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).


The ability to test the hypothesis that a set of data samples is drawn from some general n-dimensional probability distribution $f(\boldsymbol {x})$ has an interesting application in the validation of Bayesian inference analyses (indeed this application provided our original motivation for seeking to extend the K–S test to multiple dimensions). Bayesian methods are now pervasive across all branches of science and engineering, from cognitive neuroscience (Doya et al. 2007) and machine learning (Bishop 2006) to spam filtering (Sahami et al. 1998) and geographic profiling (Collins, Gao & Carin 1999; Le Comber & Stevenson 2012). In precision cosmology, Bayesian inference is the main tool for setting constraints on cosmological parameters (Hinshaw et al. 2013; Planck Collaboration XVI 2014), but very few attempts have been made to assess whether the derived posterior probability distributions are a truthful representation of the actual parameter constraints one can infer from the data in the context of a given physical model. This lack of validation has been highlighted by several authors, with the strong dependence of the inference results on the priors being of particular concern (Efstathiou 2008; Linder & Miquel 2008). There have been attempts to address this issue, ranging from the approach of Cook, Gelman & Rubin (2006), which was designed with software validation solely in mind, to a method based on the inverse probability integral transform (Smirnov transform) applied to posterior distributions that extends to spaces of higher dimensionality via marginalization (Dorn, Oppermann & Enßlin 2013). Also, validation of the Bayesian source-finding algorithm of Carvalho et al. (2012) was performed in Planck Collaboration XXIX (2014), but only point estimates deduced from the posterior distributions were actually verified. Our method for addressing this problem is based on our applying our multidimensional extension of the K–S test to sets of Monte Carlo simulations of the data and the posterior distributions derived therefrom. In particular, it can take advantage of the fact that one may typically generate simulations that are of greater sophistication and realism than may be modelled in the inference process, and thus allows for a more thorough validation of the inference than has been possible with the methods developed previously. In particular, our validation procedure enables us to test all the assumptions made (explicitly or implicitly) in the inference process, such as the statistical description of the data, model assumptions and approximations, as well as the software implementation of the analysis. Moreover, we consider the full posterior distribution, regardless of its dimensionality and form, without the need to resort to marginalization, and thereby keeping intact its n-dimensional character.


The origin of the GC VHE emission remains undetermined, due in part to source confusion and the limitations of current instruments. The source of VER J1745–290 may be Sgr A* (Atoyan & Dermer 2004; Aharonian & Neronov 2005; Ballantyne et al. 2011; Chernyakova et al. 2011; Fatuzzo & Melia 2012; Kusunose & Takahara 2012; Fujita et al. 2017; Rodríguez-Ramírez et al. 2019) or PWN G359.95-0.04 (Wang et al. 2006; Hinton & Aharonian 2007), with which it is spatially coincident (Acero et al. 2010). Other possible origins include the annihilation of dark matter particles (Bergström et al. 2005a, 2005b; Horns 2005; Profumo 2005; Aharonian et al. 2006c; Belikov et al. 2012; Cembranos et al. 2012, 2013; Gammaldi et al. 2016) or a population of millisecond pulsars (Bednarek & Sobczak 2013; Bartels et al. 2016; Guépin et al. 2018). The mechanism of gamma-ray emission may be predominantly due to hadronic processes, where relativistic protons interact with gas and subsequently produce gamma rays through neutral pion decay (Aharonian & Neronov 2005; Ballantyne et al. 2011; Chernyakova et al. 2011; Fatuzzo & Melia 2012; Linden et al. 2012; Guépin et al. 2018), leptonic processes where gamma rays are produced when electrons and positrons undergo inverse Compton scattering off a radiation field (Atoyan & Dermer 2004; Hinton & Aharonian 2007; Kusunose & Takahara 2012; Lacroix et al. 2016), or a combination of processes (hybrid scenario), where leptons produce high-energy, but not VHE, gamma rays (Guo et al. 2013). Both the correlation of VHE emission with the CMZ and the lack of a cutoff in the diffuse spectrum support a hadronic scenario, capable of explaining both VER J1745–290 and the diffuse emission (Aharonian et al. 2006b; Linden et al. 2012; Abramowski et al. 2016). Measurement of the diffuse spectrum by H.E.S.S. up to energies of tens of TeV with no evidence of a cutoff has also been interpreted as evidence for the existence of PeV protons within the central 10 parsecs of the GC, accelerated by Sgr A* (Abramowski et al. 2016). While cosmic rays are known to extend up to PeV energies (e.g., Hörandel 2003), few, if any, accelerators of PeV cosmic rays, or “PeVatrons,” have been clearly established (e.g., Abramowski et al. 2016; Abeysekara et al. 2020). Discovering the nature of PeVatrons in our Galaxy is thus a particularly important step in understanding the origins of cosmic rays.


Simulations are performed in Fig. 6 to compare the spectrum profiles of the S ii triplets for a synthetic absorption spectrum of a DLA with and without the magnetic alignment included. Fig. 6 was designed up to illustrate the effect in a realistic instrumental set-up, with a given typical pixel size and S/N for an optical spectrum taken by an 8m-class telescope. The graphical use of steps instead of curves, which is common in absorption spectroscopy, is intended to visualize the pixel-by-pixel noise variations in the data. The three transitions of singly ionized Sulfur (Sii; upper ionization potential 23.3 eV) at λλ1250.58, 1253.81, 1259.52Å represent important tracers for neutral and weakly ionized gas in the local interstellar and intergalactic medium and in distant galaxies (e.g. Richter et al. 2001; Welsh & Lallement 2012; Fox, Richter & Fechner 2014a; Kisielius et al. 2014). Being an α element, singly ionized Sulfur only has a weak depletion into dust grains (e.g. Savage & Sembach 1996). Thus the interstellar Sulfur abundance is often used as a proxy for the α-abundance in the gas. In addition, Sulfur has a relatively low-cosmic abundance (Asplund et al. 2009) and under typical interstellar conditions (in particular in low-metallicity environments) these lines are not saturated. The three lines are observed in the same wavelength region with identical S/N. The important parameters for simulations are presented in the caption, such as the assumption of S ii column density, etc. The synthetic spectra were generated using the fitlyman routine (Fontana & Ballester 1995) implemented in the eso-midas software package. Atomic data were taken from Morton (2003). To show the effect clearly, we zoom in the spectrum to the radial velocity range [ − 10km · s−1, 10km · s−1]. The enhancement and reduction of the spectral line profile due to the magnetic realignment change among the triplets. Given the fact that meanwhile optical QSO spectra reach up to a new standard of S/N of a few hundred (e.g. D’Odorico et al. 2016), the predicted effect is already VISIBLE, if the component structure of the DLA allows a detailed investigation.


The high-resolution imaging observations of TR from IRIS reveal the ubiquitous presence of network jets. We have used three different IRIS observations of the quiet sun, which are located near the disk center. On the basis of careful inspection, 51 network jets are identified from three QS observations and used for further analysis. These 51 network jets are very well resolved and are not affected by the dynamics of other jets. The study is focused on the rotating motion of network jets along with the estimation of their other properties (speed, height, and lifetime). The mean speed, as predicted by statistical distributions of the speed, is 140.16 km s−1 with a standard deviation of 39.41 km s−1. The mean speed of network jets is very similar, as reported in previous works (e.g., Tian et al. 2014; Narang et al. 2016). However, in case of their lifetimes, we found a value that is almost double (105.49 s) that of the previously reported mean lifetime of the network jets (49.6 s; Tian et al. 2014). As mentioned above, we took only those network jets that are very well resolved in space and in the time; these criteria exclude short lifetime network jets. Therefore, our statistical distribution of the lifetime predicts a higher mean lifetime. The mean length of the network jets is 3.16 Mm with a standard deviation of 1.18 Mm. In the case of CH network jets, Tian et al. (2014) have reported that most of the network jets have lengths from 4.0 to 10.0 Mm. However, the mean length for QS network jets is smaller (3.53 Mm; Narang et al. 2016). So, the mean length for QS network jets from the present work is in good agreement with Narang et al. (2016). In addition, the apparent speed and length of these network jets are positively correlated, which is very similar to what has already been reported in previous works (Narang et al. 2016). Finally, we can say that these networks jets are very dynamic features of the solar TR, as revealed by their estimated properties.


Many studies indicate a much more complicated star formation history (SFH; e.g., Pagel & Tautvaišvienė 1998; Carrera et al. 2008; Harris & Zaritsky 2009; Rubele et al. 2012; Meschin et al. 2014; Palma et al. 2015; Perren et al. 2017) or AMR for star clusters (e.g., Olszewski et al. 1991; Hill et al. 2000; Dirsch et al. 2000; Rich et al. 2001; Leonardi & Rose 2003; Kerber et al. 2007) in the LMC than in the SMC. Early studies of the cluster AMR in the LMC (e.g., the spectroscopic study of Olszewski et al. 1991) revealed a mysterious gap in the star cluster formation between 10 and 3 Gyr ago (with the sole exception of cluster ESO121-3 and very recently also KMHK1592, as reported by Piatti 2022), which follows the initial formation of old, metal-poor globular clusters, and precedes more recent active formation of intermediate-age clusters. This so-called “age gap”, which is simultaneously a metallicity gap, was later reported also by other authors (e.g., Harris & Zaritsky 2009; Sharma et al. 2010; Kerber et al. 2007, and references therein), even though it has not been fully confirmed when only the field stars are considered. For example, Piatti & Geisler (2013) reported that no clear age gap in the field star formation is observed in their study of fields in the LMC main body, based on Washington photometry. Furthermore, based on their spectroscopic analysis of four fields to the north of the LMC bar, Carrera et al. (2008) maintain that the disk and cluster AMR are similar. However, unlike clusters, there is no age gap in the field population. On the contrary, Harris & Zaritsky (2009) claim that such a gap is evident in the field population of the bar, mostly omitted in previous studies. Also Meschin et al. (2014), using optical photometry, identified two main star-forming epochs with a period characterized by a lower activity in between; however, in their work that interval of time was shorter and lasted from ∼8 up to ∼4 Gyr ago. All the aforementioned authors agree that star formation in the LMC is continuing to this day.


The first direct proof of the existence of magnetic fields in large-scale extragalactic environments, i.e., galaxy clusters, dates back to the 1970s with the discovery of extended, diffuse, central synchrotron sources called radio halos (see, e.g., Feretti et al. 2012 for a review). Later, indirect evidence of the existence of intracluster magnetic fields has been given by several statistical studies on the effect of the Faraday rotation on the radio signal from background galaxies or galaxies embedded in galaxy clusters (Lawler & Dennison 1982; Vallée et al. 1986; Clarke et al. 2001; Johnston-Hollitt 2003; Clarke 2004; Johnston-Hollitt & Ekers 2004). On scales up to a few Mpc from the nearest galaxy cluster, possibly along filaments, only a few diffuse synchrotron sources have been reported (Harris et al. 1993; Bagchi et al. 2002; Kronberg et al. 2007; Giovannini et al. 2013, 2015). Magnetic fields with strengths on the order of 10-15 G in voids might be indicated by γ-ray observations (see Neronov & Vovk 2010; Tavecchio et al. 2010; Takahashi et al. 2012, 2013; but see Broderick et al. 2014a,b for alternative possibilities). Nevertheless, up to now, a robust confirmed detection of magnetic fields on scales that are much larger than clusters is not available. Stasyszyn et al. (2010) and Akahori et al. (2014a) investigated the possibility of statistically measuring Faraday rotation from intergalactic magnetic fields with present observations, showing that only the Square Kilometre Array (SKA) and its pathfinders are likely to succeed in this respect. By comparing the observations with single-scale magnetic field simulations, Pshirkov et al. (2015) infer an upper limit of 1.2 nG for extragalactic large-scale magnetic fields, while the Planck Collaboration XIX (2016) derived a more stringent upper limit for primordial large-scale magnetic fields of B 0.67 nG from the analysis of the Cosmic Microwave Background (CMB) power spectra and the effect on the ionization history (but see also Takahashi et al. 2005; Ichiki et al. 2006).


On the other hand, PT has experienced many developments in recent years (Matsubara 2008a; Baumann et al. 2012; Carlson et al. 2013), in part because it can be useful to analytically understand different effects in the power spectrum and correlation function for the dark matter clustering. These effects can be confirmed or refuted, and further explored with simulations to ultimately understand the outcomes of present and future galaxy surveys, such as eBOSS (Zhao 2016), DESI (Aghamousa et al. 2016), EUCLID (Amendola et al. 2013), and LSST (LSST Dark Energy Science Collaboration 2012). Two approaches have been used to study PT: the Eulerian standard PT (SPT) and Lagrangian PT (LPT), which both have advantages and drawbacks, but they are complementary in the end (Tassev 2014). The nonlinear PT for MG was developed initially in (Koyama et al. 2009), and has been further studied in several other works (Taruya et al. 2014a,b; Brax & Valageas 2013; Bellini & Zumalacarregui 2015; Taruya 2016; Bose & Koyama 2016, 2017; Barrow & Mota 2003; Akrami et al. 2013; Fasiello & Vlah 2017; Aviles & Cervantes-Cota 2017; Hirano et al. 2018; Bose et al. 2018; Bose & Taruya 2018; Aviles et al. 2018). The LPT for dark matter fluctuations in MG was developed in Aviles & Cervantes-Cota (2017), and further studies for biased tracers in Aviles et al. (2018). The PT for MG has the advantage that it allows us to understand the role that these physical parameters play in the screening features of dark matter statistics. We here study some of these effects through screening mechanisms by examining them at second- and third-order perturbation levels using PT for some MG models. To this end, we build on the LPT formalism developed in Aviles & Cervantes-Cota (2017), which was initially posited for MG theories in the Jordan frame, in order to apply it to theories in the Einstein frame. Because of the direct coupling of the scalar field and the dark matter in the Klein–Gordon equation, the equations that govern the screening can differ substantially from those in Jordan-frame MG theories. In general, screening effects depend on the type of nonlinearities that are introduced in the Lagrangian density. We present a detailed analysis of screening features and identify the theoretical roots of their origin. Our results show that screenings possess peculiar features that depend on the scalar field effective mass and couplings, and that may in particular cases cause anti-screening effects in the power spectrum, such as in the symmetron model. We perform this analysis by separating the growth functions into screening and non-screened parts. We note, however, that we do not compare the perturbative approach with a fully nonlinear simulation. We refer to (Koyama et al. 2009), for instance, for investigations like this at the level of the power spectrum.


The Λ cold dark matter (CDM) model has had tremendous success in providing a cosmogony that links the state of the very high redshift Universe as inferred from cosmic microwave background observations and big bang nucleosynthesis considerations, with the observed large-scale distribution of galaxies and its evolution at later times. In this paradigm, the observed structures grew due to gravitational amplification of initially small perturbations, perhaps seeded by inflation (Guth 1981), with most of the gravitating mass in the form of ‘dark matter’, an as of yet unknown type of matter that apart from gravitationally, interacts only very weakly if at all with baryonic matter and radiation, see e.g. Frenk & White (2012) for a recent review. Observations on smaller, sub galactic scales, have proven more problematic for ΛCDM, with suggestions that the low abundance (e.g. Klypin et al. 1999; Moore et al. 1999) and shallow density profiles (e.g Gilmore et al. 2007; de Vega & Sanchez 2010) inferred for haloes hosting Milky Way satellites are inconsistent with the numerous substructures with cuspy density profiles that form in Milky Way-like CDM haloes. Indeed the substructures in the haloes of the Aquarius project (Springel et al. 2008) and the GHALO project (Stadel et al. 2009) may be difficult to reconcile with those inferred to host Milky Way satellites. Even if many of the smaller DM substructures may not have a sufficiently deep potential well to form stars after reionization (e.g Efstathiou 1992; Benson et al. 2002) and so may well be dark, some of the more massive ones are probably too big to be affected (Okamoto, Gao & Theuns 2008) and would hence appear to be ‘too big to fail’ (Boylan-Kolchin, Bullock & Kaplinghat 2011, 2012). However, the jury on this is still out: the Milky Way's halo may simply be of lower mass than those of the Aquarius haloes and hence have fewer massive satellites (Wang et al. 2012). The status of the density profiles of the satellite's haloes – cored versus cuspy – is similarly unresolved. Strigari, Frenk & White (2010) claim that the stellar dynamics observations of the satellites do not imply cores at all and hence may be consistent with CDM cusps. But even if the satellites had cored profile, this might result from the action of baryonic feedback processes (Governato et al. 2012; Pontzen & Governato 2012), and hence still be consistent with CDM.


The energy in the inner crust is largely influenced by the properties of the neutron gas and, therefore, the EoS of neutron matter of the different calculations plays an essential role. The NV calculation (Negele & Vautherin 1973) is based on a local energy density functional that closely reproduces the Siemens-Pandharipande EoS of neutron matter (Siemens & Pandharipande 1971) in the low-density regime. The Moskow calculation (Baldo et al. 2007) employs a semi-microscopic energy density functional obtained by combining the phenomenological functional of Fayans et al. (2000) inside the nuclear cluster with a microscopic part calculated in the Brueckner theory with the Argonne v18 potential (Wiringa et al. 1995) to describe the neutron environment in the low-density regime (Baldo et al. 2004). The BBP calculation (Baym et al. 1971a,b) gives the EoS based on the Brueckner calculations for pure neutron matter of Siemens (Siemens & Pandharipande 1971). The LS-Ska (Lattimer & Swesty 1991; Lattimer 2015) and DH-SLy4 (Douchin & Haensel 2001) EoSs were constructed using the Skyrme effective nuclear forces Ska and SLy4, respectively. The SLy4 Skyrme force (Chabanat et al. 1998) was parametrized, among other constraints, to be consistent with the microscopic variational calculation of neutron matter of Wiringa et al. (1988) above the nuclear saturation density. The Shen-TM1 EoS (Shen et al. 1998b,a; Sumiyoshi 2015) was computed using the relativistic mean field parameter set TM1 for the nuclear interaction. The calculations of LS (Lattimer & Swesty 1991; Lattimer 2015) and Shen et al. (Shen et al. 1998b,a; Sumiyoshi 2015) are the two EoS tables in more widespread use for astrophysical simulations. The BSk21 EoS (Pearson et al. 2012; Fantina et al. 2013; Potekhin et al. 2013; Goriely et al. 2010) is based on a Skyrme force with the parameters accurately fitted to the known nuclear masses and constrained, among various physical conditions, to the neutron matter EoS derived within modern many-body approaches which include the contribution of three-body forces.


As illustrated in this paper, the variation of spectral line intensity induced by GSA varies among different spectral lines. Thus, such influence could be precisely analysed if multispectral lines for the same element is achievable. In addition, the alignment on the ground state is transferred to the levels of atoms on the excited states through absorption process, as illustrated in Section 3 and 4. The intensity of the ultraviolet-pumped fluorescence lines, which are derived from successive decays to different levels of atoms and applied to the modelling of reflection nebulae (Sellgren 1984, 1986), are dependent on the initial upper levels, and thus is influenced by GSA through the scattering process. The influence of collision is neglected in this paper, which applies to most diffuse ISM and IGM. Collisionsreduce the alignment efficiency (see Hawkins 1955). The collision effect can become important in the case of higher density medium where the collision rate $\tau _c^{-1}$ (either inelastic collision rate or Van der Waals collision rate) dominates over optical pumping rate $B_{lu}\bar{J}^0_0$ (see Yan & Lazarian 2006 for details). The focus of our paper is on the GSA effect that is a saturated state. When the magnetic precession rate is comparable to the optical pumping from the ground state ($2\pi \nu _L\sim B_{lu}\bar{J}^0_0$), the ground-level Hanle effect is applicable (see Landolfi & Landi Degl’Innocenti 1986), which means the magnetic field influence on the spectrum is not limited to the change of direction but also the magnetic field strength. As demonstrated in Yan & Lazarian (2008), the effect becomes saturated when $2\pi \nu _L\rightarrow 10B_{lu}\bar{J}^0_0$. Therefore, we set rHanlewhere $2\pi \nu _L\simeq 10B_{lu}\bar{J}^0_0$ as the boundary between the ground-level Hanle regime and the GSA regime (see Fig. 12a):
(10)
\begin{eqnarray*}
r_{Hanle}=r_\ast \sqrt{\frac{A_{ul}[J_u]}{1.76(B/{\mu }G)(\exp (h\nu /(k_BT))-1)[J_l]}}
\end{eqnarray*}
We calculate the boundary rHanle for C ii λ1037 Å in the presence of stars with different effective temperature Teff and radius r* for the magnetic field with the strength range from μG to mG in Fig. 12(b). The rHanle increases as the magnetic field becomes weaker and as the effective temperature Teff and radius r* increase. Nevertheless, even in the most optimistic scenario with Teff = 4 × 104K, r* = 10r⊙, B = 1μG (though rarely applies), the boundary rHanle ≃ 180Au = 8.5 × 10−4pc, which is a thousand times smaller than the normal H ii Region which is in pc scale. All the analysis with GSA and their observational implication in this paper is applicable to most of the ISM, except when performing very high resolution spectral analysis on regions very close to the very bright O-type star.


The focus of this work is to address the proplyd lifetime problem from a modelling perspective using recent developments in both the theory of photoevaporating discs and observations of the star and disc population in the ONC. The most similar study to this work is that of Scally & Clarke (2001), who produced an N-body model of the ONC coupled with theoretical photoevaporative mass-loss rates. The aim was to test the idea put forward by Störzer & Hollenbach (1999) that radial orbits of stars could result in shorter periods of exposure to strong FUV flux close to θ1C, thus increasing the PPD dispersal time-scale. The models demonstrated that such dynamical orbits alone are insufficient to produce significantly extended disc lifetimes. However, since that study a number of developments mean that the problem is due to be revisited. First, thermodynamic calculations of conditions in photodissociation regions (PDRs) have been coupled with self-consistent equations for the thermally driven disc wind (Adams et al. 2004; Facchini et al. 2016; Haworth et al. 2018; Haworth & Clarke 2019). In particular, the recent Fried grid (Haworth et al. 2018) can be interpolated across FUV flux, disc outer radius, and disc mass to calculate an instantaneous mass-loss rate for a given PPD. Secondly, the recent sub-mm survey of disc masses and radii by Eisner et al. (2018, hereafter E18) offers further observational constraints on a successful model of PPD evolution, and the host mass-dependent disc properties permit a test of theoretical predictions. Thirdly, Beccari et al. (2017, see also Jerabkova et al. 2019) recently found evidence of three stellar populations with distinct ages in the ONC. This has multiple consequences for models of a PPD population, most obviously that a subset of discs has evolved for a shorter period than the oldest stars in the region. Additionally, since stars are formed from gas, during the evolution of the cluster prior to the most recent formation event there may have been considerable intracluster extinction of UV photons. Understanding how such a scenario influences the disc population would also represent a test of the hypothesis that discrete epochs of star formation occurred in the ONC.


Fast radio bursts (FRBs), a promising new and mysterious astrophysical phenomenon, are a class of bright transients with millisecond durations detected at ∼GHz (Lorimer et al. 2007; Keane et al. 2011; Thornton et al. 2013; Burke-Spolaor & Bannister 2014; Spitler et al. 2014; Masui & Sigurdson 2015; Petroff et al. 2015, 2016; Ravi et al. 2015, 2016; Champion et al. 2016; Keane et al. 2016; Caleb et al. 2017). Although the physical origin of these intense short pulses has not been figured out so far, most of them have been detected with relatively large dispersion measures (DM) (greater than the maximum produced by the Milky Way). This suggests that they may occur at a cosmological distance, probably at distances of the order of gigaparsecs (Ioka 2003; Inoue 2004; Deng & Zhang 2014; Zheng et al. 2014; Zhang 2018). So far, the cosmological origin of these kinds of mysterious flashes has been confirmed by successfully localizing several bursts (including two repeaters and several apparently nonrepeating ones; Spitler et al. 2016; Scholz et al. 2016; Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017; Bannister et al. 2019; Prochaska et al. 2019; Ravi et al. 2019; Wang & Zhang 2019; Zhang & Wang 2019; Marcote et al. 2020). The confirmation of their cosmological origin allows them to be widely proposed as promising tools for studying the universe and fundamental physics, e.g., the distribution of baryons in the intergalactic medium (IGM) or diffuse gas (Deng & Zhang 2014; Mcquinn 2014; Muñoz & Loeb 2018; Li et al. 2019, 2020; Walters et al. 2019; Wei et al. 2019; Macquart et al. 2020), dark energy (Gao et al. 2014; Zhou et al. 2014; Walters et al. 2018; Wei et al. 2018; Zhao et al. 2020), cosmic ionization history (Deng & Zhang 2014; Zheng et al. 2014), the large-scale structure of the universe (Masui & Sigurdson 2015), the Einstein’s equivalence principle (Wei et al. 2015; Nusser 2016; Tingay & Kaplan 2016), the rest mass of the photon (Wu et al. 2016; Shao & Zhang 2017; Xing et al. 2019), the cosmic proper distance measurements (Yu & Wang 2017), and constraints on the magnetic fields in the IGM (Akahori et al. 2016), as well as measuring the cosmic expansion rate (Wu et al. 2020). Because the event rate of this phenomenon inferred from observations is very high (∼103–104 per day all sky; Thornton et al. 2013; Champion et al. 2016) and due to their extragalactic origin, these bursts are likely to be gravitationally lensed by intervening objects with a different magnitude of mass. Moreover, high time resolutions or short durations of these bursts (∼(1–10) ms) (Lorimer et al. 2007; Keane et al. 2011; Thornton et al. 2013; Spitler et al. 2014, 2016; Petroff et al. 2015, 2016; Ravi et al. 2015; Champion et al. 2016) guarantee that the difference between arrival times of two images can be resolved even if they are lensed by an object as small as 10  M⊙. Therefore, different scale lensing effects of FRBs have been proposed as a probe of compact dark matter (Muñoz et al. 2016; Wang & Wang 2018; Liao et al. 2020; Ranjan 2020), motion of the FRB source (Dai & Lu 2017), and precision cosmology (Li et al. 2018; Liu et al. 2019; Wucknitz et al. 2020). Recently, taking into consideration the theoretical prediction that a portion of FRBs might be associated with GWs, Wei et al. (2018) proposed the joint measurements of DM from FRB observations and DL from GW detections in the same FRB/GW association system (i.e., the combination DM · DL as a function of redshift) as a complementary cosmic probe. The most striking merit of this probe is the independence of the Hubble constant H0, which is a fundamental cosmological parameter now under intense debate because of the well-known Hubble constant tension (Planck Collaboration et al. 2018; Riess et al. 2019). In this paper, we propose that the combination DM · DL as a function of redshift could be further extended to the independent measurements of DMs of localized FRBs and DL of other distance indicators (e.g., SNe Ia) at a similar redshift according to the fundamental assumption that luminosity distance monotonously increases with increasing redshift. Following Masui & Sigurdson (2015), who call brief broadband radio impulses like FRBs “standard pings,” the extended combination DM · DL as a function of redshift can be easily constructed from independent measurements of standard pings and standard candles at similar redshifts (Wei et al. 2018). Moreover, in the CPL framework, we investigate the constraining power on the equation of state of dark energy from this extended combination.


GRB 100814A is another special event with an early-time shallow decay phase and late-time significant rebrightening in its optical afterglow light curve (Nardini et al. 2014). The power-law (

) temporal index of the early shallow decay is

, which is inconsistent with expectations from the external shock model. It is argued that the shallow decay phases come from energy injection. Usually, the injection luminosity is assumed as

 (Nousek et al. 2006; Zhang et al. 2006; Yu & Huang 2013), which may naturally come from the magnetic dipole radiation of a newborn millisecond magnetar (Dai & Lu 1998; Zhang & Mészáros 2001; Dall’Osso et al. 2011). As a result, magnetars have been suggested as the central engines for some GRBs, including both long and short events (Zhang & Mészáros 2001; Troja et al. 2007; Metzger et al. 2011; Bernardini et al. 2012; Rowlinson et al. 2013). In both Dai & Lu (1998) and Dall’Osso et al. (2011), works considering a strongly magnetized neutron star as the central engine of a GRB, the energy injection power is more realistically derived as

, where T is the spin-down timescale and

 is the initial luminosity. In particular, while considering the exact form for the energy injection power of a spinning down magnetar due to magnetic dipole radiation, Dall’Osso et al. (2011) found that the luminosity of the X-ray afterglow naturally has a shallow decay phase with a temporal power-law index of

. Recently, a nearly constant dipole radiation luminosity (

) during the spin-down timescale was favored by observations from GRBs, such as 050801 (de Pasquale et al. 2007), 060729 (Grupe et al. 2007), and 080913 (Greiner et al. 2009). However, some observations of GRB afterglows with rebrightenings or bumps (i.e.,

) require that the injection luminosity increases with time (i.e.,

). Additionally, there is a plateau phase in the X-ray band of GRB 100814A between 103 and 105 s (Nardini et al. 2014) indicating continuous energy injection from the central engine during this prolonged period.


Outside the TC, the MHD simulation behaves similarly to its HD counterpart, with baroclinicity being the dominant contribution. In terms of balance between left- and right-hand sides of (13), panels A and D show a very good agreement in most of the CZ. Differences are only evident inside the TC in the top layers and slightly near the poles. We attribute these differences to the upper and polar boundary conditions for the magnetic field. The difference observed between panels 12A and D in the top boundary might be due to the radial magnetic field we enforce at the surface. This means that when we have a strong poloidal field located near the surface, it will be forced over a couple grid points by the boundary condition into the radial direction. This is exactly the case of the poloidal field configuration during cycle maximum (see Fig. 3E). This problem might be alleviated in future simulations by introducing more realistic top boundary conditions like that used in Warnecke et al. (2016). The differences between the left- and right-hand sides during cycle minimum (not shown here) are much smaller. It is the radial derivative of the poloidal field present in the second magnetic term of Eq. (13) that is responsible for this “artificial” contribution. There are two other possible sources of error that can explain the minute differences we find in this balance calculation (and in the previous section as well). The first is numerical diffusivity, which we cannot measure directly. The other issue is related with the different numerical methods used to compute derivatives and other composite quantities in the main code during the simulation and a posteriori. During the simulation run, EULAG numerics computes central cell values and fluxes across the cell borders, while the type of analysis that we perform a posteriori assumes values computed in the cell corners using centered finite differences. Differentiation across the poles can also introduce some artifacts. Nevertheless, the very good match obtained in the HD case indicates that these two sources of error are in fact very small, and that the main issue here seems related to the magnetic field boundary conditions. Despite these possible sources of uncertainty, we consider that there is a general good agreement between left- and right-hand sides for most of the CZ.


In analogy to the hydrodynamic case, the nonlinear coherent vortex structure also plays an important role in plasma dynamics and transport processes (Hasegawa & Mima 1978; Shukla et al. 1985; Petviashvili & Pokhotelov 1992; Horton & Hasegawa 1994). These vortices tend to have a long lifetime and are widely observed in space, laboratory, and numerical simulation of plasma (Chmyrev et al. 1988; Burlaga 1990; Volwerk et al. 1996; Stasiewicz et al. 2000; Sundkvist et al. 2005; Alexandrova et al. 2006; Alexandrova 2008; Alexandrova & Saur 2008; Vianello et al. 2010; Servidio et al. 2015). An essential subset of these plasma vortices is known as Alfvén vortices, which can be viewed as the cylindrical analog of the nonlinear Alfvén wave (Petviashvili & Pokhotelov 1992). The Alfvén vortices have an axis that is nearly parallel to the unperturbed magnetic field, along which the shape is generally invariant. Thus, these vortices are quasi-2D structures. The associated perpendicular magnetic fluctuations are linearly related with the perpendicular velocity fluctuations, but their relative amplitudes are not obligatorily equal (as is the case in an Alfvén wave):

, where ξ is not necessarily equal to 1. In addition, Alfvén vortices do not propagate along

 in the plasma frame, and they hardly propagate in the perpendicular plane when the axis of the vortex is inclined with respect to

 that are in contrast with Alfvén wave (Wang et al. 2012). After first being reported in the Earth’s magnetosheath (Alexandrova et al. 2006; Alexandrova 2008), multiscale quasi-bidimensional Alfvén vortices (with

) have been identified in numerous space environments: in slow solar wind (Perrone et al. 2016; Roberts et al. 2016), in fast solar wind (Lion et al. 2016; Perrone et al. 2017), and in Saturn’s magnetosheath (Alexandrova & Saur 2008). It seems that the intermittent structures in fast solar wind are dominated by Alfvén vortices (Perrone et al. 2017), which agrees with the 2D MHD turbulence model (Zank et al. 2017).


Given the time-consuming nature of simulating energetic particle acceleration in multiple contracting and reconnecting (merging) small-scale magnetic islands produced by magnetohydrodynamic (MHD) turbulence and particle-in-the-cell (PIC) models on large spatial scales and in three dimensions, there has been an increased effort to develop statistical kinetic transport theories that capture the essential physics of particle acceleration in multi-island regions (de Gouveia dal Pino & Lazarian 2005; Drake et al. 2006, 2010, 2013; Bian & Kontar 2013
). Recently, Zank et al. (2014) developed a comprehensive kinetic transport theory for small-scale flux ropes (helical magnetic field structures) which, for the first time, unified three different mechanisms currently thought to play an important role in the energization of suprathermal particles traversing solar wind regions containing numerous contracting and reconnecting (merging) small-scale flux ropes. It was argued that magnetic flux ropes in the solar wind can be viewed to first order as being quasi-two-dimensional (quasi-2D) helical contracting and merging flux-rope structures based on three-dimensional (3D) simulations of compressible MHD turbulence that included a strong guide magnetic field (Dmitruk et al. 2004). The three acceleration mechanisms include (i) curvature and grad-B drift acceleration in the motional electric fields generated by the plasma flow at the endpoints of contracting magnetic flux ropes (Drake et al. 2006), (ii) curvature and grad-B drift acceleration in the motional electric fields induced by x-point plasma outflows generated near the center of merging neighboring magnetic islands (Drake et al. 2013), and (iii) acceleration due to field-aligned guiding center motion in the secondary reconnection electric field formed in the x-point region at the center of reconnecting islands (Oka et al. 2010). Interestingly, in Zank et al. (2014) contracting flux ropes were modeled to behave in a compressible manner so that betatron acceleration by the increasing flux-rope field strength also contributes to particle energization. Both Drake et al. (2013) and Zank et al. (2014) consider reconnecting (merging) magnetic islands/flux ropes as incompressible phenomena so that negative betatron acceleration from the decreasing flux-rope field strength reduces particle energization. Zank et al. (2014) found that the mean electric field induced by contracting and merging flux ropes naturally accelerates suprathermal test particles to hard power-law distributions in the supersonic, slow solar wind. The results show that the power-law index depends on the Alfvén Mach number and on the ratio of the diffusion timescale to that for island contraction.


As for Mg, one would expect to find a positive correlation between δ29Si and initial 18O/16O ratios for the parent stars of category A grains if these quantities represent GCE. Indeed, there is a correlation between δ29Si and initial 18O/16O, as can be seen from Figure 10. The observed trend is a bit shallower than predicted by the GCE model of Timmes & Clayton (1996), which was inferred and extended to higher metallicities from predicted 18O/16O ratios at [Fe/H] = −0.3 (18O/16O = 0.56 × solar) and [Fe/H] = 0 (18O/16O = 1.26 × solar); corresponding δ29Si values are −500‰ ([Fe/H] = −0.3) and +260‰ ([Fe/H] = 0), respectively. It is interesting to note that astronomical observations of a variety of sources have not detected a variation in Si isotope ratios with galactocentric radius, which stands in contrast to the monotonically decreasing trend for the 18O/16O ratio (Monson et al. 2017) and our observations for category A grains. Following the approach of Nguyen et al. (2010), we have inferred a GCE relationship between δ29Si and initial 18O/16O, which is based on a tight correlation between 29Si/28Si and 46Ti/48Ti ratios in SiC mainstream grains (Figure 11, left; data from Hoppe et al. 1994; Alexander & Nittler 1999; Huss & Smith 2007) and a correlation between 46Ti/48Ti and initial 18O/16O ratios in presolar oxide grains (Figure 11, right; data from Choi et al. 1998; Hoppe et al. 2003), interpreted to represent largely GCE. Error-weighted linear regressions yield δ46Ti = (0.98 ± 0.08) × δ29Si-(11 ± 7) for SiC mainstream grains and

 for Group 1 oxides. Combining the two equations gives

, which is labeled as “SiC-Oxides GCE” in Figure 10. Note that this equation is different from the one presented by Nguyen et al. (2010), which was recognized to be in error (L. Nittler 2018, private communication). Gyngard et al. (2018) recently reported a slightly shallower slope for the δ46Ti versus δ29Si of SiC mainstream grains than inferred here, which would make the δ46Ti versus

 relationship a bit steeper. Our data for category A grains fall to the right of the SiC-Oxides GCE line but are fully compatible with this line if errors for the SiC-Oxides GCE line are taken into account (Figure 10). Finally, the good correlation between δ25Mg and δ29Si (Figure 12) rounds out the picture emerging from presolar silicate category A grains, suggesting that a subset of presolar Group 1 silicates carries resolvable signatures of the GCE of O, Mg, and Si isotopes.


In the first scenario, we assume that neither the characteristic velocity dispersion (σ*) nor the number density (n*) of galaxies evolves with redshifts (νn = νv = 0). Given the redshift coverage of the lensing galaxies in the lens sample (0.06  zl  1.0), if we constrain a non-evolving VDF using the lens data, then, assuming the VDF evolution with redshift is smooth, the fits on the VDF parameters may represent the properties of ETGs at an effective epoch of z ∼ 0.5. Such non-evolving VDF has been extensively applied in the previous studies on lensing statistics (Chae & Mao 2003; Ofek et al. 2003; Capelo & Natarajan 2007; Cao et al. 2012a). By applying the above-mentioned χ2 – minimization procedure to Sample A – we obtain the best-fitting values and corresponding 1σ uncertainties (68.3 per cent confidence level): $\alpha =0.66^{+2.13}_{-0.66}$, $\beta =2.28^{+0.24}_{-0.18}$. It is obvious that the full sample analysis has yielded improved constraints on the high-velocity exponential cut-off index β, compared with the previous analysis of using the distribution of image separations observed in CLASS and PANELS to constrain a model VDF of ETGs (Chae 2005). Suffering from the limited size of lens sample, such analysis (Chae 2005) found that neither of the two VDF parameters (α, β) can be tightly constrained, due to the broad regions in the α − β plane. Consequently, the image separation distribution is consistent with the SDSS measured stellar VDF (Sheth et al. 2003) and the Second Southern Sky Redshift Survey (SSRS2) inferred stellar VDF (Chae & Mao 2003), although the two stellar VDFs are significantly different from each other concerning their corresponding parameter values. We also consider constraints obtained for the Sample B (defined in previous section), with the likelihood is maximized at $\alpha =1.00^{+2.38}_{-1.00}$ and $\beta =2.34^{+0.26}_{-0.24}$, from which one could clearly see the marginal consistency between our fits and recent measurements of three stellar VDFs (especially the SDSS DR5 VDF of ETGs).


Cosmic rays (CRs) play a vital role in the chemistry of cold (10–30 K), dense (>102 cm−3) molecular clouds as they can pierce deep into them, unlike interstellar UV radiation (for a review see Indriolo & McCall 2013). These high-energy interstellar particles primarily consisting of protons can be heavier elements and electrons, and have large energy ranges, up to zetaelectronvolt energies (Blandford et al. 2014). Although the energies can be high, it is the lower energy CRs (≤1 TeV) that affect the dense interiors (Viti et al. 2013; Padovani et al. 2020). In these regions, CRs have a wide variety of effects, one of the most important is being a producer of atomic hydrogen through the dissociation of H2 (van der Werf et al. 1988; Montgomery et al. 1995; Li & Goldsmith 2003; Goldsmith & Li 2005; Padovani et al. 2018a). Other important effects are being the dominant source of ionization; regulating the degree of coupling of the gas and the magnetic field; having an important role in the dynamics and the collapse timescale of collapsing clouds (e.g., Padovani et al. 2013, 2014); providing heating and energy to dust grains (de Jong & Kamijo 1973; Shingledecker et al. 2018; Kalvāns & Kalnin 2019; Sipilä et al. 2020, 2021; Silsbee et al. 2021) producing internal UV photons (Prasad & Tarafdar 1983); may have a role on the charge distribution on dust grains (Ivlev et al. 2015); influencing disk growth (Kuffmeier et al. 2020); and affecting deuteration (Caselli et al. 2008). For example, each species ionized by a CR releases an electron. This secondary electron can cause further collisions, which in turn, depending on the energy, can induce more ionization and heating (Ivlev et al. 2021). If a secondary electron does not have enough energy to ionize a species, the species may become excited (Shingledecker & Herbst 2018). Excited species produced by CR bombardment have energy levels higher than their base counterparts, allowing these excited species to overcome some reaction barriers that would otherwise be difficult in cold environments. These species have been shown to drive more complex chemistry from reactions that can form interstellar complex organic molecules (Abplanalp et al. 2016).


We now turn to constraining c1. First, note that $\mathcal {Z} \gt 0$ for all x. In practice, we ask that $\mathcal {Z} \gt \mathcal {Z}_{\rm {min}}$ for some fiducial $\mathcal {Z}_{\rm {min}} \approx 0.01$. For x ≫ 1, this gives
(42)$$\begin{eqnarray*}
c_1 \gt \left(\mathcal {Z}_{\mathrm{min}}-\frac{\mathcal {S}}{\mathcal {A}}\right)\, x^{-\frac{1}{2}\left[\sqrt{\mathcal {P}^2+\, 4\mathcal {A}} - \mathcal {P}\right]}_{\mathrm{max}},
\end{eqnarray*}$$where xmax is the outer radius of the disc at which we apply this condition.3 Secondly, the total metal flux into the disc across the outer boundary cannot exceed that supplied by advection of gas with metallicity $\mathcal {Z}_{\rm CGM}$ into the disc, since otherwise this would imply the presence of a metal reservoir external to the disc that is supplying metals to it, which is true only in special circumstances, e.g. during or after a merger (Torrey et al. 2012; Hani et al. 2018), or due to long-term wind recycling through strong galactic fountains (Grand et al. 2019). Mathematically, this condition can be written as
(43)$$\begin{eqnarray*}
-\underbrace{\frac{\dot{M} \mathcal {Z}}{2\pi x}}_\text{adv. flux} - \underbrace{\kappa \Sigma _g\frac{\partial \mathcal {Z}}{\partial x}}_\text{diff. flux} \ge -\underbrace{\frac{\dot{M} \mathcal {Z}_{\rm {CGM}}}{2\pi x}}_\text{CGM flux}.
\end{eqnarray*}$$For x ≫ 1, this translates to,
(44)$$\begin{eqnarray*}
c_1 \le \frac{2\mathcal {P}\left(\mathcal {Z}_{\mathrm{CGM}} - \mathcal {S}/\mathcal {A}\right)}{\mathcal {P}+\sqrt{\mathcal {P}^2+\, 4\mathcal {A}}}\, x^{-\frac{1}{2}\left[\sqrt{\mathcal {P}^2+\, 4\mathcal {A}} - \mathcal {P}\right]}_{\mathrm{max}}.
\end{eqnarray*}$$Thus, we find that c1 is bounded within a range dictated by the two conditions above. Given a value of c1, we can also calculate the Σg-weighted and $\dot{\Sigma }_{\star }$-weighted mean metallicity in the model,
(45)$$\begin{eqnarray*}
\overline{\mathcal {Z}}_{\Sigma _\mathrm{ g}} = \frac{\int ^{x_\mathrm{max}}_{x_{\mathrm{min}}} 2\pi x \Sigma _{\mathrm{ g}0} s_\mathrm{ g} \mathcal {Z} dx}{\int ^{x_\mathrm{max}}_{x_{\mathrm{min}}} 2\pi x \Sigma _{\mathrm{ g}0} s_\mathrm{ g} dx},
\end{eqnarray*}$$
 (46)$$\begin{eqnarray*}
\overline{\mathcal {Z}}_{\dot{\Sigma }_{\star }} = \frac{\int ^{x_\mathrm{max}}_{x_{\mathrm{min}}} 2\pi x \dot{\Sigma }_{\star 0} \dot{s}_{\star } \mathcal {Z} dx}{\int ^{x_\mathrm{max}}_{x_{\mathrm{min}}} 2\pi x \dot{\Sigma }_{\star 0} \dot{s}_{\star } dx}.
\end{eqnarray*}$$Finding $\overline{\mathcal {Z}}$ is helpful because we can use it to produce a mass-metallicity relation (MZR) that can serve as a sanity check for the model. We show in a companion paper that our model can indeed reproduce the MZR (Sharda et al. 2020b).


As in G18, we compiled existing constraints on the molecular gas fraction fgas of quiescent and pSB galaxies from recent literature, namely: local QGs consisting of the ATLAS3D (Young et al. 2011; Cappellari et al. 2013; Davis et al. 2014) and HRS (Boselli et al. 2014; Lianou et al. 2016) ETG samples as well as the samples of pSB galaxies (hereafter, the “low-z pSB” sample) of French et al. (2015) and Alatalo et al. (2016); at low and intermediate redshift, the ETG sample of Spilker et al. (2018) and the pSB sample of Suess et al. (2017); at intermediate and high redshift, constraints from Hayashi et al. (2018) on gas in z ∼ 1.46 cluster ETGs, as well as on individual galaxies from Sargent et al. (2015), Bezanson et al. (2019), and Rudnick et al. (2017). Given its size, we divided the ATLAS3D sample into high- and low-mass subsamples, choosing 5 × 1010 M⊙ as the cut-off mass. In addition, we also included fgas estimates derived from the (median) stacked FIR spectral energy distributions of ETGs at z ∼ 1.8 (G18; 977 galaxies), z ∼ 1.2, z ∼ 0.8, and z ∼ 0.5 (1394, 1536, and 563 galaxies, respectively; Magdis et al. 2020, hereafter M20). Finally, at higher redshift (z ∼ 3), we converted star formation rates (SFR) estimated from spectroscopy (Schreiber et al. 2018a; D’Eugenio et al. 2020) into gas masses assuming the star formation efficiency found by G18. As a consequence of our zmax = 3.5, we did not include higher-redshift quiescent galaxies (Glazebrook et al. 2017; Schreiber et al. 2018b; Tanaka et al. 2019; Valentino et al. 2020) in the analysis and considered z ∼ 3 galaxies as pSB. The dust-based estimates of G18 and M20 (and, by extension, the z ∼ 3 semi-constraints) assume a gas-to-dust ratio (G/D). It is dependent on metallicity, which is presumed to be solar or higher owing to both the relatively high gas-phase metallicity of MS galaxies at z ≲ 1 (e.g., Mannucci et al. 2010) and the already high stellar metallicities of QGs at z >  1 (Onodera et al. 2015; Estrada-Carpenter et al. 2019). Here we adopted an intermediate value between the solar and supersolar G/Ds used in M20, and we increased the error bars of these points to include both the solar and supersolar confidence estimates. These various samples, which are summarized with their selection criteria in Table B.1, combine into a nonhomogeneous dataset: some were specifically selected as ETGs, and others were based on varying degrees of quiescence. In particular, pSB galaxies are not necessarily truly quiescent and could, in principle, resume normal star formation. However, as a possible precursor of QGs, they provide useful, though not constraining (see Sect. 4), comparison samples for the model. Here we refer to all equally as either QGs or pSB galaxies, and we make the assumption that, on average, these different samples are not otherwise significantly biased with regard to their gas content compared to the full population, given each mass limit and type.


The 3D view provided in Figure 1 can be completed through the time evolution of the cross section of the same tube presented in Figure 2 for times t = 13, 35, 48 and 62. In the figure, magnetic field strength (top row) and vorticity (bottom row) maps are drawn on a vertical cut that coincides with the midplane of the box perpendicular to the initial axis of the tube. We confirm that most of the magnetic flux is concentrated at the very top of the structure. The general appearance and evolution of the structures shown are strongly reminiscent of those described by Moreno-Insertis & Emonet (1996), Emonet & Moreno-Insertis (1998) and Cheung et al. (2006) using 2D simulations. We recall a few salient features obtained in those papers and reproduced here: in the initial stages (two leftmost panels), the rising portion of the flux tube has a mushroom shape, akin to an air bubble rising in water (e.g., Davies & Taylor 1950; Collins 1965; Parlange 1969; Wegener & Parlange 1973; Hnat & Buckmaster 1976; Ryskin & Leal 1984; Christov & Volkov 1985). Within the ascending portion of the tube, vorticity is generated at the boundary layer between the tube and the ambient flow, in fact with opposite sign on the either side of the head (bottom row in the figure). Magnetic flux is dragged along the sides of the tube into a trailing pair of counter-rotating vortices. Later in time the wake is fragmented into smaller and smaller vortices because the Reynolds number increases in time due to the expansion of the tube (Cheung et al. 2006). The AMR device allows to see this fragmentation in much more detail than could be obtained in the original work of Emonet & Moreno-Insertis (1998). Inside the head of the tube, on the other hand, the plasma and magnetic field are executing a twisting oscillation, with small velocity compared to the global rise speed. This description of the time evolution of the flux tube structure generally applies to other simulations with comparable levels of refinement even when the twist and curvature vary. The cases with untwisted flux tubes, instead, follow a somewhat different pattern and their evolution is described in the following.


Figure 3 shows the metallicity distribution of the LS. We divided the whole region into square bins with size 3 km s−1 and used color to represent the mean metallicity of each bin. The most prominent feature is the Hyades–Pleiades structure, which includes the Hyades stream (peak 0) and Pleiades stream (peak 1). It exhibits a very metal-rich and consistent distribution; it is well known that the Hyades cluster has a high metallicity ([Fe/H] ∼ 0.13, Heiter et al. 2014). We speculate that some stars of the Hyades stream might be remnants of the Hyades cluster, while others may be the dynamical effect of non-axisymmetric components (Famaey et al. 2008). Though not in sharp contrast, there is a valley with lower metallicity between the horizontal V = −50 km s−1 structure and center. To better present the gap, we picked up a window in the metallicity distribution. We divided the window into parallelogram bins, and the edge parallel to the V axis of each bin equals 1 km s−1. The right panel of Figure 4 shows that the mean metallicity of each bin distributes along with the V direction. In Figure 3, the Hercules I (peak 6) and Hercules II (peak 4) have different metallicities, and peak 4 is definitely different from peak 12 (NEW2). That the Hercules stream has a wide metallicity range (Table 1) is in accord with the purported effect of Galactic bar resonances (Kalnajs 1991; Dehnen 2000; Fux 2001). The upper left side of this plot contains stars moving faster than the LSR. In this region the kinematics suggest peak 13 and peak 14 may be part of the thin disk, although their metallicities are quite low. The Sirius stream (peak 2) is a little more metal-poor than its surroundings and has a very small metallicity scatter. This is similar to the results of Klement et al. (2008), and this supports the notion that the Sirius stream (or at least a part of it) comprises remnants evaporated from a cluster. According to De Silva et al. (2007), HR1614 (peak 11) may be dispersed remnants of star-forming events. New1 (peak 9) is close to peak 11 in both kinematics and metallicity; part of its member stars may be remnants of star-forming events, too. New2 (peak 12) has a large metallicity scatter. It is possible to be induced by other dynamical n:1 resonance of the bar. The γ Leo stream (peaks 7, 10) is more metal-poor than our Sun, and it is rapidly moving toward the center of the Galaxy. Considering that there is a radial metallicity gradient in the Galaxy, we conjecture this structure has drifted inwards from beyond the solar Galactocentric radius.


The numerous historical radial-velocity (RV) measurements of Capella have been discussed at length in our T09 study, which highlighted how challenging it has been to determine accurate values for the rapidly rotating secondary star (

 km s−1), whereas those of the sharp-lined primary (

 km s−1) have been quite consistent over the last century. T09 presented 162 new RV determinations for both components based on spectra obtained at the CfA covering only a very narrow wavelength range (45 Å). The RVs were measured using the two-dimensional cross-correlation algorithm TODCOR (Zucker & Mazeh 1994), with synthetic templates appropriate for each star. Because of the limited wavelength coverage, those measurements are susceptible to systematic errors resulting mostly from lines shifting in and out of the spectral window as a function of orbital phase. Therefore, an effort was made to control those biases by performing numerical simulations to determine corrections to the velocities, which were at the level of the final uncertainties in the individual measurements for the secondary, and slightly larger for the primary. Final errors in the RVs as measured from the scatter in the orbital fit were about 0.44 km s−1 for the primary and 0.89 km s−1 for the secondary. A sign that systematic errors remained at some level in the CfA velocities was evident in the residuals of the secondary star shown in Figure 2 of T09, in which a pattern can be seen as a function of orbital phase, with a peak semi-amplitude of about twice the typical error. Possible explanations for this, as discussed by T09, include the presence of spots on the active secondary star, or template mismatch.6

6
In particular, due to limitations in the available library of synthetic spectra they used, the macroturbulent velocity of the templates (

 km s−1) was not quite as large as appropriate for giant stars. This also resulted in an overestimation of the rotational velocities of the components, as discussed by T09.
 An additional indication of possible biases was the fact that a small offset (0.267 ± 0.079 km s−1) was found between the primary and secondary velocities in the global orbital fit of T09 that could not be accounted for by differences in the gravitational redshift between the stars, and was ascribed to similar reasons as the secondary residual pattern.


The recent detection of X-ray pulsations in PSR J1023+0038 during the accretion active state (Archibald et al. 2015; Bogdanov et al. 2015) implies that channelled accretion, similar to that seen in higher luminosity accreting millisecond X-ray pulsars, is occurring at a much lower accretion rate, implying that the inner edge of the accretion disc lies close to rcor and a propeller does not form. In this case material can accumulate near rcor and non-stationary accretion can occur as matter piles up around the intrinsically unstable magnetospheric boundary. Accretion discs accreting on to the magnetosphere of a rotating star can end up in a trapped state, in which the inner edge of the disc stays near rcor. The captured material can form a quasi-spherical shell (Pringle & Rees 1972; Shakura et al. 2012) or a new disc structure, known as a ‘dead-disc’ and episodic accretion can occur (D'Angelo & Spruit 2010, 2012). As noted by Patruno et al. (2014), variations in the mass accretion rate of the accretion disc can move the inner disc radius by a factor of 2 or 3. The viscous time-scale defines the time-scale for this drift and to reach a viscous time-scale of 10–100 s requires a region with an annulus of radius ∼10–100 km. The thermal time-scale is given by ttherm ∼ (H/R)2 tvis (where H and R are the height and radius of the disc) and for an inner thin disc with H/R  0.02 implies ttherm few seconds, much lower than the time-scales observed. Thus, in principle, fluctuations in the mass accretion rate can move rm outside rlc ,which allows the radio pulsar to turns on, triggering a transition to the passive state, where the lower X-ray luminosity is produced by a shock between the pulsar wind and innermost accretion flow. An increase in the mass accretion rate pushes the inner edge of the disc back inside the light cylinder and turns off the radio pulsar. The X-ray pulsations are only observed in the active state and not in the passive-state light curves, which suggests that the switching between the passive and active states results in transitions between a non-accreting pure propeller mode and an accreting trapped-disc mode (Archibald et al. 2015).


First, we discuss the discrepancy of the distribution scale of the radio core positions based on the discussions of the internal shock model (Koyama et al. 2015; Niinuma et al. 2015). As is discussed there, the radio cores in Mrk 501 and Mrk 421 observed at 43 or 22 GHz can usually be considered as the internal shocked regions owing to the convex shape of the radio spectrum peaking around 10 GHz (Giroletti et al. 2008; Sokolovsky et al. 2010; Lico et al. 2012; Blasi et al. 2013). The standard internal shock model of blazars considers that the discrete ejecta with higher speeds (with Lorentz factor Γf) catch up with the preceding slower ejecta (with Lorentz factor Γs), and the collision leads to the nonthermal emission (e.g., Spada et al. 2001; Tanihata et al. 2003; Guetta et al. 2004; Kino et al. 2004; Ghisellini et al. 2005). Based on the model, the distribution scale of the internal shocks (ΔDIS in Figure 7, defined as the difference between the largest distance between the internal shock and the central engine DIS,max and the closest one DIS,min) can be explained as the variation of the Lorentz factors of the ejecta (Koyama et al. 2015; Niinuma et al. 2015), by assuming the Lorentz factor ratio (Γf/Γs) and the initial separation of the ejecta (IIS). The core stable within 200 μas constrained by the VERA can be explained by Lorentz factors within a factor of two variation for the slower ejecta, i.e., 8 ≤ Γs ≤ 17, by assuming a minimum value of 8 (e.g., Kino et al. 2002), Γf/Γs ≤ 1.01 (Tanihata et al. 2003), and IIS ∼ 1 Rs (Koyama et al. 2015). This time we refined the distribution scale of the radio core within 42 μas along its main jet axis, or 4.6 × 103 Rs deprojected (see Figure 7). Based on the same assumptions as in Koyama et al. (2015), to explain the further stable distribution scale of the internal shocks, the variation of Lorentz factors of the slower ejecta is constrained to be much smaller, within 30% or 8 ≤ Γs ≤ 10. On the other hand, the radio core wandering of ΔDIS ∼ 2.6 × 105 Rs in Mrk 421 can be explained by the maximum value as Γs ∼ 60 (with different assumptions; Niinuma et al. 2015). Even by applying the same assumptions to Mrk 421 as those for Mrk 501, the maximum of the slower Lorentz factor is estimated to be Γs ∼ 50, which is still a few times as large as that of Mrk 501. Therefore, even during the X-ray and VHE γ-ray active states in 2012, the maximum Lorentz factors that explain the stability of Mrk 501's core are roughly a few times smaller than those for Mrk 421's wandering core, based on the internal shock model.


Star formation from molecular cloud cores collapse is a chaotic process that is expected to result in initially misaligned and warped discs (Bonnell & Bastien 1992; Bate, Lodato & Pringle 2010; Offner et al. 2010; Bate 2018). We expect this to be the general case for circumbinary discs as well as circumstellar ones. Indeed, some misaligned circumbinary discs have already been reported (e.g. KH 15D Chiang & Murray-Clay 2004; Winn et al. 2004; Lodato & Facchini 2013; Smallwood et al. 2019; Fang et al. 2019; Poon, Zanazzi & Zhu 2021, GG Tau A Köhler 2011; Andrews et al. 2014; Aly, Lodato & Cazzoletti 2018, IRS 43 Brinch et al. 2016, L1551 NE Takakuwa et al. 2017, and HD 98800B Kennedy et al. 2019). Misaligned discs around binaries experience a gravitational torque that leads to radially differential precession, which causes disc warping and twisting. For the gas component, warps are expected to propagate in a wave-like manner in thicker discs with low viscosity that are more relevant in protoplanetary contexts (Papaloizou & Lin 1995; Nelson & Papaloizou 1999; Facchini, Lodato & Price 2013), as opposed to the diffusive warp propagation that occurs in more viscous, thinner discs expected around black holes (Papaloizou & Pringle 1983; Ogilvie 1999; Lodato & Pringle 2007; Lodato & Price 2010). The final state of gas discs precessing around binaries depends on the binary parameters; for circular and low-eccentricity binaries, the disc will align with the binary plane, either in a prograde or retrograde sense depending on the initial misalignment (King et al. 2005; Nixon et al. 2011). For eccentric binaries, discs with high initial misalignments will align in a polar configuration around the binary eccentricity vector (Farago & Laskar 2010; Aly et al. 2015; Martin & Lubow 2017; Zanazzi & Lai 2018). Either way, the disc is susceptible to breaking when the binary torque is larger than the viscous torque and the disc communicates the precession more slowly than it occurs (Nixon, King & Price 2013; Doǧan et al. 2018).


The spectral evolution of an ensemble of stars of various masses can be combined through an IMF and followed over time. Such spectral synthesis codes are valuable tools for ultraviolet spectral libraries (Robert etÂ al. 1993; Rix etÂ al. 2004; Leitherer etÂ al. 2014) and stellar population synthesis codes such as Starburst99 (Leitherer etÂ al. 1999). FigureÂ 12 shows a series of spectra for a cluster starburst containing 105âM in which the stars follow a Salpeter IMF (0.1 â©½ m â©½ 120). The two panels show model spectra at times t = 0, 1, 3, 5, and 7âMyr after a coeval burst of star formation, using evolutionary tracks with rotation, at solar metallicity (Z = 0.014 from EkstrÃ¶m etÂ al. 2012) and at sub-solar metallicity (Z = 0.014 from Georgy etÂ al. 2013). These models were created by a Monte Carlo sampling of stars for each mass for which we have an evolutionary track (m = 20, 25, 32, 40, 60, 85, 120). For the Salpeter differential mass distribution, Î¾(m) = KmâÎ± with Î± = 2.35, the fraction of stars above mass m is given by
8The constant K is normalized to the total cluster mass M = 5.862âK, and the total number of stars N = 16.582âK for mass limits mmin = 0.1 and mmax = 120. For M = 105âM, we expect a mean number  stars and mean stellar mass â©mâª = M/N = 0.354 M. In many of our Monte Carlo samples the numbers fluctuate about this value, with small numbers at the high-mass end of the IMF. The mean numbers of stars at the high-mass end are N(> 60âM) â 30 and N(> 100âM) â 5. If the upper mass was extended to mmax = 200, we would expect 40 stars above 60 M and 15 stars above 100 M. As discussed earlier, the existence of these very massive stars is controversial, owing to resolution effects. Moreover, the LyC from stars at m > 100 M may not escape the embedded cloud from which they formed, or from the dense gas produced in mass-loss episodes or binary mergers (Smith 2014). We therefore do not consider stars above the 120 M track.


Fast radio bursts (FRBs) are bright, cosmological origin, and millisecond-duration bursts in radio wavelengths (Lorimer et al. 2007; Thornton et al. 2013; Bassa et al. 2017; Macquart et al. 2020). After the discovery of the first FRB (Lorimer et al. 2007), a number of dedicated facilities have been conducted to search FRBs, such as the Parkes telescope (e.g., Bhandari et al. 2018), the updated Molonglo Observatory Synthesis Telescope (e.g., Farah et al. 2018), the Australian Square Kilometre Array Pathfinder (e.g., Shannon et al. 2018), the Canadian Hydrogen Intensity Mapping Experiment (CHIME; The CHIME/FRB Collaboration et al. 2018), the Deep Synoptic Array (Kocz et al. 2019; Ravi et al. 2019), the Green Bank Telescope (Masui et al. 2019), Arecibo (Spitler et al. 2014; Patel et al. 2018), and the Five-hundred-meter Aperture Spherical radio Telescope (FAST; Nan et al. 2011; Li et al. 2019). All these efforts result in an increasing rate of new FRB detections. Among them, more than 20 repeating FRBs have been reported. Particularly, the physical origin of the repeating FRB 20121102A was identified to be with a low-metallicity star-forming dwarf galaxy at a redshift 0.19273 (Bassa et al. 2017; Tendulkar et al. 2017). Another repeating FRB 20190523A was found to be associated with a more massive but low specific star formation rate (Ravi et al. 2019). The identification of the counterpart of the brightest radio bursts from SGR 1935+2154 as a magnetar in our Galaxy by HXMT (Li et al. 2021) and INTEGRAL (Mereghetti et al. 2020) with short-duration X-ray bursts suggests that at least a fraction of FRBs are connected with newborn magnetized neutron stars (e.g., Weltman & Walters 2020; Zhang 2021). More bursts with similar characteristics need to be detected in the future to confirm this conclusion. Recently, a new large sample with 535 FRBs was presented by CHIME/FRB Collaboration et al. (2021) that were detected by the CHIME survey, including 61 bursts from 18 previously reported repeating sources and 474 one-off bursts. Though an increasing catalog of theories and models is developing to explain the physical nature of FRBs (e.g., see the review of Platts et al. 2019; Xiao et al. 2021), the origin of FRBs remains a mystery.

Background Background Background Background


The last column in Table 1 reports the number of OB stars minus the “diffuse” population estimated from their density in the Reference field (22.5 stars per square degree): as is immediately seen, the M-star statistics is much larger than the OB star statistics. This can hardly be considered surprising, if an ordinary IMF (e.g., that from Weidner et al. 2010) is assumed for the star-formation region. In Fig. 19 we show the density ratio between M and OB stars, which provides a consistency test between our results and a plausible IMF: this ratio varies however by a large factor, close to 20, among our subregions. This might reflect differences in the respective IMFs, but also differences in completeness among the stellar samples considered for the various regions. We first note that the ratio between M and OB stars in NGC 6231 is dramatically lower than anywhere else in Sco OB1. We can indeed expect that M stars are detected less efficiently in the inner parts of NGC 6231, where the density of bright stars is very large, and their diffuse glare raises the limiting magnitude locally. As already discussed above in Sect. 4.1, this causes our sample of M stars in NGC 6231 to be highly incomplete. Moreover, we determined above that NGC 6231 is significantly more extincted, by almost half a magnitude in V, than Tr 24, and this implies a higher minimum detectable mass among NGC 6231 M stars compared to Tr 24 (see the MDA diagrams in Fig. 5); this effect reduces the completeness of the M-star sample in NGC 6231 more than in Tr 24. If Tr 24 is also slightly younger than NGC 6231, as we argue below, our M-stars in Tr 24 will reach down to lower masses than in NGC 6231, with a steep increase in the detected M-star population: adopting the IMF from Weidner et al. (2010), the predicted number of cluster M stars doubles considering the mass interval 0.25–0.5 M⊙ rather than 0.35–0.5 M⊙. If Tr 24 is younger than NGC 6231, moreover, its stars in the mass range 2.5–3 M⊙ might not have yet reached their ZAMS position as B stars, and therefore would not be counted among OB stars; this would further raisethe M/OB star ratio there by up to 30%. Therefore, the proportions of both M and OB stars that are detected in a young cluster will depend on their age and extinction, in accordance with the MDA diagrams, even for a fixed, spatially uniform photometric sensitivity. We estimated using the Weidner et al. (2010) IMF the expected range for the observed M/OB number ratio. Siess et al. (2000) predict that the latest-type B stars have a mass of ~ 3.5 M⊙ at 2 Myr, and ~ 2.2 M⊙ at 10 Myr, that is, in the range of ages expected for Sco OB1 clusters. The MDA diagrams of Fig. 5 predict that the lowest-mass stars we are able to detect using the available Sco-OB1 data have ~ 0.2 M⊙, even assuming the most favorable (and unlikely) circumstances of an age less than 2 Myr and negligible reddening. The extreme values found for the M/OB ratio are then ~ 3.8 for a minimum M-star mass as high as 0.35 M⊙ and an old age of 10 Myr, and ~ 20 for a minimum M-star massas low as 0.2 M⊙ and age of 2 Myr. These extremes are also shown as horizontal lines in Fig. 19. We note that the M/OB ratio in NGC 6231 falls well within this range; however, both Tr 24 regions are significantly richer of M stars than expected, by more than a factor of two and well above (statistical) errors. If true, then paradoxically this part of the OB association would form preferentially lower-mass stars. Of course, more detailed studies are needed to confirm this result. In the G345.45+1.50 region the M/OB ratio is highest, and far above predictions from the IMF: we may tentatively explain this since this region is very young, and some of its most massive members, like IRAS 16562-3959, are still in formation, thus decreasing the number of optically revealed OB stars. The lowest M/OB ratio in NGC 6231 is unlikely to be real, since as discussed above our M-star sample in this densest subregion is likely incomplete.


We then allow the synthetic single or binary system to evolve until present time, adopting for our reference model a thin disc age of 10 Gyr (Cojocaru et al. 2014) and a thick disc age of 12 Gyr. This is motivated by the findings of Feltzing & Bensby (2009), who presented a sample of very likely thick disc candidates with ages, on average, well above 10 Gyr and of Ak et al. (2013), who found that thick disc cataclysmic variables have ages up to 13 Gyr. If the synthetic star is single and has time to become a white dwarf, it evolves following the cooling tracks detailed in the following section. If that is the case, the mass of the white dwarf is obtained from the initial-to-final mass relation (IFMR) according to the prescription from Hurley, Tout & Pols (2002). If the object is member of a binary system and the primary star has time to become a white dwarf, then the pair can evolve through two different scenarios. In the first scenario, the binary evolves without mass transfer interactions as a detached system and the primary star evolves into a white dwarf that subsequently cools down following the cooling sequences described in the next section. In this case, the mass of the white dwarf is also calculated from the IMFR of Hurley et al. (2002). The second scenario involves mass transfer episodes and the evolution of the binary is obtained following the prescriptions of the bse package (Hurley et al. 2002), following the parameter assumptions detailed in Camacho et al. (2014). If the system evolves though the common envelope phase, we use the α-formalism as described in Tout et al. (1997), with αCE being the efficiency in converting orbital energy into kinetic energy to eject the envelope (assumed to be 0.3 in our reference model). This implementation also takes into account the αint parameter (assumed to be 0.0 in our reference model), first presented in Han, Podsiadlowski & Eggleton (1995), describing the fraction of the internal energy (thermal, radiation and recombination energy) used to eject the envelope. As described in Camacho et al. (2014), the αint parameter is used to include the effects of the internal energy in the binding energy parameter λ, which is thus not taken as a constant, but computed using a specific algorithm (Claeys et al. 2014) in bse. In the current version of the code, provided that a positive value is used, the parameter αint represents the fraction of recombination energy that contributes to eject the envelope. It is important to note that the thermal energy of the envelope is always taken into account (using the virial theorem) even if αint is set to zero. For a more detailed discussion on how this is implemented in the latest version of BSE and important comments on the correct use of BSE and the notations used in the code itself, we direct the reader to Zorotovic, Schreiber & Parsons (2014a), mentioning that the notations αint or αrec are, in our case, equivalent.


The most general third-order statistics is the three-point correlation function (hereafter referred to as 3PCF), which is defined in configurations space. Alternatively, one can study its Fourier space counterpart, the bispectrum. These two statistics contain, in principle, the same information. However, their analyses implicate different limitations and challenges, which can affect the physical interpretation of the results. A main advantage of the bispectrum is that an analysis in Fourier space allows for a clear exclusion of high-frequency modes in the density fluctuations, which are difficult to interpret theoretically due to their highly non-linear evolution. In configuration space, these high-frequency modes contribute to the 3PCF, in principle, at all scales. In practice, one therefore needs to restrict the analysis to large scales, where their contribution is negligible, lavishing a lot of valuable data. Another advantage of the bispectrum is that its covariance is diagonal for Gaussian density fluctuations. This approximation works well, even for evolved density fields, while deviations from Gaussianity can also be taken into account (Scoccimarro 2000; Sefusatti et al. 2006; Chan & Blot 2017). The covariance of the 3PCF, on the other hand, is not diagonal, even for Gaussian fluctuations, which makes the modelling more difficult (Srednicki 1993; Slepian & Eisenstein 2015; Byun et al. 2017; Gualdi et al. 2017). An additional difference in the analysis of the bispectrum and the 3PCF lies in the fact that the computation of the latter is more expensive. However, this aspect can be tackled by employing advanced algorithms and appropriate computational resources, as done in this work (see also, Barriga & Gaztañaga 2002; McBride et al. 2011a; Jarvis 2015; Slepian & Eisenstein 2015, and references therein). Besides its disadvantages, there are some arguments that speak for the 3PCF. One of them is the fact that the amplitude of the 3PCF (but not its errors) is not affected by shot-noise, whereas the latter affects the bispectrum amplitude at all scales and hence needs to be modelled for correcting the measurements. In addition, an analysis in configuration space has the advantage that complicated survey masks can be easily taken into account in the analysis of observational data, while in Fourier space such masks impose complicated effects on the measured bispectrum, which are difficult to model (e.g. Scoccimarro 2000). A more general consideration is that it is easier to interpret effects such as redshift space distortions or baryon acoustic oscillations (BAOs) on the statistics in configuration space, since that is where the physical processes that cause these effects happen. Studies of third-order correlations in the literature usually focus on either Fourier or configuration space (e.g. McBride et al. 2011b; Marín et al. 2013; Gil-Marín et al. 2015). However, it is worthwhile studying both statistics and cross-check the results, since their different advantages and disadvantages are quite complementary.


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


Early 15 GHz images using the very long baseline interferometry (VLBI) technique obtained from the very long baseline array (VLBA) indicated a twisted morphology with jet bending on a scale of ∼20 mas (Kellermann et al. 1998), and multi-epoch 43 GHz observations show jet knots with complex kinematics involving a mixture of apparent superluminal motion and as well as stationary components (Jorstad et al. 2001, 2005). Multi-frequency (15 and 43 GHz) multi-epoch observations during the 2006 radio flare (Fromm et al. 2013a) inferred a possible association between a jet component ejection event at the end of 2005 and a strong radio flare in 2006 April. The authors interpreted the 2006 radio flare as a result of the interaction between a propagating shock and a stationary shock at a de-projected distance of 18 pc from the core. Temporal variability studies during the 2012 September–October multi-band flaring period found a near-simultaneous γ-ray and optical flaring behavior, inferring a co-spatial origin (Cohen et al. 2014; Larionov et al. 2016). The latter study in addition suggested that the measured Stokes parameter variations is consistent with a bright jet knot moving along a helical trajectory. A multi-wavelength polarimetric study during the same flaring phase (Casadio et al. 2015) detected a co-spatial origin from the near-simultaneous variability and identified the passage of a superluminal radio knot coincident with the γ-ray flare. Further evidence, including an intra-day optical polarization variability and clockwise rotation of the electric vector position angle (EVPA,

, where U and Q are Stokes components) during the flaring phase, is consistent with a jet knot passing a region hosting helical magnetic fields. A study of γ-ray–optical variability in flux density and polarization between prominent flares during the end of 2016 (Larionov et al. 2017) finds no time lag between the light curves indicating co-spatial origin of synchrotron (optical) and inverse-Compton (γ-ray) flux, and a smaller viewing angle (more energetic, emission closer to jet base) compared to a flare in 2012 when interpreted in terms of a blob or shock wave on a helical trajectory. From the above studies, the γ-ray flares may be associated with optical flux and polarization variability, and the jet kinematics and polarization properties may be described in terms of a helical jet.


On 2017 November 24 the MMS orbit allowed us to collect measurements in the pristine solar wind, well outside the Earth's magnetosheath and the bow shock, for a long period (i.e., a few times longer than the typical correlation scale) of approximately 1 hour from 01:10 to 02:10 UT. Figure 1 (upper panel) displays an overview of the magnetic field measurements collected by the FIELDS instrument suite (Torbert et al. 2016) on board of MMS1 with a temporal resolution Δt = 128 samples s−1 (Russell et al. 2016). The period of interest is a typical example of slow solar wind stream (V ∼ 377 km s−1), with an average magnetic field 〈B〉 ∼ 6.6 nT and a mean plasma density 〈n〉 ∼ 9 cm−3 (Roberts et al. 2020a, 2020b). This means that the average Alfvén speed is VA ∼ 50 km s−1, while the ion inertial length and gyroradius are di ∼ 76 km and ρi ∼ 96 km, respectively (Chhiber et al. 2018), with the corresponding timescales τd ∼ 1.3 s and τρ ∼ 1.6 s, respectively. As reported in previous works (Bandyopadhyay et al. 2018; Chhiber et al. 2018; Roberts et al. 2020a, 2020b) this interval is characterized by two different spectral scalings: a typical inertial range ∼ τ5/3 is found at large scales (i.e., τ > τb), while a steeper scaling ∼ τ7/3 is found at small scales (i.e., τ τb), with τb ∼ 2.4 s (Roberts et al. 2020a). Furthermore, the magnetic field spectrum flattens near τnoise ∼ 0.2 s, due to the instrumental noise floor near ∼5 Hz (Russell et al. 2016). Finally, a decrease at shorter timescales (e.g., τ ∼ 0.1 s) is due to an anti-aliasing filter of nonphysical origin (Russell et al. 2016; Roberts et al. 2020a). Taken together, this interval is particularly suitable for testing our formalism with respect to processes of both physical and nonphysical origin. The presence of an instrumental noise floor allows us indeed to assess our formalism with respect to purely stochastic processes, while the existence of two spectral regimes (i.e., the MHD/inertial and the kinetic/dissipative) allows us to investigate small- versus large-scale processes and their possible coupling in a dynamical system framework.


One of the key model ingredients that determines the TDE rates is the distribution of stars in galactic nuclei (Magorrian & Tremaine 1999; Wang & Merritt 2004). Depending on the merger history of the galaxy and the efficiency of feedback on star formation, the stellar density profile can develop either a core or a cusp. For simplicity, we adopt a singular isothermal sphere density profile ρ(r) = σ2/2πGR2 with σ being the constant velocity dispersion and R the halo virial radius. For a galaxy of halo mass Mh, the relation between the halo mass and the velocity dispersion is simply Mh = 2σ2R/G; while the velocity dispersion can be directly related to the black hole mass using the MBH–σ relation (Kormendy & Ho 2013; McConnell & Ma 2013; Baldassare et al. 2015; Saglia et al. 2016; Thomas et al. 2016)
(1)
\begin{equation}
M_{{\rm BH}} = 0.309\times 10^9\times \left(\sigma /200\ \rm{km\,\,s}^{-1}\right)^{4.38}\ \mathrm{M}_{\odot },
\end{equation}
which holds for a wide range of black hole masses from 5 × 104 M⊙ (Baldassare et al. 2015) to 1.7 × 1010 M⊙ (Thomas et al. 2016) in galaxies with a bulge (Guillochon & Loeb 2015). Assuming the isothermal stellar distribution, Wang & Merritt (2004) derived TDE rates for galaxies with a single central black hole, while Chen et al. (2009) report the rates in a case of a black hole binary. As we discuss in Section 2.2, for MBH with masses in the range MBH ∼ 105–108 M⊙, the TDE rates per halo computed using the isothermal stellar distribution are similar (within tens of percent) to more realistic estimates based on a large galaxy sample (Stone & Metzger 2016), which justifies our assumption. The error in the rate estimation due to the idealized stellar density profile is small compared to other uncertainties, e.g. introduced by the poorly constrained occupation fraction of IMBHs in low-mass galaxies, which amounts to one–two orders of magnitude uncertainty in the derived volumetric TDE rates.


A number of fundamental results have been rigorously proved in the mathematical literature concerning the asymptotic behaviour in time of some of the solutions of the porous medium equation and related equations (e.g. Kamin & Vázquez 1991; Bernis et al. 1993; Hulshof et al. 2001). What is of interest for us here is, primarily, the results that can be applied to the cylindrically symmetric case with diffusion coefficient which is proportional to the square of the dependent variable (n = 2, m = 3 in the notation of Eq. (7)). The most basic result, already mentioned in Sect. 4.4.1, is that initial conditions which have a finite nonzero flux integral (called ‘the mass’ in the mathematical literature for the PME) converge toward the ZKBP solution with the same flux integral (‘mass’) asymptotically in time (Vázquez 2007, Theorem 18.2); here, allowance is made for either a positive or negative flux integral by globally changing the sign of the ZKBP solution; also, ‘convergence’ is meant in the sense that the Lp norm of the difference between the actual solution and the ZKBP function tends to zero as t → ∞ faster than a negative power of the time with an exponent which is a function of n, m, and p (e.g. −1/3 for n = 2 and m = 3 in the L2 norm; see details in the book by Vázquez 2007). A complementary result is the following: when the initial condition has positive net flux and its negative part has compact support, then the whole solution evolves into a positive function after a finite time (Vázquez 2007, Theorem 18.29). Since we are dealing with signed functions which have zero flux integral, these results are of interest mainly because they impose a strict condition on the possible flux imbalance caused by numerical errors (as discussed in Sect. 4.4.1, final paragraph): if it is not small, the numerical solutions will approach the ZKBP solution in a comparatively short time. However, the flux imbalance in all the Bifrost experiments discussed in the present paper is small enough that they have not shown this behaviour even though they have been run until a very long diffusive time.


In addition to the above problem in the UV spectroscopy of disk-dominated systems, X-ray data of CVs have also been in strong disagreement with theoretical expectations for more than three decades (Ferland et al. 1982). The culprit has been the difficult to study boundary layer (BL) between the accretion disk and the WD star. About half of the disk accretion energy (in the form of kinetic energy) is expected to be dissipated in the BL between the Keplerian disk and slowly rotating stellar surface (Pringle 1981). Because of its small size, the BL was predicted to emit in the X-ray band: at low accretion rates (when the WD is dominant in the UV), the BL was expected to be optically thin and emit hard X-rays (Pringle & Savonije 1979; Tylenda 1981; Narayan & Popham 1993); at large accretion rates, typical of NLs in a high state and DNe in outburst, the BL was expected to be optically thick and emit soft X-rays (Pringle 1977; Narayan & Popham 1993; Popham & Narayan 1995). Systems in the low state indeed reveal optically thin hard X-ray emission (Szkody et al. 2002; Pandel et al. 2005; Mukai et al. 2009). However, systems in a state of high mass accretion often do not show an optically thick soft X-ray component; instead, many exhibit optically thin hard X-ray emission (Patterson & Raymond 1985a, 1985b; Mauche et al. 1995; van Teeseling et al. 1996; Baskill et al. 2005; Balman et al. 2014), with an X-ray luminosity much smaller than expected, i.e., much smaller than the disk luminosity. While optically thin hard X-ray emission from high mass accretion rate systems was unexpected, it is, however, not especially inconsistent with the theoretical work: optically thin BLs can occur in high mass accretion rate systems, since the transition to being optically thin depends not only on the mass accretion rate, but also on the WD mass, the WD rotation rate, and the (unknown) alpha viscosity parameter (Popham & Narayan 1995). Simulations of optically thin BLs (Narayan & Popham 1993; Popham 1999) show that the inner edge of the Keplerian (and optically thick) disk starts at an actual radius R0 = Rwd + δBL, where the size δBL of the BL is of the order of the stellar radius Rwd: δBL ∝ Rwd (the BL is actually geometrically thick). The direct consequence of having an optically thin and geometrically thick BL is that the optically thick Keplerian disk will appear to have an inner hole of size δBL (possibly of the order of the radius of the WD Rwd). Two decades ago, it had already been pointed out that optically thin BL can explain the inner hole observed in circumstellar disks around young stellar objects (T Tauri stars; Godon 1996). In other words, optically thin BLs are consistent not only with the X-ray data, but also with the UV data, as truncated optically thick disks produce a UV continuum with a shallow slope in better agreement with the UV observations than non-truncated disks.


We have employed a simple grey-body fitting method to model the infrared SED of individual sources. In the LMC, which has the most numerous list of LBVs, large amounts of dust are observed (∼10−3 − 10−2 M⊙), similar to Galactic LBVNe. We stacked the infrared images of the LMC LBVs and extracted the photometry of the resulting source, detected up to 160−250 μm. The integrated SED from the stacks resembles that of LBVs with a strong ionised stellar wind and an extended dusty nebula. The SED can be fitted with only two components: a power-law describing the free-free spectrum of ionised stellar winds and the stellar photosphere, and a single-component grey-body for the dust. For the grey-body we adopted two different values of the κ parameter, including the value determined by Gordon et al. (2014) to fit the integrated ISM SED of the LMC. A significant contribution to the stack SED comes from a few sources, the most important one is RMC143. This was already identified as a massive nebula (Agliozzo et al. 2019). We obtain an integrated present dust mass of

0
.

11

−
0.03

+
0.06

M
⊙

$ 0.11^{+0.06}_{-0.03}\,M_{\odot} $

. We have repeated a similar analysis on the sample of AGBs and RSGs by Riebel et al. (2012). We obtain a detection in the stacked images only when considering the “extreme”-AGBs. We find that the integrated 160 μm emission of 1342 extreme-AGBs is of the same order of magnitude as that of 18 LBVs. The integrated dust mass from these sources is

1
.

2

−
0.4

+
0.3

×

10

−
4

M
⊙

$ 1.2^{+0.3}_{-0.4}\times 10^{-4}\,M_{\odot} $

. We do not find any correlation between the dust masses and the stellar luminosities. This could be due to the fact that such stars have different evolutionary histories or that the dust production mechanism does not depend on the initial mass of the star. Most likely we are also unable to detect the lowest-mass nebulae. To estimate the total dust mass produced by LBVs in the LMC during its full lifetime, we consider two cases: constant number of LBVs across time and a case accounting for IMF and SFH. The uncertainty on the duration of LBV phase in the first case, or on the population of LBVs in the second case, add a significant uncertainty in the total mass produced by LBVs.