the general trend we find when using the [O I ] 6300 Å line is similar to the reported by Franchini et al. (2021), who use also that oxygen indicator. All thick-disk stars present negative [C/O] ratios and when using the oxygen line at 6158 Å thin-disk stars with [Fe/H] ≲ –0.2 have [C/O]  0 as well. Thick-disk stars and low-metallicity thin-disk stars at the same metallicity have similar [C/O] ratios, meaning that the balance between different production sites for oxygen and carbon is the same among both populations, despite [C/Fe] and [O/Fe] being systematically higher for thick-disk stars at a given metallicity.


In this section, we consider the plasma properties under the propagation of strong waves. In strong waves, the motion of electrons in the plasma becomes relativistic. However, different from free electrons that have a relativistic drift velocity in the direction of the incident electromagnetic wave (see Section 3.1), in plasma the space-charge potential is important in preventing the drift of electrons (Waltz & Manley 1978). For nonrelativistic electrons in plasma, if the wave duration τ is much larger than c/ωp, where

 is the plasma frequency, the drift velocity would be close to zero (Waltz & Manley 1978; Sprangle et al. 1990b). In this case, electrons in plasma under a strong wave would have a typical Lorentz factor (

) similar to that (γ) in the laboratory frame, so that

 is satisfied. Due to the relativistic and magnetic force effects, the propagation and dispersion properties of an electromagnetic wave depend on its amplitude. For a circular polarized wave, the dispersion relation in the laboratory frame is given by (e.g., Gibbon 2005; Macchi 2013; Macchi et al. 2013; see the Appendix)
11

The dispersion relation of strong electromagnetic waves is altered due to the effective electron mass increased by the relativistic effect (e.g., Sarachik & Schappert 1970; Gibbon 2005; Macchi 2013). One can define the effective plasma frequency as
12

so that the wave can propagate in the region where

. With respect to the nonrelativistic linear case, this is known as relativistically self-induced transparency. We note that since the dispersion depends on the electromagnetic field amplitude in the nonlinear case, the dispersion relation must be taken with care. The propagation of a pulse will be affected by the complicated effects of nonlinear propagation and dispersion, and finally the spatial and temporal shape of the pulse itself would also be modified. In particular, for linear polarization, the relativistic factor γ is not a constant (see Section 3.1). The propagation of the linearly polarized wave with a relativistic amplitude would lead to generation of the higher-order harmonics. Sprangle et al. (1990b) proved that the propagation of the first harmonic component, i.e., of the “main” wave, is still reasonably described by Equation (11) with

. Thus, we will directly adopt Equation (11) in the following discussion.


It is still debated if and how all the stellar debris efficiently circularizes via the stream–stream collision. Some hydrodynamical simulations show that the TDE disk retains a significantly elliptical shape because the orbital energy is not dissipated efficiently enough to reduce the eccentricity of the entire disk to zero in a reasonable time (Guillochon et al. 2014; Shiokawa et al. 2015; Sądowski et al. 2016). Lu & Bonnerot (2020) show that a significant fraction of the debris can become unbound causing an outflow from the self-interaction region. Nevertheless, the debris that remains bound eventually contributes to the accretion flow around the SMBH. This part of the debris stream will finally be circularized by energy dissipation, leading to the formation of a small, initially ring-like, accretion disk around the black hole (Hayasaki et al. 2013; Bonnerot et al. 2016; Hayasaki et al. 2016). Note that, in an inefficient debris circularization case, the subsequent fallback material interacts with the outer elliptical debris so that their combined effect on the subsequent evolution of the initial ring is negligible. Angular momentum conservation allows us to estimate the circularization radius of the stellar debris, which is given by
2

r

c

=
(
1
+

e

*

)

r

p

,
=

1
+

e

*

β

r

t

,

where e* is the orbital eccentricity of the stellar orbit, rp = rt/β is the pericenter distance radius. If debris circularization takes place only through dissipation at the self-interaction shock, the circularization timescale for the non-magnetized, most tightly bound debris can be estimated based on the ballistic approximation (Bonnerot et al. 2017) as
3

t

circ

≈

8.3

η

−
1

β

−
3

M

bh
,
6

−
5

/

3

t

mtb

∼

0.93

η

1.0

−
1

β

−
3

M

bh
,
6

−
7

/

6

m

*
,
1

−
1

r

*
,
1

3

/

2

yr
,

where the orbital period of the stellar debris on the most tightly bound orbit:
4

t

mtb

=

π

2

1

Ω

*

M

bh

m

*

1

/

2

≈

0.11

M

bh
,
6

1

/

2

m

*
,
1

−
1

r

*
,
1

3

/

2

yr
,

Ω

*

=

Gm

*

/

r

*

3

 is the dynamical angular frequency of the star, and we introduce η(≤1) as the circularization efficiency which represents how efficiently the kinetic energy at the stream–stream collision is dissipated and the most efficient (η = 1) case corresponds to that of Bonnerot et al. (2017). Note that tcirc is not the circularization timescale of all the stellar debris. Our interest here is in the circularization timescale and radius of the most tightly bound debris because the accretion of this debris contributes most to the delayed X-ray peak luminosity in terms of the emitted energy.


Some previous observations have shown that surface motions (e.g., shearing motions and converging motions) acting on preexisting coronal fields to form filaments and filament channels always act over a short period of a few days (Gaizauskas et al. 1997; Schmieder et al. 2004; Wang & Muglach 2007; Yan et al. 2015; Yang et al. 2015) to a period of months (Gaizauskas et al. 2001). However, in this event, the filament is formed within about 20 minutes. On this timescale, only the reformation of the filament in the same filament channel after the partial filament eruption has been observed (Tripathi et al. 2009a; Joshi et al. 2014). The rapid formation of the filament is a rare observation that has not been previously reported. Even though our event follows the “head-to-tail” scenario, there is a little difference. It is worth noting that the “head-to-tail” scenario seems to support the idea that the filament is supported by sheared arcades (Litvinenko & Wheatland 2005; Welsch et al. 2005). However, our event may present a picture in which the filament is supported by a twisted flux rope. Therefore, we could not fully exclude any other explanations, although we are inclined toward the “head-to-tail” scenario of Martens & Zwaan. Twisted structures in filaments have been previously observed during their eruptions (Wang et al. 1996; Bi et al. 2012, 2015; Li & Zhang 2013; Yan et al. 2014; Yang et al. 2014; Filippov et al. 2015; Raouafi 2009). In particular, Bi et al. (2012, 2015) found that the erupted filaments appear to be composed of two intertwined twisted flux ropes. However, in the present case, it is found that the flux rope appeared in the course of the formation of the filament, and as reported by Li & Zhang (2013; Yang et al. 2014), it can be observed in all seven AIA EUV lines formed from 0.05 MK to 11 MK. The AIA 304 Å observations (see panel (e) in Figure 2) also indicate that the filament may be composed of two sets of intertwined dark threadlike structures. These observations may be direct evidence that the magnetic structure of the formed filament is a flux rope. As mentioned before, the sigmoids and the elongated EUV channel-like structures are evidences for the existence of flux ropes in the higher solar atmosphere. More recently, observational evidence of the detailed structure and evolution of flux ropes in the lower solar atmosphere has been presented by Wang et al. (2015) and Kumar et al. (2015). The flux rope in our case can be observed in all of  the AIA EUV passbands, but it is still in the lower solar atmosphere. Our observation is quite similar to the formation of a chromospheric flux rope as a result of magnetic reconnection between two sheared cool Hα loops (Wang et al. 2015) in the lower atmosphere, but sheds more light on the formation of the filament. This event may also be a distinct example revealing the evolution of the flux rope in the lower solar atmosphere.


It has been thought that QPOs originate from the innermost part of an accretion disc, which is associated with strong gravity, so that we might detect general relativistic effects. Miller et al. (1998) proposed beat-frequency models and estimated the parameters of NSs using this model. Stella & Vietri (1999) developed the relativistic precession model. In the last 20 years, disc-oscillation and resonance models and wave models have been proposed (e.g. Osherovich & Titarchuk 1999; Abramowicz & Kluźniak 2001; Abramowicz et al. 2003; Zhang 2004; Li & Zhang 2005; Erkut, Psaltis & Alpar 2008; Shi & Li 2009, 2010; Shi 2011; Shi, Zhang & Li 2014, 2018; de Avellar et al. 2018). Shi & Li (2009, 2010) obtained the twin modes of MHD waves in LMXBs (including NS LMXBs and black hole LMXBs), which are considered as the sources of high-frequency QPOs. Shi, Zhang & Li (2014, 2018) also considered the waves produced by the two MHD oscillation modes at the magnetosphere radius as the origin of kHz QPOs. A relationship between the frequencies of the twin-peak kHz QPOs and the accretion rate, in which parallel tracks can be explained, was obtained (Shi, Zhang & Li 2018). Recently, many simulations on the oscillations of accreting tori in the accretion process of NS LMXBs (e.g. Kulkarni & Romanova 2013; Parthasarathy, Kluźniak, Čemeljić 2017) have been performed, and almost every model can reproduce some of the observed characteristics of QPOs. However, most models cannot fit the observed data perfectly, the observed data. Belloni, Méndez & Homan (2005) suggested that the twin kHz QPOs showed no intrinsically preferred frequency ratio, and this weakened support for the resonance models. Morsink & Stella (1999) were able to fit the overall NS data with different masses and spins of NSs using the relativistic precession model; however, Belloni, Méndez & Homan (2007) found that there were deviations between the expected and the observed trends. Recently, Török et al. (2016b, 2018) identified the observed QPO frequencies with the frequencies of the epicyclic modes of torus oscillations, and suggested that the relationship between the strong modulation of the X-ray flux and high values of QPO frequencies is connected to the orbital motion in the innermost part of an accretion disc. In addition, there are studies that compared a large set of models with the data of many sources in a complex manner (Lin et al. 2011; Török et al. 2012, 2016a).


Solar flares are powerful eruption events on the Sun associated with a rapid and violent release of magnetic free energy through a reconnection process. A typical flare can radiate at almost all wavelengths constituting the solar spectrum, ranging from radio through optical and ultraviolet (UV) to soft/hard X-ray (SXR/HXR) and even γ-rays (e.g., Benz 2017; Tan et al. 2020). Only a small part of the flare radiation is emitted at the shortest wavelengths in the X-ray and extreme-UV (EUV) ranges (Emslie et al. 2012). The quantitative estimation of the radiated flare energy partition suggested about 70% in white light (WL) for solar flares (e.g., Kretzschmar 2011) and 55%–80% in WL for stellar flares (e.g., Kuznetsov & Kolotkov 2021). In other words, most of the flare energy is radiated in the longer wavelengths (Kleint et al. 2016). Between those extremes, the solar UV spectrum from 1000 to 3000 Å, which can be further split into the far-ultraviolet (FUV), the mid-ultraviolet (MUV), and the near-ultraviolet (NUV), is thought to provide an important contribution to the flare radiation (Woods et al. 2006; Milligan et al. 2014; Dominique et al. 2018). For instance, the Lyα spectral line produced by the chromospheric neutral hydrogen, which is centered at 1216 Å (in the FUV spectrum), is among the spectral lines in which flares radiate the most (Allred et al. 2005; Curdt et al. 2001; Lu et al. 2021a). The hydrogen Balmer continuum emitted during flares, which is thought to be generated during the recombination of flare-produced free electrons in the chromosphere, is often detected in the MUV and NUV ranges, as well as close to the Balmer recombination edge at 3646 Å (Heinzel & Kleint 2014; Kotrč et al. 2016; Dominique et al. 2018). Both the Lyα and hydrogen Balmer continuum emissions during solar flares are expected to be nonthermal profiles, i.e., similar to the HXR radiation that is produced by the beam of electrons that are accelerated by the magnetic reconnection during the solar flare (e.g., Avrett et al. 1986; Rubio da Costa et al. 2009; Heinzel & Kleint 2014).


Outflows are ubiquitous in both luminous AGN and in local Seyfert galaxies, and occur on a wide range of physical scales, from highly ionised semi-relativistic winds and jets in the nuclear region at subparsec scales to galactic scale outflows seen in mildly ionised, molecular, and neutral gas (Morganti et al. 2016; Fiore et al. 2017; Fluetsch et al. 2019; Lutz et al. 2020; Veilleux et al. 2020, and references therein). In some cases molecular and ionised winds have similar velocities and are nearly co-spatial, suggesting a cooling sequence scenario where molecular gas forms from the cooling of the gas in the ionised wind (Richings & Faucher-Giguere 2017; Menci et al. 2019). Other AGN show ionised winds that are faster than the molecular winds, suggesting a different origin of the two phases (Veilleux et al. 2020, and references therein). The molecular phase is a crucial element of the feeding and feedback cycle of AGN because it constitutes the bulk of the total gas mass and it is the site of star formation activity. On galactic scales massive molecular winds are common in local Seyfert galaxies (e.g. Feruglio et al. 2010; Cicone et al. 2014; Dasyra et al. 2014; Morganti et al. 2015; García-Burillo et al. 2014, 2017, 2019); these winds likely suppress star formation (i.e. negative feedback) as they reduce the molecular gas reservoir by heating or expelling gas from the host-galaxy ISM. In late-type AGN-host galaxies, the gas kinematics appears complex at all scales, showing several components such as bars, rings, and (warped) discs, with high velocity dispersion regions (e.g. Shimizu et al. 2019; Feruglio et al. 2020; Fernández-Ontiveros et al. 2020; Alonso-Herrero et al. 2020; Aalto et al. 2020; Audibert et al. 2020). Accurate dynamical modelling of the molecular gas kinematics reveals kinematically decoupled nuclear structures, high velocity dispersion at nuclei, trailing spirals, and evidence of inflows and AGN-driven outflows. (e.g. Combes et al. 2019; Combes 2019, 2021). The outflow driving mechanism (wind shock, radiation pressure, or jet), their multiphase nature, and their relative weights and impact on the galaxy ISM are still open problems (Faucher-Giguère & Quataert 2012; Zubovas & King 2012; Richings & Faucher-Giguere 2017; Menci et al. 2019; Ishibashi et al. 2021). To date, far different outflow phases have been observed only for a handful of sources. Atomic, cold, and warm molecular outflows have been observed in radio galaxies (e.g. Morganti et al. 2007; Dasyra & Combes 2012; Dasyra et al. 2014; Tadhunter et al. 2014; Oosterloo et al. 2017). The nuclear semi-relativistic phase and the galaxy-scale molecular phase have been observed simultaneously in less than a dozen sources, with varied results: in some cases data suggest energy driven flows (Feruglio et al. 2015; Tombesi et al. 2015; Longinotti et al. 2018; Smith et al. 2019), in other cases data suggest momentum driven flows (e.g. García-Burillo et al. 2014; Feruglio et al. 2017; Fluetsch et al. 2019; Bischetti et al. 2019; Marasco et al. 2020). Fiore et al. (2017), using a compilation of local and high redshift winds, showed that there is a broad distribution of the momentum boost, suggesting that both energy- and momentum-conserving expansion may occur. Enlarging the sample of local AGN-host galaxies with outflows detected in different gas phases is important to understand the nature and driving mechanisms of galaxy-scale outflows.


An example of these implications is the so-called ‘dust budget crisis’ introduced in Section 4.4: the dust masses currently estimated at z > 5 are not compatible with standard dust production channels and require an overhaul in our models of the initial mass function for star formation, of supernova production rates, or of dust growth in the ISM. Overall, the dust production rate would need to increase by one to two orders of magnitudes, as shown by Rowlands et al. (2014). The growth of dust grains through accretion in the ISM has been proposed as a solution (e.g. Mancini et al. 2015; Michałowski 2015; Popping et al. 2017), but there are doubts on the efficiency of accretion at high z, where high dust temperatures due to the CMB (see Section 3.3) keep the desorption time-scale for accreted materials short (Ferrara et al. 2016). The dust budget crisis is not only a problem at high redshift; it is observed, e.g. in the Magellanic Clouds (SMC, LMC). As explained in Srinivasan et al. (2016) using the dust mass fits by Gordon et al. (2014), the dust replenishment time-scale in the SMC from stellar sources alone is expected to be larger than the dust destruction time-scale and, in the worst-case scenario, longer than the lifetime of the Universe. Similarly, the ratio between the best LMC dust mass estimate by Gordon et al. (2014) and the dust injection estimates by Riebel et al. (2012) results in an LMC replenishment time-scale of 34 ± 8 Gyr, exceeding the age of the Universe. Both the high redshift and the local Universe, therefore, show a dust budget crisis that could be alleviated – and, in the best case scenario, fully resolved – if the actual dust masses turned out to be lower than currently estimated, as our results suggest. More specifically, Rowlands et al. (2014) mention that dust opacity needs to be increased by just a factor of 7 to solve the high-redshift crisis (provided dust destruction by SNe is not efficient); in the LMC, the aforementioned replenishment time-scale would decrease to less than 2 Gyr if the dust mass were decreased by a factor of 20.