The populous blue LMC cluster NGC 1866 is already known to host an exceptionally rich sample of more than 20 Cepheids (Musella et al. 2006; Welch & Stetson 1993). One of these was identified by Musella et al. (2006) in a preliminary analysis of the proprietary BVI Very Large Telescope (VLT) data. It is unquestionable that such a unique sample of Cepheids – likely all members of the cluster and at the same distance, chemical composition and age – would constitute a milestone in our understanding of the Cepheid pulsational scenario. Indeed, it offers an unprecedented opportunity to investigate both empirical and theoretical estimates of the luminosity and colour of the pulsating structures and their relation with the observed periods. For this reason, many authors have studied the NGC 1866 Cepheids (Welch 1991; Welch & Stetson 1993; Gieren, Richtler & Hilker 1994; Walker 1995; Gieren et al. 2000; Storm et al. 2005; Testa et al. 2007) in both the optical and near-infrared bands, and have tested different methods to calibrate the PL relations in different filters. In Brocato et al. (2004), we already discussed the sample of the 23 known Cepheids in NGC 1866, concluding that unfortunately only 4–6 Cepheids had light curves accurate enough to allow a meaningful determination of their luminosities and colours. On the basis of such a tantalizing situation, we took advantage of assigned observing time at the ESO VLT to perform an accurate photometric investigation of the cluster field, with the aim of securing suitable data constraining the light curves of all the member Cepheids. Moreover, to get accurate information about radial velocities and chemical abundances of the stars in NGC 1866, we have performed FLAMES@VLT (Fibre Large Array Multi Element Spectrograph mounted on VLT) spectroscopic observations for 30 stars (19 belonging to the cluster and 11 to the LMC field), including three Cepheids (Mucciarelli et al. 2011; Molinaro et al. 2012). Mucciarelli et al. (2011) found that, as far as the chemical composition is concerned, the cluster stars are reasonably homogeneous. Indeed, they appear to share the same abundances within the uncertainties, and this property is independent of the evolutionary status. The average iron abundance is [Fe/H] = −0.43 ± 0.01 dex, with a dispersion σ = 0.04 dex. For the three spectroscopically investigated Cepheids Molinaro et al. (2012), adopting the same procedure used in Mucciarelli et al. (2011), found values fully consistent with the average iron content. Moreover, Molinaro et al. (2012) applied the CORS Baade–Wesselink (BW) method (Ripepi et al. 1997) to a sample of 11 Cepheids, using radial velocities obtained both from our FLAMES investigation and from literature data, and light curves based on a part of the UBVI data used in this paper complemented with K data by Testa et al. (2007). In this way, they obtained a direct estimate of the distance modulus of NGC 1866, μ0 = 18.51 ± 0.03 mag (see Molinaro et al. 2012, for details).


Here we demonstrate that, over a fixed time interval, the planar three-body problem can be solved by means of a multilayered deep artificial neural network (ANN; e.g. see LeCun, Bengio & Hinto 2015). These networks are designed for high-quality pattern recognition by mirroring the function of our brains (McCulloch & Pitts 1943; Rosenblatt 1985) and have been successfully applied to a wide variety of pattern recognition problems in science and industry, even mastering the game of Go (Silver et al. 2016). The abundance of real-world applications of ANNs is largely a consequence of two properties: (i) an ANN is capable of closely approximating any continuous function that describes the relationship between an outcome and a set of covariates, known as the universal approximation theorem (Cybenko 1989; Hornik 1991); and (ii) once trained, an ANN has a predictable and a fixed computational burden. Together, these properties lead to the result that an ANN can be trained to provide accurate and practical solutions to Newton’s laws of motion, resulting in major improvements in computational economy (Lee, Sode-Yome & Park 1991) relative to modern technologies. Our proof-of-principle method shows that an ANN can accurately match the results of converged solutions found using the arbitrary precision numerical integrator that, for computationally challenging scenarios, e.g. during multiple close encounters, can offer numerical solutions at a fraction of the time cost and CO2 expense. We demonstrate the importance of training an ANN on converged solutions. This enables the trained ANN to accurately predict particle locations even when a conventional ‘double-precision’ numerical integrator fails dramatically. By training an ANN that can accurately compute particle trajectories during close encounters, our work extends previous work training neural networks on an n-body-type problem (e.g. Quito, Monterola & Saloma 2001; Battaglia et al. 2016). Our findings also add to the growing body of literature that supports machine learning technologies being developed to enrich the assessment of chaotic systems (Pathak et al. 2018; Stinis 2019) and providing alternative approaches to classical numerical solvers more broadly (Hennig, Osborne & Girolami 2015).


Large-scale magnetic fields with strength of the order of 1–10 μG have been observed in disk galaxies (e.g., Beck et al. 1996; Fletcher 2010; Beck 2012; Beck & Wielebinski 2013; Van Eck et al. 2015). The origin of these fields can be explained through mean-field dynamo theory (Ruzmaikin et al. 1988; Beck et al. 1996; Brandenburg & Subramanian 2005a; Kulsrud & Zweibel 2008). The conservation of magnetic helicity is one of the key constraints in these models, and also leads to the suppression of the α-effect. The operation of the mean-field dynamo automatically leads to the growth of magnetic helicity of opposite signs between the large-scale and small-scale magnetic fields (Pouquet et al. 1976; Gruzinov & Diamond 1994; Blackman & Field 2002). To avoid catastrophic suppression of the dynamo action (α-quenching), the magnetic helicity due to the small-scale magnetic field should be removed from the system (Blackman & Field 2000, 2001; Kleeorin et al. 2000). Mechanisms suggested to produce these small-scale magnetic helicity fluxes are: advection of magnetic fields by an outflow from the disk through the galactic fountain or wind (Shukurov et al. 2006; Sur et al. 2007; Chamandy et al. 2014), magnetic helicity flux from anisotropy of the turbulence produced by differential rotation (Vishniac & Cho 2001; Subramanian & Brandenburg 2004, 2006; Sur et al. 2007; Vishniac & Shapovalov 2014), and through diffusive flux (Kleeorin et al. 2000, 2002; Brandenburg et al. 2009; Mitra et al. 2010; Chamandy et al. 2014). The outflow of magnetic helicity from the disk through dynamo operation leads to the formation of a corona (Blackman & Field 2000). According to Taylor's hypothesis, an infinitely conducting corona would resistively relax to force-free field configurations under the constraint of global magnetic helicity conservation (Woltjer 1960; Taylor 1974; Finn & Antonsen 1983; Berger & Field 1984; Mangalam & Krishan 2000). In this paper, we include advective and diffusive fluxes in a simple semi-analytic model of a galactic dynamo that transfers magnetic helicity outside the disk and consequently builds up a force-free corona in course of time. We first solve the time-dependent dynamo equations by expressing them as separable in variables r and z. The radial part of the dynamo equation is solved using an eigenvector expansion constructed using the steady-state solutions of the dynamo equation. The eigenvalues of the z part of the solution are obtained by solving a fourth-order algebraic equation, which primarily depends upon the turbulence parameters and the magnetic helicity fluxes. Once the dynamo solutions are written out as parametric functions of these parameters, the evolution of the mean magnetic field is computed numerically by simultaneously solving the dynamical equations for α-quenching and the growth of large-scale coronal magnetic helicity. Since the large-scale magnetic field lines cross the boundary between the galactic disk and the corona, the magnetic helicity of the large-scale magnetic field in the disk volume is not well defined. Hence we use the concept of gauge-invariant relative helicity (Finn & Antonsen 1983; Berger & Field 1984; Berger 1985) to estimate the large-scale magnetic helicity in the disk and the corona. Here the gauge-invariant relative helicity for the cylindrical geometry is calculated using the prescription given in Low (2006, 2011). We then investigate the dependence of the saturated mean magnetic field strength and its geometry on the magnetic helicity fluxes within the disk and the corresponding evolution of the force-free field in the corona.


The characterization of the dynamics of galaxies just after (4 ≲ z ≲ 6) and within the Epoch of Reionization (z ≳ 6) is still in its infancy as spatially resolved observations targeting emission lines are available only for about ten of sub-mm sources (e.g., Rizzo et al. 2020; Lelli et al. 2021; Rizzo et al. 2021) and a handful of Lyman break galaxies (e.g., Jones et al. 2017; Fujimoto et al. 2021). Instead, the number of marginally resolved observations aiming at inferring the integrated properties of early galaxies has increased in the last five years (Smit et al. 2018; Le Fèvre et al. 2020; Bouwens et al. 2022). In their pioneering work, Smit et al. (2018) used marginally resolved Atacama Large Millimeter/Submillimetre Array (ALMA, Wootten & Thompson 2009) observations of the [C II]-158 μm emission line to show that two star-forming galaxies at z ∼ 7 have smooth velocity gradients and interpreted them as being rotationally supported disks. Similar results are obtained in studies of individual galaxies at z ∼ 6–8 (Bakx et al. 2020; Harikane et al. 2020). However, given the low angular resolution of these data, they can not rule out the possibility that one or more merging [C II]-bright satellites are mimicking the smooth gradient observed in the velocity maps (see discussion in Smit et al. 2018; Simons et al. 2019). Instead, a gradient in the velocity fields of two Lyman break galaxies at z = 6.1 and 7.1, combined with the identification of two compact components, has been interpreted as evidence of mergers (Jones et al. 2017; Hashimoto et al. 2019). Recently, Le Fèvre et al. (2020) and Romano et al. (2021) performed the first systematic morpho-kinematic analysis of a statistically significant sample of z ∼ 4–6 main-sequence galaxies from the ALMA Large Program to INvestigate [C II] at Early times (ALPINE) survey. After combining the morphological and kinematic analysis of the [C II] observations with the rest-frame UV and optical data, Romano et al. (2021) conclude that 23 out of the 75 ALPINE galaxies are merging systems. This large fraction of mergers in the ALPINE sample could imply a significant contribution of major mergers to the mass assembly in the early Universe (Le Fèvre et al. 2020; Romano et al. 2021), confirming previous results based on the count of close-pair galaxies in photometric surveys (Mantha et al. 2018; Duncan et al. 2019). However, due to the sensitivity and angular resolutions of the ALPINE observations, the kinematic characterization and, consequently, the merger fraction may be uncertain.


Low- and intermediate-mass stars are formed by the gravitational collapse of the parental giant molecular cloud (GMC), followed by the accretion process (Palla 1996). During the accretion phase, material is ejected as well via collimated bipolar jets. However, when a YSO reaches 8 M⊙, the radiative flux becomes so intense (using ϕ = L/4πd2, the ratio between the radiative fluxes of an O5 and a B3 star – masses of ∼40 and ∼8 M⊙, respectively – is ≈250) that it may interrupt the accretion flow. A process that constrains the outcoming radiation field to narrower angles may leave some room for the accretion process to continue in some directions. This seems to be the case for the outflows driven by young stars from a very broad mass range, as previous reported by several authors (Bachiller 1996; Bontemps et al. 1996; Shepherd & Churchwell 1996; Beuther et al. 2002; Wu et al. 2004). Outflows associated with high-mass objects are expected to be more energetic than the outflows observed in lower mass YSOs (Beuther et al. 2005; Zhang et al. 2005; López-Sepulcre et al. 2009), with velocities greater than ∼100 km s−1 (Martí, Rodríguez & Reipurth 1998). Some authors have found evidences that outflows associated with massive stars are scaled up versions of their low-mass counterparts (Vaidya et al. 2011; Codella et al. 2013) while other works have reported that no well-collimated outflows have been found towards MYSOs (Shepherd, Testi & Stark 2003; Sollins et al. 2004). Massive YSO outflows mapped in high-velocity CO lines have collimation factors R = length/width ∼2.05 ± 0.96 as compared to R ∼ 2.81 ± 2.16 for low-mass stars (Wu et al. 2004), indicating a weak tendency that outflows associated with massive stars are less collimated than those from low-mass stars as previously thought (Richer et al. 2000). Besides the degree of collimation, these massive outflows would work removing mass from the plane of the accretion disc, lowering the density on the plane and, therefore, facilitating the accretion flow to reach the stellar core as shown in the recent 3D simulations presented by Krumholz et al. (2009). Although these authors have not included the outflow activity on their simulations, they argue that the presence of outflows would decrease the star formation efficiency from 70 per cent (considering purely radiation effects) to 50 per cent.


Next we chose an optimum cut-off radius for the detector FOV. The PSPC has a circular FOV with a radius 57′. The PSPC entrance window has a rib support structure with an inner ring at a radius corresponding to 20′ (Pfeffermann et al. 1987; Hasinger & Zamorani 2000). Both the ROSAT telescope angular resolution and its vignetting function are roughly constant within the inner 20′ ring, but degrade significantly towards larger off-axis angles. The combined detector and telescope PSFs are described in detail in Boese (2000). To the first order, the PSF at each off-axis angle can be approximated by a Gaussian function with a half power radius (HPR) of 13, 22, 52, 93, 130, and 180″, at off-axis angles of 0, 12, 24, 36, 48, and 57′, respectively (at 1 keV). The vignetting function at 1 keV drops almost linearly to about 50% at an off-axis angle of 50′. Taking into account all these effects, the HPR of the overall RASS PSF is 84″ (Boese 2000). This means that the classical confusion limit (40 beams per source) is reached at a source density of about 15 sources deg−2, which is exceeded in the high-exposure areas of our survey. In addition, we need to optimally discriminate between extended and point-like X-ray sources, calling for an angular resolution that is as high as possible. We therefore have to reduce the detector FOV. The sharpest imaging is achieved within the inner 20′ of the PSPC FOV, corresponding to the inner ring-like rib of the PSPC support structure (see Fig. 1). However, there is a trade-off between image sharpness and the number of photons required for detection and image characterization. In particular in the outer areas of our survey, where the RASS exposure times drop significantly, a 20′ FOV radius does not provide sufficient exposure time. Taking into account the various competing factors in this trade-off, we made a few tests varying the FOV cut-off radius, and finally decided on an optimum FOV radius of 30′. The PSPC detector coordinates have a pixel size of 0.934″. We thus removed all X-ray events from the dataset, which are further than 1925 pixels from the PSPC centre pixel coordinate [4119,3929]. A similar cut had to be applied to the modified PSPC instrument map (MOIMP), which is used later for the construction of the survey exposure map.


One of the most intriguing results from RV surveys is the observed scarcity of relatively close-in (a ≲ 0.5 AU) planets around post-MS stars. This observational trend has been attributed to the strong tidal torque exerted by the star as its radius grows during the giant phase. As a result, planets are expected to lose orbital angular momentum, thus moving inward until they are evaporated in the stellar atmosphere (Livio & Soker 1983; Sato et al. 2008; Villaver & Livio 2009; Kunitomo et al. 2011). On the other hand, the majority of the giant stars targeted by RV surveys are intermediate-mass stars (M⋆ ~ 1.5–3.0 M⊙), thus they are the post-MS counterpart of A and early F main-sequence stars. Therefore, their companions should not be directly compared to those orbiting solar-type stars. Based on this analysis, known planets orbiting field giant stars are expected to be born in different conditions from those around low-mass stars. In particular, these planets are formed in more massive disks (since Md ∝ M⋆; Andrews et al. 2013), from which they can efficiently accrete a significant amount of gas, becoming gas giants (e.g., Kennedy & Kenyon 2008). In addition, due to the higher gas accretion rate (Muzerolle et al. 2005) and higher irradiation, these disks have shorter dissipation timescales (Currie 2009; Kennedy & Kenyon 2009) and the snow line is located at a greater distance from the central star (Kennedy & Kenyon 2008). As a consequence, these planets are most likely formed at greater orbital distances and, due to the shorter disk timescale, inward migration is halted; they thus reach their final position at a relatively large distance from the parent star. For comparison, Currie (2009) predicted that only ~1.5% of intermediate-mass stars host giant planets with a ≲ 0.5 AU, while ≳7.5% of them host at least one gas giant at a ≳ 0.5 AU. Fig. 9 shows the mass versus the orbital distance of planets detected around giant stars (log g ≲ 3.5), via RV measurements (black dots) and by the transit method (red open circles). We note that values of the RV detected systems correspond to the minimum planet mass (Mp sini). The dotted line represents a radial velocity semi-amplitude of K = 30 m s−1 for a 1.5 M⊙ star, (corresponding to a 3-σ detection; e.g., Hekker et al. 2006). As can be seen, there is only one companion detected via RVs interior to 0.1 AU, and the rest of them reside at an orbital distance a ≳ 0.4 AU. As discussed above, this observational result might be explained by the engulfment of the innermost planets as the parent star evolves off the MS and becomes a giant star. However, since a similar trend is observed in less evolved subgiants whose radii have not yet reached a value where tidal interactions are strong enough to affect the orbits of their companions, Johnson et al. (2007) argued that this is probably explained by a different formation scenario between planets around low-mass stars and those formed in more massive disks. From Fig. 9 it is also evident that planets residing interior to ~0.1 AU are significantly less massive (Mp ≲ 1 MJ) than those orbiting at a greater distance. In fact, two of these transiting planets are well below the 3-σ detection threshold, thus they are not detectable via radial velocities. A similar trend is also observed in MS stars (Zucker & Mazeh 2002), which might be caused by a decrease in the type II migration speed with increasing planetary mass, i.e., d a∕dt ∝ M

$_P^{-1}$
P−1
 (Mordasini et al. 2009). This theoretical prediction naturally explains why the most massive planets are found at a ≳ 0.4 AU. On the other hand, the mass distribution of the parent stars of these two populations of planets are different. While the mean stellar mass of the RV detected planets is 1.78 M⊙, this value is only 1.38 M⊙ for the transiting systems and thus two distinct planet mass distributions are expected to be found. Moreover, a similar result is observed between the mass of planets orbiting subgiant and giant stars (planets around giant stars being significantly more massive than those around subgiants; see Jones et al. 2014). In fact, the mean mass of the subgiant parent stars is 1.5 M⊙, significantly lower than giant host stars. These results provide further observational support of a different formation and migration scenario for planets at different host star mass. This result suggest that the observed lack of planet around giant stars is mainly due to the primordial distinct formation scenario proposed by Johnson et al. (Johnson et al. (2007)).


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

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

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


Several mechanisms have been suggested to explain the formation of moving groups. A common explanation is that these velocity structures are the remnants of open clusters, or formed by interactions with a bar (Eggen 1965; Dehnen 2000). One problem with the cluster formation idea is that stars in moving groups can have a variety of different ages and compositions, so it is unlikely that they all came from the same cluster (Eggen 1965; Kushniruk et al. 2020). Analysis of GALAH data (Quillen et al. 2018) indicates that some moving groups, such as the Hercules moving group, may be due to a resonant bar. It has also been suggested that moving groups could have been formed from perturbations due to the Magellanic Clouds via gravitational interactions (Dehnen 1998). Recent work also finds that transient spiral structure (Hunt et al. 2018) may lead to the formation of moving groups, as well as perturbations due to spiral arms in the MW (Michtchenko et al. 2018). Moving groups in Gaia data have also been identified and analysed in action space. In the (JR, Jz) plane there are at least seven overdensities that follow lines of constant slope in this plane, which correspond to known moving groups in the solar neighbourhood (Trick, Coronado & Rix 2019). It is likely that there may be multiple causal mechanisms at play in the formation of moving groups in the Milky Way. The analysis of Gaia DR2 data has revealed many facets of a Galaxy that are clearly out of equilibrium, including the so-called phase-space spiral (Antoja et al. 2018), and the Enceladus merger (Helmi et al. 2018), that have been interpreted as arising from interactions with dwarf galaxies. Analysis of Gaia DR2 data also led to the discovery of a new dwarf galaxy (Torrealba et al. 2019) that likely interacted with the Milky Way (Chakrabarti et al. 2019). However, the formation of moving groups due to dwarf galaxy interactions has not yet been studied with full N-body simulations. Motivated by these earlier works that indicate that the MW may has been perturbed by dwarf galaxies, we focus our study here in trying to understand if some of the moving groups in the Galaxy may have arisen from dwarf galaxy interactions.


Of the almost 4000 exoplanets known today, more than 3700 were discovered via the radial-velocity (RV) or transit methods1. According to the database, only three of them (V830 Tau b, Donati et al. 2016; K2-33 b, David et al. 2016, and TAP 26 b, Yu et al. 2017) are younger than 100 Myr. The main reason for this is the strong stellar activity of young stars, which makes it hard to find the subtle planetary signal in the large stellar variations. This is unfortunate for two reasons: first, planet formation takes place in young systems and at least gas giants need to form before the disk has dissipated after less than a few tens of millions of years (e.g., Ercolano & Pascucci 2017). Second, planets at large orbital distances (≳50 AU) are almost exclusively detected via direct imaging (DI), which is best applicable to young systems where the planets are still hot from their formation. Thus, in order to discover all planets in a system, one either needs to image old stars – which seems currently impossible given the already low detection rate around young stars probed by large DI surveys (e.g., Desidera et al. 2015; Lannier et al. 2016; Tamura 2016; Stone et al. 2018) – or try to minimize the impact of the stellar activity of young stars. A lot has been done to understand and characterize stellar activity (e.g., Dumusque 2018). Lindegren & Dravins (2003) further estimate the effects of stellar activity such as oscillation, granulation, meridional flow, long-term magnetic cycle, surface magnetic activity and rotation, gravitational redshift and many more on the RV measurement. Meunier & Lagrange (2019) then try to model the effect of this kind of activity signal on RV data of mature stars. With our data probing activity timescales of days to years, we are mainly probing the combined effect of stellar rotation, reconfiguration of active regions, and long-term magnetic cycles. Still, large uncertainties remain in the prediction and interpretation of any RV signal, in particular for young pre-main sequence stars. But since this is what we measure, knowledge about the typical RV variability is important, for example for developing and testing RV activity models or planning RV surveys. In this paper we therefore derive a model-free analytic relation between stellar jitter, stellar age, and lag, where lag denotes the timescale on which the jitter is measured.


Recently, polarized (sub)millimetre emission has been detected in an increasing number of discs by Atacama Large Millimeter/submillimeter Array (ALMA) with its high sensitivity and angular resolution. However, the origin of disc polarization remains uncertain, since grains do not have to be aligned with just the magnetic field (Kataoka et al. 2017; Yang et al. 2019). They may also be aligned in the direction of the radiative anisotropy (Lazarian & Hoang 2007a; Tazaki, Lazarian & Nomura 2017) or the drift velocity of the grains relative to the ambient gas (Gold 1952; Lazarian 1995; Lazarian & Hoang 2007b). Furthermore, even spherical grains can produce polarized emission by self-scattering of large grains in an anisotropic radiation field (Kataoka et al. 2015; Yang et al. 2016, 2017; Stephens et al. 2019). The scattering interpretation of the disc polarization is favoured in several targets (e.g. Stephens et al. 2014, 2017; Kataoka et al. 2016; Bacciotti et al. 2018; Dent et al. 2019; Girart et al. 2018; Harris et al. 2018; Hull et al. 2018; Lee et al. 2018). One way to gauge the effects of scattering and identify polarization from aligned grains would be to observe at multiple wavelengths since the efficiency for scattering for grains of given sizes decreases rapidly with the wavelength in the optically thin and small-particle (or Rayleigh scattering) limit. Indeed, in the disc of Class I protostar BHB 07-11, Alves et al. (2018) detected polarization with ALMA at three wavebands (Bands 3, 6, and 7 or ∼ 3, 1.3, and 0.87 mm, respectively) with consistent polarization orientations across three bands and increasing polarization fraction with wavelength, which is generally not expected for scattering-induced polarization. The rather high mean polarization fractions (∼7.9, 5.3, and 3.5 ${{\ \rm per\ cent}}$ for Bands 3, 6, and 7 respectively) are also higher than those typically produced in models of scattering-induced disc polarization ($\sim \!1 {{\ \rm per\ cent}}$). At least for this well-studied source, scattering is unlikely the main mechanism for producing the observed multiwavelength disc polarization and aligned grains are favoured.


Blazars are a subclass of active galactic nuclei (AGNs) with relativistic jets of high-energy particles pointing near our line of sight (e.g., Urry & Padovani 1995). Their nonthermal emission is generally detected across the entire electromagnetic spectrum from radio to γ-ray bands. Blazars are subclassified into flat-spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs), according to the equivalent width of the emission lines in their optical spectrum (Stickel et al. 1991; Stocke et al. 1991; Marcha et al. 1996). These two subclasses of blazars are thought to be intrinsically different, perhaps based on their accretion mode (Dermer & Giebels 2016). FSRQs have high luminosity and a thin and radiatively efficient black hole accretion disk (Malkan & Moore 1986), while BL Lacs are powered by an advection-dominated, low radiative efficiency accretion flow (Dermer & Giebels 2016; Blandford et al. 2019). The jet emission is relativistically beamed (Ghisellini 2019), with a Doppler boosting factor corresponding to a bulk Lorentz factor of several to greater than 10 (Pushkarev et al. 2009). In both cases, the broadband spectra consist of two broad humps, one peaking in the IR-to-X-ray regime and the other peaking in the γ-ray regime. The low-energy peak is believed to be due to synchrotron emission, while the high-energy peak is likely due to inverse Compton scattering of low-energy photons of either the same synchrotron photons (for BL Lacs) or external photons from the disk/BLR (for FSRQs) (e.g., Dutka et al. 2017). However, some blazars might not necessarily be detected in γ-rays (e.g., Paliya et al. 2017). Indeed, a recent study showed that blazars undetected in γ-rays are likely to have relatively smaller Doppler factors and more disk dominance (Paliya et al. 2017). In the case of strong Compton scattering, the beaming of γ-rays could be larger than, e.g., that seen in the radio (Dermer 1995), leading to the possible nondetection (or reduced detection efficiency) of γ-rays from sources not seen exactly pole-on.


The pioneer of spectroscopic surveys, the Radial Velocity Experiment (RAVE), produced its first data release (DR) 15 years ago (Steinmetz et al. 2006) and its final one, DR6, last year (Steinmetz et al. 2020a,b). In the meantime, other surveys have been operated, and survey designers have developed new methodologies, learning progressively from their own and one another’s experience on how to reduce biases and uncertainties in the automated determination of APs and abundances from massive datasets of stellar spectra with various resolutions and spectral coverages. Besides RAVE, other surveys have published successive DRs that are available for public use. At the time of writing, there is open access to the DR16 of the Apache Point Observatory Galactic Evolution Experiment (APOGEE; Jönsson et al. 2020), the DR3 of the Galactic ArchaeoLogy with HERMES project (GALAH; Buder et al. 2021), the DR6 of RAVE (Steinmetz et al. 2020a,b), the DR5 of the Large sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST; Luo et al. 2015, 2019), Sloan Extension for Galactic Understanding and Exploration (SEGUE; Yanny et al. 2009), and the Gaia-ESO Survey (DR3; Gilmore et al. 2012; Randich & Gilmore 2013). Additional versions of APs, based on different methods, are also provided for RAVE DR6 (Guiglion et al. 2020) and for LAMOST DR5 (Xiang et al. 2019). The next generation of optical and near-infrared spectrographs, wide-field and massively multiplexed, is in preparation and will soon provide even larger catalogues of APs and abundances, such as the WHT Enhanced Area Velocity Explorer (WEAVE; Dalton et al. 2012), the Multi-Object Optical and Near-infrared Spectrograph (MOONS) on ESO’s Very Large Telescope (Taylor et al. 2018), the 4-metre Multi-Object Spectroscopic Telescope (4MOST; de Jong et al. 2019), the Prime Focus Spectrograph (PFS; Takada et al. 2014), and the Maunakea Spectroscopic Explorer (MSE; McConnachie et al. 2016). In terms of numbers of stars, the most revolutionary survey will certainly be that of the Gaia space mission (Gaia Collaboration 2016), which will deliver in its DR3 in 20221 estimates of the physical properties, including metallicities, for millions of stars obtained with various methods through an astrophysical parameter inference system (Bailer-Jones et al. 2013).


where

D
ϵ

Dt

$ \frac{D \epsilon}{Dt} $

 is the advective derivative of the energy density,

κ

=

κ
0

T

5
/
2

b

̂

$ {\boldsymbol{\kappa}} = \kappa_0 T^{5/2} \hat{{\boldsymbol{b}}} $

 is the Spitzer conductivity along magnetic field lines, Qheat and Qcool are heating and radiative cooling rates, and ρ, v, T, P and g are the plasma density, velocity, temperature, pressure, and gravitational acceleration respectively. Such user-defined heating term usually has a form of an exponentially decreasing function along the vertical coordinate y;

Q
heat

=

c
0

exp

(
−

y
λ

)

$ Q_{\mathrm{heat}} = c_0 \exp (-\frac{\mathit{y}}{\lambda}) $

, where c0 is the peak heating rate and λ is the heating scale height. The user defined heating is therefore highly stratified, spatially smooth and steady (e.g. Müller et al. 2003; Fang et al. 2013, 2015; Xia et al. 2017). The need for the user-defined heating in the previous coronal rain simulations arises from the fact that they typically do not include any self-consistent dissipation mechanisms. They also do not cover complete lower solar atmosphere including chromosphere, photosphere, and convection zone, therefore omitting key physical processes in the lower atmosphere, such as magneto-convection, associated magnetic buffeting, braiding, and flows. Another drawback of several coronal rain simulations is the commonly used approximation that all of the plasma cooling (i.e. the process essential for modelling the thermal instability and catastrophic cooling) occurs via optically thin radiative losses. This approximation is perfectly valid in the corona but ceases to apply for cool plasma (below temperatures of a few 100 000 K). Such assumption means that regardless whether the radiative loss function is calculated from CHIANTI (Dere et al. 2019) or using scaling law approximations (e.g. Rosner et al. 1978), there is a cut-off temperature for the radiative cooling of the plasma condensations. Once electron recombination starts during the cooling process, the energy gained (which depends on the ionisation potential and thus also on the ionisation degree of the plasma) is expected to slow down the cooling rate. The thermal evolution of the plasma condensations is therefore not modelled correctly at low temperatures in the previous coronal rain simulations.


In the case of systems detected via transits, passage through an inclination resonance is likely to prevent a planet from being observed in transit, assuming the natal planetary system is edge-on in the first place. Such excitations may explain inner holes in otherwise densely packed systems observed by Kepler. An example is the Kepler-11 system (Lissauer et al. 2011b), which shows six planets, five of which are between 0.09 and 0.3 au, but none interior to 0.09 au. If we examine the secular architecture of this system assuming the presence of a seventh, 1 M⊕ planet located in the interior hole,1 we find avoided crossings in both eccentricity and inclination near semi-major axes ∼0.065 au, and other secular inclination resonances at 0.0233 and 0.0763 au. Thus, it is possible that Kepler-11 contains a low-mass planet in the observed inner ‘hole’, but which is inclined relative to the outer planets. The inclination could be substantial if the current period is 8 d. The Kepler planet sample contains a large number of multiple planet systems with potential for secular resonances. KOI-94 (Weiss et al. 2013) is interesting because of the presence of a Saturn mass planet in a similar orbit to 55 Cancri. The innermost planet lies just outside a pair of secular resonances at 3.0 d (eccentricity) and 3.4 d (inclination) and so is consistent with a lack of inclination excitation. If we allow for the existence of a 1 M⊕ planet interior to this, eccentricity resonances are found at 0.9 and 2.1 d (and a mean motion resonance at 1.9 d) and inclination resonances at 1.2 and 2.1 d. Thus, a planet that has tidally migrated to orbital periods 2 d in this system could have had its inclination pumped up and have avoided transit. Another compact multiplanet system containing a massive planet is Kepler-30 (Fabrycky et al. 2012). However, the only resonances encountered by a tidally migrating 1 M⊕ inner planet lie at 14.8 d (both inclination and eccentricity) and at 13.9 d (a mean motion resonance). These may be too far out to be strongly affected by tides. Kepler-9 (Holman et al. 2010) shows two Saturn-class planets and an interior low-mass planet, that we will assume to be 1 M⊕ again. Since this inner planet does indeed transit, we must assume little inclination excitation has taken place in this system. Indeed, the sole inclination resonance for this configuration lies at 7.5 d, so that the planet would not experience any excitation if it started the migration interior to this point. Eccentricity resonances are found at 1.0 and 2.5 d, bracketing the observed period. Kepler-51 (Steffen et al. 2013) is an example of a compact Kepler planetary system with a large inner hole but no Jovian-class planets. In this case, an inner 1 M⊕ planet experiences no secular resonances. There is a frequency commensurability at 2.9 d, but no avoided crossing because the modes are almost completely decoupled (one is driven primarily by the relativistic precession).


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


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


Here, for the first time, we have identified highly dynamic non-potential activity on QS-like supergranular network scales. These events overlie mixed polarity network flux elements near the spatial resolution of HMI, and are the first non-potential structures to be unassociated with strong concentrations of bipolar magnetic flux. One event (2011b August 05) shows eruptive activity in the form of jets, which is similar to larger-scale micro-sigmoids (Raouafi et al. 2010) and even QS bright point sigmoids (Chesny et al. 2015). The existence of flaring non-potential fields in the QS-like mixed network immediately shows that supergranular-scale magnetic fields can support processes similar to sigmoid formation (Chesny et al. 2013). Strong non-potential field arcades have been observed in hot X-ray sigmoids on scales of hundreds of arseconds (Moore et al. 2001; Gibson et al. 2002; Liu et al. 2010), micro-sigmoids in soft X-ray imaging on scales of ∼50″ (Mandrini et al. 2005; Raouafi et al. 2010) and small-scale AR EUV (Zheng et al. 2012, 2013), and now at EUV temperatures in QS-like mixed network fields with lengths down to ∼10″. This range of lengths over a range of temperatures and magnetic field topologies points directly to self-similar mechanisms influencing plasma and magnetic field dynamics at a range of scales. Our findings suggest that the processes driving some large-scale eruptions (i.e., flux emergence, helicity build-up, and flux cancellation leading to non-potential field heating (Chen et al. 2014)), can also manifest in a range of configurations on sub-network size scales in QS-like magnetic field configurations. These QS flaring non-potential fields are similar to their large-scale counterparts, but not as strict in their evolution. The diversity in the observed topologies may scale with the diversity of magnetic configurations that exist in the supergranulation network. QS non-potential fields evolve in multi-polar environments, and are not restricted to strongly bipolar dominated regions as in larger-scale, higher temperature events. Despite this, one of the presented events (2011a August 05) results in a post-flare potential loop arcade, which is similar to some observed AR sigmoid fields (Moore et al. 2001).


H2O ice on Pluto has long escaped spectroscopic detection, and based on initial New Horizons data appears to be exposed only in a number of specific locations, usually associated with red color, suggestive of water ice/tholin mix (Grundy et al. 2015; Cook et al. 2015). Nonetheless, water ice is likely to be ubiquitous in Pluto’s near subsurface, given its cosmogonical abundance, Pluto’s density, and its presence on Charon’s surface6. Absorption coefficients for pure water ice (kH2O) at sub-mm-to-cm wavelengths are discussed extensively by Mätzler (1998), who also provides several analytic formulations to estimate them as a function of frequency and temperature along with illustrative plots. We use the Mishima et al. (1983) formulation (see Appendix of Mätzler 1998). Its applicability is normally restricted to temperatures above 100 K, but Fig. 2 of Mätzler (1998) indicates the trend with temperature. Absorption coefficients extrapolated to 50 K (estimated as half the values at 100 K) are shown in Fig. 5. At 500 μm, our best estimate is kH2O = 0.25 cm-1, comparable to the above values for CH4 and N2 ices. The corresponding penetration length is therefore comparable to the diurnal skin depth but remains negligible compared to the seasonal skin depth, even for seasonal Γ = 25 MKS. According to these calculations, the seasonal layer would be probed only at a wavelength of ~4 mm and beyond. We also remark that the expression from Mishima et al. (1983) would give a penetration depth of 56 m at 2.2 cm, which is an order of magnitude larger than indicated by the laboratory measurements of Paillou et al. (2008). In addition, small concentrations of impurities can dramatically reduce the microwave transparency of water ice (e.g., Chyba et al. 1998 and references therein). Therefore, the above calculations likely indicate upper limits to the actual penetration depth of radiation in a H2O ice layer, from which we conclude that the seasonal layer is not reached at the Herschel wavelengths.


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


It is as well of wide interest and diverse implications (e.g. Abadie et al. 2010; Mandel & Farmer 2017) to consider how and under which conditions NSs and BHs would pair up in tight-enough binaries so that they can spiral in by emitting GW radiation and merge within the Hubble time. Recent numerical studies based on analytical (Hénon 1975), direct N-body integration (Aarseth 2003), and Monte Carlo approach (Hénon 1971; Joshi, Rasio & Portegies Zwart 2000; Hypki & Giersz 2013) show that the retention of BHs in dense stellar clusters of wide mass range, beginning from low-/medium-mass young and open clusters (e.g. Banerjee, Baumgardt & Kroupa 2010; Ziosi et al. 2014; Mapelli 2016; Banerjee 2017, 2018a, b; Park et al. 2017; Di Carlo et al. 2019; Kumamoto, Fujii & Tanikawa 2019; Rastello et al. 2019) through globular clusters (e.g. Breen & Heggie 2013; Morscher et al. 2013; Sippel & Hurley 2013; Arca-Sedda 2016; Hurley et al. 2016; Rodriguez, Chatterjee & Rasio 2016; Wang et al. 2016; Askar et al. 2017; Chatterjee, Rodriguez & Rasio 2017a; Chatterjee et al. 2017b; Fragione & Kocsis 2018; Rodriguez et al. 2018; Antonini & Gieles 2020; Kremer et al. 2020) to galactic nuclear clusters (e.g. Antonini & Rasio 2016; Hoang et al. 2018, 2019; Antonini, Gieles & Gualandris 2019; Arca-Sedda & Capuzzo-Dolcetta 2019; Arca Sedda 2020), comprise environments where BHs can pair up through close dynamical interactions, which, furthermore, lead to general-relativistic (hereafter GR) coalescences of these BBHs. Being much more massive than the rest of the stars, the BHs, which remain gravitationally bound to a cluster after their birth, spatially segregate and remain highly concentrated in the cluster’s innermost (and densest) region (e.g. Banerjee et al. 2010; Morscher et al. 2015) due to dynamical friction (Chandrasekhar 1943; Spitzer 1987) from the stellar background. This is essentially an early core collapse of the cluster leading to its post-core-collapse behaviour (Hénon 1975; Spitzer 1987; Heggie & Hut 2003), i.e. energy generation in the ‘collapsed’ BH core leading to an overall expansion of the cluster with time (Breen & Heggie 2013; Antonini & Gieles 2020). Inside this core, BHs undergo close binary–single and binary–binary encounters giving rise to compact subsystems (triples, quadruples, or even higher order multiples) whose resonant evolution can lead to GR inspiral and merger of their innermost binaries (Leigh & Geller 2013; Samsing, MacLeod & Ramirez-Ruiz 2014; Geller & Leigh 2015; Banerjee 2018b; Samsing 2018; Zevin et al. 2019), through the binaries’ eccentricity pumping. The breakup of such subsystems or simply close, flyby encounters may also lead to a sufficient boost in eccentricity of a BBH such that it merges either promptly, in between two close encounters (e.g. Kremer et al. 2019) or within a Hubble time if it gets ejected from the cluster as a result of the interaction (e.g. Rodriguez et al. 2015; Park et al. 2017; Kumamoto et al. 2019). Note that such GR mergers can also happen in hierarchical systems, containing NSs and BHs, in a galactic field that derive from field massive-stellar multiplets (e.g. Toonen, Hamers & Portegies Zwart 2016; Antonini, Toonen & Hamers 2017; Fragione & Loeb 2019; Fragione, Loeb & Rasio 2020b).

Background Background Background Background


To derive the equipartition magnetic field of J1146+4037, we predict the rest-frame 8.4 GHz (redshifted to 1.4 GHz at z = 5.0059) flux density from our spectral model. However, there is no source size measurement at 1.4 GHz. We make use of the full width at half maximum (FWHM) source size of 0.74 ± 0.01 mas derived by the Gaussian fit from 5 GHz VLBI mas angular resolution observations (Frey et al. 2010). We note that in our calculations, we assume a source size that is 1.8 times larger than the FWHM, following the approach of Readhead (1994) and Orienti & Dallacasa (2008). The derived equipartition magnetic field is

34

−
7

+
8

$ 34^{+8}_{-7} $

 mG. This is within the range of the equipartition magnetic fields of 17 HFP radio sources (7–60 mG; quasars and galaxies at 0.22   z   2.91; Orienti & Dallacasa 2012) and 5 HFPs at 0.084   z   1.887 (18–160 mG; Orienti & Dallacasa 2008). The magnetic field calculated from the turnover information listed in Table 4 is

1
.

8

−
2.7

+
2.3

$ 1.8^{+2.3}_{-2.7} $

 G assuming an SSA origin with Eq. (3), however the uncertainty is very large. The large uncertainty is caused by the fact that we only have four data points to constrain the turnover information and we do not have source size measurements at the turnover frequency, but rather we assume the source size measured at another frequency. More data taken in other wavelength bands are needed to meaningfully constrain the turnover peak, and mas resolution observations at the peak frequency are needed to give reliable magnetic field strength measurements. This may indicate that the turnover is not caused by SSA, by comparing the large magnetic field strength measured from the spectral turnover (

1
.

8

−
2.7

+
2.3

$ 1.8^{+2.3}_{-2.7} $

 G) with the equipartition magnetic field strength (

34

−
7

+
8

$ 34^{+8}_{-7} $

 mG). As J1146+4037 is a strong blazar, the turnover may be caused by its strong variability. Another possible explanation for the spectral turnover is that high-density plasma in the nuclear region attenuates the radio emission from the central active BH. High-resolution, interstellar medium observations of the nuclear region of this target may address the latter issue.


In our model, we used a non-hydrodynamical approach based on assumptions of the motion of separate clouds under gravity and the action of radiation pressure acting on dust. This has considerable limitations but they are justified as the first approximation for modeling the LIL part of the BLR. As discussed in the classical paper of line-driven wind model (for HIL part of BLR) by Murray et al. (1995), the optical depth of the emitting region must be moderate (column density of order of 1023 cm−2) and the local density is high (for LIL part it is higher than that of HIL, many authors argue for a local density about 1012 cm−3, for instance, Adhikari et al. 2016; Baskin & Laor 2018; Panda et al. 2018), while the BLR is extended. There are two possibilities to support a consistent picture: it is either to assume a very narrow stream of material flowing out, with the cross-section on the order of 1012 cm, as in Murray et al. (1995), where they assume lower density so the size is actually larger 1014 cm; or to assume considerable clumpiness of the medium. We followed the second approach since there are natural thermal instabilities in the plasma, such as instability caused by X-ray irradiation (Krolik et al. 1981). In this case, the plasma spontaneously forms colder clumps (at a temperature of ∼104 K, cooled through atomic processes) embedded in a hotter medium (at temperature ∼107 K, set at an inverse Compton temperature value). The presence of highly or fully ionized medium is aptly supported by the observations as well as by the theory. Then the two media of density contrast of order of 103 provide the rough pressure balance. Blandford et al. (1990) discussed the typical values for ionization parameter in AGN clouds covering the range from 10−3 to 1, that is, up to four orders of magnitudes. The precise description of the structure of the clumpy medium is very difficult. Even in the case of a single cloud exposed to irradiation, in plane-parallel approximation requires a radiative transfer to be performed, which would then show the gradual change in the density, temperature, and ionization parameters (see e.g., Baskin & Laor 2018; Adhikari et al. 2018), with the low density first and the temperature roughly at inverse Compton temperature (depending on the shape of the incident spectrum), and then a relatively rapid decrease at the subsequent ionization fronts. A proper description of this transition, calculated under constant pressure, actually requires inclusion of the electron conduction (e.g., Begelman & McKee 1990; Różańska & Czerny 2000). Deep within the cloud, there is a further drop in temperature and a rise in density due to a decrease in the local flux as a result of absorption. As emphasized by Baskin & Laor (2018), radiation pressure also plays a dynamical role in this process. The picture is further complicated if plane-parallel approximation is abandoned. The presence of the numerous clouds of complex shapes can be fully consistent with simple estimates of the cloud number based on line shape properties as done by Arav et al. (1998). Of course, there are also certain processes that can lead to cloud destruction, such as the action of tidal forces (Müller et al. 2022), Kelvin–Helmholtz instabilities, and cloud ablation), however, the destruction rate can be strongly affected by the magnetic field (e.g., McCourt et al. 2015). The relative importance between the condensation rate and destruction rate depends on the cloud size, as it is set roughly by the field length (Field 1965). Cloud formation in AGN has been seen in the numerical simulations from Waters et al. (2021), but at distances much greater than the BLR distance which was likely related to the numerical setup and the requested spatial resolution of the computations. The issue is thus extremely complex, and simple order-of-magnitude estimates based on a single density and single temperature of the cloud and intercloud medium are not fully adequate and cannot reproduce the full ionization parameter range. However, addressing this point in detail is beyond the scope of the current paper.


Unfortunately, since the study by Onions et al. only used a single dark matter halo, albeit at exquisite numerical resolution, the comparison is limited to the relatively low-mass end of the subhalo mass function (SHMF), where the cumulative mass function N( > m), exceeds unity. In order to study the massive end of the SHMF, where N(>m)  1, one needs to average over large numbers of host haloes. The abundances of these rare but massive subhaloes has important implications for, among others, the statistics of massive satellite galaxies (e.g. Boylan-Kolchin et al. 2010; Busha et al. 2011) and the detection rate of dark matter substructure via gravitational lensing (e.g. Vegetti et al. 2010, 2012). In this paper, the second in a series, we use subhalo mass functions (SHMFs) and subhalo catalogues from a variety of numerical simulations that are publicly available, and that have been obtained using different subhalo finders, to compare SHMFs, focusing on the massive end. We confirm the findings by Onions et al., that the SHMFs are consistent at the 20 per cent level at the low-mass end. At the massive end, though, different subhalo finders yield subhalo abundances that differ by more than one order of magnitude! By comparing the simulation results with a new, semi-analytical model (Jiang & van den Bosch 2016, hereafter Paper I), we demonstrate that subhalo finders that identify subhaloes based purely on density in configuration space, such as the popular subfind and bdm, dramatically underpredict the masses, but not the maximum circular velocities, of massive subhaloes. We also show that the model predictions are in excellent agreement with the simulation results when they are analysed using more advanced subhalo finders that use phase-space and/or time domain information in the identification of subhaloes. We discuss a number of implications of our findings, in particular with regard to the power-law slope of the subhalo mass and velocity functions.


Similarly, according to Equation (2), angular diameter distances can be calculated from observed angular sizes as
7

and we also treated the length scale lm as a free parameter. In order to test the CDDR using different samples one should use a redshift matching criterion Δz  0.005 (Li et al. 2011; Liao et al. 2016). However, it turned out difficult to fulfill this criterion in order to compare distances derived from QSO [XUV] and QSO [CRS] directly. Therefore, proceeding in a similar manner as Li & Lin (2018), we reconstruct QSO [CRS] angular size as a function of redshift from the binned data. The angular size of intermediate-luminosity quasars was grouped into 20 redshift bins of width Δz = 0.1 starting from the smallest redshift of this sample. Median values of angular size plotted against the mean redshift in each bin are shown in Figure 4. The python package GaPP based on Gaussian Processes (Seikel et al. 2012) was used for the reconstruction process that depends on the mean function and the covariance function

. In order to discuss the influence of the choice of these two prior functions, we studied four mean functions and three covariance functions. The prior mean functions that we discussed are the following: zero mean function, the theoretical function of angular size calculated from the angular diameter distance under the assumption of three cosmological models: flat ΛCDM with Ωm = 0.27, so-called Rh = ct universe (Melia & Shevchuk 2012), and a Mirage model with Ωm = 0.27, w0 = −0.7 and w1 = −1.09 (Shafieloo et al. 2012). The linear size scaling factor lm = 11.42 pc was assumed as calibrated with SN Ia in Cao et al. (2017a). These functions (except the zero mean function) are shown in Figure 2. There are many possible covariance functions and we studied the three most popular ones. They comprise: the squared exponential function
8

the

9

and Cauchy covariance function
10

where σf and ℓ are hyperparameters that control the amplitude of deviation from the mean function and the typical length scale in x-direction, respectively. It is instructive to discuss the effects of the mean function and covariance function selection (prior assumptions) on the reconstruction. In order to show the impact of covariance function choice, we fixed the zero mean and preformed reconstruction with three different covariance functions mentioned above. The result obtained under the assumption of squared exponential covariance function is illustrated in Figure 4, where the green solid line represents the reconstructed θ(z) relation and the green region around it represents a 1σ uncertainty band. The blue dashed line and black dashed–dotted line represent the reconstructed θ(z) relation with the Matérn and Cauchy covariance functions, respectively. Their uncertainty bands are not shown in order to not blur the picture since they are similar to the one displayed. One can see that differences between reconstructions performed with different choices of covariance function are insignificant. Similarly, we checked sensitivity of reconstructions with respect to the choice of the mean function fixing the covariance as a squared exponential one and using three main functions mentioned above. It turned out that the impact of mean function choice on the reconstruction was even smaller than that of the covariance function. Therefore for further calculations we assumed the zero mean function and the squared exponential covariance function to get the reconstructed θ(z) function (i.e., the green line and region in Figure 4). Using this reconstructed relation we were able to have a one-to-one matching between the QSO [CRS] angular diameter distance and the QSO [XUV] luminosity distance at the same redshift.


The leading edges of the transients normally leave bright traces in the images of visible light, inspiring many methods that were developed to derive their locations and velocities, such as the icecream cone model (Fisher & Munro 1984), the graduated cylindrical shell (GCS) model (Thernisien 2011), geometric triangulation methods (Liu et al. 2010), mask-fitting methods (Feng et al. 2012), and trace-fitting methods including the point-p, fixed-Φ, harmonic mean, and self-similar expansion fitting methods (e.g., Sheeley et al. 1999; Howard et al. 2006; Davies et al. 2012; Möstl & Davies 2013). To derive the velocity distribution inside one transient rather than only at its leading edge, some other techniques have been proposed. Colaninno & Vourlidas (2006) applied an optical flow tool to extract the velocity vector of a coronal mass ejection (CME) in digital images. Feng et al. (2015) derived the radial velocity profiles of the whole CME from the spatial distribution of its density given by the mass continuum equation. A cross-correlation method was applied to derive continuous 2D speed maps of a CME from coronagraphic images by Bemporad et al. (2018). In their work, the radial shift pixel by pixel is determined by maximizing the cross correlation between the signal in a radial window at one frame and the signal in a radial shifted window at the previous frame, and the radial speed just equals the radial shift over the time interval between the two frames. Ying et al. (2019) improved this cross-correlation method by analyzing data in three steps: forward step (FS), backward step (BS), and average step (AS). In the FS (BS), the 2D velocity map between the current and the previous (next) frame is constructed with almost the same method as Bemporad et al. (2018). In the AS the average, velocity is obtained from the FS and BS. The velocities derived by all these methods are the component of the flow velocity vector projected onto the POS. This may underestimate the velocity especially for transients that do not propagate in the POS. Methods such as the polarizaition ratio technique (Moran & Davila 2004; DeForest et al. 2017) or the local correlation tracking (LCT) method (Mierla et al. 2009) can derive the 3D geometric information of the whole transients, but not the velocity distribution. Bemporad et al. (2018) chose the propagating direction averaged over the whole CME derived by the polarization ratio technique to correct the radial speed in the 2D maps, but the key information along the LOS is still lacking.


Given the number densities within the mass-dissociation index plane of Fig. 8, we now ask ourselves whether known dissociated clusters, such as the Bullet cluster, are expected in L210N1024NR? The Bullet Cluster has a mass of $\sim 1.5 \times 10^{15} \, {\rm M}_{\odot }$ (e.g. Clowe et al. 2004; Bradač et al. 2006; Clowe et al. 2006) and we estimated a dissociation index of SBullet ∼ 0.335 ± 0.06. As seen in Fig. 8 there are no Bullet cluster analogues (structures of approximate mass and dissociation) in L210N1024NR, this is unsurprising as a simulation requires a significantly larger volume than that of L210N1024NR ((210cMpc h−1)3) to expect such an object (e.g. Lee & Komatsu 2010; Thompson & Nagamine 2012; Bouillot et al. 2015; Kraljic & Sarkar 2015; Thompson et al. 2015). From the distribution presented in Fig. 8, it is trivial to estimate the required cosmological volume (the effective volume, Veff) to expect structures of a given mass and dissociation index. By separating the 2D distribution on the mass-dissociation index planes into the component 1D distributions of mass and dissociation the effective volume is computed as
(12)$$\begin{eqnarray}
V_\text{eff}~^{-1} &=&\int \int \,{\rm{ d}} S \, {\rm{ d}} M \phi (S, M) \\
&=& \int _{S_\text{a}}^{S_\text{b}} \, {\rm{ d}} S \phi _S(S) \int _{M_\text{a}}^{M_\text{b}} \, {\rm{ d}} M \phi _M(M)~,
\end{eqnarray}$$where ϕS(S) is the number density function associated with S and $\phi _\mathit {M}(\mathit {M})$ is the mass function presented in Fig. 7. Assuming a probable range of S = 0.335 ± 0.06 and $1 \lt M \lt 2 \times 10^{15} \, {\rm M}_{\odot }$ we estimate a number density ∼4.92 × 10−10 Mpc−3 or that an effective volume of ∼2.03 Gpc3 would be required to observe a single Bullet-like cluster. This result is inline with the number density estimate of the order of ∼10−10 Mpc−3 by Thompson et al. (2015), which improves on previous estimates (e.g. Lee & Komatsu 2010; Thompson & Nagamine 2012; Bouillot et al. 2015) due to more sophisticated halo finding methods (e.g. Behroozi, Wechsler & Wu 2013). Conversely, it was estimated by Kraljic & Sarkar (2015) (utilizing the same halo finder as Thompson et al. 2015) that given an effective volume of ∼14.6 Gpc3, no Bullet cluster analogues are expected, however as indicated by a pairwise velocity distribution it would be expected that present binary halo–halo orbits have the potential to form a Bullet-like object.


To better visualize the jet behaviour in this phase, we show in Fig. 10 the radio light curve at the core location for the first 25 d of the outburst, taken from Fig. 2. The compact jet emission peaks on MJD 58520 and then starts to decay on MJD 58521, 1 d before the system enters in the soft state and 2 d after the inferred RK1 ejection date (see Section 3.3). This whole evolution is accompanied by a smooth transition between the optically thick and the optically thin regimes of synchrotron radio emission. Compact jets are usually observed to quench at the transition from the hard to soft state (e.g. Fender et al. 1999; Corbel et al. 2000). This phenomenon is observed to start at higher frequencies (where the emission is produced closer to the compact object) and terminates as the jet break evolves through the radio band (Russell et al. 2013b, 2014; Russell et al. 2020b). It is not yet clear if compact jets switch off before or during the launch of discrete ejections, and how (or if) the two events are linked (Russell et al. 2020a). We do not have a MeerKAT observation on MJD 58521, but, as the flat spectrum obtained with ATCA suggests, the flux density would have been at the ∼50 mJy level. The following observations show instead a quick rise in flux density and a subsequent decay until MJD 58531, when the source is steadily in the soft state. With our data, we cannot conclude on the origin of the radio emission on MJD 58521, which could be produced either from the compact jet that is quenching (with no evidence of the spectral break in the radio band), or by the first self-absorbed part of the radio flare observed to peak on MJD 58523. The latter scenario implies that the jet significantly quenched in less than 2 d, a shorter time-scale with respect to what observed for other sources (e.g. Russell et al. 2013b, 2020a). Therefore, we are not able to precisely order in time the ejection of RK1 and the quenching of the compact jets, and thus we cannot draw conclusions on a potential link between the two events.


Long-duration gamma-ray bursts (GRBs) give rise to a synchrotron afterglow, detectable at optical wavelengths if sufficiently rapid and deep follow-up observations are made. A substantial fraction, however, lack such emission even when it would be expected from extrapolation of the X-ray spectral slope (Groot et al. 1998; Fynbo et al. 2001). When the X-ray to optical spectral slope, βOX, is below the recognized threshold of 0.5, the event is classified as ‘dark’ (Jakobsson et al. 2004). This is typically evaluated at 11 h post-burst to avoid contamination from early-time effects including X-ray flares and plateaus. An alternative method uses βOX  βX − 0.5 to define darkness (van der Horst et al. 2009). There are two primary causes for darkness in GRBs: attenuation by dust, or rest frame ultraviolet H i absorption at high redshift (e.g. Fruchter 1999; Levan et al. 2006; Perley et al. 2009,2013; Greiner et al. 2011; Svensson et al. 2012; Zauderer et al. 2013; Chrimes et al. 2019; Higgins et al. 2019). The number of GRBs known at high-redshift (z > 5, in the epoch of reionization) is small (∼15, from around 500 GRBs with a known or estimated redshift, Cenko et al. 2006; Grazian et al. 2006; Jakobsson et al. 2006; Kawai et al. 2006; Ruiz-Velasco et al. 2007; Salvaterra et al. 2009; Greiner et al. 2009; Tanvir et al. 2009; Cucchiara et al. 2011; Afonso et al. 2011; Castro-Tirado et al. 2013; Laskar et al. 2014; Jeong et al. 2014b; Chornock, Fox & Berger 2014b; Tanvir et al. 2018), and each one is valuable, as they provide insight into star formation in the low mass, low luminosity galaxies which power the epoch of reionization. Because they have small projected offsets from their hosts, high-redshift GRBs with a detected afterglow uniquely allow us to place accurate, deep upper limits on the luminosities of the faintest, undetected galaxies, probing fainter galaxies than deep field studies (Berger et al. 2007; Tanvir et al. 2012; Trenti et al. 2012; McGuire et al. 2016). For those with the brightest afterglows, insight into the burst environment can be gained from absorption lines in their spectra (e.g. Kawai et al. 2006; Chornock et al. 2014a; Sparre et al. 2014; Hartoog et al. 2015).


The assumptions of the equilibrium of pressures and equality of velocities along the contact discontinuity lead to

 and

, respectively. With the jump condition for the shocks and the equilibrium of pressures, we can obtain
1

The Lorentz factor of the reverse shock

 could be approximated as
2

as long as

 and

. Substituting Equation (2) into (1), we can obtain the following equation:
3

Because

 ,

, and

, we ignore the constant 1/2 term in Equation (3) and thus we can obtain the solution of this equation (ignoring the negative solution, also see Panaitescu & Kumar 2004),
4

Here we obtain the relation between the Lorentz factor of the shocked fireball shell

 and the initial Lorentz factor

 (

), which depends on the ratio of these two comoving densities. The number density of the ambient medium is assumed to be

 (Dai & Lu 1998; Mészáros et al. 1998; Chevalier & Li 2000; Wu et al. 2003, 2005; Yi et al. 2013); such a circumburst medium is a homogeneous ISM for k = 0, and a typical stellar wind environment for k = 2. The fireball shell is characterized by an initial kinetic energy Ek, initial Lorentz factor

, and a width Δ in the lab frame attached to the explosion center, so the number density of the shell in the comoving frame is

. The ratio of the comoving number density of the relativistic shell

 to the number density of the ambient medium

 defined in Sari & Piran (1995) is
5

where

. The difference between the lab frame speed of the unshocked fireball shell and that of the reverse shock is (Kumar & Panaitescu 2003),
6

Considering the thin shell case

, we can calculate the radius

 where the reverse shock finishes crossing the fireball shell,
7

The substitution of Equations (5) and (6) into (7) leads to
8

So the comoving density ratio at

 is
9

Substituting Equation (9) into (4), we can obtain the Lorentz factor of the reverse shock as it finishes crossing the shell:
10

Therefore, the relation between

 and the initial Lorentz factor

 is
11

and
12

For the thin shell case, the reverse shock crossing time

 almost corresponds to the deceleration time Tdec, i.e.,

. Therefore, we can derive the initial Lorentz factor in the ISM and wind-type cases (also see Panaitescu & Kumar 2004). For k = 0 (ISM),
13

and for k = 2 (wind),
14

With the isotropic-equivalent energy

 and the peak time of the afterglow onset

, we can estimate the initial Lorentz factor of GRBs, where

. Liang et al. (2010) discovered a tight correlation between

 and

 using 20 GRBs which show a deceleration feature in the early afterglow light curves. Other work also confirmed this correlation, but with different methods and power-law indices (Ghirlanda et al. 2012; Lü et al. 2012). Using the data on

 and

 from Liang et al. (2010, 2013) and Lü et al. (2012), we re-constrain the initial Lorentz factor, and also discover a tight

 and

 correlation for the ISM and wind cases. The

 and

 correlation in the wind case is even tighter than that in the ISM case, as shown in Figures 4 and 5.


The emission in the diffuse interstellar medium of these bands, dominated by an aromatic vibrational character, also called AIBs (aromatic infrared bands), has led to the so-called polycyclic aromatic hydrocarbon (PAH) hypothesis. Under this theory, the observed emission is related to the infrared fluorescence emission mechanism of PAH-like molecules (Leger & Puget 1984; Allamandola et al. 1985), following energetic photon absorption, although no unique PAH has been identified in the mid-infrared so far. Apart from the recent attribution of a few specific infrared emission bands to the C60 and possibly C70 fullerene molecules in some sources (Sellgren et al. 2009, 2010; Cami et al. 2010), the carriers of the AIBsremain elusive. The emission bands have been categorised in different classes (

$\mathcal{A}$
A
 to

$\mathcal{D}$D
) following the ascertainment of band profiles and center position variations, ensuing from a phenomenological deconvolution of the observations. (Peeters et al. 2002; van Diedenhoven et al. 2004; Matsuura et al. 2007; Sloan et al. 2007; Keller et al. 2008; Boersma et al. 2008; Pino et al. 2008; Acke et al. 2010; Carpentier et al. 2012; Gadallah et al. 2013). In the late classes of infrared emission spectra (so-called

$\mathcal{C}$C
 and

$\mathcal{D}$D
), a mix between an aromatic and aliphatic character is observed. Class

$\mathcal{A}$A
 sources are dominated by aromatic bands, whereas classes

$\mathcal{C}$C
 and

$\mathcal{D}$D
 harboura pronounced aliphatic character. Most of the interstellar hydrocarbons are injected through the late phases of stellar evolution, post asymptotic giant branch (AGB) and protoplanetary nebula (PPN), which often display the class

$\mathcal{C}$C
 and

$\mathcal{D}$D
 spectral character, and are eventually processed later in the ISM. In absorption, bands at 3.4, 6.85, and 7.25 μm are also observed in the diffuse interstellar medium (ISM) of our Galaxy, as well as in extragalactic ISM, and can be well represented by a material with a significant amount of aliphatic character, also called HAC or a-C:H, which is a family of hydrogenated amorphous carbons (e.g. Allen & Wickramasinghe 1981; Duley & Williams 1983; Mason et al. 2004; Risaliti et al. 2006; Dartois et al. 2007; Dartois &Muñoz-Caro 2007; Imanishi et al. 2010, and references therein). However, the AIB emission spectra are not a simple linear combination of aliphatic (such as the a-C:H observed in absorption in the diffuse ISM) and aromatic (class

$\mathcal{A}$A
 observed in emission) spectra. They show a spectral evolution of vibrational modes, including the shift of the C=C mode from 6.2 to 6.3 μm from class

$\mathcal{A}$A
 to

$\mathcal{C}$C
 (e.g. van Diedenhoven et al. 2004), and broadening and shifting back to lower wavelengths in class

$\mathcal{D}$D
. Unlike classes

$\mathcal{A-B}$A−B
 with two bands in the 7.6 to 8.2 μm range, classes

$\mathcal{C-D}$C−D
 show a broad band (Szczerba et al. 2005; Matsuura et al. 2014), with the class

$\mathcal{D}$D
 one peaking at a shorter wavelength. The out of plane, predominantly aromatic, CH vibration patterns (from 11 to 13 μm) are generally difficult to reproduce with dust analogues in the laboratory. In addition to chemically pure PAH studies, a large range of dust grain analogues have been tailored using lasers, flames, VUV continuum sources, and/or plasmas in the laboratory to tackle this identification issue (Carpentier et al. 2012; Schnaiter et al. 1999; Jäger et al. 2006; Mennella et al. 1999; Dartois et al. 2005; Furton et al. 1999; Biennier et al. 2009). Among the issues faced by laboratory synthesis of interstellar dust analogues is the ability to produce an environment that is not homogeneous for the entire batch of analogues produced (i.e. localised modifications), as many processes in the interstellar medium will only affect grain (surfaces) moieties considering, for example, hydrogen or radical accretion/addition, cosmic ray impact, grain-grain shocks, etc. As a consequence, a large inhomogeneity at the nanometre scale between adjacent constitutive elements making the grain may be expected, introducing many local defects that will influence the nature and spectroscopic properties of such small dust grains. In this study, we address a new non-homogeneous, non-bottom-up shock approach. We used a mechanochemical synthesis method under a pressurised hydrogen atmosphere, to produce laboratory interstellar dust grain analogues to explain the infrared emission spectra of the remotely observed aliphatic and aromatic mixed interstellar dust grains observed through infrared emission bands.


The kinetic energy stored in the CO shell can be estimated as $E_{\rm kin} = 0.5\, M_{\rm shell}\, V^2_{\rm exp}$, where Vexp is the expansion velocity of the shell and Mshell is the total (molecular, atomic, and ionized) shell mass. Adopting an expansion velocity equal to half the velocity interval where the structure is observed, Vexp = 7.0 ± 1.3 km s− 1 , the molecular mass given in Table 1 and the atomic and ionized masses estimated by Cichowolski et al. (2001), 1450 and 3000 M⊙, respectively, we obtain Ekin = (2.5 ± 1.0) × 1049 erg, assuming a 40 per cent error for the masses.. Although Cichowolski et al. (2001) concluded that WR 130 could have alone created the observed structure, it is important to note that they did not take into account the molecular mass present in the shell, which considerably increases the kinetic shell energy. Thus, we can compare now the new value obtained for Ekin with the mechanical energy deposited in the ISM by the wind of the WR star, Ew = (0.7–2.2) × 1050 erg (Cichowolski et al. 2001). We obtain ϵ = Ekin/Ew = 0.007–0.5. The ratio ϵ measures the energy conversion efficiency in the shell, and according to evolutionary models ϵ ≤ 0.2 (Koo & McKee 1992). Thus, not all the possible values of ϵ are compatible with the scenario where the energy injected during the WR phase is enough to create the structure. In this case, the contribution of the energy injected during the O-star phase and/or other massive stars, should be considered. As mentioned in the Introduction, WR 130 is a WNH star, and according to Smith & Conti (2008) its age would be of about 2–3 Myr and its initial mass of at least 60 M⊙. A rough estimation of the energy injected by such a star during its main sequence yields Ew = (2.5–3.5) × 1050 erg (de Jager, Nieuwenhuijzen & van der Hucht 1988), which would be enough to create the observed structure. We have nevertheless looked for the presence of other massive stars in the region. We queried the available catalogues such as the Galactic O-Star Catalog (Maíz Apellániz et al. 2013), the Early-Type Emission-Line Stars Catalogue (Wackerling 1970), the Catalogue of Be stars (Jaschek & Egret 1982), the H-alpha Stars in the Northern Milky Way Catalogue (Kohoutek & Wehmeyer 1997), and the Catalog of Galactic OB Stars (Reed 2003), for early-type and emission stars. No stars were found in any catalogue. The only massive star located nearby is, as mentioned by Cichowolski et al. (2001), an OB star, which has an uncertain spectral type and no distance estimate (Stock, Nassau & Stephenson 1960). It is located in projection not in the centre of the structure but on to the shell (there is a second OB star mentioned by Cichowolski et al. 2001 but its location is actually outside the structure, see fig. 1 of Cichowolski et al. 2001). Although we cannot completely rule out the possibility that the OB star may be playing a role in creating the shell structure, we think that the action of WR 130 is sufficient and most likely dominant in the region.


After the discovery of hard X-ray lag relative to soft X-rays in binary system (e.g. Miyamoto et al. 1988; Nowak et al. 1999), similar lags were also observed in AGN (Papadakis, Nandra & Kazanas 2001; McHardy et al. 2004). The origin of hard lag is not clearly known. One possible explanation is provided by the propagation fluctuation model in which fluctuations associated with accretion flow propagate inwards in an accretion disc and thus resulting in the emission of the soft photons from relatively outer regions earlier than the hard photons from the innermost regions (Lyubarskii 1997; Kotov, Churazov & Gilfanov 2001; Arévalo & Uttley 2006). Recently, for example, Swift monitoring of the radio-loud NLS1 galaxy PKS 0558−504 for ${\sim } 1.5{\rm \ {\rm yr}}$ has revealed that optical leads UV and UV leads soft X-rays on short time-scales of about a week (Gliozzi et al. 2013) possibly favouring the propagation model. A new type of lag has emerged from recent studies where soft photons lag to the hard photons. This is termed as the reverberation lag which is used to constrain the X-ray emitting region in AGN. Fabian et al. (2009) discovered the reverberation lag ∼ 30 s for the first time in a NLS1 galaxy 1H0707−495. Since then, such lags have been observed in dozen of Seyfert galaxies (Zoghbi et al. 2010; de Marco et al. 2011; Emmanoulopoulos, McHardy & Papadakis 2011b; Zoghbi & Fabian 2011; Zoghbi, Uttley & Fabian 2011; Cackett et al. 2013; De Marco et al. 2013; Fabian et al. 2013; Kara et al. 2013). The most of above cases reveal reverberation lag ∼100 s supporting the compact nature of X-ray-emitting region within few gravitational radii of a supermassive black hole (SMBH). In case of strong illumination, such as that implied by observation of strong blurred reflection, UV/optical emission from AGN may be dominated by the reprocessed emission and the variations in the optical/UV band emission lag behind the X-rays (e.g. McHardy et al. 2014). About a five year long campaign of Seyfert 1 galaxy Mrk 79 using six ground-based observatories for optical and RXTE for X-ray observations, Breedt et al. (2009) have shown zero lag between optical and X-rays on time-scale of about a day. Their study of correlated X-ray and optical emission suggests X-ray reprocessing on short time-scale of days and the changes in the optical emission on long time-scale of ∼ few years can be attributed to the variations in the accretion rate.


The spatial distribution of OB stars and associations, young long-period Cepheids and open clusters, star-forming regions, H ii regions, interstellar dust, and giant molecular and neutral gas clouds in the solar vicinity that have been in existence generally τ ≲ 108 yr is known to correlate with the location of the inner Sagittarius, the closest Orion, and outer Perseus spiral arm segments. (The distances for the vast majority of these spiral tracers have been determined in the literature with trigonometric or photometric methods.) The Sun is situated at the inner edge of the Orion arm (Levine et al. 2006; Hou & Han 2014; Nakanishi & Sofue 2016; Xu et al. 2018, 2021; Lallement et al. 2019; Reid et al. 2019; Skowron et al. 2019; Cantat-Gaudin et al. 2020; Fig. 2 above).3 These three spatial features nearby to the Sun appear to form part of the global spiral structure in the Galaxy. Contrary, the objects of older population with larger random velocities, for instance, main-sequence A–K stars or the oldest Cepheids and open clusters, do not currently follow the exact location of those arms (e.g. Cantat-Gaudin et al. 2020, fig. 8 therein; Griv et al. 2020, fig. 7 therein). The latter can be explained by the difference in rotation velocity between the spiral density waves and the objects. Investigating the velocity field of Xu et al.’s (2018) O and early B-type stars in the framework of the Lin–Shu density-wave proposal, we also found that the Sun lies within the Orion arm, at the inner edge of this spiral. The radial distance from the Sun to the centre of the Orion arm is ≈0.2 kpc in the direction of the Galactic anticentre, the centre of the Sagittarius arm is ≈1.8 kpc from the Sun in the direction of the GC, and the width of the arms is ≈0.5 kpc. The radial distance between the centres of the Orion and Sagittarius arms near the Sun is λrad ≈ 2 kpc (cf. Hou & Han 2014; Wu et al. 2014; Bovy et al. 2015). As for us, the nearest Orion spiral arm forms part of the dominant density-wave structure of the system.


The core/cusp problem is a clear example of this controversy: on the one hand, cosmological dark matter only N-body simulations predict cuspy dark halo density profiles; on the other hand, the rotation curves of low surface brightness disc and gas-rich dwarf galaxies favour shallower or cored dark matter density distributions (de Blok 2010 and references therein). Also for dSphs, for which the determination of the dark matter density distribution is more difficult, there are indications that cored dark matter density profiles may be favoured with respect to cuspy profiles (Kleyna et al. 2003; Goerdt et al. 2006, Battaglia et al. 2008; Walker & Peñarrubia 2011; Salucci et al. 2012; Amorisco, Agnello & Evans 2013; Zhu et al. 2016), though this finding is still debated (Richardson & Fairbairn 2014; Strigari, Frenk & White 2017). It must be stressed, however, that cored dark haloes in dSphs do not necessarily imply a failure of ΛCDM: dark matter only cosmological simulations may not reliably predict the present-day dark matter distribution in dSphs because, by definition, they neglect the effects of baryons on the dark haloes. Even in a galaxy that is everywhere dark matter dominated today, baryons must have been locally dominant in the past to permit star formation. Therefore, the effect of baryon physics on the dark halo is expected to be important also in dSphs. For instance, Nipoti & Binney (2015) showed how, due to the fragmentation of a disc in cuspy dark halo, dynamical friction may cause the halo to flatten the original cusp into a core even before the formation of the first stars (see also El-Zant, Shlosman & Hoffman 2001; Mo & Mao 2004; Goerdt et al. 2010; Cole, Dehnen & Wilkinson 2011; Arca-Sedda & Capuzzo-Dolcetta 2017). Moreover, the results of hydrodynamical simulations suggest that, following star formation, supernova feedback can also help to flatten the central dark matter distribution, by expelling the gas (Navarro, Eke & Frenk 1996a; Read & Gilmore 2005) and thus inducing rapid fluctuations in the gravitational potential (Mashchenko, Couchman & Wadsley 2006, Pontzen & Governato 2012, Tollet et al. 2016).


We obtain computationally credible samplings of the posterior probability (equation 8) by removing the burn-in steps of the random walk according to the autocorrelation time. We can then create synthetic data sets by drawing a parameter sample $\pmb {\theta }_k$ from the posterior and using it to draw from the likelihood to create a new data set, i.e. drawing new σDj from the probability distribution for all galaxies in the original data set using equation (6). We then assess the validity of the model by comparing synthetic data with the observed (i.e. original) data. This comparison is done by using a discrepancy measure $\mathcal {D}(\sigma _\mathrm{ D}|\pmb {\theta }_k)$ between data and model-derived expected values for the same data $e=\lbrace e_j(\pmb {\theta }_k)\rbrace$, where $\boldsymbol{\theta}_k$ is drawn from the posterior distribution and σD can be the observed errors or the model-generated synthetic errors. The discrepancy can be calculated using a statistic like χ2 (de la Horra 2008; de la Horra & Teresa Rodriguez-Bernal 2012), but here we will work with the Freeman–Tukey discrepancy since it is weight independent (Brooks, Catchpole & Morgan 2000; Bishop, Fienberg & Holland 2007),

\begin{eqnarray*}
\mathcal {D}(\sigma _\mathrm{ D}|\pmb {\theta }_k)=\sum _j^m \left(\sqrt{\sigma _{\mathrm{ D}j}\vphantom{e_j(\pmb {\theta }_k)}}-\sqrt{e_j(\pmb {\theta }_k)}\right)^2.
\end{eqnarray*}
For each parameter draw k, it is possible to compare the simulated discrepancy with the observed discrepancy. If the model is representative of the data, then for many parameter draws, the simulated and observed discrepancies should be similar. We can then calculate a Bayesian ‘p-value’ as the ratio of ‘draws when the observed discrepancies are larger than the synthetic discrepancies’ to ‘total draws’. If this Bayesian p-value is too close to 0 or to 1 we can reject the model, otherwise it is generating synthetic data that are similar to the original data. This is better visualized using a discrepancy plot, where for each draw k, a synthetic discrepancy is paired with its corresponding observed discrepancy. If the discrepancy points are roughly equally distributed about the $\mathcal {D}_\mathrm{obs}=\mathcal {D}_\mathrm{sim}$ line, then we cannot reject the model. As mentioned above, we expect that galaxies with the largest number of measurements are sampling more completely the ‘true’ distribution of the distance. Therefore, we need to find the minimum number of measurements per galaxy for which the Bayesian p-value shows an agreement between on the partitioned data set and the model predictions.


For the radio galaxies in our sample, we estimate the minimum jet power, Pj following Wójtowicz et al. (2020):
(4)$$\begin{eqnarray}
P_{\rm j} &\sim & 1.5 \times 10^{45} \times \left(\frac{{\rm LS}}{{\rm 100 \, pc}} \right)^{9/7} \left(\frac{\tau _{\rm j}}{{\rm 100 \, yr}} \right)^{-1} \\
&&\times \, \left(\frac{L_{\rm 5 \, GHz}}{10^{42} \,{\rm erg \, s^{-1}}} \right)^{4/7} {\rm erg\, s^{-1}} ,
\end{eqnarray}$$where LS is the linear size, τj is the source age, and L$_{\rm 5 \, GHz}$ is the luminosity at 5 GHz. In equation (4), we consider the linear size and the luminosity at 5 GHz reported in Tables A1–A3. We assume ages between about 100 yr, for the most compact sources, and 105 yr for sources with LS of several kpc, as derived from radiative and kinematic ages of sources (e.g. Murgia et al. 1999; Fanti & Fanti 2002; Murgia 2003; Polatidis & Conway 2003; Giroletti & Polatidis 2009). We end up with minimum jet powers for galaxies between 1040 and 1046 erg s−1. The higher values are obtained for sources at higher redshift, and for 3C 346, which is among the marginally detected sources from our analysis. However, the majority of the galaxies have L$_{\rm \, 5 GHz} \lt 10^{43}$ erg s−1 and estimated minimum jet power Pj  1044 erg s−1. As shown in Fig. 11, this requires a UV luminosity above 1045 erg s−1 in order to detect a cumulative signal by the stacking analysis. This is far from the expectation from Stawarz et al. (2008), in which the optimal conditions occurred for sources with jet power ∼1046 erg s−1, LS  100 pc, and at redshift 0.2 (∼1 Gpc). For our estimated jet power, the highest expected gamma-ray luminosity for sources with LS  100 pc is 1044 erg s−1. The fact that young radio galaxies are faint emitters of gamma rays is also suggested by the results of the stacking analysis, which set the upper limit to their emission, as a whole population, an order of magnitude below the Fermi-LAT threshold. This indicates that only the closest sources could be detected by Fermi-LAT, while if we consider objects at higher and higher redshift, boosting effects are necessary for their detection.