placed in very dense environments near the Galactic center. The number of BH-LCs decreases with increasing natal kick magnitudes, as larger kicks disrupt a larger fraction of primordial binaries. For the same reason, natal kicks also affect the distribution of P orb of BH-LCs. While BH-LCs are disrupted in the Zero-kick model only if the change in mass during BH formation is severe enough to reduce the orbital binding energy to the point of unbinding, the NS-kick model leads to disruption of BH-LC progenitors with . We find a significant population of BH-LCs with P orb longer than the nominal Gaia mission in all of the models except the NS-kick model. This population exclusively contains BH-LCs that have not experienced a common envelope evolution. The population with orbital periods in the Gaia -observable range and lower contain BH-LCs that have experienced a common envelope, as well as BH-LCs that have not. Our NS-kick model does not allow formation of BH-LCs with and , while other models with lower natal kicks allow formation of these systems (Figure 2 ). By extending the Gaia mission beyond the nominal , our condition could be relaxed, and Gaia may constrain highly uncertain BH natal kicks from the properties of detected BH-LCs at longer orbital periods. 3.2. Gaia-observable Population Figure 3 shows the number of systems observable by Gaia () as a function of limiting G magnitude. The number of observable systems is inversely related to the BH natal kick strength because the likelihood of unbinding a BH-LC progenitor increases with kick strength. Figure 3. Mean number of observable BH-LCs as a function of limiting G magnitude computed from 500 Galactic realizations. Shades show regions above and below the mean denoted by the lines. Dashed (solid) lines denote our estimates using pessimistic (optimistic) astrometric cuts (Section 2.4 ). Blue, red, and black shading denote Zero-kick, FB-kick, and NS-kick models, respectively. Table 1 shows the result of our astrometric cuts. The total number of BH-LCs formed in our models is roughly consistent with the estimates in Mashian Loeb ( 2017 ). However, the majority of the systems formed lie above our upper cut (Figure 2 ). In addition to the natal kick prescription, the Galactic component has a noticeable effect on the number of BH-LC systems. Because the thick disk is modeled as a single star burst ago, all of the LCs in the thick disk at present are low mass. Large natal kicks for BHs such as our NS-kick model affect BH-LCs with low-mass LCs more (Figure 2 ; Section 3.1 ). As a result, in the thick disk, the NS-kick model yields a very low . The adopted continuous SFR in the thin disk keeps relatively high even for the NS-kick model by forming high-mass luminous stars very close to the present epoch. Table 1 Average Number of BH-LCs Observable by Gaia from 500 Galactic Realizations with for Our Optimistic and Pessimistic Cuts (Section 2.4 ) We estimate 3800–12,000 BH-LCs in the MW are

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AO1 Guest Observer program. NuSTAR consists of two co-aligned X-ray telescopes, with the corresponding focal plane modules FPMA and FPMB that provide 18′′ FWHM imaging resolution over a 3–79 keV X-ray band and a characteristic spectral resolution of 400 eV FWHM at 10 keV (Harrison et al. 2013 ). The reconstructed NuSTAR coordinates are accurate to at the 90% confidence level. The nominal timing accuracy of NuSTAR is ∼2 ms rms, after correcting for drift of the onboard clock, with the absolute timescale shown to be better than ms (Mori et al. 2014 ; Madsen et al. 2015 ). Although this is insufficient to resolve the 3.05 ms period of PSR B1821-24, we devise a method, presented herein, to correct the photon arrival times to better than accuracy. NuSTAR data were reprocessed and analyzed using FTOOLS 09May2016_V6.19 ( NUSTARDAS 14Apr16_V1.6.0) with the NuSTAR Calibration Database (CALDB) files of 2016 July 6. The resulting data set provides a total of 140.7 ks and 55 ks of net good exposure time for the two pointings, respectively. For the timing analysis, the nominal NuSTAR clock correction was applied to the photon arrival times which are converted to barycentric dynamical time (TDB) using the DE405 solar system ephemeris and radio timing coordinates of Ray et al. ( 2008 ). For all subsequent analyses we merged data from both FPM detectors. Although the globular cluster M28 hosts at least 11 other pulsars (Bégin 2006 ), they are all over a factor of ∼100 fainter than PSR B1821-24 in X-rays (Bogdanov et al. 2011 ) and thus cannot be studied effectively with NuSTAR . 2.1.2. RXTE PSR B1821-24 has been observed extensively in hard X-rays with RXTE during the mission lifetime, primarily for the purpose of absolute spacecraft time calibration and to measure the X-ray-to-radio phase lags (Rots et al. 1998 ; Ray et al. 2008 ). In anticipation of the NuSTAR observations, we collectively analyzed all available archived RXTE data on PSR B1821-24 for the first time. A total of 180 ks of exposure times were accumulated during three programs: P20159 (1997 February 12–13, 16.9 ks), P90053 (2005 February 28–March 1, 14.9 ks), and P92008 (2006 July 6–8, 48.4 ks; 2006 October 9–10, 51.2 ks; 2007 April 25-26, 48.5 ks). These observations were made with the Proportional Counter Array (PCA; Jahoda et al. 1996 ) on board RXTE . The PCA consists of five collimated xenon proportional counter units (PCUs), each having a front propane anticoincidence layer, with a total effective area of ∼6500 cm −2 over a 1° field of view (FWHM). Each PCU is sensitive to photons in the energy range of 2–60 keV, with an energy resolution of 18% at 6 keV recorded in 256 channels. Data were collected in GoodXenonwithPropane mode with two to five PCU modules in use at any given time, with an average of 3.7 PCUs active overall. In this mode, photon arrival times are recorded with resolution. We analyzed events from the top xenon layers of each PCU only. Including the second layers resulted

B-Grant I-Grant I-Grant I-Grant B-Telescope O O O O B-Wavelength O O O O O O O B-Instrument O B-Instrument O O O O O O O O B-Wavelength I-Wavelength B-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength B-Citation I-Citation I-Citation I-Citation O O O B-Telescope 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 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 B-CelestialObject I-CelestialObject 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-Software B-Identifier O B-Software B-Identifier O O B-Database I-Database I-Database B-Database O O O O O O O O O O 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 O O O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Wavelength O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Instrument I-Instrument O O O O B-CelestialObject O O O O O O B-Citation I-Citation O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Wavelength B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Telescope O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O B-Wavelength O B-Telescope O O O O O O O O O O O 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 I-Citation O O O O O B-Telescope 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 B-Proposal O O O O O B-Proposal O O O O O O O B-Proposal O O O O O O O O O O O O O O O O O O O O O B-Instrument I-Instrument I-Instrument B-Instrument B-Citation I-Citation I-Citation I-Citation O O O B-Telescope O O B-Instrument O O O O B-Instrument I-Instrument I-Instrument I-Instrument 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 B-Instrument O O O O O O O O O B-Wavelength I-Wavelength 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 B-Instrument 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 O O O B-Instrument O O O O O O


of our problem. Since a series of earlier works have studied this topic (Rees 1988 ; Kochanek 1994 ; Khokhlov Melia 1996 ; Strubbe Quataert 2009 ; Kasen Ramirez-Ruiz 2010 ), we only summarize here the main results that are most relevant to our work, including the kinetic energy, internal velocity dispersion and the opening angle with respect to the central SMBH. An UDS originates from the part of a star which at the time of tidal disruption has a positive energy. This unbound part contains about half of the mass of the star if the star initially approaches the SMBH along a parabolic orbit with a pericentre distance R p comparable to R t – an probable case in real galaxies (Rees 1988 ; Magorrian Tremaine 1999 ). In this case, the unbound material distribute in a range of specific energy between and (Guillochon Ramirez-Ruiz 2013 ; Stone, Sari Loeb 2013 ). As a result, the total kinetic energy E k of the UDS is where m * ≡ M * /M ⊙ , r * ≡ R * / R ⊙ . Here we have assumed an even distribution of unbound material in specific energy, i.e. , which was proposed by Rees ( 1988 ) and later confirmed by smooth-particle hydrodynamics simulations (Evans Kochanek 1989 ). Although we made a couple simplifications to derive equa-tion ( 2 ), the result agrees well with the median value computed from a statistically representative ensemble of UDSs (Guillochon et al. 2015 ). Using this equation and the fact that for main-sequence stars at m * > 1 (Demircan Kahraman 1991 ; Gorda Svechnikov 1998 ), we further derive . This scaling relation indicates that more massive stars generally give rise to more energetic UDSs. It is worth noting that equation ( 2 ) does not include the additional gain of kinetic energy during the compression and bouncing of the star at the orbital periapsis (Brassart Luminet 2008 , and references therein). We neglected it because the bouncing energy , about β 2 GM * / R * per unit mass (Guillochon et al. 2009 ), is usually much smaller than , where β ≡ R t / R p is the so-called penetration factor. Take tidal disruption of a solar-type star ( M * = M ⊙ and R * = R ⊙ ) for example. When M • = 4.3 × 10 6 M ⊙ which is the mass of the SMBH in the Milky Way (MW; Genzel, Eisenhauer Gillessen 2010 ), the penetration factor β cannot be greater than 4.3; otherwise, the star will enter a radius smaller than the ‘innermost bound circular orbit’ (twice the Schwarzschild radius for a non-rotating black hole) and eventually be swallowed as a whole by the SMBH. Because of such a limit to β, the ratio is small, in fact in the range (0.0063, 0.12) for the SMBHs in the mass range of 10 6 M • /M ⊙ 10 8 . We notice that our E k differs by a factor of β 2 from that derived by Khokhlov Melia ( 1996 ), Strubbe Quataert ( 2009 ) and Kasen Ramirez-Ruiz ( 2010 ). This is because those authors calculated using the formula , i.e. they assumed that the energy spread is determined at R p instead of at R t . However, recent studies of the tidal-disruption

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


by most of the theoretical equations of state (e.g., Özel Freire 2016 ). Therefore, it is unlikely that this system is observed at a very large inclination angle. On the other hand, assuming that binaries are randomly inclined with respect to the observer, i.e., assuming a flat distribution of , we find a median NS mass (corresponding to ) of (see the right panel of Figure 6 ). Such a value would place M13F on the highest-mass side of the known NS mass distribution (e.g., Antoniadis et al. 2016b ; Cromartie et al. 2020 ). All of this suggests that either M13F is observed at quite small inclination angles () or it hosts a massive NS. Indeed, under the assumption of a flat distribution of , there is a probability that the compact object mass is larger than , thus favoring the case of a massive NS. Finally, we further investigate the possibility of having a standard NS mass coupled with a low-inclination angle. To this aim, we used a Markov Chain Monte Carlo sampler (Foreman-Mackey et al. 2019 ) to explore the combination of NS masses and orbital inclination angles able to reproduce the observed mass function. We defined a standard Gaussian likelihood function to minimize the difference between the left and right sides of Equation ( 2 ). We assumed a uniform prior on the distribution of and also a prior on the NS mass distribution following that empirically derived by Antoniadis et al. ( 2016b ), which is roughly a double Gaussian with a main component centered at and with a dispersion of about , and a secondary component centered at and with a dispersion of about . The posterior distribution is shown in Figure 7 , and the results based on the 16th, 50th, and 84th percentiles show that M13F could host an NS with an almost standard mass of if orbiting with an inclination angle °. Figure 6. Left panel: companion mass as a function of the NS mass. The solid horizontal line marks the best-fit value of the companion mass () and the light-gray region delimited by the two horizontal dashed lines marks its estimated uncertainty. The dark-gray shaded area is the region forbidden by the binary mass function (see Equation ( 2 )), while the dotted–dashed colored lines are the curves obtained assuming different inclination angles. The vertical line represents a canonical NS mass of . Right panel: NS mass as a function of the cosine of the orbital inclination angle, for the estimated mass of the companion star: the solid curve marks the combination of values allowed by the best-fit value of the companion mass (), while the light-gray region delimited by the two dashed curves marks the combinations allowed within the uncertainty (see the legend). The rightmost panel shows, as reference, the NS mass distribution empirically derived by Antoniadis et al. ( 2016b ). Figure 7. Corner plots showing the 1D and 2D projections of the posterior probability distribution of the NS mass and inclination angle of M13F. The 1D histograms are the marginalized probability

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


the latter is also unknown for putative extraterrestrial organisms, in principle a comparatively robust estimate is probably set by biophysical considerations: at least one photon must be incident on the area of the photosynthetic apparatus (e.g., photosystem I or II) over a single temporal cycle involving electron transfer. The minimum empirical bound of ∼6 × 10 11 photons cm −2 s −1 has been explained through charge recombination in photosystem II, protein turnover, and H + leakage (Raven et al. 2000 ). Therefore, combining these results, the critical PAR energy flux becomes Ψ 0 ∼ 2.1 erg cm −2 s −1 . Substituting this value into ( 5 ), and solving for the cutoff distance ( d P ) yields the extent of the “photosynthesis zone.” By doing so, d P is given byA couple of important points are worth highlighting regarding the above formula: (i) the net primary productivity of biospheres at these low light levels is potentially several orders of magnitude smaller with respect to modern Earth, and (ii) photoautotrophs require not only sufficient fluxes of photons at appropriate wavelengths, but also reactants (e.g., CO 2 ), nutrients (e.g., ), and electron donors (Lingam Loeb 2018a , 2019b ). The photosynthesis zone has been plotted as a function of M BH in Figure 1 . Substituting the parameters for the Milky Way, we obtain d P ≈ 0.34 kpc. This distance is fairly “large” because it approximately equals the scale height of the thin disk of our Galaxy. If we consider black holes with masses on the order of 10 9–10 M ⊙ , we find d P ≳ 10 kpc. In such instances, large swathes of galaxies would fall within the domain of the photosynthesis zone. As noted in Sections 2 and 4 , UCDs are characterized by very high stellar densities, indicating that the photosynthetic zone calculated above could also encompass most stars in these galaxies. 3.2. Timescales for the Evolution of Photosynthesis Finally, we must also examine the issue of timescales. Analyses of isotopic ratios on Earth favor the existence of anoxygenic photosynthesis by ∼3.5 Ga (Tice Lowe 2004 ; Olson 2006 ; Butterfield 2015 ), or possibly even earlier; recall that we posited life had originated by ≳3.7 Ga. The origin of oxygenic photosynthesis remains less tightly constrained, with some proxies (e.g., U–Th–Pb isotopic ratios) placing it as far back as ≳3.7 Ga (Rosing Frei 2004 ), although unambiguous evidence from microfossils appears to date back only to 1.9 Ga (Lyons et al. 2014 ; Fischer et al. 2016 ). Despite this uncertainty, there are multiple grounds for contending that oxygenic photosynthesis may have evolved by ≳2.7 Ga, based on isotopic ratios, phylogenetics, and microfossils (Buick 2008 ; Schirrmeister et al. 2016 ), thus amounting to its emergence by at least several 100 Myr prior to the Great Oxidation Event at ∼2.4 Ga; see also Knoll et al. ( 2016 ), Lingam Loeb ( 2019c ), and Cardona et al. ( 2019 ). In toto , it is not altogether unreasonable to surmise that both anoxygenic and

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


absent in the previous models, – the neutron transfer reactions, consisting in ‘hopping’ (quantum tunnelling) of a neutron from weakly bounded state in one (donor, ‘d’) nucleus to another (acceptor, ‘a’) nucleus. Neutron transfer reactions are well known in nuclear physics for near-barrier and sub-barrier energies (e.g. von Oertzen et al. 1987 ; Zagrebaev 2003 ; Zagrebaev, Samarin Greiner 2007 ; Canto et al. 2015 ; Karpov, Rachkov Samarin 2015 ) and the multinucleon transfer reaction can be used to synthesize previously unexplored superheavy elements (Zagrebaev, Karpov Greiner 2013 ; Wuenschel et al., 2018 ). The specific feature of neutron transfer reactions in NS envelope would be that the main contribution to the neutron transfer rate comes from nuclei located at distance l pk , which can be as large as 100 fm, strongly exceeding nuclei radii. As for thermonuclear reactions, this distance is determined by two competing factors: decrease of the tunnelling probability with increase of internuclear distance and decrease of number of nuclei, which can approach closer, because of Coulomb barrier. In some sense, the neutron transfer reactions are similar to a hopping transition between localized states of electrons, which is the basis of hopping conductivity (see e.g. Gantmakher 2005 ). Note, as shown by Zagrebaev, Samarin Greiner ( 2007 ) neutron transfers can also increase fusion rates due to modification of internucleus potential, but I do not discuss this effect here, limiting myself to neutron transfer reactions only, and, specifically, to order-of-magnitude estimates, which demonstrate that neutron transfer is applicable for astrophysics matters. Namely, I start from calculation of the transfer probability (per unit time) between static nuclei at given distance, following the approximative approach by Bonaccorso, Piccolo Brink ( 1985 ) and Monaco Brink ( 1985 ). Then I estimate the reaction rate λ (number of reactions per one donor nuclei per unit time) for certain plasma conditions by averaging transfer probability over realistic internuclei separation in plasma. A simple approximating expression is suggested. Surely, the model (especially its nuclear part) should be improved, and at the end of the letter I briefly discuss its further refinement, leaving detailed calculations for the future. The neutron transfer rate strongly depends on S n of the donor nuclei, charges Z d and Z a of both donor and acceptor nuclei, temperature T , and electron chemical potential μ e (which determines density ρ for given composition). Typically, neutron transfer is relevant for donor nuclei with S n ≲ 3 MeV, which may exist in envelopes of accreting NS. In particular, it can burnout strongest among URCA pairs – specific pairs of nuclei which can coexist at respective electron chemical potential μ e (density) and cools down the envelope by neutrino emission associated with cycles of beta-captures/decay (see Schatz et al., 2014 ; Meisel et al., 2015 for

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


Figure A1. A convergence study of the evolution of the mass of dense gas with n ≥ n cloud /3. For all cloud sizes the three highest resolution levels show good agreement, especially at early times t ≲ 5 t cc , implying that these simulations are well converged. At the same time, the low-resolution simulations with R cloud /Δ x = 76 (dashed lines) systematically differ from the other simulations. Thus, only the dense gas survival mass fraction in the 2D simulations with R cloud /Δ x ≥ 152 are converged. The evolution of the number of FoF groups for the 2D simulations is shown in Fig. A2 . We confirm the conclusion that clouds with R cloud ≫ l cool undergo excessive fragmentation for a resolution of R cloud /Δ x ≥ 152. Only the low-resolution simulations with R cloud /Δ x = 76 are not capturing this result in 2D. Figure A2. The number of FoF groups in 2D simulations with a resolution of 607, 304, 152, and 76 cells per cloud radius (from left to right). For R cloud /Δ x ≥ 152 the 10 and 100 pc cloud simulations have a higher peak number of FoF groups in comparison to the non-radiative and 1 pc cloud simulations. A resolution of R cloud /Δ x = 76 is evidently not enough to cause fragmentation of a 100 pc cloud since the peak N FoF value is comparable to the 1 pc simulation with radiative cooling. We conclude that a resolution of R cloud /Δ x ≥ 152 for our 2D simulations is sufficient to capture fragmentation in 1, 10, and 100 pc simulations with cooling. A2 3D simulations Because 2D and 3D clouds may be affected differently by instabilities, we carry out the same resolution studies for the 3D simulations. First, we remind the reader that the dense gas mass survival fraction studied in Fig. 1 behaves very similar for our low- and high-resolution 3D simulations, with the only notable difference being that the dense gas in the 100 pc simulation at low resolution survives longer in comparison to the high-resolution simulation. To further study convergence properties we show the evolution of N FoF in Fig. A3 . The figure confirms that the 10 and 100 pc simulations are more fragmented than the non-radiative simulation and the 1 pc simulation with radiative cooling. Overall, this confirms our result that clouds larger than the cooling length shatter to smaller cloudlets. However, the figure also shows that fragmentation of the 100 pc cloud in 3D is not fully captured, because the 10 and 100 pc clouds here have a similar number of FoF groups throughout the simulations. This conclusion is further supported by the evolution of the power spectral slope of the low-resolution simulations shown in Fig. A4 because the peak value of the slope is not larger in the 100 pc simulation in comparison to the 10 pc simulation. Excessive fragmentation of the 100 pc cloud in comparison to the 10 pc cloud is thus only seen in our high-resolution 3D simulations. Figure A3. N FoF as a function of time for the low-resolution 3D simulations (with R cloud /Δ x = 80). The

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zones leads to the depletion of stars in the regions around these resonances. These results do not strongly depend on the choice for the gravitational potential assumed for the Milky Way, provided this is generic enough (and not integrable, e.g., of Stäckel form). We also find that in two other commonly used Galactic potentials ( Piffl et al. 2014 ; Bovy 2015 ), very similar substructures can be observed in R max vs. z max , as is shown in Appendix A , and that they are associated with resonant families, and hence with the presence of non-integrability in the system. Our analysis (as also concluded by Amarante et al. 2020 ) therefore directly shows that not all substructures are related to accretion, as is often considered in the literature, nor to the settling of the gravitational potential after major merger activity as had been suggested by Haywood et al. (2018) . Nonetheless, the characteristics of substructures from merger events may sometimes be related to the presence of resonances. For example, we see that some of the stars in the Helmi streams appear to be close to a resonance, Ω z : Ω R ∼ 1:2. This perhaps explains why its stars are distributed asymmetrically in velocity space (the stream with v z 0 has more stars than that with v z > 0). This asymmetry was used to constrain its time of accretion to approximately 5-8 Gyr ago, which is a puzzlingly low value given that these stars are on relatively bound orbits. Since stars near a resonance take longer to spread out in space ( Vogelsberger White 2008 ), this can lead to an underestimation of their time of accretion. This means that chaotic dynamics may have to be considered when modelling the evolution of tidal debris, as already hinted by Price-Whelan et al. (2015) in their modelling of cold thin streams farther away in the halo. Probably the best way to tell whether substructures are related to accretion events is via a chemical tagging analysis. Stars that originate from a satellite will follow characteristic tracks in chemical abundance space. On the other hand, stars that group together because of dynamical resonances have no reason to be chemically distinct from other stars. How stars populate different resonant families depends on their distribution function and the gravitational potential in which they move. In this work, we only considered that of the Milky Way, but a recent analysis of nearby thin-disc stars has shown that the Sagittarius dwarf also plays a role in their dynamics (see Antoja et al. 2018 ). Interestingly, if we integrate the orbit of Sagittarius in the McMillan (2017) potential for 40 Gyr with the initial conditions derived by the Gaia Collaboration (2018c) and apply the SuperFreq code to obtain its frequencies (i.e. we repeat the analysis done for the stars in our sample), we find that Sagittarius falls on the Ω z : Ω R = 1:2 and Ω z : Ω φ = 1:1 resonance. The stars in our sample on the Ω z : Ω R = 1:2 resonance (green points in Figs. 7 – 9 ) define a

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


be coincidental. The small number of galaxies in our sample and the substantial uncertainties on the measurements of the dust mass and gas metallicity preclude us from drawing any solid conclusion regarding a possible physical link between metallicity and circumgalactic dust mass fraction. Our data do not allow us to determine unambiguously the origin of the circumgalactic dust. The best case in our sample for dust entrained in a galactic wind is NGC 1569, but we cannot rule out the possibility that cold dust features that appear to lie in the zone of influence of the galactic wind are instead dust features of external (e.g. tidal) origin in front or behind the wind. Future studies that not only consider the distribution of the circumgalactic dust with respect to the wind and tidal features but also the kinematics of dust tracers such H 2 , CO, Na I D, and H i will help shed light on the origin of this far-flung dust in NGC 1569 and other dwarf galaxies. APPENDIX A: NOTES ON INDIVIDUAL GALAXIES For the discussion in these notes, please refer to Figs 1 (a)–(f) and Figs 2 (a)–(f). He 2-10 : broad cold dust emission regions surround He 2–10 out to ∼1.5 kpc in all SPIRE bands as well as the 160 m PACS map. The 160 m emission map appears similar in extent and morphology to the H i distribution observed by Kobulnicky et al. ( 1995 ). The 70 m PACS map exhibits a bit more anisotropy in its distribution of circumgalactic emission with tapering features extending to the north–northeast, southwest, and east-southeast. The north–northeast feature extends furthest to ∼2.6 kpc. The orientation of these tapering features appears similar to the smaller 0.5 kpc-scale Hα bubbles (Méndez et al. 1999 ; Zastrow et al. 2013 ), but appears less correlated with the shape of the X-ray emission (Kobulnicky Martin 2010 ). Several knots of emission separated from He 2–10 by ∼3–4 kpc to the north-west and southwest in the PACS maps cannot be definitively ruled out as background sources given the signal-to-noise levels or foreground Milky Way dust clouds, but coincident emission in the SPIRE maps seems connected to the disc via colder streams bridging their separation. The most prominent emission in the SPIRE maps towards the knots north-west of the disc seems similarly oriented to an X-ray feature observed by Kobulnicky Martin ( 2010 ). NGC 1569 : a dense complex of filamentary and broad circumgalactic cold dust features stretch out from all along the disc of NGC 1569. The prominent Hα, 8.0, and 24 m filament (Waller 1991 ; Hunter et al. 1993 ; Westmoquette et al. 2008 ; McCormick et al. 2013 ) extending southwest from the western edge of the disc also shows up in the PACS maps, but becomes fainter and disappears in the SPIRE bands. A clump of cold dust directly south of the galactic nucleus resides ∼1.8 kpc from the disc mid-plane and appears to have made a clean break with the other circumgalactic cold dust emission in the PACS maps. Some faint 8.0 m emission

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-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-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 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 B-Instrument O O O O O B-Wavelength I-Wavelength B-Telescope O O B-Wavelength I-Wavelength 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-Wavelength I-Wavelength 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 B-Wavelength 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 B-Wavelength 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 B-Instrument 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 B-Instrument 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 O O B-Wavelength O O O B-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Wavelength B-Wavelength O B-Wavelength I-Wavelength O B-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 I-Citation 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 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 B-Instrument O O O B-Wavelength I-Wavelength O


weak correlation of the slope with observed for the full sample is due to a second-order age effect, since the correlation of with age was negligible. We should also note that the sample plotted in the right panel of Fig. 10 consists of only 18 stars, which is less than the half of the whole sample, thus it suffers (more) from low number statistics. 7. Summary and conclusion To explore the effect of the star formation place (birthplace) on the chemical abundance trend with the condensation temperature, we selected a sample of main-sequence stars that have similar stellar parameters and expected similar ages as our Sun. The sample consisted of 39 stars, for 25 of which we carried out new observations with HARPS and UVES spectrographs. The spectra of the remaining 14 stars were taken from HARPS and UVES archives (ESO archive). We performed a detailed differential chemical abundance analysis and derived stellar ages for this sample of stars. When analysing the effect of the GCE on the [X/Fe] elemental ratios, we found that many elements display a strong dependence on stellar metallicity and most of the elements show a statistically significant trend with stellar ages. The latter result ([X/Fe] – age correlation) agrees well with that of Nissen (2015) . To minimize the GCE effect on trend that is due to time or age, we selected a sub-sample of stars that lie in a narrow range of ages from 2 to 4.5 Gyr. The stars were also selected to have errors on ages smaller than one Gyr. For this sub-sample of stars we found that the [X/Fe] abundance ratio of some elements correlates with the mean of the apo- and pericentric distances, . These Galactic radial abundance gradients qualitatively agree with those obtained from Cepheids (e.g. Lemasle et al. 2013 ) and from open clusters (e.g. Yong et al. 2012 ; Magrini et al. 2015 ). When considering only stars that are younger than 7 Gyr, we found no significant dependence of the trend on stellar age, as was observed in previous works (e.g. Adibekyan et al. 2014 ; Nissen 2015 ). As in Adibekyan et al. (2014) , when using the full sample of stars, we also observed some hint that the slope depends on , although not in a simple way. However, this dependence completely vanished when a sub-sample of stars with similar ages (the sub-sample mentioned in the previous paragraph) was considered. Here, we should note that this sub-sample consists of only 18 stars. With this small sample we still cannot firmly conclude whether the trend depends on the formation place of stars. A larger sample size and improvement in the data quality (higher S/N) as well as the use of very similar stars in terms of stellar properties (e.g. metallicity, temperature, age) that are observed at a wide range of Galactocentric distances will help to finally answer this question. In this context, the role of large surveys and missions, such as Gaia ( Perryman et al. 2001 ), Gaia -ESO ( Gilmore et al. 2012 ), and APOGEE ( Ahn et al. 2014 ), is

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Instrument O B-Instrument O O O O O O O O O O O B-Instrument O B-Archive I-Archive B-Archive I-Archive O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-TextGarbage I-TextGarbage O B-CelestialObject 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-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 B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation 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 O O O O O O O O O O O O O O O O O O O O O O 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-Citation I-Citation I-Citation I-Citation O B-Survey I-Survey O B-Citation I-Citation I-Citation I-Citation O O B-Survey O B-Citation I-Citation I-Citation I-Citation O O


the distance of the star–planet system in parsecs (PC), and Hill radius is expressed in (au). Finally, we calculate the RV semi-amplitude, K , of each planet given its estimated mass:We further assume an orbital inclination of ∼90° and e = 0. Table 2 includes each of the parameters used in our calculations which have been extracted from the HZ catalog (Kane et al. 2016 ), as well as the NASA exoplanet archive. Table 3 presents our calculations of planet mass, Hill radii (HR), estimated RV semi-amplitudes and angular separations of the planet–star systems and potential planet–moon systems at both the full HR and Hill radii ( HR). Table 2 Habitable Zone Candidates with R p > 3 R ⊕ Note. a Semimajor axis. Table 3 Radial Velocity, Hill Radius, and Angular Separation Calculations for HZ Candidates with R p > 3 R ⊕ Tables 4 and 5 then present our calculations of HR, angular separations of a potential planet–Moon systems at the full Hill radius and RV semi-amplitudes for each exoplanet with a radius of 10 R ⊕ or greater with our chosen quantified masses: one Saturn mass ( M sat ), one Jupiter mass ( M J ), and 13 Jupiter masses (13 M J ). Table 4 Radial Velocity Semi-amplitude Calculations for Category 4 HZ Candidates with R p > 10 R ⊕ Table 5 Hill Radii Calculations for Category 4 HZ Candidates with R p > 10 R ⊕ Notes. a Angular separation of exomoon at full Hill radius for . b Angular separation of exomoon at full Hill radius for M p = M J . c Angular separation of exomoon at full Hill radius for . We plot a histogram of the effective temperatures of Kepler host stars to determine if there is a similar distribution of temperatures among both the HZ candidates and the full catalog. Figure 11 shows the stellar temperature distributions for both the HZ Kepler candidates (green), as well as the full Kepler catalog (gray). The histograms show that there is a similar distribution of temperatures among both the HZ candidates and the full catalog, with the HZ host star temperatures dropping off (around) 7000 K. As the HZ of stars with greater effective temperatures will lie further away from the star, planets in this zone are harder to detect. Thus, this drop is likely a false upper limit. Figure 11. Stellar temperature distributions. Habitable zone Kepler candidates in green overlays the distribution of the full Kepler catalog in gray. The histograms show that there is a similar distribution of temperatures among both the HZ candidates and the full Kepler catalog. While the distribution of the habitable zone candidates drops off at 7000 K, this could be a false upper limit as the habitable zone of stars with greater effective temperature lies further away from the star and current transit detection methods are less sensitive to planets at these longer orbits. Using the calculations from our Tables above, we plot the Kepler magnitude of the host star of both the unconfirmed and confirmed HZ planets and their expected RV signatures to determine the

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 B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O B-Archive I-Archive I-Archive O O O O O O O O B-Model I-Model B-Model 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 B-Model O O B-Dataset I-Dataset I-Dataset 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-Dataset I-Dataset 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 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 B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Model I-Model O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Model I-Model O O O O O O O O O B-Model I-Model 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-Telescope O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O B-Dataset I-Dataset O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O 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-Dataset I-Dataset I-Dataset I-Dataset O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O B-Dataset I-Dataset I-Dataset O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O


to the formula J = n e e ( V i − V e ), where n e is the electron number density and e is the elementary coulomb charge, respectively. Inside the first current sheet, both MMS2 and MMS4 observed mainly out-of-plane electron current J M ′ (Figures 2 (d), (n)). The current densities associated with the reversal of magnetic field component B L ′ peak at 0.7 μ A m −2 and 0.4 μ A m −2 , respectively. Similar trends to electron velocities mean that the current is mainly carried by electrons. Multispacecraft timing analysis (Schwartz 1998 ; Dunlop et al. 2002 ) indicates that the current sheet moves sunward in the current sheet normal direction (where N = L ′ M ′ N ′ (0.3747, 0.0649, 0.9249)) with a speed of 52 km s −1 . The derived normal direction is identical with the N ′ calculated from minimum variance analysis. The duration for the current sheet is ∼0.18 s (∼05:57:08.6 and ∼05:57:08.78 UT). Thus, the thickness of the current sheet is calculated to be 52 km s −1 × 0.18 s = 9.36 km ∼5.5 d e , where d e is the electron inertial length. In the turbulent magnetosheath, similar intense small-scale current sheets are favorable sites for electron-only reconnection (Stawarz et al. 2019 ). Four-spacecraft timing analysis can also be applied to gain quantitative information about the orientation and motion of the flux rope. In the case of a cylindrically symmetric flux rope, it can be shown that the times at which each spacecraft observes the peak field strength define a plane that is perpendicular to the tetrahedron direction of motion and contains the axis of the flux rope. The flux rope is found to be moving at v ∼ 43.5 km s −1 along n = (0.83, 0.26, 0.49) GSM relative to MMS. The velocity of the flux rope is found to be mainly in the x – z plane, indicating that the axis of the structure is nearly in the y direction. This orientation is also consistent with the result calculated by MVA also shown previously. As mentioned above, the flux rope crossing time is 4 s during 05:57:07 UT and 05:57:11 UT. This corresponds to a crossing distance of 174 km based on the normal speed of 43.5 km s −1 of the flux rope, or 2.26 d i . This indicates that the spacecraft are observed an ion-scale flux rope. As previously analyzed, the spacecraft are slightly off-center when crossing the flux rope. This means that any estimate of the flux rope size is a lower bound. MMS2 observed well-defined parallel electric fields (Figure 2 (f)), which indicated the invalidation of the ideal electron frozen-in condition ( E ′ = E + V e × B = 0). The electron dissipation measure ( J · E ′) is calculated in Figure 2 (j) based on current densities and electric field in the electron frame. J · E ′ is positive within the current sheet and is close to zero when MMS crossed outside the exhaust region. The electric field structure is parallel to the background magnetic field at MMS2. The dissipation is positive and basically all dominated by . The energy dissipation is mainly

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


that there is a background and foreground sources contamination. Fig. 9 shows the stellar position of the proper motion members and Fig. 10 the parallax histogram. It can be seen that they are distributed throughout the analysed region. Figure 8. Histogram of stellar proper motion probability in the left overdensity region. The dotted line corresponds to 1σ a members, the full line indicates the lowest value of the probability of the members. Figure 9. Stellar positions of the proper motions members of the left overdensity (black dots). The red squares represent the location of the known O members. Figure 10. Histogram of parallaxes of 1σ a proper motion members in the left overdensity region. Table 7. Left overdensity proper motion members up to G = 17.5 mag. The columns gives the position (α, δ), the Gaia DR2 identifier, the parallax π and its error σ π , the proper motion components and their errors μ α cosδ , , μ δ , and the G magnitude. Full table is available online. 5.3 Open cluster (bottom overdensity) We now focus on the bottom overdensity. We first eliminate the Cygnus OB2 proper motion members obtained in Section 4.2 . After applying the procedure of Section 3 , the parameters of the new group are obtained (see Table 5 ). The membership probability of the stars is calculated by equation ( 4 ). 25 proper motion members of the new stellar group are identified (see Table 8 ). The index E takes the value of 0.94, this indicates that the group and field stars are almost completely separated. Fig. 11 shows the stellar position of the proper motion members. It can be seen that this group is an open cluster, whose centre is (α, δ) = (306.99820°, 40.54240°).The radius of 9 arcmin is determined from the radial surface density profile of the members (Fig. 12 ). Figure 11. Stellar positions of the proper motions members of the open cluster (black dots). The red squares represent the location of the known O members. Figure 12. Stellar density profile of the open cluster (bottom overdensity) as a function of the distance from the cluster centre. Then, we calculate the mean parallax of the members π a and estimate the distance d π a . Both values are presented in Table 5 . The estimated distance d π a = 1457 ± 11 pc indicates that this group is also located in the foreground of the main Cygnus OB2 group. Besides, an inspection of the magnitudes in Table 8 reveals that this open cluster is formed by faint stars. Recently, Castro-Ginard et al. ( 2020 ) found 582 new open clusters distributed along the Galactic disc performing a blind search for open clusters using Gaia DR2 data. We noticed that one of these objects, UBC585, corresponds to this open cluster. Our mean parallax and mean proper motion are in good agreement with the values obtained for UBC585, while we find eight more members for UBC585. Table 8. 25 proper motion members of the open cluster (bottom overdensity). 6 CONCLUSIONS In the Cygnus OB2 region, we analyse a sample of stars from

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Telescope B-Dataset O O O 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 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 O O O B-CelestialRegion I-CelestialRegion I-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-CelestialRegion I-CelestialRegion O O O O O O O O B-Telescope B-Dataset 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 B-CelestialObject 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-CelestialObject I-CelestialObject O O O O O O O O


The two measurements are not inconsistent taking into account the large errors in both of them, however they suggest that weak CIV systems may have lower bias than strong ones, which would help explain why the individually detected CIV systems have a higher bias than the overall population. Next, we comment on the result of Blomqvist et al. ( 2018 ), who have simultaneously completed a similar analysis to ours including eBOSS data. Apart from having a larger data set (especially at low redshift), there are other differences in the two analyses. One is the different continuum-fitting methods. Blomqvist et al. ( 2018 ) apply the same continuum-fitting method used in other BOSS analyses of the Lyα forest, hence the need for a distortion matrix in their work (i.e. their Section 5.2 ). They also use an upgrade from the Baofit code we use called picca , 4 although this should not affect the results. Their fitting range extends to a larger scale than ours: from 10 to 180 , compared to our range from 5 to . We focus on comparing their result for the bias and redshift distortion factors when restricting the rest-frame wavelength range to 1420 Å Å, the same one we have used which avoids the contribution from SiIV lines. The mean result of Blomqvist et al. ( 2018 ) for their entire redshift range is β c = 0.35 ± 0.21, and b c (1 + β c ) = −0.019 ± 0.002. In comparison, we obtain a higher value of the redshift distortion factor, β c = 1.09 ± 0.56, although the large errors do not make the two values incompatible. Our value for the bias b c agrees with theirs: from our equation ( 20 ), if we use their central value β c = 0.35, we find that our measurement b c (1 + 0.44β c ) = −0.017 implies b c (1 + β c ) = −0.0199 ± 0.0015 for β c = 0.35, in full agreement with Blomqvist et al. ( 2018 ). The low central value of β c found by Blomqvist et al. ( 2018 ) is not consistent with our constraint derived from the bounds in from measurements of the incidence rate of CIV absorbers as a function of their equivalent width in Section 7.1 , implying 0.6 β c 1. Taking into account the measurement error, however, their result can be compatible with this constraint. It is worth noting that their result for β c is substantially different when dividing the data set into two redshift intervals. For low redshift ( 2.2 and a mean effective redshift eff = 1.69), they find β c = 0.05 ± 0.19, whereas for high redshift ( > 2.2 and eff = 2.41, closer to our measurement with eff = 2.29) their result is β c = 0.67 ± 0.29. Their result at high redshift is therefore fully consistent with ours, and with our derived constraint from the value of . The very low value of β c at low redshift derived by Blomqvist et al. ( 2018 ) suggests that the absorber bias factor b τ c may be increasing rapidly with decreasing redshift, perhaps because CIV is being destroyed in low-mass haloes and being formed in massive haloes. However, more data at low redshift that allow reduced error bars and

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


of prominence evolution involving B oriented across the 2.5D plane, or in full 3D (e.g., Hillier et al. 2012a , b ; Terradas et al. 2015 ; Keppens et al. 2015 ; Kaneko Yokoyama 2018 ; Popescu Braileanu et al. 2021 ). Nevertheless, we expect the misalignments between the pressure and density gradients to be present in 2.5D and therein induce motions inline with the RTI but on scales of the order of a flux tube. Evolution of the plasma in this way may also be characterised using baroclinicity, a component of the evolving vorticity ω that arises in a flow with anisotropic temperature, This baroclinicity considers that a misalignment between the pressure and density gradients gives rise to the rotation of plasma as it slides down a constant-pressure gradient. In general, plasma unstable to the RTI in a direction other than vertical may be identified using this approach. The fourth row of Fig. 3 details the evolution of the baroclinicity within the condensing material. At t = 7264.6 s, the signature of baroclinicity covers a broad region but one that is focussed at the periphery of the condensation region and forms a ‘U’ shape. As the evolution progresses, this ‘U’ shape is amplified in magnitude, but also becomes flat and increasingly localised to the bottom of the forming monolithic structure. In the density panels, at t = 7436.2 s we see that the two early condensations develop a curvature as they move towards the concave-up portions of the field topology. This is consistent with the baroclinicity-induced rotation. It is important to emphasise here that vorticity maps quantify, generally, the rotation of the velocity field (induced by some physical process), whereas the baroclinicity maps show the source of vorticity and represent a physical process in itself. Hence, maps of the vorticity would likely also show this rotation but would not offer information on the driving process. 3.1.3. Evolution of the monolithic prominence Figure 5 details the evolution of the now-formed prominence after the initial condensation period. Initially, between 7607.8 t 8294.2 s, it can be seen that the condensation region significantly deforms the field topology of the host flux rope as the concentration of highest density falls from an altitude of ≈9–6 Mm with an average velocity of 5 km s −1 . The largest magnetic field strength is recorded within the condensation at this time, with a value of ≈4.2 G. The falling material drags the associated field lines down and those field lines that lie directly below the condensation are compressed. Shortly after t = 8294.2 s, reconnection is triggered just above the densest portion of the falling monolith, creating a magnetically isolated region of high density that continues to fall, now with a velocity of ≈10 km s −1 . Behind this falling ‘plasmoid’ 3 , another high-density region forms in its wake, and at t ≈ 8500 s reconnection is again triggered behind this new region. Once the new density enhancement is isolated

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( 1996 ) and find thatThis implies that the disk is not very thin. From the vales in Equations ( 7 )–( 9 ) we can calculate the typical neutrino cooling timescale in the disk. Taking the neutrino cooling rate as (Brown Weingartner 1994 ) , we calculate the neutrino cooling timescaleWe calculate the radial velocity in the disk from mass accretion rate and find it to be . The inflow time of the gas in the disk from is . Namely, neutrino cooling in the disk is important. From the above results we can safely conclude that our derivations are self-consistent and that neutrino cooling plays a major role in the accretion disk of our scenario. Neutrino cooling lowers the temperature of the accreted gas, reducing the pressure and ensuring a mildly electron degenerate state, as in the scenario of Siegel et al. ( 2019 ). From Beloborodov ( 2003 ) the electrons become degenerate below the characteristic degeneracy temperature ofThis is somewhat higher than the typical temperature we find in Equation ( 9 ), implying that we have a mildly degenerate electron gas as Siegel et al. ( 2019 ) find in their scenario. Moreover, according to Figure 1 of Beloborodov ( 2003 ), the baryonic matter in the accretion disk of our scenario is in nuclear statistical equilibrium and is dominated by free nucleons with an electron-to-baryon ratio of Y e 0.3, i.e., a large fraction of neutron to protons. Overall, we conclude that the accretion disk conditions in the CEJSN r -process scenario resemble the conditions from the studies of Kohri et al. ( 2005 ) and Siegel et al. ( 2019 ). We expect then the outflow properties to be quite similar. 5.3. Nucleosynthesis in Jets As we mention above, at the stage of high mass accretion rate the core mass inner to the orbit of the NS is ≈1 M ⊙ . A large fraction of this mass is expected to be accreted at this high rate. As described in Section 2 , Papish et al. ( 2015 ) estimated that the jets in the CEJSN r -process scenario are expected to synthesize a mass of ≈0.001–0.01 M ⊙ of heavy r -process elements per event. This is based on the assumption that a fraction of ≈10% of the accreted mass is launched in the jets and ≈10% of the mass in the jets is transferred to heavy r -process elements. If we had used the recent results of Siegel et al. ( 2019 ), the mass of the r -process elements would have been even higher. Siegel et al. ( 2019 ) find the ejected mass to be ≈30% of the accreted mass. Under this assumption, we can take a more optimistic yield than what Papish et al. ( 2015 ) used, namely, we can take the yield per CEJSN r -process event to be ≈0.01–0.03 M ⊙ of heavy r -process elements. As we mentioned in Section 2 , Chevalier ( 2012 ) estimated the rate of events in which an NS enters the envelope of a giant star to be ≈1% of the total rate of CCSNe. Papish et al. ( 2015 ) found that if this is also the rate of CEJSN r -process events, then combining it with the yield of r -process elements that they derived, we obtain the

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


) shows that these photodissociation products are likely to dominate the gas mass, since molecules including CO are photodestroyed very rapidly after release (in less than 100 years) due to the strong UV radiation environment found around an A0V star. The same radiation field is able to ionise these atomic species, explaining the detection of CII around several debris discs (e.g. Riviere-Marichalar et al. 2014 ). HR 4796 A was also searched for atomic oxygen (OI) and ionised carbon (CII) emission by Herschel ( Meeus et al. 2012 ), which set upper limits of 6.3 and Wm, respectively. Due to the low gas densities expected in this scenario, species with high critical densities such as CO and OI are expected to be out of LTE, meaning that fluxes are dependent on the density of collider species and on the radiation field as well as on the gas temperature and OI mass (Eq. (7) in Matrà et al. 2015 ). This means that the derivation of OI mass upper limits from the observed fluxes is highly degenerate; the Herschel observations therefore cannot tightly constrain the oxygen mass in these low-density discs. On the other hand, taking electrons as the dominant collider species (as shown to be likely in secondary gas discs, Matrà et al. 2015 ), CII has a much lower critical density of order 5–10 electrons cm ( Goldsmith et al. 2012 ). The electron density predicted by this exocometary model in HR 4796 A is much higher (of order at the location of the belt, Kral et al. 2017 ). This means that LTE is a good approximation for CII, yielding a mass upper limit between for the same 20100 K temperature range considered above for CO. However, our secondary gas model predicts an average ionised fraction in the disc that is very low (), meaning that we expect a much higher total carbon mass in HR 4796 A ring of 0.19 . Then, assuming the dominant molecule released by the comets to be CO, yielding a C/O abundance of one, we can estimate the total gas mass in the secondary origin scenario to be 0.44 . However, we note that the Kral et al. (2016) model shows that the bulk of the gas will rapidly expand viscously to form an accretion disc. This means that only 6% of this total gas mass (0.03 ) will be co-located within the AU wide dust ring. In summary, the secondary model predicts a gas mass co-located with the ring of 0.03 , whereas in the primordial scenario we expect a total gas mass of at most 0.11 . On the other hand, SED modelling of the ring’s infrared excess ( Milli et al. 2015 ) yields a total dust mass in the ring of 0.5 distributed among particles between 1 m and 10 mm. Regardless of the gas origin scenario, we therefore derive an upper limit on the gas to dust ratio at the ring’s location of 0.2, which rules out the presence of gas in sufficient amounts to explain the observed ring morphology in the HR 4796 A system. A similar conclusion was recently achieved for the narrow HD 181327 debris ring ( Marino et al. 2016 ), where very low levels of

O O O O O O O O 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 O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O B-Telescope 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 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 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 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 B-Citation I-Citation I-Citation I-Citation 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 B-Model I-Model O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation 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 B-TextGarbage O O O O O O O O O O O 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 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 B-Wavelength 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 B-CelestialObject I-CelestialObject I-CelestialObject 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 O O O O O O


Via the introduction of a physical model of gravitational structure growth, the inference of the non-linear dark matter distribution today can be posed as a statistical initial condition problem. To do so, the borg algorithm seeks to infer the cosmic initial conditions from which present 3D structures in the distribution of galaxies have formed via non-linear gravitational mass aggregation. The borg algorithm explores the posterior distribution of large-scale structures, consisting of a Gaussian prior for the initial density field at an initial scale factor of a = 10 −3 and a choice of bias model and likelihood metric at scale factor a = 1. The evolution of the initial density fields can be related to the present galaxy (or halo) distribution via a first- or second-order Lagrangian perturbation theory (LPT or 2LPT) or a full particle mesh model of gravitational structure formation (for details, see Jasche Wandelt 2013 ; Jasche Lavaux 2019 ). By exploiting non-linear structure growth models, the borg algorithm naturally accounts for the filamentary structure of the cosmic web typically associated with higher order statistics induced by non-linear gravitational processes. Furthermore, the posterior distribution accounts for systematic and stochastic uncertainties, such as survey geometries, selection effects, unknown noise, and galaxy biases as well as foreground contamination (see e.g. Jasche Wandelt 2013 ; Jasche et al. 2015 ; Lavaux Jasche 2016 ; Jasche Lavaux 2017 ). In this work, we use the borg algorithm to sample the initial conditions of the dark matter density field using LPT to evolve the field to the dark matter conditions today and then rely on the neural bias model to sample the parameters of the NPE to provide a field that is most informative about the halo mass distribution function inferred from a halo catalogue. A detailed schematic of the interconnection between parts of the model is shown in Fig. 1 . The stochastic uncertainties are assumed to be Poissonian using the likelihood in equation ( 16 ), and further study into direct learning of the deviations from Poissonity will be conducted in an upcoming work. Figure 1. Schematic of the borg algorithm with the neural bias model. Initial conditions for the density field in Fourier space, , are drawn from a prior given a cosmology , . These are then evolved forward using a deterministic prescription, in this example using LPT. The evolved field is then transformed further using the NPE that requires parameters that are drawn from a prior . This provides a field from which the halo mass distribution function can be described using the MDN with parameters drawn from a prior . This halo mass distribution function is then compared to the masses of haloes M obs from the observed halo catalogue. 5 RESULTS To examine the techniques developed in this paper, we will consider a relatively simple mock run using a simulated dark matter distribution and minimal working neural bias model.

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for density ρ , velocity , the specific entropy S , and the magnetic vector potential are given by where P is the pressure with , which is defined up to some additive constant, c p and c v are the specific heats at constant pressure and density, respectively, T is the temperature with being the ideal gas equation of state, K is the thermal diffusivity, ν is the kinematic viscosity, η is the magnetic diffusivity, is the magnetic field with b 0 being the imposed field, and is the current density that was already defined in the introduction. We adopt a polytropic stratification with background temperature , so T = 0 at z = 0. We fix d and choose to set the degree of stratification. The smaller the , the stronger the stratification, i.e., the stronger the temperature contrast. In the following we choose , so the temperature changes by a factor of 10; see Hurlburt et al. ( 1984 ) for a similar setup. We choose to be either negative or positive, corresponding to the northern or southern hemispheres, respectively. A vertical magnetic field is imposed and tangled by this velocity field. The simulation setup is similar to that of Hurlburt Toomre ( 1988 ), except they did not include rotation, which makes our present simulations therefore closer to those of Brandenburg et al. ( 1990 ), which did include rotation. In the following, we denote by the density at . Some of the parameters are listed in Table 3 . The imposed magnetic field points in the z direction and is given by , where is the thermal equipartition field strength. We use in all cases. The Rayleigh number is defined as , where and are density and the specific entropy of the hydrostatic solution in the middle of the domain. Table 3 Parameters for Convection Simulations Cross sections of , , and near the surface are shown in Figure 10 for the results of such a simulation. One sees cyclonic convection in the northern hemisphere as viewed from the top, so all converging inflows attain a counterclockwise swirl, and all diverging outflows have a clockwise swirl. A similar appearance is also attained by the magnetic field. If we viewed this pattern from beneath, we would see a mirror image of the original pattern and therefore the opposite sign of the B polarization. The consequence of this can be seen in Figure 11 , where we plot and for north (red) and south (blue) for Runs A and B, whose parameters are summarized in Table 3 . There is now a systematic EB correlation, so is positive in the north and negative in the south; see Table 4 . This is very promising and agrees with our intuition. Figure 10. Magnetic field vectors and the line-of-sight component (color-coded; left) as well as E -mode (middle) and B -mode polarization (right) in rotating convection viewed from the top onto the convecting layer. Figure 11. (top) and (bottom) for convection simulations corresponding to the northern (red) and southern (blue) hemispheres for Runs A (thin lines) and B (thick lines). The gray shades indicate

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). Despite this wide range of applications, the nonlinear dynamics of eccentric discs remains poorly understood. In the Solar system, several of the rings of Saturn and Uranus are observed to be elliptical, and a theory for narrow eccentric rings (with small eccentricities but arbitrary eccentricity gradients) composed of weakly collisional particles has been developed (Borderies, Goldreich Tremaine 1983 ; Chiang Goldreich 2000 ). For small eccentricities, eccentric discs can be described as slowly precessing one-armed density waves (or shocks, e.g. Okazaki 1991 ; Papaloizou, Savonije Henrichs 1992 ; Lee Goodman 1999 ; Goodchild Ogilvie 2006 ; Ogilvie 2008 ; Saini, Gulati Sridhar 2009 ). A particular solution for a global uniformly eccentric (and apsidally aligned) Keplerian disc has also been studied by Statler ( 2001 ). A general secular theory for three-dimensional eccentric discs with arbitrary eccentricities and eccentricity gradients has been developed (Ogilvie 2001 ), and we aim to explore (and test using numerical simulations) some of its implications in this work. Secular theory describes dynamical processes that occur on time-scales that are much longer than the orbital time-scale. In the more familiar context of a system of multiple planets around a star, the Keplerian orbit of each planet is subject to perturbing forces that are proportional to the planet-to-star mass ratio; secular theory accounts for the long-term behaviour of eccentricity and inclination resulting from the orbit-averaged perturbing forces. For a thin gaseous disc around a star, however, pressure provides the most important departure from Keplerian orbital motion, being of second order in the aspect ratio H / R of the disc. Other subdominant forces, such as those due to the self-gravity of the disc (Tremaine 2001 ) and viscous or turbulent stresses (Ogilvie 2001 ), can also be important. In this situation, secular theory describes the long-term evolution of eccentricity due to gas pressure (and other forces) in thin discs, but neglects contributions that are of fourth order or higher in H / R . In recent work, we developed a local model of an eccentric disc (Ogilvie Barker 2014 ), which is similar to the shearing box that is commonly used to study the dynamics of circular astrophysical discs (Goldreich Lynden-Bell 1965 ; Hawley, Gammie Balbus 1995 ). We used this model to analyse the vertical oscillatory flows that are driven by the variation in the vertical gravity around a non-circular orbit (Ogilvie Barker 2014 ), and subsequently studied the local linear stability of these discs and their vertical flows (Barker Ogilvie 2014 ). We found that eccentric discs are generically unstable (in three dimensions), being subject to a small-scale instability in which inertial waves are driven by a parametric resonance. This instability is expected to lead to wave activity or turbulence, and to damping of the disc eccentricity and eccentricity gradients (Papaloizou 2005b

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


; Goicoechea et al. 2017 ). Because it is easily observable at millimeter wavelengths, HCO + is an excellent candidate to investigate the gas dynamics within the cavities of nearby transition disks. Our observations suggest that the HCO + /HCN ratio is a useful chemical tool in determining the structure of transitional disks. In the case of unresolved disks, a high HCO + /HCN intensity ratio might be associated with the existence of inner dust cavities full of dense gas. We note that, together with AB Aur, HD 142527 and AA Tau were the disks with a higher HCO + J = 3–2/HCN J = 3–2 ratio in the compilation of Pacheco-Vázquez et al. ( 2015 ). Both systems show filamentary gas emission through their dust cavities (Casassus et al. 2013 ; Loomis et al. 2017 ). The other disk with a high ratio, GM Aur, has not been studied in detail. This result is also in line with recent results by Alonso-Albi et al. ( 2018 ) toward the Herbig Be star R Mon. 4.3. Gas Kinematics We show in Figure 3 (a) the first moment map of HCO + emission. The outer parts of the disk resemble those of a Keplerian disk. However, twisted isophotes are observed toward the center ( r 06). The aforementioned twisting can be reproduced by two misaligned Keplerian disks as proposed by Hashimoto et al. ( 2011 ). A radial outflow would produce a similar effect (Rosenfeld et al. 2014 ). We adopt the misaligned disks as the most plausible explanation, as it is consistent with near-IR observations by Hashimoto et al. ( 2011 ). In Figure 3 (b), we show the residual map that results from subtracting a two Keplerian disks model from the HCO + first moment map. The model consists of an outer disk extending from 350 to 45 au, with i = 26° and PA = 37°, and an inner one extending from 45 au inward with i = 43° and PA = 65° (Hashimoto et al. 2011 ), around a 2.4 M star, and the final image is convolved with the observational beam. This simplistic model provides a reasonable fit across the disk, except in the southwestern region, where strong residuals are observed. Visual inspection of Figure 3 (a), shows that there is an anomaly (blue circle) in the velocity field that is spatially coincident with the bridge (BR, Figure 3 (b)). This anomaly is due to high-velocity wings in the line profile of HCO + at the bridge position (see Figure 2 (b)). The redshifted peak is compatible with Keplerian rotation, while the blueshifted one requires a different mechanism, most likely infalling gas. To test the possibility that the blueshifted wing is tracing infalling material, we built a pure free-falling velocity field following equationwhere r start is the starting position for free-fall accretion, and we adopted r start = 120 au (accretion starts at the inner edge of the dust ring, r ∼ 07). We assumed that infall stops at 40 au. Cuts in the first-moment map, and in our convolved models are shown in Figure 3 (c). The free-falling model has a velocity of 5.5 km s −1 at the bridge position, compared to the 5.3

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