for excision of data affected by radio frequency interference. After the initial data editing, and gain and bandpass calibration using observations of the compact source 0204+152, a standard self-calibration procedure was carried out for each target source, using line-free channels. The imaging was done with the task tclean , with the w-projection algorithm (Cornwell et al. 2008 ). In the case of J0229+0053, where the observing bandwidth was large, we additionally used the MT-MFS algorithm (second-order expansion; Rau Cornwell 2011 ) to image the target field. The synthesized beams of the final continuum images have full widths at half maximum (FWHMs) of (J0229+0044) and (J0229 + 0053). Both sources were found to be unresolved in our continuum images, with flux densities of mJy (J0229+0044, at ≈640 MHz) and mJy (J0229+0053, at ≈657 MHz). The quoted errors include the error measured from a two-dimensional Gaussian fit to a small region around each source and the typical error in the GMRT flux density scale, ≈10% at these frequencies. The task uvsub was then used to subtract out all detected continuum emission from the self-calibrated visibilities of each field. For each source, a spectral cube was made from these continuum-subtracted visibilities in the barycentric frame, using natural weighting. The H i 21 cm spectrum was extracted by taking a cut through the cube at the AGN location. The final spectrum for each source was then obtained by fitting a second-order polynomial to the line-free regions of each spectrum, and subtracting this out. We further smoothed the spectra by seven channels (J0229+0044) and two channels (J0229+0053), to velocity resolutions of ≈26.6 km s −1 and ≈11.2 km s −1 , respectively. The final optical depth rms noise values are ≈0.12 per 26.6 km s −1 (J0229+0044) and 0.0068 per 11.2 km s −1 (J0229+0053). Our final H i 21 cm spectra toward J0229+0044 and J0229+0053 are shown in Figures 1 (left) and (right), respectively. The velocity-integrated H i 21 cm optical depths are km s −1 (J0229+0044) and km s −1 (J0229+0053). Assuming K and a covering factor, f = 1, these integrated H i 21 cm optical depths imply H i column densities of N hi = (1.35 ± 0.14) × ( T s /100) × 10 22 cm −2 (J0229+0044) and N hi = (1.429 ± 0.015) × ( T s /100) ×10 22 cm −2 (J0229+0053). Note that these are conservative assumptions: a lower covering factor or a higher spin temperature would imply even higher H i column densities. The observational results are summarized in Table 1 . Figure 1. The H i 21 cm spectra toward (left) J0229+0044 and (right) J0229+0053, at velocity resolutions of ≈26.6 km s −1 and ≈11.2 km s −1 , respectively. The solid horizontal line in each panel indicates the AGN continuum flux density at the redshifted H i 21 cm line frequency. The 1 σ rms noise values on the spectra (the horizontal dashed lines) are 0.41 mJy per ≈26.6 km s −1 (J0229+0044) and 0.29 mJy per ≈11.2 km s −1 (J0229+0053). The velocity axes in both panels are


processes. Interestingly, the relation between av Q C and av Q O of the ICME materials is different from that of the carbon-cold materials inside the ICMEs reported by Feng et al. ( 2018 ). The carbon-cold materials are defined by the carbon ionic charge state, which should be relatively lower than the solar wind mean value by three times standard deviation, with the ionic temperature lower than 10 6.05 K. It supports the distinct source or heating process of prominence materials and coronal materials in ICMEs. Further, there are three types of charge distributions inside the studied ICMEs excluding the cold material. Similar results are reported by Rivera et al. ( 2019 ) from their case study, who identified and studied the different types from the different sources in depth. Our statistical results support the stable existence of the different types of charge distributions of heavy ions during most years of a solar cycle. And the fractions of these charge distributions show clear dependence on the solar cycle. This means that the combination of the materials from different sources is common in ICMEs, but the fraction of them would change according to the solar activity. It also implies that the acceleration, the heating process, and the freeze-in height in CMEs may be different between the solar maximum and solar minimum. The source of heavy ions and the heating processes are possibly under modulation by the status of the solar magnetic field. As the most abundant heavy ions in the solar atmosphere (Asplund et al. 2009 ), oxygen ion flux in the solar wind has been systematically studied by von Steiger et al. ( 2010 ). It distributes from 10 4 to 10 6 cm −2 s −1 in the slow solar wind, and from 10 4.5 to 10 5.5 cm −2 s −1 in the fast solar wind. The peaks of the two distributions are a bit less and a bit more than 10 5 cm −2 s −1 , respectively. In the 281 ICMEs, the oxygen flux varies from 2 × 10 4 to 1.5 × 10 5 cm −2 s −1 . And it fluctuates around 10 5 cm −2 s −1 during the solar maximum, when the annual mean SN is more than 60. It could be identified that the oxygen flux in ICMEs is similar to that in the solar wind. Since the SN represents the status of the solar magnetic field, the quasi-linear relation between the O flux in the ICMEs and the SN, when the SN is less than 60, indicates that the O flux is possibly related to the status of the solar magnetic field. 5. Conclusion This statistical work analyzes the abundances and the charge states of heavy ions in the 281 ICMEs from 1998 to 2011. The mean proton speed of the ICMEs and the average charge states of heavy ions are highly related to the solar cycle. Since the SN represents the state of the solar magnetic field by nature, the proton speed and heavy ion composition of ICMEs are possibly related to the solar magnetic field. Another interesting result is the classification effect of the speed on the abundances and charge states of heavy ions. Similar to the solar wind, ICMEs could


one typically utilizes higher resolution grid elements in regions of higher stellar volume emissivity and dust density. Although reasonable to expect a more rapid variation of the radiation field in those regions, there is no guarantee that the spatial resolution is adequate everywhere on the grid. In the absence of iterative procedures to increase the spatial resolution during the RT calculation, the effect of the grid discretization on the numerical accuracy can only be checked by repeating the calculations at progressively higher spatial resolutions. Similarly, the finite number of angular directions of the rays that are cast from each radiation source, as well as the discretization of the scattering source function, also affect the calculation accuracy. Apart from these factors, common to all 3D dust RT codes although in different forms 6 , in DART-Ray the numerical accuracy is also affected by the estimate of the extent of the source influence volumes. This is because DART-Ray calculates the RFED contributions from each source only within this volume surrounding the source itself, thus neglecting the contributions outside it. Since this is a core characteristic of DART-Ray, we are interested in quantifying the accuracy error due to the cut off of the rays when they reach the boundary of the estimated source influence volume. We note that this accuracy error is systematic, since it will always produce RFEDs which are underestimated compared to the correct value. The ray cut off occurs when the RFED contribution carried by a ray satisfy the criterion 1 . So, once a lower limit to the RFED distribution has been estimated, the input-defined parameter is the key factor affecting the numerical accuracy of the calculation. In N14, we stated that should be low enough to preserve energy balance, in the sense that, at the end of the calculation, the total radiation luminosity that has been neglected because of the ray cut-off should be only a small fraction of the total luminosity of the model. In this case, the effect of cutting the rays is minimal because almost all the radiation luminosity has been followed in its propagation throughout the model. However, since this energy balance can only be checked at the end of the RT run, potentially several attempts had to be made to find the appropriate value for . Instead, in the following, we show how the value of the parameter can be set before an RT run such to guarantee the desired level of accuracy. We have been able to find a relation between and the accuracy of the derived RFED distribution by making a minor change in the definition of in formula 1 , compared to NA14. This is now defined as: where is the average specific intensity of the ray within the ray-cell intersection path, is the area of the emitting cell originating the ray, is the solid angle subtended by the intersected cell, is the linear size of the intersected cell, its volume and the light speed. This formula differs only slightly


that the former can be used as a reliable proxy for the latter. The random theoretical 1-NNS distribution derived for an infinite media (see Appendix B for the full development) is given by: where is the mean intensity of the random process (the average surface density of stars) and is the 1-NNS. We want to compare this theoretical distribution to the 1-NNS histogram obtained from 10 000 random Monte Carlo samplings within the finite and irregularly-shaped window (see Fig. 3 ). Their respective moments are similar (see Table 2 ), with theoretical moments computed from the following expressions: where , , , , and are the mean, median, variance, standard deviation, and skewness factor (see Appendix B for their derivation). The moments from random Monte Carlo samplings were obtained from: where is the ordered sequence of the 1-NNS from the smallest to the highest value, the standard errors of the moments and the standard error of the median are taken respectively from Ahn Fessler (2003) and Kendall Stuart (1977) . Fig. 3 Probability distribution function (PDF) of the 1-NNS distributions observed in Taurus (solid blue histogram: within the limited window , that is, the three main filaments; dashed blue histogram: the entire Taurus region) and predicted for a random distribution (black histogram: for a Monte Carlo sampling within the window ; red curve: unlimited theoretical random 1-NNS distribution, see Eq. ( 3 )). Both random populations are drawn using the same mean surface density . Increasing the mean density by a factor of two leads to a shift towards the left whose amplitude is represented by the green arrow. The Monte Carlo sampling in a finite window and the unlimited theoretical random 1-NNS do not significantly differ. The 1-NNS distributions computed either in the whole Taurus or within do not significantly differ either. However, the Taurus and random 1-NNS distributions are statistically highly inconsistent, with a -value of 10 for the KS test. The spatial distribution in Taurus is clearly far from being random. We thus conclude that finite size and edge window effects do not significantly alter the distribution of 1-NNS from that obtained in an infinite medium at the level of our observational uncertainties. To prove this statement quantitatively, we use a bootstrapping technique, comparing 10 000 realizations of a) a sample of 1-NNS obtained from standard rejection sampling technique using analytical random 1-NNS distribution (Eq. ( 3 )) to b) the 1-NNS associated to random Monte Carlo samplings of 252 stars enclosed within . The two-sample Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) statistical tests indicate that the two 1-NNS distributions cannot be statistically distinguished with a mean -value and standard deviation respectively of for the KS test and for the AD test. We have therefore shown that the random theoretical 1-NNS probability density function (Eq. ( 3 )) may be used as a reliable proxy to describe a 1-NNS


in the center of the sheet, and decays toward the rear of the current layer. This transpires from the shape and separation of the red equipotential contours in Figure 1 . Beyond the center, there is more energy in the stress-release motion than in the residual shear component. This means that a Poynting flux, , builds up behind the leading edge of the current front that subsequently converges, thus feeding the electromagnetic energy into the particles by maintaining . Associated with the first increasing and then decreasing stress-release motion is an inertial current, j in , pointing away from the sheet's center line. This component leads to concentrations of the current along the leading and trailing edges of the progressing sheet. Such refinements have not been incorporated in the cartoon of Figure 1 . Underlying this figure is the assumption that the current front propagates in a stationary fashion. In this case, the perpendicular potential drop equals and can be approximated byTwo aspects differ from the mechanical analogy of the fracture process: the magnetic field lines become newly connected, and part of the converted energy remains inside the fracture zone as thermal energy. However, a fraction, say about 10% of the electrons, is running through much of the parallel potential drop without losing its energy to the wave field and escapes along B , carrying perhaps up to 50% of the converted energy out of the sheet. These are the runaway electrons, candidates for producing the hard X-rays in solar flares. Figure 1. Oblique current sheet propagating with normal velocity, v n . Red lines represent equipotential contours for a quasi-stationary situation. The transverse electric field is related to the ongoing reduction of the shear field. The associated inertial current, j in , leads to current concentrations along the leading and trailing edges (not shown). The dashed magnetic field lines indicate the presence of a shear component, B ⊥ , that decays upon transit through the current sheet. An interesting problem is that of the reverse current. The question whether it is cospatial with the flux of runaway electrons has occupied many authors (Holman 1985 ). Emslie Hénoux ( 1995 ), for instance, argue against cospatial return currents. Figure 2 shows that the obliqueness of the current sheet facilitates maintaining charge neutrality. Charge neutrality requires that the flux of high-energy and low-density runaway electrons is balanced by counter-flowing low-energy background electrons. This begins outside the current sheet and thus outside the strong electric field, which would have a repulsive action on the return current, and continues all along the respective magnetic field lines. Because of the transverse propagation of the emitting current sheet, this lasts for a very limited time, τ CS , to be defined in Section 4.3 . It must be noted that the individual fluxes of runaway and reverse current electrons are by orders of magnitude higher


left panel but for galaxies split into redshift-independent mass bins. Bottom left panel: evolution of the logarithmic derivative of the half-mass radius r 0.5 with respect to the mass ratio m m / M s ( m is the stellar mass gained between two consecutive outputs) for non-mergers or m the mass ratio of the merger, for all galaxies with M s 510 9 M and all time outputs between z 5.2 and 1.2. Filled blue symbols indicate resolved mergers, red and light green ones represent non-merger diffuse mass acquisition, with unresolved mergers (in red) or not (in green). Whilst the evolution is linear in each case, the dependence of radius growth on the mass ratio is found to be much steeper for mergers. Bottom right panel: same plot as the top left panel, except that we have split the merger sample according to different pre-merger morphologies: discs (light blue) and spheroids (red). The steepness of the radius versus m relation appears mainly caused by minor merger disruption of the discs. The top panels show the evolution of the average half-mass radius r 0.5 as a function of redshift when we split our galaxy sample in bins of final mass (i.e. galaxy masses at redshift z 1.2; top left panel) and in bins of constant stellar mass (i.e. independent of redshift; bottom right panel). These results are consistent with the overall evolution of the sizemass relationship from observations such as Huertas-Company etal. ( 2013 ) (though our simulated galaxies are a factor of 2 larger): galaxies of a given stellar mass display much larger radii at z 1.2 than their counterparts of similar stellar mass at z 5.2; most of the growth taking place between z 3 and 1.2. More specifically, at z 1.2, galaxies with M s 10 10.5 M display an average half-mass radius two to three times bigger than their counterparts at z 3 (see bottom right panel), while one can see on the bottom left panel that galaxies reaching M s 10 10.5 M at z 1.2 have also seen their half-mass radius grow by a factor of 2 to 3 since z 3 and by a factor of 4 since z 5.2. To better investigate the role of different stellar acquisition processes in growing the size of galaxies, we compute the logarithmic derivative of the half-mass radius r 0.5 with respect to M s as a function of the merger mass ratio m for post-merger structures (bottom left panel, blue curve) and the mass increase ratio m m / M s log 10 M s , where m is the stellar mass accreted between two consecutive outputs, for galaxies only subjected to diffuse stellar accretion (green curve for pure in situ formation and red curve when including possibly unresolved mergers). This is computed for all galaxies and all time outputs between z 5.2 and 1.2 with M s 510 9 M . Note that M s or M 0.5 are equivalent for the purpose of this comparison so we use them interchangeably. The result is shown in the top left panel of Fig. 8 . Note the similarity between results for the pure star formation sample (in green) and the full sample of galaxies


The gray scale is linear from zero to erg cm −2 s −1 sr −1 Å −1 , reversed (black is the maximum intensity) to reveal fainter emission between bright regions. The right panels show the average spectra (black) superposed with the average quiet Sun from the IRIS data studied here in red. Although the R-MHD calculation is for a network region, the region is dominated in area by non-network emission; thus, we compare with the quiet IRIS data. Note that the computed line k 2 peak separations are substantially smaller than observed. Significantly, in comparison with plage observations, these k 3 and K 3 components are far deeper in the computations. This suggests that the source function is on average larger where this feature is formed than is captured in the calculations. In Figure 9 we show spatial autocorrelation data computed from the 3D and 1D intensities of the k line, together with spatially averaged intensity profiles. Solid lines show observations, while dashed lines show computations. The autocorrelation characteristic lengths show three kinds of behavior depending on the wavelengths relative to line center. In the line wings, the images have the smallest spatial scales, with FWHM values of about 750 and 610 km. In 3D, these scales steadily increase toward the line cores, peaking near the observed peaks k 2 V and k 2 R . (In 1D these lengths decrease toward the line center before showing a small increase at k 2 V wavelengths.) Within the line core, the widths computed in 3D drop dramatically to below 700 km. Figure 9. Average profiles (top panel) are shown with autocorrelation full widths for the IRIS network and quiet regions (red and gray, respectively) in the bottom panel. The same data for the 3D and 1D calculations of Sukhorukov Leenaarts ( 2017 ), are also shown as dashed lines. The comparison of 3D and 1D images in Figure 8 is instructive in ways complementary to the interesting points made by Bjørgen et al. ( 2018 ). Figure 9 shows clearly that 3D transfer effects are important within ±1 Å of line center: the FWHM length scales are larger on average by a factor of two for the k line (15%–20% for the K line) in 3D than in 1D. The 3D features all appear “fuzzier” to the eye than their 1D counterparts, on scales of a few hundred kilometers in Figure 8 . We will speculate further on what is missing in these calculations in Section 5 . The second and fourth rows of Figure 8 show how the calculated model might be observed through filter instruments, with FWHM values of 0.1, 1, and 3 Å. Wider FWHM instruments have been used in solar physics for many years to record the Ca ii lines, and other, narrower filter widths have been developed. 11 11 See, e.g., https://www.su.se/isf/ . These panels suggest that 3D smearing effects are biggest at wavelengths within the emission cores. The differences are probably lower limits, given that the source functions for strongly scattering lines are smeared by the use of the short characteristics, which


WD resulting from the merger of two equal-mass WDs by assuming the extreme situation of complete mixing between the two WDs and a CO core for the merged remnant. To this end, we adopted the internal distribution of carbon, oxygen, and He of the 0.58 M ⊙ WD model computed in Miller Bertolami (2016) from the evolutionary history of its progenitor star. Assuming complete mixing between both WDs, the mass fraction of He, carbon and oxygen throughout the interior of our UMCO WD model results X He = 0.017, X C = 0.363 and X O = 0.62. We mention that in this study we are interested in the advanced evolutionary stages of the WD and not the evolutionary stages shortly after the merger event, which cannot be adequately followed by our stellar evolution code. Finally, in this work we did not consider the possibility that WD mergers lead to ultra-massive ONe-core WDs, as predicted in Schwab (2021) . The reason is that we do not expect such ONe WDs to present marked differences in their pulsational properties as compared with the ONe-core models resulting from the single evolution that is explored in the present paper. 4. Evolutionary properties In this section, we compare the evolutionary properties of our UMCO WD models resulting from the two single-evolution scenarios and from the binary merger we described previously, with those predicted by the ONe-core WD models resulting from off-centre carbon burning during the single evolution of the progenitor star ( Siess 2010 ). To isolate the impact of the internal composition predicted by the possible scenarios, we adopted the same stellar mass of 1.159 M ⊙ for all of our WD sequences 4 . In addition, we assumed the same H content for all of our sequences except the merger case. In particular, it was taken as nearly the maximum allowed by evolutionary considerations for such stellar mass, log( M H / M ⋆ )≈ − 6. This limit is imposed by the occurrence of unstable nuclear burning on the cooling track for the WD mass considered. Finally, the ONe-core WD-evolutionary and -pulsational models were taken from Camisassa et al. (2019) and De Gerónimo et al. (2019) . Such models were computed with the same codes we used to compute the evolution and pulsations of our UMCO WD models (see Sect. 2 for details). The evolutions of all of our WD sequences were followed from very high luminosities, through the domain of the pulsating ZZ Ceti stars, down to very low surface luminosities. In Fig. 5 , we display the internal chemical profile of our UMCO WD models (see Sect. 3 for details), together with the chemical profile of the ONe-core WD model studied in Camisassa et al. (2019) , top panel. Specifically, the second and third panels illustrate, respectively, the chemical profiles resulting from reducing the mass-loss rates of an initially 7.8 M ⊙ progenitor star, and those from considering core rotation ( f = 0.1) in the AGB phase of an initially 7.6 M ⊙ progenitor star. The bottom panel shows the chemical profile we


counts and fluxes from the eregionanalyse task. They have less than 35 counts in the full band; for three of them, we even only get upper limits. The EPIC net source counts and fluxes in the full (0.2–10 keV) energy range (0.2–0.5, 0.5–2.0, 2.0–5.0, and 5.0–10.0 keV energy ranges obtained for these quasars) are summarized in Table 3 . The full-band XMM 100 arcsec EPIC mosaics of these six quasars are shown in Fig. 1 (rows 2 and 3). Table 3. X-ray net counts and fluxes for the new z > 6.4 X-ray quasars. Note . The upper limits quoted in this table correspond to the 3σ upper limits and are used only if the net counts or fluxes are smaller than their 3σ error. But only the quasars with a given value in the ‘Det.’ column have an X-ray detection. For the two X-ray detected quasars in this work, the source counts and fluxes presented in Tables 2 and 3 are from the edetect_chain task. 4 DISCUSSION For the full sample of nine quasars, we derived a set of optical and X-ray properties (see Table 4 ): The absolute magnitude at the rest-frame wavelength of λ = 1450 Å ( M 1450 ) was computed by extrapolating the J -band magnitude to the monochromatic magnitude at ( m 1450 ) assuming a power-law fit of the continuum in the UV–optical with α ν = −0.3 (Selsing et al. 2016 ). The monochromatic flux at the rest-frame wavelength of () was derived from m 1450 assuming also a power-law slope of α ν = −0.3. The X-ray luminosity in the rest-frame 2–10 keV band was computed from the count rate obtained by the eregionanalyse task on the 2–10 keV image (as done in Section 2 for the other energy ranges) and then K -correction was applied assuming a photon index Γ = 1.9. For the two detected quasars VDES J0224−4711 and PSO J159.2257−02.5438, the flux was obtained directly from their X-ray spectrum. Upper limits are used if the luminosity is smaller than its 3σ error (which is the case for all the non-detected sources) and are at the 3σ level. The hardness ratio is defined as HR = ( H − S )/( H + S ), where S and H are the number of counts in the soft (0.5–2.0 keV) and hard (2.0–5.0 keV) bands, respectively. The optical-to-X-ray slope (α ox ) is defined as where is the monochromatic flux at the rest-frame wavelength of as defined above, and the rest-frame monochromatic flux at 2 keV () was inferred from the flux in the 0.2–10 keV range assuming a photon index Γ = 1.9. Upper limits for α ox are used if the flux at 2 keV is smaller than its 3σ error. Table 4. Summary of X-ray and optical properties of the DES and new z > 6.4 X-ray quasars. Note . a Reminder: VDES J0020−3653 and PSO J036+03 have a measurement of α ox with errors because their 2 keV flux (from the 0.2–10 keV flux) is at a 3σ significance, but they are not significantly (>4σ) detected as VDES J0224−4711 and PSO J159−02. We used the compilation of X-ray observations of quasars at z > 5.5 by Nanni et al. ( 2017 ) plus the new analysis by Nanni et al. ( 2018 ) for SDSS J1030+0524 ( z = 6.28) and the observation


wings. If we compare epochs I and II, we can see that the magnified blue wing fades while the red wing enhances. This supports that the separation between the approaching and receding parts of the microlensed region of the BLR is of about a few light-days. In several cases, mainly affecting the shelf-like feature at ∼ λ 1610 and several blends of Fe ii and Fe iii , the impact of microlensing and microlensing variability (see Tables 2 – 5 ) is mag, comparable to the typical microlensing magnification amplitudes observed in the X-ray continuum. It is important to notice that the spectral features mainly affected by intrinsic variability are also the C iv wings, some blends of Fe ii and Fe iii , and the shelf-like feature at ∼ λ 1610. In contrast with microlensing, we have not observed marked asymmetries induced by intrinsic variability in the line profiles. This result supports (within the statistical significance of our relatively limited sample) the hypothesis that the small region sensitive to microlensing is not systematically affected by extinction, beaming, or any other mechanism that may selectively enhance one part of it. Consequently, the asymmetric enhancements observed, in either the blue or red parts of microlensed line profiles, likely originate from an anisotropic distribution of microlensing magnification in the source plane. 3. Discussion 3.1. Microlensing and Kinematics The main result of the previous sections is that the cores of C iv , C iii ], and Mg ii and the wings of the last two lines are only weakly affected by microlensing. However, the wings of C iv can be strongly affected. In other high-ionization emission-line features, such as the Fe iii λλ 2039–2113 blend and the shelf-like feature at ∼ λ 1610, the whole feature may be globally affected by high-magnification microlensing. To discuss these results with more detailed kinematic information than the separation between core and wings, in Figure 6 we have overlapped the averaged line profiles corresponding to C iv , C iii ], Mg ii , and Fe iii λλ 2039–2113. The wings of Mg ii correspond to relatively low velocities as compared with other lines. This is consistent with the weak impact of microlensing on this line. Regarding C iii ] and C iv , the red parts of these lines match very well, except at the lowest intensity level, where the shelf-like feature at ∼ λ 1610 is present in C iv (the blue side of C iii ] is blended with Al iii , Si iii , and Fe ii , and no reasonable comparison can be made). In spite of this remarkable kinematic coincidence, the C iv line can be strongly affected by microlensing, whereas C iii ] seems to be rather insensitive to this effect (see, e.g., the cases of SDSS J1004+4112 and SDSS J1339+1310). In addition, they have a different degree of ionization (C iv is of high ionization and C iii ] of low ionization). These results suggest that both lines are mostly generated in the same region, but that there is also a contribution, exclusive to the


red giant clump (RGC), with its known de-reddened color and magnitude, in the CMD serves as a reference for the color and magnitude calibration. In the first step of the method, we estimated the combined (instrumental) flux from the source stars, F S, p = F S 1 , p + F S 2 , p , and the companion/primary flux ratio, q F, p , by fitting the KMTC photometry data set processed using the pyDIA code. Here the subscript “ p ” denotes the passband of observation. The measured values are ( F S, I , F S, V ) = (0.07751 ± 0.00060, 0.00909 ± 0.00032) and ( q F, I , q F, V ) = (0.0471 ± 0.0023, 0.0110 ± 0.0084). We note that the value of q F, I , which is derived from the pyDIA reduction, is slightly different (less than 1 σ ) from the value presented in Table 3 , which is derived from the pySIS reduction, because they come from different reductions of the data. Then, the flux values from S 1 and S 2 were estimated by respectively. From the measured flux values in the I - and V -bands, we estimated the colors and magnitudes of S 1 and S 2 to be ( V − I , I ) S 1 = (2.289 ± 0.043, 20.827 ± 0.010) and ( V − I , I ) S 2 = (3.868 ± 0.831, 24.143 ± 0.053), respectively. In Fig. 9 , we mark the locations of S 1 and S 2 with respect to the RGC centroid in the instrumental CMD that was constructed using the pyDIA photometry of the KMTC data (gray dots). Fig. 9. Locations of the primary and companion source stars with respect to the centroid of the RGC in the instrumental CMD (gray dots). The determinations of the source positions and the construction of the CMD are based on the pyDIA photometry of the KMTC data set. We also present the Hubble Space Telescope CMD ( Holtzman et al. 1998 , brown dots) to show the source locations on the main-sequence branch. Although the V -band flux of S 2 has a relatively large fractional (magnitude) error, it is strongly constrained to be faint in an absolute sense. In particular, we find (i.e., a 4.4 σ difference). This serves as a second line of evidence that the 2L2S solution is correct. If, for example, we had somehow missed a 3L1S solution, and the derived 2L2S solutions were merely mimicking it, then q F, I − q F, V should be consistent with zero because there is only one source with just one color. However, according to Eq. ( 4 ), this possibility is excluded at 4.4 σ . In the second step, we calibrated the color and magnitude. With the measured instrumental color and brightness of the source, ( V − I , I ), and the RGC centroid, ( V − I , I ) RGC = (2.473, 16.918), together with the known de-reddened values of the RGC centroid, ( V − I , I ) RGC, 0 = (1.060, 14.509) ( Bensby et al. 2013 ; Nataf et al. 2013 ), the reddening and extinction corrected color and magnitude of the source were estimated by where Δ( V − I , I ) denote the offsets in the color and brightness of the source from the RGC centroid measured on the instrumental CMD. This results in In Table 5 , we summarize the values of ( V − I , I ) RGC , ( V − I , I


HWHM versus albedo. The modeled HWHM for 67P is located close to satellites of Mars, Phobos and Deimos, suggesting a similarity in the optical properties and structure of their surfaces, such as size of particles and porosity. The variation of the enhancement factor, against geometric albedo does not follow a recognizable trend (lower panel of Fig. 5 ). However, it shows two collections of data points, the values of the enhancement factor for all bright bodies studied in Rosenbush et al. (2002) are lower than 1.6, while for all dark objects they do not go beyond 1.8. It is notable that Phobos and Deimos including 67P do not resemble any group of data points. Another comparative study can be done for 67P and existing data for Centaurs and Kuiper belts objects (KBOs) ( Belskaya et al. 2008 ). Since these bodies are observed in the small phase angle range, for the comparison we only adopt the local phase curve of 67P at nm in the same phase angle range as those objects. The best-fit value of the phase coefficient in mag/deg for 67P is plotted together with the measured phase coefficients of KBOs and Centaurs (Table 1 of Belskaya et al. 2008 and references therein) with respect to the albedo (see Fig. 6 ). Fig. 6 Phase coefficient versus albedo for Centaurs, classical Kuiper belt objects (KBOs), and trans-Neptunian objects in band together with comet 67P/CG at nm. Figure 6 shows that 67P with the phase coefficient, , is located in Centaurs regions. The two particular Centaurs with similar phase coefficient to 67P are (54 598) Bienor and 1999 TD10. Centaurs are believed to have a greater steepness in their phase curve than other solar system bodies ( Bauer et al. 2003 ; Rousselot et al. 2005 ). 5.2. Comparison with laboratory measurements of the phase curve at small phase angles We also compared our results with laboratory measurements that study the dependence of the photometric characteristics of particulate surfaces on the surface properties, that is, albedo, grain size distribution, and porosity. There are not many experimental studies at very small phase angles. The measurements that are made in the opposition-effect region were carried out by Jost et al. (2016) for icy particles with a high-albedo close to unity and by Psarev et al. (2007) for low- and high-albedo samples. We found that the laboratory reflectance data from carbon soot at the wavelength of 630 nm in the phase angle range of 0.008–1.5 ( Psarev et al. 2007 ) show a behavior that is very similar to the measured local phase curve from the Imhotep-Ash region of 67P at nm. The local phase curve we extracted was normalized at to make it comparable with normalized laboratory measurements (see Fig. 7 ). In Fig. 7 two samples, the ultra-dispersed quartz and the carbon soot, behave similarly for the phase angles >0.2 (the limit we determined for the local phase curve of 67P from OSIRIS close-flyby images). However, light scattering by quartz sample is defined by its very special


taken with the Broadband Filter Imager, and the chromosphere, using the H α 6563 Å line obtained with the Visible Imaging Spectrometer (VIS). The passband of the TiO filter is 10 Å, centered at 7057 Å, while its temporal resolution is about 15 s with a pixel scale of 00343. Concerning VIS, a combination of 5 Å interference filter and a Fabry–Pérot etalon is used to get a bandpass of 0.07 Å at the H α line. Its field of view (FOV) is about 70′′ with a pixel scale of 00323. To obtain more spectral information, we scan the H α line at 11 positions with a 0.2 Å step following this sequence: ±1.0, ±0.8, ±0.6, ±0.4, ±0.2, 0.0 Å. The time used in each scan is 23 s. Both high-order adaptive optics (Cao et al. 2010 ) and a speckle reconstruction postprocessing method (KISIP speckle reconstruction code; Wöger von der Lühe 2007 ) are employed to achieve diffraction-limited resolution images. IRIS data are taken in a very large, coarse 64-step raster mode, with a step size of 035, step cadence of 31.8 s, and a raster cadence of 2036 s. The exposure time at each step is 29 s. Several strong emission lines are recorded, such as Si iv 1402.77 Å, Si iv 1393.7 Å, C ii 1334.53 Å, and Mg ii k 2796.35 Å. The cadences of the SJI in the passbands of 1400, 2796, and 2832 Å are all 112 s. The spatial resolution is about 033, and the FOV is 224 × 175′′ and 175′′ × 175′′ corresponding to the 64-step raster mode and the SJI. All of the spectra and images used in the analysis are level 2 data, including dark current subtraction and flat-field and geometrical corrections (De Pontieu et al. 2014 ; Tian et al. 2014 ). Hinode/SP data are acquired in a fast mode with an FOV of 81′′ × 81′′, a step size of 0297, and a raster cadence of 1080 s. Additionally, in order to analyze the corona response to our event, we have used SDO data, including the EUV and UV passbands from the AIA (Lemen et al. 2012 ) with cadences of 12 s and 24 s, respectively. The spatial register between the GST and SDO is first done by coaligning the AIA 1600 Å images with the GST/TiO data, and then coaligning the TiO data with the H α images at different wavelength positions. We use the AIA 1600 Å, IRIS 2832 Å, and SP continuum images to align the SDO , IRIS, and SP data. The uncertainty of the coalignment is about one AIA pixel (06). Figure 1 contains the context image of the LBs and fan-shaped jets on the sunspot of AR 12127 that we analyze in the subsequent section. The figure shows data from GST, IRIS, and SDO /AIA after the alignment process. Figure 1. Context images of the LBs and fan-shaped jets on the sunspot of AR 12127 obtained on 2014 August 1 around 17:27 UT. The different panels show (a) the photosphere through TiO images from GST, (b) the chromosphere via Ha LOS velocity obtained with GST/VIS, (c) the TR through SJI 1400 Å images from IRIS, and (d)–(i) the corona response observed with the different filters of SDO /AIA. R1 illustrates the LB region, to be analyzed in Figure 3 . L1 represents


the outermost regions of clusters. For example, Fig. 8 shows a snapshot of the plane ( J z , R ) for model KF05, with the potential escapers highlighted in green. It is evident that almost all of the outer cluster stars beyond approximately 0.5 r J are potential escapers. This result is in general agreement with the findings of Küpper et al. ( 2010 ), who first showed that the behaviour of the surface brightness and velocity dispersion radial profiles are significantly affected by the presence of an outer population of potential escapers. Figure 8. The same as the last panel of Fig. 5 , but with the unbound particles highlighted in green. The fraction of unbound particles as a function of the fraction of the mass remaining in the cluster for all our locked models is illustrated in Fig. 9 . As discussed in Fukushige Heggie ( 2000 ) and Baumgardt ( 2001 ), initial conditions set by sampling, for example, a King model may include a fraction of stars which are unbound when an external tidal field is added. This is the case for our KF1 model. For this model, we show the evolution of the fraction of potential escapers both with and without this population of primordial potential escapers. All the other models do not have any primordial potential escapers. Figure 9. The fraction of unbound particles as a function of fraction of mass remaining in the cluster for all locked models. For KF1, there are two separate lines for including primordially unbound particles and excluding them. Fig. 9 also shows that the fraction of potential escapers increases significantly over time reaching a fraction of the remaining mass equal to about 15 per cent by the end of our simulations. Such a behaviour is qualitatively consistent with the findings reported by Baumgardt ( 2001 ), in which tidally perturbed models (initialised as critically filling King model W 0 = 3, i.e. characterized by a much higher fraction of primordial potential escapers, about 15 per cent) possess approximately 20 per cent of potential escapers during the final stages of evolution. We have examined the evolution of the fraction of prograde and retrograde particles within the populations of bound and unbound particles in models KF1 and KF025, as representative of the two filling regimes (see Figs 10 and 11 , respectively). The larger fraction of retrograde particles in the population of unbound particles is the result of the two effects discussed in Section 3.2 : the preferential escape of prograde particles and the fact that, in the transition from bound to unbound, some particles move on more eccentric/radial orbits becoming retrograde as they move outwards. Figure 10. The time evolution of model KF1 of the fraction of retrograde stars within the entire system, and within energetically bound and unbound particles. The bottom panel is a zoomed in version. The red line shows the fraction of unbound particles that do not include the primordially unbound stars. Note that the black solid lines


with ) based on their line properties alone. We compare the results from the simulations to what is expected when using the negative image FDR as reference. For the latter, we estimate the probability of an emission line with of being false by taking , where is the number of emission lines with found in the negative data, while is the number found in the positive (“real”) data. This means that for equal numbers, there is a 100% probability that the line is produced by noise. Alternatively, if there there are twice as many positive lines as negative ones, we assume that there is a 50% probability that each positive line is real. A comparison between simulations and negative image FDR probabilities for each cluster is shown in Fig. 4 . We see that using the negative image FDR as a reference for estimating the probability of a line being false returns a much noisier and sharper probability distribution for all of the clusters. At a given S/N value, it can strongly under- or overestimate the probability since it is dominated by small number statistics. Using simulations allows us to more accurately assign probabilities and understand the true detection rate at low . To confirm weaker real emission lines, we investigate the NIR and 1.1 mm counterparts of all of our candidates. Fig. 4 Probability of an emission line with a given of being produced by noise for the five clusters. Solid curves correspond to the probabilities given by the simulations, while dashed curves correspond to the probabilities estimated from the negative data. 3.3. Association with optical/NIR counterparts Clearly galaxies with high SFRs are likely to be detectable both with HST and/or ALMA in continuum and in molecular emission lines. Because of this, we expect that an emission line candidate co-spatial to a NIR detected galaxy should increase the probability of this candidate being real and thus use the NIR images for this purpose. However, the lack of a NIR or 1.1 mm counterpart does not necessarily imply that the emission line is noise. A galaxy that remains undetected both in HST/160 and ALMA continuum at could still have a at and at , based on spectral energy distribution fitting. In such cases, it should still be possible to detect CO, [NII], or [CII]lines. For instance, we find that CO can be detected at from CO lines when a high CO excitation level () is present in combination with strong dust extinction (). Detections of [NII]and [CII]lines can be made to lower SFRs and extinctions () at . Thus, some weak line candidates that lack NIR counterparts could still be real. Unfortunately, in such cases, confirmation and identification of line candidates will only be possible with deeper mm observations, which is beyond the scope of this study. Thus, we restrict our analysis of weak emission line candidates to those with nearby optical/NIR detected sources. Based on the number and spatial distribution of NIR detections in the FFs, a given random position within any of


a L ~10 3 L ⊙ low-mass YSO cluster (e.g. Estalella et al. 1993 ; Quanz et al. 2007 ). Four of the YSOs, namely VLA1-4, are known to be associated with thermal radio jets ( Anglada et al. 1994 ). Based on the previous, lower angular-resolution observations of C 18 O (1–0), Umemoto et al. (1999) estimated the enclosed gas mass over the ~ 0.5-pc-scale parent molecular gas structure to be ~120 M ⊙ . Contrasting to the aforementioned OB cluster-forming regions, the bolometric luminosity of L1287 is not contributed by the nuclear burning of OB stars, but instead is contributed by the accretion luminosity of the low-mass protostars, including two known FU Orionis objects 1 RNO 1B/1C( Staude Neckel 1991 ; Kenyon et al. 1993 ; McMuldroch et al. 1995 ; Quanz et al. 2007 ), and a L ~ 600 L ⊙ Class 0/I YSO IRAS 00338 + 6312 ( Anglada et al. 1994 ). Earlier observations resolved bipolar CO outflow(s) aligning in the NE-SW direction ( Snell et al. 1990 ; Yang et al. 1991 ; Fehér et al. 2017 ), although the identification of the powering source(s) remains uncertain. Table 1 SMA observations summary. In this paper, we report the ~ 1′′ (~ 930 AU)-angular-resolution observations with the Submillimeter Array (SMA) 2 , covering the inner ~0.25-pc region of L1287. By examining the resolved 1.3 mm dust continuum emission and various molecular line tracers of dense gas, warm gas, shocks, and outflows, our aim is to elucidate how the gas flows converge from parsec scales (e.g., due to gravitational collapse) down to ~ 1000 AU scales, and how they are dynamically forming the low-mass YSO cluster. We also provide a more definite identification of the powering source of the previously reported molecular outflows. The details of our observations are presented in Sect. 2 . The observational results from the dust continuum and molecular line emission are given in Sect. 3 . In Sect. 4 we present an analysis of fragmentation, dense gas kinematics, and powering sources of the previously reported CO outflows. Our conclusions are given in Sect. 5 . 2 Observations and data reduction 2.1 Submillimeter Array observations We performed five tracks of SMA observations at ~1.3 mm wavelengths towards the inner ~0.25-pc region of L1287 between August 2013 and July 2014, which are summarized in Table 1 . The pointing and phase referencing centers of the observationsare RA(J2000) = 00 h 36 m 46.65 s , Dec(J2000) = + 63°28′57.90′′. The projected baselines of these observations range from ~4 to ~ 140 k λ . The system temperature T sys for all the observations was around 150 K. The observations were obtained with the 230 GHz receiver, with 4 GHz bandwidth in each of the upper and lower sidebands. The correlator consisted of 48 chunks, each with a bandwidth of 104 MHz. The frequency was centered at 231.3 GHz at chunk 35 of the upper sideband (USB). This configured the correlator with the lower sideband (LSB) covering from 216.48 to 220.46 GHz and the USB from 228.46 to 232.43 GHz. These


misalignment angles in 3C sources are known to extend up to values of about 12° or more ( Leahy Williams 1984 ). We thus consider the interpretation of the S5 0836+710 large-scale morphology as an FR II radio galaxy at a small viewing angle as realistic. 4.3. Derivation of the jet parameters The total jet power for a relativistic jet is parameterized as (see, e.g., Perucho et al. 2017 ): where is the jet specific enthalpy, Γ j is the jet Lorentz factor, ρ j is the jet rest mass density, v j is the jet velocity, c is the speed of light, B φ is the toroidal field in the observer’s frame, γ j is the ratio of specific heats of the jet gas, and A j is the jet cross-section. The first term in Eq. ( 2 ) includes the kinetic, internal, and rest-mass energy contributions, and the second term stands for the magnetic energy of the jet. We can assume (see Appendix B ) that the jet is kinetically dominated and in the cold regime (i.e., its magnetosonic Mach number is high). Furthermore, taking into account that the advance speed as measured by LOFAR is also so slow, the velocity of the bulk plasma flowing into the hotspot can be assumed to be only mildly relativistic at most (see also Appendix C for the relativistic derivation). Under these conditions, the magnetic and pressure terms can be neglected and Eq. ( 2 ) simplifies to Across a strong shock, the hotspot pressure is (e.g., Landau Lifshitz 1987 ) In this equation, we need to define the jet radius at the hotspot, R j, h , which can be approximated as the hotspot radius. Because we do not know whether the southern radio structure includes the hotspot and part of the lobe or if it is the hotspot, we considered half of the whole region as the jet radius (4.5 kpc) and half of the fitted component, which represents the brightest region within the hypothetical lobe (2.5 kpc). Although the polarization seems to favor the latter interpretation (see the previous section), we study both cases here. Furthermore, we need an estimate for v j at the hotspot, v j, h . As we show in the following paragraphs, any reasonable input value is sufficient because we define an iterative method to derive its convergence value. When the hotspot pressure is obtained, we can use equipartition between the nonthermal particles and the magnetic field, as reported for FR II hotspots (see e.g., Hardcastle Worrall 2000 ), to obtain a value of the magnetic field at the interaction site. The value of the magnetic field prior to the reverse shock can be constrained by assuming conservation of the magnetic flux from the 1.6 GHz jet to the interaction site 3 . Applying the magnetohydrodynamcis (MHD) jump conditions at the reverse shock that give rise to the hotspot, allows us to derive a new estimate for v j, h . Setting a convergence criterion for this parameter at a precision of 10 −3 , we can determine the relevant parameters of the problem. We applied this method to four different sets of hotspot velocity and radius: the possible


different cell sizes in the CMDs, they are fixed each time, i.e. they do not vary across the CMDs. From our experience in cleaning the field star contamination in the cluster CMDs, we have identified some situations which still need our attention. It frequently happens that some parts of the CMDs are more populated than others, so that fixing the size of the cells in the CMDs becomes a difficult task. Small cells do not usually carry out a satisfactory job in CMD regions with a scarce number of field stars, while big cells fail in populous CMD regions. Thus, relatively bright field red giants with small photometric errors could not be subtracted and, consequently, the cluster CMD could show spurious red giant features. A compromise between minimizing the residuals left after the subtractions of field stars from the cluster CMDs and maximizing the cleaning of field stars is always desirable. In this paper, we present a comprehensive multiband photometric analysis of Haffner9 from BVI and Washington CT 1 photometry. The cluster has been previously studied from 2MASS and BVI photometry. However, a relatively lower photometric accuracy and shallower limited magnitude, in combination with a not much pinpointed treatment of field decontamination led those studies to discordant results. In Section 2 , we describe the collection and reduction of the available photometric data and their thorough treatment in order to build an extensive and reliable data set. The cluster structural and fundamental parameters are derived from star counts and CMD and colourcolour (CC) diagrams, respectively, as described in Section 3 . The analysis of the results of the different astrophysical parameters obtained is carried out in Section 4 , where implications about the stage of its dynamical evolution are suggested. Finally, Section 5 summarizes the main conclusion of this work. 2 DATA COLLECTION AND REDUCTION We used CCD images 1 obtained on the nights of 2004 December 18 and 19 with a 2048 2048pixel Tektronix CCD attached to the 0.9-m telescope (scale 0.396 arcsecpixel 1 ) at Cerro Tololo Inter-American Observatory (CTIO, Chile). Its field of view (FOV) is 13.6 13.6arcmin 2 . We used the Washington C (Canterna 1976 ) and KronCousins R filters. The latter has a significantly higher throughput as compared with the standard Washington T 1 filter so that R magnitudes can be accurately transformed to yield T 1 magnitudes Geisler ( 1996 ). We used a series of bias, dome and sky flat-field exposures per filter to calibrate the CCD instrumental signature. We also utilized images for the Small Magellanic Cloud cluster Lindsay106, which was previously observed at La Silla (ESO, Chile) with the C and T 1 filters (Piatti etal. 2007 ). Lindsay106 was used here only as control cluster, i.e. to verify the quality of the present CTIO photometry. Table 1 shows the log of the observations with filters, exposure times, airmasses and seeing estimates. A large number (typically 20) of


and the lack of a large additional population of variable sources beyond those identified with the CRATES catalogue. 4.4 Differential source counts Differential source counts have been intensively studied (e.g. de Zotti et al. 2005 ; de Zotti et al. 2010 ; AMI Consortium et al. 2011 ), having important implications for models of cosmic evolution (e.g. Condon 1984 ; Massardi et al. 2010 ) and provided the first clear evidence against the Steady State Theory (Pooley Ryle 1968 ). However, for the purposes of C-BASS analysis, we calculate the differential source counts to provide an additional cross-check on the statistical properties of our bright-source population. Given that we are concerned with the statistical properties of our catalogue at flux-densities , where the C-BASS catalogue is largely complete, we do not consider correction factors to the differential source counts necessary at lower flux densities to account for catalogue incompleteness and varying flux-density sensitivities across the sky. Sources are put into flux-density bins of width . For each bin, source counts are determined against a central flux density, S c , taken to be the geometric mean of flux densities in that bin. The differential source counts for a given flux-density bin are then given by where N c is the number of sources in the flux-density bin, A is the sky area to which the catalogue corresponds, and Δ S is the bin width. Errors are estimated for the source counts by assuming Poisson statistics. We only consider bins with N c ≥ 9 so as to maintain a SNR ≥ 3 for each bin. The differential source counts for C-BASS North in total intensity, applying the CG30 mask, are plotted in Fig. 12 , alongside the differential source counts for the GB6 catalogue. We can see that at low flux densities the C-BASS source counts are significantly lower. This is to be expected from the increasing catalogue incompleteness at these lower flux densities. From the differential source counts, we can again see the catalogue completeness level from the point at which the source count turn-off ends and we have good agreement between the C-BASS and GB6 counts, i.e. over flux densities . It is worth noting here that the errors obtained from assuming Poisson statistics are likely to be somewhat optimistic. These do not account for false positives in the source catalogues, or errors arising from incorrect binning due to uncertain flux-density estimates. These effects become significant at lower flux densities, where the proportion of false positives in the catalogue is greater, and our photometry methods become less reliable. Figure 12. Differential source counts calculated using the C-BASS northern sky total intensity catalogue, plotted alongside the differential source counts for the GB6 catalogue. Source counts were calculated applying the CG30 mask. We also plot as grey-scale the fitted model source counts corresponding to 100 posterior samples for the C-BASS and GB6 models. At low-flux


generating these artificial leakage parameters, it was ensured that no off-diagonal elements of the Mueller matrix exceeded 1.5 conversion between Stokes parameters at any frequency channel, while reaching this maximum percentage in at least one frequency channel. This limit was chosen based on the following test. From our sample, we chose all pulsars that did not show any RM(φ) variations (e.g. J1048–5832 and J1709–4429). Our assumption was that the imperfections of the receiver could not generate more RM(φ) variations than were already observed. The limit was therefore chosen as the maximum value that did not create additional apparent variations in RM(φ). The systematic uncertainties because of receiver imperfections were quantified for each pulsar by randomly generating 100 receiver imperfections obeying the above description, which then were used to distort the pulsar signal in the process of polarization calibration. 3 For each of these 100 different distortions, we calculated values for RM profile , RM scatt , RM(φ), and Δ( V / I )(φ). The standard deviation of these values was quoted as the systematic uncertainty in Table 1 . The systematic uncertainties determined for the RM(φ) and Δ( V / I ) are displayed for each pulsar, for example in Fig. 1 , in the third and fourth panels as a 1σ contour grey-shaded region overplotted over the RM(φ) values (green in the online version). 4 RESULTS Pulsars described in Sections 4.1 and 4.2 and in Table 1 , for which the phase-resolved RM profile had never been investigated previously in the literature, are marked with an asterisk (*). The plots for the 98 pulsars have been included in the online supplementary material , in Figs A1–A26 and an example is displayed in Fig. 1 . All plots were aligned, so that the total intensity peaked at pulse longitude 180°. For the six pulsars from the sample, which had both a main pulse (MP) and an interpulse (IP), we aligned the MP peak at pulse longitude 90°, and hence the IP peaked around pulse longitude 270°. The results from the analysis described in the Section 3 can be found in Table 1 . The resulting RM(φ) profiles allowed us to classify the pulsars as follows. Pulsars that had were classified as showing significant RM(φ) variations, and they are discussed in more detail, on a case to case basis, in Section 4.1 . Six pulsars, which were initially classified as showing significant RM(φ) variations, were removed from this section based on their high systematic uncertainties. A total of 42 pulsars, of our sample of 98, were classified as showing significant RM(φ) variations. For all cases where we saw RM(φ) variations, we observed deviations from the expected λ 2 dependence (as quantified by , but also by inspecting PA versus λ 2 directly), implying that Faraday law fails to describe the full frequency dependence of the PA, and there must be another frequency and pulse longitude dependent effect present. Therefore, the results obtained from the panel, where


structure than in the spot. Moreover, the smearing due to the point-spread function of the telescope may dilute into the bright structure part of the magnetic signal belonging to the surrounding umbra and penumbral filaments. However, the average estimated errors for the vertical and horizontal component of the field in the structure are about ±35 G and ±70 G, respectively. Thus, the strong horizontal field of about 2000 G appears to be a robust result. The presence of such a significant horizontal component of the magnetic field allows us to exclude the field-free configuration typically observed in LBs. Conversely, it supports the possibility that the magnetic field of the flux rope forming the filaments around the sunspot may have a counterpart in the middle of the sunspot umbra, where it appears to end in the cospatial chromospheric images. This would also provide an explanation for the detection of opposite polarity in the structure with respect to the umbra, as well as an alternative interpretation for the opposite polarity of an apparently filamentary LB with respect to its hosting umbra observed by Bharti et al. ( 2007 ). Note that this phenomenon is different from the magnetic reversals found by Lagg et al. ( 2014 ) and Felipe et al. ( 2016 ) in LBs, due to the bending of the field lines along the atmospheric stratification of the structure. Moreover, this scenario is confirmed by the portions of the bright feature characterized by both upward and downward plasma motions along it, which are compatible with plasma flows along the helical field lines of a flux rope. Actually, the length of each portion of the filamentary structure with coherent plasma motion is of about 5′′, which could correspond to the pitch of a helically twisted flux tube. Therefore, we guess that this bright filamentary structure is not a filamentary LB, but it is due to the accumulation of plasma coming from higher solar atmospheric levels into the photosphere. The presence of stably inclined fields belonging to the flux rope, which are touching part of the sunspot umbra, would set in radiatively driven penumbral magneto-convective mode (Jurčák et al. 2014 , 2017 ). Indeed, high-resolution observations have shown that plasma fallen from chromospheric layers can lead to the formation of penumbra in sunspots (Shimizu et al. 2012 ; Romano et al. 2013 , 2014 ). Furthermore, penumbral-like structures, i.e., so-called orphan penumbrae, have been reported as a photospheric counterpart of a chromospheric filament (Kuckein et al. 2012a , 2012b ; Buehler et al. 2016 ). Observations of similar bright filaments inside the umbra, called UFs, have been recently reported by Kleint Sainz Dalda ( 2013 ). They found that UFs were characterized by a horizontal flow opposite to the Evershed flow and they also found from coronal images the presence of bright coronal loops that seemed to end in UFs. To interpret such phenomena, Kleint Sainz Dalda ( 2013 ) conjectured two different


2010 ; Bahramian et al. 2021 ). In addition, very hard spectra in a bright hard state have been observed in a couple of NS LMXBs (Parikh et al. 2017 ) and BH LMXBs including V 404 Cyg (e.g., Kimura et al. 2016 ), V4641 Sgr (e.g., Maitra Bailyn 2006 ), and Swift J1858.6−0814 (Hare et al. 2020 ). All these sources have a strong short-term flux variation by a more than order of magnitude (e.g., Kitamoto et al. 1989 ; Revnivtsev et al. 2002 ; Wijnands et al. 2017 ), also in agreement with the present case. In addition, partial covering absorption is sometimes required in these sources, especially in high-flux phases (Kimura et al. 2016 ), consistent with Swift J0840.7−3516 in a bright phase. We note that the photon index of Swift J0840.7−3516 after the decay, ≲2, is reminiscent of a BH rather than an NS LMXB, which usually shows a larger value, Γ ∼ 3 (Wijnands et al. 2015 ), although the NS LMXB EXO 1745−248 showed a similar hard spectrum in quiescence (Rivera Sandoval et al. 2018 ). On the other hand, an NS LMXB would rather be favored, in that a very small radius of the thermal component, 1–2( D /8 kpc) km, consistent with emission from an NS surface, was obtained from the NuSTAR+Swift/XRT+NICER spectrum in a faint phase. This holds even if the observed power-law component was entirely produced by Comptonized disk photons. Assuming conservation of the number of photons and an isotropic scattered emission (Kubota Makishima 2004 ), and using the intrinsic disk photons estimated from the best-fit diskbb+cutoffpl model, we obtain the inner disk radius as ∼3( D /8 kpc) km (where i = 60° is assumed). Similar thermal emission likely emitted by the NS surface can be seen in NS LMXBs at low X-ray luminosities below ∼10 35 erg s −1 but not in a more luminous hard state, where the power-law component dominates the X-ray flux (e.g., Sakurai et al. 2014 ; Shidatsu et al. 2017 ). The thermal component in Swift J0840.7−3516 was observed at a consistent luminosity; the second rapid decay from 10 5 s since the discovery, in which this component was found, started at ∼10 35 ( D /8 kpc) 2 erg s −1 . Even if an LMXB scenario works for all the properties raised above, the observed absorption feature is difficult to explain. Although ionized Si and S lines from disk winds are sometimes seen at similar energies (e.g., Ueda et al. 2009 ), they are much weaker than the present case and always with strong iron K α lines at ∼7 keV. Assuming the observed feature to be the redshifted Fe K edge, we obtained z = 1.1. If this is the gravitational redshift caused by a Schwarzschild stellar-mass BH, the feature should be produced at ∼1.3 R S , where R S is the Schwarzschild radius. It is unclear that an absorber with such a small size can exist in the vicinity of a BH. Another possibility is that the observed feature is a different blueshifted line produced by an outflow at a relativistic speed, like baryonic jets observed in SS 433. However, it is unclear whether such


the position angle and distance modulus measurements with no further adjustment. In the Appendix, we discuss the validity of the approximation used in equation ( 8 ) by comparison to models which explicitly employ the posterior samples from Martin etal. ( 2016c ). We find that the posterior distributions on the model parameters are very similar when using the posterior samples compared to the approximation employed throughout the paper. Additionally, in the absence of similar sets of posterior samples for the other M31 dSphs and all of the MW dSphs, we are forced to employ an approximation. We now detail each of the models employed. The first model assumes the dSphs are randomly aligned with respect to the observer, so does not utilize the position angle information. The second model assumes the major axis of the dSphs are preferentially aligned with the galactocentric radial vector. Our final model assumes the major axes of the dSphs preferentially lie in a plane. 3.2 Random alignment We attempt to infer the distribution of T and E given the observed shapes and associated uncertainties under the assumption that there is no preferred orientation for the dSphs. Implicitly, this assumes the distribution of observed position angles is uniform. Our model is described as where is the modified normal distribution to account for the asymmetric uncertainties, and subscript i corresponds to the i th of N dSphs. is a uniform distribution between a and b and is a Cauchy distribution centred on a with scale parameter b . From dark-matter-only simulations, the triaxiality and ellipticity distribution are well approximated by Gaussians (Allgood etal. 2006 ; Knebe etal. 2008a ). The natural choice for the prior on a scale parameter would be a Jeffreys prior of 1/. Instead, we heed the advice of Gelman ( 2006 ) and adopt a half-Cauchy distribution as a weakly informative prior. Very large values of are disfavoured, whilst very small values are not strongly favoured. This matches nicely our expectation that the intrinsic triaxiality and ellipticity distributions will not be -function spikes nor will they lack any structure at all. The choice of the scale as unity is inspired by the finite range of both the triaxiality and the ellipticity. One complication is that some of our variables have a discrete range so the pdf that describe them must be normalized over this range. In the case of the uniformly distributed variables, the normalization is a constant factor but for the normally distributed variables (i.e. T i and E i ) the normalization depends on the parameters and that we are attempting to constrain. Therefore, any distributions f ( x ) for a variable x constrained to lie in the interval ( a, b ) that are quoted in this paper implicitly contain a factor of N 1 where We additionally introduce the pdf given by For clarity, the log-likelihood for this model is given by where p (cos, , T, E ) is a -function related to equation ( 2 ). 3.2.1 Results The


into comoving space. The underlying halos are the same. Left panels: a slice of the HCN temperature cube, in transverse (on-sky) directions. Middle panels: slices of the HCN and CO temperature cubes, with one axis along the line of sight. Note the difference in comoving volume and location. Right panels: a slice of the CO temperature cube, again in transverse directions. 2.3. Power Spectra from Observations We work with the spherically averaged, comoving 3D power spectrum P ( k ) and the average of all 2D sky angular power spectra from each frequency channel in the data cube. We calculate P ( k ) by calculating the full 3D power spectrum of the temperature cube in comoving space, binning it in radial “shells” of , and averaging the full spectrum in each bin. We obtain in each frequency channel through similar binning of the 2D power spectrum in space. As Figure 2 shows, the comoving volume is not perfectly rectangular, but we approximate it as such for P ( k ) calculations here. As in Li et al. ( 2016 ) and Breysse et al. ( 2015 ), we present the spherical 3D power spectra as and the averaged 2D power spectra in the conventional form, using a flat-sky approximation as in Chiang Chen ( 2012 ) and Breysse et al. ( 2015 ). While the 3D analog to is just P ( k ) rather than the rescaled , and the 2D analog to is the suitably rescaled rather than just , we use these rather disparate presentations to follow the conventions used in previous literature. 3. Results 3.1. Simulated Temperature Cubes For 496 different pairs of CO and HCN light cones, we generate temperature cubes for CO emission and HCN emission, which we then add together in observed voxels to obtain a simulated map for CO emission plus HCN contamination. Approximately halos in the CO light cones and approximately halos in the HCN light cones typically fall above the cutoff mass of 10 10 and within a patch. Since we used only 100 light cones populated with CO emitters and 100 light cones populated with HCN emitters, there is some redundancy in the light cones used. But since temperature maps were generated anew for each new pairing, even two maps from the same light cone are subject to some differences due to reapplication of halo-to-halo scatter in SFR and luminosity. On top of the halo-to-halo scatter in each light cone, sample variance in large-scale structure between light cones results in light cone–to–light cone scatter in temperature and power spectra. Figure 2 shows slices of a sample pair of temperature data cubes for a given pair of light cones. Note that these slices are generated with the fiducial model (the turnaround model with 0.3 dex log scatter in luminosity). The power-law model maps appear much sparser than maps from any of the other models, especially when only of sufficiently massive halos host emitters. Table 3 lists the mean line temperatures over each simulated CO map and HCN map. Variations in do not appear to significantly affect , and any differences present


we cannot predict how the vapour production mass will scale with impact energy for very energetic giant impacts. Although the amount of mass in vapour is relatively independent of b and μ there is a strong dependence on these parameters when determining which body (target or projectile) the vapour comes from (Fig. 3 ). As expected for equal mass impacts, the vapour mass is produced equally from both the target and projectile, which is true for all impact angles (parameter b ). However, as μ decreases, the fraction of vapour production from the projectile is higher than that of the target. This behaviour is also sensitive to the impact angles as shown in Fig. 3 . Figure 3. The difference between the escaping vapour mass from the projectile normalized by projectile mass and the escaping vapour mass from the target normalized by the target mass versus the specific energy of the collision normalized by the catastrophic disruption criteria. Symbols and colour bar are same as in Fig. 2 . 3.2 Initial vapour velocity distribution Planetary embryo impacts produce a variety of post-collision dust distributions, none are isotropic as has been assumed in previous work (e.g. Jackson Wyatt 2012 ; Jackson et al. 2014 ; Wyatt et al. 2017 ). To be able to compare different distributions, we need to characterize the anisotropic nature of the collisionally generated dust. We separate the velocity of each escaping particle into two velocity components, the velocity parallel ( v ∥ ) and perpendicular ( v ⊥ ) to the direction of the impact. For example, the direction of the impact is the same (along the x -axis) for the two collisions shown in Fig. 1 . To understand and compare different distributions of particles, we choose to take the range between the 16 per cent and 84 per cent percentiles to compute the velocity dispersion for both velocity components in each collision (Δ v ∥ and Δ v ⊥ ). This allows us to compare the velocity distribution of different distributions without assuming a given distribution for any collision. We choose to take the 16 per cent and 84 per cent percentiles as this would be comparable to taking the 1σ values either side of the mean of a normal distribution. In Fig. 4 , we show the absolute velocity distributions of vapour condensates parallel (blue) and perpendicular (orange) to the direction of the collision and absolute velocity (black) for three different giant impacts. Fig. 4(a) shows sim. 168, a collision between 1.11 and 0.1 at 15 and b = 0.4. We choose to show sim. 168 as it is the closest simulated impact we have to the Moon-forming impact. Comparing to fig. 1 of Jackson Wyatt ( 2012 ), we have an overall similar distribution of absolute velocities and we find only a small difference between the direction vapour condensates are launched (close to isotropic). Fig. 4(b) shows sim. 8, a collision between two 0.1 at 10 and b = 0. Fig. 4(c) shows sim 88, a collision between two 0.1 at 10 and b = 0.8. There is a stark difference


cut through the current sheet in turbulence, the parameters depend on time due to the turbulent interaction of larger scale magnetic eddies, so wide variation in parameters across different current sheets and at different times is expected. 7 DISCUSSION AND CONCLUSIONS As a result of the weak collisionality of many space and astrophysical plasmas, several physical processes can locally contribute to the dissipation of energy and the heating of the plasma. During magnetic reconnection, magnetic energy is efficiently converted to directed plasma flows, thermal energy, and energetic particles (Burch et al. 2016 ; Torbert et al. 2018 ). During turbulence in a magnetized plasma, the cascade provides the global amount of energy needed for fluctuations to be dissipated at small scales (Marino et al. 2008 ). However, which dissipative mechanisms are dominant and under which conditions is still not sufficiently understood (Vaivads et al. 2016 ). Therefore, a multiplicity of dissipation surrogates have been adopted in the literature to identify potential sites of dissipation. Owing to the strongly non-linear dynamics and the importance of physics at kinetic scales in such systems, numerical simulations of the Vlasov–Maxwell system (or, including collisions, the Boltzmann–Maxwell system) are the decisive tool to address the long-standing issue of plasma heating and energy dissipation in magnetized plasmas, e.g. Parashar et al. ( 2009 ). For example, the ‘turbulence dissipation challenge’ (Parashar et al. 2015 ) motivated numerical work to compare different algorithms on the solution of similar problems to assess the nature of the dissipation in magnetized turbulence (e.g. Pezzi et al. 2017 ; Perrone et al. 2018 ; González et al. 2019 ). In the same spirit, we have conducted a survey of a number of dissipation surrogates with three different kinetic plasma codes – the fully kinetic particle-in-cell vpic , the fully kinetic Eulerian Vlasov–Maxwell Gkeyll , and the Eulerian Hybrid Vlasov–Maxwell codes – to perform numerical simulations of two important physical phenomena in plasma physics. The first class investigates reconnection in an isolated current sheet, and the second concerns plasma turbulence at kinetic scales. We have calculated and compared eight distinct dissipation proxies, delineated in Section 2 . For the sake of clarity, we have categorized them in terms of (i) energy-based parameters, whose definition describes energy transfer and conversion; and (ii) VDF-based parameters, that are directly related to kinetic signatures in the particle VDF. Energy-based parameters considered here are the power density by electromagnetic fields on charged particles (Zenitani et al. 2011 ), the pressure–strain interaction (Yang et al. 2017a ), and a local proxy of the turbulent energy transfer (Sorriso-Valvo et al. 2018a ). The VDF-based parameters are the local pressure agyrotropy (Swisdak 2016 ) and three measures describing how different a local


be generated at levels within the cloud deck, where the background continuum emission originates. It must be generated well above this layer, where the absorbing gas is cooler and therefore absorbs more efficiently than it emits. The narrow width of the absorption line also suggests that it was formed at pressures substantially less than those of the cloud top (70 km, ∼3000 Pa). Figure 4. Radiance mixing ratio Jacobians ( dI ν / dr mix ) as a function of layer pressure for the radiance streams at 21 degrees zenith angle, for the 266.94 GHz feature for SO 2 (left) or PH 3 (right). Absorption line strengths (lower panels in each case) are given in units of cm −1 /(molecule cm −2 ). The continuum originates from SO 2 at ∼6 × 10 4 Pa (∼54 km, within the cloud deck), while the line cores for either species do not originate in the clouds (48–70 km) but at over 400 Pa (over 80 km) in the mesosphere. In the right panels in both plots, the temperature structure is given as a black line, while the colored lines denote SO 2 (green) or PH 3 (purple) mixing ratios. On the right, the evenly mixed 20 ppb PH 3 profile is shown with a solid line and the photochemical PH 3 profile is shown with a dashed line. 3.3. ALMA Line Dilution While the nondetection of prominent SO 2 spectral features in the ALMA wideband data could indicate a low abundance, as argued by Greaves et al. ( 2020a ), the estimation of this abundance was done without correcting for line dilution as a result of the ALMA observing geometry (Greaves et al. 2020a ). Significant line dilution is likely, especially considering the global distribution of SO 2 in the Venus atmosphere, and the exclusion of the short baseline ALMA measurements. Greaves et al. ( 2020a ) estimated line dilution (filtering losses) of 60%–92% depending on position on the disk. To determine the disk-averaged line dilution for the SO 2 reference line search, we simulated observations of Venus using the ALMA configuration of Greaves et al. ( 2020a ) by imposing an appropriate resolution spectrum (0.00003 cm −1 , 1 MHz) of our Case D atmospheric model over a limb-darkened disk model. The Fourier Transform of this model was resampled to match the ALMA configuration and reimaged using the imaging routines of Greaves et al. ( 2020a ), as provided in their Supplementary Software 3. As shown in Figure 5 , line dilutions on the order of 95% at the line core are observed for the full disk. We observe similar dilutions when the spectrum is only imposed on one hemisphere (∼8′′ extent at the time of observation). This line dilution suggests that the SO 2 reference features produced by our best-fit SO 2 distribution (Case D) would be heavily suppressed by line dilution in the ALMA data, which would cause them to mimic smaller features below the ripple detection limit of −0.0006. Figure 5. Modeled line dilution for the ALMA observations: disk-averaged line/continuum ratios for our nominal Case D atmosphere model containing SO 2


our best guess associates the sunquake location (taken to be the 5 mHz signal in Figure 1 ) to the connectivity domain indicated as Q 4 in Figure 12 . That connectivity domain corresponds to the domain 4 in Figure 13 , i.e., to the envelope bundle of field lines (in green in Figure 14 ). Given its structure, such a field line bundle has two interesting properties. First, since it is surrounding the flux rope in almost its entire length it is then directly impacted when the eruption occurs. Second, such wide-spread field lines between the erupting flux rope and the above layered structure are all connected on one side to a very small area in the photosphere (where the sunquake occurs), in this way, providing a sort of magnetic focusing. In light of this, we argue that the sunquake is a consequence of the energy release associated with the reconnection between the erupting flux rope and the envelope surrounding it. Furthermore, the envelope structure helps focus the released energy into a small area in the lower atmosphere ( Q 4 in Figure 12 ), which produces the sunquake pulse. The scenario offered by our analysis and the interpretation of the observations clarifies the dynamics of the two flares and supports the association of the sunquake to the larger, M-class flare, rather than to the first C-class flare. Moreover, it indicates how a sunquake may require a special magnetic field configuration to occur, which may help understand their erratic occurrence. 6. Conclusion In this work, we study the evolution of active region 11675 on 2013 February 17, in the hours leading to the production of two flares (a C-class flare and an M-class flare), the ejection of filament material and a sunquake. We investigate the configuration of the active region magnetic field through the construction of an NLFFF extrapolation just over one hour before the sunquake. The aim of the work is to shed light on the configuration of the coronal magnetic field in which the sunquake was produced. The sunquake occurs when active region 11675 is undergoing a period of rapid evolution during its emergence phase, and when there are strong motions of the photospheric magnetic field elements. In particular, two emerging bipoles collide and shear past each other. The magnetic field along the polarity inversion line that separates these bipoles is highly sheared and at low altitudes bald patch field lines are present. At higher altitudes, an HFT and a flux rope are present, which have been highly modified by the motions of the photospheric field in which these magnetic structures are rooted. From a time–distance analysis, we determine that the sunquake was initiated in the time period between 15:50:41 to 15:51:26 UT on 2013 February 17 in a region of positive polarity field. This finding indicates that the sunquake is initiated during the M-class flare, which is associated with the eruption of the flux rope and ejection of filament material. However, this timing determination


more spheroidal, as in the Milky Way halo, rather than being represented by a disc-like structure. One would therefore expect intermediate-aged populations traced, for instance, by RC stars to be somewhere between those. This is exactly what we found. Subramanian Subramaniam ( 2012 ) also used RC stars but found a very small inclination, i = 0.5°. Their methodology however was to use axes ratios and to limit the sample to a small depth range, which may have inadvertently introduced a bias. The PAs we found are nearest to those determined by Subramanian Subramaniam ( 2015 ), who used Cepheids, and Rubele et al. ( 2015 ), who fitted entire CMDs and found a mean i = 39°. In conclusion, while the SMC does not exhibit an obvious disc component there is a clear trend for the system to become more inclined with respect to the plane of the sky the younger the (stellar) tracer is. Since the RC stars already show this progression this means that the dynamical events that led to this time-dependent morphology must have occurred on Gyr time-scales. Indeed, while most of the stellar mass in the SMC was generated >4 Gyr ago (Sabbi et al. 2009 ; Rezaeikh et al. 2014 ; Rubele et al. 2018 ) there is evidence in the SFH of the SMC of a strong interaction with the LMC to have occurred ∼2 Gyr ago (Harris Zaritsky 2004 ; Noël et al. 2009 ; Rezaeikh et al. 2014 ; Rubele et al. 2015 ). A more recent interaction ∼0.2 Gyr ago coincident with the closest approach to the Milky Way, also seen in the SFH (Harris Zaritsky 2004 ; Noël et al. 2009 ; Sabbi et al. 2009 ; Rubele et al. 2015 ), may have caused further distortion of the gaseous disc, from which the younger population traced by the Cepheids formed. 6.3 Distortions and other structures The outer regions of the SMC have been transformed by interactions leading to stripping and spatially stretched populations. Dias et al. ( 2016 ) described the SMC as being composed of three bodies: the eastern Wing, the central main body, and the western halo. We find the intermediate-age main body traced by RC stars to be structurally simple and symmetrical, consistent with a spheroidal component. The eastern Wing, on the other hand, is split into two distinct populations (both traced by intermediate-age RC stars), of which the brighter one clearly lies in front of the main body (Fig. 9 , bottom right). We have been able to comprehensively trace these two components across the VMC footprint, allowing us to relate them to what other observational and theoretical studies have found. We find a fourth structure, in the north (Fig. 9 , top left); this has previously been described as an extension to the Wing (e.g. Gardiner Hawkins 1991 ), but Sun et al. ( 2018 ) showed this area to contain a shell of young stars which is clearly separated from the general Wing area. The intermediate-age population of RC stars in the north need not be related to these young stars, though. We compare our results with previous analyses of Cepheids, RR Lyræ,


of the cluster will change with the cluster properties. For a massive cluster or for a compact cluster the binding energy will be higher and hence the formation of the SMS would be more favourable. On the 5 Myr time-scale explored in our simulations, supernova feedback is also expected to become relevant. With the typical energy of 10 51 erg for a core-collapse supernova, it is clear that one such event will expel the gas and terminate the accretion, if it has not stopped already (either due to gas expulsion by stellar winds or as the accretion process may have depleted the gas). In future work, it will be important to study detailed gas dynamics where the kinetic energy deposition of winds as well as the supernova feedback is taken into account. 5 SUMMARY AND DISCUSSION In this work, we explored the effect of mass-loss due to stellar winds on the final mass of the SMSs, which could be formed via runaway stellar collisions and gas accretion inside NSCs. We find that an SMS of mass could be formed even in a high-metallicity environment for high accretion rates of . For an accretion rate of , the final mass of the SMS for . Whereas for solar metallicity, no SMS can be formed for f Edd = 0.9 and SMSs of masses can be formed for f Edd = 0.7 and 0.5, respectively. For the case of Eddington accretion it will not be possible to form an SMS in the metallicity regime . Finally, for the Bondi–Hoyle accretion scenario, we find that the formation of a SMS will not be possible in the high-metallicity regime of . The interaction of the stellar wind and the gas inside the cluster might play an important role in the evolution of the SMS. The winds from the SMSs have high velocities ∼10 3 km s −1 (Muijres et al. 2012 ), which might exceed the escape velocity from the centre of our modelled star cluster. The SMS in the cluster is close to the centre which results in a high collision rate near the centre due to a shorter relaxation time in the core and an increased collisional cross-section. If the SMS is displaced by collisions, it rapidly sinks back close to the centre via dynamical friction where it may eventually decouple from the remainder of the cluster. This is also known as the Spitzer instability (Spitzer 1969 ). Interestingly, Krause et al. ( 2016 ) have found that for a Salpeter -type mass function the stellar wind cannot remove the gas inside the cluster. Hence, we do not expect the stellar wind to remove gas from the cluster. One of the main caveats of this work is the neglect of the kinetic energy deposition through stellar winds, which could contribute significantly to expel the gas. The latter is likely to create a regime where the growth of a massive object is still inhibited, even though the mass-loss itself from the winds is no longer significant. Below a critical metallicity in the range (, this effect is no longer expected to be relevant; however, supernova feedback may lead to the expulsion of the gas. Another relevant caveat is the


astrometry between the initial image and each of the later sets using Astroalign . To uncover or place limits on transient emission on these timescales, we perform digital image subtraction using the HOTPANTS software package (Becker 2015 ). If an optical or NIR source is detected within the GRB localization regions (Evans et al. 2009 ; Lien et al. 2016 ), we perform aperture photometry on the residual images to determine the source brightness using the IRAF/ phot package. We determine zero-points by using point sources in common with the SDSS Data Release 12 (Alam et al. 2015 ), Pan-STARRS DR1 and DR2 (Chambers et al. 2016 ), and 2MASS (Skrutskie et al. 2006 ) or USNO-B1 (Monet et al. 2003 ) catalogs. If there are no significant residuals at or near the localization region, we obtain an upper limit from sources detected at 3 σ significance in the field. As we obtain the limit from the initial image, our values do not include additional noise from the subtraction. In total, we present 52 unpublished upper limits and three unpublished detections of 23 bursts, 10 with known redshifts (Table A1 ). 2.3. Observations from GCNs We search the GCNs of all short GRBs detected between 2013 and 2020 for constraining upper limits to ensure that our catalog covers all Swift short GRBs to date (2005–2013 is covered by the literature sample in Section 2.4 ). Considering that the range of peak kilonova luminosities based on models is ν L ν ≈ (1−5) × 10 42 erg s −1 (∼10 times the peak luminosity of AT 2017gfo; see Figure 1 ), we only include data that could feasibly fall into the range of kilonova luminosities, or fainter than a given depth, corresponding to griz ≳ 22.5 mag and JHK ≳ 21.0 mag (AB system) at z ≈ 0.25. We choose z ≈ 0.25, as 80% of known redshifts in our sample fall above this cut. Observations significantly brighter than this are not included, as they are likely dominated by afterglow emission. In total, we collect deep optical and NIR limits of 18 bursts, seven with confirmed redshifts, from the GCNs (Table A1 ). Figure 1. Light curves of AT 2017gfo and four diverse kilonova models considered in this work in the griJ bands. Light curves are shown in ν L ν (left y -axis) and absolute magnitude (right y -axis) units, and rest-frame time in days. In each panel, we show the interpolated, smoothed AT 2017gfo data set (solid dark-blue line; Villar et al. 2017 ), an NSBH kilonova model (blue dotted line; Kawaguchi et al. 2020b ), an infinite-lifetime NS remnant kilonova model (red dashed line; Kasen et al. 2015 ), an extreme optimistic magnetar-boosted kilonova model (turquoise solid line; Fong et al. 2021 ), and a model for an optimistic scenario motivated by GW190425 (magenta dashed–dotted lines; Barbieri et al. 2020 ). Black and open triangles denote upper limits of short GRBs with known and unknown redshifts, respectively, in the respective rest-frame bands at δt rest = 0.1–10 days. The most constraining limits of bursts with known redshifts


the exact influence of radiation pressure on nitrogen atoms would require a dynamical model, such a low value for β indicates that it should have a negligible effect. Despite these very weak effects from radiation pressure, we do not find an overabundance of nitrogen in the β Pictoris disc. Other papers have also shown no clear deficiency in radiation-sensitive species such as Na and Ca ( Lagrange et al. 1998 ), which also seems to disfavour the preferential depletion scenario. The preferential production scenario, also proposed by Xie et al. (2013) , suggests that the overabundance is the result of gas being produced from materials rich in C and O. This includes photodesorption from C and O-rich icy grains ( Grigorieva et al. 2007 ) and collisional vaporisation of the dust in the disc ( Czechowski Mann 2007 ). Unlike preferential depletion, this scenario does not infer high N abundances and is thus more compatible with the near-solar NI abundance measured here. 5.2 Origin and dynamics of the NI gas The absorption components that we used to model the shape of the NI absorption lines may provide us with hints at both the origin and dynamics of the NI gas. The two absorbing circumstellar components, CS 0 and CS X , were not constrained to particular radial velocity values, but provided the best fit when their radial velocities matched that of the star. This indicates an absence of detectable amounts of NI falling in towards the star. The clearly asymmetric HI Ly- α line presented in Wilson et al. (2017) , on the other hand, indicates that HI gas is falling in towards the star. The HI gas is thought to be originating from the dissociation of water molecules from evaporating exocomets. The fact that we do not see similar asymmetric line profiles in NI is perhaps not surprising. In solar system comets, nitrogen-bearing molecules are only minor constituents ( Eberhardt et al. 1987 ), which means that if β Pic comets are similar, we would not expect the NI lines to be asymmetric. This does not exclude the possibility that NI might originate from exocomets, however, as we argue in Sect. 5.3 . 5.3 Role of exocomets The NI column density measurements are dominated by the narrow absorption feature, CS 0 , which has a line width of km s −1 . With a similar line width observed in FeI ( Vidal-Madjar et al. 2017 ), this absorption feature likely traces a stable band of circumstellar material that orbits β Pic. The origin of the broad absorption feature, CS X , with km s −1 , which absorbs seven times less strongly, but clearly governs much of the absorption line shape, is less clear. The feature might consist of a number of individual absorption components that are beyond the resolving powers of the COS instrument. The broad component could be related to NI absorption lines in the stellar atmosphere of β Pic. This would certainly explain that the bulk velocity matches that of the star. However, we would then expect the lines to be much broader and shallower


coupling, resonance, and weak inner planet coupling. We elaborate on these in the following subsections. Figure 2. RMS values of the eccentricities of the planets in a fiducial two-planet system under the influence of an eccentric perturber. The perturber has initial eccentricity e p = 0.1 and has an orbit co-planar with the inner system. The other parameters are the same as Fig. 1 . The solid coloured points represent the results of numerical integrations using secular equations, while the solid coloured curves are calculated using linearized theory; the dashed coloured lines show the forced eccentricity (equation 26 ). The dashed vertical line corresponds to ε 12 = 1. 2.3.1 Resonance When ω 1 ≃ ω 2 (note that this is approximately equivalent to the condition ε 12 ∼ 1), a potential resonance feature arises where large eccentricities can be excited in the inner planets, even for small e p . If we take ω 1 = ω 2 , equations ( 34 ) and ( 35 ) becomes We see that the eccentricity of planet 1 is boosted while the eccentricity of planet 2 is dampened near the resonance. The resonance feature is most pronounced when L 2 ≫ L 1 . To illustrate this, let L 1 = 0, then the forced eccentricity on the inner planet becomes If both the inner planets have zero initial eccentricity, then their eccentricity evolution is given by In this limiting case, according to the linear theory, at ε 12 ≃ 1 the eccentricity of the inner planet can become arbitrarily large, even for small initial values of e p . In reality, the linear theory breaks down as e 1 becomes too large, and higher order terms will keep e 1 to a modest value. An illustration of the resonance behaviour can be seen in Fig. 2 . Systems with larger ratios of m 2 / m 1 tend to exhibit pronounced resonance features, whereas for systems with more comparable masses, the feature is notably reduced. 2.3.2 Strongly and weakly coupled regime In the case where the mutual precession rates of the inner planets dominate over the influence of the perturber (i.e. ω 12 ≫ ω 1 p and ω 21 ≫ ω 2 p ), the general expressions ( 34 ) and ( 35 ) can be significantly simplified; we refer to this as the strongly coupled regime. In this regime, the two planets attain very similar forced eccentricities (), and as a result equations ( 34 ) and ( 35 ) are dominated by their first terms. Explicitly, in this regime the RMS eccentricities are approximately given by where α 12 = ( a 1 / a 2 ) and β 12 = β( a 1 / a 2 ) (see equation 8 ). In the other limiting case (i.e. weak coupling), when the precession rates of the inner planets driven by the perturber dominate the their mutual precession rates (i.e. ν 1 p ≫ ω 12 and ν 2 p ≫ ω 21 ), the terms of order ν 12 /ω 1 p and ν 21 /ω 2 p in equations ( 34 ) and ( 35 ) can be dropped, and the final inner planets’ RMS eccentricities are again given by . In this case, the planets precess independently of one another, and their eccentricities are given by equations ( 11 ) and ( 12 ), i.e. Note