Apellániz 2013 ). In Section 2 , we present the characterization and definition of the GALANTE photometric system. In Section 3 , we analyse in detail the set of GALANTE standard stars. Transformation equations between the SDSS and the GALANTE photometric system are obtained in Section 4 , setting up the comparison between the synthetic magnitudes of 378 stars from the Next Generation Spectral Library (NGSL) plus 122 stars from a new catalogue from MAW in both photometric systems. Next, in Section 5 we estimate zero-points from the GALANTE photometry using the SDSS and RefCat2 . Finally, in Section 6 we summarize the results of this paper. 2 THE GALANTE PHOTOMETRIC SYSTEM 2.1 DESCRIPTION Deriving physical stellar (or galaxy) parameters is the main goal of every photometric system. In our case, we achieve this aim with a new filter set designed to plug some gaps found in previous surveys. The GALANTE photometric system has been defined using four J-PLUS filters (Cenarro et al. 2019 ) and three purpose-built filters. This composition of intermediate- and narrow-band filters covers the optical range from 3000 Å to 9000 Å. This filter set has been selected with a direct goal: deriving stellar effective temperatures for hot stars in an optimal way (Maíz Apellániz Sota 2008 ; Maíz Apellániz et al. 2014 ; Maíz Apellániz Barbá 2018 ). We will also be able to obtain the gravity and metallicity (the latter for stars cooler than 10 000 K), allowing us to discriminate between giants, supergiants, dwarfs, and between solar or SMC metallicities. In Figs 1 and 2 , we present the set-up of the filters. Figs 1(a) and (b) show the layout of the J-PLUS and GALANTE filters, respectively, leaning on the Vega spectrum. Note that in this case we only plot the transmission curves of the filters, allowing to the reader a visual location of each one. Focusing on GALANTE, Fig. 1(b) plots the configuration of the selected GALANTE filters, which enable us to measure the key zones in the spectra to derive physical parameters. In Fig. 2 , we plot the response functions of the J-PLUS and the GALANTE photometric system filters. In discontinuous grey, we represent the response curve of the CCD and mirror, while discontinuous green is the CCD, mirror and atmospheric transmission at 1.3 airmass curve. The SDSS response curves are represented in continuous black, and the other continuous curves are the GALANTE and J-PLUS filter response curves convolved with CCD, mirror and atmospheric transmission at 1.3 airmass. Using Fig. 3 , we can also explain the function of each GALANTE filter with greater accuracy. Filter F348M is a u -like filter dispose to measure the continuum from the left of the Balmer jump. We can combine this one with F420N and F450N to measure the Balmer jump on both sides and derive the T eff of stars. Both filters of our design, F420N and F450N, are the most original (compared to other large-scale recent photometric surveys) of the set-up. They have been


(See Also Section 2.7 ) Note. The motivation behind each assumption is described in the corresponding sections. We briefly summarize (column 4) the effect produced by each assumption to help the reader distinguish the genuine trends from those induced by our prior hypotheses. The reference BHAR/SFR trends are taken from Mullaney et al. ( 2012 , M12), Rodighiero et al. ( 2015 , R15), and Aird et al. ( 2019 , A19). (∗∗) The relative normalization is chosen to fit the corresponding data points of Figure 4 . 2.3. Galaxy Stellar Mass Function The first step displayed in Figure 1 consists in setting the input galaxy M ⋆ function at different redshifts. The prescription is taken from Davidzon et al. ( 2017 ), who exploited the latest optical to infrared photometry collected in the COSMOS field over the UltraVISTA area (1.5 deg 2 , see Laigle et al. 2016 ). They provide the M ⋆ function separately between SF and passive galaxies, based on the [NUV − r ]/[ r − J ] colors (Ilbert et al. 2013 ). Throughout this work, we consider only the M ⋆ function relative to the SF galaxy population (i.e., MS and SB galaxies). Next, we split the M ⋆ function of SF galaxies among the MS and SB populations. Given that the relative fraction of the two populations has been shown not to vary with M ⋆ (e.g., Rodighiero et al. 2011 ; Sargent et al. 2012 ), we only consider how their relative fraction evolves with redshift by following the prescription of Béthermin et al. ( 2012 ). The fraction of SB galaxies ( f SB ) appears to evolve linearly, from f SB = 0.015 ( z = 0) to f SB = 0.03 ( z = 1), while it stays flat at higher redshifts. By simply scaling the SF galaxy M ⋆ function down by f SB (or 1 − f SB ), we end up with the M ⋆ function of MS and SB galaxies at each redshift (Figure 2 ). We note that the input M ⋆ function of Davidzon et al. ( 2017 ) is already corrected for the Eddington bias, which might have some impact at the high- M ⋆ end. After dissecting among MS and SB galaxies, we interpolate the corresponding M ⋆ function at redshift z = 0.5, z = 1, z = 2, and z = 3 across a M ⋆ range of 10 8 M ⋆ 10 12 M ⊙ . Figure 2. Galaxy M ⋆ function at various redshifts for SF galaxies, taken from Davidzon et al. ( 2017 ). Solid and dashed lines mark MS and SB galaxies, respectively. 2.4. The MS and the SB Loci We use the MS prescription presented in Schreiber et al. ( 2015 ), which incorporates a redshift evolution and a bending toward higher M ⋆ . Schreiber et al. ( 2015 ) studied a sample of Herschel -selected galaxies out to z ∼ 4, dissecting the observed distribution of MS offsets (=SFR/SFR MS , see their Equation (10) and Figure 19) among the MS and SB populations. By fitting that distribution via a double lognormal function, MS galaxies are centered at 0.87 × SFR MS , while SB galaxies are centered at 5.3 × SFR MS (Schreiber et al. 2015 ). Both relations were rescaled to a Chabrier ( 2003 ) IMF. The 1 σ dispersion for both the MS and SB loci is assumed to be 0.3


≤ 2 redshift range. (b) The same as (a) for 10 examples of each of our three stochastic SFH families. Except for the two very lowest metallicities, increases with redshift in a nearly linear fashion, with little overlap between the tracks for the three SFH families. 5. Summary We combined SSP SEDs from the Starburst99 and BC03 codes for young and old stellar ages, respectively, and generated arrays of CSP SEDs for a large variety of exponentially declining and stochastic SFHs by stacking SSP SEDs as a function of the adopted time-dependent SFRs. For our large suite of models with stochastic SFHs, we take the average metallicity evolution as a function of cosmic age into account. We calculated the intrinsic flux ratios between rest-frame visible–near-IR and rest-frame mid-IR 3.5 μ m on a grid of six metallicities and 16 ages of stellar populations for 13 different SFH families, for 29 visible–near-IR filters, and 5 mid-IR filters with central wavelengths near 3.5 μ m. We find that taking metallicity evolution into account results in significantly higher values at higher redshifts and for higher-metallicity stellar populations at those redshifts. We also provide the range of and for different Hubble types of nearby galaxies, and confirm that our models agree with the observed values. We demonstrated that the method can infer to a level of fidelity that is comparable to that of more established methods. We conclude that the method and its extension, the method presented here, are valid as a first-order dust-correction method, when using the morphology and size of a galaxy as broad a priori constraints to its SFH and redshift, respectively. The method will be applicable to large samples of galaxies for which resolved imagery is available in one rest-frame visible–near-IR filter and one rest-frame λ ∼ 3.5 μ m bandpass. We make our CSP synthesis models and all our results available via a dedicated website in the form of FITS and PNG maps, and ASCII data tables of values as a function of age and metallicity. Appendix A A Set of Distinguishable SSPs We construct our combined model SSP SEDs for a manageably small, yet comprehensive set of parameters that span the range of representative stellar population properties and dust extinction. We consider six metallicity and 16 age values of the stellar population, four extinction laws, and eight V -band extinction values (see Table 6 ). We adopt a grid of 16 stellar population ages between 8 8 In Planck 2015 cosmology (Planck Collaboration et al. 2016 ), the Hubble time t H ≃ 13.8 Gyr. Hence, t = 13.25 Gyr corresponds to 0.55 Gyr after the Big Bang or , consistent with the Planck Collaboration (Paper 2016) results of τ = 0.058 and , so this maximum age is a deliberate choice. 3 Myr and 13.25 Gyr in steps that roughly double for ages between 10 Myr and 10 Gyr. This allows us to follow the rapid evolution of massive stars, without reserving too many SEDs for the slow evolution of low-mass stars. Because the MW


spectrum would be expected from diffusive shock accelerations, but harder spectra are possible when the particle energy is converted from the electromagnetic energy of the sources, for example, in pulsar magnetospheres (Arons 2003 ), or in reconnection processes (Guo et al. 2014 ). Besides, an interplay between the acceleration and dissipation processes at the acceleration site could also lead to a hard injection spectrum (Dermer et al. 2012 ; Murase et al. 2012 ). Murase et al. ( 2016 ) and Bechtol et al. ( 2015 ) demonstrated that a pp scenario with a spectrum normalized to the IceCube datapoint at ∼10 TeV would overproduce γ rays above the flux of the non-blazar component of the isotropic γ -ray background. In this work, we will instead consider the case of an injection spectrum index equal or smaller than two, but normalized to the IceCube datapoint at 30 TeV (which is about three times lower than that at 10 TeV). Specifically, in Section 3 , we focus on a benchmark case of an injection spectrum and maximum energy . We discuss the effects of different choices of dN / dE and in Section 4 . 2.2. Cosmic-Ray Diffusion Galaxy clusters are known as efficient containers of cosmic rays (Völk et al. 1996 ; Berezinsky et al. 1997 ). Depending on the particle’s Larmor radius r L and the field’s coherence length l c , the propagation of cosmic rays in the cluster turbulent magnetic field can be divided into semi-diffusive () and diffusive () regimes (Kotera Lemoine 2008 ). When a particle’s energy reaches , the particle enters the semi-diffusive regime with a diffusion coefficient ofHowever, most particles have energy much lower than , and thus stay in the diffusive regime. The diffusion coefficient in this regime can be written as (Brunetti Jones 2014 )where is the spectrum index for Kolmogorov diffusion, and are the typical amplitude and scale of magnetic field fluctuations in massive clusters (Brunetti Jones 2015 ). The diffusion time of cosmic rays in the cluster magnetic field can be estimated asCosmic rays with diffusion time longer than the Hubble time would be completely confined by the cluster magnetic field, defining the lower limit of the energy of cosmic rays we expect to observe from the cluster as Particles with would have a better chance to escape from the cluster magnetic field. To estimate this escape probability, we first write down the probability that a particle with diffusion coefficient D reaches radius r after time t :assuming that the particle starts from r = 0 and the field is homogenous. The probability that a particle can successfully escape from a cluster of size within the Hubble time is then (Kotera Lemoine 2008 ) Cosmic rays accelerated in the strong shocks in the accretion region are expected to have a spectrum index . The flux of high-energy cosmic rays from a cluster with mass at redshift z = 0.1 can be calculated aswhere d L is the luminosity distance, and f π is the pion production rate that will be introduced


(denoted by the cyan diamond) where the spectra of Si iv , C ii , and Mg ii are fairly bright and show some blueshifts. Figure 1. Observation overview of the B1.6 flare event on 2016 December 6. Top left panel: an AIA 131 Å image during the rise phase of the flare. The green dotted line and the purple contours mark the IRIS slit position and the flaring footpoints, respectively, which are the same as in the SJI 1400 Å image (bottom left panel). Top right panel: light curves of GOES 1–8 Å (black), AIA 131 Å (blue, arbitrary scale), and SJI 1400 Å (red, arbitrary scale) from the flaring region. The vertical black line represents the peak time of the flare. The blue and red dashed lines indicate the times when the AIA 131 Å and SJI 1400 Å images were observed, respectively. The two red dotted lines mark the time range (10:37:00–10:39:00 UT) as shown in Figures 3 and 4 . Bottom right panel: spectra of Si iv , C ii , and Mg ii along the slit at 10:37:45 UT. The vertical dotted line represents the reference wavelength for each of the spectral lines. The cyan diamond denotes the footpoint location that is cut across by the slit, as also shown in the SJI 1400 Å image. The two horizontal lines mark the space range (10′′) that is plotted in Figure 3 . 3.1.2. Blueshifts in the Si <sc>iv</sc>, C <sc>ii</sc>, and Mg <sc>ii</sc> Lines Figure 2 shows the Si iv , C ii , and Mg ii line profiles at the footpoint location denoted by the diamond (see Figure 1 ) for two times. It is seen that all the three lines are blueshifted, especially at an earlier time (10:37:34 UT, top panels). The blueshift velocities are measured to be 20, 13, and 6.3 km s −1 for the Si iv , C ii , and Mg ii lines, respectively, namely decreasing as the line formation temperature decreases. A few seconds later (10:37:45 UT, bottom panels), all the velocities decrease nearly by half. Note that some of the Mg ii (and also C ii ) line profiles are almost singly peaked, i.e., without a central reversal, which is a common feature for flaring footpoints. Figure 2. Si iv , C ii , and Mg ii line profiles at the footpoint location denoted by the cyan diamond as shown in Figure 1 for two times. The dotted line in each panel represents the reference wavelength of the spectral line. For the Si iv line, we adopt a single Gaussian fitting (the red curve) to derive the Doppler velocity ( V g ). For the optically thick C ii and Mg ii lines, we make a moment analysis to obtain the Doppler velocity ( V m ). The spatio-temporal variations of the total intensity, Doppler velocity, and line width of the Si iv and Mg ii lines are plotted in Figure 3 . From the intensity map we can see that the bright footpoints (the contours) show some apparent motion toward the north. Interestingly, at the footpoints (e.g., the ones denoted by the diamond), both of the Si iv and Mg ii lines exhibit blueshifts particularly at an early time. Such a blueshift in cool lines has scarcely been reported before. A similar


To estimate the effective area correction between the different detectors of Fermi and BAT, we adopted the following methodology (Page et al. 2009 ): (i) To find the best-fitting model: Each of the time-resolved spectra of all detectors (BAT, GBM: n6, n7, n9, and b1) was jointly analysed using different spectral functions such as power law, cutoff-power law, and Band, without including the constant factor correction. In this exercise, the time-averaged and time-resolved spectra were found to be better fitted by cutoff-power-law function. (ii) Finding the normalization-offsets: We then simultaneously fitted the five bright time-resolved spectra with the best-fitting model, cutoff-power law, by multiplying the model by a constant of normalization for each instrument. The constant of BAT data was frozen to unity, while those of the GBM detectors were kept free. In addition, these constants were also tied across the respective data sets of the different time-resolved intervals. The relative normalizations were found as, for n6 detector, for n7 detector, for n9 detector, and for b1 detector. These constant factors are consistent with the expected intercalibration uncertainty of up to and hence were used for the subsequent spectral analyses. 3.2 Parameters evolution For characterizing the spectra, we fit the brightest time intervals in both episodes with several phenomenological models including power law with an exponential cutoff (cutoff-power law), blackbody + cutoff-power law, and traditional Band functions. In Fig. 2 , the reduced χ 2 obtained for the different models in the different time intervals are shown. We note that in the brightest time interval (bin 5) of the first episode, blackbody + cutoff-power law model is statistically favoured (i.e reduced χ 2 ∼ 1). On the addition of blackbody component to the cutoff-power law, we also find an improvement in Akaike Information Criterion (AIC; Akaike 1974 ) and Bayesian Information Criterion (BIC; Schwarz et al. 1978 ) by 15 and 7, respectively (see Appendix B ). At the same time, in all other time intervals, this model is also found to give comparable statistics when compared with other models. The Monte Carlo (MC) simulations for estimating the statistical significance of the additional blackbody component with the cutoff-power-law model are performed for the brightest time interval. The difference of χ 2 -statistics between cutoff-power-law model and blackbody + cutoff-power-law model is found to be 19. Here, we assume cutoff-power-law model as the null hypothesis model and blackbody + cutoff-power-law model as the alternate model and simulations are run for realizations. We find that the probability of observing Δ χ 2 > 19 is 10 −2.52 , which corresponds to ∼3 σ significance level. Thus, by considering the above fit statistics and MC simulations obtained in the brightest time bin, we consider blackbody + cutoff-power-law model to be the best-fitting model for all the time-resolved intervals


only 13 SNe are removed as a result. The latter is much more impactful, however, removing ∼700 spectra (mostly from the SED Machine; Blagorodnova et al. 2018 ) and nearly as many SNe. In fact, this amounts to the most severe cut on SNe and the second most severe cut on spectra (after the ‘overlap’ cut). With future improvements to deepSIP it may be possible to revisit and lessen the extremity of this cut, but this will have to wait until those improvements can be realized. Irrelevant photometric passbands: After the cuts described above, the 60 140 photometric points that remain are in one of 62 different passbands, some of which are irrelevant to our analysis and must therefore be removed. By far, the primary factor in rendering observations in a given passband ‘irrelevant’ is if that band is too broad. For example, 4923 Gaia G -band (Jordi et al. 2010 ), 2104 Pan-STARRS w -band (Chambers Pan-STARRS Team 2018 ), 612 ATLAS c -band or o -band (Tonry et al. 2018 ), and many other wide-band observations are removed, along with a further 5061 unfiltered photometric points. A second factor in determining the relevance of a passband is the wavelengths of light that it passes. As our SDM uses a WLR implementation that is trained at optical and near-infrared wavelengths, we must discard passbands that lie outside this domain, thereby removing some Swift observations (Brown et al. 2014 ) among others. All told, 15 624 photometric points are removed, and those that remain are homogenized down to 21 distinct passbands. Photometric uncertainties: Of the remaining 44 516 photometric points, 10 429 do not include a symmetric uncertainty in the reported magnitude. We are able to ‘fix’ 6120 of these by taking the larger of their (asymmetric) lower and upper magnitude uncertainty estimates, but 4393 remain which have no identifiable quantitative uncertainty indicator. We take the conservative approach of removing these data, leaving us with just under 1200 SNe covered by ∼2900 spectra and ∼ 40 000 photometric points. 2 + Distinct passbands: As the SDM requires, at a minimum, two photometric points in different passbands (and a spectrum), we remove SNe that fail this requirement. This reduces the sample to 747 SNe with ∼2400 spectra and ∼ 35 000 photometric points. Literature search for photometric systems: While we have taken care above to homogenize the passbands in our sample (in the sense of resolving the differing names given to identical filters), we must also understand the photometric systems to which they belong. To do so, we visit the primary source provided by the OSC for each remaining point in our photometry sample (amounting to 248 sources) and attempt to determine the photometric system. We find that 26 SNe (covered by 5719 photometric points) are either not an SN Ia or are not an SN Ia subclass that can be handled by our WLR. These are dropped, as are an additional 7220 photometric points whose photometric system is either not clear or not


3. Enclosed gas mass profiles as a function of radius for the known potential Galactic YMCs precursors. The transition point from solid to lower opacity indicates the beam resolution of the data. The triangle markers correspond to the initial conditions used by Banerjee Kroupa ( 2013 ) to simulate the monolithic formation of the R136 YMC. The circular markers correspond to the initial conditions used by Banerjee Kroupa ( 2014 ) to simulate the monolithic formation of the NGC 3603 YMC. The star markers correspond to the initial conditions given in the fifth row of table 1 in Assmann et al. ( 2011 ) to simulate a monolithically forming star cluster. As these simulations are 3D models, and the data that we discuss in this paper are 2D projections, we must take care to ensure that we are making a fair comparison to these models. In Banerjee Kroupa ( 2013 , 2014 ), they present their initial conditions as initial cluster mass, initial gas mass and half-mass radius. Using their assumed ∼33 per cent SFE, we combine these masses to obtain the total gas mass prior to star formation. We also use the following relation to convert the half-mass radius to the Plummer radius (Heggie Hut 2003 ) – Assmann et al. ( 2011 ) already provide the Plummer radius. We then take the total 3D gas mass for all of the models and convert this to a projected mass (Heggie Hut 2003 ) – Overall Fig. 3 shows that the initial projected masses within a given radius for these various models generally too high, particularly at smaller radii. We note that the Banerjee Kroupa ( 2014 ) model for NGC 3603 appears consistent with observations at radii > 0.5 pc. However, it is overdense on smaller spatial scales. Whilst the models generally predict mass surface densities that are greater than the observed clouds, this is not unambiguously the case. The highly star-forming clouds such as Sagittarius B2, W49 and W51 all lie very close to these models. Though we reiterate that the mass estimates, particularly in the highly star-forming clouds, are upper limits. Accounting for temperature effects would create an even larger disparity between the observations and simulations. We also note that the simulated models assume that both the gas and stars are initially distributed according to a Plummer distribution, which is not true for the gas in the clouds in our sample (see Section 4 ). This is a critical difference between the true conditions in the ISM and those assumed in these models. The gravitational potential of the gas is dominant until a significant number of stars have formed within the gas, and it is clear that the initial potential does not arise from a Plummer-like gravitational potential. 4 DISCUSSION In Walker et al. ( 2015 ), we used a sample of YMCs and possible YMC precursor clouds at the Galactic Centre to attempt to distinguish between two commonly argued modes of YMC formation – popping or conveyor-belt modes. We did this simply by comparing their mass surface density


5 . Nevertheless, the resulting statistical errors of shear, as opposed to its bias, are larger by roughly the factor 1.7 with the uniform prior, which reflects the larger a-priori uncertainty on . This implies that a shear bias due to an incorrect prior is negligible in this experiment. 5. Discussion Unlike galaxy ellipticities defined in terms of unweighted moments, GLAM provide a practical definition of ellipticity with useful properties for any chosen adaptive weight : they (i) are identical with the ellipticity of isophotes of elliptical galaxy images; (ii) under the influence of shear they behave exactly like defined with unweighted moments; (iii) they are unbiased estimators of reduced shear; (i)(iii) assume ideal conditions without pixellation, PSF convolution, or pixel noise. These effects fundamentally limit our ability to determine the GLAM ellipticity (see below). Under ideal conditions, see SS97, it is known that for unweighted moments already obeys (iii). However, unweighted moments are formally infinite for realistic galaxy images because of pixel noise so that weighting is a necessity which is done adaptively for GLAM . For adaptive weights, we have shown (i) and (ii) in Sects. 2.2 and 2.3 . Their relevance as unbiased estimators for reduced shear, statement (iii), follows thereafter from (ii) and the conclusions in SS97 for unweighted moments. We emphasise that GLAM do not require “deweighting” in a lensing analysis ( Melchior et al. 2011 ): the ellipticity in term of the weighted moments is actually unbiased. In particular, we do not have to devise the same weight function for all galaxies as any weight function equally obeys (ii). If the minimum of the functional ( 3 ) uniquely exists, the moments of the best-fitting template will, for a constant scalar , be the adaptively weighted moments of the galaxy light profile for the weight (Appendix A ). This means that the radial profile of the weight at separation from the centroid is for a template profile . We exploited this relation between adaptive moments and moments of a best-fitting elliptical profile to analyse limits to adaptive moment-based methods under the loss of image information. Namely, for realistic images subject to pixellation and PSF effects, underfitting bias can be present for GLAM if estimated from the post-seeing image. It potentially arises because of a loss of information on the pre-seeing galaxy image, fundamentally due to pixellation. To explore the limitations, we assume noise-free images and express in Sect. 2.4 the mapping between a pre-seeing and post-seeing image by the linear operation . The pre-seeing adaptive moments are equivalent to a least-square fit with residual of an elliptical profile to the pre-seeing image . If estimated from the post-seeing image whilst ignoring residuals, the inferred ellipticity is biased for and a singular . The latter indicates a loss of information on the pre-seeing image. For noise-free images and the estimator


unaffected by the Milky Way potential. We see in the lower panel that even for V peak the relation is broadly flat with significant scatter. For example, without including pericentre information, the V peak error bars for CVenI and Draco overlap, suggesting that these two galaxies with similar stellar masses could have formed within haloes of the same size. The revised estimate suggest that Draco I likely formed within a halo that was more massive. Improving constraints on the dynamical mass of the dSphs at present combined with better constraints on their pericentres from Gaia can provide deeper understanding of the past evolution and the mass of the progenitor haloes of MW dwarf spheroidal galaxies. Among the galaxies we have considered, Draco and Ursa Minor are affected the most by including pericentre information. These, the densest dSph satellites, have relatively small pericentres ( r peri ≲ 50 kpc) and this pushes their inferred V max values lower than they would have been without pericentre information. We find that both Draco and Ursa Minor appear to inhabit subhaloes with V max ≃ 20–27 km s −1 (at one sigma) and are less dense than would be expected for the most massive subhaloes with small pericentres (see Fig. 1 ). This exacerbates the traditional TBTF problem (Boylan-Kolchin et al. 2011 , 2012 ). The inferred range of V peak values is also tightened for these galaxies. Both galaxies have inferred peak maximum circular velocities (prior to infall) that put them in range to have been significantly affected by reionization, V peak ≃ 25–37 km s −1 (e.g. Bullock, Kravtsov Weinberg 2000 ; Gnedin 2000 ; Benítez-Llambay et al. 2017 ; Fitts et al. 2017 ). It is worth considering whether our results are affected by numerical resolution. There is a fair amount of evidence to suggest that simulations such as ours do not accurately track halo survival after they have lost a significant () amount of mass, with ∼3000 particles appearing to be a critical limit below which haloes disrupt unphysically (e.g. Van den Bosch Ogiya 2018 ; Errani Navarro 2020 ). In terms of halo counts alone, this effect should not affect current cosmological zoom simulations significantly, since accreted halo mass functions rise steeply with increasing mass, d n /d M ∼ M −1.8 (e.g. Giocoli, Tormen van den Bosch 2008 ). This means that for every halo of mass M that was accreted long ago and has lost 99 per cent of its original mass (100 M ), there are as many as ∼4000 haloes of mass M that were accreted more recently and that can be accurately counted. As discussed in Kelley et al. ( 2019 ), convergence tests suggest that our halo counts are mostly complete down to V max > 4.5 km s −1 , or only ∼150 particles. Of particular relevance to this work is whether or not internal halo structures, e.g. V max / V peak distributions at fixed V max , are tracked appropriately. We have limited our analysis to haloes with V max > 10 km s −1 , or ≳ 3000 particles. According to the


(Hindmarsh 1983 ). We regularly reconverge the PT structure with the radiative/convective code during the time integration until we reach a steady-state for both the chemistry and energy conservation/hydrostatic equilibrium. We then get a coherent atmospheric structure with out-of-equilibrium chemical abundances. As in Tremblin et al. ( 2016 ), we propose that a process similar to fingering convection is responsible for this turbulent mixing and leads to the out-of-equilibrium abundances. Fingering convection can be triggered in the atmospheres of brown dwarfs and exoplanets because of the gradient of mean molecular weight induced by the chemical transitions CO/CH 4 and N 2 /NH 3 at chemical equilibrium. Similar to convective motions that lead toadiabatic PT structure in the deep atmosphere, fingering convection could impact the PT structure of the atmosphere. As in Tremblin et al. ( 2016 ), we propose that fingering-convective motions can induce a temperature gradient reduction, leading to a reddening in the modeled spectrum. This temperature gradient reduction in the atmosphere is modeled using an artificially reduced adiabatic index γ in between two pressure levels on top of the convective zone. In Figure 1 , we show the influence of various model parameters at effective temperatures of 1700 and 1300 K, representative of the objects studied in this study. At 1700 K we can see a degeneracy between gravity and metallicity on the spectral shape of the H and K bands. Decreasing gravity or increasing metallicity tends to form a triangular H band and a strong and flatter K band. Decreasing the temperature gradient in the atmosphere by decreasing the effective adiabatic index γ increases the reddening in J − K . Increasing the surface gravity at an effective temperature of 1300 K has a strong effect on the out-of-equilibrium chemistry and the quenching of CO in the deep atmosphere. At higher gravities, the CO/CH 4 transition is deeper in the atmosphere, hence it is more difficult to prevent the formation of CH 4 (strongly appearing at log g = 4.5) for a given mixing coefficient K zz . Thus, varying K zz would approximately lead to the same effect as varying gravity for T eff = 1300 K. For the fitting of the observations we have probed a similar parameter space, log g in [3.5, 4.5], [M/H] in [0, 0.5], T eff in [1200, 1800], and γ in [1.01, 1.1]. Figure 1. Effects of varying different parameters on the spectral models obtained with the ATMO code. 2.1. DENIS J1425279-365023 DENIS J1425279-365023 was first identified as an L5 dwarf (Kendall et al. 2004 ) with a trigonometric distance of ∼11.57 ± 0.11 pc (Dieterich et al. 2014 ). A SpeX Prism obtained by Gagné et al. ( 2015a ) indicates that the object is a L4 INT-G object based on the index-based classification of Allers Liu ( 2013 ). Based on the galactic position and space velocity of the object, Gagné et al. ( 2015a , 2015b ) also concluded that DENIS J1425-36 is a bona fide member of the


constraints and verification of the TeV emission mechanism in 30 Dor C. Hadronic and leptonic mechanisms predict different flux for the (10 GeV–1 TeV) band, both currently below Fermi -LAT ( Atwood et al. 2009 ; Ackermann et al. 2011 ) sensitivity and between Fermi and H.E.S.S. covered energy ranges, but within the capabilities of C.T.A. An example of the prospects of C.T.A. in this regard is shown by Acero et al. (2017b ) as applied to the SNR RX J1713.7–3946. The recent H.E.S.S. paper on RX J1713-3946 ( H.E.S.S. Collaboration 2018 ) demonstrates the possibilities for C.T.A. observations of LMC objects. H.E.S.S. revealed the TeV shell of RX J1713.7–3946 in unprecedented detail, and was found to extend further than the X-ray shell. This allowed the probing of particle escape, while the GeV and TeV spectra were covered to a level of accuracy that allowed a very detailed comparison of leptonic and hadronic emission mechanisms, including a magnetic field map for the leptonic case. 5 Conclusion We have presented an analysis of new Chandra observations of 30 Dor C in the Large Magellanic Cloud, the first superbubble detected in TeV γ -rays. These observations provided the sharpest view of the synchrotron X-ray shell of 30 Dor C, allowing us to perform a detailed morphological study and estimate the B -field in the superbubble, a key discriminator in assessing the dominant TeV γ -ray emission mechanism with a low B -field ~ 15 μ G required for a purely leptonic origin, a B -field of ≳ 50 μ G for a completely hadronic-dominated origin, or B -field of ~ 20 μ G for a 50–50 contribution of the leptonic and hadronic mechanisms to the TeV γ -rays. Using the new Chandra data, MCELS2 H α , and 6 cm radio continuum images we found an anti-correlation between the synchrotron X-ray and H α /6 cm shells. In addition, we discussed how long-slit spectroscopy of various regions of the 30 Dor C shell has shown no evidence of the high velocities necessary to explain the synchrotron X-rays (≳ 3000 km s −1 ). Rather the H α expansion velocities are more typical of an expanding superbubble ( 100 km s −1 ). We suggested that the SNR responsible for the synchrotron X-rays has reached the 30 Dor C supershell and has stalled in some regions, but continues through gaps in the shell in others. This is similar to the observed anti-correlation seen in RCW 86 ( Vink et al. 2006 ; Helder et al. 2013 ), which is attributed to the SNR evolving into a wind-blown cavity and encountering density gradients, though the velocity differences between synchrotron X-ray and H α shells are not as extreme. This may be a result of more pronounced density gradients and/or the fact that the shock radius is so much larger in 30 Dor C, meaning the shock energy has been distributed over a large area, making it more sensitive to these density gradients. We estimated the downstream B -field from the synchrotron X-ray shell. This was achieved by fitting the observed radial profile in sectors around 30


map in configuration space with a specific measurement error given. As possible noise sources, we take into account background fluctuations due to undetected clusters and the instrumental noise of ALMA itself. Furthermore, we include an estimate of the fluctuations introduced by primary CMB anisotropies. Since the different noise contributions are not correlated with each other, the total uncertainty is then given by Gaussian error propagation. 4.1 Background fluctuations due to undetected clusters Galaxy clusters unresolved by the telescope beam contribute a background noise level y bg that needs to be taken into account in all following calculations. Clusters are unresolved if they appear (much) smaller than the beam size. The background signal is thus dominated by low-mass clusters (Bartelmann 2001 ). Since an ideally homogeneous background could be removed from the data, we only have to consider the average background fluctuation level Δ y bg . The mean background level contributed by unresolved clusters with the differential mass function n ( M , z ) is where the mass- and redshift-integrations have to be carried out over that area in the mass–redshift plane where clusters are unresolved. The integrated Compton- y parameter Y ( M , z ) of a cluster with mass M at redshift z is where the angular diameter distance D d to the cluster appears and the total number of electrons N e . We choose the cluster mass function n ( M , z ) described by Sheth Tormen ( 1999 ), The number of background sources follows a continuous probability distribution. If one neglects any spatial cluster correlations, the rms due to background fluctuation is 4.2 Instrument noise Another relevant source of noise is the measurement noise of the telescope and the detector. We assume that the measurement error of the telescope has a Gaussian distribution with standard deviation σ Tel around zero, with σ Tel depending on the telescope and detector configuration and the setup of the specific observation. Since we need a resolution for our simulation that is comparable to other observations of massive galaxy clusters, we choose the characteristics of the Atacama Large Millimeter/submillimeter Array ( ALMA ) as an example telescope for achieving sufficiently precise thermal SZ observations. For all simulations presented in the remainder of this work, we use the following configuration of ALMA : All N ant = 32 antennas are assumed to be available with a baseline to be set. This configuration has then a given angular resolution in a frequency band chosen to be centred on ν = 116 GHz. Its bandwidth is assumed to be Δν = 7.5 GHz. According to the ALMA user manual (Lundgren 2012 ) the point-source sensitivity σ Tel of ALMA is where T sys is the temperature of the system, 1 η q the quantum efficiency (η q = 0.96), η c the correlator efficiency (η c = 0.88), A eff the effective area of an individual antenna, n p the number of polarization states ( n p = 2), and t int is the


et al. 2014 ). When the collision velocity is assumed to be the velocity difference between the stem and bowl region (∼1.5 km s −1 ), the kinetic energy released by the collision is estimated to be erg cm −3 , where the H 2 density of 10 3 cm −3 is arbitrarily assumed. This can be balanced by the cooling rate by Goldsmith Langer ( 1978 ), and the equilibrium temperature is roughly estimated to be 100 K. Here, we assume the cooling time of 10 4 yr according to Inoue Inutsuka ( 2012 ). This temperature is comparable with that observed in the interacting region. Thus, this rough estimate implies that the slight temperature enhancement is possible in the interacting region. Warm gas with a temperature of 74 K, heated by the filament–filament collision, is cooled down mainly by rotational line emission of CO for an H 2 density of 10 3 cm −3 (Inoue Inutsuka 2012 ). As mentioned above, the cooling time () is estimated to be around 10 4 yr according to Koyama Inutsuka ( 2000 ) with a cooling rate of 10 −26 erg cm −3 s −1 (Inoue Inutsuka 2012 ). Assuming that this warm gas expands with a constant velocity of 1.5 km s −1 , which corresponds to the typical velocity difference between the blueshifted and redshifted components of the Pipe nebula of 3.0 km s −1 and 4–5 km s −1 , respectively, the gas can extend in space by about 0.015 pc within the cooling time. This length corresponds to a typical spatial scale of a shocked warm region, and hence we hereafter denote it as (=0.015 pc). On the other hand, the extent of the cloud along the line of sight () can roughly be estimated from the H 2 density ( n (H 2 )), the OH column density ( N (OH)), and the OH fractional abundance ( X (OH)) relative to H 2 with the following equation:We employ an H 2 density of 10 3 cm −3 and an OH column density of 5.3 × 10 13 cm −2 , the latter of which is determined for the (0′, 32′) position (Table 5 ). Assuming an OH abundance of 10 −7 , which is a typical value in diffuse clouds (Wiesemeyer et al. 2012 ), is estimated to be ∼0.2 pc. If the OH fractional abundance is one order of magnitude higher ( X (OH) = 10 −6 ), which could be possible in a shocked gas according to Draine Katz ( 1986 ), could be 0.02 pc. The latter value ( ∼ 0.02 pc) for the shocked gas case ( X (OH) = 10 −6 ) is comparable to the of 0.015 pc. This result suggests the presence of shock-induced compression at the interface of the two filaments, and supports the filament–filament collisions picture in the Pipe nebula. 7. Summary We have observed the four hfs components of the OH 18 cm transitions toward the Pipe nebula along the two strip lines and determined the gas kinetic temperatures accurately. Along strip-1, the gas kinetic temperature is found to have no relation to the apparent distance to the nearby star, θ -Oph. This result suggests that the heating effect from θ -Oph does not significantly contribute to the heating of the Pipe nebula, although a possibility that θ -Oph illuminates the


on the impurity, physical structure, and the temperature of ice grains. In turn, we may be able to set constraints on the processes experienced by 2I and its formation conditions. There are three objectives of our study: (1) to search for diagnostic absorption features of water ice and other compounds (such as hydrated minerals or organic materials) in the coma or on its surface; (2) to compare water ice features in 2I to other Solar System icy objects (e.g., comets, Centaurs, and Kuiper Belt objects) and set constraints on its birth conditions; (3) to search for the ultra-red organic materials that may be present on the surface of 2I or in the coma as a result of long-term exposure to Galactic cosmic rays or UV bombardment. 2. Observations and data reduction We observed 2I on three separate occasions. The first two observations were made using director discretionary time allocations. The first observation was carried out in the morning twilight on 2019 September 19 UT using the SpeX spectrograph at the 3m IRTF telescope. The second observation was made on September 24 UT with the Gemini Near-InfraRed Spectrograph (GNIRS) spectrograph on the 8m GEMINI telescope. The last observation was made in service mode, again with the SpeX/IRTF, on October 9. The observing geometry is shown in Table 1 . Table 1. Journal of the 2I observations. Notes. (†) r h and Δ are the heliocentric and geocentric distances, respectively; (‡) α is the phase angle; (§) χ is the airmass. The SpeX observations were made with the high-throughput prism mode (0.8–2.5 μ m) and a 0.8′′ × 15′′ slit that provides a spectral resolving power of R ∼ 150 ( Rayner et al. 2003 ). The GNIRS observations were taken with the short camera (plate scale: 0.15′′ pix −1 ). The SXD cross-dispersed mode, 32 l mm −1 grating and 1.0′′ slit, provides an averaged resolving power of R ∼ 1700 over 0.9–2.5 μ m ( Elias et al. 2006 ). At least one nearby G-type star was observed together with the comet during each run, which was used both as the telluric correction standard star and the solar analog. The SpeX data were reduced using the reduction pipeline SpeXtool ( Cushing et al. 2004 ). The GNIRS data were reduced using the Image Reduction and Analysis Facility (IRAF) software and the Gemini IRAF package. Taking advantage of the similarity between the SpeX data and GNIRS data, we used the SpeXtool xtellcor routine for telluric correction and the xmergeorders routine for merging the GNIRS SXD spectra. 3. Results The NIR observations of 2I are presented in Fig. 1 . Although the three spectra were taken on different dates and with different instruments, they consistently appear featureless and show moderately red spectral slopes. We did not detect the diagnostic absorption bands of water ice at 1.5 and 2.0 μ m. Among the three NIR observations, the Gemini dataset has the highest signal-to-noise ratio (S/N). Thus, we use the Gemini spectra for further analysis. Fig. 1. Near-IR spectra of 2I/Borisov,


of least massive black holes (shown as green points) in order to compare the properties of the descendants of the haloes with massive quasars with those of haloes whose black hole mass is much lower, around that of the seed black hole mass. While matching haloes in the BT hydrodynamic and BTMassTracer simulation, we note that the halo mass is different in the matched haloes due to baryonic effects and there can be differences, which are mostly within 20 per cent. Figure 4. Two-dimensional histogram of the logarithm of halo mass (log 10 ( M halo [M ⊙ ])) and the black hole mass (log 10 ( M BH [M ⊙ ])) in the BT Hydrodynamic simulation at z = 8. The 23 most massive black holes are shown by scatter points in purple, while the least massive blackholes are shown in green. We also show as cyan points, the blackholes in the progenitor haloes of the most massive haloes at z = 0. Note that, for the cyan points, the halo masses shown correspond to that of the progenitors at z = 8. In the left-hand panel of Fig. 5 , we plot the halo mass function of the dark matter haloes at z = 8, 5, 2.5, 1.2 and 0. We also plot as points on the halo mass function at z = 8 and 0 the hosts of the most massive black hole tracer particles (corresponding to the purple points in Fig. 4 ). We find that at redshift z = 0, the halo masses are distributed over a wide range, compared to that at z = 8. From the figure, we can therefore clearly see that the descendants of the most massive black holes at z = 8 are not all the most massive haloes at z = 0. However, the descendants are still relatively massive haloes with mass above M * = 5.0 × 10 12 h −1 M ⊙ at z = 0. The individual tracks for the halo masses of the four most massive black hole tracers are also shown in the figure. We can see that these objects approximately keep their rank order over all redshifts. Figure 5. Left-hand panel: halo Mass function at z = 8, 5, 2.5, 1.2 and 0, and the location of the tracked hosts of the four most massive BH's on the mass function. Right-hand panel: normalized histogram of the halo masses at z = 0 of the descendants of the most massive black holes at z = 8 compared with the histogram of most massive haloes at z = 0. To further understand the distribution of the halo masses, we plot the normalized histogram of the host halo masses of the descendants of the z = 8 massive black holes, compared with the 23 most massive haloes at z = 0 in the right-hand panel of Fig. 5 . While the most massive haloes are distributed above 3.2 × 10 14 h −1 M ⊙ with a peak in the distribution at ∼1.8 × 10 15 h −1 M ⊙ , the quasar descendants have a peak distribution in the mass at ∼2.24 × 10 14 h −1 M ⊙ . These 23 most massive haloes and 23 quasar descendants have a space density of 4 × 10 −7 h 3 Mpc −3 . The massive haloes are all Coma-like objects, whereas the quasar descendants have masses an order of magnitude smaller, comparable to galaxy groups with a space density ∼ 10 − 5 h 3 Mpc − 3 . We can


molecules can be due to their reaction to form NH 4 SH in the deposited ice. The most intense band of NH 4 SH and other ammonium salts that may be present is the transition located at 1470 cm −1 . It is Unfortunate that this band overlaps with the methanol band in the same region and complicates the estimation of the column density of . Figure 3. IR spectra of H 2 S–CH 3 OH–NH 3 –H 2 O (3:3:3:1) deposited on a cold substrate (50 K) before (blue) and after (red) electron irradiation with a 10 keV electron beam (fluence = 2 × 10 21 eV cm −2 ). Figure 4. Left: Gaussian deconvolution for the 2556 cm −1 absorption corresponding to H 2 S. Band A corresponds to CH 3 OH, bands B and C correspond to H 2 S (Jiménez-Escobar et al. 2014 ), and band D corresponds to H 2 O. Right: Gaussian deconvolution for the 3367 cm −1 absorption corresponding to NH 3 . The red curve in Figure 3 shows the post-irradiation spectrum of the “with H 2 S” mixture. We note a significant decrease in band intensity of the initial constituents, and particularly a total disappearance of the H 2 S band. This is interpreted as complete destruction of the H 2 S molecules by electron irradiation. Among products of irradiation we observe the same molecules produced after irradiation of the “without H 2 S” mixture. Additionally, in the “with H 2 S” mixture alone, after irradiation an intense band appears at 2040 cm −1 . This band is assigned to the C = O stretching mode of OCS, which is observed at 2019 cm −1 in pure OCS ice and shifted to 2046 cm −1 in OCS-H 2 O and OCS-CO 2 mixtures (Ferrante et al. 2008 ). Previous experimental studies reported the formation of OCS after irradiation of ices containing CO or CO 2 as the carbon source and H 2 S or SO 2 as sulfur source (Ferrante et al. 2008 ; Chen et al. 2015 ). Our assignment of the 2040 cm −1 band to OCS is supported by the absence of this band in the infrared spectrum of the irradiated “without H 2 S” mixture, where no sulfur was present. OCS can be a product of reactions between a CO (which is produced from the dissociation of methanol as explained above) and an S atom or HS radical, as illustrated here: The column density of OCS after irradiation at 50 K (before heating) is 1.7 × 10 16 molecules cm −2 . This corresponds to a ratio of N (OCS)/ N (H 2 S) = 8.5%. In Figure 3 it is noteworthy that the absorption band at 1477 cm −1 assigned to methanol is in the same position as the strong absorption band of () (Howett et al. 2007 ). This makes it difficult to detect any production of the ammonium salt during the irradiation and heating of the sample. Thus, to assess the chemistry, we have performed an irradiation experiment using 15 N labeled ammonia. Figure 5 compares the infrared spectra of labeled and unlabeled mixtures after irradiation. We observe a shift of 16 cm −1 of the OCN − band in agreement with the isotopic shift 17 cm −1 measured in the matrix-isolated molecule (Maki Decius 1959 ). This confirms our assignment of this


probability distributions are shown in the right-hand panel of Fig. 2 , and lead to a strong bias towards Solar and super-Solar metallicity systems as potential terrestrial planet hosts, while substantial tails of potentially habitable terrestrial planet systems remain down to low metallicities. Given the flat evolution in the mean gas phase metallicity of the Universe over cosmic time, this probability distribution has relatively little effect on the evolution of habitabilities at any lookback time 10 Gyr, but instead acts primarily as a scaling factor, truncating the maximum habitable fraction at any redshift at ∼50 per cent. It will, however, tend to suppress the habitable fraction in low-mass galaxies (log(M/M ⊙ ) 8.5), since these have lower typical metallicities as shown in Fig. 3 . We note that since Lineweaver ( 2001 ) defined this probability distribution, the number of exoplanetary systems known has grown exponentially (see e.g. Winn Fabrycky 2015 , for a recent review). None the less, the metallicity dependence of planet hosting remains a complex subject, with relatively poor observational constraints, particularly at the low end of the planetary and stellar-mass functions where observational selection effects are significant. None the less, the distribution above remains broadly consistent with current observations. Wang Fischer ( 2015 ) examined the distribution of planets selected by the Kepler telescope. They lacked the sample size to determine the detailed metallicity distribution of the planet hosts, but found that the median metallicity for terrestrial planet hosts is log( Z /Z ⊙ ) = 0.04 (with Z ⊙ = 0.018), which matches the peak of the Lineweaver ( 2001 ) distribution in P HE ( Z ). By contrast, other studies such as that of Adams Kane ( 2016 ) and Buchhave et al. ( 2012 ) have found no statistically significant difference in metallicity between stars hosting terrestrial planet candidates and the whole Kepler sample, even when only planets in their habitable zones are considered. Hobson et al. ( 2017 ) also suggest that, while M-dwarf stars hosting massive planets are biased towards high metallicities, no such bias is seen for the hosts of low-mass planets. At the lowest metallicities, Zackrisson et al. ( 2016 ) have studied the history of metallicity enhancement in the Universe and its effect on terrestrial planet formation. They apply a gradual iron enrichment cut-off between 10 per cent and 1 per cent of Solar metallicity, extending well below the Lineweaver ( 2001 ) cut-off we apply, although, as Fig. 2 demonstrates, the vast bulk of galaxies have a mean gas phase metallicity above 10 per cent Solar for the entire lookback interval being considered. It is likely that future exoplanet telescope missions such as TESS and PLATO , together with extensive ground-based follow-up, will be required to clarify this issue. However, as noted above, given the relatively slow evolution of mean metallicities in cosmic time, we


view of Event 1 observed simultaneously with AstroSat SXT, Chandra LETG/HRC-S, and HST binned to 100s each. The vertical lines in brown and yellow represents the UV and X-ray flare peaks, respectively. The amplitudes of the HST light curves are scaled to be consistent with X-ray light curves. A detailed analysis of the HST data will be presented in Schneider et al. (in preparation). However, a visual inspection of Fig. 4 indicates that the HST FUV light curve precedes the X-ray light curves by about a few minutes. This may indicate that the presence of Neupert effect (Neupert 1968 ) during the rising phase of Event 2. Using the Z-transformed discrete correlation function (ZDCF) (Alexander 1997 ), the cross-correlation of the HST and the X-ray light curves produces a time-lag between the peaks to be 300–400 s, indicating that due to electron impingement the UV light curve precedes the thermal radiation. Figure 4. Event 2 simultaneously observed with the AstroSat SXT in 0.5–4.0 keV (top panel), Chandra LETG/HRC-S zeroth order in 0.1–12.0 keV (middle panel), and HST FUV in 120–170 nm (bottom panel). 3.4 Comparison The quiescent emission measured with Einstein , EXOSAT, and ROSAT in 0.3–4.0 keV energy ranges are 7.2 × 10 −12 , 4–16 × 10 −12 , and 2.6–7.2 × 10 −12 erg s −1 cm −2 , respectively. The quiescent X-ray flux observed by Einstein and EXOSAT was reported by Haisch Linsky ( 1980 ) and Haisch et al. ( 1990 ), respectively. We used the ROSAT observations listed in the ROSAT Positive Sensitive Proportional Counter source catalogue (Voges et al. 1996 , 2000 ). Assuming a coronal temperature of 1 keV, sub-solar abundance of 0.5 and using WebPIMMS, we computed energy conversion factor ECF PSPC = erg cm −2 counts −1 to convert the observed ROSAT counts to flux in canonical 0.1–2.4 keV ROSAT energy band. Furthermore, we use the X-ray flux reported by Fuhrmeister et al. ( 2011 ) based on XMM–Newton observations. In Fig. 5 , we plot quiescent flux measured between 1979 and 2017 as a function of time. Furthermore, we assume similar temperature and abundance as before and used WebPIMMS to estimate the flux in the 0.3–4.0 keV energy range. The flux obtained from AstroSat is consistent with the previous estimates of quiescent flux for Proxima Centauri. Figure 5. Quiescent emission flux as a function of time in 0.3–4.0 keV energy range observed by several X-ray missions between 1979 and 2017. An arbitrary sine curve with a period of 7 yr is represented as the blue dashed curve. While a clear cyclic behaviour with a period of ∼7.0 yr is observed for Proxima Centauri in ASAS observations, a similar variation in the X-ray data is not apparent immediately. However, Wargelin et al. ( 2017 ) reported the evidence for a 7 yr stellar cycle using observations from several X-ray missions, principally Swift . The temporal coverage of X-ray observation, the energy resolution and uncertainties due cross-calibration restrict a detailed analysis of the cyclic


no selection criteria are employed – dashed line and open squares – and the luminosity function obtained when the selection criteria previously described are used – solid line and filled squares. It is worthwhile mentioning here that the theoretical luminosity functions have been normalized to the bin of bolometric magnitude , which corresponds to the magnitude bin for which the observed white dwarf luminosity function has the smallest error bars. Thus, since this luminosity bin is very close to the maximum of the luminosity function, the normalization criterion is practically equivalent to a number density normalization. As clearly seen in this figure, the theoretical results match very well the observed data, except for a quite apparent excess of hot white dwarfs, which is discussed in detail below. We note as well that the selection criteria employed by Limoges et al. (2015) basically affect the low luminosity tail of the white dwarf luminosity function, but not the location of the observed drop-off in the white dwarf number counts, nor that of the maximum of the luminosity function. This can be more easily seen by looking at the bottom panel of Fig. 2 , where the completeness of the simulated restricted sample is shown. We found that the completeness of the entire sample is . However, the restricted sample is nearly complete at intermediate bolometric magnitudes – between and 15 – but decreases very rapidly for magnitudes larger than , a clear effect of the selection procedure employed by Limoges et al. (2015) . Nevertheless, this low-luminosity tail is populated preferentially by helium-atmosphere stars and by very massive oxygen-neon white dwarfs. The prevalence of helium-atmosphere white dwarfs at low luminosities is due to the fact that stars with hydrogen-deficient atmospheres have lower luminosities than their hydrogen-rich counterparts of the same mass and age, because in their atmospheres collision-induced absorption does not play a significant role and cool to a very good approximation as black bodies. Also, the presence of massive oxygen-neon white dwarfs is a consequence of their enhanced cooling rate, owing to their smaller heat capacity. Fig. 3 Top panel : probability test as a function of the age obtained by fitting the three faintest bins defining the cut-off of the white dwarf luminosity function. Bottom panel : white dwarf luminosity function for the best-fit age. Fig. 4 Top panel : probability test as a function of the age of the burst of star formation obtained by fitting the nine brightest bins of the white dwarf luminosity function. Bottom panel : synthetic white dwarf luminosity function for a disk age of Gyr and a recent burst of star formation (black line), compared with the observed white dwarf luminosity function of (red lines). 4.2. Fitting the age Now we estimate the age of the disk using the standard method of fitting the position of the cut-off of the white dwarf luminosity function. We did this by comparing


lighthouse effect. Comparing the neutrino modulation frequency with the GW frequency, we found that the neutrino peak frequency is m /2 times larger than the GW peak frequency, where m represents the deformation mode of the PNS (around the rotation axis) induced by the low- T /| W | instability. In the non-rotating model, the neutrino modulation frequency, as previously identified, is almost the same as the SASI frequency, and it is half of the GW frequency. We discussed the detectability of these neutrino modulations by IceCube and HK at a source distance of 10 kpc. If we observe our rotating models along the equatorial direction, the characteristic peak in the power spectra for the event rates at IceCube would be above the noise level. The characteristic peak due to the SASI in our non-rotating model observed along either polar or equatorial directions is also above the noise level. This analysis indicates that the joint observation of the GW and neutrino signals would be useful to extract information on growing multiple modes of non-axisymmetric instabilities such as the low- T /| W | instability and the spiral SASI activity, which is otherwise inaccessible by electromagnetic-wave observations. Finally, we shall discuss several major limitations of our work. Even when the pre-collapse core is a slow rotator (Heger et al. 2005 ) but with a sufficiently weak initial magnetic fields, the magnetic fields may be amplified to a dynamically relevant strength by the magnetorotational instability (MRI, e.g. Akiyama et al. 2003 ; Obergaulinger et al. 2009 ; Masada, Takiwaki Kotake 2015 ; Rembiasz et al. 2016 ). Although the role of the MRI on the post-bounce dynamics remains still unclear, the magnetic fields should play a key role in the angular momentum transport especially in the vicinity of the (differentially rotating) PNS surface. Possible impacts on the growth of the low- T /| W | instability with magnetic fields remain to be clarified. If progenitors have both strong initial magnetic fields ( G) and rapid rotation, the magnetic fields are amplified due to the field wrapping to dynamically relevant strength shortly after bounce, leading to the formation of magnetohydrodynamic (MHD) jets. This would also make the post-bounce dynamics such as the shock evolution and the post-shock flow very different comparing to those of the non-magnetized models. Such difference should also affect the angular momentum transport, not to mention the onset criteria of the low- T /| W | instability. Although there are a few recent 3D MHD core-collapse simulations, the development of the low- T /| W | instability has been only reported in Scheidegger et al. ( 2010 ) and suggested in Kuroda et al. ( 2020 ) (see, however, Mösta et al. 2014 ). Also, the convective dynamo may amplify the magnetic fields and modify the dynamics in the late phase of the core collapse (e.g. Raynaud et al. 2020 ). To answer these questions, one apparently need to perform 3D GR-MHD


1. Hipparcos AGB stars. This list is biased toward nearer stars, mostly within 1 kpc. McDonald et al. ( 2012 ) obtained effective temperatures and luminosities of 107,619 Hipparcos stars (ESA Publications Division 1997 ) based on photometric data covering from the UV to the far-IR. Our search was limited to Hipparcos stars restricted to T eff 3500 K and 1000 ( L / L ⊙ ) 54,000, which eventually reduced the number of candidates to 801 objects. The maximum temperature and minimum luminosity (Salaris Cassisi 1997 ) are criteria intended to reduce the contamination of the sample by RGB stars, while the upper limit of luminosity corresponds to the tip of the AGB ( M bol = − 7.1; Wood et al. 1983 ) that separates massive, evolved AGB stars from supergiants in the HR diagram. Eventually, the spectral classifications of the selected stars were examined in the SIMBAD database to exclude symbiotics, CVs, and other stars that had their temperature and/or luminosity miscalculated by McDonald et al. ( 2012 ). 2. Mass-losing AGB stars. In order to include stars with a circumstellar dust shell ( yr −1 ), a sample of 563 mass-losing AGB stars detected with the Japanese experiment IRTS (Murakami et al. 1996 ) was selected (Le Bertre et al. 2003 ). 3. S-type AGB stars. This class of objects represents the link between O- and C-rich AGB stars and includes two distinct subcategories: (i) the intrinsic S stars are genuine AGB that have their atmosphere enriched by s-process elements brought from the bottom of the convective shell by the third dredge-up process; (ii) the extrinsic S stars have accreted mass from a former AGB companion (now a WD) and generally do not show overabundances of s-process elements. We used the 2nd. edition of the General Catalog of S Stars (Stephenson 1984 ), that contains 1347 stars exhibiting ZrO or LaO bands in their spectra, obtained after various sky surveys. 4. C-rich AGB stars. C-rich stars are the result of the third dredge-up during the thermal-pulse AGB phase, which causes the abundance of carbon to exceed that of oxygen. We adopted a corrected, machine-readable version of the original General Catalog of Galactic Carbon Stars by Stephenson et al. ( 2001 ) available on the Vizier platform, containing 6891 objects. 3. AGB Counterparts in the 4XMM Catalog We cross-correlated the sample of AGB stars described above with the 4XMM-DR9 catalog 3 3 http://xmm-catalog.irap.omp.eu and assumed an X-ray source in this catalog to be a candidate counterpart of an AGB star if their angular distance is within three times the individual positional uncertainty (i.e., 3 × σ pos ). After discarding symbiotic systems, other X-ray emitters, and misclassified AGB stars, we were left with a sample of 19 AGB stars with candidate X-ray counterparts. We then proceeded to download all available XMM-Newton EPIC data of these sources from the XMM-Newton Science Archive (XSA). 4 4 http://nxsa.esac.esa.int/nxsa-web/#home A preliminary inspection revealed that


the mass, metallicity, effective temperature and individual frequencies in the same manner as before and computed masses and radii using the scaling relations iteratively with the reference functions described in equations ( 3 ) and ( 4 ). The results are shown in Figs 3 and A3 . These figures show that the reference functions that we present here indeed remove trends that are present in the mass and radius determinations as a function of max and mitigate the biases to a large extent. However, the standard deviations from using either equation ( 3 ) or ( 4 ) are very similar and significantly larger than in the ideal case, showing the impact of uncertainties in the observations. Figure 3. Same as Fig. 2 but now with the initial values of T eff , max and [Fe/H] perturbed with typical uncertainties (see the text for more detail). 4.1 Hare-and-hound validation We have thus far demonstrated that the new functions improve mass and radius estimations for the data they fit. In order to prove the general efficacy of our new corrections, we appraise the performance of the functions on data to which they have not been exposed. We perform a series of hare-and-hound validations using stellar models calculated with two additional evolution codes. We used garstec (Weiss Schlattl 2008 ) and mesa (Paxton etal. 2011 ) to compute grids of stellar tracks from which we draw testing data. Over 1000 models from each grid were extracted to serve as hares with the requirement that each selected model falls within the verified mass, metallicity and max ranges of the correction functions. The choice of microphysics and modelling assumptions differ between these two codes and they also differ from the code used to calculate the training data in yrec . Thus, mass and radius predictions of the hares will indicate the degree of stellar evolution code dependence in our function. Both the garstec and mesa grids were computed using OPAL opacities (Iglesias Rogers 1996 ) in conjunction with Ferguson etal. ( 2005 ) tables at low temperatures. We utilized the updated 2005 tables for the OPAL equation of state (Rogers Nayfonov 2002 ) along with the assumption of an Eddington plane-parallel atmosphere. We have taken the recommended nuclear reaction rates from the NACRE collaboration in combination with those from Caughlan Fowler ( 1988 ). The treatment of convection follows the mixing length theory as described by Kippenhahn, Weigert Weiss ( 2012 ) and models were computed with the respective solar-calibrated mixing lengths. Radial mode frequencies were calculated with the adipls pulsation package (Christensen-Dalsgaard 2008 ) for the garstec models and with the gyre pulsation package (Townsend Teitler 2013 ) for the mesa models. The large frequency separation was determined using a weighted median around the scaled max as per Bellinger etal. ( 2016 ). The grids differ in that the garstec models assume Asplund etal. ( 2009 ) solar abundances and were computed without


row: particles reach a quasi-steady-state scale height in which downward settling is balanced by upward turbulent diffusion: . Across the columns, we vary the nondimensional turbulent viscosity: . Simulations with radiative damping: we have simulated ZVI with the initial Run_Brunt_Step profile with vertical domains from z = −3 H to z = +3 H for 2000 orbits with nonuniform vertical cooling. The simulations were local at one radius—that is, the radial extent of the computational domain was . Because of additional damping near the vertical boundaries (for stability), the effective domain is . Thick, blue vertical dashed lines indicate simulations that did not develop ZVI, whereas thick, red vertical solid lines indicate that ZVI was triggered and developed into full zombie turbulence. The vertical extent of these lines indicates the effective size of the computational domain. Note that when no part of the computational domain or only a small part of the computational domain extends into the gray region, ZVI is not triggered, while simulations in which the stratified regions of the computational domain have significant overlap with the gray regions do develop ZVI and turbulence. 4.4. Simulations of ZVI with Radiative Damping If the photon mean free path is longer than the largest scale of interest within a flow, then all scales are optically thin and radiative transfer can be modeled with a size-scale independent Newton cooling (i.e., radiated power proportional to temperature difference with respect to some equilibrium value). On the other hand, if the photon mean free path is shorter than the smallest scale of interest within a flow, then all scales are optically thick and radiative transfer can be treated as simple diffusion. A significant challenge is the case where, at the same location in space and time in the flow, small fluctuations are optically thin and large fluctuations are optically thick; using Newton cooling will overdamp large fluctuations, while using a diffusion operator will overdamp small fluctuations. Monte Carlo techniques for radiative transfer are ideal for cases that do not fit safely within the optically thin or optically thick regimes, but these methods involve a substantial investment in additional coding and computational resources. Our goal here is more modest: we want to determine how much radiative damping baroclinic critical layers can withstand and still support the development of ZVI. In Appendix A , we report on simulations with uniform stratification that confirm that ZVI requires Newton cooling times longer than a few orbital periods. In real PPDs, gravity and stratification are not uniform. Not only that, the radiative relaxation time can vary in position within the disk and is especially sensitive to how much the dust has settled. The baroclinic critical layers are most sensitive to radiative cooling; their radial extent is much less than a gas scale height and is expected to be optically thin. Thus we


is small compared with the speed of the flow, and the inertial term in Equation ( 2 ) can be ignored. This eliminates the variable, v θ from the induction equation, so it can be rewritten as: Therefore, expansion of the plasma () corresponds to attenuation while compression () corresponds to amplification of b θ . From the preceding discussion it follows that the latter situation is more likely to occur in downdrafts. Equation ( 4 ) looks remarkably similar to the vorticity equation for an incompressible fluid:where ω denotes vorticity in the z direction. The right-hand side of Equation ( 5 ) represents the well-known vortex stretching effect: the angular velocity of a vortex tube increases when it is stretched and decreases when it is compressed (Spurk Aksel 2008 , p. 100). In contrast, Equation ( 4 ) shows that the θ component of the magnetic field increases when the flow decelerates and decreases when it accelerates. The fact that the twist amplification stems from the induction equation emphasizes the magnetic nature of the process. Twist amplification may be thought of as a magnetic analogue of vortex stretching. In a more general situation, when the magnetic field is not weak, the twist evolution requires a more detailed treatment. In the following sections we study the twist amplification analytically as an eigenvalue problem and numerically, and analytically. We also discuss the energy source, and the physics of the amplification process. 3. UNSTABLE TWISTS IN AN EXPONENTIALLY DECAYING DOWNDRAFT The set of governing Equations ( 1 ) and ( 2 ) can be reduced to a single second order PDE for the magnetic field perturbation, b θ (Taroyan 2008 ):where , and . In deriving the above equation, we have used the equation of mass conservation. In a static medium with , the well-known wave equation can be derived from Equations ( 1 ) and ( 2 ): Equation ( 7 ) was first analyzed by Ferraro ( 1954 ) in the context of Alfvén wave propagation in the solar atmosphere. Solutions in terms of the Bessel functions were constructed for an isothermal atmosphere with an exponential density profile:where z 0 is the scale height. The corresponding Alfvén speed is given by:Note that in our notations z denotes depth. Equation ( 7 ) with the profile ( 8 ) was subsequently analyzed by different authors. They found standing wave solutions in terms of the Bessel functions (Ferraro 1954 ; An et al. 1989 ) or propagating wave solutions in terms of the Hankel functions (Hollweg 1978 ; Cally 2012 ) depending on the imposed boundary conditions. An advantage of the smooth exponential profile is the absence of any model dependent artificial reflections that may arise due to discontinuities in the Alfvén speed or its derivatives (Cally 2012 ). One of the drawbacks is the finite Alfvén travel time to (An et al. 1989 ). In studies of the convection zone, it is common to adopt a polytrope with a linear temperature. Here, we adopt the exponential profiles ( 8 ) and ( 9 ) with


(see Section 2.4 ). For solar metallicity μ converges to ∼2.4 in the CNM, to ∼1.2 in the WIM, and to ∼0.6 in the CGM. As we deal with only low-density environments in the simulations the assumption of an ideal gas is legitimated. Hence, γ = 5/3 in all simulations. 2.2 Self-gravity The effect of self-gravity for an arbitrary matter distribution is described by solving the well-known Poisson’s Equation for the gravitational potential ( G denotes Newton’s gravitational constant). Each mass element of contributes to . If the total mass becomes a significant fraction of the (temperature-dependent) Jeans mass M J for a particular , self-gravity cannot be neglected as it relevantly affects the dynamics of all mass elements. However, it has been shown in Sander Hensler ( 2019 ) that self-gravity has a measurable effect on the evolution of even clouds with masses well below their Jeans mass. If self-gravity is stronger than the acceleration of gas in the opposite direction (e.g. due to evaporation or advection by a streaming CGM), the net acceleration at the cloud surface points in the same direction like the density gradient. Hence, the condition for Rayleigh–Taylor instability (RTI) is not given. So, it is expected that self-gravity significantly suppresses RTI at the cloud surface (see Murray et al. 1993 ) and the typical finger-like structures do not form. According to Murray et al. ( 1993 ), the minimum mass of self-gravitating clouds above which they are stable against Kelvin–Helmholtz instability (KHI) when moving with a certain speed v rel through an ambient medium with density ρ CGM reads with being the average mass density in the cloud. To ensure that self-gravitating clouds, which are embedded in a medium with pressure P CGM , are gravitationally stable initially, their total mass must be below their Bonnor–Ebert mass (Ebert 1955 ; Bonnor 1956 ) Here, and are the average temperature and mean molecular weight in the cloud, respectively. 2.3 Metallicity Observations indicate a common metallicity for CHVCs of 0.1 to (Wakker 2001 ). In order to study extreme cases, we consider two different initial cloud metallicities of and . We note that the CGM has constant metallicity in all runs (see Section 4.3 ). The ratio of the abundances of hydrogen and helium is assumed solar and so are the ratios of metal abundances. 2.4 Dissociation and ionization of elements The simulations contain cool clouds and hot plasma, so a temperature range from a few hundreds up to several 10 6 K is covered. For T 900 K molecules exist, but they successively dissociate and finally become totally ionized with rising temperature. In our simulations, we semi-analytically calculate both dissociation and ionization for hydrogen in thermodynamic equilibrium. The fraction of H ii is then related to He and metals. We note that we do not trace single species but rather have one fluid for which the composition is calculated based upon a given metallicity. The fractions of


becomes more highly extinct and fainter. Fig. 4 LUCI J -band image of T Tauri from November 2016 displayed on a logarithmic scale. The dotted arc indicates the northern arm of the reflection nebulosity discussed by Stapelfeldt et al. (1998) . The yellow arrows indicate the tadpoles. The red arrows mark bow-shaped features. Extended features around an apparent cavity are enclosed within the black dotted lines. The green, orange, and red dots identify the locations of T Tauri N, T Tauri Sa, and T Tauri Sb, respectively. The inset shows the tadpoles and the cavity with a different gray-scale cut. Fig. 5 Zoom of Fig. 4 on the area southwest of T Tauri N showing the coiling structure and the reflection nebulosity features R2 and R3 (as shown in Fig. 2 of Kasper et al. 2016 ) in the vicinity of T Tauri S. The dotted line connects the inflection points of the coil. The red arrow indicates T Tauri Sa. Fig. 6 T Tauri H 2 line emission map obtained with LBT-LUCI in October 2016. Only features which cannot be explained by image processing artifacts are indicated and discussed in the text. The positions of the stars are indicated by colored dots (T Tauri N, Sa,and Sb are in green, orange, and red, respectively). Fig. 7 IRDIS J -band image of the coiling structure taken in December 2018. The white dotted line indicates the coiling structure’s position in December 2014 (see Kasper et al. 2016 ). The relative distance to the T Tauri stars can be determined from the orbital solution and radial velocity (RV) measurements. According to the best-fit T Tauri N-S orbit ( Schaefer et al. 2020 ), T Tauri S went through the line of nodes in 1936 with an RV difference to T Tauri N of about ± 6.5 km s −1 . The sign of the RV difference cannot be determined from an astrometric orbit solution. Duchêne et al. (2005) found that the heliocentric RVs of T Tauri Sa and Sb appear to be larger than the one of T Tauri N ( Hartmann et al. 1986 ) by 2–3 km s −1 . These values are compatible with the orbit prediction within the error bars if T Tauri S previously went through the ascending node. Since then, T Tauri S has been moving behind T Tauri N for about 80 yr and would now be located roughly 90 AU behind T Tauri N. However, the N-S orbit is still only loosely constrained and orbital solutions may exist for which T Tauri S is located in front of T Tauri N, as suggested by Beck et al. (2020) . Fig. 8 IRDIS J -band image of the R4 structure taken in December 2018. The white dotted line indicates R4’s position, and the “+” symbols indicate the positions of T Tauri Sa and Sb in December 2014 (see Kasper et al. 2016 ). Fig. 9 IRDIS J -band image of the R3 structure taken in December 2018. The white dotted line indicates R3’s position in December 2014 (see Kasper et al. 2016 ). Figure 1 also shows a bow shock at ~1′′.6 west of T Tauri. This bow shock is a signpost of the east–west outflow ( Böhm Solf 1994 ; Herbst et al. 1997 ) with the western part approaching (blue-shifted) at about


fully convective (e.g., Spada et al. 2017 ). A radius of ≈0.4 R ⊙ leads to cycle times of about 4–5 years, provided that η T remains unchanged. If η T of cooler stars is reduced with respect to the solar value, as implicitly suggested by the stellar models of Spada et al. (2017) , then the cycle time from Eq. ( 1 ) basically exceeds this limit. Mixing-length estimates of 0.33 u rms l corr for M dwarfs provide about 30% of the solar value, so that the dynamo cycle times of M dwarfs should slightly exceed the duration of the solar cycle. Indeed, the cycle time of one star in Fig. 1 slightly exceeds the 11 yr (≈4000 d) limit but this turned out to be a K2 dwarf star with a rotation period of 22.8 d ( Johnson et al. 2016 ). Figure 1 shows the observed cycle lengths for M dwarfs to be generally shorter than ≈3000 d. GJ270 (spectral type M2) shows the longest known cycle period of any M star with 2700 d (7.4 yr), while the shortest cycle is seen in GJ476 (M4) with 1060 d (2.9 yr) ( Robertson et al. 2013a ). GJ328 as an M0 dwarf with 0.69 M ⊙ shows an activity cycle of ≈2000 d, while a Jupiter-mass planet orbits it with a 4100 d period ( Robertson et al. 2013b ). We note that in the solar system the orbital period of Jupiter coincides with the length of the solar activity cycle, accidental or not. GJ581 possesses an entire planetary system with four or possibly six planets and shows an activity cycle with a length of 1600 d ( Gomes da Silva et al. 2012 ; Robertson et al. 2013a ). The rotation period of the latter star is surprisingly long with 130 ± 2 d ( Robertson et al. 2014 ). Fig. 1. Cycle times vs. rotation period (both in days) for early M stars (rhombic symbols). Among the group of rapid rotators with rotation periods less than a day ( Vida et al. 2014 ) are also the early M dwarfs EY Dra and V405 And ( Vida et al. 2013 ). The remaining data are from Suárez Mascareño et al. (2016) ; Wargelin et al. (2017) , and Distefano et al. (2017) . For GJ476 the cycle time is known but not the rotation period (blue bar). The Schwabe cycle of the Sun is indicated as an asterisk denoted SUN. Gomes da Silva et al. (2012) give further examples of M dwarfs with cycle lengths between three and five years. It is of particular importance to know the relation of the cycle time with the rotation time for a homogeneous sample of stars. Savanov (2012) used light curves of cool M dwarfs to find long-term cycle variations up to timescales of years. His sample is mixed by M stars with outer convection zones and also fully convective stars. There seems to be no systematic dependence of the cycle time on the rotation time even for subsamples with slow and fast rotations. An independent study about the coupling of rotation and cycle times for fast-rotating cool stars has been published by Vida et al. (2014) . Among the analyzed 39 Kepler light curves of late-type stars with rotation periods shorter than one day, nine examples showed cyclic variations with periods between


by comparing the observed spectral energy distribution (SED) to the sum of the SEDs of both models. The reddening is modeled following Cardelli et al. ( 1989 ). From the total luminosity, the fixed light ratio and the other determined parameters, we then derive the individual luminosities. The final spectral fit is shown in Figure 4 . The inferred parameters are given in Table 4 . The spectroscopic mass for the O-type secondary is derived from , where G is the gravitational constant. The spectroscopic mass of the WR primary stems from mass–luminosity relations calculated for chemically homogeneous WR stars by Gräfener et al. ( 2011 ), which for a H-free star depends only on L . The error on the spectroscopic mass of the WR star is propagated from the error in L , and is possibly underestimated in its lower limit given that the WR primary may be chemically inhomogeneous, which would lower M for a given L . Figure 4. Upper panel: comparison between the observed SED (FUSE, IUE, and UBVJHK photometry, blue lines and squares) and the synthetic PoWR SED modeled for WR 133 (red dotted line). The synthetic SED is the sum of a PoWR model for the WR primary (black solid line) and O-type secondary (green dashed line). Lower panels: comparison between the observed and synthetic normalized spectra. See the text for details. Table 4 Inferred Physical Parameters of WR 133 from the PoWR Spectral Analysis Note. a The transformed radius should not be confused with the physical radius of the star; see the text for details. 5. Discussion Sana et al. ( 2012 ) found that about 75% of massive OB stars should have their evolution impacted by a companion. The general assumption used in the analysis of binary populations is that two stars will interact if the system is born with a semimajor axis of less than ∼10 au and the star evolves into a red supergiant (e.g., Sana 2017 ). During a binary interaction, the system initially becomes more compact until the mass ratio is unity and then increases until mass transfer eventually ceases. This has been seen in an active phase for binaries that could become similar to WR 133 with systems such as the LBV-like binaries MWC 314 (Richardson et al. 2016b ) and HDE 326823 (Richardson et al. 2011 ). The supergiant companion to the WR star in WR 133 could have a larger rotation rate than an average field star. The inferred value of of 96 km s −1 is similar to the average value found for O supergiants by Simón-Díaz Herrero ( 2014 ). However, if there were indeed past binary interactions, the O9I star should be rotating in the orbital plane, and thus would likely be a rapid rotator. In fact, if we use our derived value for and the inclination of the system imply a modestly fast rotation rate of 280 km s −1 , which is a reasonably fast rotation for a supergiant O star. However, if the O star had a magnetic field from the interaction, then the rotation rate could be drastically less than critical in a short time (Potter et al. 2012 ).


0.04 and 0.03 mag for and distance modulus, with metallicities from Rey et al. , and 0.06 and 0.05 mag for the and values derived with the metallicities from Sollima et al. At the same time, results from previous studies (e.g., Cacciari Clementini 2003 ; Del Principe et al. 2006 ; Feast et al. 2008 ; Gran et al. 2015 ) suggest that distances derived on the basis of such relations can probably not yet be considered accurate to better than about mag. Thus, we adopt the latter value as a conservative estimate of the systematic error affecting the distance moduli provided in our study. Fig. 8 Fundamental (in purple circles) and first-overtone (in yellow triangles) SX Phe as classified by Olech et al. (2005) , based on their frequency spectrum. The dotted lines are the PL relations derived previously. As can be seen, there are some claimed first-overtone pulsators that also follow the relations defined by fundamental-mode pulsators. Table 4 Adopted distance moduli based on T2Cs and RRL stars. 4.2. Pulsational distance modulus from T2Cs and RRL stars Table 4 lists the weighted-average distance modulus values from T2Cs and RRL stars and sigma as the dispersion around the mean. The weights for each individual distance modulus values were defined as the inverse of the squared of the total error, including statistical and systematic errors (summed up in quadrature). As expected, the error is lower for the PL relations compared to , since the former have lower intrinsic dispersion, and is based on a more extensive dataset. Comparing both variability types, T2Cs lead to a less accurate distance modulus, probably because they are few in number, which increases the statistical error, and they have higher systematic errors since the effect of metallicity and pulsation mode separation in the derivation of the PL relations is not completely established. Considering the distance modulus values found using T2Cs stars as well as RRL stars, both in and , the weighted-average distance modulus to Cen was determined, leading to a final value of mag, where the error bar corresponds to the standard deviation of the mean. Given the discussion of systematic errors in the previous section, our final adopted distance modulus is mag, where the error bars correspond to the adopted statistic and systematic errors, respectively. Adding the errors in quadrature (as recommended, for instance, by Barlow et al. 2002 ), this corresponds to a heliocentric distance of kpc. Table 5 Distance modulus for Cen from the literature. Notes. (a) Most of the error estimates do not consider the contribution of systematic errors, which in the case of T2Cs and RRL stars can reach values of order 0.10 mag. (b) The listed error was determined considering the internal uncertainties of the mean distance modulus value, 0.02 mag, and the observational error, 0.1 mag, as described by McNamara (2000) . (c) Based on the -[Fe/H] relation from Bono et al. (2003) and mean magnitudes from Kaluzny et al.


environment theoretically depends on the Mach number or wave amplitude (e.g., Priest 2014 ; Vršnak et al. 2016 ). As the shock wave propagates, its amplitude decreases due to the geometric expansion of its front and the fact that its leading edge is propagating faster than its trailing edge. The amplitude decrease results in the deceleration of the shock wave. Additionally, a power-law fit is a better representation of the distance–time curves of freely propagating shock waves (Warmuth Mann 2011 ) according the the Sedov solution (Sedov 1959 ). On the other hand, we found that the speed of the reflected wave (243.1 ± 30.8 km s −1 ) is much smaller than the speed of the incident wave (which approximately equals that of the unaffected wave at around 05:13 UT: 591.0 ± 25.9 km s −1 ). In other words, the reflected wave was slower than the primary wave. There may be several possibilities for this feature. The primary wave is a nonlinear wave or shock, which is propagating with a Mach number larger than unity, while the reflected wave had changed to be a linear one traveling at the characteristic speed (Liu Ofman 2014 ; Warmuth 2015 ). The reflected wave has to propagate in the rest frame of the flow field associated with the primary wave, which may result in an apparently lower speed (Kienreich et al. 2013 ). Additionally, it may be related to the properties of the medium (magnetic separatrix) causing the shock wave to reflect. The compressible magnetic separatrix may cause the decrease in the reflected wave speed, when the shock wave reacted with the magnetic separatrix. Moreover, it is interesting to find that the refracted and diffracted waves decelerated with a bigger nonconstant deceleration rate. We find that their deceleration rate was decreasing with time, as in a damping motion. These phenomena would help us to further understand the nature of the interaction between magnetohydrodynamic shock waves and the local magnetic field. Appendix Error Estimations In this study, we determine the dynamic characteristics of EUV waves by using the method in Liu et al. ( 2010 ). The track technique ignores the heights of EUV waves by assuming that all emission originates from the spherical solar surface, and this distance measurement was corrected by the sphericity of the solar photosphere. This method would cause an overestimation in the measurement of some parameters (e.g., distance, velocity). We will supply some error estimations in subsequent sections. As Figure A1 (a) shows, a coronal structure B at a height of Δ h above the solar surface is projected onto the solar surface A observed by an observer on Earth (E). We set a new “north pole” at the disk center ( D ) and make AC = d , AC ⊥ OE, OA = OD = R ⊙ , OB = OF = R ⊙ + Δ h , and DE = 1 au. Defining θ = ∠AEO, we can get We make OG ⊥ EG and get Then, we getAccording to the law of sines, in △BOE, we can get Based on the above equations, we can get whereFirst, if the EUV wave originally


between 3D and 1.5D calculations might be quite influenced by ingredients of the models that require improvement. We show the relevance of these ideas commenting on two of the few papers existing on this topic. The key is that the influence of the 3D radiation field on the Hanle polarization has been posed since the beginning in terms of inhomogeneities, and not in terms of velocity gradients. The initial conclusions in this regard were obtained by Manso Sainz Trujillo Bueno ( 2011 ) with an analytical/numerical study approaching the problem in the static regime. In more recent numerical studies the action of 3D velocities in the polarization is included, but the contributions of vertical and horizontal velocity gradients are not separated. As the effects of horizontal velocity gradients and inhomogeneities are in turn blended, it is also unknown how large their relative contributions are to the differences between 1.5D and 3D calculations. Consider, for instance, Figure 3 of Štěpán Trujillo Bueno ( 2016 ), which points out that the Hanle effect at disk center tends to depolarize when it is accompanied by other symmetry breaking effect, as happens in the well-known case of the solar limb. The figure also confirms that macroscopic motions are yet an efficient polarizing source for the spectral line considered (as known from 1D simulations of the Ca ii IR triplet; Carlin et al. 2013 ). However, it is unknown whether the polarization introduced by velocity in that figure is due to the horizontal or to the vertical velocity gradients already considered in 1D calculations. This matters not only for comparing with 1D calculations but also because horizontal velocity gradients can potentially compensate and exceed the polarization created by horizontal inhomogeneities (as explained in previous paragraphs). Thus, the dominant symmetry breaking and polarizing effect at disk center could not be the inhomogeneities, as always affirmed, but the horizontal velocity gradients. Hence, we think that is key quantifying the sensitivity of such results to the model horizontal distribution of velocity gradients , which also can help detach the conclusions from the eventual lack of realism in them. Finally, it should also be noted that when those radiative transfer simulations do not add microturbulent velocity, they use the temperature distribution of MHD models that are known to represent a chromosphere cooler than the solar one. Thus, as temperature influences significantly the effect of the velocities in the LP (see Section 4.2 ), the relative strength of the polarizing effects discussed here might change accordingly. In summary, horizontal velocity gradients compete with other physical mechanisms, such as inhomogeneities and the Hanle effect, for polarizing and depolarizing the light; hence, we need to explicitly quantify whether they are relevant for inferring magnetic fields with the Hanle effect. We also need to find theoretical and observational


ratio moving in circular orbits, the filling of the Roche lobe is followed by violent mass transfer by the primary to the secondary and the formation of a common envelope. On the other hand, an equal-mass binary may evolve into a fully contact system of two stars, both of which fill their Roche lobes simultaneously. In the limiting case, violent mass transfer may occur on the hydrodynamic time-scale, with an amount of released energy of the order of the thermal energy of the components or the common envelope: GM 2 / a 510 48 ( M /M ) 2 ( a /R ), where M is the mass of the component or of the common envelope and a is the radius of the common envelope. For 3M common envelope mass and energy from Table 1 , the envelope radius is about 10R . The high energy of the studied explosion is a consequence of compactness and the relatively high mass of the parent system. From this point of view, it is very important to obtain observational evidence for such processes in binary systems and to determine the corresponding physical properties: the released energy, ejected mass and angular momentum. Ultraluminous red novae may be observational manifestations of violent mass transfer in the binary (Blagorodnova etal. 2017 ). We already pointed out that observation of the progenitor of V1309 Sco in the form of a contact binary with a period of 1.7d (Tylenda etal. 2011 ) provides strong evidence for the merger model of LRNe. This conclusion is also corroborated by the approximate occurrence rate of such events one event in 2040yr per galaxy like the Milky Way or M31. To show this, let us use the distribution of binary semimajor axes a (Masevich Tutukov 1981 ). This distribution has the form d N 0.2dlog( a /R ). Systems with 10 a /R 1000 can pass through the common-envelope stage during the second mass exchange stage. According to the initial distribution of binaries over semimajor axes of Masevich Tutukov ( 1981 ), N 0.2dlog( a /R ), the frequency of formation of common envelopes with radii 610R is about 0.04 per year. Systems with larger component separations are less energetic. 5 CONCLUSIONS In this article, we described the discovery of the nova M31LRN and long-term observations of its light curve with the MASTER network of robotic telescopes. It is an important point that the entire observational part of the study was performed on identical telescopes equipped with identical photometers. The resulting light curve agrees fairly well with the independent light curve published by Kurtenkov etal. ( 2015b ). An analysis of the photometric observations and velocity data for the nova M31LRN made using radiative hydrodynamic modelling showed that the rather long observable plateau (50d) could be achieved only with a higher merged stellar mass M 3M and common envelope radius R 10R . The corresponding explosion energy should be lower, 310 50 erg, whereas the total kinetic energy of the ejected envelope is lower by three orders of magnitude. The proposed