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

B-Wavelength I-Wavelength 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 B-Survey O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O B-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 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 B-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 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 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 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 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 B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Software 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-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O 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 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


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

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Model I-Model 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 B-Formula I-Formula I-Formula 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 B-Formula I-Formula I-Formula 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 O O O O O O O O O O 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 B-EntityOfFutureInterest I-EntityOfFutureInterest B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey B-TextGarbage O O O O O O O O O O O O O O O O O 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 I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O


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

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


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

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

O O O O O O O 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 B-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O B-Model I-Model I-Model 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 B-Model I-Model I-Model 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 O B-Citation O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O B-Model I-Model I-Model 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 B-Citation I-Citation O B-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O B-Model 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


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

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 B-CelestialObject I-CelestialObject B-CelestialObject I-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 B-CelestialRegion O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-Survey B-Instrument O O O O O O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset B-Dataset O B-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation O O O O O O O O O B-Wavelength O B-Wavelength O O O B-Survey O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Survey O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject I-CelestialObject O O O B-CelestialRegion O O O B-Instrument B-Wavelength I-Wavelength O O O O O O O O O O O B-Instrument B-Wavelength I-Wavelength O O O B-Wavelength I-Wavelength O O O O O B-Wavelength I-Wavelength I-Wavelength B-Telescope O O O O O O O B-Wavelength O O O B-Survey O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O B-Survey O O O O B-Telescope O O O O O O O B-Wavelength I-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength B-Wavelength I-Wavelength I-Wavelength O O O O B-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope B-Telescope O B-Mission O O O O O O O O O O O O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset O O B-Survey 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 B-Wavelength I-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength B-Wavelength I-Wavelength I-Wavelength O O O O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Dataset O B-Dataset I-Dataset O O O O O O O O O O B-Survey O O O O O O O O B-Telescope O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion B-Citation I-Citation I-Citation I-Citation I-Citation O B-Organization B-Telescope O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O O O O O O O


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

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


we apply the formalism of Brogi Line ( 2019 ). We compute the log-likelihood function of each tested model by combining the data and model variances, and the cross-covariance of data and model according to their equation (9). We assume the same total observing time as for the data set presented in Benneke et al. ( 2019 ), which is 8.4 h. Since HRS does not necessarily need out-of-transit data, this integration time would allow us to observe four full transits of the exoplanet. Our results are shown in Fig. 5 . We can successfully and confidently (>5σ) recover the H 2 O signal for all instruments at the expected values of K p and V sys , as shown by the fact that the injection values (white crosses) are within or just outside the 1σ confidence interval. Given the relatively strong detection achieved, we conclude that the best-fitting atmospheric scenario for GJ 3470b would be easily detectable with current HRS with a relatively modest investment of telescope time. The size of the confidence intervals also suggests that GIANO and SPIRou are, respectively, the least and the most sensitive instruments based on our simulations. Note that CARMENES does not cover the K band unlike the other two spectrographs but does still show a strong H 2 O detection because of its high throughput. Figure 5. Detectability of the best-fitting atmospheric scenario for exoplanet GJ 3470b as derived by Benneke et al. ( 2019 ), based on 8 h (4 transits) of simulated observations with three near-infrared high-resolution spectrographs. Shown in the plot are confidence intervals on the rest-frame velocity and radial-velocity amplitude of the planet after the likelihood of Brogi Line ( 2019 ) has been applied to the simulated data. The white cross indicated the simulated values. 3.2 Constraining H2O and cloud decks – GJ 436b We now discuss how HRS may be able to constrain abundances and cloud decks on GJ 436b using the grid of model spectra discussed in Section 2.1.2 . We chose this planet as observations of the primary transit have revealed a largely featureless spectrum (Knutson et al. 2014a ). The only constraints on H 2 O thus far have been derived from low-resolution secondary eclipse observations (Stevenson et al. 2010 ). In Fig. 6 , we show three representative high-resolution transmission models with varying H 2 O and cloud top pressures but which resulted in similar muted HST WFC3 features. We studied their detectability with CARMENES, GIANO, and SPIRou, and the ability to discriminate cloudy scenarios from high-metallicity scenarios as shown in Fig. 2 . Figure 6. Model transmission spectra and binned HST WFC3 spectral points for GJ 436b. These are shown for three cases with varying cloud pressures and H 2 O abundances. The high-resolution spectra have been binned to the Knutson et al. ( 2014a ) observations, which are also shown with their associated error bars. We offset the models in the figure to highlight the spectral features. Fig. 6 shows the three

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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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-Instrument O B-Instrument O O O O O O O O O O O O O O O B-Instrument O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject 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-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-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-Tag O O O O B-CelestialObject I-CelestialObject O O O O B-Instrument 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 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-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument 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 B-Telescope B-Instrument 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 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 B-Citation I-Citation I-Citation I-Citation O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Formula I-Formula I-Formula 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-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


compensate for attenuation; a narrower attenuation region would imply a less steep radial dependence of the CO(3–2) effective emissivity. (vi) Values of the intensities integrated over the source size but not corrected for the loss of short spacing have been given for both CO(3–2) emission, 7.1 ± 0.4 Jy km s −1 , and continuum emission, 11.4 ± 0.5 mJy. Crude estimates of the short-spacing correction have been given at the respective levels of 1.6 Jy km s −1 and 5 mJy. The gas value is in good agreement with the value measured earlier by Dent et al. ( 2005 ) but the dust value is larger than that measured by Song et al. ( 2004 ) and close to the predictions of the disc models proposed by Hughes et al. ( 2008 ) and Roberge et al. ( 2013 ), according to which the CO mass of the disc is between ∼2 × 10 −4 and ∼10 −3 Earth masses and the 0.86 mm SED value is ∼19 mJy. The crudeness of our estimated short-spacing correction prevents a reliable resolution of this apparent disagreement, which will require new lower resolution observations. The use in this work of an effective emissivity to model the detected emission mixes the effects of temperature and density and the new results do not justify repeating the detailed analysis that Hughes et al. ( 2008 ) and Roberge et al. ( 2013 ) have made of the disc structure and chemistry for both gas and dust. (vii) Kinematics has been found to be Keplerian over a very broad range of distances from the star, typically 20–200 au, with a power index of and a rotation velocity of 5.1 ± 0.1 km s −1 at a distance of 60 au (1 arcsec) from the star, corresponding to a mass of 1.73 ± 0.10 solar masses for the central star. The observed line width was observed to be dominated by Keplerian shear over most of the disc volume and upper limits to the intrinsic line width (FWHM) have been evaluated, decreasing from 1.4 ± 0.3 km s −1 for | x | 1 arcsec to 0.8 ± 0.1 km s −1 for 2 | x | 3 arcsec. 6.2 Implications on the disc structure and chemistry (Hughes et al. <xref ref-type="bibr" rid="bib9">2008</xref>) The very detailed analysis of the chemistry that governs the morphology and temperature of the disc that was made by Hughes et al. ( 2008 ) remains essentially valid. However, some of the assumptions made by these authors need now to be modified; the inclination angle with respect to the line of sight is definitely non-zero; the dust extends to significantly larger distances from the star than the gas does, both radially and axially; the inner radius of the gas disc is nearly equal to the gravitational radius, ∼20 au, rather than twice this value; the lumpiness of the dust disc contrasts with the smoothness of the gas disc; the apparent warping of the gas disc suggested by the SMA data is not confirmed by the ALMA data; the central attenuation, for which the ALMA data gives strong evidence, can no longer be ignored. Yet, the analysis made by Hughes et al. ( 2008 ) of the factors governing the physics and chemistry of the gas

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 B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-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-EntityOfFutureInterest I-EntityOfFutureInterest 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-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O 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 B-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Citation I-Citation I-Citation B-Tag I-Tag I-Tag 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 B-Telescope O O O O O O B-Observatory O O O O O O O B-Observatory 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


and φ is longitude. This is an important property for the analysis to follow. Because the eigenfunctions are a complete set (with zero Lagrangian-pressure perturbation on the outer boundary, e.g. Christensen–Dalsgaard 2003 ), a general wavefield may be written as where ω is temporal frequency, a q are coefficients representing the phase and amplitude of the q th mode and we write for the sake of notational convenience. The governing equation for driven, undamped oscillations is given by where is the unperturbed source, the zeroth-order wavefield and is the unperturbed operator (whose properties are derived from a basic solar model such as model S; Christensen-Dalsgaard et al. 1996 ) as defined in equation ( 2 ). Note that the MDI and HMI missions provide time series of spherical-harmonic coefficients of oscillations, i.e. a k (ω), measured at the solar photosphere. In practice, because we only observe the side of the Sun that faces us, it is not possible to precisely spatially decompose the measured oscillation signals into individual spherical harmonics. This partial visibility leads to leakage between modes across l, introducing systematical errors in eventual inferences (e.g. Larson Schou 2015 ; Woodard 2016 ). Further, at a given frequency (between ridges for instance), many different radial orders contribute, leading to diminished ability in distinguishing between them. Temporal gaps in observations, due to instrumental or other issues, lead to mixing across temporal frequencies (and hence radial orders), thereby exacerbating this issue. To mitigate the problem of radial-order leakage, we use measurements at frequencies within a linewidth of resonance, ensuring that the associated radial order contributes a dominant fraction of observed mode power. Nevertheless, leakage across radial orders and spherical-harmonic degrees will systematically influence the results of mode-coupling analyses and needs to be carefully considered. 2.1 Perturbation theory The effects of rotation, flows, magnetic fields or thermal asphericities are modelled by adding a perturbation to the operator to in equation ( 5 ), in turn leading to a perturbation to the wavefield, . Ignoring perturbations to the source , we obtain for the wavefield in a perturbed Sun. For time-varying perturbations, behaves as a convolution operator. To the first order, we have therefore where the convolution over frequency captures the time variability of the perturbation operator (equation D4 of Appendix D ). Perturbing equation ( 4 ) while assuming that the eigenfunctions are unmodified by the perturbation, we may express the change to the wavefield as a linear combination of the original set of eigenfunctions, Using equation ( 2 ), the left hand side of equation ( 6 ) may be simplified thus Solar modes are subject to a small amount of attenuation γ k ≪ ω k that perturb the eigenfunctions and eigenfrequencies (obtained from equation 1 ) by introducing small imaginary components. However,

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


determined by the resonant conditions ( 52 ) and ( 53 ) are of order unity (see ( 42 ). In the opposite short-length limit | k | ≫ 1, the eigenvalue relation ( 30 ) yields: , k, n ≠ 0, n = 1, 2, ..., i.e. the linear growthrates are also small in the radially short-length limit, , e.g. | k | ≫ 1 for n ∼ 1 and | q | ∼ 1. In the latter case the eigenfrequency is diminished together with the growthrate. Now, let us discuss the three-wave resonant interaction in the presence of the short-length modes. First, note that a resonant triad cannot be formed by three short-length modes because of the frequency resonant condition ( 53 ) that is degenerated with all three small frequencies which can be included in the mismatch value, δω. Second, the resonant condition ( 53 ) is also degenerated for two short-length modes with | k | ≫ 1 interacting with the single long-length mode with | k | ≪ 1, since all they have zero frequencies. Finally, evidently that the radial wavenumber resonant condition ( 52 ) cannot be satisfied by the single short-length term with | k | ≪ 1 and two long-length terms with | k | ≫ 1 (at axial wavenumber n ∼ 1). Thus, resonant triads cannot be formed by interacting short- and long-length modes. In particular, this means that energy transfer from very short- to very long-length modes and vice versa cannot be realized with three-wave resonant interaction. Rigorous analytical solutions of the reduced weakly non-linear VSI problem discovered in the present study can serve as a test for numerical modelling of the full non-linear hydrodynamic problem, since the intermediate non-linear stage of VSI should be inevitably crossed before saturation. Predictions of the reduced VSI model allow to identify and discover physical mechanisms that are driving the VSI and its transition to turbulence, as well as identify tuning parameters of the physical and numerical models of the full problem. Figure 1. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class I. The two-mode regime of the resonant triad evolution. n (1) = 1, n (2) = 1, n (3) = 2 (see Table 1 ); R = +1; S = +1; , φ 0 = π/4, δω = +1. The scaled resonant frequencies and radial wavenumbers are in Table 1 with and , , ( j = 1, 2). Dashed and solid lines are solutions of the linear and non-linear systems ( 71 )–( 74 ), respectively, for triad amplitudes. Figure 2. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class II. The two-mode regime of the resonant triad evolution as in Fig. 1 , but for S = −1. Figure 3. Triad amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class III. The three-mode regime of bounded oscillations of all interacting modes as in Fig. 1 , but for R = −1. Figure 4. Amplitudes α ( j ) versus slow time θ for the triad of class I as in Fig. 1 ( S = +1, R = +1), but at small initial amplitudes . Figure 5. Amplitudes α ( j ) versus slow time θ for the triad of class II as in Fig. 2 ( S = −1, R = +1), but

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


X-ray peak and a separation from continuum X-ray dominant region (DeLaney et al. 2004 ; Helder Vink 2008 ). On the other hand, we defined “south west,” “inner”, and “forward shock” regions as the “non-thermal dominant” region (light blue regions in Figure 2 ). Helder Vink ( 2008 ) and DeLaney et al. ( 2004 ) show the area dominated by a harder spectrum. As a matter of fact, these regions manifest themselves in hard X-rays. Our image also shows the inner region is bright with hard X-rays. In addition, the forward shock has a featureless non-thermal emission (e.g., Hughes et al. 2000 ; Bamba et al. 2005 ). In the forward shock, it was found that the average proper motion of Cassiopeia A is 030 yr −1 (Patnaude Fesen 2009 ), so we corrected the forward shock region of each year with this value (see the Appendix ). 3.2. Spectra With the regions defined in the Section 3.1 , we extracted spectra using a custom pipeline based on specextract in CIAO script. Here, the background spectra were extracted from outside of the whole SNR region defined in Figure 2 . Since X-ray emission from Cassiopeia A is very strong, the background contribution is almost negligible for a time variation estimation. The background fraction of the whole SNR is only ∼3% in the 4.2–7.3 keV band and this is almost constant value from 2002 to 2013. In 2000, this fraction shows a larger value (∼7%) presumably because of an increase in the charged particle flux experienced during this observation. After background subtraction, we fitted the 4.2–7.3 keV band spectra at each epoch with a power-law model and a Gaussian line with XSPEC version 12.8.2. The best-fit parameters are summarized in Table 2 . Table 2 Best-fit Parameters of the ACIS Spectra a Note. a The errors are at 1 σ confidence level. Figure 3 shows the spectra in the 4.2–7.3 keV band taken from the six regions. As described in Section 3.1 , the two bright non-thermal dominant regions south west and inner are remarkable in the hard X-ray continuum flux (Figures 1 and 3 ). Accordingly, the photon indexes of these regions are ∼2.6–3.0 while those of the thermal regions are 3.0–3.4. In the forward shock and whole SNR, photon indexes and evolutions are slightly different from the results in Patnaude et al. ( 2011 ). This is probably due to a difference in the energy band used in the spectral fitting. The fit residuals are larger for the thermal dominant regions than for the non-thermal regions. This is because weak thermal lines such as Cr–K (5.6 keV) appeared in the band. Figure 3. Time variation of the equivalent width of the Fe–K line (plot in the middle) and observed spectra in the 4.2–7.3 keV band (on both sides) at each region (whole SNR: WS, east: E, inner: I, north west: NW, south west: SW, forward shock: FS). In the central plot, the solid lines show best-fit linear models. Individual spectra were fitted with a model composed of a power law and a Gaussian. The error bars of all figures are 1 σ . As shown in Figure 3

B-Wavelength O O O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O B-Wavelength 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 B-CelestialObject I-CelestialObject 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 B-Software O B-Software 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-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-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 B-Wavelength I-Wavelength O O O O O O O B-Model I-Model O O B-Model I-Model O B-Software B-Identifier I-Identifier O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-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 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 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 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 B-Model I-Model O O B-Model O O O O O O O O O O O O O O O


by myriads of DM particles. This is the reason why two convergence criteria are applied to the CDM numerical simulations to declare their results reliable (e.g. Power et al. 2003 ). First, the finite mass of the DM particles used in the simulation artificially shortens the two-body relaxation time-scale; and this artificially-set time-scale is asked to be longer than the age of the Universe, t U . This requirement prevents the thermalization of the resulting DM haloes. Secondly, the artificial thermalization starts in the centre of the DM haloes so the simulated centres are discarded within the so-called convergence radius. However, when these two restrictions are relaxed then cores with polytropic shape appear in the DM haloes produced in the numerical simulation. In this paper, we analyse two types of numerical simulations showing such cores. If the DM is assumed to be self-interacting (SIDM) then the effective relaxation time-scale t SIDM is shortened by an amount that depends on the SIDM cross-section. As soon as , t SIDM ≲ t U and the simulations produce cores. We show in Section 4 how different SIDM from several authors produce cores which after normalization are very similar and equal to the core of a polytrope within error bars. The simulations comprise haloes that go all the way from dwarf galaxies to galaxy clusters. This agreement between simulated haloes and polytropes holds even when baryons are included in the simulation (Section 4.1 ). If standard CDM numerical simulations are analysed within the convergence radius, they also show cores (e.g. Wang et al. 2020 ). These cores are also indistinguishable from polytropes, a result which holds even with halo masses differing by as much as 10 orders of magnitude (Section 5 ). Outside the convergence radius, the profiles follow the canonical NFW or Einasto shape. Thus, because the profiles within the convergence radii are excluded from the standard analysis, it is easy to understand why the profiles resulting from these numerical simulations seem to be set by the initial conditions during the big bang (Cen 2014 ; Nipoti 2015 ; Brown et al. 2020 ). As a by-product of this work, we have shown how unresolved DM clump–clump collisions cannot thermalize the DM haloes within a time-scale comparable to the age of the Universe (Appendix A ). We conclude that the presence of cores in simulated or observed density profiles can used as evidence for systems in thermodynamic equilibrium. Cores are theoretically predicted and also generated in self-gravitating system known to be in thermodynamic equilibrium. APPENDIX A: TWO-BODY COLLISION CROSS-SECTION FOR A CONTINUOUS DISTRIBUTION OF UNRESOLVED DM CLUMPS Starting from the primordial density fluctuations, the gravitational collapse produces a full distribution of DM haloes with masses going from galaxy clusters to Earth-mass clumps and even smaller (e.g. Diemand, Moore Stadel 2005 ; Wang et al. 2020 ). However, numerical simulations cannot treat all

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


regression with an objective datapoint rejection, which models the outlier distribution. The method infers the parameters of the linear fit together with the mean and variance of the distribution of outliers as well as the number of outliers at the same time. The model is run through 50 000 MCMC realizations, taking the maximum of the posterior distribution and the standard deviation as the best values of the slope, intercept, and their uncertainties. We did not attempt to perform the fits when the number of stars with abundances is smaller than 15 or when the uncertainties in the abundances are greater than 0.05 dex. In Fig. 10 we plotted some examples of the differential element abundance pairs δ X 1 vs. δ X 2 . The full matrix of differential abundances for the three clusters are found in Figs. B.3 – B.5 . We also overplotted the obtained fits. For the Hyades, we obtain statistically significant correlations (larger than 3 σ ) of all pairs of abundances that we analyze. Some elements with larger intrinsic uncertainties (e.g., Ba, Nd, Cu) tend to present less significant correlations. For the other two clusters, we have a smaller sample of stars selected as most probable members and larger uncertainties in the abundance measures. The less significant fit is the one of Na I vs. Mn I in NGC 2632, which has a posterior with two peaks. For Ruprecht 147, two fits involving Si I have lower significances than 3 σ . In Fig. 10 we see no dependence of the differential abundance pairs with respect to T eff , showing that all types of stars correlate in the same way. Fig. 10. Left : δ Ti vs. δ Si. Right : δ Mg vs. δ Fe, rows correspond to the three clusters. Color represents effective temperature. Translucid lines are the resulting linear fit according to the obtained posterior distribution in the intercept and the slope. The slope and its uncertainty is indicated in each panel. For the Hyades, a large number of slopes were retrieved, so we computed the compatibility among all of them within the quoted uncertainties: (i) The fits involving Cu I and Ba II that have non-Gaussian posterior distributions tend to not be incompatible with the others. However, in these cases, uncertainties are not well represented by the standard deviation. (ii) In the case of Nd II , all the slopes are larger than 3 σ when compared to most of the other cases. We attribute this to the larger uncertainties of this element in comparison with the others, which makes the dispersion in its dimension increase, and thus increase the value of the slope. (iii) We find compatible slopes for all other pairs of elements. This indicates that the differences in star-by-star abundance are due to a zero-point difference in the overall metallicity among the stars in the cluster, with no variation depending on the chemical element. 6.3.2. Comparison with synthetic data To better interpret our results, we generated a set of simulated spectra of stars in two cases: a chemically homogeneous

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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


is time independent. This is the case for GRB 160625B as presented in the right panel of Figure 8 . Moreover, typically is observed to increase over a pulse (Ryde Pe’er 2009 ); this is the case for most bursts in our sample, with two prominent cases of GRB 140329B and GRB 150330A. To derive the parameters in the rest frame, the selected bursts should have known redshift. A redshift measurement of 2.73 is found for GRB 140206B (Malesani et al. 2014 ), and 1.406 for GRB 160625B (Zhang et al. 2018 ). However, GRB 140329B and GRB 150330A are without known redshift. On the other hand, Iyyani et al. ( 2013 ) investigated how the different values of redshift affect the results and pointed out that, for different values of z , the estimated outflow parameters change within a factor, but the time evolution of the behavior of the parameters remains the same. We therefore assume an average value of GRBs (Bagoly et al. 2006 ), z = 2, for the calculation (see also Iyyani et al. 2013 ). 3.1.3. Lorentz Factor, Γ The coasting values of the Lorentz factor ( r ph > r s ) are derived by (see also Equation (4) in Pe’er et al. 2007 )where Y is the ratio of total fireball energy to the energy emitted in gamma-rays. As depicted in Figure 9 , temporal evolution of the Lorentz factor shows either an approximately constant (e.g., GRB 140206B) or a monotonic decay behavior (e.g., GRB 140329B exhibits a monotonic decline throughout the pulse), or both (initially approximately constant and then decaying faster, e.g., GRB 150330A, GRB 160625B). The best fitting gives power-law indices of 0.01 ± 0.07 for GRB 140206B and −0.51 ±0.03 for GRB 140329B. We fit GRB 150330A and GRB 160625B with a smoothly broken power-law function, which gives best-fitting power-law indices of 0.69 ± 0.52 (before the break) and −0.52 ± 1.44 (after the break) for GRB 150330A and 0.37 ± 0.27 (before the break) and −0.89 ± 0.33 (after the break) for GRB 160625B. Figure 9. Temporal evolution of the Lorentz factor, Γ. Color conventions are the same as in Figure 8 . GRB 140206B presents a constant behavior with best-fitting power-law index 0.01 ± 0.07, while GRB 140329B shows monotonic decay with index −0.51 ± 0.03. GRB 150330A and GRB 160625B exhibit a smoothly broken power-law decay feature, with power-law indices 0.69 ± 0.52 and −0.52 ± 1.44 before and after break for GRB 150330A and 0.37 ± 0.27 and −0.89 ± 0.33 before and after break for GRB 160625B. The global view on the evolution of the Lorentz factor decays with time from a few hundred Y 1/4 down to below 100 Y 1/4 (e.g., Fan Wei 2011 ; Peng et al. 2014 ). The decreasing Lorentz factor may have several implications (Iyyani et al. 2013 ). Since typically is observed to increase over a pulse, the decrease of the values of Γ = Γ( t ) is not surprising by realizing the fact that ; such a decrease must, therefore, be a common characteristic form over individual pulse structures in GRBs (Ryde Pe’er 2009 ). This is because, during the rising

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


of the two mini-BAL outflows in our sample. The Si iv trough is quite narrow and the S iv trough is unblended with other absorptions. In this case, the excellent template fit of the S iv trough is evident by the value f ( χ 2 ) = 0.4%. The S iv * absorption trough is heavily contaminated by a strong, likely Ly α , absorption trough from an intervening system. SDSS J0941+1331 has a multicomponent Si iv absorption trough. Following the procedure outlined in Section 4.2 , we use only the marked velocity interval as the template (see Figure 2 ). The Si iv template fits the S iv and S iv * absorption troughs well with f ( χ 2 ) = 6.8% in our survey. In order to account for possible systematic differences due to the Ly α forest, we compare the derived N ion from template fittings to the pure AOD extractions that includes the Ly α absorption features. In Figure 3 , we show the comparison of the observed S iv 1062.66 Å and S iv * 1072.97 Å absorption troughs. The AOD column densities integrated within the vertical dotted lines are N(S iv ) AOD = 1620 and N(S iv *) AOD = 740 (hereafter, all N ion are in units of 10 12 cm −2 ), while the template fitting results integrated over the same velocity range are N(S iv ) tmpl = 1450 and N(S iv *) tmpl = 540. Note that the AOD method overestimates the N ion because it includes the contributions from Ly α forest absorptions. These values yield a density sensitive ratio, N(S iv *)/N(S iv ), of 0.46 and 0.37 for the AOD method and the template fitting method, respectively. Using the same methodology described in Section 4.2 , the AOD method leads to a decrease of R by 25%. This difference is similar to our errors for the determined R value in Table 4 . Therefore, possible Ly α forest contaminations affect our derived R only moderately. Figure 3. Comparison of the normalized flux of S iv 1062.66 Å(blue) and S iv * 1072.97 Å(red) absorption troughs for SDSS J0941+1331 in velocity space. The vertical dotted lines are the range of Si iv template corresponding to Figure 2 . For SDSS J1135+1615, the Si iv absorption troughs are saturated and rather flat. We choose the two strongest local maxima to construct the Si iv template (see Figure 2 ). Fitting the S iv and S iv * absorption troughs with this template shows that the S iv * absorption trough is deeper than that of S iv . Including the many narrow Ly α troughs in the N ion extractions increase the N(S iv ) and N(S iv *) by 16% and 14%, respectively, and only marginally affects the derived R (6%). 5. Photoionization Analysis In order to derive the physical conditions of the outflow, we run the spectral synthesis code Cloudy (version c17.00, Ferland et al. 2017 ) to generate a grid of photoionization simulations. Cloudy self-consistently solves the photoionization and thermal equations necessary to model the physical conditions in the gas. The absorbers are modeled as plane-parallel, thin slabs of constant density gas with solar metallicity. They are irradiated by

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


12 . Panel (a) shows that current sheet disruption takes place earlier with a lower growth rate in Run H5, where the initial noise level is three orders of magnitude higher than that of Run S5. Because current sheet disruption marks the onset of fast reconnection, the onset also occurs earlier in Run H5. However, as can be seen from panel (b), the reconnection rates after nonlinear saturation are essentially the same for the two runs. Therefore, we can conclude that reconnection rate after saturation is nearly unaffected by the noise level and the condition at the onset. This conclusion is also in line with the earlier study by Huang Bhattacharjee ( 2010 ), where the dependence of the averaged reconnection rate on the external forcing amplitude is found to be fairly weak. Figure 12. Comparison of time histories of (a) the linear growth rate and (b) the reconnection rate for Run S5 (with an initial noise level ) and Run H5 (). The initial noise level affects the disruption time and the linear growth rate at disruption, but the reconnection rate after nonlinear saturation remains approximately the same. From an observational point of view, the period between current sheet disruption and nonlinear saturation is especially interesting. During this period, the outflow profile changes from a nearly monotonic Sweet–Parker-like profile to a highly fluctuating profile, as shown in Figure 13 (a). Accordingly, the histogram of the outflow profile changes from double-peak-shaped to triangle-shaped, as shown in Figure 13 (b). A triangle-shaped histogram is a natural consequence of the outflow profile being highly fluctuating. To understand this, we consider an outflow profile of the form , where and the first term mimics the Sweet–Parker-like outflow profile, and the second term mimics short-wavelength fluctuations. If the fluctuation is much smaller than the background, i.e., , the histogram is double-peak-shaped. Conversely, if the fluctuation is on the order of the background, i.e., , which is the case in the saturated phase, then the histogram is triangle-shaped. The transition from a double-peak-shaped histogram to triangle-shaped histogram can lead to observable effects through spectroscopy of emission spectral line profiles, because the observed spectral line profile is a convolution of the Doppler-shifted line profile along the line-of-sight direction. For reconnection events with outflows along the line-of-sight direction, the histogram of the outflow profile can be viewed as a proxy for the shape of the line profile. A triangle-shaped line profile has been reported in IRIS (Interface Region Imaging Spectrograph (De Pontieu et al. 2014 )) observations of the Si iv line during transition region explosive events (Innes et al. 2015 ). With the high-cadence mode of the IRIS spectrograph, it is possible to observe transitions from a double-peak-shaped line profile to a triangle-shaped line profile in transition region explosive events (L.-J. Guo

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O B-Model I-Model I-Model O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Telescope I-Telescope I-Telescope I-Telescope B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O B-Person I-Person


certain criteria. As an independent check the scatter in the DM was investigated among the models with a reduced χ 2 less than twice the minimum value. If there were five or more such models the dispersion (actually 1.48 ⋅ MAD) around the median was determined and compared with the formal error. This exercise revealed no systematic effects and the errors estimated in such a way are consistent with the formal errors. 6.2. Comparing dEBs and TRGB with Cepheid and RR Lyrae distances In Sect. 5.1 the TRGB distances are compared with the distances to 14 dEBs. One can also compare the TRGB distances with other independent distance estimates, as we did in Sects. 5.2 – 5.4 . We therefore took an identical approach as in Sects. 5.2 and 5.3 and determined the median DM and reddening value of CCs (in the LMC), and RRL (in the SMC) in the direction of the dEBs. The results are listed in Table 3 which first repeats the DM and reddening derived in the literature for the dEBs and the TRGB distance (based on the second-order derivative method) from Table 1 . Columns 5 and 6 give the DM and reddening values based on the CCs and RRL in those fields. Table 3. Comparison of distances for the dEBs. Notes. Columns 1–4 are taken from Table 1 . They indicate the name of the system, the DM and reddening based on the works listed in Col. 4 of Table 1 , and the TRGB distance using that reddening. Columns 5 and 6 list the DM and reddening of CCs (for the LMC objects) and RRL (for the SMC objects) in the direction of the EBs, and Col. 7 lists the TRGB distance using the reddening in Col. 6. Similarly, Columns 8 and 9 give the DM and reddening in Rubele et al. (2018) in the direction of the EBs, and Col. 10 lists the TRGB distance using the reddening in Col. 9. Finally, Col. 11 lists the reddening derived from Haschke et al. (2011) in the direction of the EBs, and Col. 12 lists the TRGB distance using that reddening. It is evident that the reddening estimates are smaller than adopted in the dEB analysis. In the SMC this is the case for all five objects. Although the differences are within the respective error bars it appears to be a systematic effect. In the LMC this is the case for eight out of nine objects but the differences appear to be smaller on average than for the SMC. To test the effect of reddening, the TRGB distance was derived using the E ( B − V ) from col. 6, and the results are listed in col. 7. It is clear that the effect on the DM is roughly inversely proportional to a change in E ( B − V ). Based on the definition of the sharpened magnitude, the absolute calibration equation (Eq. ( 3 )) and the reddening coefficients one expects a relation ΔDM/Δ E ( B − V )= − 1.1. The overall effect is noticeable however. The weighted mean DM of the nine LMC dEBs is shifted from 18.483 ± 0.006 mag to 18.523 ± 0.005 mag, and that of the five SMC binaries is shifted from 19.023 ± 0.007 mag to 19.051 ± 0.009 mag. In a similar way we used the data of Rubele et al. and took

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


than those within the UV/optical, and Figure 3 (top) indicates a diversity of X-ray/optical lags, with the X-rays leading the UV in three cases and lagging in the fourth. This poses a severe problem for the reprocessing model, for which no simple solution is currently apparent. Swift is the workhorse without which IDRM cannot (currently) be successfully performed on AGN, as it is the only fully operational observatory that provides coverage throughout the X-ray, UV, and optical regimes necessary for this experiment. However, it is also very helpful to expand coverage to longer wavelengths and higher spectral resolution than that afforded by the UVOT alone. We note that simultaneous ground-based optical monitoring has been performed on all four of these targets, and the first data set (on NGC 5548) has already been published (Fausnaugh et al. 2016 ). Reduction and analysis of optical photometry and spectroscopy on the other three is ongoing and will be published shortly. This should provide further constraints on the accretion disk fits (especially the overall size) and possibly on the effects of line and diffuse emission from the BLR. Note as well that HST can also play a crucial role, providing much higher spectral resolution that can allow direct detection and modeling of the BLR continuum component, and extending the UV coverage to wavelengths as short as ∼1150 Å. That has been used to resolve the excess lag around the Balmer jump in one of these targets (NGC 4593, Cackett et al. 2018 ), and it is expected to play a prominent role in future campaigns. These unprecedented data also reveal the need for improvements in our time-series analysis tools. The uncertainties on lags output by JAVELIN (Zu et al. 2011 ) are on average a factor of ∼2.5 smaller than the same quantity output by the ICCF FR/RSS technique (Peterson et al. 1998 ). The precise cause of this discrepancy has not been determined, but given the fact that the older ICCF FR/RSS technique is more conservative in its assumptions and results, it and not JAVELIN was utilized to derive the results reported above. More generally, there is no interband lag tool currently in use that allows separation of two distinct interband lag signals, as seems to be the case in the UV/optical, where short lags from the accretion disk and longer lags from the BLR appear to be present. Ultimately these techniques will need to be developed and directly compared so as to determine which are more reliable and suitable for studying AGN interband lags, but that is beyond the the scope of the current paper. These results also highlight the need for theoretical progress, especially in understanding the X-ray emission component(s) and relation to the disk and BLR that emit at lower energies. These observations present severe problems for the “lamp-post” reprocessing model. Can it be modified or must it be discarded? If it is the latter then what will take its place? The current set of reduced data are

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Long Baseline Array (VLBA) images at 43 GHz (Agudo et al. 2011 ). A quasi-stationary feature within 2 mas of the core associated with a bent jet, was analyzed using Global mm-VLBI Array (GMVA) observations at 86 GHz (Hodgson et al. 2017 ). The spectral index showed different behavior between the core and stationary component at millimeter wavelengths. The spectral index of the core at 15–43 GHz and 43–86 GHz was inverted. When the core was flaring, the spectral index changed to become steep (Hodgson et al. 2017 ). In this paper, we report results from simultaneous multiple frequency VLBI observations of OJ 287 using the KVN at 22, 43, 86, and 129 GHz, from 2013 January until 2016 December. The paper is organized as follows. In Section 2 , we present the description of the observations for both the KVN data and all archival data used in this paper. Sections 3 and 4 describe results and various analyses of the flux density variability at multiple frequencies, respectively, and Section 5 includes a discussion on the results and analysis. Finally, we summarize this paper in Section 6 . We use luminosity distance D L = 1630 Mpc at a redshift of z = 0.306, assuming H =68 , , and = 0.302 (Spergel et al. 2015 ; Hodgson et al. 2016 ). Spectral index α is defined as . 2. Observations and Data Reduction 2.1. KVN Observations OJ 287 was observed by the KVN, which is a 500 km scale VLBI network consisting of three 21 m radio telescopes located in Seoul (KVN Yonsei), Ulsan (KVN Ulsan), and Jeju (KVN Tamna), Korea. The observations were conducted at 22, 43, 86, and 129 GHz bands simultaneously with a mean cadence of roughly a month since 2012 December except for maintenance seasons from June to August. The angular resolution of the KVN is 6 mas at 22 GHz, 3 mas at 43 GHz, 1.5 mas at 86 GHz, and 1 mas at 129 GHz. We used data obtained in 31 epochs in the period from 2013 January 16 (MJD 56308) until 2016 December 28 (MJD 57750). In each epoch, the source was observed with up to eight 5 minute scans per 24 hr. The observations were recorded in left circular polarization mode at a recording rate of 1 Gbps. The observing frequencies are 21.700–21.764, 43.400–43.464, 86.800–86.864, and 129.300–129.364 GHz, with a total bandwidth of 256 MHz (thus 64 MHz in each band). Detailed descriptions of the iMOGABA observations are reported in Lee et al. ( 2016b , 2017a ). 2.2. Data Calibration The obtained data were processed with the DiFX software correlator in Daejeon, Korea (Deller et al. 2007 , 2011 ), and the data was calibrated following the standard procedures (phase and amplitude calibration, fringe fitting, and bandpass correction) in the AIPS package from the National Radio Astronomy Observatory (NRAO). For the calibration, the KVN pipeline was used (Hodgson et al. 2016 ). The visibility phases in the 22 GHz band were transferred to those in the higher frequency bands (i.e., 43, 86, and 129 GHz bands) using the pipeline in order to increase the coherence time

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