2 (Lanzafame et al. 2018 ). The extended Kepler mission, K2 (Howell et al. 2014 ), has now observed more stars than the original Kepler sample, potentially doubling the number of high precision stellar rotation periods available. Similarly, by the end of the Gaia mission, Lanzafame et al. ( 2018 ) estimate that a sample of 3–20 million rotation periods will be recovered. The era of statistical studies in stellar rotation has therefore begun. Stellar rotation has long been noted as a means to possibly age-date stars due to their constant angular momentum loss via winds (Skumanich 1972 ). While studies of open clusters give hope that this “gyrochronology” model broadly works for solar-type and lower-mass stars, many uncertainties exist about the details of this spindown and its utility as a clock. These include questions about the initial rotation period distribution for stars (e.g., Barnes 2010 ; Matt et al. 2015 ), the specific analytic prescription for modeling the spindown (Angus et al. 2015 ), and exploring the efficiency of this angular momentum loss mechanism at older ages (van Saders et al. 2016 ). One of the most compelling results from the rotating star sample in Kepler is the discovery of a bimodal period distribution. McQuillan et al. ( 2013 ) first found a bimodality in the distribution of field M dwarfs with periods between ∼10 and ∼50 days. This feature was also found in the Kepler field K dwarfs in McQuillan et al. ( 2014 ), but was not seen in the bluer stars. Using Gaia DR1, Davenport ( 2017 ) was able to remove contaminating subgiants from the rotation sample, and found the bimodality extended to the G dwarfs as well. This bimodal surface rotation period distribution is either a new short-lived transition or instability phase of rapid angular momentum loss, or a signature of star formation history imprinted in the present-day rotation period distribution. However, as Davenport ( 2017 ) notes, this feature has only been observed in the Kepler rotation period catalog, and most critically only for stars within ∼300 pc. The faint M dwarfs in the Kepler sample studied in McQuillan et al. ( 2013 ) have an average distance of 250 pc. Gaia DR1 was also only able to provide precise distances for the nearest stars, and Davenport ( 2017 ) report a median distance of 285 pc for the blue stars in their sample. In this paper, we extend the work of Davenport ( 2017 ) in studying the Kepler rotation period distribution as a function of distance from the Sun using new astrometric data from the Gaia mission. By matching the McQuillan et al. ( 2014 ) rotation period catalog to the newest data from Gaia Data Release 2 (Gaia Collaboration et al. 2018b ), we can use precise distances for essentially every star in the McQuillan catalog to select the most-likely main-sequence dwarfs out to distances >2 kpc. Importantly, this filters out both subgiants, the main contaminant noted by Davenport ( 2017 ), and unresolved binary stars. Here we demonstrate


( 2015 ). 2.1. Data Data for these studies were collected by reviewing published papers and searching both exoplanet.eu (Schneider et al. 2011 ) and exoplanets.org (Han et al. 2014 ). We start with the six planets in Table 2 of Schwartz Cowan ( 2015 ) that have thermal eclipses and phase amplitudes (at wavelengths longward of 0.8 μ m): HD 149026b, HD 189733b, HD 209458b, WASP-12b, WASP-18b, and WASP-43b. Then we add WASP-14b (Wong et al. 2015 ), HAT-P-7b, and WASP-19b (Wong et al. 2016 ) to our sample. We also incorporate new data from Zhou et al. ( 2015 ), Evans et al. ( 2015 ), and Line et al. ( 2016 ). We collect first-order phase offsets (Sections 2.2 and 2.2.1 ) from Knutson et al. ( 2009a , 2009b ) and Wong et al. ( 2015 , 2016 ). Knutson et al. ( 2009a ) concluded the offset they found was not statistically significant, so we use their largest uncertainty (72° ± 61°). Phase offsets through second order come from Cowan et al. ( 2012 ), Knutson et al. ( 2012 ), Maxted et al. ( 2013 ), Zellem et al. ( 2014 ), and Stevenson et al. ( 2014c , 2017 ). However, the second-order components in Zellem et al. ( 2014 ) were found to be unnecessary, and those at 4.5 micron in Cowan et al. ( 2012 ) are disputed, so we use neither in our fits. Also, Cowan et al. ( 2012 ) reported two sets of fit parameters for WASP-12b based on different models for detector systematics 7 7 Stevenson et al. ( 2014a ) fit eclipse depths for WASP-12b but not phase parameters, so we do not use their values in our analysis. —we use parameters from their preferred polynomial model but test the other scenario in Section 3.1 . 8 8 WASP-12 has binary companion stars that affect photometry of the system (Bechter et al. 2014 ). We use dilution factors from Stevenson et al. ( 2014b ) to correct observations of WASP-12b when appropriate. All nine planets in our study have non-zero offsets in at least one waveband. 2.2. Model We take an energy balance approach to interpreting thermal phase variations (e.g., Cowan Agol 2011b ): we compare the radiation going into and coming out from a planet to infer bulk energetics of that planet’s atmosphere. For our study, we use the energy balance model described in Section 3.1 of Schwartz Cowan ( 2015 ), which accounts for uncertainties in system parameters, as well as reflected light contamination, meridional heat transport, and other sources of uncertainty. Those authors treated stars as blackbodies; we estimate better stellar brightness temperatures for each observation by using BT-NextGen spectra (Allard et al. 2012 ). Then we use our compiled data to calculate the relative intensity of planets and their host stars at each observed wavelength. In Cowan Agol ( 2011b ) and Schwartz Cowan ( 2015 ), this nightside intensity ratio, ψ n ( λ ), is defined aswhere δ ecl is the eclipse depth, δ var is the peak-to-trough phase amplitude of the full light curve, and δ tr is the transit depth. This is exact only when an observation has no phase offset


ALMA data (Decarli et al. 2017 ; Champagne et al. 2018 ). In Figure 9 we show the postage stamps of our follow-up observations. Figure 9. Source adjacent to the quasar J2211, detected solely in the dust-continuum emission, i.e., with no secure redshift measure. Top: postage stamps (20′′ × 20′′) of our follow-up observations; labels are as in Figure 3 . Bottom: spectral energy distribution of J2211c. We assume that the source is located at the same redshift of the quasar. We report our photometric measurements/limits and, for comparison, various templates of local galaxies and the best SED fit from MAGPHYS -high- z . The labels and templates are as in Figure 6 . J2211c SED results to be intermediate between the low- z ULIRG Arp220 and the starbursting galaxy M82 (see Section 3.1 ). On the basis of our follow-up observations, and considering the predicted density of millimeter-sources, we are not able to exclude that this source is a fore/background (see the text for details, and Champagne et al. 2018 ). In analogy with the companions discussed in the main body of the paper, we compare the SED of J2211c with those of local galaxies, and we fit our photometric data with MAGPHYS -high- z (see Figure 9 ). From the latter, we find that the SED of J2211c is better reproduced by a galaxy model in between Arp 220 and M82 (i.e., a powerful local ULIRG and a starburst), with M * ∼ 3 × 10 10 M ⊙ and SFR ∼ 130 M ⊙ yr −1 . We further measure the obscured/unobscured SFR ratio of J2211c, following the procedure used for PJ167c (see Section 3.2 ). The star formation rate is dominated by the obscured contribution (SFR UV ∼ 2 M ⊙ yr −1 and f obscured ∼ 0.99). We report all these estimates in Table 6 . The lack of a secure redshift confirmation prevents us from drawing further conclusions on the nature of this source, or from placing it in the context of previous observations. Appendix B Quasar Photometry In the framework of our study of companion galaxies, we also perform forced photometry at the position of the quasars in the Spitzer /IRAC and HST /WFC3 images (see Section 2 for methodology). In Table 7 we report the derived quasars’ photometry. In Figure 10 , we show the quasars SEDs. The fluxes measured in our follow-up data are consistent with those expected from the observed optical/NIR spectra, when available, and/or from a lower- z quasar template (Selsing et al. 2016 ) rescaled to match the observed J band magnitude. Figure 10. Spectral energy distribution of the quasars in our sample. The observed photometric measurements (filled points) are obtained from our new follow-up data and from the literature (see Table 7 ; the filter responses are reported in the lower right panel). We also show the available optical/NIR spectra (light gray; see also Mazzucchelli et al. 2017b ), and a lower-redshift composite template shifted at the redshift of the quasar (black solid line; Selsing et al. 2016 ). The location of the Ly α line is highlighted with a light blue


for % of the total instantaneous flux, and % of the total maximum flux, in good agreement with the results of Lamb et al. (2013) . Here and in the next paragraph again, we assume that, the term “total instantaneous (maximum) flux” corresponds to the sum of instantaneous (maximum) fluxes in those events considered in the context. 8. Cancellation in Lamb et al. (2013) accounts for % of the total flux. If we leave out the fluxes contained in surviving features in semi and partial cancellation (as done in Lamb et al. 2013 ), we obtain % total instantaneous flux and % total maximum flux that corresponds to cancellation event, which is comparable to the results obtained by Lamb et al. (2013) . 4.5. Lifetimes of the features In Fig. 11 we plot histograms of lifetimes of the features, where the lifetime is determined as described in Sect. 3.3 . A correction factor was proposed by Danilovic et al. (2010) to avoid underestimating the lifetimes for the same observations, as the duration of the time-series is itself comparable to the lifetimes of longer-lived features. The suggested weight to be multiplied to each bin of the lifetime distribution is , where is the number of frames a feature lives, and is the total number of frames. We show lifetime histograms before (yellow-shaded) and after (gray-shaded) applying the correction factor. The plotted histograms are for the two extreme ratios of the areas of the children or parents, 10:1 and 2:1 respectively. Nearly % and % of the features for the 10:1 and 2:1 cases respectively, in the lifetime histograms live for just one time step. The more restrictive the area-ratio criterion restricting the death of features, the larger the number of long lived features, since a higher number of splitting and merging events are not counted as contributing to the birth or death of the larger feature (although for the smaller features they are involved in these events). Fig. 12 Lifetimes of features for an area ratio criterion 5:1 vs. maximum fluxes (panel a) ) and maximum, average magnetic field strength values from COG technique (panel b) ). Here maximum refers to the maximum value of the magnetic flux or the field strength, gained during the entire lifetime of the feature. In panel a) the range of plotted maximum flux of feature is limited to Mx to enhance the visibility of flux-lifetime relationship. The over-plotted red cross symbols represent the average value of the maximum flux of features at each lifetime (see main text for details). The maximum averaged over each lifetime bin are displayed in the figure and are represented by over-plotted red cross symbols. The mean and median lifetimes are and min ( and s) respectively for the 10:1 case and are and min ( and s) respectively for the 2:1 case. Similar lifetimes were also reported by Zhou et al. (2010) . These small mean and median lifetimes are clearly dominated by the large number of features that live for just one time step. Thus, the mean lifetimes we


in Section 3 and the recipe given in González Delgado et al. ( 2016 ). In a nutshell, for any given galaxy we compute the spaxel-to-spaxel mass of stars younger than 32 Myr. Then, by co-adding these masses over a given aperture we can estimate the average rate of star formation during this period. An extensive discussion about the optimal limiting age of the stellar population used to define the recent SFR is given in González Delgado et al. ( 2016 ). We tested the robustness of this assumption by repeating Fig. 1 using a limit of both 100 and 200 Myr. We find that the results are qualitatively in agreement. Our stellar population approach to compute the SFR is different from the usual H α luminosity proxy used in the literature (Kennicutt 1998 ; Catalán-Torrecilla et al. 2015 ). Nevertheless, it is most powerful for ETGs since it does not depend on the detection of emission lines. Fig. 1 also shows the SFR versus M ⋆ relation for our comparison sample of ellipticals (see Section 2 ) and for our ETGs without considering the bulge and disc components separately. It is worth noticing that, due to the optical coverage of the spectroscopic data, our SFR might be considered as upper limits for very low values of the SFR (see discussion in López Fernández et al. 2018 ; Bitsakis et al. 2019 ). This issue is more acute for our bulge SFR estimations, while it is almost negligible for the star-forming discs. However, this does not affect the main conclusions of this paper. Figure 1. Star formation rate (SFR) versus stellar mass ( M ⋆ ) distribution for our sample of ETGs. The measurements for bulges and discs are shown with red circles and blue stars, respectively. Black squares represent measurements of the same ETGs, but for the galaxy as a whole. Yellow triangles display the position of the sample of elliptical galaxies described in the text. The best fit to the SFMS from Elbaz et al. (Grey; 2007 ), Renzini Peng (Orange; 2015 ), González Delgado et al. (Navy; 2016 ), and Sánchez et al. (Green; 2019 ) are also shown for comparison. Downward arrows mark where the measured SFR is an upper limit (see text for details). It is clear from Fig. 1 that, whenever there is star formation in our ETGs, this is happening in the disc component and not in the bulge. We find that on average the specific SFR (sSFR) is smaller in bulges than in discs. The mean difference in logarithm accounts to at least 1.3 dex when integrated across the whole FoV of CALIFA. In addition, a qualitative comparison with literature values of the best fit to the SFMS at z ∼ 0 (Elbaz et al. 2007 ; Renzini Peng 2015 ; González Delgado et al. 2016 ; Sánchez et al. 2019 ) shows that several discs of our ETG sample are compatible (or over) with the SFMS, whereas bulges occupy the region of the diagram defined by retired galaxies (Crocker et al. 2011 ; Cano-Díaz et al. 2016 ; Sánchez et al. 2018 ), or even lower SFR values for a given stellar mass. In particular, we find that 30 discs and only


and E05, PSP arrived at the perihelion of 27.8 R ⊙ (~0.13 AU) on January 29, 2020 and June 7, 2020, respectively. In Sect. 3.1 , we give an overview of the PSP measurements of solar wind density, speed, and energy flux for all available encounters including E01, E02, E04, and E05. We note that E03 observationsare not considered due to the lack of SPC observations near the perihelion. For each encounter, 12-day observations around the perihelion were used for calculations. The heliocentric distance for both E01 and E02 ranges from 35.66 to about 55 R ⊙ , and it ranges from 27.8 to about 57 R ⊙ for both E04 and E05. In Sect. 3.2 , we combine the observations from E01, E02, E04, and E05 to show the histogram distributions and the evolution of the energy flux as a function of heliocentric distance. 3.1 Overview of E01, E02, E04, and E05 Figure 1 shows an overview of the PSP measurements of solar wind density, speed, and energy flux during E01 (from October 31, 2018 00:00:00 to November 12, 2018 00:00:00 UTC). The top panel presents the electron number density ( n e ) obtained by the QTN method. In the second panel, the proton bulk speed is shown. The third and fourth panels present the solar wind energy flux ( W , from Eq. ( 2 )) and its value scaled to one solar radius (, from Eq. ( 6 )), respectively. In Fig. 1 , n α and n p were computed from n e based on n α ∕ n e = 2.5% for calculating W and . Most of the time, V p varies around 300 km s −1 , and varies around 70 W m −2 . The average values of W and are 0.045 and 77.3 W m −2 , respectively. The average value of of E01 is consistent with the long-term observations from Le Chat et al. (2012) (around 79 W m −2 ). We note that does not vary much with V p when V p increases abruptly (i.e., from November 8 to 10, 2018). Figure 2 , which follows the same format as Fig. 1 , summarizes the PSP measurements of solar wind density, speed, and energy flux during E02 (from March 30, 2019 00:00:00 to April 11, 2019 00:00:00 UTC). We deduced n p and n α with the same method used for E01to calculate both W and . We note that n e shows two successive low plateaus near the perihelion of E02 (from April 3 to 8, 2019 UT), as shown in the first panel of Fig. 2 , whereas V p shows two high peaks. This is in agreement with the well-known anticorrelation between the solar wind speed and density (e.g., Richardson et al. 1996 ; Le Chat et al. 2012 ). Both and W also show two low plateaus near the perihelion of E02 (from April 3 to 8, 2019 UT), similar to the solar wind density. Elsewhere, V p remains around 300 km s −1 and varies around 70 W m −2 . The mean values of W and during E02 are 0.032 and 59.4 W m −2 , respectively. Similarly, Fig. 3 illustrates the PSP observations during E04 (from January 23, 2020 00:00:00 to February 4, 2020 00:00:00 UTC). We used n p and n α , which were deduced with the same method used for both E01 and E02, when calculating both W and . The second panel of Fig. 3 shows that V p varies


paper. A direct comparison with our kinematic estimate is also difficult because they do not specify at which radius the rotation velocity is measured. Our data however seem to rule out the possibility of a value of 150 km s −1 for the line-of-sight velocity: In such cases, the combined effect of rotation velocity, velocity dispersion, and instrumental smearing would have made the galaxy to spread its emission in channels with velocities well beyond those detected in our data cube (for example, in the first channel of Fig. 2 ). Livermore et al. ( 2015 ) included the data set from Yuan et al. ( 2011 ) in a broader study on the kinematics of a sample of lensed galaxies. Similarly to Yuan et al., they fitted the delensed 2D kinematic velocity map with an arctangent model of the rotation curve (Courteau 1997 ) without any correction for beam smearing and found a rotation velocity V 2.2 = 59 ± 3 km s −1 at R 2.2 = 1.32 kpc for an inclination i = 45°, where R 2.2 is the radius at 2.2 times the scalelength of an exponential disc. Although MUSE data have lower spatial resolution than OSIRIS data, our rotation curve seems to exclude such a low rotation velocity at 1.32 kpc (Figs 4 and 5 ), even after correcting for the different inclination ( V 2.2 ≃ 73 km s −1 for i = 35°). Since Livermore et al. do not report their asymptotic velocity nor turnover radius of the best-fitting arctangent function, we cannot compare the values of the flat part of the rotation curve. To constrain the velocity dispersion, Livermore et al. fitted an exponential model to the velocity dispersion map, which they derived applying a correction for beam smearing and spectral broadening. They quote a luminosity-weighted average of σ = 50 ± 10 km s −1 . This value is however strongly affected by the two inner points (see their Fig. 2 ), which appear to be dominated by residual beam smearing. Finally, we stress that the Livermore et al.'s kinematic model does not satisfactorily reproduce their data (see the residual velocity field in their fig. 1). This is likely due to a combination of the low S/N AO data, some beam smearing, and the uncertainties related to the lensing modelling. Mason et al. ( 2017 ) made use of the 3D fitting code galpak 3D (Bouché et al. 2015 ) on KMOS –λ5007 Å data. Like B arolo , this code fits 3D models to data cubes and properly accounts for beam smearing. The main difference between this code and B arolo is that, in galpak 3D , the rotation curve follows a pre-selected functional form and the velocity dispersion is always assumed to be constant across the disc. Mason et al. have observations of both M1 and M3 images. They could not fully model the M3 image because it is unresolved. M1 image is poorly detected with S/N ∼ 2 in their data, and the KMOS FOV is not sufficiently large to include the entire galaxy disc. They modelled the M1 image without any reconstruction on the source plane and found an asymptotic rotation velocity of 227 ± 32 km s −1 , an


half of the mass that has become unbound during the preceding orbit. The simulations used by Fujii et al. ( 2006 ) followed the orbital evolution of a self-consistent satellite galaxy (without dark matter). As they point out in their paper, since dark matter subhaloes are more extended than satellite galaxies, they typically experience more mass loss and thus should be susceptible to even stronger self-friction. In this paper we use idealized numerical experiments to study the nature of self-friction acting on dark matter substructure. Our main goal is to characterize the relative contribution of self-friction to the total drag force experienced by subhaloes as a function of orbital properties and the mass ratio between the sub- and host halo. The rest of the paper is organized as follows: in Section 2 , we discuss our numerical methods and simulations. Our results are presented in Section 3 and then discussed and summarized in Section 4 . Throughout, we adopt a Hubble parameter , which corresponds to a Hubble time of 13.97 Gyr for a flat universe with Ω Λ = 0.73. 2 METHODS To investigate the effects of self-friction, we perform numerical simulations of an N -body subhalo orbiting within an analytical host halo potential. We provide a brief description of the methods here and refer the reader to van den Bosch Ogiya ( 2018 , hereafter BO18 ) for full details on the numerical simulations and analysis methods. The host- and subhaloes are both assumed to initially be spherical NFW profiles (Navarro, Frenk White 1996 ) with concentrations c host and c sub , respectively. The host halo is represented by a static analytical potential. Throughout this paper, R and r are used to indicate radii with respect to the centre of the host halo and subhalo, respectively. The virial masses and radii of the two systems 2 are indicated by M host and R host , in the case of the host halo, and M sub and r sub in the case of the subhalo. Note that for the latter, these relate to the initial, unperturbed subhalo, prior to being introduced to the tidal field of the host halo. Positions and velocities of particles in the subhalo are initialized assuming an isotropic distribution function, f ( E ), using the method of Widrow ( 2000 ). In order to roughly account for the tidal mass loss that the subhalo would have experienced when approaching its initial position from an infinite distance, the initial distribution of particles is truncated at the instantaneous tidal radius of the subhalo (King 1962 ), given by Here, R is the (initial) distance from the centre of the subhalo to the centre of the host halo, m ( r tidal ) is the mass of the subhalo within r tidal , M ( R ) is the mass of the host halo within R , and Ω is the (initial) angular speed of the subhalo (Taylor Babul 2001 ; Zentner Bullock 2003 ; Peñarrubia Benson 2005 ; van den Bosch 2017 ). Note that we do not account for this truncation in the distribution function, which instead is computed under the


criterion, and a maximum of five features, as this set of hyperparameters performed best for V19 (see also Appendix D ). After training the random forest on the oversampled training set, we apply it to the 10 sets of features (from the 10 random posterior draws) for each transient in the test set to get 10 sets of classification probabilities. We then average the 10 sets of classification probabilities for each transient and adopt the classification with the highest probability. 4. Results and Validation Applying our classification pipeline to the test set described in Section 2 yields 1257 photometrically classified SNe Ia, 521 SNe II, 298 SNe Ibc, 181 SNe IIn, and 58 SLSNe. 26 26 To exactly reproduce our results, a seed of 0 must be used in the pseudorandom number generators during feature extraction, oversampling, and classification. Table 3 gives the full list of classifications, and Figure 5 shows a sample of photometrically classified light curves. These are among the largest samples of each of these classes of SNe in the literature from a single survey. In the remainder of this section, we assess the performance of our algorithm in general and discuss how to use these classifications in practice. Figure 5. Sample observed (circles) and model (lines) light curves of photometrically classified SNe. Downward-pointing arrows indicate 3 σ upper limits from nondetections. The left column shows the highest confidence classification in each class, and the right column shows the median-confidence classification. Notice that in three of the lower confidence classifications, the light curve is poorly sampled on either the rise or the decline because it was discovered close to the boundaries of the observing season. Table 3 Classification Results (This table is available in its entirety in machine-readable form.) Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available. 4.1. Cross-validation We first validate our classifier using leave-one-out cross-validation with the SNe in our training set. For each iteration of the cross-validation, we retrain the classifier on all but one of the SNe in our training set (still using the median parameters) and then use it to classify 10 sets of features derived from the posterior parameter distributions for that SN. We then average the 10 sets of probabilities to determine the cross-validation classifications listed in Table 4 . Figure 6 shows the resulting confusion matrices. Figure 6. Confusion matrices of our validation results. Each cell lists and is colored by the fraction of each class, with the raw number in parentheses. The top matrix aggregates by true label, so its diagonal shows completeness. The middle matrix aggregates by predicted label, so its diagonal shows purity. The bottom matrices are the same results but with all four non-SN Ia classes grouped together as CCSNe. N is the total size of the training set, A is the


(Planck Collaboration I 2016 ), is implemented in order to constrain the Λ( H ) models. Up to now, for these models only the set of the CMB shift parameters have been used in order to test the dynamical vacuum scenarios close to recombination. Then, we combine CMB in an overall likelihood analysis with other cosmological probes (SNe type Ia, BAOs, H 0 ) in order to extract the probability distribution (via Monte Carlo MCMC method) of the solutions for a large set of cosmological parameters, as well as, to search for deviations from the concordance ΛCDM model for which Λ( H ) = const. The structure of our paper is as follows: in Section 2 we provide a brief introduction of the running vacuum cosmology and we present the most popular Λ( H ) models that have appeared in the literature. In Section 3 , we discuss the methodology and the cosmological data that we utilize and we perform a detailed statistical analysis aiming to provide the corresponding best-fitting values and contour plots for the current Λ( H ) models. Finally, we discuss our results in the conclusions. 2 BACKGROUND EXPANSION IN RUNNING VACUUM MODELS In this section, we briefly describe the main features of the dynamical vacuum models for which Λ is not constant but evolves with cosmic time. This is perfectly allowed by the cosmological principle embedded in the FLRW metric (Perico et al. 2013 ). In general, if we model the expanding Universe as a mixture of perfect fluids N = 1, 2, ... then the total energy momentum tensor is given by where is the four-velocity field. In this case the components of are written as where the quantities p T and ρ T are the total pressure and energy density in the comoving frame , respectively. The next step is to apply the covariant local conservation law for the mixture, namely ∇ μ T μν = 0. Inserting equation ( 1 ) into the latter expression and with the aid of the relation (which comes from the fact that for any four-velocity vector, we have ) we arrive at (Grande, Pelinson Sola 2009 ): For a Friedmann–Lemaître–Robertson–Walker (FLRW) metric, it is easy to check that for a comoving frame (), one obtains: and thus equation ( 3 ) reduces to where is the Hubble parameter and a ( t ) is the scale factor of the Universe normalized to unity at the present epoch. For the rest of the paper we focus on the spatially flat FLRW metric. In the aforementioned discussion, we did not address the physics of the fluids involved. The total density ρ T receives contributions from non-relativistic matter (cold dark matter and baryons) ρ m = ρ b + ρ dm ( p m = 0), radiation ρ r ( p r = ρ r /3) and vacuum ρ Λ ( p Λ = −ρ Λ ). In this context, the index N specifies the specific components of the cosmic fluid, namely {dm, b, r , Λ}. Assuming that baryons and radiation are self-conserved, namely the corresponding densities evolve in the nominal way ρ b = ρ b0 a −3 and ρ r = ρ r0 a −4 the overall conservation law ( 5 ) becomes or Note that Q is the interaction term


have discussed the mechanisms that lead to the formation of instabilities in both the upstream and downstream medium when the magnetic field makes a large angle with the direction of the flow. We should note here that when we increase the angle even further beyond 70° this effect disappears since the component of the particle velocity parallel to the flow becomes smaller than the drift velocity if the thermal plasma, which eliminates the upstream current. The question remains as to why this result was not obtained with a hybrid code. By comparing our simulations with these earlier models we come to the conclusion that there are two factors that contribute to the different outcome. The number of particles . Although the absolute number of particles used in hybrid codes is far larger than what we use for the PI[MHD]C simulations, only a small fraction of the particles in hybrid simulations actually are suprathermal. Combined with the fact that for an oblique magnetic field only a fraction of the non-thermal particles are able to move upstream because of their pitch angle, we conclude that the hybrid simulations lacked the necessary number of particles to create the current that causes the disturbance of the upstream magnetic field. The size of the physical domain . The size of the simulated domain in the direction perpendicular to the shock-normal is six times larger in our model than in the original hybrid simulation. This allows us to capture the long wavelength that corrugates the shock and sets in motion the process that disturbs the upstream magnetic field. The smaller domain used by the hybrid simulation makes this impossible. We have tested our code with a physical domain similar to that used by Caprioli Spitkovsky ( 2014a ) and find that indeed the long wavelength never appears and so is the magnetic field amplification. These points demonstrate the opportunity that the PI[MHD]C method affords us. Because it is designed primarily to simulate large-scale structures, is less computationally expensive than PIC or hybrid methods and enables us to run the simulation in the frame of reference of the shock, it allows us to simulate a large volume of space, using a larger number of suprathermal particles, which in this particular case proved to be necessary. 2D3V PIC simulations of perpendicular and oblique shock configurations have been performed by Kato Takabe ( 2010 ) and Wieland et al. ( 2016 ), respectively. These studies verify that fast Alfvénic Mach (low magnetization) number shocks are mediated by the Weibel instability. These studies only report on electron thermalization but find pieces of evidence for the development of a proton non-thermal tail possibly connected with a shock surfing process in the perpendicular shock configuration. We confirm with Caprioli Spitkovsky ( 2014a ) that protons are accelerated at oblique shocks with an angle θ B = 45°. However, we argue that particle acceleration also proceeds at larger angles through


of the local inventory, nearby solar-type stars provide particular insight from high signal-to-noise, high-resolution spectroscopy, such as stellar masses and ages. Most or possibly all of the scatter in the barium-to-iron abundances among the Population II and intermediate-disc stars, as we learn in this work, is subject to former mass transfer from white dwarf companions, and, if accounted for, discloses a very different perspective on the underlying core-collapse supernovae nucleosynthesis. Likewise, non-single sources cause false stellar parameters and hence scatter or potentially even false trends in the derived elemental abundances, if their individual components are not treated as such, which is often commonplace. There are numerous examples in the literature, and sources like the intermediate-disc star HD 117635 in P1 or the Population II star HD 34297 in Fig. 1 below in this work, just two of them. On the other hand, it is frequently found that towards the metal-rich end the Population I and II disc star sequences progressively coalesce with their iron versus α-chain elemental abundances. This is – for many reasons – important to know, and it may be of some relevance that the stars of P1 and the present, extended sample, as we see, do not comply with a systematic trend of this kind in terms of their magnesium and iron enrichment. Figure 1. Modelling of the Fe i line λ6200.321 in the composite spectrum of the spectroscopic binary HD 34297 ( P ≃ 2 yr) with its Aa and Ab components being Doppler-displaced at Δ v r (Aa–Ab) =+28.9 km s −1 in this 2017 November 28 observation. Stellar parameters for the spectrum synthesis (dark blue) are as given in the legend. The light blue-dotted curves denote abundance changes of Δ[Fe/H] = ±0.10 dex. Individual line profiles of the Aa and Ab components are given by the red-dotted curves. In Sections 2 and 3 we briefly discuss the characteristics of the underlying spectroscopic observations and the model atmosphere analyses. The very centrepiece of our work follows in Section 4 with the analyses and related key aspects of the Population II and intermediate-disc stars. As in P1, these comprise about one-third of the final 42 pc ancient star sample and several findings as they become apparent at the present, intermediate stage of the survey will be the subject of Section 5 . Population I stars, which in the course of the analyses are identified as such, and hence not part of the survey, are presented in the Appendix. 2 OBSERVATIONS With reference to a Gaia -based 42 pc parent sample, our strategy to cover an unbiased, complete subset of more than a hundred ancient solar-type Population II (τ ≥ 12 Gyr) and intermediate-disc stars (τ ≃ 10 Gyr) within that volume has been described in detail in P1. The spectroscopic observations that we present in this second work on the survey were all secured in 2017 with the FEROS échelle spectrograph (Kaufer et al. 1999 ) at the La Silla Observatory in Chile, most of them


be plausible for an isolated black hole accreting from the interstellar medium (see, e.g., Maccarone ( 2005 ) and references therein). At the same time, in this context there is an indication that the observed X-ray spectrum of source XMM0838+24 does not coincide with predictions for various forms of radiatively inefficient accretion flows that are expected to have a power law spectrum with a photon index 1.4–2.5 (see, e.g., Narayan et al. 1998 ; Ball et al. 2001 ). High-resolution X-ray and deeper optical/NIR imaging and spectroscopic observations are required to shed light on the association of XMM0838+24 and its host galaxy and thus derive its X-ray luminosity. In this study we do not attempt to analyze X-ray spectra of XMM0838+24 and XMM1226+12 in detail because given their faint X-ray fluxes it would not provide any selectivity to discriminate conventional X-ray object types. However, we compare their X-ray colors from the XMM-Newton source catalog to a sample of known AGNs and ESO 243–49 HLX–1. In Figure 6 we overplotted two “corner” states of ESO 243–49 HLX–1 and our HLX candidates on hardness diagrams of the known AGN population from Lin et al. ( 2012 ). Clearly, broadband properties of ESO 243–49 HLX–1 do not allow its unambigous separation from AGNs. At the same time, selected HLX candidates are situated aside from ESO 243–49 HLX–1 and most AGNs due to systematically harder HR2 and HR3 colors. This, however, does not favor any particular interpretation of the nature of the candidates. Figure 6. The X-ray color–color diagrams for known AGN (from Lin et al. ( 2012 ), black dots), selected HLX candidates (this paper, gray stars) and two spectral states of ESO 243–49 HLX–1, high luminosity consistent with steep power law and low luminosity hard power law states (gray triangles). X-ray colors are taken from the 3XMM-DR5 catalog and are defined as HR , where f i is the X-ray flux in i -th XMM-Newton band (band 1 is 0.2–0.5 keV, band 2 is 0.5–1.0 keV, band 3 is 1.0–2.0 keV, band 4 is 2.0–4.5 keV, band 5 is 4.5–12 keV). As claimed by Lin et al. ( 2012 ) systematic uncertainties of these colors can reach 0.1 which is below symbol size for ESO 243–49 HLX–1 and HLX candidates. 5.2. Base Sample Properties We compute the number density of the HLX candidates from our base sample. From a total of 98 HLX candidates we expect to have at least 21 true HLXs. Given the searched volume and solid angle this gives us an HLX number density of 1 object per ≈2 × 10 5 Mpc 3 . This is approximately a factor of several thousand less frequent than ULXs (Swartz et al. 2011 ). We can normalize this density to galactic halos. In total there are 33,879 SDSS galaxies falling in the 3XMM-DR5 footprint (total observed part of the sky) that we included in our off-nuclear cross-match procedure. Hence, the density of our HLX candidates is about 6 × 10 −4 per galaxy. If they are all accreting IMBHs, this is too large a density for a population of “naked” IMBHs that


drive heat conduction in the disc. Fig. 6 shows that, even in 3D, the heat flux is negligible in regions in which P saturates. From our EMHD simulation, we thus deduce that the average impact of the non-ideal effects in the compact, SANE disc studied here might be close to what one would obtain by setting P 0.25 b 2 and q 0. We also note that our results critically depend on the treatment of the mirror instability in the EMHD model. We assume that when driven above the threshold of the mirror instability, P saturates at nearly exactly the instability threshold P mirror , through an increase in the effective collision rate in the plasma. Our results are only physical if this approximation captures the global behaviour of the plasma on length-scales much larger than the Larmor radius. While current particle-in-cell simulations broadly support this model (Kunz etal. 2016 ), comparisons of the EMHD model with more detailed local simulations of a plasma at the mirror instability threshold will be required to test this assumption. 4.4 Stress tensor and angular momentum transport In previous sections, we showed that the pressure anisotropy in the EMHD simulations of SANE discs is, in a time-averaged sense, P b 2 /4. In Section 2 , we also argued that the associated shear stress, , is effectively of the same form as the magnetic shear stress, . As far as the shear stress is concerned, the pressure anisotropy P and the magnetic energy density b 2 are interchangeable. Additionally, for 1, the magnetic energy density and isotropic magnetic pressure do not play a particularly important role in the dynamics of the accretion flow. Accordingly, we may expect similar steady-state accretion flows to be reached in EMHD and IMHD if . Our simulations appear consistent with this interpretation of the impact of the pressure anisotropy: in the SANE configuration, we find similar density and temperature profiles for the EMHD and IMHD models, but with a weaker magnetic field in the EMHD model, e.g. the plasma parameter satisfies EMHD (1.21.4) IMHD . We can have a more direct look at the component of the stress-energy tensor responsible for angular momentum transport in the disc, . We show radial profiles of (averaged over time and azimuthal angle, integrated over the polar angle and normalized by the gas pressure) in Fig. 10 . The first notable result is that the total stress is nearly identical for the IMHD and EMHD simulations. However, in the IMHD model, the entirety of the stress is due to the magnetic field. In the EMHD model, outside the ISCO of the black hole, (7075)percent of the r component of the stress is due to the magnetic field and (2530)percent to the pressure anisotropy. While this fraction of the stress due to the pressure anisotropy is slightly larger than one might have expected based on the simplistic model of a constant ratio P b 2 1 4, the properties of the stress tensor are broadly consistent with the view that in EMHD the pressure anisotropy


have this feature, because we masked them out during the second iteration of the PSF construction process. As our PSF model cannot model all the particular features of a given individual star, after the subtraction process, some residuals are left. They correspond mainly to the central region due to the saturated pixels and the internal reflections. In the same way, the diffraction spikes of each star will remain as residuals since the PSF models do not have diffraction spikes. All of these effects can be seen in Fig. 10 , where we show a colour composite ( g, r , and i ) image of the central part of the Coma Cluster field in three different panels. Top panel shows the original image with the scattered light of near and bright stars contaminating the central parts of the galaxy cluster. In the middle panel, we show the scattered light field model of the brightest stars. In this particular case, we have modelled all stars brighter than 13 mag in the G band of Gaia DR2 Catalogue. The bottom panel shows the stars subtracted image. After the removal of the scattered light field, the low-surface brightness features of the Coma Cluster and its intracluster light are ready to be explored. Figure 10. Colour composite image of the central region of the Coma Cluster as seen by SDSS (R.A. (J2000) = 12:59:48.658, Dec (J2000) = +27:58:49.443), using g, r , and i filters. The background of the image (in grey) corresponds to the sum of g, r , and i SDSS band images. The images have been rebinned to 2 arcsec per pixel and convoluted with a Gaussian kernel of 3 pixel width to enhance the low-surface brightness features of the data. Upper panel: SDSS original image. Note the enormous contamination produced by the brightest stars in the central part of the Coma Cluster. Middle panel: scattered light field of all stars brighter than 13 mag in the G band of Gaia DR2 Catalogue. Bottom panel: scattered light field subtracted image. 4 DISCUSSION AND CONCLUSIONS We present extended SDSS PSFs (8 arcmin in radius) for all five SDSS filters. We describe the methodology we have used to obtain these large PSFs and we show a particular example of how the PSFs can be used to remove the scattered light field produced by the brightest stars in the images. Our approach of deriving PSFs can be applied to other telescopes and instruments to obtain extended PSFs. This work shows that the PSFs have complex bidimensional structures as it can be seen in Fig. 6 . In that figure we plot the radial profiles produced by de Jong ( 2008 ) with a maximum radius of ∼46 arcsec for the g, r , and i SDSS filters together with the radial profiles of the PSFs of this work. The reader can see that the agreement between these two works is rather good. Compared to de Jong ( 2008 ), we enlarge the model of SDSS PSFs an order of magnitude in radius. In addition to this, we also provide their 2-dimensional structures. The PSFs models presented in this work can be used to remove the scattered light of


than the canonical value of 1.4. Fig. 12. Parallax over error ( π / σ π ) ve. RUWE of the three new groups of YSOs identified in this work, compared to sources with resolved protoplanetary disks (DSHARP, Andrews et al. 2018 ) and known binaries in Upper Sco ( Tokovinin Briceño 2020 ). 5. On the origin of the new YSOs with disks The previous section has shown that the new disk-bearing YSOs that are candidate members of the US and V1062 Sco comoving groups are relatively well characterized with respect to proper motion, on-sky spatial distribution, distance distribution, and age. Tables D.1 and E.1 list the source designations and disk type for US and V1062 Sco, respectively. The remaining YSOs in our sample (Table A.1 ) are likely composed of sources from both Lupus and UCL, as they present properties that are common to both of these populations; the emerging picture is that the youngest of these YSOs are likely Lupus members, while the older sources are likely part of UCL. Comparison of our new disk-bearing sources in Table A.1 with the YSOs identified by Damiani et al. (2019) yielded 37 out of 60 matches: 45% of the sources were found to belong to the UCL diffuse group D1, and 35% to the UCL diffuse group D2 (further divided into 27% from the D2a sub-group and 8% from the D2b sub-group). Of our new disks in US, 12/19 were identified as belonging to either the diffuse population in Upper Sco USC-D2 (32%), to UCL-D1 (32%), or to UCL-D2a (5%). Regarding the new disks in V1062 Sco, 14/19 were also previously identified by Damiani et al. (2019) as candidate members of the V1062 Sco/UCL-1 compact group. We discuss in this section the region morphology and kinematics to explore the origin and connection between Lupus, UCL, US, and V1062 Sco. 5.1. Evolution of the 3D spatial distribution We attempt to reconstruct the 3D distribution of the new disk-bearing YSOs and the Lupus clouds to better determine how the Lupus and UCL stellar groups are related. Figure 13 shows in the upper (middle) row the Galactic Latitude (Longitude) as a function of distance for these sources. Each row has three panels, corresponding to the three iscohronal age groups. These panels also include a schematic representation of the Lupus clouds (the central Galactic coordinates were taken from Hara et al. (1999) and the width of the filled ellipses are merely representative and not to be taken to scale). The distances to Lupus 1, Lupus 2, Lupus 3, and Lupus 4 were derived from the distances to their previously known members ( Dzib et al. 2018 ), and for clouds with no known associated YSOs we use the distances calculated by Zucker et al. (2019) (we exclude the clouds Lupus 7 and Lupus 8, due to the lack of distance estimates). To constrain the origin of a YSO, we explore its travel radius, specifically, the projected distance traversed by a source since its birth assuming a 1D velocity of 1 km s −1 . We represent the area corresponding to the travel radius as dotted-line


is a potential avenue for direct comparison of the observations to the predictions of MHD simulations. The magnetic field lower limits found in each region are consistent with the values found by H13 . This peak value is consistent with the H13 value of +8 μ G. Our data strengthen their finding that the RMs along the narrow, decelerated H α ridge in region 5 are much stronger and opposite in sign compared to the surrounding RMs, suggesting a compressed ambient field which has a preferred direction. Due to the high density of RMs on and next to the Smith Cloud, we can model the morphology of the magnetic field. As seen in Figures 5 and 6 , an arch of positive RMs extends above and around the head of the Smith Cloud. This cap of RMs suggests a magnetic field that has been compressed by the ambient field as the Smith Cloud travels toward the Galactic plane, assuming that the cloud is falling into the disk (see Section 1 ). Under the assumption that is roughly constant, since RM is proportional to , the small negative RMs over the body suggest that the perpendicular component of the magnetic field is stronger than the parallel component, while the large positive RMs surrounding the cloud indicate that the parallel component dominates. To the right of the major axis of the cloud, the RMs are predominately positive, indicating a weak magnetic field perpendicular vector component pointing toward the observer. To the left of the Smith Cloud, strong negative RMs dominate. These strongly negative RMs indicate weak magnetic field perpendicular vector components pointing away from us. This picture is qualitatively consistent with strong parallel magnetic field vector lines along the sides of the Smith Cloud. This suggests a magnetic field that is draped, or laid over the ionized gas of the Smith Cloud, where the field has been compressed near the tip of the cloud (Dursi Pfrommer 2008 ). This configuration of a draped magnetic field surrounding the Smith Cloud is consistent with the idea that the neutral gas in the Smith Cloud is shielded from the hot ISM by a magnetic barrier between the surrounding medium and the gas; this has been seen in 2D simulations by Konz et al. ( 2002 ) and in 3D simulations by Grønnow et al. ( 2017 ). Grønnow et al. ( 2017 ) found that a magnetic field of a few μ G associated with an HVC within 10 kpc of the disk corresponds to an ambient field that has been “swept up” by the cloud, which is consistent with H13 and our observations. The Grønnow et al. ( 2017 , 2018 ) models also support the idea that a magnetic field could keep the cloud intact as it travels through the Galactic halo by delaying hydrodynamic instabilities and stripping of the gas. With this draped magnetic field barrier, the Smith Cloud could travel through the ISM while keeping the gas near the head and body intact. To test if this geometry could produce the observed RMs, we create a toy model of a possible magnetic field configuration surrounding the


of the system is maintained. When a dust grain is placed on a highly eccentric orbit by radiation pressure, it may create a local pressure maximum but not linger in a nearby circular orbit where it can stabilize the gas through drag backreaction. This may provide a second possible explanation for the difference between runs D and F discussed above, especially given the higher mass of dust used in run F, which may lead to drag backforces that are strong enough to destabilize gas from a circular orbit. In Figure 12 , we show the dust surface density every 50 orbits for 400 orbits. We find that the photoelectric instability sets in quickly, but has no obvious secular effect on the global structure of the disk. The behavior is dominated by transient clumps and arcs that appear and disappear on timescales of orbits to dozens of orbits. Figure 12. Dust surface density over time for run F. The photoelectric instability arises quickly and is continuously redistributing dust into nonaxisymmetric structures throughout the disk over the full course of the simulation. Though the radial distribution of the dust varies nonsecularly over time, the 2D dust structure is consistently dominated by clumps and arcs that form and dissipate on short timescales of ≈1–10 orbits. In order to confirm that the dust structures seen in run F (Figure 8 ) are conclusively attributable to the photoelectric instability, we run two additional models: one without photoelectric heating, and one with an increased dust resolution to ensure model convergence (run I). For run I, we double the number of dust superparticles while reducing the mass of each superparticle by half in order to maintain the same overall dust mass. The run without photoelectric heating shows no clumping behavior, and instead yields a smooth, structureless dust distribution, as one would expect. Run I, on the other hand, shows the same clumping behavior as seen in run F, suggesting that the resolution we use in runs A–F is sufficient. Gas and dust surface densities after 400 orbits are shown in Figure 13 . Figure 13. Gas (left panel) and dust (right panel) surface densities after 400 orbits for run I, which is a high-dust resolution (2× superparticles) duplicate of run F (). The dashed circle indicates (100 au). Doubling the number of superparticles yields substantially similar gas and dust spirals and clumps, suggesting that 800,000 superparticles are sufficient for capturing the effect of the PeI in the presence of radiation pressure (for the chosen grain size distribution). In run H, we duplicate run F (Figure 8 ) but double the birth ring width to verify that the dust clumping behavior of run F does not require a narrow birth ring. Dust and gas surface densities after 400 orbits for run H are shown in Figure 10 . We find that the enhancement in the dust-to-gas ratio produced by the photoelectric instability is weaker in run H compared with run F (see Table 1 ), but the qualitative results—nonaxisymmetric,


in Appendix B . The final result is plotted in Fig. 5 and the derived parameters are listed in Table 5 . The final rms is small with only 90 m s −1 . Our revised orbital parameters are in rather good agreement with the literature, within 1.5 σ ( Groenewegen 2013 ; Gorynya et al. 1995 ; Evans et al. 1990 ; Abt 1959 ). Fig. 5. Result of our combined fit of the orbital and radial pulsation velocity for FF Aql. Left panel : fitted (solid lines) and measured primary (blue dots) orbital velocity. Right panel : fitted (solid line) and measured (blue dots) radial pulsation velocity. Table 5. Pulsation and orbital parameters of the FF Aql system. Notes. Parameters are the same as in Table 4 . a 1 : semi-major axis of the Cepheid’s orbit relative to the centre of mass. f ( M ): the mass function. (a) Not fitted. (b) Uncertainty not given by the authors. (c) Gorynya et al. (1995) . We noticed a slight difference in the systemic velocity between our estimate and that from Gorynya et al. (1995) . This might be linked to the wide companion, however, this might also be from other non-astrophysical effects. We should be cautious when studying long-term variations of v γ , unless a clear pattern is observed. Differences in v γ of the order of 0.5–1 km s −1 might be caused, for instance, to the way RVs are determined (cross-correlation, bisector, ...), the mask used, and the instrument zero points. In this paper we did not perform such a long-term study as this is out of the scope of this paper and would require a complete analysis of all available literature data. U Car. The companion orbiting this Cepheid is below our detection level. Our datasets enable us however to exclude any companion with a flux ratio higher than 0.4%, which would correspond to a spectral type earlier than B2V (see Table 3 ). Y Car. Our possibly detected companion has a flux ratio of ∼0.94% and would correspond to a ∼A0V spectral type, which is in rather good agreement with the ∼B9V derived by Evans (1992c) . Although additional data are still necessary to confirm, this possible detection seems in a very close orbit as we measured a projected separation of ∼2.5 mas (∼3.5 au), as for S Mus. We first estimated the 3 σ detection limit using all observables and removing the possible companion (because we have a detection), and second using only the CPs but without removing it (because there is no detection). We noticed that the closure phases alone provide a contrast limit lower than 4 mag (see Table 3 ), which explains the non-detection with only this observable. With all observables we add more constraints in this case, which provide contrast limit of 5 mag (5.1 mag within 25 mas). This enables us to exclude at 3 σ any companion with a spectral type earlier than A0V. YZ Car. The faint companion orbiting this Cepheid is not detected from our observations. Our 3 σ detection limits (see Table 3 ) show that any companion with a spectral type later than B3V would not have been detected with


tailed and, more importantly, (b) that the chosen statistical sample represents the global statistics well, at least, in terms of statistical power. Therefore, if we randomly select a pixel inside our sampling area from one observation, the probability to observe a value above the background mean is or . 10 10 is the probability to observe values outside the limits. Half of this should account for the probability to observe values that exceed only the upper limit of . Therefore, a is adopted as a realistic occurrence probability of a stochastic observation in a single experiment. Considering also that the signal at the position of the A HST star is the only consistently present signal in all our NIR images (i.e., three independent experiments), then the probability that it results from random background noise fluctuations should be or . Since this is negligible, we can safely consider that the presence of an underlying point source, which systematically “drives” the flux within the green circle above the limit in repeated observations, is the origin of the observed NIR signal of Figure 5 . However, are the photometric properties of the NIR detections consistent with the visual photometric properties of the A HST star? 4.2. Photometric Calibration of the NIR Detections The V and I magnitudes of the A HST star are measured with aperture photometry on the and HST images (Section 2.2 ). They are and mag, respectively. On the contrary, due to the low S/N of the Subaru data, aperture photometry is not an option for estimating the NIR magnitudes of the detections. Alternatively, their , and K S magnitudes are estimated by taking the ratio of the sum of the peak fluxes of G.S.1 and G.S.2 to the peak fluxes of the NIR detections, 11 11 Point sources of different brightness observed simultaneously with the same telescope are convolved with the same PSF as long as their separation is smaller than ∼30 arcsec (i.e., isoplanatic angle). Consequently, the ratio of the fluxes at any point across their PSFs and the ratio of their integrated fluxes will (ideally) be approximately equal and, therefore, any of these measurements should be representative of their true brightness difference. usingwhere is the peak flux of each guide star, is the peak flux of each detection, and is the Subaru magnitude of the G.S. that we derived in Section 2.1.1 . This yields , and mag for the detections in each NIR band. The uncertainties are calculated according to the relationwhere we assume a 10% uncertainty in the measurement of and a 30% uncertainty in the measurement of the, more uncertain, . Having magnitude estimations, we can attempt to constrain the V − I , visual–NIR, 12 12 I.e., , and . and NIR 13 13 I.e., , and . colors of the NIR detections/A HST star and, therefore, its spectral type and luminosity class. 4.3. An Extragalactic Supergiant or a Galactic Dwarf? There are two possibilities for the nature of the A HST star. It can be either a distant supergiant


the course of the observations (typically 33 d for Campaign 0). Eclipsing binaries and ellipsoidal variables of high amplitude are avoided. If we consider only stars where the harmonic is visible, then 86 out of 265 (32 per cent) 09–B9 stars are rotational variables. In addition, if we add stars with just a single isolated peak (no harmonic), then 142 out of 265 stars (54 per cent) are rotational variables. Balona et al. ( 2015 ) has already shown that rotational modulation is probably present in many late B stars in the Kepler field. This study shows that this effect is likely to be present in early B stars as well. Indeed, the large fraction of B-type rotational variables in the K2 data is probably an underestimate because these data are less precise and of much shorter duration than the Kepler data. The only test that is available for rotational modulation is to compare the rotational velocity obtained from the photometric period (using an appropriate stellar radius) with the rotational velocity derived from spectroscopic line broadening measurements. This could be done successfully for the A stars (Balona 2013 ) because of the large number (875) of stars with known periods and radii derived from multicolour CCD photometry. Unfortunately, there are too few B stars in our sample to use this statistical test. Moreover, stellar radii cannot be estimated with sufficient precision. There are only seven stars in Table 2 with known values of the projected rotational velocity. Rotational modulation implies some kind of surface brightness inhomogeneity. The nature of such a ‘spot’ cannot be determined at present. However, it is difficult to conceive of starspot formation without the presence of a magnetic field. We therefore suggest that magnetic fields are probably present in stars with radiative envelopes. According to Cantiello Braithwaite ( 2011 ), dynamo action in a subsurface convection zone in such stars might generate a magnetic field. In this case magnetic spots of size comparable to the local pressure scaleheight can manifest themselves as hot, bright spots. A theory of dynamo action in radiative envelopes which involves differential rotation has been developed by Spruit ( 1999 , 2002 ) and Maeder Meynet ( 2004 ). In this theory, a magnetic instability in the toroidal field wound up by differential rotation replaces the role of convection in closing the field amplification loop in conventional dynamo theory. Zahn, Brun Mathis ( 2007 ) has verified that an instability is generated, but concludes that a dynamo cannot be sustained by this effect. Arlt Rüdiger ( 2011 ) found that feedback by the Lorentz force diminishes differential rotation and brings field growth to a halt in a few thousand years for intermediate mass stars. It seems that we still do not have an acceptable explanation for the generation of a magnetic field in radiative envelopes. One way to circumvent the generation of magnetic fields in radiative envelopes is to suppose


NW part of RCW 57A. This could be the plausible reason behind the low-density material observed in the NW region. The observed large-scale loops in the NE possibly formed due to the newly formed material as a result of the expanding I-fronts driven by the embedded massive protostars of RCW 57A. A high-velocity outflow signature is discernible close to eastward from the center, while the same feature is absent westward as shown in Figure 11 (b). This could be attributed to the confined ionized gas flow toward the west from the center, possibly due to the presence of a dense clump (thin black contours in Figure 7 ). The high-velocity component is only seen toward the east, which implies a dearth of dense material there, and hence, the ionized gas expanded with relatively more velocity. de Pree et al. ( 1999 ) and Purcell et al. ( 2009 ) have also shown that the H ii region expands more freely toward the east and is bounded on the west. 5.2. Column Density Profiles: Filament Width and <italic>B</italic>-field Support We fitted a Gaussian to the mean column density profiles spread over ∼5 pc radii, and the resultant FWHM, corresponding to the filament characteristic width, is found to be 1.55 ± 0.09 pc. However, a constant filament width of ∼0.1 pc was determined for the central dense sections of the filaments of nearby star-forming regions in the Herschel Gould Belt survey by fitting a Gaussian to the inner sections (radii up to 0.3–0.4 pc) of the filaments (Arzoumanian et al. 2011 ). Therefore, we fit a Gaussian to the inner 1 pc width of the clubbed mean column density profiles of the NW and SE regions, and the resultant σ and FWHM are found to be 0.21 ± 0.08 pc and 0.50 ± 0.18 pc, respectively. Within the error, the filament width (0.50 pc; green lines in Figures 12 (a) and (b)) of RCW 57A is higher than the constant filament width of ∼0.1 pc (Arzoumanian et al. 2011 ). Filament widths larger than 0.1 pc (spanning over 0.1 –1 pc with typical widths of ∼0.2–0.3 pc) have also been witnessed in a number of studies (e.g., Hennemann et al. 2012 ; Juvela et al. 2012 ). However, the filament width of RCW 57A has not been resolved because of the limited resolution (364 corresponds to 0.42 pc as shown by the cyan lines) of the column density map. The mean power-law index p = 2.02 ± 0.05, derived from the Plummer-like profile fitting for RCW 57A, is consistent with other studies (Arzoumanian et al. 2011 ; Juvela et al. 2012 ; Palmeirim et al. 2013 ; Hoq et al. 2017 ). A similar value of this index ( p = 2) is expected if the filament is supported by B -fields (Hennebelle 2003 ; Tilley Pudritz 2003 ), in accordance with our observational results supporting the active role of B -fields in the formation and evolution of filaments, as described in Section 5.3 below. 5.3. Evolutionary Scenario of the Filament and Bipolar Bubble Based on the morphological correlations among the (i) dense filamentary cloud structure seen in the submillimeter (in the 0.87 mm


the matter power spectrum (left-hand panel) and reduced bispectrum (right-hand panel), defined as S ( k ) ≡ T ( k ) BCM / T ( k ) GrO for T ( k ) = { P ( k ), Q ( k )}, at z = 0. Solid, dashed, dash–dotted and dotted lines are computed with simulations of box side and 768 3 , 512 3 , 384 3 , and 256 3 particles, respectively. Colours are referred to different halo mass bins, expressed in decimal logarithm of , with which the baryon corrections have been computed, according to the legend. APPENDIX B: FOLDING OF THE PARTICLE DISTRIBUTION Measuring the three-point clustering with the classical Fourier estimators can be very expensive in terms of memory and CPU, especially when using covering larger dynamical ranges. In fact, it is easy to see that, being k Ny = π N g / L box the Nyquist frequency of the grid, increasingly large number of grid points N g are required to get a given accuracy at a fixed wavenumber, when using progressively larger simulation boxes L box . Additionally, when using “interlacing” to suppress aliasing, the number of grids used must be doubled (Sefusatti et al. 2016 ). However, since our measurements are limited by discreteness noise (and not cosmic variance), we can obtain accurate estimates of Fourier statistics on small scales by folding the density field (Jenkins et al. 1998 ; Colombi et al. 2009 ). The idea is to fold the particle distribution by re-applying the periodic boundary conditions assuming a new boxsize L ′ = L / f , where we call f the number of foldings. If L ′ is large enough to assure that the modes inside the new box are uncorrelated, we can measure in principle the clustering from a new effective fundamental wavenumber up to a new effective Nyquist frequency, given by . For instance, by folding the box four times, we will get to Nyquist frequency four times higher. In Fig. A3 , we apply this technique to our simulation, folding the particles up to six times, and reaching a with a 64 3 and a 128 3 grid for the power spectrum and the bispectrum, respectively. Even using a TSC scheme on interlaced grids, we note that it is safer to use the measurements up to in the bispectrum. Also, the measurement of the largest modes of the folded box are noisy because they are sparsely sampled; for this reason, it is convenient to discard these modes, taking for instance wavenumbers . Figure A3. Left-hand panels: Matter power spectrum measured in a 64 3 (interlaced) mesh, folding the box up to six times, following the technique explained in the text (coloured dots). For comparison, the matter power spectrum measured with a 696 3 mesh and not folding the box is plotted as a black solid lines. The equivalent Nyquist frequencies for each folded box is plotted as a solid line. In the bottom panel, we display the ratio between the power spectrum measure with a 64 3 and a 696 3 mesh. Right-hand panels: Similarly to the left-hand panels, we display the measured bispectra using a 128 3 mesh and the folding of the box, and


governed by the equilibrium with SI clathrate hydrates of CO 2 (Levi et al. 2017 ). For convenience, we have cast into a cubic polynomial form the first and second dissociation constants of carbonic acid, as derived from the revised-HKF equation of state:where P is in GPa and K is in molal. For each isotherm, the polynomial is valid for the pressure range spanning from the surface of the ocean to the ocean floor. The coefficients for the polynomials are tabulated in Table 4 . Table 4 Polynomial Coefficients to Be Used with Equation ( 55 ) In Figure 14 , we give the speciation of the carbonate system as a function of depth, for the 280 K isotherm. We find that freely dissolved CO 2 is the most abundant form of DIC, in water-planet oceans, at any depth. This is opposite to the case of Earth, where freely dissolved CO 2 is the least abundant DIC component in the ocean (Zeebe Wolf-Gladrow 2001 ). Also, we find no substantial difference between the concentrations of hydrogen complexes, H + , and bicarbonate (both are represented by the red curve). Carbonate and hydroxide ions are found to exist in the ocean only in trace amounts. We note that the errors we have suggested above for the values of K 1 and K 2 do not have any considerable effect on our results. Figure 14. Speciation of the carbonate system as a function of depth in the ocean. Here we consider the ocean thermal profile to be an isotherm, set at 280 K. The black curve is the concentration of freely dissolved CO 2 . The red curve is the concentration of H + and . The green curve is the concentration of . The cyan curve is the concentration of OH − . The curves terminate at the pressure representing the bottom of the ocean, for the given isotherm. From Figure 14 we see that the different chemical species making up the carbonate system have a concentration gradient with depth. We argue that these concentration gradients with depth are stable against vertical mixing in an ocean world. First, we wish to point out that the concentration of freely dissolved CO 2 , the most abundant component of the DIC, is constant to a good approximation throughout the ocean. This is a consequence of the nature of solubility when in equilibrium with the clathrate phase, rather than due to efficient vertical mixing in the ocean (Levi et al. 2017 ). Thus, the concentration gradients with depth are most pronounced for the chemical species that are least abundant. If a parcel of ocean water migrates from the ocean floor upward, this parcel would be supersaturated with carbonate with respect to the shallower depth. The value for K 2 decreases with decreasing pressure. Consequently, carbonate would transform to bicarbonate, and then to freely dissolved CO 2 . However, this newly formed CO 2 is added to an ocean likely saturated with CO 2 , at equilibrium levels with respect to its clathrate hydrate phase (Levi et al. 2017 ). Therefore, the additional dissolved CO 2 forms grains of clathrate hydrates, which tend to


the importance of the presented ion acceleration in these systems. In terms of the long-term evolution of the acceleration process, in the turbulent environment of accretion disks, we expect many successive δB / B ∼ 1 amplifications and decreases of the field (see also Verscharen et al. 2016 ). So a more realistic picture of this process should consider the acceleration presented in this work occurring many times as the plasma is gradually accreted. We will study these aspects of the acceleration process in future investigations. Appendix A 1D Shear Setup The goal of our 1D shear runs is to simulate a shearing plasma, assuming that its properties depend only on the direction parallel to the mean background magnetic field . This is equivalent to assuming that the wavevectors captured in the simulations satisfy . Since, in a shearing plasma, the direction of evolves with time, the orientation of the wavevectors that can be consistently resolved needs to evolve accordingly. Our 1D setup is built upon the “shearing coordinates” setup presented by Riquelme et al. ( 2012 ). This setup was designed so that the simulation domain follows the shearing flow of the plasma, which is given by the shear velocity (). Figures 1 (a) and (b) illustrate the way the shape of the domain evolves as seen by an inertial observer at t = 0 and t > 0, respectively. Formally, the shearing coordinates ( x ′, y ′, z ′, t ′) are defined in terms of the regular, inertial coordinates ( x , y , z , t ) aswhere , , 19 19 Notice that, in the limit , the expression is equivalent to v = − sx , as we assume in Figures 1 (a) and (b). Since ensures that , this expression was adopted in Riquelme et al. ( 2012 ) in order to deduce the equations that describe the dynamics of the plasma in the shearing coordinate system. However, both in Riquelme et al. ( 2012 ) and in this paper, we are interested in the plasma dynamics in the regime . c is the speed of light, and s is the shear rate of the plasma. Our 1D domain is defined by y ′ = 0, and it is shown by the blue lines in Figures 1 (a) and (b). These figures show how the length and orientation of the 1D domain change over time. This in turn changes the orientation of the wavevectors that can be captured within the domain. In addition, there is an initially homogeneous magnetic field, , pointing parallel to the 1D domain at t = 0. Magnetic flux conservation ensures that will always be parallel to the 1D domain at t > 0, as also depicted in Figures 1 (a) and (b). Therefore, if the problem of interest is dominated by waves that propagate parallel to , our 1D simulations will be able to capture the essence of the phenomenon. In order for our 1D simulations to consistently satisfy , we need to replace x ′ with a new coordinate, , which we define asIndeed, if we assume and sy ′ ≪ c , the partial derivatives with respect to the spatial coordinates in the inertial frame areThus, if we impose the 1D condition, namely, that the fields depend only


line properties.) In our initial line list, we use the former model, adopting the tuned of −0.224 from DM12 . This blended feature is addressed more fully in the next section. We use this initial line list to generate synthetic spectra of the benchmark stars without any Be to compare to the observed spectra, as shown in Figure 3 . The solar spectrum clearly requires the inclusion of Be absorption to fit the observed spectrum at the location of both Be ii lines. In the modeled solar spectrum, the contentious Mn i line at λ 3131.037 Å is clearly visible in the blue wing of the Be line at λ 3131.065 Å. The Mn i line is stronger in the red giant spectra and is well fit with the larger adopted from DM12 . All three benchmark observations show excess absorption predicted by the models, especially below λ 3130.0 Å. Most concerning is that the Arcturus model is a poor fit to the spectrum in the contentious blue wing of the λ 3131.065 Å Be line. Some tuning of the gf -values is therefore required. 4.2. Tuning the Line List The poor fit seen at λ ≲ 3129.6 Å has little effect on the Be abundance analysis, but we did trace the problem to two Fe lines: λ 3129.102 Å (Peterson Kurucz 2015 ) and λ 3129.619 (VALD K14). Removing these lines (which we achieved by changing the by the arbitrarily large factor of −10) produces a much better fit to the spectrum. The focus of the remainder of astrophysical tuning is on a narrow wavelength region around the redder Be ii line, which is the more isolated of the two lines. Despite the relative isolation, there are still significant blends including the contentious Mn i noted earlier as well as a contribution in the red wing of the line by Zr i at λ 3131.109 Å. In the solar spectrum, this line is weak (∼1.5 mÅ equivalent width), but the equivalent width is at least an order of magnitude larger in red giant stars. Furthermore, Figure 3 shows that a nearby Cr i line is poorly fit in the solar spectrum and is nearby enough to potentially affect the neighboring Be blend. The synthesized Cr i line at λ 3131.212 Å is too weak in the spectra of all three benchmark stars. Its was increased by 0.60 dex in order to match the observed solar spectrum. Figure 3. Comparison of synthetic spectra generated with the newly compiled line list (solid lines) compared to observations (dotted lines), assuming no Be in the stellar atmospheres. The dashed vertical lines show the location of the Be ii lines, and the dotted–dashed vertical line marks the Mn i line at λ 3131.037 Å, using the tuned of −0.224 from DM12 . The solar spectrum (top, archival Subaru spectrum of Vesta) is clearly not well fit when Be is absent from the model. The spectrum of Pollux (middle) shows a generally good fit at wavelengths above ∼3130.5 Å, whereas the model for Arcturus (bottom) has conspicuous missing absorption just blue ward of the λ 3131.065 Å Be ii line. As we considered additional changes to the line list in the blended Be feature, we generated spectra


Ly α power spectrum. Although the impact of inhomogeneous reionization in the Ly α forest is slightly larger than in our previous work, if one takes into account the statistical error in the eBOSS measurement (plotted in Fig. 2 ) and the systematic error in modelling reionization, it is currently challenging to use the 1D Ly α power spectrum to extract information from the reionization epoch via the impact of inhomogeneous reionization in the Ly α forest. However, in future work we will investigate this possibility. On the other hand, the expected effect of patchy reionization in the 3D Ly α forest power spectrum are significantly larger. We find that the percentage deviation is in the order of tens of per cent at z obs = 4.0 (e.g. for our model R2), and in the order of a few per cent at z obs = 2.0 (e.g. for our model T3). Our results for the survival of these thermal relics from reionization to low redshift might seem unconventional . However, the reason we obtain such strong ripples in the Ly α forest power spectra is because we are able to carefully track the small-scale structure prior to reionization. In particular, mini-voids at reionization get later compressed to mean-density while they simultaneously get reionized to high entropy. This HEMD gas gets usually heated to ∼3 × 10 4 K in our simulations and remains above the temperature–density relation even at z 4. As a result the temperature–density relation of the IGM becomes bimodal with the usual low-entropy phase and this high-entropy phase. The ultimate fate of the HEMD phase is to merge with the low-entropy mode; however, it takes cosmological time-scales for the merging to complete (see figs 4 and 5 of Hirata 2018 ). We stress here that it is really both the patchy nature of reionization , and the HEMD gas, i.e. ∂ψ/∂ z in equation ( 3 ) together that result in the memory of reionization in the Ly α forest computed in this work. To illustrate the former point, we turn-off the inhomogeneous recombination prescription of Sobacchi Mesinger ( 2014 ) in 21cmFAST , and consider the resulting, faster, reionization. Our results for the 3D Ly α power spectrum are reduced by a factor of 2 at z = 2.0 and ∼1.8 at z = 4.0. This decrease of deviations in the Ly α power spectrum is expected since a faster reionization model will have a narrower range of redshifts for the peak of that contribute the most to the integration in equation ( 3 ). Regarding the importance of the HEMD gas, on the other hand, we artificially limit the impact of the HEMD gas by only considering our transparency results for higher redshifts, say z re = {8.0, 9.0, 10.0, 11.0, 12.0}. Note that patches of the sky that reionize earlier will have the high-entropy gas merge with the low-entropy gas sooner, thus recovering the temperature–density relation of the IGM faster. In this case we obtain a decrease in the 3D Ly α results with a factor of 2.2 at z = 2.0 and 1.8 at z = 4.0. In Fig. 3 , we plot the k -dependence of both the


a complete exploration of the parameter space, we first show a comparison between the model cumulative halo distributions and the observed number density of galaxies in the illustrative case of a sterile neutrino with (the candidate origin of the potential 3.5 keV line) in the limit of small Higgs portal coupling . In such a freeze-in limit (see Section 3 ), the scalar mass m S and the Yukawa coupling y will be related for any given value of , thus reducing the parameter space to be explored. In Figure 3 (left panel) we show the cumulative halo distributions in such a regime for a scalar mass of and for different values of the Yukawa coupling y , and we compare them to the observed number density of high-redshift HFF galaxies. The strong dependence of the maximum predicted number density on y is highlighted in the right-hand panel of Figure 3 , which shows that present data allow us to set a constraint at the 2- σ confidence level. Figure 3. Left panel: the cumulative mass functions computed at z = 6 for SD sterile neutrino models with and , for different values of the Yukawa coupling y (shown by the labels on the right) in the freeze-in limit. The shaded areas correspond to the observed number density of galaxies with within within the 1- σ , 2- σ , and 3- σ confidence levels. Right panel: the maximum value (including the theoretical uncertainties) of the predicted number density of DM halos at z = 6 for the case with as a function of the Yukawa coupling y . The upper shaded areas represent the observed number density of galaxies with within the 1- σ , 2- σ , and 3- σ confidence levels. We then extend our exploration to cover the whole parameter space of SD production models for sterile neutrinos. With this aim, we consider a grid of λ and y values for six different values of the scalar mass . The latter are chosen to cover the three different regimes where different production channels are dominant for the production of scalars from their coupling to the SM particles (see König et al. 2016 , and Section 3 ): the light scalar case , the intermediate case , and the heavy scalar case (in terms of the Higgs mass m h = 125 GeV). These different regimes have a rather strong impact on the distribution function of the scalar which directly translates into that of the sterile neutrino and, hence, on the resulting power spectrum. For each value of m S , we compute the power spectrum corresponding to each point in the plane as described in Section 3 . Then we represent in Figure 4 , left panel, the regions of the parameter space consistent with the galaxy number densities measured in the HFF (). These regions clearly split into a freeze-out (for ) and freeze-in (for ) family. For the freeze-out family, decreasing the scalar mass m S leads to a tighter bound on y , while yielding an approximate lower bound of for the Higgs portal coupling. For the freeze-in family, decreasing the scalar mass m S pushes the admitted values of λ to progressively


with the wind. Fig. 15 shows the time evolution of the WIM for all simulations with self-gravity. We note that the main parameter that changes the total amount of the WIM is not the density of the cloud but mainly the magnetic field properties in the wind. All winds with a perpendicular field keep the clouds mostly intact due to the magnetic draping effect and reduce the mixing of warm atomic and hot ionized gas at the cloud–wind interface. The resulting mass in the WIM converges after about 10 Myr of evolution to give . For the non-magnetic wind, the masses converge after at a value of . The highest masses in the WIM are reached for with about . We highlight that this is 60 per cent of the initial mass of the low-density cloud. The parallel field configuration delays or completely suppresses the contraction perpendicular to the magnetic field and thereby maximizes the mixing that leads to the WIM. Figure 15. Time evolution of the warm ionized gas. The total mass mainly depends on the the wind properties rather than the cloud mass. We show only the low density up to the point of their destruction and their transport through the right-hand boundary. We find the lowest mass in , followed by the non-magnetic wind. For , the warm ionized mass is largest. 8 DISCUSSION AND CONCLUSIONS 8.1 Cloud survival For clouds exposed to a wind, the question arises whether the clouds are fed by condensation of wind material and grow, whether they are effectively unaltered, or whether the wind eventually disrupts the cloud. In absence of stabilizing mechanisms, the clouds are likely to be destroyed by the efficient growth of the Rayleigh–Taylor instability and the shear-driven Kelvin–Helmhotz instability (Cooper et al. 2009 ; Scannapieco Brüggen 2015 ; Schneider Robertson 2015 , 2017 ). Stabilizing mechanisms can be the cooling and a resulting contraction that reduces the effective interaction cross-section with the wind. It could also be the magnetic field configuration, which suppresses instabilities or – mostly for dense clouds – self-gravity, which leads to faster contraction and even collapse. Cooling can lead to fragmentation of the clouds (McCourt et al. 2018 ) but can also help to stabilize them by suppressing the mixing (Sparre et al. 2019 ). Stabilized condensations can grow in mass due to newly accreted gas from the wind (Marinacci et al. 2010 ; Armillotta, Fraternali Marinacci 2016 ; Armillotta et al. 2017 ; Gronke Oh 2018 , 2020 ). Magnetic fields help to keep the gas confined and increase the acceleration efficiency (McCourt et al. 2015 ; Banda-Barragán et al. 2016 , 2018 , 2019 ; Grønnow et al. 2018 ; Sparre, Pfrommer Ehlert 2020 ). All of the above-mentioned simulations have focused on overall lower densities and higher temperatures compared to this study. However, the surviving fragments develop strong density fluctuations, which are comparable to the initial configurations in our models. We evolve the systems for a time-scale of a few


[4.5] mean magnitudes of GM Aur taken in E4 to E8. There is a significant increase of emission in both Spitzer bands in E7. MIR emission shortward of ∼20 μ m has been attributed to optically thin dust within ∼1 au of GM Aur (Calvet et al. 2005 ; Espaillat et al. 2010 ). Espaillat et al. ( 2011 ) found that GM Aur’s Spitzer spectra varied by about 10% and attributed this to variability in the mass of dust in the inner disk. Using NASA IRTF SpeX spectra covering 1–5 μ m, Ingleby et al. ( 2015 ) found that the emission varied and also attributed this to dust-mass changes. The Spitzer spectra were not coordinated with indicators, but the SpeX data were taken within a day of HST UV data. With three epochs of HST data, Ingleby et al. ( 2015 ) found that and the IR emission are roughly the same between their first two epochs (taken about 1 week apart), but both and the IR emission decreased in the third epoch (taken about four months later). Their first and second epochs had of dust in the inner disk and the third epoch had of dust in the inner disk. Here we have only two MIR magnitudes, which is not ideal for constraining the optically thin dust model. However, we can estimate the amount of dust in the inner disk in E4 through E8 by scaling with the results of Ingleby et al. ( 2015 ) under the assumption that only the dust-mass changes (Espaillat et al. 2011 ; Ingleby et al. 2015 ). The SpeX spectra cover 3.6 μ m, but not 4.5 μ m where atmospheric telluric lines are strong. Convolving the SpeX spectra with the IRAC [3.6] bandpass, we determine that the SpeX flux in the first epoch of Ingleby et al. ( 2015 ) corresponds to an IRAC [3.6] magnitude of about 8.1, which is roughly consistent with our measurements in E4, E5, E6, and E8. Therefore, we assume that the mass of dust in the inner disk in these four epochs was about the same as that seen in the first epoch of the Ingleby et al. ( 2015 ) observations. For E7, if we scale the optically thin model from the flux seen in the first epoch of the Ingleby et al. ( 2015 ) observations to the flux seen in E7, we estimate a dust mass of , corresponding to a factor of ∼2.5 change. 3.2. Centimeter Continuum Emission In Figure 2 (left), we compare the spatial brightness distribution of our 2018 data to 2015 data from Macías et al. ( 2016 ). We combine the three epochs of 2018 VLA data in this figure to achieve a high enough sensitivity to use robust weighting ( r = −0.5) and a similar angular resolution to the 2015 data. There is a seeming change in the morphology of the jet emission between 2015 and 2018. The upper right emission that Macías et al. ( 2016 ) attributed to a jet appears to be present only in the 2015 data. The resolution and sensitivity of both maps are similar, suggesting that the difference in morphology is real. However, our resolution is limited, so higher spatial resolution observations are needed to confirm this. Figure 2. Left: superposition of 3 cm VLA observations of GM Aur from


and potentially applicable for an empirical fitting of particle distribution function in various physical situations, it turns out that such a model has a fundamental significance in the context of the self-consistent Langmuir turbulence theory as discussed by Yoon ( 2012 , 2014 ). The purpose of the present paper is to reformulate the modified κ -model of Scherer et al. ( 2017 ) in light of the quasi-equilibrium of turbulent Langmuir fluctuation and the related electron κ -like distribution. That is, the present paper aims to show that the modified κ -model of Scherer et al. ( 2017 ) generalizes the model solution discussed by Yoon ( 2012 , 2014 ), thereby resolving the undesired feature of “infrared catastrophe” associated with the steady-state Langmuir turbulence spectral intensity. The organization of the present paper is as follows: Section 2 discusses the generalized theory of turbulent quasi-equilibrium between the Langmuir turbulence and suprathermal electron distribution, which leads to the regularized κ -model of Scherer et al. ( 2017 ). In the last section we discuss the physical origin of the exponential cutoff factor and also discuss the possible application of the present finding. 2. Equilibrium State between Langmuir Fluctuations and Electrons The fundamental aspect and details of the theory of the quasi-equilibrium state between Langmuir fluctuations and suprathermal electrons are already discussed elsewhere (Yoon 2012 , 2014 ; Livadiotis 2017 ), so we will not repeat the derivations of the equations here. Essentially, the theory boils down to three equations. The first equation represents the formal solution to the steady-state particle (electron) kinetic equation,where f e ( v ) represents the isotropic electron velocity distribution function; m e is the electron rest mass; I L ( k ) corresponds to the steady-state Langmuir fluctuation intensity, k being the magnitude of the wave number, which is being replaced by by virtue of the resonance condition; and is the plasma frequency, n and e being the ambient plasma number density and unit charge, respectively. The second equation is the steady-state wave kinetic equation for Langmuir fluctuations that are in linear wave–particle resonant interaction with the particles (electrons),whereis the Langmuir wave dispersion relation,being the electron thermal spread. Here, T e represents the Maxwellian temperature for the electrons. That is, T e is a macroscopic parameter had the electron distribution function been given by the Maxwellian distribution function. Of course, when the underlying distribution function is non-Maxwellian, then T e is not the real temperature, which is well known, and has also recently been systematically discussed by Nicolaou Livadiotis ( 2016 ). Consequently, by T e , we mean the electron temperature in a loose sense. The third equation represents the steady-state turbulent interaction between Langmuir fluctuations and thermal ions (protons), Of the


to 0.015 mag. Again, we used the Kron magnitudes for the subsequent analysis. An example of the excellent HEROES image quality, and also the various artefacts produced by bright objects in the field, is given in Fig. 3 showing the shell around the planetary nebula NGC 6543 (Cat’s Eye Nebula) that was ejected by its predecessor red giant star. The actual PN in the centre of the image is completely over-exposed. The high density of faint background objects shows the excellent sensitivity of the data. The NEP lies at relatively low Galactic latitudes, and therefore contains a rather large number of overexposed foreground stars showing up in this image (in cyan). In order to obtain reliable optical photometry for brighter objects, which are saturated in the HEROES HSC images, we also made use of the SDSS DR16 ( Ahumada et al. 2020 ), the Pan-STARRS DR1 DR2 ( Chambers et al. 2016 ), and Gaia DR2 ( Gaia Collaboration 2018 ) catalogues. We also cross-correlated our samples with the far-UV and near-UV photometry from the GALEX surveys ( Martin et al. 2005 ). Fig. 3. False colour HEROES image of the shell around the planetary nebula NGC 6543 (Cat’s Eye Nebula) close to the NEP. Blue represents the HSC g -band image and predominantly shows the green [OIII] line. Green represents the HSC r -band and predominantly shows H α . Red corresponds to the 4.5 μ m Spitzer image and mainly shows dust emission. 4.2. Mid-infrared observations of the HEROES field For photometric information in the 3−25 μ m band across the whole HEROES field we used the WISE all-sky survey catalogues. For the shorter wavelength W 1 (3.4 μ m) and W 2 (4.6 μ m) bands we used the CatWISE2020 all-sky catalogue ( Eisenhardt et al. 2020 ) (see also Marocco et al., in prep.), containing about 1.89 billion objects observed by the Wide-field Infrared Survey Explorer (WISE and NEOWISE), both of which have higher sensitivity and higher angular resolution than the original ALLWISE catalogue ( Cutri et al. 2013 ). The CatWISE2020 catalogue in the HEROES field contains 2.4 million objects and gives fluxes in the W 1 and W 2 band, which we converted to the AB magnitude system. For the longer wavelength W 3 (12 μ m) and W 4 (22 μ m) bands we used the original ALLWISE catalogue ( Cutri et al. 2013 ), again converting to AB magnitudes. At the faintest magnitudes the CatWISE2020 and ALLWISE catalogues are, however, severely confusion limited. In the centre of the HEROES field, we therefore also made use of the Spitzer Observations of the Euclid Deep Field North (Moneti et al., in prep.). The ESA Euclid mission is making great progress towards its launch, scheduled in 2022. Euclid ’s main goal is to survey a large fraction of the sky and image billions of galaxies to investigate dark energy and dark matter over the history of the Universe. Roughly 10% of the observing time will be dedicated to the Euclid Deep Fields, repeatedly observing three specific areas of the sky, covering a total of


field, while higher density regions become dominated by an interplay between the action of gravity and magnetic fields. Similarly to the left-hand panel of Fig. 11 , the right-hand panel shows the HRO curves for two self-gravitating models, and . This was evaluated over the last output of these models, when t = 0.3 t ff (see the right-hand side of Fig. 1 which shows the filamentary 3D distributions for these models). Compared to the models that do not include self-gravity, there is an enhancement in the number of regions perpendicular to the magnetic field at higher densities. With the action of gravity, even the super-Alfvénic model shows a change in the number of counts of cos φ = ±1.0. The critical density (vertical green dashed line in Fig. 12 ) seems to be related to the densest bins, for models with sonic Mach numbers and 7.0. There is a jump in the values of density between the penultimate and last points in the diagrams that comes from the density range considered in the bins (as can be seen in Fig. 11 , the 20th bin has a wider range of densities compared to the 19th). In the super-Alfvénic case, the penultimate point is very close to ρ c for all times considered, while in the sub-Alfvénic case the penultimate point approaches ρ c as the system evolves. Tracing the critical density of a system using the PRS is an interesting possibility, and the exact relation between the two can be further explored following the evolution of the alignment between structures and the magnetic field at smaller scales and at latter times of the collapse. However due to the lack of an adaptive mesh with increasing resolution in the densest regions in our models, this is out of the scope of this work. Still, the results discussed up to this point are very similar to what was shown in the previous section for models without self-gravity. We note that self-gravity does increase the number of denser regions where is parallel to the gradient of density in the sub-Alfvénic models, but it has little or no effect when it comes to the simulations with . At smaller scales this is probably not true, since we expect that magnetic fields should be brought along with the fluid during collapse, at some point these cores must become sub-Alfvénic and once again would influence how the gas collapses. However, the simulations do not have enough resolution to follow the process up to this point. In order to compare the simulations with observations we need once again to integrate along a defined LOS. We will follow same directions used in Section 3.1.1 . Figs 13 and 14 show the PRS analysis ( Z 2D ) for the integrated density (column density) distribution along the three different LOSs. As in Figs 8 (for initial ) and 9 (for initial ), the left-hand column represents the LOS perpendicular to the initial magnetic field, the middle column shows Z 2D for a LOS making an angle of 45° with respect to the initial field, and the right-hand column is for a LOS parallel to the