the position angle and distance modulus measurements with no further adjustment. In the Appendix, we discuss the validity of the approximation used in equation ( 8 ) by comparison to models which explicitly employ the posterior samples from Martin etal. ( 2016c ). We find that the posterior distributions on the model parameters are very similar when using the posterior samples compared to the approximation employed throughout the paper. Additionally, in the absence of similar sets of posterior samples for the other M31 dSphs and all of the MW dSphs, we are forced to employ an approximation. We now detail each of the models employed. The first model assumes the dSphs are randomly aligned with respect to the observer, so does not utilize the position angle information. The second model assumes the major axis of the dSphs are preferentially aligned with the galactocentric radial vector. Our final model assumes the major axes of the dSphs preferentially lie in a plane. 3.2 Random alignment We attempt to infer the distribution of T and E given the observed shapes and associated uncertainties under the assumption that there is no preferred orientation for the dSphs. Implicitly, this assumes the distribution of observed position angles is uniform. Our model is described as where is the modified normal distribution to account for the asymmetric uncertainties, and subscript i corresponds to the i th of N dSphs. is a uniform distribution between a and b and is a Cauchy distribution centred on a with scale parameter b . From dark-matter-only simulations, the triaxiality and ellipticity distribution are well approximated by Gaussians (Allgood etal. 2006 ; Knebe etal. 2008a ). The natural choice for the prior on a scale parameter would be a Jeffreys prior of 1/. Instead, we heed the advice of Gelman ( 2006 ) and adopt a half-Cauchy distribution as a weakly informative prior. Very large values of are disfavoured, whilst very small values are not strongly favoured. This matches nicely our expectation that the intrinsic triaxiality and ellipticity distributions will not be -function spikes nor will they lack any structure at all. The choice of the scale as unity is inspired by the finite range of both the triaxiality and the ellipticity. One complication is that some of our variables have a discrete range so the pdf that describe them must be normalized over this range. In the case of the uniformly distributed variables, the normalization is a constant factor but for the normally distributed variables (i.e. T i and E i ) the normalization depends on the parameters and that we are attempting to constrain. Therefore, any distributions f ( x ) for a variable x constrained to lie in the interval ( a, b ) that are quoted in this paper implicitly contain a factor of N 1 where We additionally introduce the pdf given by For clarity, the log-likelihood for this model is given by where p (cos, , T, E ) is a -function related to equation ( 2 ). 3.2.1 Results The

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O B-Model I-Model O O O O O O O O O B-Model B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O


into comoving space. The underlying halos are the same. Left panels: a slice of the HCN temperature cube, in transverse (on-sky) directions. Middle panels: slices of the HCN and CO temperature cubes, with one axis along the line of sight. Note the difference in comoving volume and location. Right panels: a slice of the CO temperature cube, again in transverse directions. 2.3. Power Spectra from Observations We work with the spherically averaged, comoving 3D power spectrum P ( k ) and the average of all 2D sky angular power spectra from each frequency channel in the data cube. We calculate P ( k ) by calculating the full 3D power spectrum of the temperature cube in comoving space, binning it in radial “shells” of , and averaging the full spectrum in each bin. We obtain in each frequency channel through similar binning of the 2D power spectrum in space. As Figure 2 shows, the comoving volume is not perfectly rectangular, but we approximate it as such for P ( k ) calculations here. As in Li et al. ( 2016 ) and Breysse et al. ( 2015 ), we present the spherical 3D power spectra as and the averaged 2D power spectra in the conventional form, using a flat-sky approximation as in Chiang Chen ( 2012 ) and Breysse et al. ( 2015 ). While the 3D analog to is just P ( k ) rather than the rescaled , and the 2D analog to is the suitably rescaled rather than just , we use these rather disparate presentations to follow the conventions used in previous literature. 3. Results 3.1. Simulated Temperature Cubes For 496 different pairs of CO and HCN light cones, we generate temperature cubes for CO emission and HCN emission, which we then add together in observed voxels to obtain a simulated map for CO emission plus HCN contamination. Approximately halos in the CO light cones and approximately halos in the HCN light cones typically fall above the cutoff mass of 10 10 and within a patch. Since we used only 100 light cones populated with CO emitters and 100 light cones populated with HCN emitters, there is some redundancy in the light cones used. But since temperature maps were generated anew for each new pairing, even two maps from the same light cone are subject to some differences due to reapplication of halo-to-halo scatter in SFR and luminosity. On top of the halo-to-halo scatter in each light cone, sample variance in large-scale structure between light cones results in light cone–to–light cone scatter in temperature and power spectra. Figure 2 shows slices of a sample pair of temperature data cubes for a given pair of light cones. Note that these slices are generated with the fiducial model (the turnaround model with 0.3 dex log scatter in luminosity). The power-law model maps appear much sparser than maps from any of the other models, especially when only of sufficiently massive halos host emitters. Table 3 lists the mean line temperatures over each simulated CO map and HCN map. Variations in do not appear to significantly affect , and any differences present

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


we cannot predict how the vapour production mass will scale with impact energy for very energetic giant impacts. Although the amount of mass in vapour is relatively independent of b and μ there is a strong dependence on these parameters when determining which body (target or projectile) the vapour comes from (Fig. 3 ). As expected for equal mass impacts, the vapour mass is produced equally from both the target and projectile, which is true for all impact angles (parameter b ). However, as μ decreases, the fraction of vapour production from the projectile is higher than that of the target. This behaviour is also sensitive to the impact angles as shown in Fig. 3 . Figure 3. The difference between the escaping vapour mass from the projectile normalized by projectile mass and the escaping vapour mass from the target normalized by the target mass versus the specific energy of the collision normalized by the catastrophic disruption criteria. Symbols and colour bar are same as in Fig. 2 . 3.2 Initial vapour velocity distribution Planetary embryo impacts produce a variety of post-collision dust distributions, none are isotropic as has been assumed in previous work (e.g. Jackson Wyatt 2012 ; Jackson et al. 2014 ; Wyatt et al. 2017 ). To be able to compare different distributions, we need to characterize the anisotropic nature of the collisionally generated dust. We separate the velocity of each escaping particle into two velocity components, the velocity parallel ( v ∥ ) and perpendicular ( v ⊥ ) to the direction of the impact. For example, the direction of the impact is the same (along the x -axis) for the two collisions shown in Fig. 1 . To understand and compare different distributions of particles, we choose to take the range between the 16 per cent and 84 per cent percentiles to compute the velocity dispersion for both velocity components in each collision (Δ v ∥ and Δ v ⊥ ). This allows us to compare the velocity distribution of different distributions without assuming a given distribution for any collision. We choose to take the 16 per cent and 84 per cent percentiles as this would be comparable to taking the 1σ values either side of the mean of a normal distribution. In Fig. 4 , we show the absolute velocity distributions of vapour condensates parallel (blue) and perpendicular (orange) to the direction of the collision and absolute velocity (black) for three different giant impacts. Fig. 4(a) shows sim. 168, a collision between 1.11 and 0.1 at 15 and b = 0.4. We choose to show sim. 168 as it is the closest simulated impact we have to the Moon-forming impact. Comparing to fig. 1 of Jackson Wyatt ( 2012 ), we have an overall similar distribution of absolute velocities and we find only a small difference between the direction vapour condensates are launched (close to isotropic). Fig. 4(b) shows sim. 8, a collision between two 0.1 at 10 and b = 0. Fig. 4(c) shows sim 88, a collision between two 0.1 at 10 and b = 0.8. There is a stark difference

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O


cut through the current sheet in turbulence, the parameters depend on time due to the turbulent interaction of larger scale magnetic eddies, so wide variation in parameters across different current sheets and at different times is expected. 7 DISCUSSION AND CONCLUSIONS As a result of the weak collisionality of many space and astrophysical plasmas, several physical processes can locally contribute to the dissipation of energy and the heating of the plasma. During magnetic reconnection, magnetic energy is efficiently converted to directed plasma flows, thermal energy, and energetic particles (Burch et al. 2016 ; Torbert et al. 2018 ). During turbulence in a magnetized plasma, the cascade provides the global amount of energy needed for fluctuations to be dissipated at small scales (Marino et al. 2008 ). However, which dissipative mechanisms are dominant and under which conditions is still not sufficiently understood (Vaivads et al. 2016 ). Therefore, a multiplicity of dissipation surrogates have been adopted in the literature to identify potential sites of dissipation. Owing to the strongly non-linear dynamics and the importance of physics at kinetic scales in such systems, numerical simulations of the Vlasov–Maxwell system (or, including collisions, the Boltzmann–Maxwell system) are the decisive tool to address the long-standing issue of plasma heating and energy dissipation in magnetized plasmas, e.g. Parashar et al. ( 2009 ). For example, the ‘turbulence dissipation challenge’ (Parashar et al. 2015 ) motivated numerical work to compare different algorithms on the solution of similar problems to assess the nature of the dissipation in magnetized turbulence (e.g. Pezzi et al. 2017 ; Perrone et al. 2018 ; González et al. 2019 ). In the same spirit, we have conducted a survey of a number of dissipation surrogates with three different kinetic plasma codes – the fully kinetic particle-in-cell vpic , the fully kinetic Eulerian Vlasov–Maxwell Gkeyll , and the Eulerian Hybrid Vlasov–Maxwell codes – to perform numerical simulations of two important physical phenomena in plasma physics. The first class investigates reconnection in an isolated current sheet, and the second concerns plasma turbulence at kinetic scales. We have calculated and compared eight distinct dissipation proxies, delineated in Section 2 . For the sake of clarity, we have categorized them in terms of (i) energy-based parameters, whose definition describes energy transfer and conversion; and (ii) VDF-based parameters, that are directly related to kinetic signatures in the particle VDF. Energy-based parameters considered here are the power density by electromagnetic fields on charged particles (Zenitani et al. 2011 ), the pressure–strain interaction (Yang et al. 2017a ), and a local proxy of the turbulent energy transfer (Sorriso-Valvo et al. 2018a ). The VDF-based parameters are the local pressure agyrotropy (Swisdak 2016 ) and three measures describing how different a local

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O B-Software I-Software I-Software O O O B-Software I-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation O O O O O O O O O


be generated at levels within the cloud deck, where the background continuum emission originates. It must be generated well above this layer, where the absorbing gas is cooler and therefore absorbs more efficiently than it emits. The narrow width of the absorption line also suggests that it was formed at pressures substantially less than those of the cloud top (70 km, ∼3000 Pa). Figure 4. Radiance mixing ratio Jacobians ( dI ν / dr mix ) as a function of layer pressure for the radiance streams at 21 degrees zenith angle, for the 266.94 GHz feature for SO 2 (left) or PH 3 (right). Absorption line strengths (lower panels in each case) are given in units of cm −1 /(molecule cm −2 ). The continuum originates from SO 2 at ∼6 × 10 4 Pa (∼54 km, within the cloud deck), while the line cores for either species do not originate in the clouds (48–70 km) but at over 400 Pa (over 80 km) in the mesosphere. In the right panels in both plots, the temperature structure is given as a black line, while the colored lines denote SO 2 (green) or PH 3 (purple) mixing ratios. On the right, the evenly mixed 20 ppb PH 3 profile is shown with a solid line and the photochemical PH 3 profile is shown with a dashed line. 3.3. ALMA Line Dilution While the nondetection of prominent SO 2 spectral features in the ALMA wideband data could indicate a low abundance, as argued by Greaves et al. ( 2020a ), the estimation of this abundance was done without correcting for line dilution as a result of the ALMA observing geometry (Greaves et al. 2020a ). Significant line dilution is likely, especially considering the global distribution of SO 2 in the Venus atmosphere, and the exclusion of the short baseline ALMA measurements. Greaves et al. ( 2020a ) estimated line dilution (filtering losses) of 60%–92% depending on position on the disk. To determine the disk-averaged line dilution for the SO 2 reference line search, we simulated observations of Venus using the ALMA configuration of Greaves et al. ( 2020a ) by imposing an appropriate resolution spectrum (0.00003 cm −1 , 1 MHz) of our Case D atmospheric model over a limb-darkened disk model. The Fourier Transform of this model was resampled to match the ALMA configuration and reimaged using the imaging routines of Greaves et al. ( 2020a ), as provided in their Supplementary Software 3. As shown in Figure 5 , line dilutions on the order of 95% at the line core are observed for the full disk. We observe similar dilutions when the spectrum is only imposed on one hemisphere (∼8′′ extent at the time of observation). This line dilution suggests that the SO 2 reference features produced by our best-fit SO 2 distribution (Case D) would be heavily suppressed by line dilution in the ALMA data, which would cause them to mimic smaller features below the ripple detection limit of −0.0006. Figure 5. Modeled line dilution for the ALMA observations: disk-averaged line/continuum ratios for our nominal Case D atmosphere model containing SO 2

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O B-Observatory O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Observatory O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O B-Observatory O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O B-Model I-Model I-Model O O O O O O O O O O O B-Observatory O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O B-Model I-Model I-Model I-Model O O O


our best guess associates the sunquake location (taken to be the 5 mHz signal in Figure 1 ) to the connectivity domain indicated as Q 4 in Figure 12 . That connectivity domain corresponds to the domain 4 in Figure 13 , i.e., to the envelope bundle of field lines (in green in Figure 14 ). Given its structure, such a field line bundle has two interesting properties. First, since it is surrounding the flux rope in almost its entire length it is then directly impacted when the eruption occurs. Second, such wide-spread field lines between the erupting flux rope and the above layered structure are all connected on one side to a very small area in the photosphere (where the sunquake occurs), in this way, providing a sort of magnetic focusing. In light of this, we argue that the sunquake is a consequence of the energy release associated with the reconnection between the erupting flux rope and the envelope surrounding it. Furthermore, the envelope structure helps focus the released energy into a small area in the lower atmosphere ( Q 4 in Figure 12 ), which produces the sunquake pulse. The scenario offered by our analysis and the interpretation of the observations clarifies the dynamics of the two flares and supports the association of the sunquake to the larger, M-class flare, rather than to the first C-class flare. Moreover, it indicates how a sunquake may require a special magnetic field configuration to occur, which may help understand their erratic occurrence. 6. Conclusion In this work, we study the evolution of active region 11675 on 2013 February 17, in the hours leading to the production of two flares (a C-class flare and an M-class flare), the ejection of filament material and a sunquake. We investigate the configuration of the active region magnetic field through the construction of an NLFFF extrapolation just over one hour before the sunquake. The aim of the work is to shed light on the configuration of the coronal magnetic field in which the sunquake was produced. The sunquake occurs when active region 11675 is undergoing a period of rapid evolution during its emergence phase, and when there are strong motions of the photospheric magnetic field elements. In particular, two emerging bipoles collide and shear past each other. The magnetic field along the polarity inversion line that separates these bipoles is highly sheared and at low altitudes bald patch field lines are present. At higher altitudes, an HFT and a flux rope are present, which have been highly modified by the motions of the photospheric field in which these magnetic structures are rooted. From a time–distance analysis, we determine that the sunquake was initiated in the time period between 15:50:41 to 15:51:26 UT on 2013 February 17 in a region of positive polarity field. This finding indicates that the sunquake is initiated during the M-class flare, which is associated with the eruption of the flux rope and ejection of filament material. However, this timing determination

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


more spheroidal, as in the Milky Way halo, rather than being represented by a disc-like structure. One would therefore expect intermediate-aged populations traced, for instance, by RC stars to be somewhere between those. This is exactly what we found. Subramanian Subramaniam ( 2012 ) also used RC stars but found a very small inclination, i = 0.5°. Their methodology however was to use axes ratios and to limit the sample to a small depth range, which may have inadvertently introduced a bias. The PAs we found are nearest to those determined by Subramanian Subramaniam ( 2015 ), who used Cepheids, and Rubele et al. ( 2015 ), who fitted entire CMDs and found a mean i = 39°. In conclusion, while the SMC does not exhibit an obvious disc component there is a clear trend for the system to become more inclined with respect to the plane of the sky the younger the (stellar) tracer is. Since the RC stars already show this progression this means that the dynamical events that led to this time-dependent morphology must have occurred on Gyr time-scales. Indeed, while most of the stellar mass in the SMC was generated >4 Gyr ago (Sabbi et al. 2009 ; Rezaeikh et al. 2014 ; Rubele et al. 2018 ) there is evidence in the SFH of the SMC of a strong interaction with the LMC to have occurred ∼2 Gyr ago (Harris Zaritsky 2004 ; Noël et al. 2009 ; Rezaeikh et al. 2014 ; Rubele et al. 2015 ). A more recent interaction ∼0.2 Gyr ago coincident with the closest approach to the Milky Way, also seen in the SFH (Harris Zaritsky 2004 ; Noël et al. 2009 ; Sabbi et al. 2009 ; Rubele et al. 2015 ), may have caused further distortion of the gaseous disc, from which the younger population traced by the Cepheids formed. 6.3 Distortions and other structures The outer regions of the SMC have been transformed by interactions leading to stripping and spatially stretched populations. Dias et al. ( 2016 ) described the SMC as being composed of three bodies: the eastern Wing, the central main body, and the western halo. We find the intermediate-age main body traced by RC stars to be structurally simple and symmetrical, consistent with a spheroidal component. The eastern Wing, on the other hand, is split into two distinct populations (both traced by intermediate-age RC stars), of which the brighter one clearly lies in front of the main body (Fig. 9 , bottom right). We have been able to comprehensively trace these two components across the VMC footprint, allowing us to relate them to what other observational and theoretical studies have found. We find a fourth structure, in the north (Fig. 9 , top left); this has previously been described as an extension to the Wing (e.g. Gardiner Hawkins 1991 ), but Sun et al. ( 2018 ) showed this area to contain a shell of young stars which is clearly separated from the general Wing area. The intermediate-age population of RC stars in the north need not be related to these young stars, though. We compare our results with previous analyses of Cepheids, RR Lyræ,

O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Formula I-Formula I-Formula O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject O O O O O O B-CelestialObject O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-CelestialObject O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O B-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag I-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Wavelength I-Wavelength O O O B-Telescope O O O O O O O O O O B-Model I-Model O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-TextGarbage O O O


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of the cluster will change with the cluster properties. For a massive cluster or for a compact cluster the binding energy will be higher and hence the formation of the SMS would be more favourable. On the 5 Myr time-scale explored in our simulations, supernova feedback is also expected to become relevant. With the typical energy of 10 51 erg for a core-collapse supernova, it is clear that one such event will expel the gas and terminate the accretion, if it has not stopped already (either due to gas expulsion by stellar winds or as the accretion process may have depleted the gas). In future work, it will be important to study detailed gas dynamics where the kinetic energy deposition of winds as well as the supernova feedback is taken into account. 5 SUMMARY AND DISCUSSION In this work, we explored the effect of mass-loss due to stellar winds on the final mass of the SMSs, which could be formed via runaway stellar collisions and gas accretion inside NSCs. We find that an SMS of mass could be formed even in a high-metallicity environment for high accretion rates of . For an accretion rate of , the final mass of the SMS for . Whereas for solar metallicity, no SMS can be formed for f Edd = 0.9 and SMSs of masses can be formed for f Edd = 0.7 and 0.5, respectively. For the case of Eddington accretion it will not be possible to form an SMS in the metallicity regime . Finally, for the Bondi–Hoyle accretion scenario, we find that the formation of a SMS will not be possible in the high-metallicity regime of . The interaction of the stellar wind and the gas inside the cluster might play an important role in the evolution of the SMS. The winds from the SMSs have high velocities ∼10 3 km s −1 (Muijres et al. 2012 ), which might exceed the escape velocity from the centre of our modelled star cluster. The SMS in the cluster is close to the centre which results in a high collision rate near the centre due to a shorter relaxation time in the core and an increased collisional cross-section. If the SMS is displaced by collisions, it rapidly sinks back close to the centre via dynamical friction where it may eventually decouple from the remainder of the cluster. This is also known as the Spitzer instability (Spitzer 1969 ). Interestingly, Krause et al. ( 2016 ) have found that for a Salpeter -type mass function the stellar wind cannot remove the gas inside the cluster. Hence, we do not expect the stellar wind to remove gas from the cluster. One of the main caveats of this work is the neglect of the kinetic energy deposition through stellar winds, which could contribute significantly to expel the gas. The latter is likely to create a regime where the growth of a massive object is still inhibited, even though the mass-loss itself from the winds is no longer significant. Below a critical metallicity in the range (, this effect is no longer expected to be relevant; however, supernova feedback may lead to the expulsion of the gas. Another relevant caveat is the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


astrometry between the initial image and each of the later sets using Astroalign . To uncover or place limits on transient emission on these timescales, we perform digital image subtraction using the HOTPANTS software package (Becker 2015 ). If an optical or NIR source is detected within the GRB localization regions (Evans et al. 2009 ; Lien et al. 2016 ), we perform aperture photometry on the residual images to determine the source brightness using the IRAF/ phot package. We determine zero-points by using point sources in common with the SDSS Data Release 12 (Alam et al. 2015 ), Pan-STARRS DR1 and DR2 (Chambers et al. 2016 ), and 2MASS (Skrutskie et al. 2006 ) or USNO-B1 (Monet et al. 2003 ) catalogs. If there are no significant residuals at or near the localization region, we obtain an upper limit from sources detected at 3 σ significance in the field. As we obtain the limit from the initial image, our values do not include additional noise from the subtraction. In total, we present 52 unpublished upper limits and three unpublished detections of 23 bursts, 10 with known redshifts (Table A1 ). 2.3. Observations from GCNs We search the GCNs of all short GRBs detected between 2013 and 2020 for constraining upper limits to ensure that our catalog covers all Swift short GRBs to date (2005–2013 is covered by the literature sample in Section 2.4 ). Considering that the range of peak kilonova luminosities based on models is ν L ν ≈ (1−5) × 10 42 erg s −1 (∼10 times the peak luminosity of AT 2017gfo; see Figure 1 ), we only include data that could feasibly fall into the range of kilonova luminosities, or fainter than a given depth, corresponding to griz ≳ 22.5 mag and JHK ≳ 21.0 mag (AB system) at z ≈ 0.25. We choose z ≈ 0.25, as 80% of known redshifts in our sample fall above this cut. Observations significantly brighter than this are not included, as they are likely dominated by afterglow emission. In total, we collect deep optical and NIR limits of 18 bursts, seven with confirmed redshifts, from the GCNs (Table A1 ). Figure 1. Light curves of AT 2017gfo and four diverse kilonova models considered in this work in the griJ bands. Light curves are shown in ν L ν (left y -axis) and absolute magnitude (right y -axis) units, and rest-frame time in days. In each panel, we show the interpolated, smoothed AT 2017gfo data set (solid dark-blue line; Villar et al. 2017 ), an NSBH kilonova model (blue dotted line; Kawaguchi et al. 2020b ), an infinite-lifetime NS remnant kilonova model (red dashed line; Kasen et al. 2015 ), an extreme optimistic magnetar-boosted kilonova model (turquoise solid line; Fong et al. 2021 ), and a model for an optimistic scenario motivated by GW190425 (magenta dashed–dotted lines; Barbieri et al. 2020 ). Black and open triangles denote upper limits of short GRBs with known and unknown redshifts, respectively, in the respective rest-frame bands at δt rest = 0.1–10 days. The most constraining limits of bursts with known redshifts

O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software B-Citation I-Citation O O O B-Wavelength O B-Wavelength O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Software B-Software O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O B-Dataset I-Dataset O B-Dataset B-Citation I-Citation I-Citation I-Citation O O B-Survey B-Citation I-Citation I-Citation I-Citation O O B-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O B-Model I-Model O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Model I-Model I-Model O O O B-Citation I-Citation I-Citation I-Citation O O B-Model I-Model I-Model I-Model I-Model O O O B-Citation I-Citation I-Citation I-Citation O O B-Model I-Model I-Model I-Model I-Model O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O


the exact influence of radiation pressure on nitrogen atoms would require a dynamical model, such a low value for β indicates that it should have a negligible effect. Despite these very weak effects from radiation pressure, we do not find an overabundance of nitrogen in the β Pictoris disc. Other papers have also shown no clear deficiency in radiation-sensitive species such as Na and Ca ( Lagrange et al. 1998 ), which also seems to disfavour the preferential depletion scenario. The preferential production scenario, also proposed by Xie et al. (2013) , suggests that the overabundance is the result of gas being produced from materials rich in C and O. This includes photodesorption from C and O-rich icy grains ( Grigorieva et al. 2007 ) and collisional vaporisation of the dust in the disc ( Czechowski Mann 2007 ). Unlike preferential depletion, this scenario does not infer high N abundances and is thus more compatible with the near-solar NI abundance measured here. 5.2 Origin and dynamics of the NI gas The absorption components that we used to model the shape of the NI absorption lines may provide us with hints at both the origin and dynamics of the NI gas. The two absorbing circumstellar components, CS 0 and CS X , were not constrained to particular radial velocity values, but provided the best fit when their radial velocities matched that of the star. This indicates an absence of detectable amounts of NI falling in towards the star. The clearly asymmetric HI Ly- α line presented in Wilson et al. (2017) , on the other hand, indicates that HI gas is falling in towards the star. The HI gas is thought to be originating from the dissociation of water molecules from evaporating exocomets. The fact that we do not see similar asymmetric line profiles in NI is perhaps not surprising. In solar system comets, nitrogen-bearing molecules are only minor constituents ( Eberhardt et al. 1987 ), which means that if β Pic comets are similar, we would not expect the NI lines to be asymmetric. This does not exclude the possibility that NI might originate from exocomets, however, as we argue in Sect. 5.3 . 5.3 Role of exocomets The NI column density measurements are dominated by the narrow absorption feature, CS 0 , which has a line width of km s −1 . With a similar line width observed in FeI ( Vidal-Madjar et al. 2017 ), this absorption feature likely traces a stable band of circumstellar material that orbits β Pic. The origin of the broad absorption feature, CS X , with km s −1 , which absorbs seven times less strongly, but clearly governs much of the absorption line shape, is less clear. The feature might consist of a number of individual absorption components that are beyond the resolving powers of the COS instrument. The broad component could be related to NI absorption lines in the stellar atmosphere of β Pic. This would certainly explain that the bulk velocity matches that of the star. However, we would then expect the lines to be much broader and shallower

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coupling, resonance, and weak inner planet coupling. We elaborate on these in the following subsections. Figure 2. RMS values of the eccentricities of the planets in a fiducial two-planet system under the influence of an eccentric perturber. The perturber has initial eccentricity e p = 0.1 and has an orbit co-planar with the inner system. The other parameters are the same as Fig. 1 . The solid coloured points represent the results of numerical integrations using secular equations, while the solid coloured curves are calculated using linearized theory; the dashed coloured lines show the forced eccentricity (equation 26 ). The dashed vertical line corresponds to ε 12 = 1. 2.3.1 Resonance When ω 1 ≃ ω 2 (note that this is approximately equivalent to the condition ε 12 ∼ 1), a potential resonance feature arises where large eccentricities can be excited in the inner planets, even for small e p . If we take ω 1 = ω 2 , equations ( 34 ) and ( 35 ) becomes We see that the eccentricity of planet 1 is boosted while the eccentricity of planet 2 is dampened near the resonance. The resonance feature is most pronounced when L 2 ≫ L 1 . To illustrate this, let L 1 = 0, then the forced eccentricity on the inner planet becomes If both the inner planets have zero initial eccentricity, then their eccentricity evolution is given by In this limiting case, according to the linear theory, at ε 12 ≃ 1 the eccentricity of the inner planet can become arbitrarily large, even for small initial values of e p . In reality, the linear theory breaks down as e 1 becomes too large, and higher order terms will keep e 1 to a modest value. An illustration of the resonance behaviour can be seen in Fig. 2 . Systems with larger ratios of m 2 / m 1 tend to exhibit pronounced resonance features, whereas for systems with more comparable masses, the feature is notably reduced. 2.3.2 Strongly and weakly coupled regime In the case where the mutual precession rates of the inner planets dominate over the influence of the perturber (i.e. ω 12 ≫ ω 1 p and ω 21 ≫ ω 2 p ), the general expressions ( 34 ) and ( 35 ) can be significantly simplified; we refer to this as the strongly coupled regime. In this regime, the two planets attain very similar forced eccentricities (), and as a result equations ( 34 ) and ( 35 ) are dominated by their first terms. Explicitly, in this regime the RMS eccentricities are approximately given by where α 12 = ( a 1 / a 2 ) and β 12 = β( a 1 / a 2 ) (see equation 8 ). In the other limiting case (i.e. weak coupling), when the precession rates of the inner planets driven by the perturber dominate the their mutual precession rates (i.e. ν 1 p ≫ ω 12 and ν 2 p ≫ ω 21 ), the terms of order ν 12 /ω 1 p and ν 21 /ω 2 p in equations ( 34 ) and ( 35 ) can be dropped, and the final inner planets’ RMS eccentricities are again given by . In this case, the planets precess independently of one another, and their eccentricities are given by equations ( 11 ) and ( 12 ), i.e. Note

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


described by a modified lognormal distribution (equation A7), whose parameters are given in Table 3 . The density PDF converges after the jet breaks out (jet head ≳ 3 kpc) and decouples from the ISM, proceeding along the cleared path. Figure 7. The evolution of the density PDF as the jet evolves with time and breaks out of the dense central region after a height of z ∼ 3 kpc. The density PDF converges after the jet breakout as the jet decouples from the ISM. Table 3. Coefficients of the fit to the density PDF in Fig. 7 following equation (A7). a Mean density for the PDF. b Standard deviation of variable ρ, which is related to the standard deviation in s = ln ρ as in equation (A8). c Volume-weighted mean of density for n > 10 cm −3 . This gives a measure of the mean density of the highdensity filaments. The duration of the flood-channel phase of the jet evolution depends on the mean density and filling factor of the ISM, as previously discussed in Wagner et al. ( 2012 ). Clouds with denser cores are less ablated and more efficiently impede the progress of the jet. In Fig. 8 , we present the density for Sim B (Table 2), where a jet of power 10 45 erg s −1 passes through a medium with a lower mean density ( n w 0 = 150 cm −3 ) compared to that of Sim A. We note that the jet breaks out of the warm dense gas much faster, in comparison to Sim A. Figure 8. Density {log [ n (cm −3 )]}in the x – z plane for Sim B. 4.2 Evolution of the energy bubble The confined jet creates a nearly homogeneous high-pressured energy bubble that sweeps through the ISM (see Fig. 6 ). The expanding bubble engulfs the dense filaments, driving shocks into them. The shocks become radiative, forming a skin of heated gas (∼10 5 K) surrounding the warm filaments at ∼10 4 K. The energy bubble also shears the filaments sweeping up the ablated gas to high velocities ≳1000 km s −1 . The dense filaments are themselves accelerated by the ram pressure from the bubble. However, the efficiency of acceleration depends on the density of the gas and injected mechanical power. For lower density (Sim B), the dense filaments are accelerated to speeds of ∼100 km s −1 , similar to earlier reported results of Wagner et al. ( 2012 ). In simulations with higher ISM density (Sim A) or lower jet power (Sim C and Sim D), the outward pressure from the bubble is not sufficient to overcome the inward gravitational acceleration. In Figs 10 and 11 , we show the evolution of the mean pressure inside the energy bubble and the volume of the bubble. Initially, the bubble pressure decreases more rapidly ( p ∼ t −1 ) than an adiabatic spherical bubble (with p ∼ t −0.8 ). This is likely the result of mixing with the ISM and cooling losses. Consistent with the lower pressure, the bubble expands less rapidly (volume ∼ t 1.5 ) than an adiabatic bubble (volume ∼ t 1.8 , see Appendix B). At later stages after the breakout from the confining dense ISM, the bubble expands more rapidly. 4.3 Evolution of the phase space

O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O


the analytical solution without collisional damping is in good agreement with the results of the simulations. The timescale of dynamical friction is estimated from Equations ( 14 ) and ( 15 ). If and ≫ for m p,1 ≫ m p,2 , the damping timescale due to dynamical friction has the relation as (Ohtsuki et al. 2002 )where M disk is the total mass of a planetesimal disk, is the width of the disk, and is the Hill radius for protoplanets. Note that P DF is estimated to be ∼10 for m p,1 ∼ M ⊕ and ≈ ∼ 0.1. This estimate implies a planetesimal disk with M disk ≪ M ⊕ leads to eccentricity damping for Earth-sized protoplanets in a long timescale ≫10 4 yr via dynamical friction. In addition, dynamical friction increases of planetesimals. The increase in makes τ df longer. On the other hand, the decreasing timescale for M disk due to the collisional cascade, τ cc , is estimated from Equation ( 5 ) to be (Kobayashi Tanaka 2010 )For 100 km sized planetesimals ( m c ≈ 10 22 g) with ∼ 0.1, τ cc ≲ τ df . It should be noted that dynamical friction increases during the damping, which shortens τ cc and elongates τ df as discussed above. Therefore, even if τ cc ∼ τ df initially, τ cc eventually becomes much shorter than τ df . Equations ( 17 ) and ( 18 ) show τ df / τ cc is independent of M disk and Δ a but depends on , , and m c . For = = 0.03, P DF ≈ 60 so that τ cc ≪ τ df for g, while τ df ∼ τ cc for g. Therefore, as shown in Figures 4 and 5 , e and i of protoplanets insignificantly decrease for m c = 10 16 g because of τ cc ≪ τ df , while those are moderately damped for τ cc ∼ τ df with m c = 10 19 g. Therefore, the condition with t cc ≫ τ df is required for the e damping of protoplanets. Planets formed via giant impacts have high eccentricities and inclinations. For Earth-sized planets formed in the giant impact stages, the mean eccentricity is ∼0.1 (Chambers 2001 ; Kokubo et al. 2006 ), which is much larger than the current eccentricities of Earth and Venus. The depletion timescale of a planetesimal disk with is shorter than the e damping timescale via dynamical friction (see Equations ( 17 ) and ( 18 )). Therefore, larger planetesimals are required for e damping of Earth-sized planets. We consider such large planetesimals as being produced from a giant impact. The mass ratios of the largest giant impact ejecta to parent protoplanets are ∼0.01 (Genda et al. 2015 ). Therefore, we consider g, resulting in τ cc ≫ τ df . We perform the simulation for the orbital evolution of a high-eccentricity planet with mass M ⊕ in a swarm of giant impact ejecta. Giant impact ejecta initially have similar orbits to parent planets. However, the orbital phases of giant impact ejecta, π and Ω, are eventually distributed uniformly due to perturbation from other planets. The timescale to achieve a uniform distribution for π and Ω, which is roughly given by the precession rates for π and Ω obtained from the secular perturbation theory (Murray Dermott 1999 ), is estimated as ∼10 5

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O


BCGs at 0.5 z 1.0 to derive L TIR and the associated dust-obscured SFR, dust temperature and mass, but neither attempt to mitigate confusion noise in these bands, nor match the observed far-infrared and 24 m emission in their SED template fitting (which are not simultaneously fit for significant number of their sources).] Furthermore, the results of the template fitting show that a customized combination of components provides a better description of the BCG stacked fluxes in most instances than any single template from a pre-existing library, particularly at z 1. It is clear that an elliptical template alone is not sufficient to match any of the SEDs, as the far-infrared emission of the BCGs is one to two orders of magnitude brighter than that of a typical red and dead elliptical galaxy template. As for the star-forming fit, based on the dust scaling relations empirically derived from observations of Herschel Reference Survey galaxies of a large range of spectral type, environment and stellar mass in the local Universe (Cortese etal. 2012 ) specifically the inverse proportionality between dust-to-stellar mass ratio and the stellar mass and specific star formation rate (sSFR) in local galaxies we expect the BCG SED to exhibit a significantly lower dust-to-stellar mass ratio in comparison to a less-massive non-BCG star-forming template, which is exactly what we find. At the lowest redshifts we consider, z avg 0.185, a typical star-forming SED matches the BCG SED longward of 10m, but the BCG stellar bump rises significantly above that of the star-forming template. While these dust scaling relations were derived for local field and cluster galaxies (including those in cluster cores), we do not yet know if they apply to galaxies at higher redshifts (though we might expect them to if we assume an understanding of the physics underlying the trends). Whereas massive elliptical galaxies in the local Universe exhibit relatively low dust-to-stellar mass ratios on account of a diminished level of star formation activity with respect to local star-forming late-type galaxies, we would expect an increasing dust-to-stellar mass ratio for elliptical galaxies of a fixed stellar mass with increasing sSFR towards higher redshift, as observed in our sample (reminding the reader that we select BCGs from clusters of the same mass at every redshift, therefore the BCGs are of the same mass at every redshift based on the linear scaling of BCG stellar mass and host-cluster mass, e.g. Lidman etal. 2012 ). In fact, as we move towards higher redshift, we observe the star-forming templates begin to match the full extent of the BCG SED better and better, as opposed to just the far-IR bump at lower redshifts. Therefore, we fit the stellar emission shortward of 10m in the faint SEDs with only the Polletta 13-Gyr Elliptical model, but add to this model either of the Polletta M82 or S18 templates to fit the stellar emission in the bright SEDs. In the mid- and far-infrared

O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey I-Survey O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O B-Wavelength O O O O O O O B-Model I-Model I-Model O O O O O O O O O B-Model B-CelestialObject O B-CelestialObject O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength


i.e., α 0 = −0.5 and α a = 0. In Figure 3 , we show T 21 , as defined in Equation ( 2 ), for various values of α 0 and α a , with other cosmological parameters fixed at values consistent with a Planck 2015 cosmology (Planck Collaboration et al. 2016 ). As shown, the ΛCDM model ( α 0 = α a = 0), denoted by the star, is in tension with the EDGES measurement at a 99% CL illustrated by the hatched region. However, interacting vacuum models can in principle release the tension, namely, T 21 can be pushed into the hatched region by a large range of the α 0 and α a parameters. Figure 3. Illustration of the intensity of the 21 cm signal relative to the CMB temperature, T 21 (K), for various values of α 0 and α a . The color bar indicates the values of α a . The black solid curve corresponds to the case in which α does not evolve with time ( α a = 0). The intersect between the black solid curve and the vertical gray dashed line, marked by a yellow star, denotes the ΛCDM model. The hatched region illustrates the observed 21 cm signal from EDGES at a 99% CL. We then perform a Monte Carlo Markov Chain global fit for the parameters in Equation ( 12 ) using a modified version of CosmoMC 8 8 Available at https://cosmologist.info/cosmomc/ . (Lewis Bridle 2002 ) with a combined data set including, 1. the angular power spectra of temperature and polarization measurements of CMB from the Planck mission (Planck Collaboration et al. 2016 ); 2. the Joint Light-curve Analysis sample of supernovae (SNe) measurements (Betoule et al. 2014 ); 3. the Baryonic Acoustic Oscillations (BAO) distance measurements from 6dFGS (Beutler et al. 2011 ), Sloan Digital Sky Survey (SDSS) DR7 Main Galaxy Sample (Ross et al. 2015 ), Ly α forest of BOSS DR11 quasars (Font-Ribera et al. 2014 ; Delubac et al. 2015 ), and BOSS DR12 with tomographic information (Wang et al. 2017 ; Zhao et al. 2017 ); joint BAO and Redshift Space Distortions (RSD) measurements from WiggleZ (Blake et al. 2012 ) and from eBOSS DR14 (Zhao et al. 2019 ); and RSD measurements from 6dFGS (Beutler et al. 2012 ), 2dFGRS (Percival et al. 2004 ), SDSS LRG (Samushia et al. 2012 ), and VIPERS (de la Torre et al. 2013 ); 4. and the local H 0 measurement using the Cepheids, i.e., H 0 = 73.24 ± 1.74 km s −1 Mpc −1 (Riess et al. 2016 ). The results are summarized in Table 1 and in Figures 4 and 5 . To quantify the constraint on α 0 and α a from the EDEGS measurement, we show the 68%, 95%, and 99% CL contours of α 0 and α a using EDEGS alone in the lower left part of Figure 4 . 9 9 As EDEGS alone cannot constrain all of the parameters in Equation ( 12 ) simultaneously, we only vary α 0 and α a in this case while the other parameters are fixed to the values derived from the Planck 2015 measurement. As illustrated, the EDGES measurement favors an interacting vacuum model over the ΛCDM model at more than 99% CL, and this would induce a quintessence-like effective dark energy as shown in the lower left of Figure 1 . We then

O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Model I-Model O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-Wavelength I-Wavelength O O B-Collaboration O O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O B-Software O O O O B-URL O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Collaboration I-Collaboration B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Survey B-Citation I-Citation I-Citation I-Citation O B-Survey I-Survey I-Survey I-Survey B-Survey B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O B-Wavelength I-Wavelength O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Survey B-Citation I-Citation I-Citation I-Citation O O O B-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O B-Survey B-Citation I-Citation I-Citation I-Citation O B-Survey B-Citation I-Citation I-Citation I-Citation O B-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O B-Survey B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O B-Collaboration O O O O O O O O O O O O O B-Collaboration O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O B-Dataset I-Dataset O O B-Model I-Model I-Model O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O


( 2010 ), 8 8 The quantity f end is set to avoid higher PN tidal effects and nonlinear hydrodynamical coupling. In Hinderer et al. ( 2010 ), the fitting formula was obtained by choosing 400 Hz ≥ f end ≥ 500 Hz. The end frequency adopted in Abbott et al. ( 2017c ) is the frequency of the innermost stable circular orbit, which is above 1 kHz. and an equal mass binary with M ∼ 2.7 M ⊙ , we obtain an estimate on the tidal deformability of a star: ΔΛ 1 (1.35 M ⊙ ) ∼ 2.9 × 10 3 . Assuming a low-spin prior, GW170817 places on upper bound Λ 1 (1.4 M ⊙ ) ≤ 800 at the 90% confidence level, and an upper bound of ≤1400 if the prior is relaxed to allow for high spin (Abbott et al. 2017c ). Such constraints on Λ 1 also limit the allowed radius of NS, given the family of EOS assumed in this paper. In Figure 3 , we compare this uncertainty in the tidal deformability to the amount by the tidal deformability that has to be changed in order for a BH–NS binary to have the same leading order tidal effects as a corresponding NS–NS binary with the same masses. Figure 3. Amount by which the dimensionless tidal deformability Λ has to be increased, relative to the quantity of the corresponding NS in an NS–NS binary with the same masses, for a BH–NS binary to have the same leading order tidal effects. This is shown as a function of the mass ratio. In all of the cases, one NS is assumed have a mass of M NS1 = 1.4 M ⊙ . The different curves correspond to different choices of EOS for the NS–NS binary, and hence the different values of the radius of 1.4 M ⊙ NS in the NS–NS, as shown in the legend. For comparison, we also show the Fupper bound constraints on Λ from GW170817 assuming either a high-spin prior or a low-spin prior on the source. The highest NS radius shown (dotted red curve) saturates the former constraint on Λ, while the middle value for the NS radius (dotted blue curve) saturates the latter constraint. With N identical detections, and under the same high-spin prior, such uncertainty scales as Δ N Λ 1 (1.4 M ⊙ ) ∼ 1.4 × 10 3 N −1/2 . In reality, the component masses and source distances are different for different events, and it is possible that the best event of previous N detections dominates the constraint of NS EOS. In the above discussion, we have not accounted for the effect of spin, which is particularly important for LMBHs formed through hierarchical mergers, as they are expected to have relatively high spin ( a ∼ 0.7; Fishbach et al. 2017 ; Gerosa Berti 2017 ). 9 9 The effective spin of a most rapidly spinning pulsar known so far is ∼0.4 (Hessels et al. 2006 ). Such a spin magnitude will generate ∼0.15 mismatch between a nonspinning BH–NS waveform and a generic pretcessing waveform if the mass ratio is around 2 (see Figure 8 of Harry et al. 2014 ). The relation between distinguishable mismatch and S/N is discussed in Baird et al. ( 2013 ). 5. Multi-messenger Detection An important question is whether multi-messenger signals can help us identify an

B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-CelestialObject O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-EntityOfFutureInterest O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O


L5 dwarfs includes 10 L4.5, 24 L5, and 2 L5.5 sources with seven detections of H in emission, and three classified as variable ( Schmidt et al. 2007 , 2015 ). In their sample, Pineda et al. (2016) included three L5 with H in emission in one of them. They inferred a fraction of emitters of 20% (15–35%; 68% confidence limit), consistent with the interval from the SDSS sample ( Schmidt et al. 2007 , 2015 ) although to date there are few sources. We also note that a large number of L dwarfs show variability in their level of chromospheric activity ( Liebert et al. 2003 ; Reiners Basri 2008 ; Schmidt et al. 2015 ; Pineda et al. 2016 ), the most striking case being the L5.5eT7 binary 2MASSI J1315309264951AB (L5.5; Hall 2002a ; Gizis 2002 ; Burgasser et al. 2011 , 2013 , 2015 ). 4.3. Radial velocity and space motion We measured the position of the H line in the 1D spectrum with the splot task under IRAF. We identified the line at 6564.11 Å with an error of 0.04 Å, while the nominal air wavelength from the NIST Atomic Spectra Database is 6562.819 Å. We should add in quadrature the error on our wavelength calibration (2.3 km s, see Sect. 3.2 ) and uncertainty due to the timespan in our observations (0.2 km s). We inferred a radial velocity of 38.0 2.9 km s (Table 2 ) after correcting for the rotation of the Earth, the motion of the Earth around the Earth-Moon barycentre, the orbit of the barycentre about the Sun, and motion of the Sun relative to the specified standard of rest. Our value is in close agreement with the mean values derived independently from the ground (39.1 0.2 km s; Detweiler et al. 1984 ) and from space with H ipparcos (39.48 0.30 km s; de Bruijne et al. 2001 ). We infer a space motion of [] = [42.92, 23.96, 0.72] km s for 2M041821 (Table 2 ), in close agreement with the space motion of the cluster centre: [41.1, 19.2, 1.4] km s ( van Leeuwen 2009 ). According to Vizier, 2M041821 should have a Gaia -band of 19.096 and should be included in the Initial Gaia Source List ( Smart Nicastro 2014 ), implying that precise coordinates and proper motions will be soon available since the second Gaia data release is expected at the end of 2017. However, based on the equation 0.244 SpT 12.6332 kindly provided by R. Smart, 2M041821 might have 20.4 mag, placing it at the faint end of the Gaia catalogue. Therefore, it is unclear at this stage whether its membership to the Hyades cluster will be settled by Gaia in the near future. Table 2 Results of our spectroscopic analysis. Notes. We quote the spectral type, spectroscopic distance, radial velocity, space motion, and ratio of the (/) luminosity for 2M041821. 4.4. Lithium absorption The lithium test was proposed in the early 1990s to distinguish between very low-mass stars and brown dwarfs ( Rebolo et al. 1992 ; Magazzu et al. 1993 ). The technique relies on the detection of lithium in absorption at 6707.8 Å for late-M and L dwarfs straddling the stellar-substellar boundary. If lithium is detected,

O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject I-CelestialObject O B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O B-Software O O B-Software O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O B-Database I-Database I-Database I-Database O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O B-CelestialObject O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Telescope I-Telescope O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O B-Database B-CelestialObject O O O B-Telescope B-Wavelength O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O B-Person I-Person B-CelestialObject O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O


that are not perfectly linear too. As a consequence, the linear fittings tend to overestimate the ages of stars younger than 3 Gyr by 0.8–0.9 Gyr. By fitting the distributions of thin and thick discs with quadratic functions, we found the following solutions: From these higher order functions, we find that the average scatter is reduced to 0.5 Gyr for both [Y/Mg] and [Y/Al], which is very close to the typical uncertainties of our age determinations (i.e. 0.4 Gyr). 4 CONCLUSIONS We used high-quality HARPS spectra of 79 solar twins to obtain precise estimates of their atmospheric parameters (T eff , log g , [Fe/H] and ξ) and abundances of 12 neutron-capture elements (Sr, Y, Zr, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd and Dy). The knowledge of the stellar parameters and absolute magnitudes of these stars allowed us to determine their stellar ages through the isochrone method with typical precisions of 0.4 Gyr. Our main scientific results can be summarized as follows: The thin and thick disc populations, identified through the α-elements abundances, are separated in the [X/Fe] versus age plots showed in Fig. 5 . The [X/Fe] ratios of the thin disc distribution are highly correlated with stellar age for the s -process elements (e.g. Ba), while the [X/Fe]–age distributions of the r -process elements (e.g. Eu) are nearly flat. Indeed, the slopes of the [X/Fe]–age distributions are dependent on the s -process contribution percentage of the X element (see Fig. 6 ). This indicates that, the enrichment from AGB stars is the predominant channel of nucleosynthesis of n -capture elements during the evolution of the thin disc. Despite the flat slopes traced by the thin disc stars, the nearly pure r -process elements Eu, Gd and Dy appear more enhanced in the thick disc population. This dichotomy between the thin and thick discs resembles the [X/Fe]–age distributions outlined by the α-elements, which are more abundant in the thick disc than in the thin disc due to an intensive nucleosynthesis by Type II SNe during the early stages of the Galactic disc. The sites of production of the r -process elements are not very well known at present; however it is likely that they are mainly synthesized during the mergers of massive compact objects. On the other hand, the similarity to the distributions of the α-elements suggests that the r -process elements are also produced by Type II SNe and that they could be additional tracers of intensive star formation episodes. There is a possible direct observational test that could be performed on OB associations which are regarded as a proto-type of triggered star formation, such as Orion or Sco-Cen. If Type II SNe are producing copious r -process elements, then the youngest sub-clusters should be enriched in these elements relative to the oldest stars (Cunha Lambert 1992 ). The [X/Fe]–age distributions of the thin disc stars have been fitted by linear functions. The parameters resulting from this linear fitting, listed in Table 5 ,

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-Model I-Model I-Model I-Model O O O O O B-Model I-Model O O O O O


O O B-Software O O O O O O O B-Model I-Model O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-CelestialObject O O O O O B-Telescope O B-Telescope O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O B-Telescope O B-Telescope O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O B-CelestialObject O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O B-Telescope O O O O O O B-Telescope O B-Identifier I-Identifier B-URL O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O B-Collaboration I-Collaboration O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O


O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag I-Tag B-CelestialObject O B-Tag I-Tag B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


this model may not represent the true morphology, but may vary on smaller angular scales, the intensity scaling parameters per energy bin, , are re-determined for each sky region and the background. The celestial emission in this energy range is dominated by the bright 511 keV line emission from the Galaxy’s centre, modelled by a narrow bulge, a broad bulge, and a point-like source centred at ( l ∕ b ) = (0°∕0°). To this, a Gaussian-shaped disk is added to capture the faint and low surface-brightness emission beyond the bulge. The details of the spatial emission models are found in Siegert et al. (2016) , and in Appendix E . The instrumental background is modelled in a self-consistent way, using long-term knowledge about the detector behaviours, space-craft conditions, and physical origins of γ -ray emissions in the satellite ( Siegert et al. 2016 ). Based on the INTEGRAL/SPI BackGround and Reponse Data Base (BGRDB, Diehl et al. 2018 ), the background detector patterns (relative counts of the detectors to each other) are predicted as a function of energy and time. This data base includes spectral information of hundreds of instrumental γ -ray lines which are associated to different isotopes in the satellite, being excited by cosmic rays and de-exciting either promptly or delayed. The continuum background, mainly due to bremsstrahlung inside the spacecraft, is modelled in the same manner. Long-term trends of detector degradation from this cosmic-ray bombardment and radioactive build-ups are individually traced by this technique. In this energy range, eight instrumental γ -ray lines, on top of a power-law shaped instrumental continuum is used to model these background patterns ( Siegert 2017 ), to be distinguished from the expected sky emission patterns. The background scaling parameters are determined on a three-day interval (see also Siegert et al. 2019 ), corresponding to one INTEGRAL orbit. This amounts to 1083 background scaling parameters for each, the continuum and line backgrounds as a whole, for a total of 2166 background parameters. In summary, 6 (2) parameters are fitted for the celestial emission in a simultaneous (consecutive) fit of the regions of interest and the remaining Galaxy, and 2166 parameters for the instrumental background. The number of degrees of freedom in each individual energy bin thus amounts to 1 212 627 (1 212 631). 2.2 Spectral extraction 2.2.1 Full sky 511 keV signal The total spectrum of the 511 keV template map ( Fig. 2 , top two panels) is shown in Fig. 1 , together with a multi-component fit. We identify four spectral components: a narrow and a broad 511 keV line, an ortho-positronium (o-Ps) continuum, and the galactic diffuse γ -ray continuum. The derived spectral parameters are consistent with previous studies, although both, the broad and the narrow annihilation line, appear to be broader than previously measured ( Jean et al. 2006 ). This may arise from different line shapes in the disk and the bulge

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Database I-Database I-Database I-Database I-Database I-Database B-Database B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObjectRegion O O B-CelestialObjectRegion


SNe Ia are monitored only for a few months; if CSM is more prevalent or typically at a larger distance from the WD, we might be systematically missing the signatures of this interaction. There are additional good reasons to suspect that we might be systematically missing the interaction of CSM with SN Ia ejecta. For example, SN Ia 2006X exhibited evolution in its Na i D absorption line that was interpreted as the photoionization of more distant CSM (Patat et al. 2007 ). In a compilation of 17 SNe Ia with multi-epoch spectra, Sternberg et al. ( 2014 ) found that ∼18% exhibit time-variable Na i D. Additionally, studies of large samples of SNe Ia (20–35 events) found Na i D to be blueshifted ∼20% more often than redshifted, and argued that this was best explained by outflowing CSM from the progenitor system (Sternberg et al. 2011 ; Maguire et al. 2013b ), but see Bulla et al. ( 2018 ) for an alternative explanation. In these SNe Ia, the CSM is estimated to be 5 × 10 16 –10 18 cm away from the explosion, which means that the fastest SN ejecta would not interact until >1 yr. Additional support for late-time CSM interaction is presented by Fox Filippenko ( 2013 ), who interpreted an increase in Spitzer mid-infrared photometry for SN Ia 2005gj at 500–1000 days past peak brightness as dust heated by hot blackbody emission from new or renewed CSM interaction. Additional evidence for nondegenerate companion stars comes from detections of, e.g., “blue bumps” in the light curve in the week after explosion (Hosseinzadeh et al. 2017 ) and/or late-time H α emission from material swept off the companion star and embedded in the ejecta (Maguire et al. 2016 ; Sand et al. 2018 ). It is important to note that there is plenty of evidence that many—or even most—SN Ia progenitor systems host a degenerate binary companion and/or do not contain CSM (e.g., Hayden et al. 2010 ; Maoz et al. 2014 ; Chomiuk et al. 2016 ). In this work, we aim to put upper limits on the fraction of SNe Ia with CSM, and we are not suggesting that the systematic oversight of CSM interaction extends to all SNe Ia. There is further motivational support in searching for late-onset CSM interaction in SNe Ia from recent observations of delayed CSM interaction in some core-collapse SNe. For example, Milisavljevic et al. ( 2015 ) presented observations of SN 2014C, an SN Ib that appeared to explode into an empty cavity surrounded by CSM, which transitioned to an SN IIn as the ejecta caught up with the previously released material. Similarly, Mauerhan et al. ( 2018 ) found H α emission from the onset of CSM interaction in SN Ib 2004dk 13 years after the original explosion. Vinko et al. ( 2017 ) presented the results of a systematic narrow-band imaging survey of Type Ibc, IIb, and Ia SNe to look for H α emission due to CSM interaction years to decades after explosion. Most of their potential detections are associated with core-collapse events, but of particular relevance to this work is their

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