Resonant dynamical friction may have applications beyond the relaxation of IMBHs examined in this paper. It may affect all objects in stellar clusters much more massive than the individual constituents of the disk, if present, including massive stars, stellar mass black holes (BHs), or the center of mass of massive binaries. Furthermore, it is also expected to operate in any type of disk with a high number of particles, including active galactic nucleus (AGN) accretion disks. Previously, it has been argued that stars and BHs crossing the disk on low-inclination orbits get captured by Chandrasekhar dynamical friction into the disk (Bartos et al. 2017; Panamarev et al. 2018; Tagawa et al. 2020). An interesting implication is that, if BHs settle into the disk, they interact dynamically and form BH–BH binaries efficiently, and frequent dynamical interactions and gas effects drive the BHs to merger, producing gravitational waves (GWs) detectable by LIGO, VIRGO, and KAGRA (McKernan et al. 2014, 2018; Bartos et al. 2017; Leigh et al. 2018; Yang et al. 2019; Tagawa et al. 2020, 2021; Samsing et al. 2020). Mergers are also facilitated by Lidov–Kozai oscillations in anisotropic systems (Heisler & Tremaine 1986; Petrovich & Antonini 2017; Hamilton & Rafikov 2019). The results in this paper show that resonant dynamical friction may accelerate the capture of objects in the accretion disks by a factor proportional to the SMBH mass over the local disk mass for large orbital inclinations. Pressure and viscosity in a gaseous disk do not inhibit the orbit-averaged torque from the IMBH, which leads to realignment and the warping of the disk (Bregman & Alexander 2012). Thus, RDF may efficiently catalyze the alignment of the orbital planes of BHs even in low-luminosity AGN or Seyfert galaxies with relatively small disk masses, which may not be possible for Chandrasekhar dynamical friction. In fact, this mechanism extends the scope of the “AGN merger channel” for GW source populations even beyond low-luminosity AGN and Seyfert galaxies, as it may organize BHs into disks also in nonactive galaxies with nuclear stellar disks.


Solar-like oscillations are excited stochastically by motions in the convective envelope of stars. For low-mass stars, they are found in all evolutionary states, between the main sequence and horizontal branch of helium-core burning stars (e.g. Leighton et al. 1962; Frandsen et al. 2002; Carrier et al. 2003; Hekker et al. 2009; Chaplin et al. 2011; Huber et al. 2011; Kallinger et al. 2012; Mosser et al. 2013) and were even detected in the M5 super giant α Her (Moravveji et al. 2013). These very characteristic oscillations lead to a nearly regular spaced comb-like pattern in the power spectrum. It was shown by Deubner (1975) that this ridge structure is governed by the degree of oscillation modes and resembles the predictions made by Ando & Osaki (1975). Empirically, the frequency patterns of solar-like oscillations were described through scaling relations by Kjeldsen & Bedding (1995). Since then, these relations have been tested and revised from large sample studies based on high-precision space photometry of red giants in clusters and in eclipsing binaries besides single stars (e.g. Corsaro et al. 2012b; Frandsen et al. 2013; Kallinger et al. 2010). Indications of non-radial oscillation modes were found in the variations of the absorption lines of bright red giants (Hekker et al. 2006; Hekker & Aerts 2010), but were firmly established in a large set of red giants observed with the CoRoT satellite (De Ridder et al. 2009). The identification of dipole mixed modes extended the sensitivity of the seismic analyses towards the core of evolved stars also (Beck et al. 2011; Bedding et al. 2011; Mosser et al. 2011). The analysis of solar-like oscillations enabled us to unravel many open questions on stellar structure and evolution, such as constraining the internal rotational gradient (Elsworth et al. 1995; Beck et al. 2012; Deheuvels et al. 2012) or determining the evolutionary status in terms of nuclear burning of a given red giant star (Bedding et al. 2011; Mosser et al. 2011).


In this context, massive (ℳ>1011 M⊙) PGs (MPGs) deserve particular attention. These systems are expected to evolve mainly through (dry) mergers (e.g. Hopkins et al. 2009; De Lucia & Blaizot 2007). If this is the case, in this mass range we should detect a stronger signal of the size-growth with respect to a lower mass range. So far, because MPGs are extremely rare, there have been very few studies that have investigated the combined evolution of the number density and of the age of MPGs as a function of their compactness (Carollo et al. 2013; Fagioli et al. 2016). Carollo et al. (2013) found that the number density of massive quiescent and elliptical galaxies with Re 2.5 kpc decreases by about 30% from z~1 to z~0.2 and that their U−V colours are consistent with passive evolution. They concluded that the driving mechanism for the average size-growth of the whole population is the appearance at later epochs of larger quiescent galaxies. More recently, Fagioli et al. (2016, hereafter F16) analysed the spectroscopic properties of ~500 MPGs (defined as galaxies with absent or very weak emission lines and no MIPS detections) at 0.2 z 0.8 in the zCOSMOS-bright 20 K catalogue (Lilly et al. 2007). From the analysis of stacked spectra of small and large MPGs, they dated the stellar content of these groups and found that the two sub-populations have similar ages. The authors concluded that, in this mass regime, the size growth of individual galaxies through dry mergers is the most likely explanation for the increase in the mean effective radius of the whole population. A recent analysis by Zahid et al. (2016) on the physical properties of compact post starburst galaxies at 0.2z0.8 with ℳ>1011 M⊙ provides new insights. On the basis of both their number density and of their ages, which have been found to be 1 Gyr, the authors suggest that this class of objects are the progenitors of compact quiescent galaxies. They conclude that a substantial fraction of dense quiescent galaxies at z0.8 are newly formed.


Also addressing the evolution of a stellar cluster is work by Parker & Goodwin (2009) and Parker et al. (2009) in which N-body simulations are used to analyse the prevalence of planets susceptible to the Kozai effect from a binary companion, and the stability of binaries in a dense cluster environment respectively. Parker & Goodwin (2009) found that around $20{{\ \rm per\ cent}}$ of all exoplanets should at one point in their lives be in the presence of a binary companion that has been sufficiently inclined by its cluster environment such that Kozai cycles can occur. This is an intriguing finding that bolsters the idea that a binary companion perturbed by the cluster can have an appreciable effect on the planets, with the caveat that the authors only examined the evolution of binary orbits, as their simulations lacked planetary bodies. Additionally, they particularly focused on very dense clusters similar to Orion, with a half-mass radius of only 0.1 pc. This is not typical for embedded clusters, which have typical half mass radii of ∼0.8 pc (Lada & Lada 2003), and therefore would have less frequent interactions between cluster stars and a particular binary. Nevertheless, they showed that a dense cluster environment can significantly alter the architecture of a stellar binary which in turn can affect a protoplanetary disc or mature planet system. Parker et al. (2009) focused on similarly dense cluster environments, but instead explored the longevity of moderately wide (${\sim }10^3 \, \text{au}$) to ultrawide (${\gt}10^4 \, \text{au}$) binaries. They found that cluster environments strip away all ultra-wide binary companions, and that the denser clusters, with half mass radii of 0.1–0.2 pc do not retain any binaries with separations ${\gt}10^3 \, \text{au}$. The less dense clusters, with half mass radii 0.4–0.8 pc, do retain some of these moderately wide binaries. The authors noted that as ultrawide binaries are often stripped in only a few cluster crossing times, these very separated binaries may form in isolation An alternate channel is that ultrawide binaries are formed during cluster dissolution (Kouwenhoven et al. 2010).


Several qLMXBs have been identified in GCs and in the Galactic field (for some examples, see table 4 in Guillot et al. 2009 and references therein). While LMXBs in the field were detected following the onset of a bright accretion outburst, most qLMXBs in GCs, including all those with the highest flux at Earth, have not shown accretion activity.3 Most of these sources have only been spectrally identified based on their similarities to field LMXBs, observed during quiescence (e.g. Cen X-4 or Aql X-1). Previous works have confirmed that H-atmosphere models accurately describe the spectra of qLMXBs, with radii in the range 10–15 km, as expected for NSs, either from single sources (e.g. Heinke et al. 2006a; Webb & Barret 2007; Guillot, Rutledge & Brown 2011; Heinke et al. 2014; Bogdanov et al. 2016), or from statistical analyses of multiple qLMXBs (e.g. Guillot et al. 2013; Guillot & Rutledge 2014; Lattimer & Steiner 2014; Guillot 2016; Steiner et al. 2018). However, in some cases the accreted material may not be hydrogen, but helium (e.g. Servillat et al. 2012; Catuneanu et al. 2013; Heinke et al. 2014). One way to circumvent this is to identify the nature of the donor star, i.e. to determine the nature of the material transferred on to the NS (e.g. with the detection of an H α emission line, presumably originating in a faint accretion disc, Haggard et al. 2004). The possibility of helium (or heavier element) atmospheres is well-founded on the existence of ultracompact X-ray binaries (UCXB), with white dwarfs or helium-dominated donors4 (e.g. Zurek et al. 2009; Altamirano et al. 2010; Sanna et al. 2017; Cadelano et al. 2019). In fact, around 1/3 of the LMXBs in GCs with constraints on the companion nature, possess a white dwarf donor (Bahramian et al. 2014). Since NS He-atmosphere models have harder spectra than H-atmosphere models, using the incorrect composition for the observed thermal emission can result in biases of the inferred radii (Servillat et al. 2012; Heinke et al. 2014).


A variety of formation channels have been proposed for binary black holes (BBHs; see e.g. Mapelli 2021 for a recent review): BBH mergers can be the outcome of isolated binary evolution via common envelope (Tutukov & Yungelson 1973; Bethe & Brown 1998; Portegies Zwart & Yungelson 1998; Belczynski, Kalogera & Bulik 2002; Belczynski et al. 2008, 2016a; Dvorkin et al. 2016, 2018; Eldridge & Stanway 2016; Mapelli et al. 2017, 2019; Stevenson, Berry & Mandel 2017; Kruckow et al. 2018; Spera et al. 2019; Belczynski et al. 2020; Klencki et al. 2021; Olejak, Belczynski & Ivanova 2021; Tanikawa et al. 2021a), stable mass transfer (Giacobbo, Mapelli & Spera 2018; Neijssel et al. 2019; Bavera et al. 2021; Gallegos-Garcia et al. 2021; Shao & Li 2021), or chemically homogeneous evolution (de Mink & Mandel 2016; Mandel & de Mink 2016; Marchant et al. 2016; du Buisson et al. 2020; Riley et al. 2021). Alternatively, BBHs can form dynamically in triples (e.g. Antonini, Toonen & Hamers 2017; Silsbee & Tremaine 2017; Fragione & Silk 2020; Arca Sedda, Li & Kocsis 2021a; Vigna-Gómez et al. 2021), multiples (e.g. Fragione & Kocsis 2019; Liu & Lai 2019, 2021; Hamers & Safarzadeh 2020), young star clusters (YSCs; Banerjee, Baumgardt & Kroupa 2010; Mapelli 2016; Banerjee 2017, 2021; Di Carlo et al. 2019, 2020a; Kumamoto, Fujii & Tanikawa 2019, 2020), globular clusters (GCs; Portegies Zwart & McMillan 2000; Tanikawa 2013; Samsing, MacLeod & Ramirez-Ruiz 2014; Rodriguez, Chatterjee & Rasio 2016; Askar et al. 2017; Fragione & Kocsis 2018; Hong et al. 2018; Choksi et al. 2019; Kamlah et al. 2022), and nuclear star clusters (NSCs; Antonini & Rasio 2016; Petrovich & Antonini 2017; Antonini, Gieles & Gualandris 2019; Arca Sedda 2020; Arca Sedda et al. 2020; Fragione, Loeb & Rasio 2020). Furthermore, gas torques in active galactic nucleus (AGN) discs trigger the formation of BBHs and speed up their mergers (e.g. Bartos et al. 2017; Stone, Metzger & Haiman 2017; McKernan et al. 2018; Yang et al. 2019; Ishibashi & Gröbner 2020; Tagawa, Haiman & Kocsis 2020). Finally, primordial black holes (BHs), born from gravitational collapses in the early Universe, might also pair up and merge via gravitational wave (GW) emission (e.g. Carr & Hawking 1974; Carr, Kühnel & Sandstad 2016; Sasaki et al. 2016; Ali-Haïmoud, Kovetz & Kamionkowski 2017; Clesse & García-Bellido 2017; De Luca et al. 2021).


The only gas-phase process, which is predicted to efficiently lead to products, is the process involving ionized methanimine. According to the model by Vuitton et al. (2007), the amount in the upper atmosphere of Titan of ionized methanimine is small, but not negligible. The products of the reaction CH2NH + CH2NH+ all have a mass-to-charge ratio of 57, where an important contribution is given by the abundant carbocation C4H\hbox{$_{9}^{+}$}+9. In this condition, it is difficult to see if any of these species is present in small amounts in the ionosphere of Titan. Interestingly, we have also investigated the further reaction of 1+ with a third molecule of methanimine. The formation of the species (CH2NH)\hbox{$_{3}^{+}$}+3, starting from two molecules of methanimine and the ionic species CH2NH+, is a strongly exothermic reaction, being \hbox{$\Delta H_{0}^{\circ} = -305$}ΔH0◦=−305 kJ/mol at CCSD(T) level. The optimized structure of (CH2NH)\hbox{$_{3}^{+}$}+3 is shown in Fig. 10. We can conclude, therefore, that polymerization of methanimine in the gas-phase at low temperatures may well be initiated by the presence of an ionized molecule. As for the experiment on ice by Bernstein et al. (1995), it is well known that reaction barriers possibly present in gas-phase reactions are not significantly reduced when moving to ice-mediated reactions (see, for instance, Rimola et al. 2014). Normally, the tunneling effect is invoked to explain the observed reactivity, but in this case the reaction barriers are so high that it is difficult to think that a reaction sequence starting with dimerization of neutral methanimine molecules can account for the observed formation of hexamethylenetetramine or polymethylenimine. The reaction must start by involving a radical or an ionized species and not two neutral closed shell molecules. This was already noted by Vinogradoff et al. (2012), who suggested that the reaction between two neutral methanimine molecules is mediated by formic acid, which acts as a proton donor, and by Cottin et al. (2001), who irradiated mixed ice with protons. Notably, Vinogradoff et al. (2013) failed to see hexamethylenetetramine and polymethylenimine formation starting from pure CH2=NH ice. Since in the experiment by Bernstein et al. (1995) the ice was irradiated by VUV photons at the Lyman alpha wavelength, and we now know that methanimine can be efficiently ionized by those photons, we can also argue that ionization of several methanimine molecules can instead trigger the process in cold interstellar ices. This statement is in line with what is known for gas-phase polymerization of olefin (El-Shall 2008; Cottin et al. 2001). Alternatively, external strong energy sources that can induce local nonequilibrium conditions, could promote neutral-neutral dimerization. For instance, in a study by Zhou et al. (2010), in which acetylene ices were irradiated with energetic electrons, electronically excited acetylene molecules were invoked to account for the experimental observation of benzene formation.


While most systems with exo-Kuiper belts do not have known planetary mass companions, in a few of these it has been possible to directly image one, thus enabling the study of planet–disc interactions in more detail. There are well-known examples such as β Pic with a massive planet possibly warping the disc (Mouillet et al. 1997; Lagrange et al. 2012, 2019; Matrà et al. 2019); HR 8799 with four giant planets creating a scattered disc and possibly replenishing its warm dust closer in (e.g. Marois et al. 2010; Booth et al. 2016; Zurlo et al. 2016; Read et al. 2018; Wilner et al. 2018; Geiler et al. 2019; Faramaz et al., in preparation); HD 95086’s axisymmetric disc implying a low eccentricity of its 4MJup planet (Rameau et al. 2016; Su et al. 2017); and Fomalhaut having a narrow and eccentric planetesimal belt (Kalas, Graham & Clampin 2005; Acke et al. 2012; Boley et al. 2012; MacGregor et al. 2017), implying that its candidate companion on an eccentric orbit has a low mass (∼Earth or super-Earth) and is not sculpting the belt (Quillen 2006; Kalas et al. 2008; Chiang et al. 2009; Beust et al. 2014; Faramaz et al. 2015), or is not a compact object but rather the dusty aftermath of a recent planetesimal collision (Gaspar & Rieke 2020). Some exo-Kuiper belt host systems even have companions in the brown dwarf (BD) or low stellar mass regime, suggesting that their likely formation through gravitational instability (Boss 1997, 2003, 2011; Vorobyov 2013) is compatible with the formation of massive Kuiper belt analogues, e.g. HR 2562 (Konopacky et al. 2016), HD 193571 (Musso Barcucci et al. 2019), HD 92536 (Launhardt et al. 2020), and HD 206893 (Milli et al. 2017). The last one is the subject of this paper. For even more massive companions, Yelverton et al. (2019) found a significant lower detection rate of debris discs around binaries, with no discs detected in binaries with separations between 25 and 135 au (comparable to typical debris disc radii; Matrà et al. 2018b). This is likely due to dynamical perturbation inhibiting planetesimal formation or clearing any debris disc formed near those separations.


Amongst the oldest stellar systems known to exist in the Milky Way (MW) are metal-poor globular clusters (GCs). These accumulations of stars do not seem to have undergone substantial star formation for extended periods. Given the limited quality of the available data, for a long time color-magnitude diagrams (CMDs) of GCs appeared to be narrow and could be readily described by a single isochrone. These observations have justified the establishment of the long-lasting paradigm that considers CGs as prime examples of simple stellar populations (SSPs), that is, the results of very short bursts of star formation in their natal clouds. However, improved photometric precision indicates the presence of sub-populations in the cluster CMDs that are inconsistent with the SSP assumption, for a number of luminous GCs in a variety of bandpasses. Thus, early detections of chemical abundance variations (e.g., Cohen 1978) could be more easily explained in a scenario involving several populations. Moreover, in recent years evidence has grown supporting the statement that GCs are generally composed of two or three chemically distinct populations. These subpopulations are separated by a few tens to hundreds of Myr in age and show vastly varying abundances of light elements such as C, N, O, Na, Mg, and Al (see, e.g., Carretta et al. 2009b; Gratton et al. 2012, and references therein). Theoretical considerations (see, e.g., D’Ercole et al. 2008, 2011) imply that GCs could have lost the majority of the initial stellar content of the first population, which consequently should have ended up in the Galactic halo. In fact, numerous studies found metal-poor GCs to be consistent with the abundance trends of the MW halo at equally low metal content (e.g., Pritzl et al. 2005; Koch et al. 2009; Koch & McWilliam 2014; Villanova et al. 2016). We address this scenario by adding NGC 6426 to the short list of metal-poor clusters with available information on detailed chemical abundances. There are only two GCs in the Harris catalog (Harris 1996, 2010 edition) more metal poor than NGC 6426. At 12.9 ± 1.0 Gyr, the cluster is the oldest in the age compilation by Salaris & Weiss (2002). At a galactocentric distance of Rgc = 14.4 kpc and a galactic latitude of 16.23° it is located in the transition region between inner and outer halo. Previous studies found consistent [Fe/H]1 values: −2.20 ± 0.17 dex (Zinn & West 1984), −2.33 ± 0.15 (Hatzidimitriou et al. 1999), and −2.39 ± 0.04 dex (Dias et al. 2015). The latter value originates from the very first spectroscopic analysis of NGC 6426 at low resolution, which also stated [Mg/Fe] = 0.38 ± 0.06 dex. To date, there has been no study further addressing the detailed metal content of this cluster.


In this paper, we focus on the bright elliptical galaxy NGC 5813. This is the central dominant member of a subgroup, hereafter referred to as the NGC 5813 group with an extensive diffuse X-ray emission. While part of the well-isolated NGC 5846 group ($z$ = 0.006578; e.g. Mahdavi, Trentham & Tully 2005a; Machacek et al. 2011) with a projected separation of ∼740 kpc, the two show no signs of an interaction between them. NGC 5813 itself has been assumed to be dynamically old (Emsellem et al. 2007) with no evidence of a recent major merger in its history, as indicated by the lack of any significant disturbances to its dusty circumnuclear disc (Tran et al. 2001). Regarding its atmosphere, the NGC 5813 group has a rather well defined regular morphology, consisting of three collinear pairs of cavities and associated shock fronts that are products of three distinct outburst events in the central AGN’s history (Randall et al. 2011). Such cavities often appear as a result of AGN-ICM or IGrM interaction with the radio lobes produced by the AGN, displacing the surrounding X-ray gas (e.g. McNamara & Nulsen 2012; Barai et al. 2016; Yang, Gaspari & Marlow 2019; Gastaldello et al. 2021). It has been estimated that an energy of 1.5 × 1056 and $4\times 10^{57}\,{\rm erg}$ has been released by the most and second most recent outbursts, respectively, suggesting that the most recent outburst might still be ongoing. These features are correlated with the group’s observed radio emission (Giacintucci et al. 2011), and can serve as clear indicators of AGN feedback being present. Previous studies of the group have also found moderate turbulent velocities of ∼175 km s−1 and a 3D Mach number of the order of 0.4, using resonant scattering (e.g. de Plaa et al. 2012; Ogorzalek et al. 2017), further indicating the presence of such a mechanism. These characteristics make NGC 5813, a very promising candidate for the study of the effects AGN feedback can have on the distribution of elements in a low-mass system. Additionally, this target has some of the deepest Chandra data of any galaxy group and, similar to M49 has been found to have an explicit anticorrelation between its metal abundance and the location of its radio lobes (Randall et al. 2015). That work already suggests that this result is dependent on the Fe-bias; employing a two-temperature fit brings the abundance in the region of the extended radio lobes in better agreement to their surrounding medium, although the central region appears to remain under-enriched.


The comparison with other hot corinos is not easy since we have a limited number of lines, in contrast to well-known sources that have been more extensively observed with ALMA and NOEMA, such as IRAS 16293-2422 (Jørgensen et al. 2016), NGC1333 IRAS 2A and 4A (Taquet et al. 2015; López-Sepulcre et al. 2017). In general, we find a similar inventory ofCOMs but with lower abundances in B1b-S. We are going to discuss only some trends and ratios between them. For example, one similarity with hot corinos are the higher abundances of O-bearing COMs with respect to N-bearing COMs. We observe a similar trend in B1b-S, where the highest abundances are obtained for CH3OCOH, CH3OCH3, and CH3CHO, while NH2CN and NH2CHO show the lowest abundances. Cyanamide has been recently detected towards the low-mass protostars IRAS 16293-2422 and NGC 1333 IRAS2A (Coutens et al. 2018), with observed NH2CN to NH2CHO ratios of 0.2and 0.02, respectively. These values are in the range of those observed towards the molecular clouds in the Galactic centre but lower than in Orion KL (Coutens et al. 2018). We obtain in B1b-S anabundance ratio of 0.25, similar to IRAS 16293-2422. Of the three possible isomers of C2 H4O2, we have detected CH3OCOH and CH2OHCHO. The observed CH3OCOH to CH2OHCHO ratio in B1b-S, ~20, is similar to that observed in the low-mass protostars in NGC 1333 and IRAS 16293-2422 (Taquet et al. 2015; Jørgensen et al. 2012). Acetic acid (CH3COOH) is the most stable but the least abundant of the three isomers (Lattelais et al. 2010). It has been observed in IRAS 16293-2422 with a ratio with respect to glycolaldehyde of ~11 (Jørgensen et al. 2016), consistent with the 5–15 upper limits in B1b-S. Glycolaldehyde and its corresponding alcohol, ethylene glycol, show similar abundances in B1b-S, slightly higher for CH2OHCHO. This is in contrast to other hot cores and hot corinos (Fuente et al. 2014; Jørgensen et al. 2016; Favre et al. 2017). However, since detections of these species are based on just a few lines, in particular for glycolaldehyde, column densities may be not well constrained and it is difficult to make definite conclusions. We have detected CH3CH2OCOH for the first time in a low-mass protostar, with a relatively high abundance of 10−11. This detection shows that the molecular complexity is high in young hot corinos and other species with a larger number of atoms could be present. It is noteworthy that this species has not been previously observed in the well-known hot corinos IRAS 16293-2422 and NGC 133IRAS4A. We have checked in the ALMA spectral line survey of IRAS 16293-2422 (PILS, Jørgensen et al. 2016), and no clear features are seen, although the observed frequencies are different. It is possible that the high level of line confusion and wide linewidths prevent the detection of weak lines in these more evolved sources.


Over the past decade, direct measurements of the Hubble constant have achieved few percent in precision (Freedman 2017). Among the conducted measurements, the Supernova H0 for the Equation of State (SH0ES) team challenged the well-believed Hubble constant (H0) value inferred from the Planck cosmic microwave background (CMB) measurements assuming a flat Lambda cold dark matter (ΛCDM) model. In detail, the latest result from SH0ES is H0 = 74.03 ± 1.42 km s−1 Mpc−1 (Riess et al. 2019), which differs from thePlanck result H0 = 67.4 ± 0.5 km s−1 Mpc−1 by 4.4σ (Planck Collaboration VI 2018). Recently, the Carnegie-Chicago Hubble Program (CCHP) also presented a new and independent determination of H0 parameter based on a calibration of the Tip of the Red Giant Branch (TRGB) applied to Type Ia supernovae (SNIa) (Freedman et al. 2019, 2020). They find a value of H0 = 69.6 ± 2.5 km s−1Mpc−1, which is in the middle and consistent with the SH0ES and Planck values. A different analysis and calibration of the TRGB method was performed by the SH0ES team (Reid, Pesce & Riess 2019; Yuan et al. 2019) resulting in a slightly higher H0 value. To come to a robust conclusion, independent H0 probes with accuracy better than $2{{\ \rm per\ cent}}$ are crucial (Verde, Treu & Riess 2019). Among the possible independent probes, the time-delay strong lensing (TDSL) measurements, such as from the H0 Lenses in COSMOGRAIL’s Wellspring (H0LiCOW) collaboration (Bonvin et al. 2017; Suyu et al. 2017; Birrer et al. 2019; Wong et al. 2019), are the most precise to date. The latest constraint from a joint analysis of six gravitationally lensed quasars with measured time delays (Wong et al. 2019) indicates for a flat ΛCDM, $H_0=73.3^{+1.7}_{-1.8}$ km s−1 Mpc−1, a $2.4{{\ \rm per\ cent}}$ precision measurement, which are in agreement with local measurements from SNIa, but in 3.1σ tension with CMB. The forecasts of future 40 TDSL measurements suggest that the H0 would be constrained at $\mathcal {O}(1){{\ \rm per\ cent}}$ level (Shajib, Treu & Agnello 2018; Yildirim, Suyu & Halkola 2019). A more optimistic forecast for dark energy studies can be found in Shiralilou et al. (2019). On the other hand, one of the main obstacles for the lensing mass modelling, or to determine precise H0 value, is the mass-sheet degeneracy (Schneider & Sluse 2014; Xu et al. 2016). These issues in the H0LiCOW analysis have been frequently discussed in the literature (Sonnenfeld 2018; Kochanek 2019; Pandey, Raveri & Jain 2019). No direct evidence of bias or errors is found from a comparison of self-consistency among the individual lenses  (Millon et al. 2019; Liao et al. 2020). Considering the fact that both the Planck and H0LiCOW’s H0 values are based on general relativity (GR) plus ΛCDM model, it inspires us to question the concordance cosmology model and investigate the modified gravity (MG).


During the X-ray ‘hard’ state, the radio and X-ray behaviour of BHXBs is correlated and has been studied in depth for many sources using quasi-simultaneous observations (e.g. Corbel et al. 2003; Gallo, Fender & Pooley 2003; Coriat et al. 2011; Corbel et al. 2013), and is known as the radio:X-ray plane. It was thought that all XBs followed a relation in the form of $L_{\rm Radio} \propto L_{\rm X-ray}^{0.6}$, based upon early observations of GX 339−4 (Hannikainen et al. 1998; Gallo et al. 2003; Corbel et al. 2003, 2013). This relation extends down to very low luminosities, i.e. into quiescence (Corbel et al. 2003, 2013; Plotkin et al. 2017; Tremou et al. 2020), and has been observed in other sources such as V404 Cygni (e.g. Corbel, Koerding & Kaaret 2008) and XTE J1118+480 (Gallo et al. 2014). However, further observations revealed the presence of another population of BHXBs which are less radio luminous than this relation, a so-called ‘radio-quiet’ branch, which followed $L_{\rm R} \propto L_{\rm X}^{1.4}$, such as H 1743−322 (e.g. Coriat et al. 2011; Williams et al. 2020). In some of these ‘radio-quiet’ objects, they are then found to re-join the ‘radio-loud’ branch when they go back into quiescence (Coriat et al. 2011; Carotenuto et al. 2021). The underlying cause of the split tracks (see Gallo, Miller & Fender 2012; Gallo et al. 2014; Gallo, Degenaar & van den Eijnden 2018, for a clustering analysis into the statistical robustness of this split) for BHXBs is not known, but it may be due to differences in the radiative efficiency of the accretion flow (Coriat et al. 2011; Koljonen & Russell 2019), an inclination effect of the source (Motta, Casella & Fender 2018), differences in the accretion disc contribution (Meyer-Hofmeister & Meyer 2014) or changes in the magnetic field (Casella & Pe’er 2009). For the purposes of this paper, we will refer to the original $L_{\rm R} \propto L_{\rm X}^{0.6}$ correlation sources as ‘radio-loud’ objects, and those that diverge on to the $L_{\rm R} \propto L_{\rm X}^{1.4}$ track as ‘radio-quiet’ sources.


Historically, the study of orbits in potentials has focused on periodic orbits. In systems like disk galaxies small perturbations to closed periodic orbits (e.g., the epicyclic and vertical perturbations of circular orbits) provided a good analytic description of most orbits. Self-consistent distribution functions are thought to be “parented” by stable periodic orbits (Arnold 1978). Early works (e.g., Contopoulos & Papayannopoulos 1980) identified and characterized the stability properties of the periodic orbit families in rapidly rotating bars. The most important periodic families in two-dimensional bars were identified as the prograde x1 family, which is elongated along the major axis of the bar, and the prograde stable x2 and unstable x3 families, which are elongated perpendicular to the bar (primarily found at small radii). The retrograde x4 (stable) orbit family is also elongated perpendicular to the bar at small radii, but becomes rounder as it extends to larger radii (for detailed description of orbit families and how they are identified see Contopoulos & Grosbol 1989; Sellwood & Wilkinson 1993; Binney & Tremaine 2008; Sellwood 2014b). In the frame of reference rotating with the bar, all of these families are characterized by a 1:2 resonance between the tangential oscillation frequency (Ωϕ) and the radial or epicyclic frequency (ΩR). Indeed, studies of orbits in 2D N-body bars largely confirmed the picture arising from the study of periodic orbits and showed that many regular orbits elongated along the bar were parented by x1 orbits, a small fraction were parented by retrograde x4 orbits (Sparke & Sellwood 1987), and none were parented by prograde x2 orbits. The realization that bars can also undergo buckling instabilities (Combes & Sanders 1981; Raha et al. 1991), which makes them develop substantial vertical thickness and peanut-shaped morphologies, led to the study of periodic orbits in three-dimensional bars (Pfenniger 1984; Martinet & de Zeeuw 1988; Pfenniger & Friedli 1991; Skokos et al. 2002a, 2002b). It was shown that the appearance of specific morphological features in images of bars, such as the X-shape and peanut features seen in edge-on bars and the boxy/rectangular isophotes and “ansae” of face-on bars, could be explained by orbits trapped around specific periodic orbit families (Patsis et al. 2002, 2003, 2010). The introduction of a third dimension did not drastically change the picture of the nature of periodic orbits, and it was found that 3D bars are composed primarily of vertical bifurcations (resonances) of the x1 family and a few additional families (e.g., Pfenniger & Friedli 1991; Skokos et al. 2002a, 2002b). Most studies of periodic orbits in analytic potentials consider prograde x2 orbits (but not retrograde x4 orbits) to be another fundamental building block of bars (Skokos et al. 2002a; Binney & Tremaine 2008). In our study we do not find any orbits parented by the periodic prograde x2 orbit in our initial bar model, but we do find orbits that are parented by the periodic retrograde x4 orbit—a result that is consistent with previous studies (Sparke & Sellwood 1987; Pfenniger & Friedli 1991; Voglis et al. 2007).


The sample of DES RETs shows a preference for low-metallicity, strongly star-forming host environments. The PDF of their metallicities displays a strong similarity to the hosts of SESNe, as well as LGRBs. There is a clear difference to the PDF of SNe II, which follow SDSS field galaxies. The preference for low-metallicity systems is not as strong as for LGRBs or SLSNe, but the highest metallicities found in all three samples are very similar at around solar metallicity. This result is suggestive of a stripped-envelope, massive-star origin for RETs. The population of RET hosts lies, on average, between CCSNe and LGRBs/SLSNe in terms of both star formation and metallicity. A loose correlation exists between the luminosity and rarity of events, and the host galaxy conditions required for their formation − on average, rarer events occur in more extreme environments. The approximate rate of RETs (≥10−6 Mpc−3 yr−1; Drout et al. 2014; P18; Coppejans et al. 2020; Ho et al. 2020; Tampo et al. 2020, although the definition of RET varies in the above calculations) is ${\sim}1{{\ \rm per\ cent}}$ of the CCSN rate (Horiuchi et al. 2011; Li et al. 2011; Strolger et al. 2015), which itself is divided into the more common SNe II and sub-dominant SESNe (Kelly & Kirshner 2012; Frohmaier et al., submitted). At ${\sim}1{{\ \rm per\ cent}}$ of the CCSN rate, RETs are more common than SLSNe (${\sim}0.01{-}0.05{{\ \rm per\ cent}}$ of CCSNe; McCrum et al. 2015; Prajs et al. 2017; Frohmaier et al., in preparation) and LGRBs (intrinsically ${\sim}0.08{{\ \rm per\ cent}}$ when accounting for beaming; Graham & Schady 2016). These figures place the rate of DES RETs between extreme objects (SLSNe, LGRBs) and more common SNe (SNe II, SESNe) in terms of rate, matching the location of RET hosts in the various host galaxy parameter spaces presented in Section 5. While stressing rates are uncertain and host galaxy parameters span wide ranges for all transients, they are both linked to the respective transients’ progenitor channels. While it is likely that RETs are a heterogeneous population comprising several progenitor scenarios, it is reasonable to infer from the rates and the host properties that RETs are linked to very massive stars, potentially stripped of their envelopes, and possibly sharing some of the extreme properties of SLSN or LGRB progenitors such as rapid rotation and low metallicity. This hypothesis can be extended to posit that some RETs represent an intermediate and/or precursory step in the late stages of evolution of a massive star that is close to forming a SLSN or LGRB, whereby the initial collapse of the star occurs leading to shock breakout and subsequent cooling driving the RET light curve (P18), but the progenitor is sufficiently different to the progenitors of LGRBs and SLSNe such that the central engine either does not form or has properties that differ from the central engines of LGRBs or SLSNe, hence the lack of longer-term light curves.


Finally, we emphasize that although the parameter space covered by the DASH library is vast, it is by no means exhaustive. One obvious shortcoming, as discussed above, is that the DASH simulations are inadequate to describe major mergers with ${\cal M}\lesssim 100$. In those cases, dynamical friction due to the host, and self-friction due to tidally stripped material, cause the orbit of the subhalo to decay, exposing it to stronger tides. Another degree of freedom not covered here is the inner density slope of dark matter haloes. It is well known that observations of dwarf galaxies often suggest that their haloes have constant density cores, rather than the steep r−1-cusps predicted by dark matter-only simulations (e.g. Burkert 1995; Gentile et al. 2004; Oh et al. 2011; Hayashi & Chiba 2015). Such cores can be created within the CDM paradigm by a variety of baryonic processes (e.g. El-Zant, Shlosman & Hoffman 2001; Inoue & Saitoh 2011; Pontzen & Governato 2012; Ogiya & Mori 2014), and have a dramatic impact on the tidal evolution of subhaloes (Peñarrubia et al. 2010; Errani, Peñarrubia & Tormen 2015; Ogiya 2018). In addition, baryons modify the potentials of host- and subhaloes through the bulges and discs that they form at the halo centres, and these also strongly impact the tidal fields (Errani et al. 2017; Garrison-Kimmel et al. 2017). Finally, in the DASH simulations presented here, the host halo is assumed to be spherically symmetric, which allows us to completely specify each orbit with only two parameters (energy and angular momentum). Cosmological simulations, though, indicate that dark matter haloes are expected to be triaxial systems (e.g. Jing & Suto 2002; Allgood et al. 2006; Hayashi, Navarro & Springel 2007), consistent with the shapes of the gravitational potentials of galaxies and clusters as inferred from a variety of observations (e.g. Oguri et al. 2005; Corless & King 2007; Law & Majewski 2010). Triaxial systems have a much richer variety of orbits, which is likely to impact the tidal evolution of subhaloes.


One of the main debates in the star formation community is whether the massive young stellar objects are a scaled-up version of low-mass young stellar objects (YSOs) where disc-accretion plays the dominating role for gaining the stellar mass. For the formation of low-mass stars, bipolar outflows driven by the accretion discs are proposed to be the basic formation mechanism theoretically (Shu, Adams & Lizano 1987), and are also verified observationally (e.g. Bontemps et al. 1996; Richer et al. 2000; Arce et al. 2007, and references therein). However, on the other hand, understanding of the formation mechanism of massive stars is still elusive (Tan et al. 2014). Two major competing models for massive star formation are (i) core accretion via disc (McKee & Tan 2003) and (ii) competitive accretion (Bonnell et al. 2001). The most obvious way to distinguish between these two models might be the detection of the accretion disc around massive protostars. But a direct detection of accreting disc is difficult because the accretion disc is small and short-lived, and also because of complicated gas dynamics at that scale (Kim & Kurtz 2006). Here, the study of the properties of molecular outflows that are the manifestation of disc-accretion in young sources, could help us to improve our understanding of the underlying formation process (Shepherd & Churchwell 1996; Beuther et al. 2002; Molinari et al. 2002; Arce et al. 2007). If it is assumed that massive stars do form via an accretion disc similar to the low-mass stars, they should generate massive and powerful outflows (see de Villiers et al. 2014, and references therein). A few recent studies (e.g. de Villiers et al. 2014; Li et al. 2018) indeed found the applicability of the same scaling between outflow activity and clump masses for both low-mass and massive objects, suggesting a similar formation mechanism. However, an extensive study of molecular outflows toward massive star-forming regions is still lacking owing to their large distances and high level of clustering. Recent interferometric observations with the ALMA enable us to target such regions thanks to its high spatial resolution and sensitivity. Studies of outflows associated with Galactic massive star-forming regions may provide us clues to better understand the launching mechanism of molecular outflows, and hence, the underlying star formation mechanism.


The majority of the particles collected by Stardust are olivine and pyroxene silicates with solar isotopic compositions, which suggests an origin in our solar system rather than an interstellar provenance. These polymineralic particles dominate those made of a single mineral even down to sizes smaller than 100 nm, indicating that the dust composition is surprisingly consistent at different scales and that the smallest subunits of the dust may be as small as tens of nanometers (Hörz et al. 2006; Zolensky et al. 2006). The sizes of these smallest single mineral impactors are similar to those of the nanocrystals determined by Rietmeijer (1993). As discussed above, they might also be existing in MIDAS dust particles and might be fused into the 100 nm features. Price et al. (2010) and Wozniakiewicz et al. (2012) investigated the sizes of particles smaller than 10 μm that impacted the aluminum foils of the Stardust probe. The distribution peaks at about 175 nm, but if we assume that the particles areagglomerates of smaller subunits, as indicated by their common polymineralic nature, then the subunit size distribution would peak at sizes below 100 nm (Price et al. 2010). A study of over 450 particles that do not seem to be agglomerates, that is, those that show single mineral impactors of silicate or sulfide, found geometric mean sizes of

$532^{741}_{-310}$

532

−310

+741

 nm for the silicate particles and

$406^{491}_{-222}$

406

−222

+491

 nm for the sulfides (Wozniakiewicz et al. 2013). These sizes are notably larger than the 175 nm (or less) found for the whole dataset. This large spread of subunit sizes could indicate a size distribution with a large width. No fits of these sizedistributions are available, but the figures in Wozniakiewicz et al. (2012) and Price et al. (2010) indicate that the differential sizes may follow a log-normal distribution. When we assume that the smallest subunit sizes are possibly between tens and hundreds of nanometers, the subunit size range found for MIDAS smallest features would be encompassed. The determination of the size distributions for the small Stardust particles and a detailed comparison to the distributions obtained for comet 67P could be the work of an interesting future project.


For most of our models, we can already determine, or at least extrapolate, the final neutron star properties quite well, barring the possibility of late-time fallback. Except for the $12.5 \, \mathrm{M}_\odot$ model, mass outflow already dominates over mass accretion on to the PNS, and the PNS mass has practically stabilized at its final value. Correcting for the binding energy of the neutron stars, we obtain gravitational masses between $1.22 $ and $1.44 \, \mathrm{M}_\odot$, which is compatible with the distribution of observed neutron star masses (Özel & Freire 2016; Antoniadis et al. 2016; Tauris et al. 2017). While the neutron star kicks are still growing at the end of the simulations due to the long-range gravitational tug by the asymmetric ejecta, the subsequent acceleration of the neutron star can be smoothly extrapolated to obtain tentative final values in all but one case. The extrapolated kicks range from $11$ to $695 \, \mathrm{km}\, \mathrm{s}^{-1}$. Thus, the most extreme, ECSN-like models with the smallest helium cores can reproduce the very low kicks required to explain some double neutron star systems and pulsars in globular clusters (Tauris et al. 2017), while the models with higher He core masses are compatible with the typical kicks of young pulsars (Arzoumanian, Chernoff & Cordes 2002; Hobbs et al. 2005; Ng & Romani 2007). If the extrapolated kick of $1236 \, \mathrm{km}\, \mathrm{s}^{-1}$ for the $18 \, \mathrm{M}_\odot$ model of Müller et al. (2017a) is included, the 3D coconut-fmt models roughly span the full range of observed kick velocities. We see tentative evidence for a correlation of the kick velocity with the explosion energy as proposed by Janka (2017) and Vigna-Gómez et al. (2018, as a refinement of earlier ideas for progenitor-dependent kicks by Bray & Eldridge 2016). Our models suggest that this correlation may not be a tight one, however, and that the kicks may scatter between zero and an upper limit that scales with the explosion energy. Low kicks can be achieved in more energetic explosions if the explosion geometry is bipolar rather than unipolar, as has already been noted in 2D by Scheck et al. (2006). Such a bipolar explosion occurs in one of our seven simulations (the $12 \, \mathrm{M}_\odot$ model). Although there is some concern that the bipolarity may be connected to the grid geometry, we find unipolar models even in cases where we do not include strong aspherical seed perturbations in the convective O shell that break grid alignment; this suggests that the possibility of bipolar neutrino-driven explosions with low kicks is real in 3D. We also find a loose correlation between the neutron star mass and the kick velocity, which is in line with current observations, and partly theoretical expectations, of double neutron stars (Tauris et al. 2017), but cannot make as strong a case for this correlation based on our simulations. An investigation of a larger suite of supernova simulations of ultra-stripped stars is needed to confirm this hypothesis.


While treated as fully relativistic under strong gravity, we note that the current model is time-independent based on axisymmetric plasma. This assumption makes it impossible for us to predict any temporal nature of the soft excess considered in this work, e.g., spectral time variabilities associated with shock compression and cooling effects. The downstream plasma properties are numerically solved by considering adiabatic (nonradiative) Rankine–Hugoniot jump conditions as a pure mathematical discontinuity with no energy/mass loss. Hence, most of the heat generated at the shock front is advected with the downstream plasma. A more realistic shock process, on the other hand, is most likely accompanied by radiative cooling to some degree in which the post-shock plasma temperature may stay comparatively as cool as that of the upstream one, as in the isothermal shocks (e.g., Lu & Yuan 1998; Das et al. 2003; Fukumura & Tsuruta 2004; Fukumura & Kazanas 2007b). Radiative dissipation at the shock front could therefore drastically change the subsequent downstream plasma condition, which in turn alters the Comptonization process. In reality, furthermore, accreting plasma may be characterized by a two-temperature gas between electrons and ions (e.g., Shapiro et al. 1976; Mahadevan 1998; Manmoto 2000) unless the Coulomb coupling between the two is very efficient, whereas in this work we prescribed a single-fluid approximation for simplicity. Becker et al. (2011) have considered a particle transport process (e.g., bulk advection, spatial diffusion, and particle escape) via the effects of the first-order Fermi acceleration across a standing shock. In a more self-consistent scenario such a calculation of diffusive shock acceleration should be incorporated to reflect the energetic outflows/jets from the shock front. Although all these micro-physics should be addressed and incorporated into more sophisticated calculations of GRMHD simulations for completeness, this is beyond the scope of this paper.


The vertical structure of an accretion disk has been investigated in detail by Abramowicz et al. (1997). A general accurate expression for the vertical hydrodynamical equilibrium that is valid both for a thin and a slim disk has been derived in their work. The hydrostatic balance in the vertical direction of the accretion disk is described by
19

1

ρ

∂
p

∂
z

+

∂
ψ

∂
z

+

v

r

∂

v

z

∂
r

+

v

z

∂

v

z

∂
z

=
0
,

where p is the pressure, ρ is the density of the gas, and ψ is the gravitational potential (Abramowicz et al. 1997). As we focus on how the general properties of the disk structure is affected by the radiation-driven outflows, we simplify the expression of the hydrostatic balance in the vertical direction as
20

1

ρ

∂
p

∂
z

+

∂
ψ

∂
z

=
0
,

where we assume the terms of ∂vz/∂r and ∂vz/∂z to be negligible. It has been pointed out that the widely used approximation for the vertical component of gravity, GMz/R3, is only valid for the thin disk with H/R ≪ 1, and a more accurate expression of vertical gravity should be adopted for a slim disk (McClintock et al. 2006; Gu & Lu 2007; Jiao et al. 2009; Gu 2012; Cao & Gu 2015). As discussed in Cao & Gu (2015), there are upper limits on the radiation flux, frad, and half-thickness, H, for a slim disk, above which the radiation force will overwhelm the vertical gravity. The maximal flux can be calculated with
21

f

rad

=
q
(
H
)
=

GM
ρ
H

r

2

+

H

2

r

2

+

H

2

−

r

s

2

.

How the gas is blown away from the disk surface by the radiation force is still quite unclear, and, therefore, it is assumed that the gas at the disk surface will be accelerated into outflows by the radiation force when the radiation force overwhelms the vertical gravity. Then, the mass accretion rate in the disk decreases due to outflows. This self-adjustment mechanism leads to a maximal radiation flux/thickness of the disk (see the detailed discussion in Cao & Gu 2015). Thus, we assume that the outflows are triggered in the disk where the condition
22

Q

rad

−

≥
2

f

rad

max

,

is satisfied, where

f

rad

max

 is the maximal radiation flux and then the mass accretion rate is self-adjusted by the outflows to maintain
23

Q

rad

−

≡
2

f

rad

max

.

This relation is adopted in the energy of Equation (15) for the region in the disk where outflows are driven. Otherwise, the outflows are suppressed, and the disk is the same as the conventional slim disk without outflows, which is described by a set of equations for a slim disk (see Chapter 7 in Kato et al. 2008), i.e., the continuity equation:
24

M

˙

=
−
2
π
r
Σ

v

r

,

where

M

˙

 remains constant radially, and the angular momentum equation is
25

v

r

(
Ω

r

2

−

j

in

)
=
−
α

rc

s

2

.

Except for Equations (23)–(25), the radial-component of the momentum equation, the equation of state, and energy equation are the same as the disk with outflows (see Equations (4) and (14)–(18)).


Altogether, the approximately thirty molecules recently detected have confirmed the chemical complexity in the nebula, and generated our interest for the present study. Of these species, we will focus our attention on the seventeen species listed by molecular families in Table 1. As can be seen in this table, the WHISPER survey allowed the detection of some organic molecules in the Horsehead nebula, such as formaldehyde (H2CO) and methanol (CH3OH), which constitute key species in the likely synthesis of more complex organic molecules such as some prebiotic molecules (Bernstein et al. 2002; Muñoz Caro et al. 2002; Garrod et al. 2008). Because they are detected in a wide variety of interstellar sources – in hot cores (Sutton et al. 1995; Ceccarelli et al. 2000), dark clouds (Bergman et al. 2011), shocked regions (e.g. Sakai et al. 2012; Codella et al. 2012; Tafalla et al. 2010) and even in comets (Mumma & Charnley 2011; Cordiner et al. 2015) – it is of prime importance to understand well how these precursor molecules form. H2CO is commonly thought to form both in the gas-phase and on grain surfaces, while CH3OH is believed to be only formed on grain surfaces (Garrod et al. 2006; Geppert et al. 2006). Guzman et al. (2013) reported the observations of these two molecules toward the Horsehead nebula in both the PDR and Core positions. Unable to reproduce the observed abundances of either H2CO or CH3OH at the PDR position with only pure gas-phase models, they concluded that, for this region, both species are formed on grain surfaces and then photodesorbed into the gas phase. On the other hand, at the Core position, a pure gas-phase model can reproduce the observed H2CO abundance, while photodesorption of ices is still needed to explain the observed abundance of CH3OH. Other organic molecules were reported in the Horsehead nebula as first detections in a PDR environment, including HCOOH (formic acid), CH2CO (ketene), CH3CHO (acetaldehyde), and CH3CCH (propyne) (Guzman et al. 2014). Their abundances were found to be higher at the PDR position than at the Core, revealing that complex organic chemistry is also occurring in UV-illuminated neutral gas (Guzman et al. 2014). Of these molecules, some – HCOOH, CH2CO, and CH3CHO – have now also been detected in the Orion bar PDR (Cuadrado et al. 2016, 2017).


Plots of various characteristics of sub-DLAs and DLAs in the literature. Blue star points mark the detections presented in this work. The Xs mark detections from Péroux et al. (2011a, 2012). The points mark detections from previous studies in the literature. References: Pettini et al. 2000, Lacey et al. 2003, Junkkarinen et al. 2004, Chen et al. 2005, Burbidge et al. 1996, Chun et al. 2006, Rao et al. 2005, Boisse et al. 1998, Le Brun et al. 1997, Gharanfoli et al. 2007, Rao et al. 2006, Steidel et al. 2002, Lanzetta et al. 1997, Lanzetta et al. 1995, Bergeron & Boisse 1991, Deharveng et al. 1995, Meiring et al. 2007, Cristiani et al. 1987, Bowen et al. 2005, Schulte-Ladbeck et al. 2005, Bowen et al. 2001, Rosenberg et al. 2006, Steidel et al. 1997, Bergeron 1986, Prochaska & Wolfe 1997, Weatherly et al. 2005, Lu et al. 1993, Lu et al. 1997, Djorgovski et al. 1996, Christensen et al. 2004, Rao & Turnshek 2000, Ellison et al. 2005, Pettini et al. 1994, Francis et al. 1996, D'Odorico et al. 2002, Francis & Williger 2004, Francis et al. 2001, Kulkarni et al. 2005, Cherinka et al. 2009, Noterdaeme et al. 2009 and Fynbo et al. 2010. See table 4 of Péroux et al. (2011a) for the full list of measurements. For the SFR plots, colour indicates the emission from which the value was determined: blue for [O ii], red for Hα, green for Hβ, purple for [O iii], and black for Lyα. SFRs have not been dust corrected. The solid red line in the SFR versus zabs plot is an arbitrarily normalized Madau relationship from Madau & Dickinson (2014). Qualitatively, the trend in host galaxy SFR with absorber redshift is consistent with this trend. The blue outliers are the interacting pair in the field of Q1436-0051. The dashed vertical lines indicate the column density boundary between sub-DLAs and DLAs. The outlier in b at ∼180 kpc is a host galaxy that is part of a cluster environment reported in Francis et al. (2004) and references therein. Grey arrows indicate limits for non-detections. The error bars on Q1436-0051 zabs = 0.9281 correspond to the range of possible H i column densities and Zn metallicities. See the text for details.


In order to predict emission and transmission spectra of HJs we utilized the τ-REx (Tau Retrieval for Exoplanets) software package (Waldmann et al. 2015b,a). This package uses up-to-date molecular cross sections based on line lists provided by EXOMOL6 project (Tennyson & Yurchenko 2012) and HITEMP (Rothman et al. 2010). We additionally used the HCN line list after Harris et al. (2006) provided by the EXOCLIME project7 and generated with the HELIOS-K code (Grimm & Heng 2015). These cross sections are precomputed on a grid of temperatures and pressures and are stored in binary opacity tables that are available for a number of spectral resolutions. The continuum opacity includes Rayleigh scattering on molecules and collisionally-induced absorption due to H2 –H2 and H2–He either after Abel et al. (2011, 2012) or Borysow et al. (2001); Borysow (2002); Borysow & Frommhold (1989), respectively. We extended the public version of τ-REx8 by incorporating additional opacity sources essential for the atmospheres of UHJs. In particular, bound-free and free–free transitions of H− become one of the major continuum opacity contributors for the temperatures hotter than about 2000 K. The cross sections of H− are from John (1988). We also included opacity due to free–free transitions of He− as well as Rayleigh scattering on H I atoms and Thomson scattering on free electrons. The He− cross sections are those originally from John (1968) using the polynomial fit by Carbon et al. (1969). Rayleigh scattering on H I is calculated after Dalgarno (1962). All relevant numerical routines were extracted from the LLMODELS stellar model atmosphere code (Shulyak et al. 2004). At the temperatures of HJs the H− is the dominant continuum opacity source that impacts the observed spectra of these planets (Arcangeli et al. 2018). However, at millibar pressures the He− opacity can become comparable or even stronger than that of H− for wavelengths longer than 1.6 μm, as shown in Fig. 3 (third panel from the bottom), where we display examples of continuum opacity coefficient at different altitudes in the atmosphere of a HJ with Teq = 3000 K. At even smaller pressures, electron scattering and Rayleigh scattering on H I atoms also become important contributors to the continuum opacity at particular wavelengths (top panel in Fig. 3). However, the contribution of H I and e− on the transmission spectra is marginal because their opacity is only strong at low pressures that are hardly probed by transmission spectroscopy. Thus, among all continuum opacity sources, only H− and He− significantly contribute to the predicted amplitude of the transmission spectra, as shown in the bottom plot of Fig. 3. When both continuum and line opacity are included, the effect of He− on predicted spectra is diluted by a much stronger opacity in molecular lines while H− contribution is still significant. Nevertheless, as can be seen from the second plot (from bottom) in Fig. 3, in optically thick layers the He− opacity could still be stronger than, for example, collision-induced absorption due to H2 -H2 and H2-He. We thus conclude that accurate calculation of atmospheric opacity requires He−, H I, and e− opacity included especially at low pressures, following modern stellar atmosphere codes. However, observed transmission and emission spectra of HJs are hardly affected by these three opacity sources. Finally, we updated HELIOS with He− and e− opacity and found out that this has little impact on the atmospheric temperature structure (with a modification in local temperature of at most ΔT ≈ 10 K) and thus can be ignored; the original version of HELIOS already includes H I Rayleigh scattering.


Erwin (2018) showed that, in a sample drawn from the Spitzer Survey of Stellar Structure in Galaxies (S4G), the bar fraction is constant over a range of (g −r) colours and gas fractions. Their bar fraction does not increase, but rather decreases for stellar masses higher than ∼ 109.7M⊙. These results are in contrast to many SDSS-based studies cited above. Erwin (2018) argues that this apparent contradiction can be explained if SDSS-based studies miss bars in low-mass blue galaxies. In Figs 5 and 6, we showed that the newly detected bars in GZD (compared to GZ2) are weak bars in low-mass blue galaxies. Nevertheless, the ‘combined’ bar fraction in Fig. 6 is not constant over (g −r) colour and agrees well with Masters et al. (2011) for redder colours [(g −r) colour > 0.5]. Additionally, our ‘combined’ bar fraction remains roughly constant over stellar mass. As mentioned before, we conclude that strong bars drive the trends of bar fraction with (g −r) colour, stellar mass, and SFR observed in other studies (Nair & Abraham 2010b; Masters et al. 2011, 2012; Vera et al. 2016; Cervantes Sodi 2017). However, the addition of weak bars in low-mass blue galaxies is insufficient to resolve the apparent disagreement between Erwin (2018) and many SDSS-based studies (Masters et al. 2011, 2012; Vera et al. 2016; Cervantes Sodi 2017; Kruk et al. 2018), which instead seems likely to be due to the very different sample selection of the S4G and SDSS galaxy samples. For example, the median stellar mass of the sample used in Erwin (2018) is ∼109.6M⊙ (based on their Fig. 4 and the bins in the top left-hand panel of their Fig. 5). However, the median stellar mass of our sample is 1010.6M⊙. As stellar mass correlates with many parameters (including bar length), this can have major consequences. Additionally, as Erwin (2018) notes, there is also the issue of resolution to consider. With an r-band FWHM of 1.18 arcsec from DECaLS (Dey et al. 2019) and a mean redshift of 0.036, the mean linear resolution of our sample is approximately 834 pc, which is higher than the 165 pc of Erwin (2018). This explains why they observe many sub-kpc bars, while we do not. These differences in stellar mass and resolution will manifest themselves in the conclusions, so a more detailed analysis is needed for a proper comparison with Erwin (2018).


At a distance of 4.8 Mpc (Karachentsev et al. 2002), NGC 5408 X-1 is one of the best studied ULXs. It has been observed by several of the current generation of X-ray satellites, on a multitude of occasions. Observations include: an XMM-Newton large programme (e.g., Pasham & Strohmayer 2012); Swift XRT monitoring (Kaaret & Feng 2009; Grisé et al. 2013); and eight Chandra exposures, which we re-analyze here. The flux variability of the source rules out an X-ray supernova remnant and confirms that it is powered by accretion onto a compact object (Kaaret et al. 2003; Soria et al. 2004). It persistently displays a distinct soft ultraluminous two component X-ray spectrum in XMM-Newton data (Sutton et al. 2013) at an average 0.3–10 keV unabsorbed luminosity of

 (Strohmayer 2009; although we note that they fit the high energy spectrum with a soft power-law, so may over-estimate the intrinsic luminosity, cf. Middleton et al. 2014). Additional soft residuals have been detected in the XMM-Newton spectra which can be well modeled as thermal plasma emission (Strohmayer & Mushotzky 2009; Miller et al. 2013; Middleton et al. 2014). It has previously been assumed that these were the result of diffuse star formation related emission in the host galaxy (Strohmayer et al. 2007; Strohmayer & Mushotzky 2009; Miller et al. 2013). However, we know from observational studies of galaxies that the X-ray luminosity of such emission is correlated with star formation rate (

/

; Mineo et al. 2012), and Middleton et al. (2014) contend that the luminosity of the putative thermal plasma emission (

 calculated from Miller et al. 2013) greatly exceeds that inferred from star formation, even over the entirety of NGC 5408 (

 calculated from a 24 μm flux density of

, Dale et al. 2005). Instead, Middleton et al. (2014) show that the putative plasma emission features could actually be commensurate with broadened, blueshifted absorption in a partially ionized, optically thin medium. Such a medium would be expected to occur in a super-Eddington wind, as it becomes optically thin at large distances (

) from the central black hole.


The Hamiltonian used in the present work is the so-called RAM (rho axis method) internal-rotation Hamiltonian based on the work of Kirtman (Kirtman 1962), Lees and Baker (Lees & Baker 1968), and Herbst et al. (Herbst et al. 1984). Since rather complete descriptions of this method, which takes its name from the choice of axis system, have been presented several times (Hougen et al. 1994; Kleiner 2010) we do not repeat this general description here. The main advantage of the RAM Hamiltonian is its general approach that simultaneously takes into account the A- and E-symmetry species and all the torsional levels, intrinsically taking the intertorsional interactions into account within the rotation-torsion manifold of energy levels. This method was successfully applied to a number of molecules containing a C3v rotor and Cs frame, including the main isotopolog of acetaldehyde (Smirnov et al. 2014). As for the main isotopolog (Smirnov et al. 2014) we employed the RAM36 (rho-axis-method for 3- and 6-fold barriers) code that uses the RAM approach for the molecules with the C3v top attached to a molecular frame of Cs or C2v symmetry and having 3- or 6-fold barriers to internal rotation, respectively (Ilyushin et al. 2010, 2013). The Hamiltonian in the RAM36 program is presented by the following expression: (1)\begin{eqnarray} \nonumber  H &=& (1/2) \sum_{knpqrs} B_{knpqrs0} \left[P^{2k}P_z^nP_x^pP_y^qp_{\alpha}^r \cos(3s{\alpha}) \right.\\[2mm] \nonumber  &&\left. + \cos(3s\alpha) p_{\alpha}^rP_y^qP_x^pP_z^nP^{2k}\right] \\[2mm] \nonumber && + (1/2) \sum_{knpqrt} B_{knpqr0t} \left[P^{2k}P_z^nP_x^pP_y^qp_{\alpha}^r \sin(3t\alpha) \right.\\ [2mm]  &&\left. + \sin(3t\alpha) p_{\alpha}^rP_y^qP_x^pP_z^nP^{2k}\right] \end{eqnarray}H=(1/2)∑knpqrsBknpqrs0[P2kPznPxpPyqpαrcos(3sα)+cos(3sα)pαrPyqPxpPznP2k]+(1/2)∑knpqrtBknpqr0t[P2kPznPxpPyqpαrsin(3tα)+sin(3tα)pαrPyqPxpPznP2k]where the Bknpqrst are fitting parameters; pα is the angular momentum conjugate to the internal rotation angle α; and Px,Py,Pz are projections on the x,y,z axes of the total angular momentum P. In the case of a C3v top and Cs frame (as is appropriate for acetaldehyde), the allowed terms in the torsion-rotation Hamiltonian must be totally symmetric in the group G6 (and also must be Hermitian and invariant to the time reversal operation). Since all individual operators pα,Px,Py,Pz,P2,cos(3sα) and sin(3tα) used in Eq. (1) are Hermitian, all possible terms provided by Eq. (1) will automatically be Hermitian. The particular term to be fit is represented in the input file with a set of k,n,p,q,r,s,t integer indices that are checked by the program for conformity with time reversal and symmetry requirements, to prevent accidental introduction of symmetry-forbidden terms into the Hamiltonian.


Various surveys of SNRs in our Galaxy and nearby galaxies have been carried out at radio, X-ray, Infrared (IR), and optical wavelengths. The first extragalactic SNR candidates were identified in the LMC by Mathewson & Healey (1964) and later confirmed with a combination of radio and optical techniques by Westerlund & Mathewson (1966). To date, a total of 60 SNRs have been confirmed in the LMC with an additional 14 suggested candidates (Maggi et al. 2016; Bozzetto et al. 2017; Maitra et al. 2019). However, sensitivity and resolution limitations severely reduce the effectiveness of the past and present generations of radio and X-ray searches for SNRs in galaxies beyond the Small and Large Magellanic Clouds (MCs) (Goss et al. 1980; Long et al. 1981; Cowan & Branch 1985; Matonick et al. 1997; Matonick & Fesen 1997; Millar, White & Filipovic 2012; Galvin & Filipovic 2014; Sasaki et al. 2018; Lin et al. 2020; Sasaki 2020). As a result, optical studies have produced the largest number (∼1200) of new extra-Magellanic SNR candidates. Optical extragalactic searches for SNRs are mainly done by using an emission line ratio criterion of the form [S ii]/H α > 0.4–0.5 (Mathewson & Clarke 1973; Dodorico, Dopita & Benvenuti 1980; Fesen 1984; Blair & Long 1997; Matonick & Fesen 1997; Dopita et al. 2010b; Lee & Lee 2014; Vučetić et al. 2019b, a, 2018; Lin et al. 2020). This criterion separates shock-ionization from photoionization in SNRs from H ii regions and Planetary Nebulae (PNe) (Frew & Parker 2010). SNR radiative shocks collisionally excite sulphur ions in the extended recombination region resulting in S+, hence the larger contribution of [S ii] accounting for an increase of the [S ii] to H α ratio. In typical H ii regions, sulphur exists predominantly in the form of S++, yielding low [S ii] to H α emission ratios. Ratios from narrow-band imaging are usually verified spectroscopically, since [N ii] lines at 6548 and 6584 Å can contaminate the H α images at an unknown and variable level. Spectroscopic observations of such emission nebulae also can provide other evidence of shock heating, such as strong [O i] λ6300 emission, elevated [N ii] to H α with respect to H ii regions, or high [O iii] electron temperatures, verifying the candidate as being an SNR (Blair, Kirshner & Chevalier 1981, 1982; Long et al. 1990; Smith et al. 1993; Blair & Long 1997). Although somewhat biased as an isolated criterion, this method is proven and a good way of identifying ordinary radiatively cooling SNRs in nearby galaxies. We note that young, Balmer-dominated SNRs (Chevalier, Kirshner & Raymond 1980) would be missed by this criterion.


The stellar photospheric parameters we collected from literature for the benchmark stars are summarized in Table A.1. Although most benchmark stars have v sini  2 km s−1 (Reiners et al. 2018), there are two stars with larger values: J07558+833 (12.1 km s−1) and J13005+056 (16.4 km s−1). These stars are useful to investigate the performance of the algorithms when dealing with higher rotational velocities. The literature values were derived with different methods. These methods include: interferometry to estimate the stellar radius and Teff (Boyajian et al. 2012; Ségransan et al. 2003; von Braun et al. 2014; Berger et al. 2006; Newton et al. 2015), synthetic model fitting using BT-Settl models to determine Teff (Gaidos et al. 2014; Lépine et al. 2013; Gaidos & Mann 2014; Mann et al. 2015) and log g (Lépine et al. 2013), empirical relations to derive stellar mass in the form of mass-luminosity relations (Mann et al. 2015; Khata et al. 2020; Boyajian et al. 2012; Berger et al. 2006; Ségransan et al. 2003), along with the mass-magnitude relations (Maldonado et al. 2015), mass-radius relations (von Braun et al. 2014), mass–Teff relations (Gaidos & Mann 2014; Gaidos et al. 2014), empirical relations to derive the stellar radius in the form of mass-radius relations (Maldonado et al. 2015) and Teff–radius relations (Gaidos & Mann 2014; Gaidos et al. 2014; Houdebine et al. 2019), pEW measurements to determine Teff (Maldonado et al. 2015; Neves et al. 2014; Newton et al. 2015) and [Fe/H] (Maldonado et al. 2015; Neves et al. 2014; Gaidos et al. 2014; Mann et al. 2015), the definition of spectral indices such as the H2O-K2 index to estimate Teff (Rojas-Ayala et al. 2012), as well as the combination of the H2O-K2 index with pEWs to derive [Fe/H] (Rojas-Ayala et al. 2012; Khata et al. 2020), the stellar radius and Teff (Khata et al. 2020), and spectral curvature indices for the determination of Teff (Gaidos & Mann 2014). Additionally, [Fe/H] was derived by using color-magnitude metallicity relations (Dittmann et al. 2016), atomic line strength relations (Gaidos & Mann 2014), and spectral feature relations (Terrien et al. 2015). Terrien et al. (2015) used K-band magnitudes and the Dartmouth Stellar Evolution Program (Dotter et al. 2008) to derive the stellar radius, whereas Mann et al. (2015) employed the Boltzmann equation with Teff determined from synthetic model fits. Last, but not least, Houdebine et al. (2019) derived Teff from photometric colors. For more details on the individual methods, we refer to the descriptions in the corresponding works.


Major nearby galaxy cold gas mapping surveys (Regan et al. 2001; Wilson et al. 2009; Rahman et al. 2011; Leroy et al. 2009; Donovan Meyer et al. 2013; Bolatto et al. 2017; Sorai et al. 2019; Sun et al. 2018) have focused on observations of the molecular gas (through CO lines). Despite a few notable exceptions (e.g. Alatalo et al. 2013; Saintonge et al. 2017), these surveys observed mainly spiral or infrared-bright galaxies (i.e. galaxies with significant star formation) and have furthered our understanding of how star formation happens, rather than how it stops. This boils down to quantifying the relation between molecular gas and star formation rate (SFR), which appears nearly linear in nearby discs (Kennicutt 1998; Bigiel et al. 2008; Leroy et al. 2013; Lin et al. 2019). This relationship is often parametrised via the ratio between the SFR and the molecular gas mass (Mmol), which is called the molecular star formation efficiency (SFE = SFR/Mmol = 1∕τdep), where the inverse of the SFE is the depletion time, τdep. The depletion time indicates how much time is necessary to convert all the available molecular gas into stars at the current star formation rate. On kpc scales and in the discs of nearby star-forming galaxies, τdep is approximately constant around 1–2 Gyr (Bigiel et al. 2011; Rahman et al. 2012; Leroy et al. 2013; Utomo et al. 2017), and it appears to weakly correlate with many galactic properties such as stellar mass surface density or environmental hydrostatic pressure (Leroy et al. 2008; Rahman et al. 2012). Nevertheless, small but important deviations for a constant SFE have been noticed, which can be the first hints of star formation quenching. In some galaxies, the depletion time in the centres appear shorter (Leroy et al. 2013; Utomo et al. 2017) or longer (Utomo et al. 2017) with respectto their discs. These differences may correlate with the presence of a bar or with galaxy mergers (Utomo et al. 2017; see also Muraoka et al. 2019) and do not seem to be related to unaccounted variation in the CO-to-H2 conversion factor (Leroy et al. 2013; Utomo et al. 2017). Spiral arm streaming motions have also been observed to lengthen depletion times (Meidt et al. 2013; Leroy et al. 2015).


In the last few years, similar (though not identical) experimental developments to our method have been reported. In 2017, Wehres et al. (2017) described the design, construction, and operation of two laboratory broad-band emission spectrometers for gas-phase characterization of large molecules. The first is based on a Schottky-barrier diode heterodyne receiver spectrometer operating between 80 and 110 GHz, whilst the second uses cryo-cooled SIS technology and operates at higher frequency between 270 and 290 GHz (Wehres et al. 2018). Gas phase pyridine and methyl cyanide features were successfully detected and matching analytic simulations were presented for the observed spectral transitions. In 2018, a new technique combining a Terahertz radiometer (41–49) GHz and a vacuum chamber was developed to observe the generation of cold plasma and UV photochemistry in the gas phase under low pressure as described by Tanarro et al. (2018). However, in the context of the experiment described here, it is important to note that the spectrometer developments were not coupled to a system where the gas-phase molecules originated from the solid state. More recently, Yocum et al. (2019) reported on an experiment using a THz source and hot-electron bolometer to detect the gas-phase absorption spectra of simple molecular species such as H2O, D2O, and CH3OH desorbing into the gas-phase from ices grown in an ultra-high vacuum (UHV) chamber. This experimental set-up (SubLIME), combined with a Fourier-transform Infrared spectrometer, gave the spectroscopic insight on the ultraviolet photolysis and warm up of methanol ice sample, for the first time observed at those frequencies in a laboratory environment (Yocum et al. 2021). This showed the potential of the use of submillimetre/far-infrared technique to identify molecules in complex gas mixtures. One challenge of this technique was the sensitivity of absorption spectroscopy to the desorbing molecules. The combination of temperature programmed desorption and microwave spectroscopy techniques was discussed by Theulé et al. (2020). In these experiments, desorption of water, deuterium, methanol, and ammonia was measured using a hot electron bolometer and Si-diode technology combined with Fourier transform spectrometers. While Theule et al. observed the desorption through a waveguide cavity, Yocum et al. (2019) detected the desorption above a metal substrate as seen in standard TPD studies.


Various theoretical studies employing numerical simulations have been performed to investigate planet formation through core accretion or gravitational instability in the low-stellar-mass regime. Payne & Lodato (2007) assessed planet formation around brown dwarfs adapting models for higher stellar masses based on core accretion. Through Monte Carlo simulations, they found that Earth-like planets can form in this condition and the planet mass depends strongly on the disk mass. However, none of their simulations showed a planetary rocky core accreting a gaseous envelope to form a giant planet. A way to overcome the radial drift barrier is a rapid rocky core growth. Pebble accretion is a mechanism able to speed up significantly the giant planet formation process (i.e., Lambrechts & Johansen 2012; Bitsch et al. 2015). Liu et al. (2020) carried out a theoretical study on planet formation driven by pebble accretion in the (sub)stellar mass range between 0.01 and 0.1 M⊙. First, they calculated the initial masses of protoplanets by extrapolating previous numerical simulations conducted in previous literature. Next, they performed a population synthesis study to track the growth and migration of a large sample of protoplanets under the influence of pebble accretion. Their results show that, around a 0.01 M⊙ brown dwarf, planets can grow up to 0.1−0.2 M⊕, while, around 0.1 M⊙ stars, planets can reach a maximum mass of 2−3 M⊕. Findings from this study show that even pebble accretion does not seem to be sufficient to form gas giants around VLM stars and brown dwarfs. Miguel et al. (2020) used a population synthesis approach based on planetesimal accretion to explore planet formation in the stellar mass range between 0.05 and 0.25 M⊙. They let the synthetic population of planetary systems evolve for 108 yr. The authors find that to form planets with masses higher than 0.1 M⊕ they need stars of at least 0.07 M⊙, implying that planet formation around brown dwarf may not be a usual outcome. Then, stars with masses higher than 0.15 M⊙ are necessary to form planets more massive than the Earth. Therefore, from all of these studies, we conclude that core accretion model currently cannot explain the presence of gas giants around VLM stars or brown dwarfs. Either the core accretion theory is incomplete, or another mechanism for planet formation is needed. This is confirmed by Lodato et al. (2005), who discussed the origin of the 5 MJup planet detected around the 25 MJup brown dwarf 2MASSW J1207334-393254 (Chauvin et al. 2005). They found that the core accretion mechanism is far too slow to generate such a planet in less than 107 yr, the estimated age of the system. Therefore, the authors proposed gravitational instabilities arising during the early phases of the disk lifetime as a viable possibility for the formation of the planet.


Detections of ionic, atomic, and molecular species in exoplanetary atmospheres serve as a unique and strong diagnostic of those chemical and dynamical processes driving their formation and evolution. Their detection and abundance measurements could act as indicators of planetary formation scenarios and reveal connections to the primordial protoplanetary disc and the host star (Williams & Cieza 2011; Mordasini et al. 2016; Madhusudhan et al. 2017). Furthermore, discoveries of atmospheric chemical species allow us to better understand various thermodynamical processes and chemistry, winds in the upper atmosphere (Goodman 2009; Snellen et al. 2010; Brogi et al. 2016; Madhusudhan et al. 2016; Wyttenbach et al. 2020), and to probe planetary interiors and various bulk properties through their abundances (Kite et al. 2016; Thorngren & Fortney 2019; Madhusudhan et al. 2020). A whole host of ions, atoms, and molecules have been detected through a variety of, often complementary, techniques, such as differential spectrophotometry using low-to-mid resolution spectroscopy (e.g. Gibson et al. 2012, 2017; Deming et al. 2013; Kreidberg et al. 2014; Kirk et al. 2016; Nortmann et al. 2016), and high resolution spectroscopic techniques (e.g. Redfield et al. 2008; Snellen et al. 2008; Rodler, Lopez-Morales & Ribas 2012; Birkby et al. 2013; Hoeijmakers et al. 2015, 2018, 2020; Brogi et al. 2016; Birkby et al. 2017; Žák et al. 2019; Ehrenreich et al. 2020). To date, ionic species such as Fe ii and Ti ii (Hoeijmakers et al. 2019), atomic absorption from Na, K, H α, and He (e.g. Redfield et al. 2008; Sedaghati et al. 2016; Casasayas-Barris et al. 2017; Spake et al. 2018; Chen et al. 2020; Seidel et al. 2020), and molecules such as H2O, CH4, and CO (e.g. Konopacky et al. 2013; Brogi et al. 2014; Fraine et al. 2014; Barman et al. 2015; Sing et al. 2016) have been detected through the aforementioned techniques. Needless to say that this list of detected constituents is by no means exhaustive, nor that of methods employed to detect exoplanetary atmospheres. For instance, high-resolution imaging instruments such as SPHERE (Beuzit et al. 2019) and GRAVITY (Gravity Collaboration et al. 2017), both at the VLT (ESO’s Very Large Telescope), through combination with low-dispersion spectroscopy, have facilitated direct measurements of exoplanetary atmospheres (Samland et al. 2017; Gravity Collaboration et al. 2020).


This last, more technical aspect gave birth to a branch of research devoted to the problem of fitting a galaxy SED (Arnouts et al. 1999; Bolzonella et al. 2000; Cid Fernandes et al. 2005; Ilbert et al. 2006; Ocvirk et al. 2006; Tojeiro et al. 2007; Fritz et al. 2007, 2017; Franzetti et al. 2008; Pappalardo et al. 2010; Han & Han 2014; Leja et al. 2017; Weaver et al. 2021). Historically, models reproducing the evolution of stellar populations and dust emission have been developed separately, with reliable prescriptions to build synthetic spectra of different stellar populations on one side (Bruzual & Charlot 1993; Bressan et al. 1994; Worthey 1994; Fioc & Rocca-Volmerange 1997; Leitherer et al. 1999; Vazdekis 1999; Charlot & Fall 2000; Maraston 2005; Fioc & Rocca-Volmerange 2019), and a thorough comprehension of dust grains physics on the other (Draine & Lee 1984; Dale 2001; Takeuchi et al. 2003, 2005; Zubko et al. 2004; Draine & Li 2007; da Cunha et al. 2010; Silva et al. 2011; Asano et al. 2013; Calura et al. 2014; Zhukovska 2014; Mancini et al. 2015; Schneider et al. 2016; Popping et al. 2017; Aoyama et al. 2017; De Vis et al. 2017, 2019; Ginolfi et al. 2018; Graziani et al. 2019; Burgarella et al. 2019; Nanni et al. 2019; De Looze et al. 2020; Galliano et al. 2021). The two branches finally converged, performing simultaneous fitting of both components (Devriendt et al. 1999; Groves et al. 2008; da Cunha et al. 2008; Noll et al. 2009; Silva 2009; Graziani et al. 2019). Two of these methods, CIGALE (Noll et al. 2009; Boquien et al. 2019) and MAGPHYS (da Cunha et al. 2008), are based on the assumption that the radiation produced during the star formation process by the stellar and nebular components is partially absorbed by the dust and then re-emitted in the infrared part of the spectrum, satisfying the energy conservation. New families of codes have also been produced in an attempt to combine spectral and photometric analyses, recovering, through the implementation of photoionisation codes, emission lines feature; see for example Prospect (Robotham et al. 2020) and Prospector (Leja et al. 2017). Recently, even machine learning methods have been involved in this challenge, with neural network algorithms showing promising results (Simet et al. 2021).


The H i mass fraction of every gas particle in the baryonic runs is calculated based on the particle’s temperature and density and the cosmic UV background radiation flux while including a prescription for self-shielding of H2 and dust shielding in both H i and H2 (Christensen et al. 2012). This allows for the straightforward calculation of the total H i mass of each simulated galaxy. We create mock H i data cubes only for the 42 halos that contain

. Specifically, we create mock data cubes that mimic ALFALFA observations (Haynes et al. 2011). After specifying a viewing angle (see below), our code considers the line-of-sight velocity of each gas particle. The velocity of each particle is tracked in the simulation by solving Newton’s equations of motion, but any turbulent velocity of the gas is not taken into account. Velocity dispersions in dwarf galaxies can be on the order of the rotational velocity, ∼10–15 km s−1 (e.g., Stanimirović et al. 2004; Tamburro et al. 2009; Oh et al. 2015). Dispersions are thought to be driven at least partially by thermal velocities or supernovae (Tamburro et al. 2009; Stilp et al. 2013a, 2013b). In our simulations, supernovae inject thermal energy, and the thermal state of the H i gas needs to be considered in the mock H i linewidth for a realistic comparison to observations. To account for the thermal velocity, the H i mass of each gas particle is assumed to be distributed along the line-of-sight in a Gaussian distribution with a standard deviation given by the thermal velocity dispersion,

, where T is the temperature of the gas particle. After this thermal broadening is calculated, a mock H i data cube can be generated by specifying the spatial and velocity resolution. For all of our galaxies, we adopt a spatial resolution of 54 pixels across 2Rvir. In practice, this corresponds to a range of ∼1 kpc resolution in our lowest-mass galaxies up to ∼9 kpc resolution in our most massive galaxies. However, the spatial resolution plays no role in our study, since measurements of the VF are based on spatially unresolved H i data. For the velocity resolution, we match the ALFALFA specification of 11.2 km s−1 (two-channel boxcar-smoothed).


Pulsar timing experiments (Sazhin 1978; Detweiler 1979) allow us to explore the low-frequency (∼1–100 nHz) part of the gravitational-wave (GW) spectrum. By measuring deviations from the expected arrival times of radio pulses from an array of millisecond pulsars, we can search for a variety of GW signals and their sources. The most promising sources in the nanohertz part of the GW spectrum are supermassive binary black holes (SMBHBs) that form via the mergers of massive galaxies. Orbiting SMBHBs produce a stochastic GW background (GWB; Lommen & Backer 2001; Jaffe & Backer 2003; Volonteri et al. 2003; Wyithe & Loeb 2003; Enoki et al. 2004; Sesana et al. 2008; McWilliams et al. 2012; Sesana 2013; Ravi et al. 2015; Rosado et al. 2015; Kelley et al. 2016; Sesana et al. 2016; Dvorkin & Barausse 2017; Kelley et al. 2017; Bonetti et al. 2018; Ryu et al. 2018), individual periodic signals or continuous waves (CWs; Sesana et al. 2009; Sesana & Vecchio 2010; Mingarelli et al. 2012; Roedig & Sesana 2012; Ravi et al. 2012, 2015; Rosado et al. 2015; Schutz & Ma 2016; Mingarelli et al. 2017; Kelley et al. 2018), and transient GW bursts (van Haasteren & Levin 2010; Cordes & Jenet 2012; Ravi et al. 2015; Madison et al. 2017; Islo et al. 2019; Bécsy & Cornish 2021). We expect to detect the GWB first, followed by detection of individual SMBHBs (Siemens et al. 2013; Rosado et al. 2015; Taylor et al. 2016; Mingarelli et al. 2017) that stand out above the GWB. Detection of GWs from SMBHBs will yield insights into galaxy mergers and evolution not possible through any other means. Other potential sources in the nanohertz band include cosmic strings (Damour & Vilenkin 2000, 2001; Berezinsky et al. 2004; Damour & Vilenkin 2005; Siemens et al. 2006, 2007; Ölmez et al. 2010; Sanidas et al. 2013; Blanco-Pillado et al. 2018; Chang & Cui 2021; Ghayour et al. 2021; Gorghetto et al. 2021; Wu et al. 2021a; Blanco-Pillado et al. 2021; Lin 2021; Chiang & Lu 2021; Lazarides et al. 2021; Chakrabortty et al. 2021; Ellis & Lewicki 2021), phase transitions in the early universe (Witten 1984; Caprini et al. 2010; Addazi et al. 2021; Arzoumanian et al. 2021; Di Bari et al.2021; Borah et al. 2021; Nakai et al. 2021; Brandenburg et al.2021; Neronov et al. 2021), and relic GWs from inflation (Starobinskiǐ 1979; Allen 1988; Lazarides et al. 2021; Ashoorioon et al. 2021; Yi & Zhu 2021; Li et al. 2021; Poletti 2021; Vagnozzi 2021; Sharma 2021), all of which would provide unique insights into high-energy and early-universe physics.


A black hole mass this large in Leo I is not expected from extrapolation of any of the standard black hole−host galaxy correlations. Of course, these small systems do not necessarily need to follow the trends seen in normal galaxies, but the black hole mass reported here does stand out. Lützgendorf et al. (2015) explore extrapolations of black hole correlations down to globular cluster scales, and using a velocity dispersion of 12 km s−1, Leo I has a black hole mass a factor of 100 more than the extrapolated trends. On the numerical side, van Wassenhove et al. (2010) consider different scenarios for formation of a black hole in Milky Way satellites and place the likelihood of one of them having a black hole around the size found here to be below 1%, but this result also depends on the initial seed mass (see also Bellovary et al. 2021). Runaway mergers of stellar mass black holes are unlikely to produce such a black hole in such a small galaxy, since the required initial mass function to reach the ratios seen in the models might be more top-heavy than what chemical abundances and star formation history studies suggest. An alternative explanation for the abnormally large central black hole may come from the recent study on Leo I’s star formation history from Ruiz-Lara et al. (2020). The authors identify a period of quenching from z = 1–2 followed by reignition until almost the present day, when ram pressure stripping may have shut it down as it fell into the Milky Way. While the authors speculate that this reignition at intermediate redshifts could be due to a past merger with a smaller dwarf, this could also be consistent with gas accretion and potential active galactic nuclei feedback, lending support to the high MBH values presented here. Amaro-Seoane et al. (2014) also suggest that dwarf systems may in fact have significantly larger black holes compared to the host galaxy–black hole relationships. Having a larger sample of black hole limits measured in dwarf galaxies will be important to explore.


As seen in Fig. 6 (right), the large MOS-pn discrepancy in the Ni/Fe abundance ratio prevents us from deriving a precise measurement. This discrepancy is worrying, but can be explained by imperfections in the cross-calibration of the two instruments. Alternatively, and perhaps more likely, the high energy band around the Ni-K transitions is significanly affected by the instrumental background (as the flux of the cluster emission sharply decreases at high energies). This hard particle background (already mentioned in Sect. 3.1) has a different spectral shape in MOS and pn, which might even vary with time, thus between observations. In particular, an instrumental line (Cu Kα) is known to affect pn at a rest-frame energy of ~8 keV (Mernier et al. 2015). Despite our efforts to carefully estimate the background, that line might interfere with the Ni-K line in several observations, making a proper modelling of the Ni-K line impossible, and hence, boosting the Ni absolute abundance in pn. In this context, it can be instructive to compare our Ni/Fe measurements with those of Suzaku, which has a lower relative hard particle background. Sato et al. (2007b) (A 1060) and Tamura et al. (2009) (Perseus) reported ratios of ~1.3 ± 0.4 and ~1.11 ± 0.19, respectively (after rescaling to the proto-solar values). Although these measurements might be also be affected by further uncertainties (e.g. the choice of the spectral modelling, Sect. 4.3), they appear to be consistent with the Ni/Fe average ratio measured with MOS is this work, favouring our above supposition that MOS is more trustworthy than pn for measuring Ni/Fe. However, in order to be conservative, we prefer to retain the pn value as a possible result and, therefore, we keep large systematic uncertainties for Ni/Fe. We finally note that, unsurprisingly, Ni/Fe cannot be constrained in the cool objects (Fig. 6, left) because the gas temperature is too low to excite Ni-K transitions.


In recent years, type Ia supernovae (SNe Ia; Amanullah et al. 2010; Suzuki et al. 2012; Betoule et al. 2014; Scolnic et al. 2018) have been widely employed to test cosmic isotropy. Antoniou & Perivolaropoulos (2010) searched for the preferred direction of anisotropy for the Union2 sample by adopting the hemisphere comparison (HC) method (Schwarz & Weinhorst 2007). They found a maximum accelerating expansion rate, which corresponds to a preferred direction of anisotropy. After that, Mariano and Perivolaropoulos (Mariano & Perivolaropoulos 2012) found a possible preferred anisotropic direction at the 2σ level using the Union2 sample, but by employing the dipole fitting (DF) method. Since then, these two methods have been widely used to explore the cosmic anisotropy (Cai & Tuo 2012; Cai et al. 2013; Zhao et al. 2013; Li et al. 2013; Heneka et al. 2014; Bengaly et al. 2015; Andrade et al. 2018; Sun & Wang 2019) by investigating observational data of, for instance, the Union2.1 sample (Yang et al. 2014; Javanmardi et al. 2015; Lin et al. 2016a), the Joint Light-Curve Analysis (JLA) sample (Lin et al. 2016b; Chang et al. 2018a; Wang & Wang 2018), the Pantheon sample (Sun & Wang 2018), gamma-ray bursts (GRBs; Wang & Wang 2014), galaxies (Zhou et al. 2017), as well as gravitational wave and fast radio bursts (Qiang et al. 2019; Cai et al. 2019). Using HC and DF methods, Zhao et al. (2019) studied the cosmic anisotropy via the Pantheon sample. They found that the SDSS sample plays a decisive role in the Pantheon sample. It may imply that the inhomogeneous distribution has a significant effect on the cosmic anisotropy (Chang et al. 2018b). This opinion was also presented by Sun & Wang (2019). Their conclusions show that the effect of redshift on the result is weak and there is a negligible anisotropy when making a redshift tomography. Deng & Wei (2018a) tested the cosmic anisotropy with the Pantheon sample, but by using the following three methods: the HC method, the DF method, and Healpix1 (Górski et al. 2005). They also performed a cross check. There are two preferred directions from the HC method. In adopting the DF method and Healpix, they found no noticeable anisotropy. They also compared the HC method with the DF method by using the JLA sample (Deng & Wei 2018b) and found that the results of these two methods have not always been approximately coincident with each other. In order to better test the cosmic isotropy, the best way would be to add new samples with a relatively homogeneous distribution.


The lack of data on inelastic processes due to collisions with neutral hydrogen atoms has been a major limitation on modelling of F-, G- and K-star spectra in statistical equilibrium, and thus to reliably proceeding beyond the assumption of local thermodynamic equilibrium (LTE) in analysis of stellar spectra and the determination of elemental abundances. This problem has been well documented, e.g. see Lambert (1993); Barklem (2016a) and references therein. Significant progress has been made in recent times through detailed full-quantum scattering calculations, based on quantum chemical data, for the cases of simple atoms such as Li, Na, Mg and Ca (Belyaev & Barklem 2003; Barklem, Belyaev & Asplund 2003; Belyaev et al. 2010; Barklem et al. 2010; Belyaev et al. 2012; Barklem et al. 2012; Mitrushchenkov et al. 2017). These calculations have demonstrated the importance of the ionic-covalent curve crossing mechanism leading naturally to charge transfer processes (mutual neutralization and ion-pair production), in addition to excitation and de-excitation processes. The importance of this mechanism has allowed various simplified model approaches to be developed, which may be used in cases where suitable quantum chemistry data are not been available. In particular a semi-empirical model has been employed for Al, Si, Be and Ca (Belyaev 2013a,b; Belyaev, Yakovleva & Barklem 2014b; Yakovleva, Voronov & Belyaev 2016; Belyaev et al. 2016), and a theoretical model based on a two-electron asymptotic linear combinations of atomic orbitals (LCAO) approach, has also been employed for Ca (Barklem 2016b, 2017). Comparisons of the two methods show quite good agreement and reasonable agreement with the full quantum calculations is found, particularly for the most important processes with the largest rates (Barklem 2016b, 2017; Mashonkina, Sitnova & Belyaev 2017; Mitrushchenkov et al. 2017). Thus, the model approaches provide a useful route for obtaining estimates of the rates for these processes for many elements of astrophysical interest.