J03473-019 and J22518+317 ( Fig. 3 ) can be compared to, for example, those of fully convective J19169+051S ( Fig. 4 ) and J19511+464 ( Fig. 3 ). The only noticable feature is that fully convective stars tend to have stronger average magnetic fields represented by magnetic components of stronger strength. For instance, in the hottest partly convective star J05365+113( Fig. 3 ) we derive a very weak magnetic field with only two magnetic components (0 and 2 kG, respectively). Fig. 3 Distribution of filling factors as derived from Ti lines. The stars are sorted according to the S/N of their spectra (S/N decreases from left to the right and top to bottom). Fig. 3 continued. In our sample, we find several objects with remarkably very similar properties of their magnetic fields. For the two stars with largest υ sin i values, J01352-072 ( Fig. 3 ) and J04472+206 ( Fig. 3 ), we recover the same average magnetic fields with identical filling factors. The only difference between these stars is that J04472+206 is one spectral type cooler. Next, we find a twin of J07446+035 (very well studied M dwarf YZ CMi, Fig. 4 ), which is J23548+385 (not studiedat all, Fig. 3 ). Both stars have very similar filling factors, average magnetic field, projected rotational velocities, but slightly different spectral types. In addition, another famous M dwarf J22468+443 (EV Lac) has filling factors similar to J07446+035 and J23548+385, although its average magnetic field is noticeably weaker. We next looked for a connection between our filling factors and the geometry of large scale magnetic fields. In our sample, we have nine stars for which the geometry of their surface magnetic fields was previously derived from polarimetric measurements. In particular, Morin et al. (2010) showed that partly convective M dwarfs tend to have complex multipole fields with strong toroidal contribution, while stars that are fully convective generate dipole-dominant predominantly poloidal magnetic fields. Following this classification, in Fig. 4 we grouped M dwarfs in columns depending on whether the star has dipole-dominant magnetic field (left column) or more complex multipole-dominant field (right column). One can see that there is no obvious difference in filling factors between these two groups. A smooth distribution is found for dipole-dominant J01033+623 on one side, and multipole-dominant J08298+267 and J19169+051S on the other side. Likewise, J07446+035 and J22468+443 have almost identical filling factors but their magnetic field geometries were found to be very different. Nevertheless, there seems to be one common feature that is intrinsic to complex multipole fields as observed from filling factors, which is the appearance of strong zero-field components. This zero-field component is strongest in four out of five stars with multipole-donimant fields with an exception of J22468+443. 4.4 Magnetic field and rotation The detection of very strong magnetic fields in M dwarfs needs to be


is injected (brown) is different from the false positive when no planet is injected (in red, completely to the black curves on the left-hand side panels) for supergranulation (especially SGlow) alone, but it is similar when granulation and supergranulation are superposed. For GRA, ALL GRAhigh,SGmed , and ALL GRAhigh,SGlow , the red and brown curves are similar, that is, the percentage of false positives is the same whether a planet (of 1 M Earth ) is injected or not. However, the situation is different for SGmed and SGlow because when no planet is injected, the percentage of false positives is the same, even though they have very different rms RV. This is due to the fact that here the comparison of the power is made with the FAP. Because the shape of the power spectrum is the same between SGmed and SGlow, and because the FAP values are scaled with the rms, both power and FAP increases from SGlow to SGmed in a similar manner and the percentage of false positives is similar. In the case of ALL GRAhigh,SGlow , the signal is dominated by GRAhigh, hence, a level that is similar between GRAhigh and ALL GRAhigh,SGlow , while the situation is intermediate for ALL GRAhigh,SGmed . For GRAlow, the rate of false positives is very small in all cases. However, when added to supergranulation (either SGmed or SGlow), the latter dominates, and rates are very similar to those obtained when combining with GRAhigh, only slightly lower. The level of false positives here may be large because our analysis is too simplistic. When a peak is detected above the FAP, we should test the robustness of the detection to determine whether the peak is stable or not for example. More sophisticated methods will have to applied in this area in the future (see Sect. 6 ). Another representation of these results is shown in Fig. 18 , showing the percentage of false positives (sum of the two contributions described above) versus the detection rate (computed on the cases with an injected planet), which is similar to a ROC curve (but where each point corresponds to a spectral type). Each curve corresponds to one of the OGS configurations. Ideally, we would like points to be in the lower right corner. Points at the top have a high false positive level and points on the left correspond to poor detection rates. If we compare the global level of false positive here and the rms for each type of OGS configuration, we see that there is not a direct correspondence, because a granulation-like signal provides better performance due to their more suitable power spectrum (for a given rms). High-mass stars are to the left of each curve and lead to high rates of false positives and low detection rates except for granulation alone (for GRA low all points are in the lower right corner), and to a lesser extent ALL GRAhigh,SGlow . We also note that the highest level of false positives is obtained for SG alone. However, when granulation is added to supergranulation, the rms increases, but the level


We calculated a simple reduced-chi-square statistic for each DM model, finding for the YMW16 model and ∼10.2 for the NE2001 model. (The χ 2 statistic accounted only for uncertainties in the parallax distances, not in the electron-density models.) The similarity in these χ 2 values suggests the two models are comparable in their ability to predict distances to pulsars based on DM values, at least in the distance regime probed by millisecond pulsar timing parallax measurements (within ∼2.5 kpc of the Sun). Figure 4. Distances from timing parallax measurements (67% confidence) vs. distances from DM models for pulsars with significant timing parallax detections. The blue points show the NE2001 DM model distances, while the red points show the YMW16 model distances. The black dashed line shows a one-to-one relation. 4.2. Parallax Upper Limits The remaining 25 pulsars had timing parallax measurements with significances of less than 3 σ . Of these, 22 had positive parallaxes and only 3 had negative parallaxes in the formal timing fits. This is strong evidence that most of the non-detections have a true low-level parallax signal; otherwise, there would be comparable numbers of negative and positive parallax measurements. None of the negative parallaxes had large significance. We used 95%-confidence upper limits on parallax to compute 95%-confidence lower limits on distance for these pulsars. The details of this calculation are outlined in M16 . We have not attempted to correct for Lutz–Kelker bias (Verbiest et al. 2010 ), as we do not believe we have enough prior information on the spatial and luminosity distribution of the millisecond pulsar population to accurately correct for this bias (see M16 ). For the lower limits on distance, our 95% limit remains a conservative approach, in the sense that compensation for Lutz–Kelker bias would only push these limits higher. Among the pulsars with parallax upper limits, a few have significant discrepancies with previous measurements (PSRs J0636+5128, J1455−3330, J1640+1224, B1937+21). As with the pulsars discussed in Section 4.1 , the source of these discrepancies are not clear, but may result from differences in DM variation models or red-noise models. 4.3. Distance Constraints from Rotation and Orbital Period Derivatives Observed pulsar rotation period derivatives are a combination of intrinsic pulsar spin-down and kinematic terms due to the acceleration and transverse motion of the pulsar relative to the Sun (Shklovskii 1970 ; Nice Taylor 1995 ). By assuming that the pulsar is losing rotational energy, so that it has a positive rotation period derivative, an upper limit can be placed on the pulsar distance. See M16 Section 4.3 for details. Table 5 lists such “ distance” constraints, which we calculated following the procedures of M16 . In the table, we omitted distance limits greater than 10 kpc as physically uninteresting, and we did not calculate a constraint for PSR J1024−0719, as its observed rotation


timing solution in NGC 6624 (pulsars D, E, F, and H), as well as to validate the timing position derived for the new eccentric binary MSP (pulsar G). With the observing setup used for the two TRAPUM observations of this cluster (obs. ids 10L-trapum and 12L-trapum) reported in Section 2 , the individual beams making up the tiling have elliptic shapes with semi-axes in the range of ∼8–11 arcsec at half power, with the exact values depending on the elevation and other factors. We assume these values as the maximum uncertainty for the positions that we derive for these pulsars. The results are summarized in Table 5 , where we list the beams in which each pulsar was detected with highest S/N, and their boresight coordinates. The same is shown graphically in Fig. 9 , where the beams with localized pulsars are higlighted in grey. As can be seen, pulsars D and H are found to be the closest to the cluster centre. The eclipsing redback F and the isolated pulsar E are found at the much larger distances of ∼0.42 and ∼0.79 arcmin from the cluster centre, respectively, putting the former near the edge of, and the latter well outside, the half-power of the tied-array beam of the MeerTime observations (which used only the 1-km core antennas) of NGC 6624. This explains why NGC 6624F and E are detected with significantly lower S/N than expected in the MeerTime data as compared to the TRAPUM data, even after accounting for the different number of antennas used. Finally, the eccentric binary pulsar NGC 6624G showed up with highest S/N in beam #002. This had boresight coordinates of α = 18 h 23 m 41150 and δ = −30°21 ′ 3850. This is just ∼6.2 arcsec away from, and therefore consistent with, the corresponding timing-derived position of G. Figure 9. Tiling of the first TRAPUM observation (obs. id 10L-trapum) of NGC 6624. 60 dishes were used to synthesize 288 beams, with an overlap fraction of 80 per cent, around the nominal centre position of the cluster. The light-grey ellipses show the contours of the TRAPUM beams at their 84 per cent level of the power of their boresights, at 1284 MHz. We highlight in solid grey those beams where some of the pulsars show up with the highest S/N: the name of such pulsars are shown next to the beam number. The red ellipses show, for comparison, the L -band beam (at half power for top, centre, and bottom of the band) of one of the MeerTime observations of NGC 6624, which used only 42 antennas of the 1-km core of the array. Table 5. List of the NGC 6624 pulsars without a known position and corresponding beams where they were detected with highest S/N in the first TRAPUM observation of the cluster (id 10L-trapum). θ ⊥ is the angular distance between the beam boresight and the nominal cluster centre. 4.2.3 L-band detection of the transitional pulsar M28I M28I (also known as PSR J1824−2452I) is a redback system in M28 beloging to the rare class of transitional MSPs (see Jaodand, Hessels Archibald 2018 ; Papitto de Martino 2020 for


law may be more appropriate in these cases. Figure 9. A two parameter estimate of accreted mass fractions . The distribution of stellar mass fractions and 2D power-law slopes for the mass-matched sample of TNG100 galaxies, colour coded by the fraction of stellar mass beyond 2 half-mass radii. The full TNG100 parent sample is shown in grey points. Smaller symbols indicate galaxies for which a single power law was not an acceptable assumption, included here only for completeness (approximately ∼37 per cent of the samples require a more complex fit; see Section 6.2 for details). In general, shallower slopes correspond to higher accreted fractions, albeit with significant scatter. At fixed power-law slope, galaxies with higher stellar mass fractions in the outskirts have higher total accreted fractions. Figs 8 and 9 also raise the question of why these observations seem to be ‘missing’ stellar mass beyond 20 kpc (or, phrased differently, why TNG100 galaxies appear to be more massive and extended than DNGS galaxies at fixed stellar mass). No galaxy in DNGS has a stellar halo mass fraction above 10 per cent (when measured outside of 20 kpc), whereas the TNG100 galaxies in Fig. 9 can have up to 40 per cent of their stellar mass beyond this radius. One possible explanation is that DNGS galaxies may have low total accreted mass fractions relative to their mass-matched counterparts in TNG100. It is worth pausing to point out, however, that our capacity to ask (and answer) this question depends heavily on the validity of three assumptions: The parameters we chose to observationally estimate the accreted mass fraction (stellar halo mass fractions and power-law slopes) are able to provide an appropriate description of the outer profiles of galaxies – and, further, that these profiles contain sufficient information to characterize the assembly histories of galaxies. TNG100 reproduces observed properties of galaxies accurately enough that we should expect agreement in stellar halo properties. DNGS represents an unbiased sample of Milky Way-mass spiral galaxies. We will delve more deeply into these assumptions in Section 8 . First, however, we will proceed in Section 7 as though each of these are true, and attempt to carve out a more complete physical understanding of galaxies with low accreted mass fractions. 7 INSIGHTS FROM STELLAR PARTICLES We have seen from Figs 6 – 9 that DNGS galaxies appear to have low accretion fractions relative to their mass-matched TNG100 galaxies. The next logical question, then, is: ‘What are the assembly histories of (TNG100) galaxies with low accretion fractions?’ If, for example, this subset of galaxies shares any additional aspects of their assembly histories, we can potentially gain physical insight into our observations. In this section, we explore several different properties of our mass-matched TNG100 galaxies, combining information from the density profiles and individual stellar particles to characterize their past


location of the continent, especially for the fast-rotating case; for P = 5 days, this transport can increase from 7 gigatons per second (Gt s −1 ) in an aquaplanet to 14 Gt s −1 when the continent is located to both the northeast and northwest of the substellar point (not shown). This transport brings along with it the ozone and water vapor; it is unclear how the change of this transport will affect the ozone of the dayside. Therefore, AOGCMs are required to fully understand the ozone distribution on these exoplanets when a continent is present. 5. Conclusion We investigate the climate impact of a relatively small terrestrial continent on a synchronously rotating exoplanet by using an AOGCM. It is found that the climate is highly sensitive to the location of the continent, mainly because of the existence of a strong equatorial westerly jet in the ocean on such exoplanets. This jet is responsible for ∼90% of the heat transport from dayside to nightside, and the strength of this jet largely determines the global climate. When a continent blocks the equatorial westerly at the equator, the climate cools. The strongest cooling effect is obtained when it lies slightly to the west of the substellar point. On the other hand, the westerly jet strengthens and the climate warms when a continent obstructs the poleward branch of the oceanic gyre circulation. Enhanced oceanic heat transport to the nightside reduces the sea ice coverage there, and thereby reduces the sea-ice input from nightside to dayside. This reduces the planetary albedo and raises the GMST. The GMST can vary by more than 20° C simply by varying the location of a small continent. The outer edge of the HZ around an M-dwarf defined by stellar constant can change by 15% when a small continent is located at different places. Because the atmospheric circulation, atmospheric water vapor, and ocean upwelling all change with the location of the continent, the atmospheric chemistry and biosignatures may also change. Although major topographic features tend to align with the star-planet axis for synchronously rotating exoplanets (Wieczorek 2007 ; Turbet et al. 2016 ; Salazar et al. 2020 ), small continents or island arc chains may exist everywhere. Our results demonstrate that these small features can still change the global climate of a synchronously rotating exoplanet by as much as that of a round continent. We also demonstrate that it is possible to engineer the climate on such exoplanets by building a dam at appropriate locations. Appendix Model and Experimental Design A.1. Climate Model The simulations presented in this work were done with the fully coupled AOGCM CCSM3 (Collins et al. 2006 ). The model is the same as that used by Yang et al. ( 2019 ). The atmosphere and land share the same Gaussian grid, and the ocean and sea ice share the same grid. The resolution of the CCSM3 model is set to “T31_gx3v5.” In this setting, the horizontal grid spacing is approximately 3.75° by 3.75° for the


described via perturbations. Although isotropy has been directly tested (Blake Wall 2002 ; Bernui, Oliveira Pereira 2014 ; Bengaly 2016 ; Schwarz et al. 2016 ; Tiwari Nusser 2016 ; Bengaly et al. 2017 ; Bernal, Cardenas Motta 2017 ; Javanmardi Kroupa 2017 ), homogeneity is much harder to probe by observations (see e.g. Clarkson Maartens 2010 ; Maartens 2011 ; Clarkson 2012 ). As is well known, the smaller the scale we observe, the clumpier the universe appears. However, non-uniformities, such as groups and clusters of galaxies, voids, walls, and filaments, are expected in a Friedmann–Lemaître–Robertson–Walker (FLRW) universe according to cosmological simulations. In such a background, a transition scale is also expected, above which the patterns composed by these structures become smoother, eventually becoming indistinguishable from a random distribution of sources. This homogeneity scale r h has been identified and estimated at 70–150 Mpc h −1 , using data from several galaxy and quasar surveys (Hogg et al. 2005 ; Scrimgeour et al. 2012 ; Nadathur 2013 ; Alonso et al. 2015 ; Pandey Sarkar 2015 ; Laurent et al. 2016 ; Sarkar Pandey 2016 ; Ntelis 2017 ), although other authors have claimed no evidence for it (Sylos Labini, Montuori Pietronero 1998 ; Sylos Labini 2011 ; Park et al. 2017 ). In the context of the ΛCDM paradigm, an upper limit for the homogeneity scale was estimated by Yadav, Bagla Khandai ( 2010 ) to be r h ∼ 260 Mpc h −1 . Tests of homogeneity of the matter distribution by counting sources in spheres or spherical caps are not direct tests of geometric homogeneity, i.e. of the Cosmological Principle. Source counts on spatial hypersurfaces inside the past lightcone cannot be accessed by this method, since the counts are restricted to the intersection of the past lightcone with the spatial hypersurfaces. Instead, source counts provide consistency tests: if the count data show that the matter distribution does not approach homogeneity on large scales, then this can falsify the Cosmological Principle. Alternatively, if observations confirm an approach to count homogeneity, then this strengthens the evidence for geometric homogeneity – but cannot prove it. A test of homogeneity of the galaxy distribution that does probe inside the past lightcone has been developed by Heavens, Jimenez Maartens ( 2011 ) and Hoyle et al. ( 2012 ) – but this test is unable to determine a homogeneity scale. When a length-scale r h is used to probe homogeneity, a further assumption is made – a fiducial FLRW model is assumed a priori, in order to convert redshifts and angles to distances. In order to circumvent this model dependence, one can use an angular homogeneity scale θ h (Alonso et al. 2014 ). It was shown by Alonso et al. ( 2015 ) that the θ h determined from the 2MASS photometric catalogue is consistent with ΛCDM-based mock samples within 90 per cent confidence level. In this letter , we make tomographic measurements of θ h in the Luminous Red


ejecta, we assume the nucleosynthesis pattern following the recent numerical results for the postmerger ejecta which show significantly low production of lanthanide elements and give the mass fraction of 1.64 × 10 −2 for Y (Fujibayashi et al. 2018 ). We first propose that the degree of Eu enhancement indicated by [Eu/Fe] for individual stars (−0.7 ≲ [Eu/Fe ≲ +2) could be basically determined by the different levels of Eu enrichment of gas via associated merger events (cNSNSs or cBHNSs) with widely varying masses ( M tid ) of tidally driven dynamical ejecta (10 −4 M ⊙ ≲ M tid ≲ 0.1 M ⊙ ). Considering the [Eu/Fe] distribution of halo stars, we identify two components: a Gaussian distribution in the range of −0.7 ≲ [Eu/Fe] ≲ +1.3 with a peak around [Eu/Fe] ≈ + 0.3 and an extending Eu-rich tail up to [Eu/Fe] ≈ + 2. Then, considering the predicted mass range of dynamical ejecta, cNSNS and cBHNS are likely to be responsible for producing the former and latter distributions, respectively. Approximate mass values of the mass of tidally driven dynamical ejecta are shown in the upper part of the figure, implying that the average value for cNSNSs should be a few times 10 −3 M ⊙ . If cBHNS should produce more Eu than cNSNS due to larger tidal mass ejection, the Y/Eu plateau could be connected to such events. Considering very low production of light r -process elements in the tidally dominated dynamical ejecta of cBHNSs, owing to a low Y e (≲0.1) (Roberts et al. 2017 ), we argue that Y in r-II stars originates from the postmerger ejecta. The Y/Eu plateau implies that the production ratio of Y to Eu is quite similar in individual cBHNS events. As each element owes its origin to nontidal postmerger ejecta and tidally driven dynamical ejecta, respectively, the masses of the two type of ejecta are suggested to scale linearly with each other. This linearity may also hold even if the postmerger ejecta of cBHNSs produce Eu due to a smaller Y e resulting from lower neutrino irradiation (Fernández et al. 2017 ), though there are complexities in the influence of the neutrino treatment (e.g., Caballero et al. 2012 ; Wu et al. 2017 ). The required condition of a similar mass ratio of light r -process elements to Eu among the postmerger ejecta of cBHNSs needs to be investigated (see also Kyutoku et al. 2015 ). On the other hand, the trend of an increasing [Y/Eu] with decreasing [Eu/Fe] at the low [Eu/Fe] regime is likely to reflect the diversity of nucleosynthesis in cNSNSs with various mass asymmetry in our scheme. The tidal interaction tends to be efficient for asymmetric binaries, and M tid becomes large. At the same time, shock interactions are not very efficient for dynamical mass ejection in asymmetric binaries. Thus, the average values of Y e become lower for more asymmetric binaries (e.g., Sekiguchi et al. 2016 ). These combined effects could explain the observed Y/Eu trend. According to this view, cNSNSs with massive (≳0.01 M ⊙ ) tidally driven dynamical


orders, which yields a sufficient number of data points for our analysis and still benefits from a low-dimensional covariance. The fact that we do not find a decrease of parameter uncertainty with increasing number of orders shows that the first few orders indeed contain all the relevant information (see Fig. 7 for more details). The measured data points for both angular intervals are presented in Fig. 8 . Unlike correlation function measurements, these Υ data points cannot be interpreted easily. However, it is clear that in the large-scale interval the model (see Section 2 ) is a very good fit to the data regardless of the cosmological parameters used. 4 This is not the case for the smaller scale interval, where one of the clustering data points is several σ away from the best-fitting model. Clearly, the assumption of linear bias on these non-linear scales is not valid for the clustering data. This is partly due to the small uncertainties in these measurements as well as the fact that we neglect the model uncertainties. As can be seen in Fig. 9 , a change of b of about 5 per cent within this interval would already be enough to reconcile the data with the model. If model uncertainties had been included in this figure, it is likely that data and model would be in line again. Another possible explanation of the discrepancy between data and model in this case could be that the fiducial cosmology is wrong. Furthermore, we also investigate if an indication for a scale-dependent bias can be found in the γ t (θ) data in Fig. 10 and find no such preference. Figure 7. Left: the measured parameter values as a function of maximum Υ-order n for b and r . No significant difference in the values is visible, from which we conclude that the data compression is indeed working and only a few orders contain all the information from the measured signals. Right: the parameter uncertainty in per cent for b and r , again as a function of n . Here, we do not find a significant difference, which again shows that the data compression of Υ( n ) is robust. Figure 8. The top panels show the measured Υ gm and Υ gg and the best fit using one of the two cosmologies. The magenta and dark blue lines are the connections between the predicted data points using the Planck or the CFHTLenS cosmology. In the bottom panels, we show the residuals (Υ obs − Υ model )/ΔΥ obs , where ΔΥ obs is the uncertainty in the measured Υ. Left: measurements for the 3-20 arcmin interval. Clearly, on these scales the model we adopt to describe the galaxy bias is not a good description of the data shown here, especially the clustering data. For more details, see Section 3.4 Right: measurements for the 20-70 arcmin interval. Figure 9. Top: the angular correlation function within the small-scale interval, the corresponding model for b = 1, and the best-fitting model. This best-fitting model has been determined from a joint fit of Υ gm and Υ gg . Bottom: the square root of the ratio between the


of signal-to-noise ratio (S/N), and found it to be well described by a power-law function, . is about 0.01 mas in the innermost jet region where , and 0.1 mas downstream of the jet where , corresponding to 1/50 and 1/5 of the beam size, respectively. Therefore, we consider the upper limit of the imaging error to be because the jet radii are measured where the jet is detected when (see also Homan et al. 2002 , Appendix A for similar analysis). Note that we assume the time variability of the jet radius is negligible. 3.3. Radius Profile of the Jet Figure 2 shows the distribution of jet radius ( r ) as a function of the deprojected distance to the core ( z ), which we refer to as the radius profile of the jet. The jet viewing angle ( θ ) for deprojection is uncertain within an upper limit of 47° (Jones Wehrle 2002 ). We adopt θ = 19° derived from a detection of a sub-parsec scale counter-jet (Sudou et al. 2000 ). An independent model of the spectral energy distribution (SED) suggests a consistent result of θ ≃ 18° (Chiaberge et al. 2003 ). We use the deprojected length scale ( mas −1 ) along the jet throughout the rest of the paper. Note that the jet radius, as defined perpendicular to the jet axis, is not affected by projection effects. Figure 2. Top panel: the radius profile of the NGC 6251 jet in units of Schwarzschild radius ( r s ) and parsecs, using archival VLBA data at 15 and 5 GHz (green and blue triangles), and our EVN(-i, ii, iii) data at 1.6 GHz with three weighting schemes (violet square, cross, and circle). The single (orange solid line) and double (brown solid line) power-law models are fitted to the profile, where we show the case of the sharpness parameter n = 100 here. Statistics ( f -test) shows that the data are better fitted by double power-law models (with all n ) than by a single power law (see also Section 3 ). Furthermore, the double power-law model is also supported by the VLA data (gray plus signs) and VLBI cores (gray triangles and squares), as discussed in Section 4.1 . Note that the VLA data and VLBI cores are not included in the regression analysis. Middle and bottom panels: the relative residuals of the jet radii for the single () and double () power-law models. The single power law shows a systematic deviation at the two ends of the distance scale, while the double power laws (i.e., for all n ) show uniformly distributed relative residuals on all distance scales. We conclude that the jet structure is described by a parabolic shape upstream () and a conical shape downstream () with a transition located at a characteristic distance z 0 = (1–2) × 10 5 r s (brown dashed line), which is fairly close to the sphere of gravitational influence (SGI) of the SMBH (, black dashed line) in order of magnitude. 3.4. Collimation Profile by Power-law Models Collimation profiles are generally described by power-law functions with some local features, as a manifestation of the global evolution of jet structure. We perform


Einstein radius, and σ SIS is the velocity dispersion of the star in the periphery of the lens galaxy due to the lens mass distribution in the SIS model. In general, σ SIS does not equal the observed stellar velocity dispersion σ 0 (White Davis 1996 ). To express the difference, researchers use a phenomenological free parameter f e defined by the relation σ SIS = f e σ 0 (Kochanek 1992 ; Ofek et al. 2003 ; Cao et al. 2012 ), where (0.8) 1/2 f e (1.2) 1/2 . In this case, the systematic error is caused by σ 0 as σ SIS , the deviation of the SIS model, the effects of secondary lenses (nearby galaxies), the line-of-sight contamination (Ofek et al. 2003 ), etc. The uncertainty of Equation ( 5 ) can be written by (Liao et al. 2016 )In Equation ( 6 ), and are the fractional uncertainties of σ sis and θ E , respectively. To test the CDDR, the left term of Equation (ref5), R A ( z l , z s ), should be expressed as the ratio of luminosity distance, . We transform R A ( z l , z s ) into the ratio of comoving distance ( D C ) or dimensionless distance H 0 D C / c . In a flat space, the dimensionless distance satisfiesBy using the equation, R A ( z l , z s ) can be written as In a nonflat space, the expression of R A ( z l , z s ) is more complicated; one can refer to Räsänen et al. ( 2015 ). Fortunately, most cosmological tests today support a flat cosmic space (Planck Collaboration et al. 2016 ). In this work, we test the CDD relation in a case where we assume that spacetime is flat. To test the CDDR, Equation ( 1 ) is rewritten by the parameterization of the deviation:Combining Equations ( 9 ) and ( 10 ), R A ( z l , z s ) can be written asBy using Equation ( 4 ), of Equation ( 11 ) can be rewritten aswhere the absolute magnitude of the SNe Ia subsample is offset, z l is the redshift of the lens, and z s is the redshift of the sources. 3. Method and Results In this section, we introduce a distance-deviation consistency method of data selection and show the result. 3.1. Method of Data Selection: A Distance-deviation Consistency Method To test the CDDR, D L and D A at the same redshift z need to be provided simultaneously. However, redshifts of subsamples from the SGLS and the SNe Ia are different. To take full advantage of the data, one needs to pair the subsamples efficiently. In this section, we provide a distance-deviation consistency method to pair subsamples, which outperforms the conventional method. The redshift-difference of subsample pairs in this work is not fixed, and it decreases with the redshifts to ensure the distance deviation of the sources stays the same. The relation between Δ z and coordinate distance with a cosmology model readswhere R h = ct and flat ΛCDM model with Ω m = 0.31 (Planck Collaboration et al. 2016 ) are used. By setting Δ d c / d c equal to 5% and combining Equation ( 13 ), Δ z ( z ) can be calculated:where the distance formulas of the two cosmology model, ΛCDM and R h = ct are used. The R h = ct cosmological model was


are sub-Keplerian by ≈20%, in good agreement with other studies (e.g., Wade Horne 1988 ). Note that for q ≳ 0.25, the 3:1 resonance lies beyond ≈0.9 R L 1 and the outer disk radius is then limited by the largest non-intersecting orbits allowed by three-body interactions (Paczyński 1977 ). Under these circumstances the dependence of the outer disk radius on q is very weak (Frank et al. 2002 ) and therefore a nearly constant evolution of DP/FWHM versus q is expected for q ≳ 0.25. For practical purposes, in Figure 3 we display the variation of DP/FWHM versus q in logarithmic units. A least-squares linear fit yieldswith a Pearson correlation coefficient r = 0.95. To estimate the error in q implied by this relation, we computed the difference with respect to the true observed values for our 11 SXTs. The distribution of differences can be approximated by a normal function with σ ( q ) = 0.015. This indicates that mass ratios can be realistically obtained from Equation ( 6 ) with a typical ∼25% uncertainty. Figure 3. Correlation between q and the parameter DP/FWHM in log units and the best linear fit. 4. ORBITAL EFFECTS The spectra that we used to produce the DP/FWHM– q correlation are orbital averages. This is party because individual spectra rarely possess enough signal-to-noise for the technique to be applicable. But also because orbital means average out possible asymmetries in individual spectra from, for example, hot spots or disk eccentricities that could potentially bias the determination of q . At this point we decided to explore the impact of line asymmetries in the results of our technique. Since this can only be tested on data with sufficient signal-to-noise, we have focused on the 154 high-quality GTC spectra of XTE J1118+480 obtained along four different orbits over two years. We have performed Gaussian fits on every individual spectrum and computed mass ratios using Equation ( 6 ). The distribution of q values is found to peak at q = 0.024 (the bottom panel in Figure 4 ), with 68% of the values contained between q = 0.021 and 0.045. This indicates that if q were to be obtained from a single individual spectrum, the typical uncertainty would be about ∼56%. Figure 4. Bottom: histogram of q values derived through Equation ( 6 ) from 154 GTC spectra of XTE J1118+480. Top: variation q with the V / R parameter of the H α profile (see the text for details). In order to trace the effect of line asymmetries we also extracted the V / R parameter from each spectrum, with V / R defined as the ratio of equivalent widths between the blue and the red part of the H α profile. The two halves of the line are set from the rest wavelength till ±2500 km s −1 . For example, indicates a line with the red part stronger by 20% while implies a symmetric profile. A plot of q versus V / R (see top panel in Figure 4 ) seems to show a trend, with asymmetric profiles preferring slightly lower q values, although the large scatter prevents from drawing a firm


include in our list lines with intensity values that are above some fraction (typically 50%) of the maximum value. Of course, the choice of this threshold is arbitrary, but we find it works well for our needs, and can be easily modified as needed. In a companion paper (A. Gal-Yam, O. Yaron N. Knezevic 2019, in preparation), we describe in more detail our survey of other methods to compile line lists from the NIST databases and other tests and applications of this method. With the line lists in hand, we attempt to determine for each spectrum the expansion velocity by matching features to ions with numerous transitions (see. e.g., Figure 1 ). Once we have determined a convincing value for the expansion velocity, we inspect other ions using the fixed velocity (i.e., we assume explicitly that the photosphere is narrow). Figure 1. High S/N red spectra of the nearby SLSN-I PTF12dam (left; from Nicholl et al. 2013 ) and Gaia16apd (right; from Yan et al. 2017b ). Both spectra (black curves) have been binned and host narrow emission lines have been excised; the original spectra are shown in blue. Uncorrected Telluric features in the spectrum of PTF12dam are marked. The spectrum of PTF12dam shows pronounced lines of O i and C ii with distinct sharp absorption troughs, all at the same expansion velocity of V exp = 11,000 km s −1 ; O ii lines probably begin to shape the spectrum blueward of 5000 Å while C iii may also contribute. The same is true for Gaia16apd (right) though absorption lines are of lower contrast and have higher expansion velocity, V exp = 14,000 km s −1 . 2.1. Application to Peak Spectra of SLSNe-I We begin our examination of SLSN-I peak spectra by inspecting high signal-to-noise ratio (S/N) spectra of nearby events. In particular, we begin by focusing on the red side of the optical range ( λ > 5000 Å). This wavelength range has been relatively neglected in studies of SLSN-I spectra as it does not contain strong features, and is redshifted out of visible-light spectra of SLSNe at moderate or high redshifts, which are typical for SLSNe. Figure 1 (left) shows our analysis results for a high S/N spectrum of PTF12dam from Nicholl et al. ( 2013 ) obtained on 2012 May 24, 25 restframe days prior to peak according to the analysis of Vreeswijk et al. ( 2017 ). This and all other spectra discussed here are available from WISeREP (Yaron Gal-Yam 2012 ). 5 5 http://wiserep.weizmann.ac.il/ Multiple features consistent with O i and C ii lines all align using a single expansion velocity of V exp = 11,000 km s −1 ; these two ions explain most features in this wavelength range. C iii may explain two weak features, and may also contribute to the observed spectrum. O ii has a distinct line spectrum with a multitude of lines all with very similar intensities. While a few single lines seen around 6500 Å appear to be negligible compared to the stronger C ii lines, accumulations of multiple overlapping O ii lines begin to shape the spectrum bluewards of 5000


of star formation, with the younger and older parent stellar population dominating the brighter and fainter PNe respectively. Indeed Valenzuela et al. (2019) showed theoretically that the shape of the PNLF at the faint-end is quite sensitive to the distribution of masses of the PN central stars, which is a product of the SFH in any galaxy. In Bhattacharya et al. (2019b , hereafter Paper I), we observed a rise in the faint-end of the PNLF in the central 16 sq. deg. of M 31 conceivably linked to the SFH of M 31. Owing to the homogeneous spatial coverage and photometry of the current imaging survey, we detect PNe associated with all the stellar populations in a 54 sq. deg area centred on M31. Thanks to the sensitivity and accuracy, we can quantify the imprints of the variations of these stellar populations on the PNLFs in distinct regions. In Sect. 2 , we describe our PN survey of M 31 and set the geometrical boundaries around any regions with RGB overdensities where the PNLFs are measured. In Sect. 3 , we obtain the PNLFs, compute the parameters that best describe them, and carry out the quantitative comparisons via statistical methods. In Sect. 4 , we correlate the stellar population parameters with the properties of the PNLFs at both their bright and faint ends in the different subregions. In Sect. 5 , we discuss the implications of our results on the origin of the M31 inner halo substructures. We draw our conclusions in Sect. 6 . 2. Survey of Planetary Nebulae in M31 2.1. Survey characteristics In Paper I, we identified PNe candidates in a 16 sq. deg. uniform imaging survey of the disc and disc–halo interface of M 31 using the MegaCam wide-field imager ( Boulade et al. 2003 ) mounted on the 3.6-meter Canada-France-Hawaii Telescope (CFHT). M 31 was observed through a narrow-band [O III ] filter ( λ c = 5007 Å, Δ λ = 102 Å, on-band) and a broad-band g -filter ( λ c = 4750 Å, Δ λ = 1540 Å, off-band). An additional 38 sq. deg. were obtained in late-2019, providing uniform coverage of the inner halo of M 31 and its substructures reaching a uniform sensitivity down to fluxes equivalent to m 5007 = 26.2 mag over the entire area. Details of these observations are presented in Bhattacharya (2020) and Bhattacharya et al. (in prep.). The zero points for each of the [O III ] and g -band frames in AB magnitudes, normalised to a 1 s exposure, are Z [O III ] = 23.4 and Z g = 26.5. The photometry was calibrated using observations of spectrophotometric standard stars. The photometric uncertainty (err m , 5007 ) is well below 0.01 mag at m 5007 = 22.5 mag; it increases to err m , 5007 ∼ 0.15–0.2 mag at m 5007 = 26.2 mag. The PN identification in each of these pointings was also carried out as described in Paper I using the on-off band technique developed and validated in Arnaboldi et al. (2002 , 2003) , later optimised for large imaging surveys by Longobardi et al. (2013) . The PNe are identified as point-like sources with an excess [O III ] – g colour. The


accuracy and precision of the density and the potential depends on the choice of the parametrized density model used. Both the Gamma and the Einasto profiles produce nearly unbiased density-potential pairs when compared to observables, while the Einasto profiles are most sensitive to how the halo centers are defined. As noted in the Introduction, Serra et al. ( 2011 ) attribute the lower numerically integrated potential (Equation ( 3 )) compared to the NFW potential (Equation ( 4 b )) to mass outside the cluster that is not accounted for by the NFW profile shape. We show that the opposite is true. The NFW model density profile overpredicts the true density profile outside r 200 and thus overpredicts the true gravitational potential via the Poisson equation by 10%–15%. 3.3. Mass and Redshift Dependence Next, we examine how the edge varies as a function of halo mass and redshift. In this case, we use a new sub-set of the Millennium simulation with the 100 most massive halos halos smaller than 1 × 10 14 M . The minimum mass of this new subset is M , i.e., a factor of 10 smaller than the previous sample. We measure the spherically averaged density profiles and infer the escape edges via the potential from Equation ( 6 ). The edges are measured using the same algorithm as applied to the more massive subset studied previously. We find no statistical or systematic difference between the high mass and low mass halo data sets. The density profile predicts the escape edge via the Poisson equation to the same level of accuracy and precision for over two orders of magnitude in cluster halo mass. We study the low mass clusters at four different simulation snapshot outputs, corresponding to , and 0.75. We keep the 100 most massive halos in each snapshot as provided by the “millimil” subset of the Millennium data. At , the deceleration parameter goes from negative to positive. As the cluster density profiles evolve with redshift, we fit the profiles separately for each halo at each redshift. Instead of physical coordinates, we use radial coordinates with respect to the r 200 of each cluster for the profiles, due to the fact that the cluster sizes also evolve with redshift. In Figure 4 we show that within r 200 , the edge can be accurately predicted from the density and Poisson equation to z = 0.75. However, outside the virial radius, the edge is increasingly underpredicted compared to the model, as a function of increasing redshift. We find the same result when using the mean background density as opposed to the critical background density when defining the cluster masses and radii. Figure 4. The variation in the fractional difference between the measured v esc edge and the predictions from the density profile from redshift 0 to 0.75. The light (dark) gray bands represent the 67% (90%) scatter of the individual halos. We can explain this by the growing influence of the halo gravitational potential well on the dynamics of the infall regions around the


regions described in Section 4.3 . In Figure 6 , we present again the oxygen abundance and ionization parameter gradients and color-code the data points based on the different groups defined in Figure 5 . Figure 6 further demonstrates the azimuthal variation visible on the oxygen abundance map (Figure 3 ). The H ii regions that belong to the spiral arms have systematically higher oxygen abundances than the other regions at the same galactocentric distances. The upstream regions have oxygen abundances in between the main spiral and downstream H ii regions. A total of 73 of the 96 (76%) main spiral H ii regions are above the best-fit abundance gradient. In contrast, only 13/76 (17%) of the downstream H ii regions are above the best-fit abundance gradient. The distributions of the residuals, after the best-fit radial gradient is subtracted, are statistically different (Figure 6 histogram). Two-sample Kolmogorov–Smirnov tests suggest that the null hypotheses that the residuals of the spiral-downstream, spiral-upstream, and downstream-upstream regions are drawn from the same distributions can effectively be ruled out (, , and , respectively). Figure 6. Oxygen abundance (left panels) and ionization parameter (right panels) gradients in NGC 1365. Each color point corresponds to one H ii region in Figure 3 . The different color symbols correspond to the different groups of H ii regions shown in Figure 5 . The black solid lines are the overall best fits. The red, blue, and green dashed lines are the linear best fits to the “Spiral,” “Downstream,” and “Upstream” H ii regions, respectively. The bottom panels show the residuals, i.e., after subtracting the black solid lines. The best-fit parameters are tabulated in Table 2 . We also perform the same least-squares linear fits to the spiral arm, downstream and upstream H ii regions. The best-fit oxygen abundance gradients shown in Figure 6 demonstrate the systematic differences between the different groups of H ii regions. The best-fit parameters are tabulated in Table 2 . It is worth pointing out that the majority of the upstream H ii regions (21/27; 78%) are in between the best-fit gradients defined by the spiral arm and downstream H ii regions. In Figure 7 , we present the azimuthal variations more clearly by subtracting the best-fit radial gradient from the oxygen abundance map. If there are no azimuthal variations, the residual map should present only random fluctuations. However, Figure 7 shows that the residuals correlate strongly with the spiral pattern. Figure 7. Oxygen abundance map after subtracting the best-fit radial abundance gradient (i.e., Figure 6 ). The azimuthal variations of oxygen abundance are now clearly visible. To summarize, while overall there is a negative oxygen abundance gradient in NGC 1365, we find pronounced azimuthal variations. The H ii regions on the spiral arms present, on average, about 0.2 dex higher oxygen abundance than the downstream H ii regions. The upstream H ii


focal ratios of these telescopes, along with the use of single, photographic-film size CCDs, allow them to probe large areas of galaxy haloes while reaching surface brightness levels ∼2–3 mag deeper (μ r ∼ 28 mag arcsec −2 ) than the classic photographic plate surveys (e.g. Palomar Observatory Sky Survey) and the available large-scale digital surveys (e.g. the Sloan Digital Sky Survey). Here, we use this amateur telescope discovery approach to report a faint, elongated galaxy projected on to the halo regions of NGC 253 in the Sculptor Group. With a distance of 3.5 Mpc, NGC 253 is one of the nearest large spirals, and has been well studied (e.g. Davidge 2010 ; Bailin et al. 2011 ; Greggio et al. 2014 ; Monachesi et al. 2015 ), but this object, which we call NGC 253-dw2, had been missed. 1 2 OBSERVATIONS We first noticed a candidate satellite galaxy in a visual inspection of several images of NGC 253 available on the internet that were taken by amateur astronomers: Alessandro Maggi using a Takahashi Epsilon 180ED astrograph (18-cm diameter at f /2.8) and Mike Sidonio using a 12-in f /3.8 Newtonian (see Fig. 1 ). In both cases, the pixel scale was coarse and it was not clear if the elongated feature was real or an artefact or reflection as often present in amateur images. Figure 1. Left: amateur images of NGC 253 and its satellite NGC 253-dw2. The grey-scale image is from the 12-in telescope luminance filter, with colour image of NGC 253 from the 0.5 m Black Bird Remote Observatory, and field of view of ∼11 × 10 (67 kpc × 60 kpc). North is up and east is left. The zoom-in on NGC 253-dw2 is from CHART32 and covers ∼8 kpc × 7 kpc. Right: Subaru/Suprime-Cam image, colourized using g and R c bands, and demonstrating that the satellite is composed of red, discrete stars. We investigated the nature of NGC 253-dw2 by follow-up observations using amateur and professional facilities as follows. First, deep imaging of the field centred on the dwarf candidate was collected remotely with the Chilean Advanced Robotic Telescope (CHART32), an 80-cm f /7 corrected Cassegrain telescope located at the Cerro Tololo Inter-American Observatory, Chile. An FLI PL-16803 CCD camera was used, with a pixel scale of 0.331 arcsec over a 22 arcmin × 22 arcmin field-of-view. Two overlapping sets of 20 × 1200 s individual image frames were obtained through a Baader luminance filter over several photometric nights between 2013 September and November. Each individual exposure was reduced following standard image processing procedures for dark subtraction, bias correction, and flat-fielding (Martínez-Delgado et al. 2010 ). The images were combined to create a final co-added image with a total exposure time of 48 000 s. Part of the resulting CHART32 image is shown as an inset in Fig. 1 (left-hand panel). The candidate object looks fairly regular and diffuse, like a nearby dwarf spheroidal (dSph) galaxy rather than a background galaxy or an instrumental artefact. It is also very


The Gini coefficient is strongly dependent on the noise level of imaging data, and suffers from large uncertainties when signal-to-noise ratio (S/N) is not sufficiently high (Lisker 2008 ). For asymmetry A , noise contribution is estimated by measuring A on a nearby blank sky region with the same aperture. However, the noise correction for A overestimates the real noise contribution (see Figure 1 of Shi et al. 2009 ). Here, we present a detailed analysis on the noise effects in quantifying A , and introduce a revised correction for noise contamination. We construct a sample of model galaxies to test the morphological parameter A . The sample consists of 569 isolated galaxies from Wen et al. ( 2014 ) (see Section 3.2 for more details about model generation). An artificial image is obtained through combining a model galaxy image with a background noise image. The parameter A is measured from both the artificial image and the noise-free model image, following the definition given in Conselice ( 2003 ) aswhere I 0 is the intensity of the galaxy and I 180 is the -rotated intensity. The second term refers to the correction for noise contamination. A group of pixels are randomly selected from the background regions to form the background noise image. In the measurement, an aperture of 1.5 times the circular Petrosian aperture is adopted to estimate the noise contamination. Figure 12 presents comparison of the measure of A from noise-added and noise-free images for the 569 model galaxies. We find that noise contamination leads A to be systematically overestimated, and the significant scatter indicates the uncertainties of noise contribution—about 20% for a high-S/N galaxy, and up to 50% for a low-S/N galaxy (Shi et al. 2009 ). The noise correction introduced in Conselice ( 2003 ) overestimates the noise contamination and biases A to be smaller. This is because all background pixels within the Petrosian aperture are used to calculate the noise contamination. As discussed in next section, however, we argue that noise contamination is correlated with the number fraction of noise dominated pixels of a galaxy, indicating that only part of the noise contribution estimated by Equation ( 6 ) is real. Furthermore, the measured A from noise-added images is found to be lower than that from noise-free model images, in a few cases, even if no noise correction is adopted. This is because that noise dramatically changes the segmentation maps of the model galaxies. In our measurements, only pixels within 1.5 times the circular Petrosian aperture are counted to calculate A , and the extended structures of the model galaxies out of the aperture are ignored, leading to an underestimation in A . For a noise-free model galaxy image, all pixels associated with the galaxy are used to compute A . Figure 12. Comparison of asymmetry measurements from galaxy images, with and without noise. Asymmetry of the noise-free model image is marked as . For galaxy image with


TNG100 parent galaxy sample selection . Red dots show the positions of central disc galaxies in TNG100 with 10 10 ≤ M stell /M ⊙ ≤ 10 11 . Orange dots point to satellite disc galaxies in the same mass range; to avoid Virgo-like clusters, we require that satellite galaxies have no more than one massive () neighbour within 1 Mpc. Galaxies rejected on the basis of these environmental and morphological criteria are shown in black, for reference. In the background, the spatial distribution of all galaxies in TNG100 is shown in blue. In addition to this parent sample, we also defined smaller samples of stellar mass-matched TNG100 galaxies for each DNGS galaxy. This time we applied a more observer-friendly definition of stellar mass – specifically the integral of the stellar mass surface density profile. We denote this ‘observed’ stellar mass as M stell, obs . When calculating TNG100 stellar masses in this way, we only integrated the density profiles down to a floor of kpc −2 , as this corresponds to the typical depth of the DNGS observations (see Section 5.1 for more details). The physical radius corresponding to this surface density threshold varies between ∼20 and 70 kpc for the DNGS sample, although we note that this is an extrapolation (from a last observed point of kpc −2 ) in two of the eight galaxies. TNG100 galaxies exhibit an even larger variation in physical extent under this definition, ranging from ∼40 to 140 kpc. We can infer from these numbers that TNG100 galaxies are more extended than DNGS galaxies, and we will return to this idea in Sections 6 and 8.2 . As a consequence of the shape of the stellar mass function, if we were to simply select all TNG100 galaxies that fall within the observational errors (see Table 1 ), each mass-matched sample would remain strongly weighted towards low-mass galaxies by number. To alleviate this issue, we instead used rejection sampling (Press et al. 1992 ) to obtain a normal distribution of TNG100 galaxy stellar masses for each DNGS galaxy. We calculated the probability P ( M stell, obs ) of drawing each TNG100 stellar mass from a Gaussian distribution with a mean and standard deviation set by M stell, DNGS and , respectively, and accepted the first 50 TNG100 galaxies with P ( M stell, obs ) less than a random number drawn from the same distribution. We emphasize that, in addition to the definition of the mass-matched samples, all comparisons between observations and simulations were done using quantities measured from TNG100 galaxies (e.g. surface density profiles or stellar halo mass fractions) in a manner identical to the DNGS sample. When exploring parameter space that contains unobservable properties, however, we will occasionally use the ‘true’ stellar mass instead. In the following sections we will explicitly distinguish between the parent and mass-matched samples, and between ‘true’ and ‘observed’ stellar masses where applicable. 3 FROM STELLAR PARTICLES TO IMAGES The first step towards


with the power-law function. The fits included wavelength intervals that were free of BALs and ranged from approximately 1260–1350, 1370–1480, 1530–1800, and 1870–2500 Å. We integrated the P v BAL apparent optical depth profiles using the following equation, in Gaussian units: where f is the oscillator strength, λ is the rest wavelength, and τ( v ) is the apparent optical depth at a given velocity (Savage Sembach 1991 ). Rest wavelengths and oscillator strength values were taken from Morton ( 2003 ), and we treat the P v λλ1117,1128 doublet as a single line at λ = 1121 Å with an average oscillator strength of f = 0.705. For sources that showed non-variable P v BALs, we integrated τ( v ) in accordance with equation ( 1 ). For sources that showed variable P v BALs, we integrated the optical depth difference to generate lower limits for the change in P v column density. 4 RESULTS AND DISCUSSION 4.1 Absorption line variability We detect definite (category Y, see Section 3.1 ) variability from any BAL (i.e. from C iv , Si iv , or P v ) in 10 out of 71 BAL quasars in our sample (see Appendix A for notes on individual objects). Spectroscopic comparisons in our sample probe rest-frame time-scales between 0.5 d and 3.8 yr, and we measure BAL variability time-scales over ≤ 0.2–3.8 yr for our Y sources. Tentative (T) variations from any BAL are detected in 44 sources, and 17 objects exhibit no (N) BAL changes. Table 3 lists the associated ion, measured variability time-scale, LOS velocity of the variable region, and variability direction for each variable (Y) BAL in our sample. BALs not presented in Table 3 show no significant absorption line changes over the listed time-scales. For example, there are four cases of P v and/or Si iv BAL variations with no associated C iv BAL changes in the same time-scale comparison (see 013652 + 122501, 104059 + 055524, 104247 + 061521, and 0.5 yr for 122654−005430 in Table 3 ). We detect four cases of C iv BAL variability with non-variable P v absorption at similar velocities (see 2.4 yr for 122654−005430, 0.8 yr for 140532 + 022947, along with 2.0 and 0.8 yr for 221326 + 003846 in Table 3 ). There are five cases of C iv BAL changes with likely no associated P v absorption at similar velocities (see 102154 + 051646, 114548 + 393746, 2.4 and 0.8 yr for 122654−005430, and 2.4 yr for 140532 + 022947 in Table 3 ). In two cases (025042 + 003536 and 230721 + 011118; see Table 3 ), it is unclear if the C iv BAL variations are associated with P v absorption at the same velocity. Table 3. Variable BAL properties. Information listed for each BAL quasar includes (from left to right): ions associated with variable BALs, measured time-scales of BAL variability (upper limits that are limited by temporal sampling), LOS velocities corresponding to maximum flux changes within the BALs, and whether the BAL depths increased (+) or decreased (−). Multiple, distinct time-scales listed for the same ion are in chronological order.


also listed in Table 2 . The decay time-scale is in these cases of the order of or shorter than the decay time of the previous stages, meaning that the coalescence would supposedly happen promptly once the hard binary is formed. None the less, the hardening and inspiral time-scales we inferred in Table 2 are definitely longer than the ≈15 Myr found by Khan et al. ( 2016 ), who addressed the pairing of ∼10 8 M ⊙ MBHs in a massive early-type galaxy at z ≈ 3, which would become a BCG by z ≈ 0. The discrepancy in the time-scales has to be attributed to the large central density in their merging systems, which is nearly 10 5 M ⊙ pc −3 at 10 pc (their fig. 6), i.e. ∼100 times larger than in our case (see also the discussion in Khan et al. 2018 ). 13 Note that the system they explored is not a main-sequence galaxy as the one presented in this paper, but instead it is a more massive and relatively rare galaxy. For this reason, we expect the time-scales we obtain in this paper to be more representative for the bulk of the galaxy population. Note that the hardening and inspiral time-scales extrapolated for our system do not account for the effect of gas (which is expected to assist the orbital decay at scales larger than a pc, but could hinder it at smaller scales; Souza Lima et al. 2020 ), and the possibility that the fragmentation of a non-resolved circumnuclear disc could render even this smaller-scale decay very stochastic (Fiacconi et al. 2013 ; Souza Lima et al. 2017 ). If we neglect these complications, our estimates suggest that the whole pairing process could take as short as a few hundred Myr and probably less than a Gyr for the majority of the studied MBHs; that is, the MBH merger is likely going to occur at z = 5–3.5, except for the MBH that undergoes the bar-induced ejection. Concerning the fate of this latter object, one could wonder how likely this scattering event is. Even if our statistical sample is too limited to make strong statements here, it is worth mentioning that a test simulation with virtually the same initialization as the current one also showed evidence for a bar slingshot, suggesting that the event can be relatively common. 4.5 Caveats and limitations Our simulation has the important limitation that MBH accretion and feedback were not taken into account. However, it is reasonable to expect that feedback would render DF even less efficient by evacuating the gas mass around an MBH (see e.g. Park Bogdanović 2017 ; Souza Lima et al. 2017 ; del Valle Volonteri 2018 for a smaller scale analysis of this aspect). Each inspiralling MBH may also enhance its mass by accreting gas along the decay: in this respect, it is worth remembering that the MBH masses we adopted in the current run are relatively large for z ≳ 6, thus we do not expect this to represent a major issue. We also note that accretion on to MBHs may be used in future high-redshift surveys to sample offset or dual active galactic nuclei (e.g. Capelo et al. 2017 ; De


i.e. brighter quasars may live in a region of higher overdensities (e.g. see Guimarães et al. 2007 , and references therein). In order to check this effect, we devised a subsample of low and high luminosity. In the low- and high-luminosity bins, we have taken 25 lowest and 25 highest (∼29 per cent of the total sample) luminosity quasars of our sample. As can be seen from Fig. 7 , the excess pixel optical depth ratio is more pronounced for the subsample of higher luminosity with log L 912 > = 31.85 ergs s −1 Hz −1 as compared to the subsample of lower luminosity with log L 912 > = 31.12 ergs s −1 Hz −1 . The integrated value under the curve (e.g. see Fig. 7 ) between −6 ≤ r ≤ 0 Mpc is found to be higher by a factor of 3.70 ± 0.22 and 1.62 ± 0.15 w.r.t. the corresponding integrated value of curve for high- and low-luminosity subsamples, respectively. However, we do notice that higher luminosity bin has a higher median redshift of z em > = 4.06 in comparison to z em > = 3.85 for the lower luminosity bin (e.g. see grey shade). However, the above R τ ( r ) difference, especially in the −2 ≤ r ≤ 0 Mpc bins, cannot be merely explained by the redshift difference in these two subsamples. Figure 7. Left-hand panel : Same as right-hand panel of Fig. 6 , but for the extreme 25 low- and 25 high-luminosity quasars in our sample (e.g. see text in Section 4 ). The Lyman continuum luminosity was calculated using V -band magnitude (e.g. see Section 2.1 .) Right-hand panel : The plot shows radius versus flux (similar to Fig. 4 ) for the transmitted flux of the subsample of extreme 25 low- and 25 high-luminosity quasars. The transmitted flux of the control sample of IGM for the higher luminosity and lower luminosity quasars is shown in red-dashed and blue dot–dashed lines, respectively. Additionally, we also used our sample to check for any difference of density profile among radio-loud (RL) and radio-quiet (RQ) quasars (e.g. see Sopp Alexander 1991 ; Dunlop et al. 2003 ; Gopal-Krishna, Mangalam Wiita 2008 ; Retana-Montenegro Röttgering 2017 ) where the former is generally associated with the presence of jets. Such bimodality is quantified by a radio-loudness parameter ( i.e. ) defined as a ratio between the rest-frame flux densities at 5 GHz and 2500 Å (see also Ivezić et al. 2002 ; Kellermann et al. 2016 , and references therein), with its value ≥ 23 for RL sources as adopted by Ganguly et al. ( 2013 ). Out of the 85 quasars in our sample, 62 have information about its radio-loudness parameter with 12 being RL (i.e. 23) quasars and 50 being RQ (i.e. ) quasars (e.g. see Perrotta et al. 2016 ). We plot R τ ( r ) for these two sub-samples (i.e. RL and RQ quasars) as shown in Fig. 8 . From this figure, it is evident that their R τ ( r ) profile is quite similar in both sub-samples, though the sample size of RL quasars is small here (12 quasars). Here, we also note that the typical luminosity (median) among these two sub-samples is almost similar being 2.23 × 10 31


et al. ( 2006 ) and Kuwabara et al. ( 2009 ) in a numerical model best fitted to the measurements taken by the muon network at Earth (rigidities 50–100 GV) and was utilized to determine the orientation of the interplanetary FR. The diffusion–expansion approach was also implemented by Subramanian et al. ( 2009 ) and Arunbabu et al. ( 2013 ) in an analytical model and was compared to the GRAPES-3 muon telescope (rigidities 12–42 GV). These authors estimated the radial perpendicular diffusion coefficient to be of the order of 10 21 . However, it should be noted that their studies are adjusted to large events associated with a shock and observed at high particle rigidities. At rigidities of about 1 GV, Cane et al. ( 1995 ) estimated a diffusion coefficient of about 10 19 by utilizing spacecraft measurements and less energetic ejecta. Both are roughly in agreement with values obtained from the typical empirical expression used in numerical models (see Equations (23) and (24) in Potgieter 2013 ). Therefore, these studies outline a rough constraint on the diffusion coefficient used to model FR-associated FDs. Since both diffusion and expansion are processes that presumably start close to the Sun, FRs, that is, CME initial properties, need to be considered in the model. This ejecta-only FD model primarily describes FDs associated with ICMEs without a shock/sheath region, because a shock/sheath region presumably introduces an additional decrease via different physical mechanisms, but could also be applied to explain the ejecta part of an FD in two-step FDs. The aim of this study is to take these considerations into account in the diffusion–expansion model to improve our understanding on the cause, formation, and evolution of FDs. 2. Basic Diffusion Model First we consider an analytical FD model based on the perpendicular diffusion of GCRs into the FR, similar to Cane et al. ( 1995 ). In this basic diffusion model, the FR is regarded as a closed magnetic structure, rooted at the Sun, that is initially empty of GCRs (Figure 1 (a)). The FR is of cylindrical form (hereafter approximated by a long cylinder), it moves with constant velocity, and as it moves, it does not vary in shape or size (Figure 1 (b)). We note that the assumption of constant velocity is in general not valid, as it was shown that CMEs slower than the solar wind accelerate, whereas CMEs faster than the solar wind decelerate (e.g., Sheeley et al. 1999 ; Gopalswamy et al. 2000 ), which is attributed to the magnetohydrodynamical drag (see, e.g., Vršnak et al. 2013 ; Hess Zhang 2014 ; Sachdeva et al. 2015 , and references therein). However, since the acceleration/deceleration depends on the difference between the CME and solar wind speed, for a substantial subset of CMEs (and especially the slower ones that do not drive shocks), a constant velocity can be taken as a relatively fair approximation. We also note that the assumption of constant shape and size might not hold true, as it was


distribution ranging from 0° to 10°. Asteroids which were initialized at the beginning of the simulation with two different eccentricity distributions depending on the simulation. The first is circular asteroids, and the second are asteroids with eccentricities chosen between 0.3 and 0.7. We emphasize that we are not attempting to reproduce realistic exo-debris discs in these systems (which would require a more detailed Yarkovsky model), but rather are placing bounds and observing trends. The evolution of most initially eccentric asteroids that we simulated are not shown, unless they provide a revealing physical trait. Asteroids which were initialized at the beginning of the simulation with orbital angles chosen from a random uniform distribution across their entire ranges. Yarkovsky prescriptions according to the following values of : Our choices for Models A–C arise from our intention to place bounds on the motion by considering only extreme cases. We approximate these extreme cases by focusing on and from the analytic limits in equation ( 25 ). In terms of semimajor axis, Model A pushes asteroids outwards, Model B pushes asteroids inwards, and Model C does not push them outwards or inwards. Alternatively, Models A and B have only a minor effect on asteroid inclination evolution, whereas Model C has a relatively large effect. We actually ran additional simulations with other combinations of entries, but found that they represent intermediate cases to Models A–C, supporting the analytics and our choices. 4 SIMULATION RESULTS The asteroids in our simulations are accelerated by three major effects: (i) mass-loss from the star, (ii) perturbations from the planet, and the (iii) radiation-driven Yarkovsky effect. 4 Mass-loss always pushes the asteroids away from the star (the orbital pericentre is always increasing, even in non-adiabatic motion; Veras et al. 2011 ), whereas the planet and the Yarkovsky effect may perturb the asteroids in either direction. Although both mass-loss and planets can increase e ast to unity (causing escape; e.g. Veras et al. 2011 ; Adams, Anderson Bloch 2013 ), Yarkovsky acceleration cannot. Veras et al. ( 2015 ) investigated the relative magnitudes of stellar mass-loss and the Yarkovsky effect, but did not include a planet nor run numerical simulations. The Yarkovsky drift varies in a complex manner, and is a function of all orbital elements, even in our simple approximation where the matrix elements of remain constant over time. However, equations ( 25 )–( 26 ) reveal that for small inclinations, the key parameters are and . Although the Yarkovsky acceleration drops roughly as , the acceleration increases linearly as a function of L ⋆ ( t ). The time dependence of L ⋆ ( t ) is, in turn, non-linear, and dominates changes in the Yarkovsky acceleration. Mass-loss is largely adiabatic within about 10 3 au (Veras et al. 2011 ), meaning that within this region it feeds back into the Yarkovsky acceleration only through an


the motivation for the selection of the volume for LASr. Next, we require a galaxy parent sample highly complete in terms of galaxies sufficiently massive to host an AGN, which can then be used to select AGN from. We will see that current local galaxy samples do not fulfil this criterion so that we have to assemble our own galaxy parent sample. Finally, we describe the assembly of the galaxy properties relevant for this work, namely the coordinates, redshifts, and distances, allowing us to find the MIR counterparts of the galactic nuclei and compute their luminosities. 2.1 Selection of the volume We wish to construct a highly complete census of SMBH growth in the local Universe. The choice of volume to be used for this purpose is motivated by several factors. In order to obtain a census that is representative for the whole AGN population, the volume needs to be representative of the larger scale, low redshift Universe. It is estimated that cosmological isotropy is reached for length scales of ∼200 h −1 Mpc with h = H 0 /100 km s −1 Mpc −1 and H 0 the Hubble constant (Sarkar, Pandey Khatri 2019 ). The volume should also be large enough to sample rarer AGN sub-populations in sufficient numbers to yield statistically robust conclusions on their relevance. Here particular emphasis should be on the high luminosity AGN regime because these may dominate the integrated black hole growth and AGN feedback (e.g. Aird et al. 2010 ; Fabian 2012 ). However, high-luminosity AGN have a low space density. For example, current estimates of the AGN luminosity function in X-rays, e.g. Aird et al. ( 2015 ), let us expect a space density of Mpc −3 for AGN with an intrinsic X-ray luminosity of , e.g. ∼20 objects within a sphere of 200 Mpc radius. On the other hand, the volume should be small enough so that the depth of the all-sky surveys, used to identify and characterize the AGN, is sufficient to probe the lower parts of the AGN luminosity range. This is particularly important in the X-rays where extinction is large for obscured AGN. For example, the final all-sky maps of eROSITA and ART-XC are expected to have depths of and at 2–10 keV, respectively (Merloni et al. 2012 ; Pavlinsky et al. 2018 ), which corresponds to a distance of 150–250 Mpc for an observed X-ray luminosity of . However, CT AGN are suppressed by easily a factor of 10 to ∼100 at these wavelengths. The MIR is much less affected by extinction, but sensitivity is the key restricting factor, i.e. the WISE all-sky maps have an average depth capable of detecting an AGN with a 12 luminosity up to a distance of 220 Mpc with ≥3σ in band 3 (W3∼11.6 mag; WISE documentation 1 ). Another factor to take into account is that the completeness of our parent sample of galaxies directly restricts the completeness of our AGN search. According to a recent estimate, our all-sky redshift completeness is only for galaxies with a redshift z 0.03 (Kulkarni, Perley Miller 2018 ), and the completeness is quickly


(Figure 4 ). Furthermore, we collected emission line fluxes of SLSN host galaxies from the literature (Lunnan et al. 2014 ; Leloudas et al. 2015 ; Perley et al. 2016 ; Chen et al. 2017a , 2017b ) and corrected for foreground and internal dust extinction given by the Balmer decrement. SLSN host galaxies are located predominantly in the high ionization region with log([O iii ]/H β ) > 0.5 for any given [N ii ]/H α . This region is populated by extreme emission line galaxies as Leloudas et al. ( 2015 ) pointed out. The positions of SN 2017egm and PTF 10uhf are in a distinct region of the BPT diagram to most other SLSN hosts. This may indicate that SLSNe extend out to large metallicities than previously believed, and may cover a similar region to gamma-ray burst (GRB) host galaxies (see Figure 8 in Krühler et al. 2015 ). Improved completeness of low-redshift SN samples should indicate if SN 2017egm (and PTF 10uhf) are outliers or at the extreme end of a continuous distribution. 3.4. Stellar Ages and Progenitor Masses of GRBs and SLSNe We used the EW of H α as a tracer of the stellar population age (e.g., Leloudas et al. 2011 ; Kuncarayakti et al. 2016 and references therein). In general, a higher EW H α corresponds to a younger age, though the relation between the EW and age is dependent on the star formation history, initial-mass function, metallicity, and binary or single stellar populations (e.g., Eldridge Stanway 2009 ). We collected the EWs of H α at the explosion sites of SLSNe and GRBs from IFU and resolved long-slit spectroscopy (for references see Table 1 ), and estimated the stellar population ages using consistent models. We applied the Binary Population and Spectral Synthesis (BPASS 11 11 http://bpass.auckland.ac.nz version 2.1 models (Stanway et al. 2016 ; Eldridge et al. 2017 ) in combination with the photoionization Code CLOUDY (e.g., Ferland et al. 2013 ). These models will be described in L. Xiao et al. (2017, in preparation), and have been used to study the ages of SNe in Xiao et al. ( 2017 ). Table 1 Physical Properties Inferred from IFU and Resolved Long-slit Spectroscopy at SLSN and GRB Explosion Sites Note. For consistency, we re-calculate the stellar population age using the BPASS model. References are: (a) this work, (b) Cikota et al. ( 2017 ), (c) Thöne et al. ( 2015 ), (d) Krühler et al. ( 2017 ) (GRB 980425/SN 1998bw), (e) Izzo et al. ( 2017b ) (GRB 100316D/SN 2010bh), (f) Thöne et al. ( 2014 ) (GRB 060505/SN-less), (g) Tanga et al. ( 2017 ) (GRB 111005A/SN-less). To calculate the ages of the stellar populations, we select models that have a [O/H] value closest to that measured at the explosion site using the O3N2 calibration (Table 1 ). For each of the SLSNe we then provide two ages, one assuming the stellar population is comprised of single stars only and the second of binary stars. We see that there is a significant difference in the stellar population ages that we derive depending on the BPASS model used. The


O 2 and O 3 are also central to the design of instruments for other future ground- and space-based telescopes (Domagal-Goldman et al. 2014 ; Harman et al. 2015 ; Meadows 2017 ; Meadows et al. 2017 ). Here, we argue that O 2 and O 3 are reliable bioindicators under certain conditions. We outline what those conditions are, and discuss the secondary measurements and constraints necessary to demonstrate the presence of such conditions on another world. 2. False Positives in the Literature For the purposes of this paper, a “false positive” for life is defined as any abiotically derived O 2 concentration that exceeds that estimated to follow the GOE (Harman et al. 2015 ). If the low estimates by Planavsky et al. ( 2014 ) are correct, then that level is at or below 0.1% PAL. Possible mechanisms by which that abiotic O 2 level might be achieved have been recently reviewed by Meadows ( 2017 ) and Meadows et al. ( 2017 ). Broadly speaking, these scenarios can be divided into two categories: (1) false positives driven by changes in the redox budget of a planet’s combined atmosphere/ocean system (as defined in Catling Kasting 2017 , Ch. 8), and (2) false positives driven by photochemistry within the atmosphere. Examples of the redox-driven false positives include planets that experience substantial water loss because they are located inside the inner edge of the habitable zone (Kasting 1988 ; Kasting et al. 1993a ), or their host (M) stars are relatively bright during their pre-main sequence lifetimes (Ramirez Kaltenegger 2014 ; Luger Barnes 2015 ; Tian 2015 ), or they lack non-condensable gases like N 2 that help keep water vapor contained in the lower atmosphere (Wordsworth Pierrehumbert 2014 ). In each of these cases, hydrogen is irreversibly lost to space, oxidizing the atmosphere and oceans and, ultimately, the planet itself. The ultimate fate of this oxygen is uncertain but it could potentially be drawn down to undetectable concentrations by absorption into the mantle (Wordsworth et al. 2018 ). Other types of redox-driven false positives essentially correspond to hypothetical planetary scenarios that lack a volcanic source of reduced gases (e.g., Hu et al. 2012 ; Domagal-Goldman et al. 2014 ; Gao et al. 2015 ). If volcanic outgassing is assumed to be nearly zero, or the recombination of CO and O 2 is inefficient, or there are no surface sinks for O 2 , then O 2 can build up in the atmosphere. If reduced volcanic gas emissions are entirely neglected, O 2 can and must accumulate as H 2 O is photolyzed and the resulting hydrogen escapes to space. But we argue that a false positive driven by such low outgassing rates is unlikely if the planet is habitable, so that liquid water can facilitate reactions of O 2 with its crust, and if the planet remains volcanically active for long time periods, as the Earth has done. (We return to this last point in Section 6 .) The false positive scenario described by Hu et al. ( 2012 ) is more complicated in that it has


Plummer profile reads while the normalized projected number of stars for the same model is where φ( X ) is the angle corresponding to the effective circular section where we analyse the data (see Fig 1 ), and whose analytical expression for small cone apertures (i.e. R max ≪ 1 radian) is where R max is the maximum radius of the original cone search (usually two degrees) and d is the distance between the fitted centre and the centre from the original cone search (), set as the source centre on SIMBAD 2 by the automatic Gaia advanced query. The projected radius R is defined, in spherical trigonometry, as where () is the centre of the galactic object. The likelihood function is therefore written as Figure 1. Geometry of the centre estimation: The original cone search, centred in () is shown as the grey circle of radius R max , while the new fitted centre is (). The angles φ 1 and φ 2 represent the effective circular sections (of respective radii R 1 and R 2 ) where we consider circular symmetry. The distance between the two centres is labeled as d , and we consider this value to approach zero for the cone searches around the centres from the SIMBAD data base. We minimize with the differential_evolution routine from the scipy.optimize method and find the centres that best fit the data, later displaying them in Table C1 . For simplicity, since we expect the quantity d to approach zero, given the reliable previous measurements of the centres in SIMBAD, one can assume the case where , and thus derive Notably, for more general cases, the BALRoGO routine allows to use the more general representation of φ( X ), and thus to account for a more complicated expression for , presented in Appendix A . 2.2 Surface density The next step of our algorithm is to fit the surface density of the galactic object plus interlopers once again, using a maximum likelihood estimation (MLE), but this time with fixed centres from the previous step. It may seem as an unnecessary step given the previous one, but it actually allows the fitting routine to better explore the other parameters ranges like the scale radius and the interloper fraction, and thus derive more robust results. In this step, we also reduced the maximum projected radius whenever there was a parasite galactic object on the field or when the MW contamination was too intense, disturbing the fits. In this step, we not only tested Plummer profiles, but also Sérsic models (Sérsic 1963 , 1968 ) that can be useful for core-collapsed clusters such as NGC 6397, and the Kazantzidis model that is motivated from dynamical simulations of repeated tidal encounters (Kazantzidis et al. 2004 ). Equations ( 2 ), ( 6 ), and ( 7 ) remain valid for these two other models, while the normalized surface density and projected number of stars become for Sérsic, where is the lower incomplete gamma function and X = R / R e , R e being the effective projected radius. And for Kazantzidis, where K 0 and K 1 are modified Bessel functions of


at 200 pc. We retain the functional form of the Bovy model’s bulge, ρ( r ) = ρ s ( r / r 1 ) −α exp [−( r / r c ) 2 ], but with revised parameters α = 2.0, r c = 1.0 kpc, ρ s = 6.58 × 10 8 M ⊙ kpc −3 , and r 1 = 1 kpc. This yields a total bulge mass of 7.33 × 10 9 M ⊙ . The model roughly matches the observed circular velocity data collected in Sofue ( 2013 ), for both large and small radii. It does not match this data’s prominent peak at R ∼ 500 pc and subsequent decline out to 2.5 kpc, but Chemin, Renaud Soubiran ( 2015 ) argue that this feature is likely an artefact of perturbations from the Galactic bar. The axisymmetry of our potential is clearly an approximation, as orbits on scales of a few kpc are strongly affected by the bar. However, at smaller radii, the observed physical components are close to axisymmetric and the potential contours generated in physical models such as Bissantz, Englmaier Gerhard ( 2003 ) are more rounded. Thus, axisymmetric models continue to be used to study orbits near the Galactic centre (Molinari et al. 2011 ; Kruijssen et al. 2015 ; Perera et al. 2019 ), and we leave non-axisymmetric refinements to future work. We integrate orbits in this potential using the package galpy 3 (Bovy 2015 ). We assume Sgr A* lies fixed at the GC. We adopt a distance to Sgr A* of 8.175 kpc from Gravity Collaboration et al. ( 2019 ), and do not consider in the computation its small observed error. We first define a modified Galactic coordinate system ( l ′ , b ′ ), which uses a slight rotation around the Sun’s position to place Sgr A* exactly at l ′ = 0, b ′ = 0, then shift by the solar distance to obtain Galactocentric coordinates. We choose a solar motion in the Galactocentric frame of v X = −11.0 km s −1 , v Y = 248.5 km s −1 , v Z = 7.8 km s −1 that produces the observed PM of Sgr A*, μ α cos δ = −3.156 mas yr −1 and μ δ = −5.585 mas yr −1 (Reid Brunthaler 2020 ). We use a Galactocentric convention where X points from Sgr A* to the Sun. Fig. B1 shows orbits of the two clusters with a series of initial distances, using our best values of the PM. Orbits starting well on the near side of the GC are highly eccentric. The eccentricity first decreases with the starting distance, then increases again as the orbits become almost radial near the GC. About 20 pc further out, the orbits become retrograde. We note none of the orbits are circular. As noted in Stolte et al. ( 2008 ), circular orbits require , which implies X = ( v X / v Y ) Y , radius R = [1 + ( v X / v Y ) 2 ] 1/2 Y , and circular velocity . For Quintuplet, this implies a velocity V c = 177 ± 4 km s −1 at R = 48 ± 1 pc, where the uncertainties incorporate our estimated PM and RV errors. For Arches, it implies a velocity V c = 168 ± 7 km s −1 at R = 40 ± 2 pc. Both of these circular velocities are comfortably above those implied by our potential, which are 108 and 98 km s −1 , respectively, at these radii. They also lie well above the scatter of different observational


by the region of the disk behind the inner rim. Detailed studies have indicated that the picture is somewhat more complicated, with intermediate-temperature components of uncertain origin (e.g., Fischer et al. 2011 ). For the purposes of this study, where we are taking a first look at time variability in the excess, we restrict ourselves to blackbody models of these three basic components, following McClure13 . Each component is defined by a characteristic blackbody temperature and solid angle. We fix the hot component temperature at 7000 K, which is characteristic of accretion shock emission (the exact value is not crucial here because we are seeing only the Rayleigh–Jeans tail). We also fix the cool component to a value of 600 K, which is roughly the equilibrium temperature for dust at the predicted location of the inner edge of the circumbinary disk around DQ Tau (∼2.5 a = 0.35 au). For the warm component, we allow the temperature to vary between 1000 and 1800 K (which spans the range of typical characteristic maximum dust temperatures in TTSs) in increments of 50 K. The solid angles (relative to the stellar value assuming R = 1.5 R ⊙ and d = 196 pc) are varied from 0 to 0.1 for the hot component, 10–50 for the warm component, and 50 to 300 in the cool component. For each set of parameters, the models for all three components are combined and then compared to the observed excess spectra, with the best fit determined via chi-squared minimization. The resulting best fits for each observed spectrum are shown in Figure 20 , with the associated parameters listed in Table 8 . Given the systematic uncertainties in the template matching, these values should be taken as indicative only; however, they provide a useful first look at the gross properties of the material in the inner regions of this system. The best-fit hot component solid angles range from zero at several epochs, where no discernible emission was detected, to 0.06, which is typical for weakly to moderately accreting TTSs. The cool component solid angles range from 90 to 260, roughly as expected for a puffed inner disk rim with a temperature of 600 K. If our assumption of a constant temperature is a reasonable approximation, the variation in solid angle may be indicative of a variable scale height of the inner edge of the putative circumbinary disk. There is a weak correlation with the warm component, but the origin of such a variation is unclear; it could simply be an artifact of our overly simplistic assumptions of the dust temperatures. However, it should be noted that this cooler dust emission is not as well-constrained as the other components because it contributes an appreciable fraction of the total emission only at the long-wavelength end of our spectra. Observations in the 5–10 μ m range are required to test the significance of any variations from this region. Figure 20. DQ Tau excess spectra shown in black, in order of observation time (left to right, top to bottom), with


the values. The values are taken from Dunlop et al. ( 1993 , 2003 ). 3.1. Evolution of the bolometric luminosity with time To demonstrate how luminous progenitors of UCDs could have been, we compute the bolometric luminosity, , as a function of time. The dependency is shown in Fig. 3 . The benchmark initial mass is . For the cases of the CAN, SAL, and vDC IMF, the values are proportional to . The MKDP IMF is a function of and therefore we plot also for . The main difference between different MKDP IMFs (for different ) is the slope of the luminosity as a function of time. For a more top-heavy IMF, the decrease of the luminosity with time is steeper. Fig. 4 Time evolution of for all IMFs considered (vDC, SAL, CAN, MKDP). We assume that the system is gas free and that neither stars nor remnants are leaving (the dashed lines show the change of the value if only certain fractions of remnants are kept). The upwards-pointing blue arrow demonstrates that if we would assume more massive remnants (e.g. due to implosion directly to a BH without a SN explosion), then this would lead to larger values of . The models are computed for the case of , however, results are mass independent. The only exception is the MKDP IMF, which depends on initial object mass, , and also contains a large fraction of mass in high mass stars, and is plotted for (, bottom to top) and different fractions of remnants retained (100%, 10%, 0%). Left panel : the evolution for [Fe/H] = 2. Right panel : the evolution for [Fe/H] = 0. The grey band indicates the span of the observed present-day values, approximately five to ten for the majority of UCDs ( Mieske et al. 2008 ; Dabringhausen et al. 2009 ). The scales are identical for both panels. There are several noticeable features in this figure: (i) for the MKDP IMF, UCDs with initial stellar mass are as bright as quasars ( Dunlop et al. 1993 , 2003 ; Souchay et al. 2015 ) for the first few yr. We use low-redshift quasars because for these we were able to find bolometric luminosities, but also absolute magnitudes in the and band. We note, however, that high redshift quasars show very comparable luminosities as shown for example by Mortlock et al. (2011) . This is mainly due to the presence of a large number of O and B stars () in the system. Due to stellar evolution and core-collapse supernova explosions, these UCDs will be variable on a timescale of months. As already suggested by Terlevich Boyle (1993) , such objects might be confused with quasars, especially for example in large photometric surveys. (ii) After less than a Myr, a strong degeneracy between the IMF and appears. That is, for different and different IMFs, similar luminosities comparable to those of observed UCDs occur. (iii) The metallicity is a second-order effect. Fig. 5 Values of the ratio as a function of for three different times . The case of the MKDP IMF is not constant since the MKDP IMF is a function of initial mass. The plotted curves assume all remnants are


found that Centaurs can produce HTCs, in particular they are trapped very often in the 1:6 MMR with Jupiter (energy x ≃ 0.058 au −1 ) which coincides with the clustering in the observed energy distribution of HTCs (Fig. 3 ). It is worth noting that our sample of observed Centaurs contains a few objects in retrograde orbits. To produce both the observed HTCs and JFCs, a flattened source is required but it should contain a small fraction of high-inclination and retrograde comets. A near-ecliptic, prograde population of Centaurs ( i ≲ 25°) will essentially lead to a near-Earth population of bodies with Tisserand parameters T J > 2, typical of JFCs (Levison Duncan 1997 ). It is not the purpose of this paper to discuss the ultimate origin of Centaurs, though we can say a few words. A flat source in the outer planetary region such as the Scattered Disk will essentially produce a population of Centaurs in prograde, low-inlination orbits (Di Sisto Brunini 2007 ), though it may be possible that some SDOs scattered outwards attain Oort cloud distances where galactic tides switch their orbits from prograde to retrograde (Levison et al. 2006 ). Other possibility is that some Centaurs come from the capture of Oort cloud comets in the outer planetary region as discussed by Emel'yanenko Bailey ( 1998 ) and Emel'yanenko, Asher Bailey ( 2005 ). In the end, some HTCs may come from the Oort cloud but through a different dynamical route that direct insertion of Oort cloud comets in the near-Earth or the Jupiter region. In order to check the results found for the simulated HTCs produced from the evolution of the Centaur sample, we also studied the dynamical evolution for 1 Myr to the past and to the future of 15 observed HTCs with q 1.3 au and energies 0.04 x 0.08 au −1 (orbital periods 44.2 P 125 yr). We confirm that HTCs do not diffuse randomly in a but tend to be captured in the 1:6 exterior resonance with Jupiter generating a concentration at x ∼ 0.058 au −1 . This peak is clearly shown in Fig. 17 which is a histogram of frequencies of the average 〈 a 〉 over periods of 500 yr. Figure 17. Frequency distribution of the time-average 〈 a 〉 (over 500 yr) for the observed sample of 15 HTCs with q 1.3 au and energies 0.04 x 0.08 au −1 (periods 44.2 P 125 yr) integrated over 10 6 yr in the past and in the future. Other remarkable feature is that even though we obtain about 4.7 per cent of LPCs that end up colliding with the Sun for a purely dynamical evolution, this percentage is practically null when we consider appropriate physical lifetimes ( N phys ( q = 1) ≲ 500 revolutions). The reason is that LPCs approach the Sun in a smooth way through a multi-step process, that takes hundreds to thousands revolutions, so it is very likely that such comets will disintegrate well before reaching sungrazing states. The inclination distribution of the old LPCs shows clear departures from randomness that can be described as a slight predominance of prograde orbits that is


we study (i) a constant 29% contamination and (ii) the stellar mass and redshift-dependent [N ii ]/H α flux ratios predicted by our model (Equation ( 3 )). To derive the intrinsic H α LF using our model [N ii ] contamination (Φ model ), we start with the observed ([N ii ]+H α ) tot LF (Φ obs ( L )) measured by Colbert et al. ( 2013 ) at 0.9 z 1.5. We obtain this LF from their published H α LF (, , and α = −1.43) by dividing the luminosities by a factor of (1–0.29) to undo the constant total [N ii ] flux contamination fraction correction of 29%, which the authors applied. In the following, we treat Φ obs as the true observed LF. Importantly, this LF is not corrected for [N ii ] contamination and dust. Because of the evolution of the star-forming main sequence, Φ obs is redshift-dependent, but here we do not model this dependence across z = 1–2, as we are only interested in the effects of [N ii ] contamination and not the absolute number of galaxies. On the other hand, the stellar mass and redshift-dependent [N ii ] contamination correction will change the intrinsic H α LF across the redshift range studied here. To obtain Φ model , we choose an approach that only uses the measured ([N ii ]+H α ) tot luminosities as input and assumes the most likely underlying stellar mass distribution (robustly determined from other studies), from which we obtain the [N ii ] contamination from our model. This approach has the advantage that it enables an easy implementation and propagation of a variety of uncertainties into the final results. Furthermore, this method results in reliable intrinsic H α LFs even if the mass distribution is poorly measured due to the lack of sufficient multiwavelength data, as long as the selection function of the galaxy sample is known. Here we make use of the Schreiber et al. ( 2015 ) parameterization of the star-forming main sequence to derive the underlying stellar mass distribution (we comment below on possible shortcomings). We note that the choice of different parameterizations (e.g., Tomczak et al. 2016 ) should not change the following results. We start with a distribution of ([N ii ]+H α ) tot , which we sample from Φ obs . To obtain stellar masses for these galaxies, we use a “backward-engineering” technique. First, we convert the SFRs of the Schreiber et al. ( 2015 ) parameterization into H α luminosities using the Kennicutt ( 1998 ) prescription. Thereby, we include a dispersion of 0.3 dex measured on the SFR–versus–stellar mass main sequence. Second, we redden the H α luminosities according to the relation between A H α (H α extinction) and stellar mass robustly derived from the spectra of local galaxy samples in SDSS (Garn Best 2010 ). This relation holds for the WISPS sample at 0.8 z 1.5, as shown in Domínguez et al. ( 2013 ). Third, we add the contribution of [N ii ] to the H α luminosity by using our model. Finally, this translation between dust-reddened ([N ii ]+H α ) tot luminosities and stellar masses allows us