of the ion spectra, here taken to be 30 MeV nucleon −1 . The figure suggests that neutron production is more effective for steep ion spectra, and this is true but only if the number of ions above this normalizing energy is kept constant as the index is changed. Figure 6. Total angle-integrated neutron yield, , as a function of the accelerated-ion spectral index for accelerated α /proton = 0.1 and 3 He/ 4 He = 0.0, 0.5, 1.0, 5.0, and 10.0. Even for accelerated 3 He/ 4 He ratios as large as 1, neutron production is seen to be increased only by about a factor of 2. Only for much larger 3 He enhancement factors would the neutron yield be significantly affected. Most of the impact is for ion spectral indexes near the middle of the index range (3–6). We can understand this by considering Figure 7 , showing the fraction of the total neutron yield contributed by each of the 11 reaction types (p+ 4 He, 4 He+H, p+H, 4 He+ 4 He, p+[CNO], [CNO]+H, 4 He+[CNO], [CNO]+ 4 He, 3 He+H, 3 He+ 4 He, and 3 He+[CNO], where “[CNO]” refers to all nuclear species heavier than 4 He). For spectral indexes of 3–5, the 3 He + H reaction contributes more than 10% of the total (and the new 3 He + [CNO] reactions, shown in red, are not significant). For flatter spectra, neutron production is dominated by proton reactions so the impact of enhanced 3 He is less. For steeper ion spectra, the contribution of the 3 He + H reaction falls due to its relatively high threshold energy and the 3 He + [CNO] reactions increase to contribute up to about 10%. We note that for steep spectra, the contribution of the 4 He + [CNO] reactions and their inverse reactions [CNO] + 4 He are both strong. As noted above, the corresponding 3 He inverse reactions ([CNO] + 3 He) are not included because of the very low abundance of 3 He in the ambient solar atmosphere ( 3 He/H ∼ 10 −5 ). Figure 7. Fraction of the total neutron yield contributed by the 11 reaction types for accelerated α /proton = 0.1 and 3 He/ 4 He = 0.5. The red curve shows the fraction from 3 He + [CNO] reactions. Figure 8 shows the total angle-integrated neutron yield as a function of accelerated-ion spectral index for two values of accelerated α /proton: 0.1 and 0.5. For each value, total yields are shown for (1) coronal accelerated-ion composition and 3 He/ 4 He = 0 (solid), (2) coronal accelerated-ion composition and 3 He/ 4 He = 0.5 (dotted), and (3) impulsive-flare accelerated-ion composition and 3 He/ 4 He = 0.5 (dashed). For the α /proton = 0.1 case, the impact of the impulsive-flare accelerated-ion composition via the inverse reactions can be seen for the steeper spectra because the cross sections for neutron production by α -particle (and 3 He) reactions with heavy elements tend to have lower threshold energies than those of proton reactions; these reactions are emphasized by steep accelerated-ion spectra. The impact of the impulsive-flare accelerated-ion composition is not as large for the α /proton = 0.5 case because


for both efficient and inefficient conduction appear to follow the general shape of the data. High velocities also correspond to high q values, reflecting the effects of shocks on both the cloudlets within a cooling run and the filaments in the conduction runs to compress the gas. 5. Application: Down-the-barrel Outflow Observations As an illustration of the types of studies enabled by our results, we consider an application of our tabulated fits. We consider the observations in Chisholm et al. ( 2018 ) in particular as many of our assumptions, including radiation background and ionization parameter, are derived from these observations. Chisholm et al. ( 2018 ) report absorption profiles from down-the-barrel observations, of a lensed galaxy with z ≈ 2.9. Observations are made of both low and high ionization profiles that indicate that the two phases are cospatial, much like the wind-cloud interaction considered here. We aim to determine an appropriate scaling of our column density profiles by accounting for two factors that can influence the optical depth of the absorbing clouds. To approximate an absorption profile for these ions, we estimate the observed optical depth for a particular ion aswhere is the average number density of the ion, where x is the number of points within the column density profile and N i is the column density for the i th point. Here τ is approximated with the center of the absorption profile at the average velocity of the cloud, v 0 , with a velocity dispersion determined by the b parameter estimated for each ion, and v is the velocity bin within the absorption profile. We consider velocities between −600 and 200 km s −1 offset from line center. This average optical depth approximates a single cloud. There are then two ways to parameterize the absorption profile, we consider the two parameters independently. The first method is altering covering fraction, which describes the fraction of the area within the sightline that is obscured by the cloud. With this covering fraction parameter, the observed flux from the derived column density profiles can be expressed aswhere τ ( v ) is the average optical depth above and f is the free parameter describing the covering fraction. The second way to parameterize the absorption profile considers scaling the optical depth either to represent multiple clouds along the sightline or one cloud with scaled density. For this, we assume that the intervening cloud has the average optical depth of a single cloud from Equation ( 16 ), which is then scaled by the parameter α , which describes the number of clouds within the sightline or the scaling factor of the density for a single cloud. In this case, the observed flux can be described aswhere τ ( v ) is again the average optical depth above and α is the second free parameter we consider, describing the number of clouds. Best-fit covering fraction and α are found by performing χ 2 minimizations for each of the parameters independently


this may be the largest hurdle in adapting numerical magnetohydrodynamics software to use the proposed scheme. 6. Conclusion We have shown that self-gravity source terms, when properly evaluated, are capable of being fully conservative (e.g., Mikami et al. 2008 ; Springel 2010 ; Katz et al. 2016 ; Hanawa 2019 ). The source terms are derived by guaranteeing their equivalence to a corresponding flux (i.e., for momentum source terms, the divergence of the gravitational stress tensor, and for the energy source term, the divergence of a gravitational energy flux). As presented in this work, the fully conservative source-term scheme is formally second-order accurate in space and time, and does not increase the total number of Poisson solves needed per numerical time step compared to Athena++ v19.0 . The scheme can be implemented alongside a broad class of temporal integrators (e.g., VL2, RK2, RK3, etc.), Riemann solvers, or reconstruction methods (e.g., PLM, PPM, etc.). The three test problems presented in Section 4 exemplify the key advantages of the fully conservative source-term scheme. For the 1D Spitzer sheet advection test problems, we see that the the L 1 errors are nearly identical when the momentum equation is integrated via the divergence of the gravitational stress tensor T g and the conservative source terms. The similarities in the errors and error convergence between the two Spitzer sheet runs are a reflection of the two schemes being equivalent via Equations ( 23 ) and ( 24 ). The Jeans linear stable wave tests demonstrate that the scheme can be used in conjunction with a variety of temporal integrators (e.g., VL2, RK2, and RK3) and reconstruction methods (e.g., PLM and PPM). Although we use the HLLC Riemann solver for all test problems, any solver can be employed; the scheme only mandates that the energy source term uses the same mass fluxes applied in evolving the continuity equation (i.e., the energy source term must be consistent with mass conservation). The source terms in Section 2.3 only guarantee spatial second-order accuracy; however, the Jeans linear wave analysis shows that higher-order temporal integrators and reconstruction methods are still capable of significantly lowering errors at a given resolution. The Jeans instability test demonstrates the algorithm’s ability to conserve total energy. Despite significant changes in each component of the volume-integrated energy, total energy is conserved to round-off error. Total energy conservation (to round-off error) requires that the solution to the discretized Poisson equation (Equation ( 26 )) is of round-off error accuracy at the beginning and final stages of the numerical time step. As noted in Section 3.2 , the algorithm does not require machine-accurate evaluations of the potential at intermediate stages for total energy conservation. This could yield interesting strategies when solving the Poisson equation with iterative methods, e.g., perhaps the convergence


2018 ; Barr et al. 2018 ; Wang et al. 2017 ; Grimm et al. 2018 ), and we also find the vast majority of the possible structures for planet b have substantial quantities of water (~ 20−70%). The remaining planets are also probably water-rich, most notably planet d, where a volume fraction of H 2 O ≳ 0.25 is required by the mass and radius data. Table 2 summarises the values of T eq , F tidal , F RG , and F glob for planets b through e. Planets d and e have a total heat flux from tidal heating and insolation that is below the critical heat flux for a runaway greenhouse effect, and could be potentially habitable from a geophysical point of view. Planet d avoids a runaway greenhouse effect only if its Bond albedo is greater than or approximately equal to 0.3, indicating a surface at least as reflective as the surface of the Earth, which could be possible if the planet has a thick atmosphere with highly reflective clouds ( Turbet et al. 2018 ). In this case, planet d could have liquid water at its surface at the substellar point provided it is tidally locked in a 1:1 spin–orbit resonance ( Turbet et al. 2018 ). It is not clear whether TRAPPIST-1d is synchronously rotating because perturbations from the other planets may prevent tidal locking which results in a larger illuminated area where surface ice can melt ( Vinson Hansen 2017 ). Since the planet is close to the runaway greenhouse state due tothe received stellar irradiation, the moderate tidal heating rate that is present in the body might transform itinto a “tidal Venus” ( Barnes et al. 2013 ). We find that planet e avoids a runaway greenhouse effect regardless of its albedo, bolstering the already strong case for its habitability based solely on climate modelling ( Wolf 2017 ; Turbet et al. 2018 ). Planet e can maintain global surface oceans through greenhouse warming from H 2 O alone ( Turbet et al. 2018 ); alternatively, Earth-like temperatures can be maintained with an atmosphere of 1 bar of N 2 and 0.4 bar CO 2 , or a pure CO 2 atmosphere with a pressure of 1.3 bar ( Wolf 2017 ). If planets d or e were ice-covered, they could still sustain biospheres fueled by the planet’s interior tidal dissipation, for example clustered around hydrothermal vents at the top of the internally heated rock layer, as has been suggested for Jupiter’s moon Europa ( Vance et al. 2007 ). Since both planets have equilibrium rock mantle temperatures greater than the melting point of silicate minerals, heat dissipation from the rock layer can locally melt the high-pressure ice to provide a warmer environment where the rock interacts with water through the vents and might support simple chemoautolithotropic life ( Vance et al. 2007 ). The surface ice layer would shield the biosphere from the strong stellar flare activity of the TRAPPIST-1 star ( Vida et al. 2017 ; Garraffo et al. 2017 ), but would serve as a barrier to detection of the biosphere through telescopic observations. Hydrothermal activity would be more


and systematics), the resulting co-added maps will now also contain a simulated signal that is correlated with that present in the FIRST simulations. We have passed these co-added ‘real + simulated’ maps through our power spectrum estimation pipeline, and the results are shown in Fig. 11 . These recovered power spectra have undergone noise bias subtraction and correction of the mask in exactly the same way as before. Fig. 11 shows that the recovery of the SDSS auto-power spectrum is very heavily biased by the presence of unaccounted-for large-scale systematics in the data. 17 However, the right-hand panel of Fig. 11 shows that the recovery of the SDSS–FIRST cross-power spectrum remains unbiased, despite the presence of the SDSS systematics. Note that, although it is unbiased, the cross-power spectrum is not completely unaffected by the presence of the SDSS systematics: their presence results in a significant increase in the error bars of the cross-power spectrum recovery (compare the right-hand panel of Fig. 11 and the central panel of Fig. 10 ), due to chance correlations between the noise in the FIRST simulations and the SDSS systematic effects. We will account for these additional contributions to the errors in our final analysis. Figure 11. The recovered mean SDSS auto-power spectra ( left-hand panel) and SDSS–FIRST cross-power spectra ( right-hand panel) from 100 simulations in the presence of large-scale systematic effects in the real SDSS shear maps. For a definition of the symbols, see Fig. 10 . Systematic effects in the SDSS data severely bias the recovery of the auto-power spectrum, shown as the solid curve in the left-hand panel. The cross-power spectrum recovery is unbiased in the presence of the same systematics, though the errors are amplified. To assess the impact of the large-scale systematics in the FIRST shear maps, we have performed an equivalent analysis to what is described above but now we superimpose the FIRST signal-only simulations on to the real FIRST maps and cross-correlate these with the SDSS simulations. The results of this test are displayed in Fig. 12 where we observe similar effects to what is described above – the FIRST auto-power spectrum is heavily biased by large-scale systematics while the cross-power spectrum remains unbiased. Figure 12. The recovered mean FIRST auto-power spectra ( left-hand panel) and SDSS–FIRST cross-power spectra ( right-hand panel) from 100 simulations in the presence of large-scale systematic effects in the real FIRST shear maps. For a definition of the symbols, see Fig. 10 . Systematic effects in the FIRST data severely bias the recovery of the auto-power spectrum, shown as the solid curve in the left-hand panel. As in Fig. 11 , the systematics contribute additional errors into the cross-power spectrum measurement though it remains unbiased. 6 REAL DATA MEASUREMENTS In the previous section, we have demonstrated, using simulations, that we can extract an unbiased cosmic shear


dispersion as a Doppler shift; Huang et al. 2018 ). Battye et al. ( 2020 ), however, suggested that the line width is dominated by the neutron star’s spinning magnetosphere. We adopt this spinning-mirror model that produces, on average, a bandwidth Δ f / f ≃ Ω r c / c , where Ω is the rotation angular frequency and r c is the axion–photon conversion radius. Hook et al. ( 2018 ) showed that r c depends on the neutron star’s radius r 0 , magnetic field B 0 , angular frequency, polar orientation angle θ , and magnetic axis offset angle θ m :where . For now, we assume that θ = π /2 and θ m = 0 (we deal appropriately with these angles in Section 5 ) to obtain the expected line width:(4.1 μ eV corresponds to 1 GHz as observed). PSR J1745−2900 has a 3.76 s rotation period (Kennea et al. 2013 ) and a magnetic field of 1.6 × 10 14 G (Mori et al. 2013 ). The expected axion–photon conversion line width is thus . This corresponds to 2500 km s −1 at 1 GHz and 215 km s −1 at 40 GHz, which is generally broader than the expected dark matter dispersion, ∼300 km s −1 , except at the highest observed frequencies. Spectra have flagged channels due to RFI and due to spectral window edges that lack the signal-to-noise for calibration. Flagged channels are much narrower than the expected line width, so we interpolate across these channels when smoothing using a Gaussian kernel (Price-Whelan et al. 2018 ). In some bands, such as L , S , and K , entire spectral windows can be flagged and all information is lost. RRL emission lines are seen toward the magnetar in X -band (15A-418), Ku 12–13 GHz, Ku 17–18 GHz, and K 26 GHz. To remove RRLs, two 2 MHz channels are flagged per line, and subsequent smoothing interpolates across the flagged channels (the expected axion conversion line width is 18 MHz in X -band and 24 MHz at 26 GHz). We combine spectra obtained from multiple observing sessions using an error-weighted average, and the sky (noise) spectra are combined in quadrature. When different observing programs overlap in frequency, we select the lowest noise observation. This is effectively the same as combining overlapping spectra in quadrature because the less sensitive spectra contribute negligibly to an error-weighted mean. Table 1 lists synthesized beam parameters, channel widths, and spectral rms noise values. Appendix appendix presents the new magnetar flux, noise, and signal-to-noise spectra. 4. Results After smoothing to the expected axion–photon conversion line width, the new spectra show no significant (>3 σ ) single-channel emission features (Appendix appendix ). The exception is a 3.2 σ channel at 2.34 GHz (9.67 μ eV) in an RFI-affected region of S -band (there also two channels at 3.1 σ and 3.5 σ in previous Ka -band spectra; Darling 2020 ). We obtain single-channel 95% confidence limit flux density spectra from the sky noise spectra. Figure 1 shows the combined limits from this work and Darling ( 2020 ). These limits do depend on the assumed line width


Figure 4. The (0.3–1 versus 1–4 keV) X-ray soft time lag versus frequency (left) and (3–10 keV) energy spectra (right) fitted with the kynxilrev and kynxillver models, respectively, using the same values of the parameters obtained for spin zero (top) and rapidly spinning (bottom) from Table 2 . Both the a = 0 and the a = 0.99 best-fitting probability values are rather low. The poor quality of the model fits is mainly due to the best-fitting residuals that appear at energies below ∼3.3 keV in the LF energy spectra (possibly due to the presence of a blueshifted, S xvi absorption line at these energies, e.g. Jiang et al. 2018 ). Indeed, if we neglect the points below 3.3 keV in the energy spectra the best-fitting improves significantly, with χ 2 = 430.8 and 439.8 for 403 degrees of freedom when a = 0.99 and 0, respectively. We furthermore investigated if we can get a better fit by allowing the inclination to be a free parameter. The best-fitting inclination values are 47.5° ± 1.8 and for the a = 0.99 and 0 cases, respectively. These values are similar to the one we had originally assumed (45°). Since the fit does not improve significantly (Δχ 2 ∼ 2 − 3 for 414 degrees of freedom) and the new model parameter values are consistent (within the errors) with the previous ones, we decided to keep the original best-fitting results. Looking at the residuals of the best-fits to the time lags (left column panels in Fig. 4 ), we can see that the a = 0 model cannot fit the observed time lags as good as the a = 0.99 model. The best-fitting a = 0 model appears to be consistently overestimating the time lags at frequencies higher than ∼3 × 10 −4 Hz. As a result, Δχ 2 = 10 between the best fit of the a = 0.99 and the a = 0 model to the time lags only (Δχ 2 = 18 when we consider the energy spectra as well). It is difficult to quantify the improvement in the quality of the best fit in the high-spin case, one reason being that the number of degrees of freedom is the same in both model fits. One way to do so is to compute the ‘ratio of likelihoods’, which is equal to (Mushotzky 1982 ). In our case, this is identical to the ‘evidence ratio’, which is based on the Akaike (Akaike 1973 ) information criterion (see equation 8 in Emmanoulopoulos et al. 2016 ). This ratio is a measure of the relative likelihood of one versus the other model, and is equal to ∼150 in our case (∼8100 when we consider the energy spectra as well). This result does not mean that we can reject the hypothesis that the spin zero model is correct, but it does mean that the high spin model is much ‘more likely’ (i.e many hundreds to thousand times) to be the ‘real/correct’ model. We therefore conclude that our results strongly support the hypothesis of a rapidly rotating BH in IRAS 13224-3809. The best-fitting iron abundance and disc density values are consistent with the results from Jiang et al. ( 2018 ), who performed a detailed spectral analysis of the 2016 observations using a similar model to


which suffers from persistence and non-linearity effects, we created pixel-to-pixel maps that we used to pre-process and correct the raw images (details in Section A ). Following standard data reduction steps (Section 2.1 ) and calibration, we were able to achieve excellent image registration resulting in a sharp and stable PSF of 0.25 arcsec over the entire 4 × 4 arcmin LUCI 2 field of view. The PSF size only increases by out to the detector edges (Section B ). This superb spatial resolution enables to resolve the star cluster candidates and bring down contamination from background galaxies to a minimum. In a forthcoming paper we will present the detailed analysis of the star cluster system of NGC 6384 (cf Section 3.1 ), while in this paper we mainly focus our analysis to the central 3.6 × 3.6 kpc (∼36 × 36 arcsec) overlap region between our NIR LBT and archival optical HST data. The high spatial NIR imaging is crucial for breaking age–metallicity–reddening degeneracies when combined with the optical HST data. We reprocessed the HST /ACS and WFPC 2 data to the same plate scale resolution of 0.05 arcsec pixel −1 (cf. Section 2.2 ). We performed a 2D MCMC image fitting with imfit (Erwin 2015 ) by using a PSF model built from stars in the image for both the LUCI 2 and HST images, as well as a tinytim (Krist et al. 2011 ) PSF model drizzled in identical manner as the science images. We also used a pixel mask of the central obscuring disc-like dust lane (Section 3.2 , Fig. 5 ). Our main findings can be summarizes as follows: We uncover the presence of two inner Sersić (low n − index, disc?) components (Section 3.2 , Fig. 4 and Section 4.1 ) with effective radii of ∼100 and 400 pc in which the NSC with r eff ≃ 10 pc is embedded. NGC 6384 has a large-scale boxy bulge (∼1 kpc) and disc (∼6 kpc). The effective radius of NGC 6384 NSC increases with wavelength (cf. Fig. 11 , which suggests for a more centrally concentrated younger stellar population (see Section 4.2 ). The NSC effective radius is smaller in size by and magnitude/mass by ∼0.4 mag/dex compared to Georgiev Böker ( 2014 ) due to the unaccounted inner Sersić components. This highlights the need to resolve and model well the central regions for the correct mass and dynamical modelling of the NSC when follow-up spectroscopy is included. The NSC SED is described by an old population (10.8 ± 1.3 Gyr) with a stellar mass of , and a young population of 230 ± 140 Myr, which is of its mass. Both populations have slightly sub- and supersolar metallicity, respectively (cf. Table 2 , Sections 3.3 and 4.3 ). We obtained the SED radial surface mass density, age, metalicity, and reddening (cf. Fig 10 , Sections 3.3 and 4.3 ) We find negligible, 1 per cent fraction of the light, to be coming from a weak AGN (Section 4.4 ). If an MBH in the NSC of NGC 6384 is present, then it might not be accreting, similarly to those in the MW and M 31. All structural and stellar population evidences suggest that the NGC


geometry, but, e.g., if we assume a 3D isotropic velocity dispersion it would reduce the dynamical time by ). So far, we have no measurement of the shell’s velocity dispersion, but estimates from the literature suggest for similar shells in other galaxies (Quinn 1984 ). This would give a dynamical timescale of t dyn ≈ 192 Myr. Detailed simulations of shells in other galaxies suggest that survival time could be even larger that 1 Gyr, depending on the assumed scenario (Pop et al. 2017 ). The survival time of the shell could be used as an upper limit for the time the minor merger took place, i.e., t dyn ≥ t merg , so we estimate t merg ≲ 200 Myr. If the BNS was formed as such in a shell, then we would have expected to see evidence for recent star formation, but we find no indication of this. In the absence of star formation, it is plausible that the BNS coalescence was triggered by a dynamical process, e.g., NS–NS capture or the destabilization of a pre-existing wide-separation binary. These processes will be quite sensitive to the stellar density, which, given the Sérsic index and the luminosity from the residual image found in Section 3.2 , is high in the center of NGC 4993 and around the transient position. If this dynamical hypothesis is true, then the delay time Δ t NSM between the BNS formation and coalescence is ≲200 Myr. On the other hand, Blanchard et al. ( 2017 ) find a median delay time of under the assumption that the binary was formed through secular SF. 5.2. Galaxy Environment If the binary formation is related to dynamical processes in galaxy merging, as we are investigating here, then this is most likely to happen in galaxy groups and low-mass clusters. According to the 2MASS catalog (Tully 2015 ), NGC 4993 resides in a group of which we analyze the remaining seven galaxies. A spectral analysis shows that NGC 4993 is not the only galaxy showing AGN activity (see Figure 3 ), but it is peculiar in terms of age, metallicity, and mass-to-light ratio. It shows an older stellar population (the mean age of the other 13 galaxies is Log (Age) = 9.56 ± 0.17), lower metallicity (mean: M / H = −0.31 ± 0.11), and higher M / L r (mean: 2.41 ± 0.45) than the average. The group has a projected virial radius of R vir = 0.36 Mpc and a line-of-sight velocity dispersion (Tully 2015 ). The crossing time is therefore . If galaxy mergers are correlated to BNS coalescence, future GW studies could possibly concentrate on galaxy groups (but note that these are crowded regions and therefore matching candidates to a host could be difficult). In order to have precise measurements of H 0 , one needs to identify the host galaxy redshift clearly. When the match is clear, the properties of the type of host galaxy found could help future studies to select the right host galaxy or create galaxy catalogs of likely hosts for GW–EM follow-up and untriggered kilonova searches (Doctor et al. 2017 ). In fact, large photometric surveys such as DES, LSST, or WFIRST are


fitted separately. Posterior distributions of { M disc , γ, R c , h c , ψ} are presented in Figs 2 and 3 , while the best-fitting models are shown in Fig. 4 . Best-fit parameters are listed in Table 2 . Figure 2. Posterior distributions of each disc parameter, including their marginalized distributions for V2775 Ori, V1647 Ori, and NY Ori. The vertical dashed lines represent the 16th, 50th, and 84th percentiles. Contours correspond to 68 per cent, 95 per cent, and 99.7 per cent confidence regions. These plots were produced using the python module corner (Foreman-Mackey et al. 2013 ). Figure 3. Same as Fig. 2 , but for V883 Ori and HBC 494. Figure 4. The ALMA 1.3 mm continuum data (left), synthetic best-fitting models (middle), and the residuals (right) for the five resolved discs in our sample. Table 2. Disc parameters from radiative transfer modelling. 4.2.1 Radiative transfer results We find that, at the observed resolution of 0.2 arcsec, V2775 Ori, V1647 Ori, and NY Ori are well described by our parametric models: the posterior distributions of the disc parameters are relatively narrow and single peaked (see Fig. 2 ). However, V883 Ori and HBC 494 have disc parameters that are less well defined. In the case of V883 Ori, the posterior distributions for M disc , R c , and Ψ are double peaked (see Fig. 2 , left-hand panel). We speculate that this is due to the particular structure of V883 Ori, showing a very bright core, 0.2 arcsec in diameter, surrounded by a much more tenuous outer disc. The boundary between these two regions has been interpreted as the water snowline, resulting in a discontinuity in the disc properties (Cieza et al. 2016 ). The water snowline is typically at ∼3–5 au for solar-mass stars and would be unidentifiable at the 80 au resolution of our observations, but given the extreme accretion rate of V883 Ori, it has been displaced to a radius of ∼40 au. Detailed modelling of the V883 Ori disc at ∼0.03 arcsec (15 au) resolution will be presented in a follow-up paper. Similarly, in the case of HBC 494, the posterior distributions for M disc and R c are wider than for the other sources (see Fig. 3 , right-hand panel). This is likely due to the fact that the disc is not radially symmetric, preventing the MCMC exploration of parameters to converge to a narrower range of values. The disc asymmetry in HBC 494 can be seen in Figs 1 and 4 as an elongation of the disc on the south-east direction. We speculate that this asymmetry might indicate the presence of two barely resolved discs (e.g. the system might be a binary as V1118 Ori and FUor itself; Hales et al. 2015 ; Liu et al. 2017 ) or a non-axisymmetric disc structure. Fig. 5 shows a ∼8 mJy residual in our ALMA image after subtracting the best-fitting model, but in different scaling than Fig. 4 . This residual has the same intensity level ( ∼ 7 per cent of the total flux) as in other sources, but is compact and not radially symmetric. A particularly interesting possibility for this


of each other. Finally, we compare a total system luminosity of log ( L /L ⊙ ) = 6.38 (log ( L 1 /L ⊙ ) ≈ 6.25 and log ( L 2 /L ⊙ ) ≈ 5.8) to the tracks for Galactic metallicity from Ekström et al. ( 2012 ), which go up to 120 M ⊙ . While this comparison is much less accurate than the bonnsai method, these luminosities correspond to masses of approximately M 1 ≈ 100 M ⊙ and M 2 ≈ 60 M ⊙ at the ZAMS. This is in excellent agreement with our results from the LMC tracks. 4.3 Origin and evolutionary state Due to its projected distance of 16 arcmin from the massive Westerlund 2 cluster, it has been suggested that WR21a originates from this cluster. Roman-Lopes et al. ( 2011 ) proposed that WR21a might have been dynamically ejected from the Westerlund 2 core, together with the massive binaries WR20aa and WR20c. These stars are in a similar geometrical configuration in the plane of the sky as AE Aur, μ Col, and ι Ori, for which a binary ejection scenario has been proposed (Blaauw 1961 ). Disruption of the orbit by dynamical interactions may explain the high eccentricity of the WR21a system, which is higher than any other massive binary with similar period (Sana et al. 2012 ). The systematic RV of WR21a that follows from our orbital solution is comparable to that of Westerlund 2 (Rauw, Sana Nazé 2011 ), and thus there is no indication for a significant runaway motion in the line of sight if the system indeed originates from this cluster. At the maximum distance estimate of 8 kpc, the projected distance of WR21a from Westerlund 2 is 37 pc. WR21a would have to travel at a speed of 36 km s −1 along the plane of the sky in order to reach its current position within 1 Myr. This is well within the maximum velocity at which a massive binary can be ejected (Perets Šubr 2012 ). The derived spectral types of the components of WR21a can be used to estimate their evolutionary state. Hydrogen-rich WN stars are very massive main sequence stars that are so luminous that their winds are optically thick (de Koter et al. 1997 ). They are located close to the ZAMS in the Hertzsprung–Russell diagram (e.g. Hamann, Gräfener Liermann 2006 ), and thus the primary O3/WN5ha star is expected to be very young, likely around 1–2 Myr. The O3 secondary star is likely of similar age, as stars this massive will evolve towards later spectral types within 2 Myr unless they are rotating rapidly (e.g. Brott et al. 2011 ). The full width at half-maximum of He ii λ4541 Å (3.0206 ± 0.0996 Å) indicates a rotational velocity of v rot sin i ≈ 66 km s −1 following the Ramírez-Agudelo et al. ( 2015 ) calibration ( v rot = 77 km s −1 for an inclination i = 588). Thus, we have no indication for such rapid rotation. 5 SUMMARY AND CONCLUSIONS We have presented multi-epoch observations of the massive binary system WR21a, including a full coverage of the 2011 January periastron passage. The data allow us to obtain orbital solutions for both components of WR21a. We derive very accurate minimum masses


owned by the most massive passive galaxies (Correa et al. 2017 ; Lagos et al. 2018 ). But we note that simulated passive galaxies with are more dispersion dominated than the observed ones. It is revealing to compare our Figs 1 and 7 with the left-hand panel of fig. 1 of Wuyts et al. ( 2011 ), which visualizes the distribution of the Sersic ( 1968 ) index on the diagram. That diagram is often presented to demonstrate the connection between galaxy structure and SFR, with the passive galaxies having a index close to the de Vaucouleurs ( 1948 ) profile ( n = 4) while the star forming ones having exponential profiles ( n = 1) typical for discs. This is interpreted as galaxies generally becoming ellipticals when they quench and become passive. Our diagrams show a similar feature. However there is a crucial difference. In fact, while in the Wuyts et al. ( 2011 ) figure the region of ‘ellipticals’ span the entire mass range from just above up to the largest masses of nearly , in our diagrams there is a clear transition at a significantly larger mass of . Only above that mass being passive is associated to being slow-rotator ‘ellipticals’, while for lower masses quenching produces no more than a minor structural transformation (increasing bulges but retaining significant level of rotation). This is the key novelty we want to emphasize with this work. A transition around is not unexpected, given that only above that critical mass are slow-rotator ETGs starting to dominate, while below that mass one only finds spiral galaxies and fast rotator ETGs (e.g. Cappellari 2016 ). However, in this paper we first illustrate this fact specifically in the context of the diagram, where this fact is not yet universally appreciated. Beyond fitting the light distribution with single Sersic profiles, efforts have been made to decompose galaxies into multiple components including bulges and discs. A relevant result in Morselli et al. ( 2017 ) illustrating bulge to total ratio as a function of stellar mass and SFR shows overall consistency with what we found using the spin parameter. The same difference appears in the passive population for which the spin reveals a sharp transition among massive galaxies whereas fig. 5 of Morselli et al. ( 2017 ) by contrast indicates very close bulge to total ratio over a large range of mass. In the following section, we discuss the implications of our results in the context of a ΛCDM universe in order to understand the trend of with stellar mass and star formation level. 5.1 Understanding in the context of galaxy formation and evolution within a ΛCDM universe 5.1.1 Early/late star formation and in/ex situ mass assembly To understand the current kinematics of galaxies, we need to know what factors are key to determining their kinematic state. In principle, whether a galaxy is mainly in ordered rotation or random motion (that is disc-dominated or spheroid-dominated) will depend on (1) the angular momentum of gas used as raw material for


instability sets in (see Fig. 7 ). As predicted analytically, the main effect of increasing angular momentum ratio j for fixed e b is to monotonically decrease the relative planet-to-binary stationary tilt i s in the prograde case (where i s ≤ 90°) (see Fig. 6 ). In addition, this stationary tilt increases with increasing e b for fixed j in the prograde case. The behaviour of the stationary tilt in the retrograde case is more complicated, but agrees with the analytic predictions given in Martin Lubow ( 2019 ). In this case, the stationary inclination for non-coplanar orbits decreases with increasing j for fixed e b as in the prograde case. But i s changes from increasing with e b to decreasing at . In addition, there are no non-coplanar stationary orbits below a certain j value, denoted by j cr , that depends on e b (see Fig. 10 and equation 4 ). This property does not hold in the prograde case. Another difference between the prograde and retrograde cases is the topology of librating orbits. In the prograde case, the librating orbits are always nested in a i cos φ– i sin φ phase plane about point with i = i s and φ = ±90° (e.g. red and cyan lines in Fig. 1 ). In the retrograde case, for j > j cr described above, librating orbits are not fully nested within each other and do not orbit about the stationary point in the i cos φ– i sin φ phase plane (e.g. magenta lines in Fig. 5 ). In this phase plane, prograde librating orbits have an oval shape, while retrograde librating orbits have a crescent shape. The variation of e b in time is significantly larger for a higher mass planet. In the simulation with initial conditions e b = 0.2, f b = 0.1, and i = 170°, the binary eccentricity can be excited to values very close to 1 (see Fig. 1 ). This behaviour is similar to what occurs with Kozai–Lidov oscillations in which the outer object is a planet (e.g. Naoz 2016 ). The recent release of data from Gaia allowed us to better characterize the kinematics of the eccentric equal mass close binary HD 106906 in the Lower Centaurus Crux group (Bailey et al. 2014 ). This system hosts both a wide asymmetric debris disc and a planetary-mass companion. The formation mechanism of the planet and the stability of its orbit are still under debate (Rodet et al. 2017 ; De Rosa Kalas 2019 ). The binary components have masses and . The orbital period is and the binary eccentricity is e = 0.67 (De Rosa Kalas 2019 ). The planet, HD 106906b, was directly imaged with a projected separation of 738 au and was found to be oriented at 21° from the position angle of the disc mid-plane, suggesting that the planet’s orbit is not coplanar with the system (Kalas et al. 2015 ). The planet mass was inferred to be (Bailey et al. 2014 ). While the orbital properties of the planet are uncertain, if we assume that the semimajor axis is and the eccentricity is zero, the ratio of the angular momentum of the planet to the angular momentum of the binary is j = 1.13. According to our


at low redshifts. Despite this some uncertainty remains as to whether the smaller scatter of Aquarius haloes is mainly due to differences between the particle tagging approach and hydrodynamical simulations. The way to verify this would be to construct mock stellar halo catalogues for apostle haloes and make direct comparisons. Such mock catalogues are not yet available for apostle haloes. Nevertheless, we note that as there are only five Aquarius haloes used for our analysis, it is quite possible that these haloes happen to be good cases simply due to statistical fluctuations. To quantify the statistical significance of the difference, we perform a one-dimensional KolmogorovSmirnov (KS) test to estimate the probability that the fitted parameters of the two samples are drawn from the same distribution. This is done separately for the two parameters. The resulting p -values are 0.3656 for M 200 and 0.6196 for c 200 , This means that we have 36.56percent (61.96percent) probability of observing an equal or even larger difference in the two mass (concentration) samples, which implies that the two samples are not statistically distinguishable for either parameter. Repeating the one-dimensional KS test for the parameter combination corresponding to anticorrelation direction we find a p -value of 0.6506, again compatible with the smaller scatter of Aquarius haloes appearing by chance due to the small sample size. With the larger sample of apostle haloes, we found a significantly larger scatter in the best-fitting parameters compared to those based on Aquarius haloes. The nearly a factor of 2 scatter in M 200 and a factor of 3 scatter in c 200 is worrying for dynamical modelling of MW halo mass. As discussed above, this amount of systematic uncertainty is irreducible and applies to any steady-state models. The typical size of the statistical errors is smaller than the symbol size in Fig. 2 , and is about a factor of 0.03 of the scatter of the systematic uncertainty (blue ellipse). With an average sample size of 4.5 10 4 star particles in each apostle halo, we find the effective number of independent star particles (see Section 4 ) to be N eff 0.03 2 4.5 10 4 40. Recall that in Section 4 , we estimated the effective number of dark matter particles in MW-like haloes to be about 1000. This means, star particles in our hydrodynamical simulations are far more correlated than dark matter particles. This is not surprising, because we know stars have much higher binding energy than dark matter and are more centrally concentrated. They are less relaxed and less phase mixed than dark matter. In addition, we expect about 3040percent of dark matter particles are smoothly accreted instead of being accreted as part of a bound substructure (see details in Wang etal. 2011 , where a detailed investigation on possible dependencies on the resolution has been made), whereas we have checked that more than 80percent of star particles beyond 20kpc of the halo centre


opposite polarity (see the orange and black arrows in Figures 5 (f) and (g), respectively). Noticeably, at the end of the penumbra formation process (Figure 5 (j)), the penumbra covers only half of the umbra, the one facing the opposite polarity. 3.2. Velocity Pattern In Table 2 we report the presence of the inverse Evershed flow. In particular, in 11 spots we observe a flow of opposite sign with respect to that observed when the penumbra has formed, i.e., an inverse Evershed flow. In Figure 6 , we show images of the continuum and LOS velocity maps for the following polarities of the AR 11610 and 11630, respectively. In these cases the inverse Evershed flow was clearly detected before the penumbra formation, although this process has already started. Figure 6. From top to bottom: zoom of the continuum and LOS velocity maps of the following sunspot of AR 11610 ((a)–(h)) and of the following sunspot of AR 11630 ((i)–(p)). The boxes in all of the panels indicate the regions where we calculated the values of the velocity reported in Table 3 . Table 3 reports the mean values of the velocity before and after the penumbra formation, calculated inside the boxes shown in Figure 6 . Table 3 LOS Velocity around Sunspots In fact, in AR 11610, despite that the penumbra surrounds almost all of the umbra, there is a sector without penumbra on the eastern side of the following spot where we observe the onset of the Evershed flow in about 6 hr. The mean velocity before the onset of the Evershed flow is around −160 m s −1 , which becomes 210 m s −1 when the penumbra is completely formed. Concerning AR 11630 (Figure 6 (i)–(p)), we observe at 11:00 UT (Figure 6 (m)), when no penumbra filament is present (Figure 6 (i)), that most of the area inside the black square is redshifted, with a mean velocity of about 220 m s −1 . After 1 hr, filaments appear and the area becomes blueshifted (see Figures 6 (j) and (n)). 4. Discussion and Conclusions In this paper we studied the penumbra formation in 12 ARs. We followed the evolution of the preceding and following spots using continuum and LOS velocity maps and, depending on where the first penumbral sector started to form, we classified the following and preceding polarities of each AR. Despite the fact that not all of the spots in our sample are fully surrounded by a penumbra, we found that the last penumbral sector does not always appear on the side toward the opposite polarity (see the preceding polarity of AR 11150 in Figure 5 ), contrarily to what was reported by Zwaan ( 1992 ). We found that in eight spots the first stable penumbral sector formed on the side toward the opposite polarity, and in nine spots it formed away from the opposite polarity (see Table 2 ). We can also note that the time needed to form the penumbra seems to be related to the side where the penumbra appears. In particular, it seems that the penumbra formation in B-type spots is quicker (about 21 hr) than in the A-type spots (about 39 hr). In


in mass by up to one or two order of magnitudes from z 8.5 to z 0. Its final mass ranges between 10 6 and 10 7 M . On the other hand, when the AGN feedback energy is entirely supplied to the cold phase, the multiphase ISM is not promoted to outflow and remains close to the BH: as a consequence, the BH experiences a rapid phase of mass growth (in the redshift range 3 z 5) during which it deprives of gas the centre of the host galaxy, and then its growth is suppressed. The BH mass at z 0 can be as high as 10 9 M in this case. Therefore, a prediction of our model is that at least a share of the AGN feedback energy has to couple with the diffuse hot gas. We examined the effect of coupling the AGN feedback energy injected in a multiphase ISM to its phases according to their physical properties. We considered a model where feedback energy coupling is driven by the covering factors of the hot and cold phases, assuming that the larger is the volume occupied by the cold gas clumps, the larger is the amount of energy that the cold gas absorbs. Gas accretion is the process that contributes the most to the BH growth, rather than mergers with other BHs (Section 5.2 ). Throughout BH evolution, the total accretion rate is dominated by cold gas accretion, with respect to hot. Remarkably, this is in line with Gaspari etal. ( 2019 ), even if they derived such a conclusion by means of a phenomenological analysis focused on more massive systems. Our conclusion that gas accretion is a more viable channel for BH growth than BH mergers is true unless cold gas which is supported by rotational velocity is prevented from accreting (see below). The quiet merging history that characterizes our simulated disc galaxies also contributes to the minor role of BH mergers to the BH mass growth. The contribution to the BH growth by BHBH mergers is expected to become more significant for increasing BH mass (Fanidakis etal. 2011 ; Dubois, Volonteri Silk 2014a ; Weinberger etal. 2018 ). We find that when the BH only accretes cold gas, it experiences a growth by gas accretion that is faster than (or at most comparable to) the case in which both cold and hot gas are accreted. As for the galaxy M bulge M BH scaling relation, predictions from simulations are in keeping with observations, considering a BH seed mass as large as 10 5 M . When the accretion of cold gas that is supported by rotational velocity is reduced, the BH mass growth is delayed and the BH mass at z 0 is reduced by up to an order of magnitude with respect to the case in which both hot and cold gas accretion proceed unimpeded. Within the scenario emerging from this model, the SMBH in MW-sized galaxies is prevented from growing by gas accretion below z 2, aside from possible BH mergers. Also, the lower is the viscosity assumed for the cold gas which is accreting (i.e. the larger C visc ), the more the BH mass growth is reduced. Simulations that include AGN feedback produce spiral galaxies with a more extended stellar


always is assumed, that β is independent of r . Following the well-known spherical Jeans equation (Binney 1980 ), the radial velocity dispersion of the luminous matter in the early-type lens galaxies can be expressed aswhere β is a constant velocity anisotropy parameter. Combining the mass density profiles in Equation ( 4 ), we obtain the relation between the mass enclosed within a spherical radius r and M E aswhere by we denoted the ratio of respective Euler’s Gamma functions. Simplifying the formulae with the notation ξ = δ + α − 2 taken after Koopmans ( 2006 ), we obtain a convenient form for the radial velocity dispersion by scaling the dynamical mass to the Einstein radius: In all strong lensing measurements used, the observed velocity dispersion is reported, which is a projected luminosity-weighted average of the radially dependent velocity dispersion profile of the lensing galaxy. Its theoretical value can be calculated from Equation ( 7 ) with the assumption that the relationship between stellar number density and stellar luminosity density is spatially constant. This assumption is unlikely to be violated appreciably within the effective radius of the early-type lens galaxies under consideration. Moreover, the actual observed velocity dispersion is measured over the effective spectrometer aperture θ ap and effectively averaged by line-of-sight luminosity. Taking into account the effects of aperture with atmospheric blurring and luminosity-weighted averaging, the averaged observed velocity dispersion takes the formwhere and χ = θ ap / σ atm (Schwab et al. 2010 ); σ atm is the seeing recorded by the spectroscopic guide cameras during observing sessions (Cao et al. 2016 ). The above equation tells us that we can constrain the PPN parameter γ on a sample of lenses with known redshifts of the lens and the source with a measured velocity dispersion and the Einstein radius, provided we have reliable knowledge about the cosmological model and the parameters describing the mass distribution of lensing galaxies ( α , β , δ ). For the purpose of our analysis, the angular diameter distances D A ( z ) between redshifts z 1 and z 2 were calculated using the best-fit matter density parameter Ω m given by the Planck Collaboration assuming a flat FRW metric (Ade et al. 2014 ). Moreover, we allow the luminosity density profile to differ from the total mass density profile, i.e., , and the stellar velocity anisotropy exits, i.e., . Based on a well-studied sample of nearby elliptical galaxies from Gerhard et al. ( 2001 ), the anisotropy β is characterized by a Gaussian distribution, β = 0.18 ± 0.13, that is also extensively used in previous works (Bolton et al. 2006 ; Schwab et al. 2010 ). More recently, Xu et al. ( 2016 ) measured the stellar velocity anisotropy parameter β and its correlations with redshifts and stellar velocity dispersion, based on the Illustris simulated early-type galaxies with spherically symmetric density distributions. It is


conclusions of this paper are correct and valuable. The data presented here suggest another mechanism for the ECD instability. The LHD electric fields of 20 mV m −1 in B ∼ 15 nT produce ExB drift of 1300 km s −1 , comparable to the electron thermal speed. Since this field exists in narrow channels with a width of a few electron gyroradii, only electrons can participate, making ion–electron drift V d ∼ te . Incidentally, this instability attracted attention of researchers working with Hall ion thrusters, because it affects their performance (Ducrocq et al. 2006 ; Boeuf Garrigues 2018 ). In these devices, the instability is also caused by the ExB drift of electrons only, because ions have gyroradius larger than the drift channels. The theory of ECD instability that links electron–cyclotron harmonics with ion-acoustic waves and wavelengths at scales below the electron gyroradius provides convincing explanation for waves in Fig. 2(a) , which exhibit both large E ∥ and E ⊥ , and contain f ce harmonics and Doppler shifted structures smaller than electron gyroradius. 3 WHAT HEATS IONS AND ELECTRONS? We focus now on the fundamental question: What kinds of processes can increase the ion energy by 400 eV (from 50 to 450 eV), and the electron energy by 50 eV (from 20 to 70 eV), within a few seconds, or during one proton gyroperiod, as seen in Fig. 1 . Additional constraint is that heating of electrons is isotropic, while that of ions is mostly perpendicular. The mechanism of stochastic ion heating on LH waves related to condition (1 ) has been explained by Karney ( 1979 ), and on other wave structures by other authors: Cole ( 1976 ), McChesney et al. ( 1987 ), Balikhin et al. ( 1993 ), Mishin Banaszkiewicz ( 1998 ), Stasiewicz et al. ( 2000 , 2013 ), Stasiewicz ( 2007 ), Vranjes Poedts ( 2010 ), and Yoon Bellan ( 2019 ). The essence is that particles stressed by strong gradients lose adiabaticity and can be scattered along the electric field for direct acceleration by both DC and wave fields. The computed χ (Fig. 4b ) shows values greatly exceeding 1, needed to stochastically perturb proton orbits; however, significant ion heating is observed when χ > 10. Formally, the error of χ is equivalent to the error of the electric field measurements, i.e. ∼1 mV m −1 , or 10 per cent of large-amplitude fields, because of gain uncertainty in amplifiers (Lindqvist et al. 2016 ). However, in the case of ECD waves with wavelengths of the electron gyroradius, the gradient computations underestimate real values because the spacecraft separations are larger than wavelengths. In Fig. 4(c) , we show the difference in E y components measured by satellites 1 and 2. Similar differences are seen on any pair of the satellites. A difference Δ E y ≈100 mV m −1 on scales of the electron gyroradius r e ≈ 1 km in the field B = 15 nT would produce χ ≈ 4600, which could effectively demagnetize electrons and subject them to stochastic heating. Applying 10 per cent error to this value


each ii / jj term and ij / ji term. We start with ii / jj . For a fixed baseline that involves antenna i , there will be at most N ant additional baselines that are redundant with and do not involve gain i . From equations ( 39 ) and ( 40 ), the weighting of the covariance between two different baselines in which only one involves antenna i goes as . Thus, the presence of redundancy adds no more than N ant terms involving antenna i but not antenna j and vice versa, for each ii summand in equation ( 35 ). Multiplying this overall factor of by N ant to account for the N ant different ii sums leads to a contribution to the noise amplitude of the order of , similar to the level of the noise without redundancy. As a result, redundancy changes the modelling error amplitude by a factor of order unity in the diagonal terms. Next, we consider the ij terms in equation ( 35 ). For a given and i j , there will be at most N ant redundant baselines that do not involve the i th or j th gains, causing the weighting of each unique variance term to go as rather than in the non-redundant case. Since there are unique baselines that do not involve i or j , the overall sum of these terms goes as . As a result, the ij terms in equation ( 35 ) will have a similar magnitude as the ii / jj terms but the overall impact on the amplitude of the modelling noise described in equation ( 35 ) still changes the amplitude by a factor of order unity. We confirm these arguments with a numerical comparison between the amplitude of the modelling noise with and without redundancy taken into account for two redundant arrays of 91 and 331 hexagonally packed 14m apertures and S min equal to the naturally weighted confusion limit. We compute the off-diagonal elements of by numerically computing the beam integral in equation ( 5 ) for all with Airy beams and perform the full matrix inversions prescribed in equations ( 16 ) and ( 17 ). We compare our results to the same calculation where all off-diagonal elements of are set to zero (Fig. A1 ) and find that the difference in amplitude is essentially a factor of order unity, leading to a negligible increase in the effective . This calculation confirms our argument for HERA-scale arrays. Figure A1. We compare the amplitude of modelling noise on a short baseline when is assumed to be diagonal (orange line) and the off-diagonal terms of are explicitly included for a 91-element (dashed lines) and 331-element (solid lines) hexagonally packed array of 14m apertures. We find that even in a maximally redundant array, the effect of redundancy is to change the overall amplitude of the modelling noise by a factor of a few. This only has a small impact (10percent) on , the smallest k where the 21cm signal (black line) dominates over the modelling noise, as computed from equation ( 51 ) that ignores the effect of redundancy. APPENDIX B: The Point Source Approximation For Modelled Foregrounds For analytic tractability, we assumed that the modelled


groups as an extra fractional uncertainty, f , on all data points. 10 10 We use this statistically robust approach instead of assuming a fixed fractional uncertainty of 3%–5% at each frequency as done by Dobie et al. ( 2018 ). We use emcee (Foreman-Mackey et al. 2013 ) to perform a MCMC analysis to determine the posteriors of the model parameters. We use a logarithmic flat prior on s (−1.5 log s 1.5) because by definition s must be positive and the data do not provide any constraints once s becomes sufficiently large (corresponding to a sharp break that occurs over a time interval much smaller than our sampling time). We use flat linear priors on all other parameters: 10 t b 1000 days, 1 F ν , b 1000 μ Jy, −20 α 1 , α 2 20, and 0 f 1. Figure 1. Up-to-date X-ray, optical, and radio light curves of GW170817 (solid circles; open circles are the new data presented in Dobie et al. 2018 ). The data are clearly indicative of a decline at ≳200 days. Also shown are our structured jet models from Margutti et al. ( 2018 ); see Xie et al. ( 2018 ) for full details of the simulations. Both jets have an ultra-relativistic core with E K,iso = 6 ×10 52 erg within an opening angle θ jet = 9°. The solid lines are for a model with n = 10 −5 cm −3 , θ obs = 17°, e = 0.1, and B = 0.0005, while the dashed lines are for n = 10 −4 cm −3 , θ obs = 20°, e = 0.02, and B = 0.001. Our new radio, optical, and X-ray observations continue to support these models. We find and , with days, indicative of a clear transition from a rising to a declining light curve; see Figure 2 . We can reject a non-declining light curve (i.e., α 2 ≥ 0) at 7 σ confidence level. These results are consistent 11 11 We repeat our MCMC analysis for the subset of radio data presented in Dobie et al. ( 2018 ), with α 1 , α 2 , t b , F ν , b , s , and f as free parameters, and find that t b and α 2 are more loosely constrained: t b = 190 ± 30 days and . A value of α 2 ≥ 0 can be ruled out at 3.0 σ confidence level. with the analysis of Dobie et al. ( 2018 ), but provide a much stronger indication of a break due to the longer time baseline of our observations. Both successful structured jet models and pure cocoon models are consistent with the observed α 1 . Unlike the simplest analytic models, which predict α 2 ≈ −2.2 (Sari et al. 1999 ; Nakar Piran 2018 ), the Xie et al. ( 2018 ) successful structured jet simulations predict a shallower initial post-peak decline (Figure 1 ). Therefore, our derived α 2 does not strongly discriminate between successful structured jet and pure cocoon or dynamical ejecta models, although the latter are mildly disfavored (Lamb et al. 2018 ). Of the two Xie et al. ( 2018 ) models shown in Figure 1 , our most recent data at 273 days and 289 days mildly favor the dashed model, which in comparison to the solid model implies a higher circumbinary density of 10 −4 cm −3 , a lower fraction of the shock energy imparted to electrons ( e = 0.02), and an observer viewing angle


excluded in this work due to their CMD locations. Figure 1. Locations of the stars in the 7 deg 2 APOGEE field of NGC 6553. The solid circle indicates cluster tidal radius ( R t ). Non-members outside 1.5 R t are the grey crosses. Stars inside 1.5 R t are colour-coded, according to the colour bar shown in Fig. 2 . Cluster members and non-members are labelled by filled circles and crosses, respectively. Figure 2. [Fe/H] versus radial velocity. Symbol meanings are the same as in Fig. 1 . Figure 3. Differential reddening-corrected CMD of NGC 6553 from PSF photometry of VVV imaging, supplemented with bright stars from 2MASS. The cluster fiducial sequence is indicated as a red solid line. The grey dots are stars within 1.5 R t (tidal radius), and the black dots are stars within the half-light radius. APOGEE targets inside 1.5 R t are colour-coded as in Fig. 1 . Cluster members and non-members determined in this paper are labelled by filled circles and crosses, respectively. See text for more details. From the coordinate information, these 10 cluster members have no overlap with the spectroscopic studies of NGC 6553 published after the year 2006, which include Alves-Brito et al. ( 2006 ), Zoccali et al. ( 2008 ), Gonzalez et al. ( 2011 ), Johnson et al. ( 2014 ), and Dias et al. ( 2015 ). Most of the APOGEE spectra for our sample have signal-to-noise ratio (SNR) higher than 80, except the second star in Table 1 . Each star in our sample has a single one hour visit, so the SNR is correlated with the stellar brightness. The SNR plays an important role in estimating the uncertainties of element abundances (Majewski et al. 2015 ). Three stars are labelled as ‘PERSIST_HIGH’ and six are labelled as ‘PERSIST_MED’. Persistence, where a significant fraction of accumulated charge is released over a long period of time, is particularly strong in one of the detectors (1.51–1.58 μm) used in the SDSS III/APOGEE survey (Nidever et al. 2015 ). However, by comparing the results from spectra containing persistence pixels versus those without, Holtzman et al. ( 2015 ) suggest that persistence does not dramatically impact the parameters in DR12 data. Table 1. Basic parameters for cluster members of NGC 6553. Notes . a Differential reddening corrected magnitude. b Measurement error. c PERSIST: spectrum has a significant number (>20 per cent) of pixels in the high (or medium) persistence region. d N_WARN: parameter value is within half grid spacing of the synthetic spectrum grid edge for nitrogen. SN_WARN: SNR70. From APOGEE measurements, our 10 cluster members show a mean RV of −0.14 ± 5.47 3 km s −1 . Our mean RV agrees with the other determinations: 4 ± 7.1 km s −1 (Cohen et al. 1999 ), 1.6 ± 6 km s −1 (Meléndez et al. 2003 ), −1.86 ± 2.01 km s −1 (Alves-Brito et al. 2006 ), −2.03 ± 4.85 km s −1 (Johnson et al. 2014 ), and 6 ± 8 km s −1 (Dias et al. 2015 ). In addition, the APOGEE data yield a mean [Fe/H] of −0.15 ± 0.05. This GC has diverse metallicity results from


Sky Survey (Skrutskie et al. 2006 ) with an overall accuracy of 012–015. To account for the measured absolute offsets of ∼023 and ∼057 in the NUV and FUV image astrometry, respectively, we used a grid of reference stars detected in a ∼12′′ radius around PSR B0540−69 to register the MAMA images onto the astrometry reference frame of the WFPC2 images with an accuracy of better than 001. We used the NUV and FUV images with the recalibrated astrometry as a reference for the pulsar identification. 3. Results 3.1. Imaging and Photometry Figure 1 shows the NUV- and FUV-MAMA images of the PSR B0540−69 field obtained after the processing described in the previous section. We clearly detected PSR B0540−69 in both the NUV and FUV images at a position coincident with its optical coordinates computed by Mignani et al. ( 2010b ): (0009), (005). 21 21 The errors refer to the average of the pulsar coordinates computed on four independent WFPC2 data sets, see Mignani et al. ( 2010b ) for details. Ours is the first detection ever of PSR B0540−69 in the UV, which also makes it the fifth isolated pulsar, among the ∼2300 known, that has been detected in the near-IR, optical, UV, X-rays, and γ -rays, after the Crab and Vela pulsars, PSR B0656+14, and Geminga (the fourth among radio pulsars 22 22 Geminga (PSR J0633+1746) has not yet been unambiguously detected as a radio pulsar despite many searches, see Maan ( 2015 ) and references therein. ). We also detected the PSR B0540−69 PWN in our NUV image (Figure 1 , left), with a structure and extent similar to what is observed in the optical and near-IR (Mignani et al. 2010b , 2012 ). The PWN is at most barely visible, however, in the FUV image (Figure 1 , right). The bright emission knot in the PWN detected at ∼17 southwest of the pulsar in HST /WFPC2 images (De Luca et al. 2007 ) is also visible in the STIS/NUV-MAMA observation, aligned with the major axis of the PWN. Since our paper is focused on the pulsar, a coherent multi-wavelength spectral and spatial analysis of the PWN and its features will be the subject of a subsequent paper. Figure 1. STIS NUV-MAMA (left) and FUV-MAMA (right) images (9650 s each) of the PSR B0540−69 field taken with the F25QTZ filters. The images are aligned in R.A. and decl., with north to the top and east to the left. The image size is 9′′ × 9′′. The pulsar is the source at the center of the nebula, clearly visible in the NUV-MAMA image. The intensity scale of the two images (in counts) is color-coded in the two horizontal bars at the bottom of each image. In both cases, the minimum/maximum of the scale have been adjusted to favor the identification of the pulsar and other stars in the field. For a better visualization, both images have been smoothed with a Gaussian function using a kernel of 2 pixel radius. For both the NUV and FUV images, we computed the pulsar fluxes through aperture photometry employing the tools in the IRAF 23 23 IRAF is distributed by the National Optical Astronomy


six ARs. Each time interval is limited to a duration when the corresponding AR is between −60° and +60° with respect to the central meridian to avoid extreme magnetic field projection effects (Bobra et al. 2014 ). Table 1 includes also the onset time and the associated GOES class of the flares occurred in each AR, based on the GOES solar flare catalog. 9 9 https://hesperia.gsfc.nasa.gov/ Furthermore, Table 1 gives information on the dominant evolutionary phases of the ARs, and on how long their δ -spots were observed. Table 1 Summary Table of the Properties of the Studied Six ARs: NOAA Number and the Study Period of AR, Information about the Investigated Flares, Dominant Evolution Phase of the ARs Note. The last coulomb shows the time interval when δ -spot(s) appeared in an AR. 3. Data Analysis The emergence and shearing components of the magnetic helicity of the six ARs are calculated using Equation ( 1 ). The total magnetic helicity flux for a given AR was generated by summing the emergence and shearing components. The three helicity injection rates for each AR were obtained by integrating the helicity flux over the entire area of the AR. The three helicity flux components are further normalized by their respective largest absolute value in order to facilitate comparison on similar scales. The normalized emergence, shearing, and total helicity fluxes are shown in the top panels of Figures 2 – 3 . In the upper panels of Figures 2 – 3 , the blue line is the emergence term, the red line the magnetic helicity flux associated with shearing motions at the photosphere, and the black line is the total magnetic flux. Figure 2. Time analysis of three flaring ARs, namely (a) AR 11166 (b) AR 12192, and (c) AR 11890. The top panels show time series of the normalized emergence (EM, blue), shearing (SH, red), and total (Total, black) helicity fluxes. The red vertical lines mark the onset time of flares. The second through fourth rows show the wavelet power spectrum (WPS) of the normalized emergence, shearing, and total helicity fluxes, respectively. Rather than plotting power directly, the color bars visualize the logarithm of the ratio of the power to the expected power for Gaussian-distributed white noise and significance compared to noise. The x -axis of each WPS is the observation time, and the y -axis is the period, both in hours. On the WPS plots, the black lines bound the cone-of-influence, i.e., the domain where edge effects become important. The plots to the right of each WPS are the corresponding global wavelet spectra (power averaged over time, similar to a Fourier power spectrum). The orange dashed lines mark the one σ confidence in global wavelet spectrum analyses. Figure 3. Same as Figure 2 , but for the three non-flaring ARs, namely (a) AR 12645, (b) AR 12470, and (c) AR 11785. By inspecting the top panels of Figures 2 – 3 (a)–(c), we can see that the normalized shearing and total helicity flux components show similar evolution trends in


, d v1 ) and ( V v2 , d v2 ) (not shown here). Figure 8. Left-hand panel: 2D distribution ( V v3 , d v3 ). There is a linear trend that is correctly described by the linear relation given by equation ( 16 , dashed line). Right-hand panel: the correlation disappears after correcting the LOS displacements with this equation. This is a statistical demonstration of the v-RSD off-centring effect. From this analysis, we arrive at the third important conclusion of this work: void centres shift preferentially along the LOS when they are mapped from r - into z -space, and this displacement can be statistically quantified by means of equation ( 16 ). These results give support to the v-RSD off-centring effect postulated in Section 4.5 . 5.3 Cross-correlations Since the ratio (or equivalently δ R v ) characterizes the change of volume in voids, the statistical analysis of Section 5.1 gave support to the t-RSD expansion effect postulated in Section 4.2 . On the other hand, as the displacement and velocity characterize the movement of voids, the statistical analysis of Section 5.2 gave support to the v-RSD off-centring effect postulated in Section 4.5 . It only remains to test if both effects are statistically independent by looking for cross-correlations between these quantities. Fig. 9 shows the 2D LOS distributions (left-hand panel) and (right-hand panel). The horizontal lines are the theoretical predictions (dashed) and (solid). No correlations can be seen, giving support to the postulated independence. It is worth mentioning that the analogue POS distributions (not shown here): , , , and show a similar behaviour. These results allow us to interpret the large-scale dynamics of the whole region containing the void (v-RSD) as decoupled from the dynamics of the galaxies at scales of the void radius (t-RSD). This also suggests that potential distortion patterns in observations due to these two effects can be treated separately. This is the fourth important conclusion of this work. Figure 9. 2D distributions (left-hand panel) and (right-hand panel). The horizontal lines are the theoretical predictions (equation 7 , dashed) and (equation 10 , solid). No correlations can be seen in both panels, suggesting that the t-RSD expansion effect and the v-RSD off-centring effect are statistically independent. 6 IMPACT ON THE VOID SIZE FUNCTION This section has a double intention. On the one hand, we will finish the analysis of the last section incorporating the additional AP change of volume not treated yet. On the other hand, we will study the impact of all the z -space effects on the void size function. For this reason, we turn now to the FC void catalogues, fully affected by z -space systematics (see Table 1 ). 6.1 Generalities of the <xref ref-type="bibr" rid="bib69">VSF</xref> modelling We start with some generalities concerning the VSF modelling. The real-space VSF can be modelled using the excursion set formalism combined with the spherical expansion of matter


A MannWhitney U test of the distributions about this relation does not rule out at 95percent confidence the null hypothesis that the two redshift sub-samples have the same parent population. We do, however, acknowledge that weak evolution of the relation of equation ( 1 ) cannot be ruled out by our data, and this should be investigated with deep X-ray survey data and/or further deep X-ray observations of carefully selected samples. 2.3 Using pressure to predict X-ray luminosity We have demonstrated that we can reliably predict the ICM pressure at a single distance within the group/cluster surrounding an FRII radio galaxy. To be able to predict the overall group/cluster richness, we need to be able to translate this pressure measurement into an estimate of cluster mass and/or X-ray luminosity. We therefore examined whether the universal cluster pressure profile of Arnaud etal. ( 2010 , hereafter A10 ) could be applied to FRII environments. We first examined the sub-sample of I17 for which reliable spectral temperature estimates were available. Our pressure profiles have large uncertainties, and were obtained from either beta or double-beta model fits to the surface brightness profiles (see I17 ). The environments are also mainly galaxy groups ( L X 10 44 ergs 1 ), and so lower in mass than the sample of A10 ; however, Sun etal. ( 2011 ) have shown that the universal profile also applies on galaxy group scales. Fig. 2 (right-hand panel) shows that our scaled pressure profiles are in broad agreement with the A10 profile shape and normalization. We therefore tested whether the universal cluster pressure profile could be used to translate our single-radius pressure estimates, from our radio predictions, into cluster mass and X-ray luminosity. We used the profile of A10 (equations 1113 of that work) and our input values of distance and P ext to estimate M 500 for each radio-galaxy environment. We then applied the L X M 500 relation of Pratt etal. ( 2009 ) (using the relation for L X R 500 ) to obtain a predicted L X for each radio-galaxy environment. Finally, we compared our radio-predicted X-ray luminosities with the measured X-ray luminosities for our sample from Ineson etal. ( 2015 ). Fig. 3 (left-hand panel) shows the remarkable agreement between our radio-predicted X-ray luminosities and the measured values across more than three orders of magnitude in X-ray luminosity (from poor groups to moderate clusters) our method appears to predict ICM X-ray luminosity to within roughly a factor of 2 based only on the measured properties of the embedded radio galaxy. We have included four additional FRII sources from the sample of I17 for which external pressure distributions could not be measured, but which meet the morphological selection criteria, leading to a sample of 31 objects, as listed in Table 1 . Figure 3. Left-hand panel: the predicted group/cluster X-ray luminosities determined from the radio-measured pressures for the morphologically


data presented here from the ion trap measurements are not of the bare species itself because of the slight perturbation induced by the weakly bound helium atom; however, the shift is expected to be small. Laboratory data on the band at and the weak features near for are shown in Figure 2 and indicate that the helium causes a shift on the electronic transition of of around . Considering the width of the absorption band profiles, for comparison with astronomical data, the spectrum of (Figure 1 ) can be considered to be that of the bare cation. In Table 1 , the band maxima of the absorptions shown in Figure 1 are listed. Figure 2. Electronic spectra of below 10 K. Data are shown for two of the bands presented in Figure 1 . The similarities between the spectra indicate that the shift on the electronic transition of due to the presence of the helium is small. Therefore, the spectrum of may be regarded as that of for comparison with DIB data. A linear offset has been applied to the vertical axis of the data shown. Table 1 Absorption Maxima of the Bands Shown in Figure 1 and Compared with the Gas-phase Prediction Reported by Steglich et al. ( 2011 ) Note. a Weak absorptions with intensities lower than 20% of the band at are not listed. Steglich et al. ( 2011 ) predicted the gas-phase transition energies of based on their matrix data using a model (Gredel et al. 2011 ) to account for the polarizability of the different rare gases used. The present work enables a direct evaluation of this model and their results are also given in Table 1 . The predicted values are found to be in accord with the gas phase; however, the large error bars () indicate that the extrapolation is not quite good enough for an unambiguous comparison with DIB data, especially if the latter are weak (). For this, gas-phase measurements as done here on are required. In the context of astronomical detectability it is important to determine the oscillator strength f of the individual transitions, in particular for the strongest band in the spectrum. This is because the theoretically calculated values are not expected to be very accurate because of the nature of the method that had to be used (TD-DFT). Furthermore, these f values correspond to the the whole electronic absorption system(s). To obtain the oscillator strength for the origin band at , its Franck–Condon factor relative to all the other transitions has to be estimated, as carried out by Steglich et al. ( 2011 ). Thus, it is more reliable to measure absolute absorption cross sections directly in the ion trap, as previously demonstrated by measurements on the fullerene cations (Campbell et al. 2016a , 2017 ). Absorption cross-section measurements were carried out at the wavelength of the band maxima of the two strongest absorptions in the spectrum. Plots showing typical exponential fits to the data are shown in Figure 3 . The characteristic fluence, , was found to be 0.8 and at 8281 and , respectively. These values correspond


gravitational influences, it is possible to include the Yarkovsky drift. The orbfit has additional programs for determination of the orbits of Solar system bodies and could compute their positions on the celestial sphere. Also, orbfit has tools for the computation of Lyapunov exponents and proper elements (analytic and synthetic). 2.2 <sc>mercury</sc> The mercury package (Chambers Migliorini 1997 ) includes different algorithms, and it is written using the fortran 77 programming language. The mercury package is publicly available at https://github.com/4xxi/mercury . mercury has the following N -body algorithms: second-order mixed-variable symplectic integrator, general Bulirsh–Stoer integrator, conservative Bulirsh–Stoer, Everhart's RA15 (RADAU) algorithm and Hybrid symplectic/Bulirsh–Stoer integrator. For the purpose of our analysis, we chose the explicit Hybrid symplectic integrator (Chambers 1999 ) that could compute close encounters. Symplectic integrators (for complete theory of symplectic integrators, see e.g. Sanz-Serna 1991 ; Yoshida 1993 ) do not have a long-term build-up of energy error, and they are significantly faster with problems in which most of the mass is contained in a single body. The idea behind symplectic integrators is to split the Hamiltonian () of a system into pieces, each of which can be solved on its own. Symplectic integrator use a fixed time-step that makes calculating close encounter difficult due to the necessity of changing the step size. One of the possible solutions is to use a smaller step size or to split the perturbing terms, and to set a different step size for each term (SYMBA integrator; Duncan, Levison Lee 1998 ). Chambers ( 1999 ) described an alternative solution – a hybrid integrator that has symplectic and non-symplectic components. That alternative solution keeps the best properties of both symplectic and non-symplectic integrators. 3 METHODOLOGY AND DATA To study the limitations of the BIM, first a fictional family is going to be created based on the Veritas asteroid family. In this respect, all adopted values of the fictitious family are the same as of Veritas. Families are formed from the disruption of the target asteroid during a high-velocity impact. Escape speed ( v esc ) is much smaller than orbital speed ( v orb ≈ 15–20 km s −1 ), and, consequentially, all the orbits of family members are similar immediately after their creation. To create the initial orbital distribution of the simulated family members, we have used Gauss equations. These equations describe changes in the osculating elements due to the components of the ejection velocity: where a , e , i , f and ω are the semimajor axis, eccentricity, inclination, true anomaly and argument of perihelion, respectively; na is the mean orbital velocity; and δ V T , δ V R and δ V W are the components of ejection velocity along the direction of the motion, the radial direction and perpendicular to the orbital plane, respectively. The


value . This gives 3785 proper motion slices. A slice then contains all stars for which holds. The number of stars per slice strongly varies and depends on the total proper motion. The maximum number of 44 000 stars is at (, ) = (), whereas the number of stars is less than 150 at the borders of our square. About 90% of all slices contain less than 6000 stars. These slices represent a seven-fold overlap of the proper motion plane, with the effect that the same over-density on the sky may appear in neighbouring slices, but not necessarily with the same signal-to-noise ratio. (b) In the second step we searched for over-densities in the proper motion slices. For each star in a proper motion slice, we counted all its neighbours within a radius of 0.5 deg (called sky circle) and determined the surface density (stars/square degree). Surrounding each star’s sky circle we chose an area of 9 sq. deg, where we determined the background surface density . We only retain stars if / of their sky circle is larger than 3.0 and their is larger than 5 per square degree. If the angle between two stars is less than 0.5 deg, each of them is in the other’s sky circle. We discard the one with the lower . This cleaning goes on until only “isolated” sky circles are left, which we call Level-0 centres. As our system of proper motion slices represents a seven-fold overlap of the sky, a Level-0 centre in slice could have the same or a neighbouring Level-0 centre in slice . If they are within the radius of a sky circle and the difference of their actual proper motions is less than 1.5 mas/yr, only the Level-0 centre with higher / is retained. These survivors are called Level-1 centres; they are disjunct and are possible candidates of new open clusters. Of course, many of the Level-1 centres could be related to known open clusters. Therefore, we cross-matched them with the MWSC clusters ( Kharchenko et al. 2013 ) and with the most recent cluster list DAML02 VERSION 3.5 of 2016 Jan. 28 ( Dias et al. 2002 ). We discarded all the Level-1 centres within 0.7 deg of a known cluster, if the radius of the known cluster was larger than 0.2 deg. If the radius of a cluster in MWSC or in DAML is smaller than 0.2 deg, it cannot be identical to one of the objects found here. About 200 Level-1 centres surviving this process (called Level-2 centres) were then subject to a photometric check of their existence as clusters. Table 1 New open clusters. 3.2. Photometric steps. (a) We first made a quick and coarse check in the CMD of the stars in a sky circle around a Level-2 centre using TOPCAT 2 ( Taylor 2005 ). A Level-2 centre was selected for a more careful treatment if stars with showed a tendency towards building a part of a main sequence or a turn-off region. The Level-2 centres were immediately rejected if, on the CMDs, their stars showed a random distribution, mostly at the limiting magnitude of Tycho-2. So, we could quickly check the Level-2 centres found in the previous step. Fig. 1


to be the main reason for our different conclusions. 7.4. [α/Fe]–[Fe/H] relations As shown in Fig 7 , the intrinsic scatter of the [ α /Fe]–[Fe/H] data is large as a result of the complex galactic gas and metal distribution, and stars may form outside the main progenitor halo ( Jeon et al. 2017 ). Our mean galactic metal evolution track assumes that all the gas is always well mixed and thus cannot explain the metal abundance of individual single stars. The best-fit model that applies the IGIMF theory is comparable with the solutions given by Lacchin et al. (2020) , which are shown as the dashed and dash-dotted lines in Fig 7 , especially for metal-rich stars. The highest [ α /Fe] value at the metal-poor end of the plots from Lacchin et al. (2020) is lower than our results and fits the data not as well as our model, however. The highest [ α /Fe] value should be the IMF-weighted [ α /Fe] value of all the massive stars with Z = 0 and a lifetime shorter than the first time step, δ t = 10 Myr, that is, the IMF-weighted thin solid line within the red shaded region of Fig. 8 . Because the IMF-weighted value should be higher than the lowest value in the red region, it is clear that the initial [Mg/Fe] should be higher than 0.5, which agrees with our result and disagrees with Lacchin et al. (2020) . Fig. 8. Stellar metal yield ratios of stars with different mass and metallicity given by Kobayashi et al. (2006) . The yields from Kobayashi et al. (2006) for massive stars are connected to the yields for low- and intermediate-mass stars as given by Marigo (2001) . The yields for M * > 40 M ⊙ are the same as the yields for M * = 40 M ⊙ stars. The red shaded region with log 10 ( M *, initial [ M ⊙ ]) > 1.3073 indicates zero-metallicity stars with lifetimes shorter than 10 Myr, according to Yan et al. (2019) . The [ α /Fe] plateau can be enhanced by a more complicated gas-flow model, a larger stellar-mass bin, or a larger model time step. A non-simultaneous star formation combined with local stellar wind pollution can also complicate the model prediction, which requires more elaborate hydrochemical simulations. We note that the metal-poor part of the [ α /Fe]–[Fe/H] relations is affected by the small number of stars that are formed at the earliest time steps. This shows that the metal-poor part is not reliable because of the small number of massive stars that affect it, the uncertainty of the stellar winds of these massive stars, and whether or not they explode as supernovae. Stars that are more massive than 40 M ⊙ share the same adopted stellar yield table, which is certainly not the ideal setup to discuss the chemical abundance of the extremely metal-poor era of a galaxy. 7.5. Star formation history The tested SFT in Lacchin et al. (2020) is generally longer than our best-fit models. Because the gas flow and star formation criteria in the chemical evolution model are uncertain, the best-fit SFH is not conclusive. Here we compare it with the independently


PSFs (right). Both images are on the same logarithmic scale with a lower threshold of . The theoretical PSF is calculated for the central monochromatic wavelength of 3.94 microns. The retardance in the simulation was set to radians, thereby creating the leakage PSF “0.” The observed image is saturated on the first diffraction ring and on the central core but is corrected to the unsaturated flux level. A small asymmetry of the intensities of the wind-driven halos of the star inside the dark holes is seen between the two coronagraphic PSFs (“+” and “−”). Instrumental ghosts are indicated with arrows and are not related to the coronagraphic optic. Further investigation showed that these ghosts can be removed from the dark hole by setting the rotator to an angle of 30°. 3.1. Data Reduction To remove hot, dead, or flaky pixels in each image, we subtracted a median-filtered image with a 3 × 5 pixel box from the cleaned and centroided image cube to generate an image where the outliers clearly stand out. The 3 × 5 box is chosen because the outlying pixels tend to have structure in the direction of the readout and not perpendicular to it. The data points that deviate more than 1000 counts are replaced by the local value of the 3 × 5 median. Four sets of sky reference frames are taken between on-target observations. The sky frames were median-combined for every one of the four sets. Each of the previously taken frames on target were subtracted by the first consecutive median-combined master sky frame. After the background subtraction, the median of a cosmetically clean part of the chip is subtracted in order to remove any residual background offset. A theoretical diffraction pattern consistent with the geometry of the telescope and wavelength of observation is used as a fiducial. This theoretical PSF is then fit to the central leakage PSF by minimizing the chi-squared residuals between the theoretical PSF and the leakage PSF, with x , y , radius (), and intensity as the free parameters of the fit. All images of the data cubes are coregistered by shifting the images to the central pixel of the frame with the previously fit x and y values using a bilinear interpolator. The radius of the leakage PSF fit is used as a subframe quality indicator. The few images (39) that have fitted radii significantly smaller ( pixels) or larger ( pixels) than the diffraction limit are excluded. The best 91% (5200) of the images are selected after sorting the frames by radius from smallest to largest. Both criteria remove the frames where the seeing conditions temporarily worsened or where the AO system lost its lock. After these selections, the images are reordered to original chronological order and binned by four frames, corresponding to 4 s integration time per binned frame to reduce memory consumption and computational time. Instrumental ghosts due to internal reflection of the refractive optics are present in the image; see Figure 2 . Several of these ghosts are


on different methods. A simple radial profile (black line within gray boundaries) nicely fits our results and the previous estimates in the range 1.1−2.0 R ⊙ . 3. Coronal magnetic field The upstream Alfvénic Mach number M A and the magnetic field strength B can be derived from the standoff distance Δ r ≡ r sh − r cme between the shock front r sh and the CME leading edge r cme by employing a model adapted by Gopalswamy Yashiro (2011) but originally developed for the bow shock of the Earth. Physically, the shock Mach number M and the radius of curvature r cme of the nose of the driver control the standoff distance. The ratio δ between Δ r and r cme can be expressed as δ ≡ Δ r / r cme = 0.81[( γ − 1) M 2 + 2]/[( γ − 1)( M 2 − 1)], (e.g., Farris Russell 1994 ) where γ is the adiabatic index (we assume γ = 4/3 and γ = 5/3 as two possible values). This expression allows the applicability of the model to the low- M regime provided that M is substituted for the fast magnetosonic Mach number ( Fairfield et al. 2001 ). In the inner corona, v A substantially exceeds the sound speed meaning that M can be confidently replaced by M A . It then follows that , implying that M A (and thus the coronal magnetic field strength B , given the upstream electron density profile n e ) can be obtained at any given time by measuring δ . We point out that the above method is based on a model by Farris Russell (1994) , further extended by Russell Mulligan (2002) , which was specifically designed for the bow shock of the terrestrial magnetosphere. As such, the evaluation of the standoff distance in that model is intended to be strictly valid for the nose of a CME. It is then important to discuss how appropriate such a model is for a coronal CME-driven shock, for which, moreover, as in our case, the standoff distance is calculated not at the nose but at the flanks of the shock. In order to check the validity of the model in the coronal environment, Schmidt et al. (2016) used a 3D magnetohydrodynamic code to test the accuracy of the Gopalswamy Yashiro (2011) method by applying it to a simulation of a real CME event, where the standoff distances between the driven shock and the CME could be determined within the simulation box. As a result, the predicted magnetic field strength profile agreed very well (within 30% between 1.8 and 10 R ⊙ ) with the magnetic field strength profile found in the simulation domain, thus providing strong support for the Gopalswamy Yashiro (2011) method in a coronal environment. As for the second issue, that is, the reliability of the model when the standoff distance is evaluated at the flanks of the shock surface, it is obvious that in the general case of a fast moving CME the shock surface would be better represented by a 3D oblate surface. In that case, our model would yield unreliable results away from the CME nose. In our specific case however, since the translational motion of the CME away from the Sun is negligible with respect to its


obtained by using (green), (red), and (blue). We make the following observations from this plot: β BF versus f / b falls nicely on a straight line with slope 1 (the solid line). However, note that the relation β BF = f / b (the dashed line) does not exactly hold: we have checked that the actual value of β BF depends on the s min used, decreasing as smaller s min is used; the results here are for . The trend suggests that, roughly speaking, the reconstruction simply spits out the ‘input’ β value, and therefore has no predictive power regarding β, unless we look beyond R ( s ). The two ( f, b ) pairs that give similar β = f / b ≈ 0.375 produce very similar β BF : this is a consequence of the f – b degeneracy mentioned above. For comparison, we include the best-fitting β BF values obtained using R ′ ( s ) as filled circles, using the same colour scheme. Because there is no reconstruction and therefore no input ( f, b ), we simply plot the points at a fixed horizontal coordinate f / b = 0.376 (the theoretical value). We note that β BF obtained in this way is indeed larger than the theoretical value, but this is well within the 1σ uncertainty indicated by the grey error bar; at this stage, we cannot accurately assess the impact on β BF due to the small sample size or the simulation resolution, which is beyond the scope of this work. As mentioned above, a small (per cent-level) error in R ′ ( s ) leads to a larger error in β BF . Figure 12. Upper left : The best-fitting β BF values obtained using the estimator R ( s ) and the reconstructed galaxy catalogues, for seven choices of input parameters f and b supplied to the reconstruction pipeline (crosses). R ( s ) is measured as the weighted average between and s max = 98.9 (green), 90.2 (red), and (blue), respectively, and the error bars are the standard deviations obtained from 15 realizations for the case of as the error bars for remaining s max values are similar. The filled circles are the same, but obtained from the estimator R ′ ( s ): since there is no reconstruction in this case, these data are plotted against the theoretical value β = f / b = 0.376. The dashed line represents β = f / b , whilst the solid line is a line of slope 1.0 to guide the eyes. Lower left : , the constant value fitting ξ 2 [ x ( k ) ]( s ) between and the three s max values s max = 98.9 (green, densely dashed), 90.2 (red, dashed), and (blue, sparesly dashed), respectively, for the same seven choices of input ( f, b ). The vertical dotted lines mark the best-fitting β BF at which is closest to zero, and the error bars are again the standard deviations of the 15 realizations. Right-hand panels : the estimator E4 (upper) and E3 (lower) for the seven choices of reconstruction parameters ( f, b ), with their corresponding values of β indicated in the legend and increase with the sparsenesses of the curves [β increases from to the top (bottom) to the bottom (top) line in the upper (lower) right-hand panels]. Note that


domain to a 4 kpc × 4 kpc × 24 kpc region of the Galaxy. The Galactic magnetic field model of Jansson Farrar ( 2012a ) has azimuthal variations in strength and direction of the field. We chose the galactocentric azimuthal angle to roughly agree with that of the point of impact 70 Myr ago shown in Nichols Bland-Hawthorn ( 2009 ), Figure 3 . The minihalos are necessarily constrained by the size of the domain and fill spheres of 2 kpc radius (the domain half-width). In Figure 1 , the minihalo barycenters are indicated by the “ X ” markers, and their simulation domain half-mass radii (distinct from r e ) are shown as black circles. Figure 1. Initial conditions for each simulation along slices through the y = 0 plane. log 10 of hydrogen number density is represented by the continuous color scale. log 10 of magnetic field strength is represented by contours. The dark matter barycenter is marked with an “ X ” and a circle at the half-mass radius. From left to right, the panels show Runs 1–6, as in Table 1 . 3. RESULTS All six clouds behave similarly as they fall through the halo. This can be seen in Figure 2 which shows the clouds at epoch t 1 (35 Myr), just before impact. As they pass through the halo, the clouds lose material due to ram pressure stripping. By epoch t 1 the clouds with minihalos have lost 16%–18% of their original baryonic mass, while those without have lost ∼36% (see Table 1 ). During this stage, the effect of the weak magnetic field is merely to become swept up and wrapped around the cloud. This is similar to the behavior in the 2D MHD simulations of Konz et al. ( 2002 ). Figure 2. Same as Figure 1 , but for epoch t 1 , equivalent to t = 35 Myr for all runs. Figure 3. Same as Figure 1 , but for epoch t 2 , equivalent to t = 52 Myr for Runs 1 and 2 and t = 45 Myr for Runs 3–6. Each of the clouds approaches the disk with a similar velocity (∼300 km s −1 ). However, the dark matter confined clouds are more compact and contain more baryonic mass and so carry larger kinetic energy densities than the dark-matter-free clouds. The collision of the cloud with the Galactic disk occurs at 36 Myr for those runs with a minihalo, and 37 Myr for the dark-matter-free runs. At the moment of collision, the minihalo separates from the cloud gas and continues, unhindered, through the Galactic disk. During the collision, the gas cloud ceases to exist as it mixes with and transfers momentum to the disk gas. This results in a hole in the disk and a long-lived, bubble-shaped structure filled with moderately dense (∼0.1 to ∼3 H cm −3 ), cool gas (∼1000 K) below the disk, as shown in Figure 3 at 52 Myr for Runs 1 and 2 and 45 Myr for Runs 3–6 (epoch t 2 ). These structures are composed of a mixture of gas from both the cloud and the disk, with the majority of their mass originating from the disk. Tenorio-Tagle et al. ( 1986 ) found similar structures in 2D hydrodynamic simulations of dark-matter-free clouds colliding with the Galactic thin disk. Over