(). While p obs does not depend on the longitude, the diurnal cycle can still introduce a systematic bias. This is because the Sun will be systematically out of phase with the northern maximum of the antenna pattern (Figure 2 ). We find that this typically produces an increase in p obs of a few percent for mid-latitudes in the north, even though their analogs in the south have larger p obs because of the shorter solar exposure over an entire year. We note that p obs reflects the amount of localization probability that is observable in the limit of many detections. Outside of this limit, we consider an analogous quantity () defined for a finite number of detections ( N d ) with the correspondence (see the Appendix for more details). Statistical fluctuations in , caused by variations in which events occur, can be large, particularly with events. We expect the uncertainty in this estimate to scale inversely with , and Figure 3 reports as error bars along with the mean. Typical values are between 6% and 10%. We also note that the intrinsic distribution for single events ( N d = 1) may not be Gaussian but, in the limit of many detections, the distribution of the mean will be. 3.2. Minimum Zenith Distance While equatorial observatories may be able to access the largest integrated probability, and thereby have the largest probability of being able to image a source, this does not necessarily imply that they will have the best conditions for observing. An important consideration is the closest approach of each field to an observatory’s zenith (minimum zenith distance). Sources at large angles from an observatory’s zenith can be difficult to observe because of high air mass and mechanical limitations. In Figure 4, we present the mean minimum zenith distance, as a function of latitude, weighted by the probability that the source actually comes from each location: accounts for solar occlusion and incorporates when the EM facility will actually be able to observe. We find that observatories at extreme latitudes () will almost always have large D zen , with a gradual transition to lower values at lower latitudes. There is a difference in D zen between observatories at mid-latitudes, with northern sites preferred. This is because the diurnal cycle makes the Sun preferentially overlap the southern antenna pattern and forces southern facilities to observe closer to sunrise and sunset than their northern counterparts. This behavior is particularly evident at the northern winter solstice. Figure 4. D zen averaged over a year (black), near the northern summer solstice (blue), and near the northern winter soltice (red). Shaded regions correspond to the fluctuations from localization maps simulations (), and we note that 's distribution is quite broad. Figure 4 also shows the fluctuations in the analgous stastic defined for a finite number of detections (). We typically find –10 (see the Appendix for more details). Furthemore, D zen corresponds to the mean of

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


≈ x .). For the sample galaxies. the resulting rotation curves predicted by Newton (dashed curves) and by MOND (solid curves) are compared with the observations in Fig. 2 (lower panels), along with the surface density distributions of the baryonic components. It is striking that in several cases (DDO 87, WLM, IC 2574, NGC 3741) the observed structure corresponds quite precisely to that predicted by the baryonic mass distribution in the context of MOND. Note that for WLM, there are a large number of points on the derived rotation curve given by Oh et al. These are certainly not independent, and given their high density obscure the match of the predicted rotation curve to the observations; therefore I have taken only every fourth point in the curve presented here. Figure 2. Each galaxy is represented in two panels. The upper panel is the surface densities of gas (dashed) and stars (dotted) as a function of radius. The lower panel shows the observed rotation curve (points), the Newtonian rotation curve of baryonic components (long dashed curve), and MOND rotation curve. calculated from the Newtonian curve via equation (1) (long dashed curve). The claim here is that these rotation curves are essentially predictions with no free parameters because of the dominance of the gaseous component. But of course there is some contribution from the luminous disc so one might question the extent to which M/L of the disc does vanish as an adjustable parameter. The stellar masses of the sample galaxies (column 6 in Table 1 ) are taken from fitting the colours of the galaxies to population synthesis models in the cases of the seven LT galaxies (Oh et al. 2015 ). For the other five cases they are taken from kinematic considerations (e.g. not exceeding the observed rotation velocity). In half the cases, the implied stellar masses exceed 15 per cent of the total baryonic mass (those objects in which the ratio of gas mass to stellar mass is less than six, column 8). This can certainly affect the shape of the galaxy rotation curves, especially in the inner regions where the stellar disc often contributes more significantly to the baryonic surface density distribution. In Fig. 3 we see the calculated rotation curves for one case where the fraction of stellar mass is fairly typical, DDO 87. The lower dashed curve is the Newtonian rotation curve of baryons excluding the estimated contribution of the visible disc and the upper dashed curve is the same but including the stellar disc with a contribution of 17.5 per cent of the total baryonic mass as given in Table 1 . The upper and lower solid curves are the same but for the MOND rotation curves. While it is obvious that the contribution of the stellar disc provides better agreement with the observations, it is also clear that, with no stellar contribution at all, the rotation curve is qualitatively similar: The overall form of the calculated rotation curves is quite independent of the mass-to-light ratio of the stellar

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


of BAT 677; Right: CO(2–1) spectra (detection) of BAT 677. Figure 2.138. BAT 679 Left: Pan-STARRS gri color cutout of BAT 679; Right: CO(2–1) spectra (detection) of BAT 679. Figure 2.139. BAT 682 Left: Pan-STARRS gri color cutout of BAT 682; Right: CO(2–1) spectra (nondetection) of BAT 682. Figure 2.140. BAT 687 Left: Pan-STARRS gri color cutout of BAT 687; Right: CO(2–1) spectra (nondetection) of BAT 687. Figure 2.141. BAT 694 Left: Pan-STARRS gri color cutout of BAT 694; Right: CO(2–1) spectra (nondetection) of BAT 694. Figure 2.142. BAT 696 Left: Pan-STARRS gri color cutout of BAT 696; Right: CO(2–1) spectra (nondetection) of BAT 696. Figure 2.143. BAT 707 Left: DSS cutout of BAT 707; Right: CO(2–1) spectra (nondetection) of BAT 707. Figure 2.144. BAT 710 Left: DSS cutout of BAT 710; Right: CO(2–1) spectra (nondetection) of BAT 710. Figure 2.145. BAT 717 Left: Pan-STARRS gri color cutout of BAT 717; Right: CO(2–1) spectra (detection) of BAT 717. Figure 2.146. BAT 719 Left: Pan-STARRS gri color cutout of BAT 719; Right: CO(2–1) spectra (detection) of BAT 719. Figure 2.147. BAT 723 Left: Pan-STARRS gri color cutout of BAT 723; Right: CO(2–1) spectra (detection) of BAT 723. Figure 2.148. BAT 733 Left: Pan-STARRS gri color cutout of BAT 733; Right: CO(2–1) spectra (detection) of BAT 733. Figure 2.149. BAT 735 Left: Pan-STARRS gri color cutout of BAT 735; Right: CO(2–1) spectra (detection) of BAT 735. Figure 2.150. BAT 751 Left: DSS cutout of BAT 751; Right: CO(2–1) spectra (detection) of BAT 751. Figure 2.151. BAT 754 Left: Pan-STARRS gri color cutout of BAT 754; Right: CO(2–1) spectra (detection) of BAT 754. Figure 2.152. BAT 760 Left: Pan-STARRS gri color cutout of BAT 760; Right: CO(2–1) spectra (detection) of BAT 760. Figure 2.153. BAT 772 Left: Pan-STARRS gri color cutout of BAT 772; Right: CO(2–1) spectra (detection) of BAT 772. Figure 2.154. BAT 783 Left: Pan-STARRS gri color cutout of BAT 783; Right: CO(2–1) spectra (detection) of BAT 783. Figure 2.155. BAT 795 Left: DSS cutout of BAT 795; Right: CO(2–1) spectra (nondetection) of BAT 795. Figure 2.156. BAT 801 Left: DSS cutout of BAT 801; Right: CO(2–1) spectra (nondetection) of BAT 801. Figure 2.157. BAT 836 Left: Pan-STARRS gri color cutout of BAT 836; Right: CO(2–1) spectra (detection) of BAT 836. Figure 2.158. BAT 841 Left: Pan-STARRS gri color cutout of BAT 841; Right: CO(2–1) spectra (detection) of BAT 841. Figure 2.159. BAT 898 Left: DSS cutout of BAT 898; Right: CO(2–1) spectra (detection) of BAT 898. Figure 2.160. BAT 970 Left: DSS cutout of BAT 970; Right: CO(2–1) spectra (detection) of BAT 970. Figure 2.161. BAT 981 Left: Pan-STARRS gri color cutout of BAT 981; Right: CO(2–1) spectra (nondetection) of BAT 981. Figure 2.162. BAT 986 Left: DSS cutout of BAT 986; Right: CO(2–1) spectra (detection) of BAT 986. Figure 2.163. BAT 988 Left: DSS cutout of BAT 988; Right: CO(2–1) spectra (nondetection) of BAT 988. Figure 2.164. BAT 995 Left: DSS cutout of BAT 995; Right: CO(2–1)

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capture times for the SG model are always shorter, due to higher gas densities; thus, an SG disc captures all stars with a 10 4 R g at i 0 = 45 ○ , while a TQM disc captures only red giants at most radii, and captures other stars only at very small disc radii. Fig. 3 shows T cap as function of i 0 for different stellar types assuming a 0 = 10 4 R g . The curves in Fig. 3 correspond to the analytic expression in equation ( 4 ), which is very weakly dependent on i , hence the flatness of the curves. However, Fig. 3 also shows that numerical integration is required for highly inclined orbiters, and for disc models with highly variable radial height profiles (e.g. TQM ). Counter-intuitively, highly inclined stellar orbiters are captured faster by the SG disc since relative velocity is largest for high i 0 and therefore F GEO is largest. The TQM disc model is less efficient than the SG disc model at star capture at a 0 = 10 4 R g for all stellar types, with only red giants ending up within the disc for plausible AGN lifteimes. For simplicity, we assume no change in stellar mass due to disc crossings. For red giants in particular, this is likely incorrect since Kieffer Bogdanović ( 2016 ) find that RG can lose up to 10 per cent of their mass during a disc crossing (though for column densities at least two orders of magnitude greater than any considered here). Further complicating the expected capture time of giants, Armitage, Zurek Davies ( 1996 ) find that mass-loss due to repeated impacts with the disc leads to an expansion of the star and a subsequent increase in cross-section. Stars with a larger surface area are captured by the disc faster since they experience a stronger drag from equation ( 1 ), and therefore including the mass-loss in our calculations could reduce capture time for giants in both SG and TQM . In addition, Armitage finds that a main-sequence star can survive repeated disc passages while RG are destroyed much more quickly; this could explain the observed absence of RG in the NSC of the Milky Way. Some variations in T cap as a function of i for the TQM disc appear to be due to changes in a across the sharp (unphysical) density and scale height changes at a ∼ 5 × 10 3 R g . Figure 3. Capture time T cap due to geometric drag as a function of initial orbital inclination ( i 0 ) with notation as for Fig. 2 . We assumed Keplerian prograde orbits with a 0 = 10 4 R g . Curves correspond to the analytic estimate for T cap from equation ( 4 ) and star symbols indicate T cap from numerical integration of equation ( 10 ). Conservation of energy causes higher orbital inclinations to produce large changes in a , as the number of orbits needed to reach i crit increases. This results in a lower T cap for higher inclinations at the same initial semimajor axis. Changing a also causes the anomalies seen in the TQM panel, due to orbiters crossing the (unphysical) density discontinuity at approximately 5 × 10 3 R g . From the top panels of Figs 2

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of the field defined by the generating cylinder, which was used for the construction of the flux rope shown in panel a) , i.e., the field ( 9 )–( 11 ). In contrast to panel a) , they are shown in the whole field of view because the magnetic field from all space contributes to the solution displayed in panel a) . The cross section of the generating cylinder is plotted by the dashed circle, and the tilde coordinates are the local coordinates defined in Fig. 1 . The magnetic-field magnitude scale is the same, but the values of are different and correspond to the maxima in the displayed fields in panels a) and b) . The thick lines indicate magnetic surfaces on which the axial magnetic field is zero. This method is able to generate toroidal flux ropes with another cross sections. Figure 6 displays shapes very close to ellipses, a prolate shape in Fig. 6 a and an oblate one in Fig. 6 b. Only ten coefficients were used to define these fields (it was , and half of the coefficients were close to zero, as in the previous case). The axial field does not vanish at the boundary shown, but it is small, as is shown in Fig. 3 . There are limitations of our procedure. One has been mentioned: it was not commonly possible to obtain the desired shape of the flux rope for low aspect ratios. But even if the desired shape is obtained, the flux rope may be multipolar, that is, with the axial field reversed inside. For instance, the generating toroid of the flux rope shown in Fig. 6 a has an oblateness 2:3, but the flux rope has a dual polarity. Therefore we limited it to the inner unipolar part, displayed in Fig. 6 a, which has a smaller oblateness and a different aspect ratio (if defined as ) from 5. If the generating toroid has an oblateness or prolateness larger than about two, unipolar flux-rope cross sections deviate from desired elliptic shapes (they more resemble a rectangular shape). Fig. 6 Magnetic field magnitude distribution inside toroidal flux ropes with practically elliptic cross sections. The closed lines are cross sections of magnetic surfaces. The values of free parameters are a) , , , , , , , , , , , and ; b) , , , , , , , , , , , and . The other coefficients are (numerical) zero. 3.1. Calculating the helicity It is relatively easy to (numerically) calculate the helicities of these flux ropes. Introducing toroidally curved cylindrical coordinates , , and in our global system, the helicity for toroidal flux ropes with an axially symmetric linear force-free field ( Vandas Romashets 2015 ) is where determines the boundary; and are polar coordinates in a cross-section plane (like in Figs. 5 and 6 , where we should have for the case of Fig. 5 a and for Fig. 6 , where and are semi-axes of the boundary ellipses; for a more general boundary, would be defined numerically from the results). To compare different flux ropes, Vandas Romashets (2015) used the unitless quantity derived from the helicity by where is the volume of the toroidal flux rope and

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


reversals are also sometimes seen in other regions of the disk, potentially complicating the extraction of the planet position. The rotation maps presented in Figure 3 were generated with a perfect knowledge of the velocity field and taking into account the radiative transfer effects at the emitting surface. For actual data, the velocity field and altitude of the emitting layers have to be estimated. Uncertainties in the Keplerian velocity model that is subtracted may limit the detectability of Doppler flips to high signal-to-noise cases. In particular, the predicted Doppler flip for the orientation of HD 163296 (labeled panel in Figure 3 ) shows an amplitude twice smaller (≈150 m s −1 ) than the residuals in the differential rotation map presented in Casassus Pérez ( 2019 ) (≈300 m s −1 ). This explains why they did not detect the counterparts of the kink as a Doppler flip in the first moment map. The absence of visible Doppler flip corresponding to the kink in HD 97048 is also not surprising given the residuals in the differential rotation maps presented by Casassus Pérez ( 2019 ). Because of the tentative nature of the detections and the need for higher-resolution observations, we are not yet confident enough to estimate planet masses from the candidate kinks seen in the DSHARP data. Detailed hydrodynamical and radiative transfer modeling of the kink candidates is currently the only way to determine planet masses from velocity deviations in a quantitative manner but would result in poorly constrained estimates given the low signal-to-noise ratio and poor velocity resolution of the data. We estimated the velocity deviations from the number of channels in which the kinks are detected. The limited spectral resolution prevents us from measuring velocity deviations smaller than 640 m s −1 . By comparing with the expected location of the isovelocity curves if the disks were in Keplerian rotation, we estimate the observed perturbations to be of the order of 10%–15% of the local Keplerian velocity (Table 1 ). From our previous modeling of velocity kinks, we expect masses of the order of 1–3 M Jup . In HD 143006, the velocity deviation appears significantly larger, pointing toward a more massive planet. With the exception of HD 143006, such masses are larger by a factor of 4–10 than the masses derived from the continuum gaps by Zhang et al. ( 2018 ) or Lodato et al. ( 2019 ). We found a similar discrepancy for HD 97048 (Pinte et al. 2019 ), where to get a coherent match to both the continuum and line data, our models required the dust grains dominating the emission to have a Stokes number of a few 10 −2 , suggesting very porous and/or fluffy aggregates. The high detection rate of velocity kinks at several tens of astronomical units in the DSHARP data set may appear in contradiction with the occurrence rate of known giant extrasolar planets at large distance from their host stars: in the range of a few up to 5% for massive planets ( M > 2 M Jup )

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


sample into three metallicity bins ([Fe/H] −1.5, −1.5 [Fe/H] −0.7, and [Fe/H] > −0.7) and plot the difference between CaT based and literature [Ca/H] as a function of half-mass relaxation time for each bin in Fig. 26 . For the low-metallicity bin, Kendall’s test provides weak evidence (τ = 0.26 and p = 0.07) for a correlation. Using pymc3 , we fit a significant slope () and find that a linear relationship is favoured over a constant value (ΔBIC = 9.2) for 24 GCs in the low-metallicity bin. The nuclear remnant NGC 5139 appears to be a significant outlier from the low-metallicity GC relationship. For both the 20 GCs in the intermediate-metallicity bin and the eight GCs in the high-metallicity bin, Kendall’s test provides no evidence (τ = 0.18, p = 0.27 and τ = 0.00, p = 1.0, respectively) for a correlation. The pymc3 fits finds an insignificant slope for the intermediate-metallicity subsample (), but a marginally significant slope for the metal-rich subsample (). Neither subsample shows strong evidence that a linear relationship is preferred over a constant value (ΔBIC = −5.7 and 2.3, respectively). A significant relationship between the CaT and the half-mass relaxation time only at low metallicity is consistent with the behaviour of the CaT with the slope of the stellar mass function. For both the mass function slope and the relaxation time relations, the relationship for all GCs in our sample is driven by the low-metallicity GCs. Figure 26. Differences between literature and our CaT-based [Ca/H] as a function of the half-mass radius relaxation time. From top to bottom, we show the difference for GCs with [Fe/H] −1.5, −1.5 [Fe/H] −0.7, and [Fe/H] −0.7. The black line in the top panel shows our fitted relationship between Δ[Ca/H] and the relaxation time for the lowest metallicity GCs. The colours, shapes, and size of the points are the same as in Fig. 18 . Since the slope of the mass function and the relaxation time are anticorrelated, the relaxation time increases from right to left to match the trend seen between the CaT and the mass function slope. As in Fig. 19 , the dashed line shows Δ[Ca/H] = 0, while the dotted lines show Δ[Ca/H] = ±0.1. The lowest metallicity GCs with longer relaxation times show stronger CaT strengths than GCs with shorter relaxation times and the same metallicity. However, intermediate- and high-metallicity GCs show no relationship between CaT strength and relaxation time at fixed metallicity. The observed qualitative behaviour of the CaT with the stellar mass function slope and metallicity is consistent with the predictions of stellar population synthesis models. The Vazdekis et al. ( 2003 , 2012 ) models predict that the slope of the relationship between the CaT strength and the slope of the stellar mass function as a strong function of metallicity, with a strong anticorrelation between the two at low metallicities, little or no mass function dependence at intermediate metallicities and a strong correlation at solar

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


and one, two, or three Faraday-thin components, each with a contribution to the polarized emission as given in equation ( 4 ), approximated to be independent of frequency over our band. We did not fit for any models with Faraday-thick components, as LOFAR HBA is minimally sensitive to these: from equation ( 3c ), the maximum scale recoverable is φ max-scale = 0.46 rad m −2 , so any polarized source with Faraday thickness exceeding this would not be properly recovered, although the edges in its Faraday spectrum might be visible if they were sufficiently sharp (Van Eck et al. 2017 ). In order to evaluate the quality of the fits across the knot and the lobe we calculate for each pair of models the Bayes factor, K , which is given by where each model M i is defined in terms of parameters θ i , is assigned a prior Pr(θ i | M i ), fits the data with likelihood Pr( D |θ i , M i ), and is supported by Bayesian evidence Pr( D | M i ). Following Kass Raftery ( 1995 ), we evaluate the Bayes factor based on the derived value 2ln K : a positive value supports model 2, constituting weak (values 2), positive (2–6), strong (6–10), or very strong (>10) evidence. Negative values of 2ln K , similarly, support model 1. Fig. 15 shows the results for the model with one Faraday-thin source. The evidence of the models with two or three Faraday-thin sources is less than the evidence for all other tested models and so is not shown. Fig. 15(a) shows the Faraday depth found for the one-source model for each pixel, where it is favoured over the null hypothesis with K > 1. Fig. 15(b) shows 2ln K > 2, where white to blue (2ln K > 2) indicates support for the one-source model and red (2ln K 2) indicates support for the null hypothesis. We find an average Faraday depth in the knot of −50.8 rad m −2 . Despite the fact that there is a significant polarized region detected at 5σ in the Faraday spectrum of the lobe, only 10 pixels are detected in the lobe with the presence of a source favoured over the null hypothesis with K > 1. The average Faraday depth of these pixels is −53.6 rad m −2 , in agreement with that measured from the Faraday spectra, with a standard deviation of 0.7 rad m −2 . The difference of 2.8 rad m −2 between the Faraday depths in the knot and the lobe is significant against this uncertainty. Figure 15. Results of the QU fitting. (a) shows the Faraday depth of the component found for a single Faraday-thin screen plus instrumental polarization. (b) The Bayes factor when comparing the fits for a single thin screen plus instrumental polarization with just instrumental polarization. Red indicates support for the null hypothesis while blue indicates support for the single Faraday-thin screen. White indicates inconclusive. There are small variations in Faraday depth of order Δφ ∼ 0.2 rad m −2 . Fig. 16 shows the distribution of the Faraday depths in the knot. The variance in the Faraday depth is rad 2 m −4 . If the variation in Faraday depth is due solely to

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


estimate of 5.36 ± 0.30 kpc was obtained by Thompson et al. ( 2001 ) from the detached eclipsing binary OGLEGC 17 (Kaluzny et al. 1996 ). A once challenging aspect of using Gaia parallaxes from the earlier DR2 release (Gaia Collaboration et al. 2018a ) to directly measure the parallax of ω Cen was the systematic uncertainty related to the parallax offset found by Lindegren et al. ( 2018 ), and which has been demonstrated to vary with source color, magnitude, and position (e.g., Arenou et al. 2018 ; Riess et al. 2018 ; Zinn et al. 2019 ). The DR2 offset was calibrated from quasars, which are bluer than the bulk of ω Cen stars and only cover a small fraction of their magnitude range. The DR2 parallax estimates for ω Cen by Gaia Collaboration et al. ( 2018b ) and by Shao Li ( 2019 ) are 0.1237 ± 0.0011 and 0.1368 ± 0.0015 mas, respectively, and are substantially smaller than other determinations, presumably because the parallax offset estimated by Lindegren et al. ( 2018 ) on the basis of quasar data is much smaller than the values found for brighter stars (Arenou et al. 2018 ; Riess et al. 2018 ; Zinn et al. 2019 ). Our own analysis of the DR2 parallax data for stars in ω Cen revealed a large trend of parallax with star magnitude and color, leaving us to conclude that a parallax from DR2 was not reliable without further characterization of its parallax offset. Thanks to the great improvement realized in EDR3, such a measurement is now feasible, and we exclude the DR2 parallax-based estimates from further consideration. Both ground-based and HST observations have shown that ω Cen has a complex structure, with many separate color–magnitude sequences (Bellini et al. 2009 , 2010 , 2017 ) that can have different structural (Bellini et al. 2017 , 2018 ; Calamida et al. 2017 ) and metallicity (Milone et al. 2017 ; Zennaro et al. 2019 ) properties, including an unusually broad range in helium abundance for a globular cluster. We would not expect this large range to influence the peak of the TRGB in the I -band as the mean helium abundance in ω Cen is not much affected by the high and low sub populations, and the TRGB is known to be quite insensitive to star formation and chemical abundance in general (McQuinn et al. 2019 ). The complex kinematic structure of ω Cen does not affect its parallax measurement, but needs to be taken into account in kinematics-based distance measurements (Watkins et al. 2015 ; Baumgardt et al. 2019 ). It is also worth recognizing that the distribution of abundances in the halos of galaxies where TRGB is often measured is largely unknown making comparisons difficult; it is not unlikely that the multiple populations of ω Cen could more closely resemble the complexity of a galaxy’s stellar halo than a simpler, single-population system. 5.2. Comparison with the TRGB Calibration Anchored to 47 Tuc The determination of the parallax to a globular cluster as massive as ω Cen represents a milestone of distance measurements and

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


2003 ) and have been called “GRB-SNe” (see Hjorth Bloom 2012 ; Woosley Bloom 2006 ; Cano et al. 2017 , and references therein). While most GRB-SNe are Type Ic-BL, not all SNe Ic-BL are associated with GRBs. The most luminous SNe Ic are Type Ic superluminous supernovae (SLSNe; Quimby et al. 2011 ; Gal-Yam 2012 ); their peak luminosities are erg s −1 (Gal-Yam 2012 ), tens of times greater than those of ordinary SNe. In this paper, we study a very luminous SN Ic, SN 2011kl, which is associated with an ultralong ( s) GRB 111209A (see, e.g., Golenetskii et al. 2011 ; Palmer et al. 2011 ; Gendre et al. 2013 ; Stratta et al. 2013 ; Gompertz Fruchter 2017 ) at redshift z = 0.677 (Vreeswijk et al. 2011 ). By using WMAP ΛCDM cosmology (Spergel et al. 2003 ) and adopting km s −1 Mpc −1 , , and , Kann et al. ( 2016 ) derived a luminosity distance ( D L ) of 4076.5 Mpc for GRB 111209A/SN 2011kl. With a peak luminosity erg s −1 (Kann et al. 2016 ), 8 8 Greiner et al. ( 2015 ) found a slightly lower peak luminosity ( erg s −1 ) since they ignored the near-infrared contribution. SN 2011kl is the most luminous GRB-SN yet detected and is significantly more luminous than all other GRB-SNe, whose average peak luminosity is erg s −1 with a standard deviation of erg s −1 (Cano et al. 2017 ). The time of peak brightness of SN 2011kl is ∼14 days, slightly larger than the average (13.0 days with a standard deviation of 2.7 days; Cano et al. 2017 ) of other GRB-SNe. The peak luminosity of SN 2011kl is comparable to that of some not-quite-superluminous “SLSNe” (e.g., PTF11rks, PTF10hgi, iPTF15esb, iPTF16bad, and PS15br) and commensurate with the SN-SLSN gap transients observed by Arcavi et al. ( 2016 ); moreover, Liu et al. ( 2017c ) pointed out that the spectrum of SN 2011kl is very different from those of GRB-SNe, but consistent with that of SLSNe. So, the energy sources powering its LC might be similar to those powering the SLSN LCs instead of the ordinary SNe Ic. The most popular prevailing models for explaining SLSNe are the magnetar model (Maeda et al. 2007 ; Kasen Bildsten 2010 ; Woosley 2010 ; Chatzopoulos et al. 2012 , 2013 ; Inserra et al. 2013 ; Chen et al. 2015 ; Wang et al. 2015a , 2016b ; Dai et al. 2016 ; Moriya et al. 2017 ) and the ejecta plus circumstellar medium (CSM) interaction model (Chevalier 1982 ; Chevalier Fransson 1994 ; Chugai Danziger 1994 ; Chatzopoulos et al. 2012 , 2013 ; Liu et al. 2017a ). 9 9 Almost all SLSNe (except maybe pair-instability SLSNe—e.g., SN 2007bi Gal-Yam et al. 2009 ; but see Dessart et al. 2012 and Nicholl et al. 2013 ) cannot be explained by the radioactive heating model (Quimby et al. 2011 ; Inserra et al. 2013 ; Nicholl et al. 2013 , 2014 ). Some SLSNe I exhibiting late-time rebrightening and H α emission lines (Yan et al. 2015 , 2017 ) might be powered by a hybrid model containing contributions from magnetars and ejecta-CSM interaction (Wang et al. 2016c ). There have been several papers modeling the LCs of SN

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


index increases and the energy distribution becomes steeper, limiting the number of higher energy electrons. This leads to spectra becoming more ‘thermal’ in shape, with the peak generated from the non-thermal particles flattening out. This is seen in the spectra of Fig. 10 compared to the Z LOS spectra from Fig. 7 . At earlier times, the peaks are similar, with the power-law index not increasing noticeably until later times where the system relaxes to a thermal state much faster than when the index was a constant value. The addition of an increasing power-law index therefore results in the spectral peak being reduced faster and pushed to lower frequencies as the contribution of higher energy particles is reduced. 3.2.2 Distribution along the loop The distribution of non-thermal particles within the loop is defined by equation ( 2 ). As described in Section 2.2 , l is the distance from the loop top to one footpoint, or the loop half length. The factor that controls the distribution within the loop, s , is initially set as this half loop length. As is seen in Fig. 4 the density is always highest at the apex of the loop, dropping off to a minimum at the footpoints. The parameter s was varied to investigate the effect on spectra of different spreads in non-thermal electrons within the loop. The results from using two values of s are shown in Fig. 11 . The top two panels show the spectra along the Z LOS for s parameter values of 0.3 times the half loop length (same as the Z LOS spectra in Fig. 7 ) and 1.5 times the half loop length. Figure 11. Spectra along the Z LOS resulting from changing the non-thermal electron density spatial distribution (top two plots). Two different configurations ( s = 0.3 and 1.5) are considered over the seven time-steps. The results for s = 1.0 are seen in the Z LOS spectra of Fig. 7 . Bottom images are 3D visualization of sample flux tube field lines at 4112 s and their electron distributions. Colour bar units are cm −3 . Also shown in Fig. 11 is a 3D visualization of these two different distributions along the field lines (and throughout the total flux tube) at 4112 s (bottom two panels of Fig. 11 ). For a smaller value of s , the particles are more constrained to the loop top, while a larger s value leads to a more even distribution from the apex down towards the foot points. The point of highest non-thermal density is found in the s = 0.3 plot as the total mass is conserved before the time decay component is added (equation 4 ). For the more compact distribution ( s = 0.3), in Fig. 11 it is clear the system relaxes to a thermal state quicker than for the more extended distribution of electrons and has a lower peak flux overall. However, the peak in emission occurs at 1790 s, later than the 169 s seen for the other distributions and LOSs. Overall, however, the pattern of the spectral evolution is similar. After the peak flux is reached, the decay of the non-thermal particles leads to the decrease in peak flux

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


density and, through a gradient descent, iterate in small steps until the solution has converged. The total electron density is given by the sum of the free electrons from hydrogen, helium, and metals. We approximate this using only the first 30 elements, as others are too scarce to contribute significantly:where Z is the atomic number, A Z is the fractional abundance of element number Z , k is the ionization stage ranging from singly to fully ionized (e.g., k = 2 corresponds to doubly ionized), and Y k is the fractional population of ionization stage k of element Z . For trace elements, we solve for the ionization fractions Y k either using an equilibrium calculation that is a function of temperature or by solving a continuity equation for nonequilibrium ionization states, as detailed in, e.g., Bradshaw Mason ( 2003a , 2003b ). The nonequilibrium solver can be used with all, some, or none of the elements, as determined by the user and appropriate for the study at hand. The code makes an initial guess for the H ii fraction, calculates the rate coefficients, solves Equations ( 1 ) and ( 16 ) for the level populations, and then recalculates the total electron density. The process iterates in small steps until the electron density has converged within a defined relative tolerance (default of 10 −6 ). 2.5. Radiative Losses In previous versions of HYDRAD, the hydrogen and electron densities were essentially equal in the corona (with only a small correction for trace elements), so that the optically thin radiative losses E R (erg s −1 cm −3 ) were approximated by setting n e = n H . We drop that approximation now and use the more accuratewhere is the sum of the emissivity over all ions and all transitions for each ion (more precisely, all in the current version of CHIANTI; Del Zanna et al. 2015 ), and the minus sign indicates that the energy is lost from the system. The loss function Λ also depends on the ionization fractions and therefore also utilizes a nonequilibrium ionization calculation when desired. In the chromosphere, we calculate radiative losses following the prescription of Carlsson Leenaarts ( 2012 ), which calculates the loss of energy from hydrogen, calcium, and magnesium. We now use the ionization fraction of hydrogen determined by the above prescription in this calculation. 2.6. Performance The calculation of the NLTE level populations is a significant computational task. A few simple tests with various parameters have found that the code slows by a factor of 10–20 compared to previous versions. To offset these losses, HYDRAD has been recently parallelized with OpenMP. In Figure 3 , we briefly show a timing test of the code. We have run an example simulation of electron beam heating (see the next section) with energy flux F 0 = 3 × 10 9 erg s −1 cm −2 for 10 s of heating. The simulations were only run for 100 s of simulation time. The timing of the simulation run with the older version of the chromosphere is shown in red, compared

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Database B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


by observations of polarised optical radiation. However, polarization signatures are only detected in a limited number of sources, and at present there is no compelling evidence for a simple obscuring torus geometry, possibly suggesting a more complex picture (see the recent review by Netzer 2015 , and references therein). The different AGN obscuration models discussed so far may be broadly divided into two categories: static and dynamical. For instance, analytic calculations suggest that the vertical support for a geometrically thick obscuring torus can be provided by infrared radiation pressure (Krolik 2007 ). The different nature of smooth and clumpy dust tori in different AGN luminosity regimes has been analysed by Hönig Beckert ( 2007 ). Circulation of gas leading to the development of radiation-driven ‘fountains’ has also been proposed as a possible mechanism for the formation of the obscuring torus (Wada 2012 ). The detailed evolution of dusty gas clouds illuminated by the central AGN has been followed through 3D radiation hydrodynamic (RHD) simulations (Namekata, Umemura Hasegawa 2014 ); while the dynamics of dusty gas subject to anisotropic disc emission, X-ray heating, and reprocessed IR radiation, has also been analysed via axisymmetric RHD simulations (Namekata Umemura 2016 ). Another class of models consider the obscuring toroidal structure as supported by winds driven by IR radiation pressure, due to the relatively high opacity of dust in the infrared (Dorodnitsyn, Bisnovatyi-Kogan Kallman 2011 ). A further refinement of the model based on multi-dimensional RHD simulations confirms that the geometrically thick obscuration on pc-scales can indeed be provided by outflows driven by IR radiation pressure on dust (Dorodnitsyn, Kallman Bisnovatyi-Kogan 2012 ). On the other hand, 3D time-dependent RHD simulations of dusty tori suggest that UV radiation pressure on dust can drive powerful winds with considerable mass loss rates (Chan Krolik 2016 ). The most recent multidimensional simulations of dusty gas acceleration in star-forming galaxies suggest that dusty clouds are able to survive longer when accelerated by radiation pressure rather than entrained in a hot outflow (Zhang et al. 2018 ). Here, we envisage a simple picture in which AGN radiative feedback, driven by radiation pressure on dust, strongly affects the distribution of dusty gas in the host galaxy. We assume a spherical shell geometry throughout the paper, while we separately discuss the case of anisotropic emission from the accretion disc in Section 5 . Our results are in agreement with the observational data based on the 70 month Swift /BAT AGN catalogue, which is the most complete sample of X-ray selected AGNs, with measured N H and λ values, currently available. This suggests that our model assumptions are reasonable and sufficient for the purpose of comparison with X-ray observations. In summary, the overall picture is entirely consistent with AGN radiation pressure

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


IRIS detected quasi-periodic fluctuations in the relative Doppler velocities of the C i , Si iv , and O iv lines from its pointing position about 40′′ east of the EIS slit's position, within the same ribbon. This may be because IRIS observed that portion of the ribbon to undergo quasi-periodic fluctuations before the onset of explosive chromospheric evaporation, when the amplitudes of velocity fluctuations may simply have been more readily detected. Other contributing factors may be that IRIS has higher spectral resolution, and spatial resolution nearly 10 times better than that of EIS. The IRIS observations revealed, however, that quasi-periodic fluctuations in the redshifted relative Doppler velocities were not as pronounced as those in the intensities (Brosius Daw 2015 ). The observed O iv , Mg vi , Mg vii , Si vii , Fe xiv , and Fe xvi fluctuations provide evidence for quasi-periodic energy injections into the chromosphere (at the location of the flare ribbon observed by EIS) by accelerated, nonthermal particles. This is similar to the conclusion drawn by Brosius Daw ( 2015 ) regarding fluctuations in C i , Si iv , and O iv at the location of the flare ribbon observed by IRIS , all of which occurred before the appearance of Fe xxi . In that work, each episode was thought of as a failed flare because no emission at temperatures characteristic of flares (Fe xxi emission) was produced. The quasi-periodic injections of nonthermal beam particles could be produced by MHD oscillations in a magnetic trap at the site of the initial flare energy release (by reconnection). In this scenario (Zaitsev Stepanov 1982 ), MHD oscillations in the trap are generated by the impulsive energy release itself, and modulate the periodic escape of particles therefrom. Alternatively, the injections could be driven by MHD oscillations in a nearby, nonflaring magnetic loop (Nakariakov et al. 2006 ). In this scenario, the oscillating loop perturbs the plasma external to the loop, which produces variations of the electric current density in the flaring region. These induce plasma micro-instabilities and, hence, anomalous resistivity in the flaring region, which can quasi-periodically trigger the magnetic reconnection events that accelerate nonthermal particles. Yet again, the injections could be supplied by quasi-periodic particle acceleration episodes during dynamic magnetic reconnection in a large-scale current sheet dominated by repeated formation of magnetic islands (Kliem et al. 2000 ). In this scenario, particles are presumably accelerated by DC electric fields near the reconnection X points. Recently, Guidoni et al. ( 2016 ) extended the approach of Drake et al. ( 2010 ; see also Karpen et al. 2012 ) to investigate electron acceleration in sunward-moving magnetic islands (plasmoids) formed by spatially and temporally variable reconnection in a flare current sheet. Their 2.5D simulations of an eruptive flare revealed that energy gains in a single island

B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O B-Telescope O B-Telescope O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


estimated using the total luminosity derived from spectral energy distribution fitting. Traditionally, MYSOs have been found to have luminosities similar to those of OB stars. Spectral types have rarely been determined directly in MYSOs, and not at all in the case of RMS MYSOs. Previously, Kendall, de Wit Yun ( 2003 ) observed H -band spectra of three MYSOs, IRAS 171753544, IRAS 174412910 and IRAS 180791756 at intermediate resolution ( R 5000). They found photospheric absorption lines in IRAS 180791756 and IRAS 171753544, and assigned spectral types of B3V and O7-8V, respectively, to these two objects. Most of our MYSOs show strong, broad H i emission, due to a stellar wind (see also Section 4.3 ). However, two of the objects in this sample (G015.128800.6717 and G034.050000.2977_A) display absorption in the Br10, Br11 and Br12 lines. There are also hints of absorption in Br and Pa, alongside the dominant emission in these two objects. The RMS survey quotes G015.1288 as having L bol =13000 L . This corresponds to a spectral type of B0.5 for a zero-age MS star, using the values from Straizys Kuriliene ( 1981 ). The absorption profiles in the H band of G015.1288 were compared with spectra of normal stars from the surveys of Hanson etal. ( 2005 ) and Meyer etal. ( 1998 ). A spectral type of F or later would be hard to reconcile with the MYSO data, as F stars have a large number of metal lines. In addition, early O spectral types do not match either, as their Brackett lines are weaker and narrower than observed for G015. Late O and B1-5 stars show a He i line at 1.70 m, at comparable or higher strength than the hydrogen absorption lines, and is not present here. We note that the NIR spectral surveys of Hanson etal. ( 2005 ) and Meyer etal. ( 1998 ) do not have complete spectral and luminosity type coverage. The only B-type sample spectra available are those for B1, B5 and B8. Therefore, given the presence and absence of diagnostic absorption lines, a spectral type later than B5 and earlier than A9 would be expected for G015.1288. However, the Br10-12 absorption in the B5-A9 stars is much stronger than observed for G015.1288. It is possible that a significant amount of excess could veil the H i absorption to a comparable level to our observations. This is illustrated in Fig. 2 , which zooms in on the Brackett lines in the H -band spectra of a number of normal B8-A7 stars with added excess continuum emission. Overplotted on these are the GNIRS data for G015.1288. A B8 star with added excess does not match the observed H i profiles, as the lines in this star are considerably narrower than in G015.1288. In addition, the He i line is still strong enough to be observed. The A0Ib profiles match the observations in strength, but are narrower. An A0V star has broader wings in the Brackett lines than G015.1288. In the absence of more observational MS star spectra to compare our data with, we conclude that G015.1288 has the spectrum of an early A giant or

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Wavelength I-Wavelength O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O B-Wavelength B-Wavelength O B-Wavelength O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O B-Survey I-Survey O B-CelestialObject O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength I-Wavelength O B-CelestialObject O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O B-CelestialObject O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-Wavelength O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O B-Instrument O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-CelestialObject O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O


by disk emission, even in the optical/near-IR, and changes at optical/near-IR wavelengths would reflect changes in the superheated disk rather than the central protostar. FU Ori ( D = 450–500 pc; Hartmann Kenyon 1996 ) is a wide (05) binary system consisting of a outbursting young stellar object modeled as a 0.3 M ⊙ with a 0.04 M ⊙ circumstellar disk (Zhu et al. 2007 ) at an A V of ∼1.8 (Zhu et al. 2010 ) and a more massive young companion ( M > 0.5 M ⊙ ; Reipurth Aspin 2004 ; Wang et al. 2004 ; Hales et al. 2015 ). The companion (FU Ori S) is at 227 au projected separation (Pueyo et al. 2012 ) at an A V of 10 ± 2 (Pueyo et al. 2012 ). In this work, we present new mid-IR photometry and spectroscopy of FU Ori obtained with the Stratospheric Observatory for Infrared Astronomy (SOFIA) in 2016, and we compare with previous observations with the Spitzer Space Telescope in 2004. We find that the continuum flux density at wavelengths shorter than 8 μ m (arising from the hot inner disk) has decreased by 12%–14%, dominating the change in bolometric luminosity. The continuum at wavelengths greater than 10 μ m has decreased only 7% within uncertainties. Additionally, we find that the H 2 O vapor absorption resulting from the superheated disk photosphere remains present. Finally, we note no significant changes in the 10 μ m silicate emission feature, which shows no evidence of crystallinity. Taken together, these observations suggest that the decrease in viscous dissipation energy over this period has occurred predominantly in the innermost section of the disk, which is changing rapidly. 2. OBSERVATIONS 2.1. SOFIA-FORCAST Spectroscopy We observed FU Ori with the Faint Object infraRed CAmera for the SOFIA Telescope (FORCAST; Herter et al. 2012 ) on board the SOFIA (Young et al. 2012 ) on two separate flights (on 2016 February 4 and 18, flights 272 and 279, respectively); see Table 1 for the Observing Log. The observations were performed in C2N (two-position chop with nod) mode, using the slit, and the NMC (nod-match-chop) pattern in which the chops and nod are in the same direction and have the same amplitude. In each case, an off-source calibrator was selected from an ensemble of observations from cycles 2–4, using the observation closest in zenith angle and altitude to the science target (A. Helton 2016, private communication). We did not use dithering. The FORCAST grism was used in three spectral bands, G063 (∼5–8 μ m), G111 (∼9–13 μ m), and G227 (∼18–25 μ m), from 5 to 25 μ m to achieve a spectral resolving power of R ∼ 200. The achieved signal-to-noise ratios (S/Ns) for the individual grism segments were 30 (G063, measured from 5.7 to 6.3 μ m), 46 (G111, measured from 11 to 12 μ m), and 22 (G227, measured from 21 to 26 μ m). The total observation time was 12,904 s (3.58 hr), including imaging, which totaled approximately 0.33 hr (see below). Table 1 Observing Log Notes. Observations were performed over two different flights, 272 (2016 February 04) and

O O O O O O B-Wavelength O O O B-Wavelength O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Wavelength O O O O B-CelestialObject I-CelestialObject O O O B-Telescope I-Telescope I-Telescope I-Telescope I-Telescope B-Telescope O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O B-Telescope O O O B-CelestialObject I-CelestialObject O O B-Instrument I-Instrument I-Instrument I-Instrument O O B-Telescope I-Telescope B-Instrument B-Citation I-Citation I-Citation I-Citation O O O O B-Telescope B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Instrument O O O O O O O B-Wavelength B-Wavelength I-Wavelength I-Wavelength B-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


et al. ( 2018 ) and Stiele Kong ( 2018 ), but due to sparser observations, the correlation is not as tight as seen in this work. The trend observed by Mereminskiy et al. ( 2018 ) resembles the correlation observed for GRS 1915+105 (Shaposhnikov Titarchuk 2007 ). Such correlation has been previously observed in several BHBs (Vignarca et al. 2003 ) and provides a tight link between the QPO and the hard-component emission. In that work, the correlation was found on time-scales of minutes in GRS 1915+105 and on scales of days-months for other transient BHBs. Here, we follow the QPO over 5 d. Fürst et al. ( 2016 ) also observe an extremely tight correlation of the QPO centroid frequency and power-law index in GX 339–4 in the frequency range of 0.6–1 Hz and Γ of 1.66–1.8. The authors estimate the mass of the BH using a scaling relation provided by Shaposhnikov Titarchuk ( 2007 ). The typical evolution of the outburst of a transient BHB is described best in the HID, where a counterclockwise path is followed. In the right ‘vertical’ branch, corresponding to the LHS, the flux increases, the power-law index is well correlated with it and the frequencies of the timing features, although no peaked QPO is usually observed, also increase (see Motta, Belloni Homan 2009 ; Motta et al. 2011 ). Since the spectrum gradually softens, this branch is not really vertical. In the top ‘horizontal’ branch, the spectrum softens further, with the power-law index Γ increasing in absolute value. The flux usually continues to increase, although not as fast as before, and the QPO frequency is correlated with both flux and Γ. Given the shape of the PDS and the values of the power-law index, our observations are clearly on the HIMS branch. As described before, usually both the LHS and HIMS branches see a correlation between frequencies of variability components, the flux, and the power-law index. In our AstroSat data, we see the absence of a correlation with flux, although in the NICER data following our observation this correlation is present. This clearly shows that the QPO frequency is related to the energy spectral shape and not to the flux in either of the two main components. In a thermal-Comptonization scenario, the index of the hard component steepens as the population of electrons cools due to the increase in soft photon input (see e.g. Motta et al. 2009 ). However, from Fig. 4 we can see that this is not the case here, as the secular increase in disc flux is not followed by Γ. Another parameter that determines Γ is the optical depth of the cloud τ 0 : for a spherical cloud and input photon energy much lower than the temperature of the electron cloud kT e , we have , where (Sunyaev Truemper 1979 ; Sunyaev Titarchuk 1980 ). Therefore in order to keep Γ from varying when the cloud temperature decreases is to increase τ 0 in a very specific way. Fluctuations in this relation could in principle give rise to the observed fluctuations in Γ, but it would be very ad hoc.

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well as for the PSF models, which are mandatory for the subsequent modeling. Excluding variance maps from our modeling process would lead to an underestimation of errors, since the flux contribution of the central point source is significantly higher than that of the sky. We create 2D two-component models of our objects utilizing the same PSFMC algorithm 8 8 See https://github.com/mmechtley/psfMC for additional information, examples, and documentation. (Mechtley 2014 ) as described and used in Mechtley et al. ( 2016 ). We use a point-source model and a Sérsic profile to simultaneously fit the central AGN component and underlying host galaxy respectively. Although the host galaxies may not be perfectly described by a Sérsic profile, it is a sufficient approximation, as we can determine potential asymmetries or tidal features. Furthermore, the advantage of using this two-component approach is twofold: First we ensure that the contributing flux of the central source is not overestimated and avoid an eventual PSF oversubtraction. Second, we also keep the computational time needed for the fits within reasonable limits. Contrary to most of the tools adopted to fit galaxies in 2D, our algorithm allows the user to input a prior probability distribution for the fitted model parameters of the different components, e.g., position in x and y , total magnitude, Sérsic index, position angle, etc. As a result, posterior probability distributions of all the fitted parameters with the corresponding covariances are produced. Additionally, the algorithm does not assume an error-free PSF, but implements the respective PSF variance maps during the convolution process. To that end, we deploy the HST focus model (Cox Niemi 2011 ) to determine for each source the time-dependent telescope focus and find the best matching PSF within our PSF library. For the actual fit we mask and exclude neighboring galaxies, whose central flux peaks do not coincide with the central source in the respective images. The resulting point-source component is then subtracted from the HST /WFC3 image to remove the AGN, and just display the host galaxy beneath. In Figure 2 we show from left to right for one of our 21 AGN sources the original, drizzled HST /WFC3 image, the convolved point-source model, and the point-source-subtracted model showing the host galaxy, once with the original sampling, and once smoothed by a σ = 2px Gaussian. All four images are shown in the same arcsinh scaling. Comparable plots for the remaining 20 AGNs are shown in Appendix A . Figure 2. Different steps of our AGN/host galaxy modeling process shown for one sample source: in the first column we show the original, processed HST /WFC3 image, the second column displays the PSF convolved point-source model, and the third and fourth columns show the host galaxy after point-source subtraction with the original sampling and smoothed by a σ = 2px Gaussian, respectively. This particular source is one of the five AGN host

O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O B-URL O O O O O O B-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Model O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O


locations of the clusters on the sky are used to correct the individual distances for the inclination of the disc, thereby obtaining a more accurate absolute distance for the LMC. By doing so, we obtain an estimate of the distance to the centre of the LMC that is more accurate by accounting for the geometry of the galaxy. This process is outlined in van der Marel Cioni ( 2001 ) and Grocholski et al. ( 2007 ), and also presented in the equations ( 2 ) through ( 5 ) below. The distance to the LMC centre is denoted by D 0 . D is the distance to any location on the disc plane; in our case, this is the distance to each cluster. We assume the LMC disc to be inclined to the plane of the sky by the angle i around an axis at the position angle θ. θ is measured anticlockwise from the west; to align with usual astronomical convention, Θ is defined the position angle from the north (Θ = θ – 90). These values are adopted from van der Marel Cioni ( 2001 ); we note that Grocholski et al. ( 2007 ) explored a variety of geometries from other past works, finding differences in the resulting distance of ≲0.2 kpc for varying i and Θ. The right ascension and declination are referred to, respectively, by α and δ in the equations below, To calculate the distance to the LMC centre (D 0 above), it is necessary to calculate ρ and sin (φ − θ). The former is calculated as in equation ( 3 ): Via a typical trigonometric expansion, sin (φ − θ) may be determined as sin (φ − θ) = sin (φ) cos (θ) – cos (φ) sin (θ). These components are detailed in equations ( 4 ) and ( 5 ): Through these equations, we determine an average, error-weighted distance modulus to the LMC centre of D 0 = 18.20 ± 0.06, equivalent to 46.42 ± 0.45 kpc. However, NGC 1841 is a clear outlier in these data, lying more than 3σ from the mean D 0 distance to the LMC centre, leading to several possibilities. One interpretation of these results suggests that NGC 1841 may not be a member of the LMC disc or may have been pulled out of the LMC disc (Grocholski et al. 2007 ). Another view is that an uncertain estimate of the reddening could be affecting the distance measurement; if the reddening of NGC 1841 is doubled from E ( B − V ) = 0.18, it is brought into agreement with the LMC mean distance. Regardless of the reason(s) for the discrepancy, we remove NGC 1841 as an anomaly and continue determining the distance to the LMC centre. Leaving out the extreme outlier of NGC 1841, the error-weighted mean is 18.41 ± 0.04 (48.29 ± 0.38 kpc). This is within the range of the result from Section 4.1 from the HB analysis, as seen in Fig. 9 , where the subdwarf distances and the subdwarf distances adjusted for the LMC geometry are compared to those derived from the HB analysis. The subdwarf distances tend to be shorter by 0.14 mag on average than those of the HB analysis; the adjusted subdwarf distances are also shorter than the HB distances by 0.12 (leaving NGC 1841 out). The offset appears to be consistent regardless of

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


of residuals on a cluster by cluster basis when compared to the grid of Kroupa IMF clusters. This increase in scatter is especially noticeable in mass, and originates from the M 3 degeneracy described above. As the mass of the cluster and its IMF slope are highly dependent on each other, the wide scatter in mass is further reflected in the inaccurate values inferred for 3 . Indeed, given the flat and broad posterior PDFs for the IMF slope, the medians of 3 have no physical meaning and simply reflect the median of the prior on 3 [in this case, a flat distribution between 3 (3.0, 1.5)]. Figure 12. Comparisons of the medians of the inferred posterior PDFs for all physical parameters to the true values adopted when simulating the mock clusters. Results for individual clusters are shown with crosses, while the median residuals and the corresponding 10th and 90th percentiles are shown with solid points. We plot analysis including broad-band photometry alone with (red, circles) and without (purple, diamonds) a variable IMF. We also show results of the variable IMF case when including H (blue, squares). Points are shifted along the x -axis for visualization purposes, with the leftmost point situated at the true x -axis value. Given this result, it is clear that obtaining refined constraints on the upper-end slope of the IMF requires a different approach. It is readily apparent that the ability to apply a prior to the cluster mass that is independent of the broad-band photometry may prove very useful in analysis such as this. Independent constraints on the masses of clusters that may then be used as priors may come, for instance, from dynamical measurements (e.g. Ho Filippenko 1996 ; Gerhard 2000 ). The era of 30m telescopes will make such measurements more feasible. To illustrate this point in more quantitative terms, we present in Fig. 13 a study of the accuracy with which 3 can be recovered as a function of the width of a prior on the mass. Results of three choices of IMF are presented, when leaving the mass unconstrained, and when applying a top-hat prior with width 0.5, 0.2 and 0.05 dex about the target cluster mass, for both broad-band filters only and broad-band filters combined with H. This analysis shows that, as expected, the medians of the PDFs on 3 converge to the true value as the precision with which the mass is constrained increases. Due to the difference between logarithmic intervals in mass and the linear scale in the IMF upper-end exponent, there is a weak scaling of 3 with the prior width, with the underlying values being recovered to within 3 0.2 when the mass is constrained to better than 10percent. However, more realistic constraints on the mass (to within a factor of 2), for instance via dynamical measurements, appear to be useful to rule out extreme variations in the IMF upper-end slope through multiwavelength broad-band photometry, given the assumptions of our model. The inclusion of H results in a wider scatter towards zero for

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