but slightly lower mean ellipticity ⟨ε⟩ = 0.38 ± 0.05. Clampitt Jain ( 2016 ) used the technique to measure the quadrupole weak-lensing signal, and applied it to a sample of SDSS clusters. They corrected anisotropic PSF following Reyes et al. ( 2012 ) and Huff et al. ( 2014 ). They considered the noise from the intrinsic shape of galaxies and measurement on each background galaxy following Mandelbaum et al. ( 2013 ) and Sheldon et al. ( 2012 ). They obtained the best-fitting value of the mean ellipticity of ε = 0.19 with 1σ uncertainty of ∼0.05. They ascribed this smaller value to the misalignment between major axis of the BCG and that of cluster halo which is implicitly assumed to be aligned. Given the large uncertainty, their result is broadly consistent with our result. Shin et al. ( 2018 ) applied the quadrupole technique to SDSS clusters. They estimated anisotropic PSF, measurement noise, and noise from the intrinsic shape following Clampitt Jain ( 2016 ). The resulting mean ellipticity value with a prior of the satellite galaxydistribution is ⟨ε⟩ = 0.45 ± 0.09 after correcting for Poisson sampling. They also measured the ellipticity of satellite galaxy distribution as ⟨ε⟩ = 0.42 ± 0.04, and that derived from stacked weak lensing with a prior of the CG major axis as ⟨ε⟩ = 0.25 ± 0.06. By comparing these ellipticity values, they also estimated the rms misalignment angle of 30° between the CG and DM halo and 18° between satellite galaxies and DM halo. These misalignment values are in good agreement with our result (see also Section 6.2 ). van Uitert et al. ( 2017 ) used an estimator similar to Clampitt Jain ( 2016 ) and constrained the average ellipticity of galaxy groups obtained from Galaxy And Mass Assembly (GAMA) survey combined with the weak-lensing signal measured by Hildebrandt et al. ( 2017 ) from the Kilo Degree Survey (KiDS). They did not consider anisotropic PSF, but consider the intrinsic shape noise. They compared different priors for stacking analysis of weak-lensing signals at different scales. Their resulting values of the mean ellipticity are ε = 0.38 ± 0.12 (40 R 250 kpc) and ε = 0.05 ± 0.13 (250 kpc R 750 kpc) for the BCG prior, whereas ε = −0.04 ± 0.11 and 0.349 ± 0.13, respectively for the prior of the satellite galaxy distribution. They concluded that the BCG major axis (satellite galaxy distribution) is aligned (misaligned) with the DM halo orientation on small scales (250 kpc), whereas the BCG major axis (satellite galaxy distribution) is misaligned (aligned) with DM on large scales (>250 kpc). This result appears to be inconsistent with our result which indicates that the distribution of satellite galaxies is aligned better than the major axis of the CG at all scales, 100–1000 kpc. This discrepancies are partly because they use galaxy groups with M 200 ∼ 10 13 M ⊙ rather than galaxy clusters we considered in this paper, M 200 ∼ 10 14 M ⊙ . Nevertheless, further work is needed to explain this inconsistency, for

O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Survey O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Survey O O O B-Model I-Model O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Model I-Model O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Survey I-Survey I-Survey I-Survey B-Survey O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Survey I-Survey I-Survey B-Survey O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


resonant stripes, where comets have experienced secular eccentricity pumping. This sometimes goes in the direction of circularizing the orbits, but very often leads into the sungrazing end state. This structure would be smeared out in the plot by Levison et al. ( 2006 ; their Fig. 1b) but, most likely, in reality it does not exist. The small nudges given to the comets in their integrations would inhibit the long-term action of these resonances in most cases (see Valsecchi et al. 1995 ), while we introduced the artefact of leaving the comets entirely in the hands of the giant planets. We find that deflected comets get dynamically ejected only if their starting semi-major axis is larger than 2.6 au. For smaller values, if Jupiter-crossing occurs, it is only for very high eccentricity, so encounters are rare and inefficient in changing the orbits. In these cases, collision with the Sun is very likely, and many of the survivors are “saved by the bell”, since our integrations stop after 100 000 revolutions. Looking at the statistics of end states for the deflected comets, we find a remarkably high probability (more than 12%) of collision with the Sun, and most of these belong to the above-mentioned category. We derived the steady-state number of comets in the Encke-like domain and made a comparison with the results of Levison et al. (2006) , which are quite different. In fact, using our first approximation to the transfer mechanism, we predict only of the number of Encke-like comets found by Levison et al. (2006) . Thus, our method is severely lacking in transfer efficiency, and several reasons for this are easily identifiable. First, consider the limitation caused by cutting the integrations to a total of 100 000 revolutions. Compare this to the example provided by Levison et al. (2006) of a comet meeting their requirement for an Encke-like comet. Before this happens, the comet spends a million years (roughly 250 000 orbits) in a decoupled state waiting for the completion of its transfer. If this situation is typical, we missed most of the transfers by stopping the integrations too early. Moreover, as in the case mentioned, most transfers may require several interventions by terrestrial planets – something that our procedure forbids. It is true that more than one encounter as close as those that we modelled is not likely to occur for the same object. However, significant perturbations may be caused by the more distant and more frequent encounters that our procedure neglects. The main reason for the inefficiency of our transfer model appears to be the premature cutting of the integrations and the limitation to the closest encounters, which are too rare to have a large effect. If we had integrated all the comets until ejection or collision, some more deflections would have occurred. However, the increase would not be large, as seen from the fact that more than 80% of the deflections that we followed up by further integrations happened before the 50

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


difference for the XIS detector normalization constants between both observations. We attribute this difference to the different data extraction modes and the lacking attitude correction for the first observation, as the simulation of the effective area of XIS, which is performed by the FTOOL xissimarfgen in the course of data reduction, is highly sensitive to the source position and extent. Possible deviations of the apparent source position due to thermal wobbling might therefore introduce an unknown systematic uncertainty to the flux measured by XIS. For this reason, we normalized the detector constants with respect to PIN. The fitted fluxes in the 15–50 keV band are about 10% higher than the corresponding NuSTAR observations, which agrees with flux calibration uncertainties ( Madsen et al. 2015 ). For future reference, we also determined the continuum flux for other energy bands than those given in Table 2 . For the first observation, the measured fluxes are and in the 2–10 keV and 3–60 keV band, respectively, and for the second observation and for the corresponding energy bands. Fig. 7 a) Unfolded phase-averaged spectra of Obs. I (red) and Obs. II (blue) and b) ratio of the phase-averaged spectra of Obs. I and Obs. II. For clarity, only XIS3, PIN and GSO data are shown in both panels. The ratios of the individual detectors have been corrected for the respective cross-calibration constants given in Table 2 . Fig. 8 Panels a) and b) : phase-averaged spectrum of XIS0 (blue), XIS1 (red), XIS3 (magenta), PIN (green), and GSO (brown) of Obs. I with best-fit model and residuals. Panels c) through g) : count rates of the respective phase-resolved spectra (1) to (5), divided by the count rate of the phase-averaged spectrum. Figure 7 shows the unfolded spectra of Obs. I and Obs. II in comparison and the photon ratio of both observations. We observe a slight change in spectral slope below 2.5 keV, while the ratio is mainly flat above this energy with slight differences in the Fe band. Above the cyclotron line energy the spectra are very similar in shape. Comparing the fit parameters of both Suzaku observations, we find a decrease of the photon index and folding energies toward higher flux. Fürst et al. (2014) observed the opposite trend regarding the first and second NuSTAR observation. There is, however, an artificial correlation between these two parameters, which is not taken into account here, so we cannot conclude that the Suzaku and NuSTAR results contradict each other. The excess of soft photons in Obs. I below 2.5 keV is at least partly explained by the increase of the blackbody temperature from Obs. I to Obs. II, while the relative normalization of the blackbody component is very similar in both observations. This behavior is consistent with NuSTAR . See Fürst et al. (2014) for a more detailed investigation of the parameter evolution over the outburst. The fitted value is comparable to the value of given by the Leiden/Argentine/Bonn

O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O B-Software B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O B-Instrument O O O O O B-Wavelength I-Wavelength O O O O O O O O B-Telescope O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument B-Instrument O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O B-Instrument O B-Instrument O B-Instrument O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O B-Telescope O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O B-Telescope O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Collaboration


). CSM densities and radii of well-known Type Ia SNe and confirmed Type Ia SNRs studied in radio or X-rays, along with SN 1885A and G1.9+0.3 from this work (shown in red). The blue dashed and dotted lines indicate different CSM profiles of the form ( is mass-loss rate, u w is wind velocity, μ = 1.4) created by isotropic mass loss at constant rates and constant velocities given in the legend box. The horizontal gray bars show examples of cavity sizes created by accretion-driven outflows from the progenitor due to fast and slow winds (Badenes et al. 2007 ). The ruler on top shows the timescales on which these CSM profiles formed in the pre-supernova stage. The gray region shows the typical extent of the warm ISM phase (Ferrière 2001 ). The red dashed line indicates the average ISM density of the M31* circumnuclear region where SN 1885A is located (Li et al. 2009 ). Object references: SN 2011fe (Chomiuk et al. 2012 ), SN 2014J (Pérez-Torres et al. 2014 ), Tycho (Slane et al. 2014 ), SN 1006 (Raymond et al. 2007 ). Our upper limit on density of 0.04 ± 0.03 (sys) cm −3 derived in Section 5.1 suggests that, on average, SN 1885A has interacted with very little CSM material. A significantly denser CSM, if present, would have been captured in our measurements of n 0 in Figure 3 (a), and could have produced detectable radio emission. A low-density CSM (≲0.1 cm −1 ) around SN 1885A was also inferred from the non-detection of X-rays from SN 1885A (Perets et al. 2011 ). Our inferred density is lower than that of well-known Type Ia SNRs, and well-studied, recent nearby Type Ia SNe such as SN 2011fe and SN 2014J, although the density values are in the range of the warm ISM phase (Figure 5 ). The strong density limit rules out some portion of the SD parameter space for SN 1885A. For example, if the progenitor binary emitted a wind, the mass-loss rate, , would have been 10 −4 M ⊙ yr −1 (assuming wind velocity u w = 1000 km s −1 ) or 10 −6 M ⊙ yr −1 (assuming u w = 20 km s −1 , see Figure 5 ). Modeling the shock dynamics in such an r −2 profile can lead to even deeper constraints on , because the SNR would have evolved in much denser CSM in the past, and this would have left an observable effect on the radius and velocity of the SNR. However, as noted in Badenes et al. ( 2007 ), it is likely that the CSM structure on parsec scales resulting from an SD configuration is more complex than either a uniform or r −2 profile. Accounting for realistic CSM would require detailed modeling to predict the radius, velocity, and radio luminosity of the SNR, and is beyond the scope of this paper. In the future, it would be productive to revisit the observational constraints from this paper and Fesen et al. ( 2015 ) when a grid of models become available, predicting the CSM structure out to ∼10 pc for a variety of SN Ia progenitors (C. Badenes et al. 2019, in preparation). Such an analysis would allow us to make definitive statements about the progenitor of SN 1885A. As an

O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O B-CelestialObject O O O B-CelestialObject I-CelestialObject O O B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O B-Wavelength O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O


the interaction between the outgoing shock and the CSM. The X-ray data of SN 1978K prior to 2006 have been analysed; however, whether its flux has remained constant since the first detection by ROSAT or started to decline was still under debate (e.g. Lenz Schlegel 2007 ; Smith etal. 2007 ). It is worth noting that there are another two ULXs (X-1 and X-2) in NGC 1313. Thus, this region has been frequently visited by X-ray missions in order to explore the spectral evolution of these sources (e.g. Feng Kaaret 2006 ; Bachetti 2013 ; Weng, Zhang Zhao 2014 ). In the past decade, some additional long observations were carried out with XMMNewton (Table 1 ), which is a focusing X-ray telescope having a large effective area in the soft X-ray band. In this paper, we study the XMMNewton observations covering the period from 2000 to 2015 (spanning 5277d), aiming to explore the evolution in X-ray emission of SN 1978K at very late times. The XMMNewton data reduction is described in the next section, and the results are presented in Sections 3 . Discussion and conclusion follow in Section 4 . Table 1. XMMNewton observations of SN 1978K. Net exposure: clean exposure after background flares excluded. 2 DATA REDUCTION The detailed study presented here was on data from the main camera on board XMMNewton , the European Photon Imaging Camera (EPIC), which consists of two MOS arrays and one PN CCD array (Strder etal. 2001 ; Turner etal. 2001 ). We analyse all of the available XMMNewton observations of SN 1978K made from 2000 October to 2015 March; however, we discard those data that were affected by the strong background flares or when the source falls in the CCD gaps. The observational details of the data used in this paper are listed in Table 1 . The XMMNewton observations of SN 1978K are quite clean except two short data observed in 2003 December (ObsIDs = 0150280401, 0150280501), and only less than 12percent of data on average are excluded. The data are reduced with the Science Analysis System software ( sas ) version 14.0.0. We create the light curve above 10keV, and exclude background flares with a count-rate cut-off criterion. The data from good time intervals are filtered by setting FLAG = 0 and PATTERN 4 for the PN data, and PATTERN 12 for the MOS data to ensure the best quality spectrum (Snowden etal. 2016 ). The source photons are extracted from a circle aperture with a radius of 30arcsec, while the background is taken from the same CCD chips as the source and at a similar distance from the readout node. Since the source is relatively faint ( flux 10 12 erg cm 2 s 1 ), the pile-up effect can be neglected. We produce the response files with the sas tasks rmfgen and arfgen to facilitate subsequent spectral analysis. The task specgroup is carried out to rebin all spectra to have at least 20 counts per bin and not to oversample the instrument energy resolution by more than a factor of 3. We fit the spectra in the 0.510keV range using xspec with a 2percent

O O O O O O O O O O B-Wavelength O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Telescope O O O O O O O B-Wavelength O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O B-Telescope O O O O O O O O O O O O O O O O B-Wavelength O O B-CelestialObject I-CelestialObject O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O B-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O O O B-Instrument O O O B-Instrument I-Instrument I-Instrument B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O B-Telescope O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O B-CelestialObject I-CelestialObject O O O O O O O O O O O B-Identifier I-Identifier I-Identifier B-Identifier O O O O O O O O O O O O O O O O O B-Software I-Software I-Software I-Software O B-Software O B-Identifier I-Identifier O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Instrument O O O O O O B-Instrument O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O B-Software O B-Software O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Software O O O


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

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


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

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


this 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

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


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

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


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

O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O B-Software I-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Survey O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O B-Survey O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O B-Survey O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O B-Survey O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


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

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


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

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


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

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Instrument I-Instrument O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength B-Telescope O O B-Wavelength B-Telescope O O O O B-Wavelength O O O O O O O O O O O B-Wavelength B-Telescope O O O O B-Telescope B-Telescope O B-Telescope I-Telescope O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O B-Software B-Citation I-Citation O O O O B-Model I-Model O O O O O O O O O B-Instrument I-Instrument O B-Telescope O O O O O B-Software B-Citation I-Citation I-Citation I-Citation O B-Model I-Model O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O B-CelestialObjectRegion


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

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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

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


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

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


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

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


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

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


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

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


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

O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O B-EntityOfFutureInterest O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Identifier I-Identifier O B-Identifier O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Identifier I-Identifier O O O O O O O B-Identifier I-Identifier O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Identifier I-Identifier O O O B-EntityOfFutureInterest O O O O O B-EntityOfFutureInterest O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Identifier I-Identifier O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Identifier I-Identifier O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


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

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