effect of photoionization feedback may be further reduced for realistic accretion disc spectra, compared to the case of the single power-law spectrum usually assumed in most works (Takeo et al. 2019 ). As mentioned in Section 3 , for accretion rates satisfying the DPT condition, photon trapping due to electron scattering is also naturally realized. In the case of DPT, radiation feedback at radii larger than the dust sublimation radius is effectively suppressed, while the entire accretion disc radiation may be efficiently trapped. Under such extreme trapping conditions, most of the infalling matter may be accreted (Volonteri et al. 2015 ). Since the accretion rate leading to DPT is much larger than the Eddington rate, it should be associated with super-Eddington accretion and a rapid build-up of the BH mass. For instance, adopting the DPT accretion rate as the minimal value, the enhancement factor could be of a few orders of magnitude compared to the Eddington-limited value. It is known that photon trapping also modifies the emergent spectral energy distribution. Higher energy photons originating in the inner accretion flow are more efficiently trapped than lower energy photons produced from the outer regions. As a result, the peak frequency of the spectral energy distribution shifts towards lower energies, and the active galactic nucleus (AGN) spectrum becomes softer with increasing accretion rate (Ohsuga, Mineshige Watarai 2003 ). In the case of DPT, the whole disc radiation can be efficiently trapped, and the overall luminosity output may be considerably reduced, leading to faint AGNs. This may not be the case for photon trapping due to electron scattering, where only the inner parts of the accretion disc remain trapped (unless the accretion rate is extremely high). Since UV/optical photons are trapped within the dusty accretion flow, the central AGN will be obscured. In this context, X-ray photons would be even more readily trapped than UV/optical photons, and thus super-Eddington accretion will be likely obscured at both X-ray and UV/optical wavelengths (Wyithe Loeb 2012 ). In our picture, DPT accretion is indeed associated with significant obscuration, with large column densities expected at early times. Cosmological hydrodynamic simulations also suggest heavy obscuration on different physical scales around the growing BHs in the early Universe (Trebitsch, Volonteri Dubois 2019 ; Ni et al. 2020 ). Assuming a standard radiative efficiency of ε ∼ 0.1, it has been estimated that the obscured fraction at z > 7 could be of the order of (Davies, Hennawi Eilers 2019 ). In fact, much of the early growth of massive BHs could be obscured, and a large number of such obscured quasars may be missed in current optical surveys (Hickox Alexander 2018 ; Inayoshi et al. 2020 , and references therein). On the other hand, obscuration by dusty gas may cause higher energy photons to be reprocessed into lower energy photons, leading to an increase in 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 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 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 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 B-Wavelength 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 B-Wavelength O O O O O O O O O O O B-Wavelength 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 O O O O O O O O O O O O O O O O O O O B-CelestialObject 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-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula 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-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


projected orbital arrangements in motion in Figure 17 . Figure 5. Smoothed representation of the pole vision image of the circularized orbit distribution shown in Figure 4 (f). We subtracted the distribution expected from 39 randomly oriented orbits. Here the 39 orbits are generated, assuming a sin i uniform distribution for the inclinations, and a circular uniform distribution of the longitude of ascending node after setting e = 0 and a = const. As in Figure 4 (f), the black lines indicate the directions of the disks and the white line the direction of the Galactic plane. The two orbital disk systems are well separable (see above) but rather thick. Furthermore, the orthogonal X-shaped disk structure becomes apparent only in alternating zones in the position angle histogram (see Figure 8 ) and in Ω diagrams as well (see Figure 6 ). This leads to the fact that they cannot easily be recognized in polar diagrams as used by, e.g., Gillessen et al. ( 2017 ) in their Figure 12 or Yelda et al. ( 2014 ) in their Figure 21. In Figure 6 , we show the inclination of the stars as a function of the longitude of the ascending node Ω. The color indicates their membership in either the red or the black disk. It becomes clear that the two disks cannot easily be identified as the angular momentum vectors of the disk members point in opposite directions. Comparing Figure 8 and Figure 6 , one can also see that the two disks can be better separated by evaluation in the position angle histogram instead of evaluating the longitude of the ascending node Ω. Figure 6. The inclination as a function of the (LOAN) longitude of the ascending node Ω. However, compared to Figure 6 , the inclined and face-on representations of the disk members as shown in Figure 4 are better grouped together, because the direction of the angular momentum vector is not relevant in this representation. Here, looking at circularized orbits as described above is more successful in searching for face-on orbits that bunch close to the circumference of the sky-projected distributions as in Figures 4 (g) or (h). In Figures 7 (a) and (b), we show the inclinations of the two stellar systems. On the right-hand side of Figure 7 , we show the distribution of inclinations for all stars within the central acrseconds for which we can provide Newtonian orbital fits. In particular, the distributions for the two disks do not follow a sin i distribution as one might have expected for a fully randomized scenario. There is a clear clustering of inclinations around a mean value of ∼90° with the bulk of the higher inclined stellar orbits contained within an interval width of 50° (red disk with bulk between 80° and 130°) or even a width of only 40° (black disk with bulk between 70° and 110°). In comparison to the expected width of about 100° for a sin i distribution, this implies that the two separate disk are highly biased toward high inclinations. There are also no stars with inclinations in the intervals 0°–20°

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


every system, the highest mass is generally achieved when pebble accretion is included. While the systems in which pebble accretion is neglected fail to build super-Earths with our input parameters, the formation of such planets becomes possible when pebble accretion is included. While the number of active embryos decreases when pebble accretion is included (see Fig. 6 ), their masses increase drastically. 4.3 Active number and total mass Figure 6 shows the total number of embryos and the total mass that is in embryos over time for the setups from Figs. 2 – 4 . We also give the fraction of total embryo mass M Emb over the mass that was given to the planetesimaldisk after 1 Myr ( M D ) for each setup. The first embryos always form in the systems in which pebble accretion is enabled. However, the number of active embryos during the simulation is almost a factor of 2 below the number of embryos in the systems without pebble accretion. The mass in embryos differs even more strongly than the active number of embryos for the corresponding systems. The fraction M Emb ∕M D consistentlyincreases for higher total masses and steeper Σ P -profiles, respectively. In the systems in which pebble accretion is included, it can exceed unity. This means that the mass in planetary embryos can be higher than the mass that is transformed into planetesimals through pebble accretion. 4.4 Orbital separation In Fig. 7 we compare the mean orbital separation of embryos over time for the systems from Figs. 2 – 4 . The orbital separation is expressed in units of the embryos Hill radii. The mean orbital separation after 1 Myr converges to ≈ 10R Hill for each setup. The simulations in which pebble accretion is included show a smoother and more stable behavior over time than the systems in which pebble accretion is neglected. The explanation of these differences lies in the fact that the first embryos can start growing farther apart from each other in the runs that only consider planetesimal accretion. Therefore numerous embryos are needed in order to converge for a characteristic orbital Hill spacing. Fig. 2 Time over semimajor-axis evolution of the N -body simulation in LIPAD. The time and location at which an object has first reached lunar mass is indicated by the black dots in the plot. The subsequent growth of the embryo is tracked and connected with the gray lines, and its mass is given by the color bar. The mass in planetesimals after 1 Myr is given by 6 M ⊕ in these runs, the planetesimal surface density slope is varied (Σ P ∝ r −1.0 , Σ P ∝ r −1.5 , and Σ P ∝ r −2.0 ). Left panels : system without pebble accretion. Right panels : system in which pebble accretion is included. The red line indicates the time after which the analytic model presented in Voelkel et al. (2021) states that embryo formation is possible. Fig. 3 Time over semimajor-axis evolution of the N -body simulation in LIPAD. The time and location at which an object has first reached lunar mass 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 B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model 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 B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O


scales linearly with the gas mass density and is proportional to the logarithm of the ratio of the spatial extents of the forming wake and the shock’s stand-off distance. 7.2. Minimum spatial interaction scale The numerical experiments allow us to compute the total dynamical friction acting on the body and also to investigate the individual contributions to the dynamical friction induced as a function of the distance to the perturber. The outcome of this analysis reveals that the region within the stand-off distance of the shock does not contribute to the dynamical friction of a gaseous medium (see Fig. 5 and associated main text for details). At radii smaller than the stand-off distance, a spherically symmetric stratified atmosphere forms around the moving body, which (because of its symmetry) induces no net gravitational pull or pressure force. Only at radii above the stand-off distance does a wake of higher density form behind the moving body and induce a gravitational drag force. Hence, in the well-known formulae by Ruderman Spiegel (1971) or Ostriker (1999) , it is the stand-off distance of the shock that determines the minimum length scale required in the so-called Coulomb logarithm : The formation of a spherically symmetric atmosphere (with radius ) around a gravitating object implies that the hydrodynamical drag force vanishes in contrast to the non-gravitating case, but for very high Mach numbers the shock moves very close to the object so that . In this case, no spherically symmetric envelope around the object forms and the hydrodynamical drag becomes important again. 7.3. Relating the stand-off distance to the accretion radius In a second step, to allow an easy computation of the stand-off distance for general dynamical friction problems, we derived the relation of the shock’s stand-off distance to the common definition of the accretion radius, depending only on the mass and velocity of the perturber. The stand-off distance is directly proportional to the accretion radius and the proportionality constant only depends on the Mach number and the adiabatic index of the gas: Additionally, the geometrical radius of the moving body yields a lower limit for the stand-off distance . Via an analytical estimate of the stand-off distance and further fitting of the remaining free parameters, we gave an approximate relation of the dependence of the shock’s stand-off distance on the adiabatic index of the gaseous medium with and values of and . 7.4. Updated formula of dynamical friction Finally, we combined the derived scaling relations and the finding on the stand-off distance to give an update of the well-known formula of dynamical friction acting on a body of mass moving at supersonic speed through a gaseous homogeneous medium of mass density , where denotes the extent of the wake and can be obtained using the derived formulae above. Appendix A: Hydrostatic equilibrium stratification As mentioned in the text, behind the shock front 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


ejecta and a shock-heated ISM (Park et al. 2003 , 2012 ; Uchida et al. 2015 ). It is also notable that N49 is in an overionized state (Uchida et al. 2015 ) and is interacting with dense molecular clouds on the eastern side (Banas et al. 1997 ; Yamane et al. 2018 ). Due to the interaction with molecular clouds, the thermal X-ray emission of N49 is particularly bright on the southeastern rim. N49 is an attractive object since Kaastra Mewe ( 1995 ) pointed out that the effect of RS can show up if an SNR has such a spherically asymmetric structure. Here, we present a high-resolution X-ray spectroscopy of N49 with the Reflection Grating Spectrometer (RGS) on board XMM-Newton. The obtained G-ratio of O vii He α and other line ratios as well imply a nonnegligible contribution of RS in N49. Throughout the paper, errors are given at a 68% confidence level. We assume the distance to N49 (LMC) to be 50 kpc (Pietrzyński et al. 2013 ). 2. Observation and Data Reduction N49 was observed with the XMM-Newton satellite (Jansen et al. 2001 ) in 2001 (Obs.ID 0113000201) and 2007 (Obs.ID 0505310101). The observation in 2007 was performed with a roll angle which placed N49 and a nearby SNR, N49B, along the dispersion direction of the RGS, making it difficult to extract RGS spectra of N49. We thus analyzed only the data obtained in 2001. For our spectral analysis, we used the RGS (den Herder et al. 2001 ) and the European Photon Imaging Camera MOS (Turner et al. 2001 ) data. We reduced the data using XMM Science Analysis Software version 16.1.0. The RGS data were processed with the RGS pipeline tool rgsproc . To discard periods of background flares, we applied good time intervals based on the count rate in CCD9, which is the closest to the optical axis of the mirror and has the least X-ray counts from the source and most affected by the background flares. The resulting effective exposure time is 11 ks for both RGS1 and RGS2. Since the second-order spectra are of low statistical quality, we analyzed only the first-order spectra. Figure 1. EPIC-MOS image (0.4–8.0 keV) of N49. The cross-dispersion width of the RGS (5 arcmin) is in between the white solid line. The spectral extraction region is enclosed by the white dashed line. N49B is another SNR near N49. 3. Analysis We analyzed the spectra using version 3.04.0 of the SRON SPEX package (Kaastra et al. 1996 ) with the maximum likelihood C-statistic (Cash 1979 ; Kaastra 2017 ). The RGS spectra were fitted simultaneously with those of MOS1 and 2. To account for the spatial broadening of the source, we multiplied spectral models with the SPEX model lpro , to which we input the MOS1 image of the source. Our models have two absorption models: one for the Milky Way and the other for the LMC. The column density of the former was fixed to (Dickey Lockman 1990 ) whereas that of the latter is left free. The elemental abundances for the LMC absorption were fixed to values found in the literature (∼ 0.3 solar; Russell Dopita

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final abundances at some specific values of A that differ by up to two orders of magnitude from the other networks. This shows the importance of using detailed balance to compute inverse rates. We again see good agreement in Figure 9 between the three networks in the proton and helium abundance evolution (left panel of Figure 9 ). But there are a handful of mass numbers at which SkyNet with detailed balance produces final abundances that deviate from the other networks by up to two orders of magnitude (right panel of Figure 9 ). This indicates that it is vital to compute the inverse reaction rates correctly with detailed balance in this scenario. However, the average fractional abundance errors shown in Table 1 are still at the few percent level, and the differences between SkyNet and WinNet or XNet are comparable to the differences between WinNet and XNet. We note that the errors between SkyNet without detailed balance and WinNet or XNet are smaller if screening is turned off. This indicates that screening is somewhat important in this case and we expect the discrepancy between the different codes to increase if screening is turned on due to the different screening implementations. The effect is especially noticeable when comparing SkyNet to XNet with self-heating turned on, because in this case XNet uses the temperature computed from SkyNet, but if screening is switched off, the error between the two codes decreases from 1.1% to 0.086%. SkyNet’s screening implementation is presented in Section 4 . WinNet uses a single fit of the two-body screening factor by Chugunov et al. ( 2007 ) across all screening regimes. XNet computes the two-body screening function provided by Graboske et al. ( 1973 ) for weak and intermediate screening and the one provided by Dewitt Slattery ( 1999 ) for strong screening. XNet then uses a selection rule to select one of the three screening regimes without interpolating between them. 6.2.3. Hydrostatic Carbon–Oxygen Burning In order to compare the self-heating methods implemented in the three networks, we perform a hydrostatic burn at constant density. We keep fixed and start with T = 3 GK and the initial composition consists of half and half (by mass). As a baseline comparison, we also perform non-self-heating runs, where we keep the temperature fixed at 3 GK. The increase in the errors between the networks with self-heating enabled compared to with it disabled must then be due to the difference in the self-heating implementations in the codes. We use a mid-size network containing all nuclides from the full network (Section 6.2.1 ) with . This results in a network with 1530 species and 20,000 reactions. Figure 10 shows the results of this test case with self-heating and screening turned on. We find good qualitative agreement between the three networks, but quantitatively, the differences between XNet and the other two codes are much larger than in the previous test cases. These discrepancies come from the different

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O B-Software O B-Software O O O O O O B-Software O B-Software O O O O O O B-Software 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 O O O 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 B-Software O O O O O B-Software O O O O O O O O O O O O O O O O O B-Software O O O O O O O O B-Software 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-Software 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 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 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 B-Software O O O O O O O O O O O O O O O O O O O O