DR2 is designed to observe stars in the Milky Way, it does not always provide accurate measurements of sources that are extended or in very crowded regions. We urge caution and follow-up observations of the objects in Table 5 , especially those within 10′ of the center of NGC 5128. B.2. Conflicting Measurements Thirty-two of the identified GCs in NGC 5128 have conflicting classifications in the literature. A majority of these conflicting classifications occur because the target was measured to have a radial velocity within the standard acceptable range of ≈250–1000 km s −1 by one group but outside this range by another. Additionally, two objects were identified as GCs based on their radial velocities but later found to be stars based on HST/ACS imaging (see note 9 in Table 6 ). Another object was subsequently identified as a star because of its spectrum (see note 5 in Table 6 ). These objects with conflicting measurements have been matched between studies based on position and, in many cases, by their ID as well. When examining these objects with PISCeS data, several of them can be visually identified as background galaxies. We believe seven of these objects are indeed true GCs because a majority of their measurements classify them as such and they do not visually look like galaxies; thus, they are included in Table 4 . For each object with conflicting measurements, Table 6 first lists the R.A. and decl. in degrees (J2000). The position measurements are from the PISCeS data set to provide a uniform astrometric system. Next, we list the conflicting radial velocities and errors (kilometers per second) along with their sources, if applicable. We also include our classification of each object, as well as further information indicated by the “Notes” column. Many sources experience a discrepancy between Woodley et al. ( 2007 ) and Beasley et al. ( 2008 ). It appears that between sharing data with the Woodley group and publishing their own paper, the Beasley group updated measurements on several targets. We note that one of these objects has a mismatch in both its velocity measurement and coordinates. Woodley et al. ( 2007 ) gave an incorrect position for HH-048 as α = 20068900, δ = −43119111, with a radial velocity of 856 ± 56 km s −1 , citing Beasley et al. ( 2008 ) as the reference for this radial velocity measurement. Beasley et al. ( 2008 ), however, listed HH-048 as a background galaxy ( v r = 16,000 km s −1 ) with the same R.A. but a decl. of δ = −43011917 (note the single difference in the digit following the decimal point). Based upon visual inspection, the position listed in the Beasley catalog is correct and listed in Table 6 . Appendix C An Object with Too Many Names During our literature review, we found an instance where two confirmed GCs with different names shared the same position. The objects identified most recently as GC 0462/T17-1560 and GC 0200/T17-1561 have the same position in the Sinnott et al. ( 2010 ) catalog and positions


the system cannot fulfil thermal and mechanical equilibrium conditions simultaneously unless some further assumptions are implemented. Since the initial state is in mechanical equilibrium and the upper and the lower layers have different densities, their temperatures cannot be the same. Thermal equilibrium, however, implies that both layers have the same temperature. In order to resolve this contradiction, JK proposed that one layer is optically thin and the second layer is optically thick. They also assumed that the system is adiabatic. In the optically thin regime, where opacity on one or both sides of the interface is negligible, continuity of E r holds on whereas T g is independent of E r . There are, however, further alternative ways to specify background state self-consistently. Jiang et al. ( 2013 ), for instance, argued that even when the background state is not strictly in thermal equilibrium, its evolution affects subsequent thermal evolution very slightly so long as the thermal time-scale is much longer than the instability time-scale. Under these circumstances, therefore, the initial configuration can be imagined to be in thermal equilibrium for exploring RT instability. Jiang et al. ( 2013 ) constructed an initial state with zero absorption opacity and effectively infinite thermal time-scale. As we mentioned earlier, JK assumed that one side of interface is in the adiabatic regime, and the other side is in the optically thin regime. Here, we follow JK assumptions for specifying the initial equilibrium state, which are applicable to some astrophysical systems. The momentum and the energy equations are written as where . Equation ( 34 ) is written based on the radiative equilibrium assumption. The parameter E represents the Eddington limit of the background state. We note that the introduced parameter E is then replaced by E 1 and E 2 for the lower and the upper layers, respectively. Also, we have Introducing the ratio of the radiation pressure to the gas pressure as r = p r / p g , we can then write and We note that the Eddington limit parameter E is a fixed input parameter; however, the parameter r is not a constant under our imposed conditions. 4.2 Linear perturbations After deriving the above equilibrium solutions, we can now linearly perturb RMHD equations. Following JK , we also apply adiabatic approximation by which we mean perturbed energy flux is neglected. It means that we require F ́ ≪ E t u ́. Using this approximation, we can assume that the opacity is a constant. Therefore, the parameter E becomes a constant, which is actually our implemented assumption. This parameter, however, may be different in the upper and lower layers. We thereby arrive at the following equations: As before, we assume that all the perturbed quantities are proportional to exp (ω t + i k x x ), and the above equations after lengthy but straightforward calculations reduce to the following equations: where parameters B and C depend on the input


with redshift in terms of the density parameters of both SF and AGN components of the RLF. In terms of luminosity evolution, parameters of both the SF and AGN components are consistent with being constant with redshift. We note that the normalization of the radio luminosity function of optically selected quasars by Kimball et al. (2011) and Condon et al. (2013) , shown in the first panel of Fig. 6 , is more than an order of magnitude lower than that constrained by fitting our data with the analytic RLF form of Eq. ( 8 ). The main cause of this difference is most likely different sample selection. While optically-selected quasar samples by Kimball et al. and Condon et al. contain only the unobscured quasars, sampling quasars with very high bolometric luminosities, our XQSO sample contains a combination of both obscured and unobscured quasars. By examining the hardness ratios (HRs) of XQSOs ( Civano et al. 2016 , Marchesi et al. 2016 , Pierre et al. 2016 ) and applying a threshold of HR = −0.2 to separate obscured from unobscured sources (e.g., Civano et al. 2012 ), we found that most of XQSOs (190/274) are obscured based on their X-ray emission and most likely would not be included in samples of optically-selected quasars. 5. Discussion In this section, we discuss the possible contributions to the detected radio emission: star-forming processes within the host galaxy in Sect. 5.1 and AGN activity in Sect. 5.2 . In Sect. 5.3 we define thresholds in 1.4 GHz radio luminosity below and above which more than 80% of the XQSOs are dominated by the star formation and AGN-related radio emission, respectively. This enables us to disclose the true composite nature of the XQSOs in which both star formation and AGN activity make a significant contribution to the radio luminosity. 5.1. Host galaxies of XQSOs The sample of XQSOs studied in this work was drawn from the moderate-to-high radiative luminosity AGN sample in COSMOS and high-excitation radio galaxy sample of AGN in XXL-S. Host galaxy properties of these AGN have been studied by Delvecchio et al. (2017 ; COSMOS) and in Paper XXXI (XXL-S). Both studies found that the host galaxies of these AGN have star formation rates (SFRs) and stellar masses consistent with typical main-sequence galaxies. The same conclusion was reached by Stanley et al. (2015) from an analysis of a sample of ∼2000 X-ray selected AGN (10 42 erg s −1 L 2−8 keV 10 45.5 erg s −1 ) detected in the Chandra Deep Field North and South and the COSMOS field. They studied the distribution of star formation-related IR luminosity and X-ray luminosity over 0.2 z 2.5, as proxies of SFRs and AGN activity, respectively. They found a strong evolution of their average L IR, SF with redshift, similar to that observed for the main-sequence galaxies. These results were confirmed by multiple other studies of quasar and AGN host galaxy properties (e.g., Rosario et al. 2012 , 2013 , Lanzuisi et al. 2017 , Stanley et al. 2017 , Suh et al. 2017 , 2019 ).


from isolated cold globules to rich clusters. There are two common observational measures of environment. First, there are the environmental conditions set by the properties of the natal molecular gas, particularly the dense gas. Second, there are environmental variations traced by the protostars, including the densities of young stars and the systematic changes in their properties. These protostars and their properties have been quantified with Spitzer data (Megeath et al. 2012 , 2016 ) and Herschel data from HOPS (Furlan et al. 2016 ). With Herschel observations at longer wavelengths and of wider fields than included in HOPS, Stutz Kainulainen ( 2015 ) measured the column density ( N H ) and mass distributions across Orion A, quantifying environmental differences. They found that the column density probability distribution functions vary with location. Furthermore, Stutz Gould ( 2016 ) and Stutz ( 2018 ) found that the mass per unit length ( M / L ) varies significantly across Orion A, with the Orion Nebula Cluster (ONC) having a higher M / L than the Integral-Shaped Filament (ISF), and the ISF having a higher M / L than LDN 1641. These column density and mass variations imply variations in the volume density and gravitational potential across the cloud. Environmental variations are also found in the radial velocities of the gas. González Lobos Stutz ( 2019 ) showed that gas radial velocities in both high- and low-density tracers show significant variations within the ISF. In particular, the northern portion has more centrally concentrated gas with stronger overall velocity gradients compared to the southern portion, which transitions to the LDN 1641 region. Systematic variations of the temperatures, line widths, and densities of cores across Orion A were also found by Wilson et al. ( 1999 ). Turning to the stellar content, the observed properties of the protostars vary with environment. Kryukova et al. ( 2012 ) found variations in the Orion clouds, where protostars are more luminous in dense regions. Stutz Kainulainen ( 2015 ) found that the fraction of protostars in the young Class 0 phase varies systematically with changes in environment as indicated by variations in the slope of the dust column density probability distribution function ( N -PDF) across the region. The variation in this fraction may result from differences in the star formation history between these regions, as also suggested by Fischer et al. ( 2017 ), the influence of feedback (Sadavoy et al. 2014 ), or variations in infall rates of the protostars due to variations in the densities of their birth environments (Kryukova et al. 2012 , 2014 ; Dunham et al. 2014 ). This paper is part of a series describing results from HOPS. These papers include evolutionary studies of protostars via modeling of their SEDs (Furlan et al. 2016 ; Fischer et al. 2017 ), the analysis of far-IR protostellar spectra (Manoj et al. 2013 , 2016 ), the discovery and characterization of the


both the expansion history, due to the scalar field, and the evolution of the density perturbation amplitudes, due to the coupling. 3 SIMULATIONS For our study, we produced simulations of the following three cosmologies: a reference CDM, an uncoupled quintessence model and a CDEdark matter model with a coupling parameter 0 . Following Elahi etal. ( 2015 ), we choose ( h , m , b , 8 )(0.67, 0.3175, 0.049, 0.83), ensuring all cosmologies have parameters consistent with z 0 CDM Planck data (Planck Collaboration XVI 2014a ; Planck Collaboration XIII 2016a ). The coupling parameter, 0 0.05, in our simulation is chosen to test the boundaries of allowed coupling (see Pettorino etal. 2012 ; Xia 2013 ) in order to maximize any observational differences that may exist between this cosmology and the standard CDM model. The linear power spectrum and growth factor f dln D ( a )/dln a are calculated using first-order Newtonian perturbation equations and the publicly available Boltzmann code cmbeasy (Doran 2005 ). We note that although these simulations have neglected star formation and feedback physics, baryons do not appear to significantly change the void population. Large underdense regions remain underdense, regardless of whether the simulation includes dark matter only, or accounts for hydrodynamic or full baryonic physics (Paillas etal. 2016 ). Initial conditions are generated by perturbing particles placed on a Cartesian grid with the first-order ZelDovich approximation using a modified version of the publicly available n-genic code. The modified code uses the growth factors calculated by cmbeasy to correctly calculate the particle displacements in the non-standard cosmologies. All the simulations are started at z 100 with the same phases in the density perturbations. Choosing the same initial phases results in underdense regions forming in the same locations in all simulations. Each underdense region in a given simulation has a corresponding underdense region in other simulations (perhaps with different properties such as density or size, depending on cosmology). This is the case despite potentially different exact profiles of density perturbations between cosmologies with different power spectra, such as CDM and CDE. We use dark-gadget , a modified version of the p-gadget -2 N -body code (for more details, see Carlesi etal. 2014a ). The key modifications are the inclusion of a separate gravity tree to account for the additional long-range forces arising from the scalar field, and an evolving dark matter N -body particle mass that models the decay of the dark matter density. The code requires the full evolution of the scalar field , the mass of the dark matter N -body particle and the expansion history. Our simulations are 500 h 1 Mpc boxes containing 2512 3 particles (dark matter and gas) giving a mass resolution of m dm ( m gas )6.9(1.3)10 10 h 1 M . 1 Though the mass resolution is poor for identifying haloes, this resolution is more than enough to


between the two models, though the BHWR model has a slightly higher duty cycle than the onyWind model that matches the correspondingly higher SMBH mass growth (compare Fig. 3 ). Figure 5. Time distribution of different accretion rates in units of the Eddington limit (so-called BH duty cycles) for the different models (see legend) compared to observations (square: Ho 2009 , circles: Greene Ho 2007 , upward-pointing triangles: Kauffmann Heckman 2009 , downward-pointing triangles: Heckman etal. 2004 , all taken from Novak etal. 2011 ). Both simulated models fall above all of the observed values at Eddington ratios of 1 and 10percent. 4 BH-GOVERNED ISM EVOLUTION IN THE GALACTIC CENTRE In the previous section, we have shown that feedback from the central SMBH is both necessary to, and capable of keeping an isolated ETG quiescent. The more physically complete feedback model, including both wind and radiative feedback, is more efficient at this than the pure wind-feedback model. In this section and the next, we will now examine how the feedback achieves the quiescence through its influence on the galactic gas, and how it creates metal-rich, large-scale outflows that enrich the CGM. We begin with Fig. 6 , which shows colour maps of the surface density (top row) and the density-weighted average temperature (central row) in a 10kpc thick, 10 10kpc wide region around the galactic centre (and therefore around the central SMBH), as well as the temperaturedensity phase diagrams of all the galactic gas (bottom row). In the maps, the line of sight is chosen to be parallel to the initial angular momentum of the hot halo. The solid vertical and horizontal lines in the phase diagrams mark the star formation thresholds in density and temperature, respectively, i.e. gas in the bottom-right corner of the diagrams is star forming. The individual temperaturedensity bins are colour-coded for gas particle number density, which is rising from blue to red colours. The dashed line marks the resolution of the simulation limit following the Jeans mass criterion. Above the dashed line, the Jeans mass of the gas is resolved, i.e. it is larger than the kernel mass M ker N ngb m gas 10 7 M , while it is unresolved below the line, meaning that gas in this region could collapse under self-gravity due to numerical noise. Almost all of the gas particles with unresolved Jeans mass are in the star-forming region of the phase space, and will be turned into star particles if they stay in this region. Hence, while this resolution does not allow us to make predictions about the substructure of the dense, cool gas, it is high enough to investigate the global and larger scale effects of AGN feedback that interest us in this paper. The three left columns show the state of the three models (from left to right: no-BH, BH-W, BH-WR) at the end of the simulation time, while the last column to the right shows the state of the BH-WR model about 30Myr before the end of the simulation. Figure 6. Maps


relative to the forcing from m p is Even with the uncertainties of various parameters, the inner two planets are weakly coupled to the stellar spin. The stellar obliquity (relative to ) therefore varies between |θ ⋆ p − θ 12, p | and θ ⋆ p + θ 12, p (see equations 12 – 13 ; note that since L p ≫ L 12 = L 1 + L 2 for this system, is aligned with ). The observed stellar obliquity (λ ≥ 37°) then implies θ ⋆ p + θ 12, p ≥ 37° (see Li et al. 2014 , for a previous analysis). 5 DISCUSSION The main goal of this paper is to inform the readers that there is a simple way to assess the dynamical role of external perturbers on the spin–orbit misalignments of planetary systems. The key is to evaluate the star–planet spin–orbit coupling parameter (equation 9 ) in response to the differential precession induced by the external companion. We have presented analytic results in various limiting regimes, and discussed how the general results (see Fig. 1 ) scale with different physical parameters (Section 2). Similar analysis can be done for multiplanet systems (Section 4), with the generalised spin–orbit coupling parameter given by equation ( 31 ). As alluded to in Section 1 , continuing observations on spin–orbit misalignments and external companions in HJ and other compact planetary systems may shed light on the formation mechanism of close-in planets. Several recent works of such systems have either neglected some key physical ingredients (spin–orbit coupling) or made such a relatively simple problem unnecessarily complex or obscure. We hope that the results of this paper (see also Lai 2016 ) will make future analysis simpler and more transparent. A recent analysis on spin–orbit alignments of HJs around cool stars concluded that the exterior companions are coplanar (Becker et al. 2017 ). While this may be the case of a few systems with gas giant companion at ∼1 au, similar conclusions cannot be drawn for systems with a slightly more distant companion (see Section 3 , especially Fig. 2 ) since the spin–orbit coupling parameter ε ⋆1 depends sensitively on the perturber’s semimajor axis (; see equation 24 ). Certain HJ formation mechanisms (e.g. LK migration) require an external companion with a high inclination (>40°). As a result, constraints placed on the mutual inclination via the method discussed in this paper may in principle lead to a constraint on the migration mechanism. However, we emphasize that the inclination of giant planet companion may only be constrained when it is relatively close (at ∼1 au, so that ε ⋆1 ≳ 1, see equation 24 ), such as for WASP-41 and HAT-P-13. These systems clearly could not have formed in the classic LK migration picture (with the HJ originating at ∼1 au or beyond), because the initial configuration would be unstable. LK migration requires a much more distant companion (at, say ∼5 − 10 au, so that the initial configuration is stable). Such a companion will satisfy ε ⋆1 ≪ 1, so that it is unable to affect the star-HJ


Chandraobserved 4C +74.26for 70 ks in total using the HETGS in combination with the ACIS detector. The total exposure time was split into two observations that were taken two days apart. For both Obs-ID 4000 and 5195 we retrieved the Medium (MEG) and High Energy Grating (HEG) spectra and their respective response matrices from the tgcat 1 archive. We further treated these spectral products with the CIAO tools (version 4.6). For each observation and for both HEG and MEG, we combined the first positive and negative spectral order using the CIAO script add_spectral_orders. Hence, we fitted jointly HEG and MEG (allowing a free intercalibration factor) with a simple phenomenological power law to check for variability between the two observations. The fitted slopes (, ) and normalizations (Norm, Norm) were very consistent with each other (see below for a physically motivated fit). Therefore, we were able to sum the spectra of individual observations into a single spectrum to improve the signal-to-noise ratio. We did this using the CIAO script add_grating_spectra. We reduced the raw XMM-NewtonObservation Data Files (ODF), available at the ESA archive 2 , using the Science Analysis Software (SAS, version 13) and the HEASARC FTOOLS. We created calibrated EPIC-pn event files selecting only the unflagged single events. To check the time stability of the background we used the light curve in the hard 10–12 keV band, which is background dominated. A high level of background light due to soft proton contamination is evident towards the end of the observation. Thus, we cleaned the event file using a time filter, following the same procedure explained in Di Gesu et al. (2013) . For RGS-1 and RGS-2 we created calibrated event files and background light curves taking the background from CCD 9. The RGS background light curve was quiescent. Next, for all the instruments, we extracted the source and background spectra and we created the spectral response matrices. Finally, we extracted the source count rate in all the available OM filters, namely ( Å), ( Å) and ( Å). Using the interactive SAS tool omsource, we computed the source count in a circular region centered on the source coordinates and with a radius of 6 pixels. For the background we used another circular region of 12 pixels, free from other sources and instrumental contamination. We converted the count rates to fluxes using the standard conversion factors provided in the SAS watchout web page 3 . Hence, assuming , we corrected all the fluxes for the Galactic reddening ( =0.39, Schlafly Finkbeiner 2011 ). For the correction, we used the IDL routine ccm_unred, which dereddens a user-defined vector of fluxes using the Galactic extinction curve of Cardelli et al. (1989) . Fig. 1 Spectral energy distributions for 4C +74.26. Filled circles: OM fluxes corrected for the Galactic extinction. Open squares: X-ray intrinsic continuum obtained from a phenomenological fit of the EPIC-pn spectrum. Fig. 2


suppression of gas inflow into a massive core (Ikoma Hori 2012 ). For example, high opacities of dust grains in a disk reduce accumulation of a disk gas by a massive core (Lee et al. 2014 ; Lambrechts Lega 2017 ). Gas dynamics around a planet embedded in a disk would be another adjustment knob for the gas flow. Ormel et al. ( 2015 ) found that an isothermal gas flow from high altitude might escape from the gravitational sphere of a planet before accreting onto it. A rapid recycling of a disk gas regulates gas accretion onto a massive core, whereas in a non-isothermal case this recycling process would be limited by an entropy gradient between the ambient disk gas and the atmosphere (Cimerman et al. 2017 ; Kurokawa Tanigawa 2018 ). According to recent magnetohydrodynamic (MHD) simulations, the gas flow structure in a disk has two layers, which is similar to the layered accretion model (e.g., Gammie 1996 ). There can be a rapid gas flow driven by a disk wind near the disk surface due to the effect of magnetic braking (e.g., Bai Stone 2013 ; Suzuki et al. 2016 ), while the accretion flow would be weaker at the midplane of the disk. In fact, global ideal MHD simulations observed a supersonic accretion flow in the upper layer of a disk (Zhu Stone 2018 ). A rapid gas flow near the disk surface may put a brake on runaway gas accretion onto a planetary core. In this paper, we have investigated how a time-dependent layered accretion in a disk affects formation of super-Earths having H 2 /He atmospheres. We have performed N -body simulations for formation of close-in super-Earths, coupled with disk evolution based on MHD simulations and gas accretion onto a planetary core. Note that hydrogen can be generated from secondary processes such as degassing via oxidation of metallic iron by water (Abe et al. 2000 ) and H 2 O photolysis and escape from a planet via stellar X-ray and extreme ultraviolet (XUV) irradiations (e.g., Lammer et al. 2013 ) and interactions between the stellar plasma and a hydrogen atmosphere (e.g., Kislyakova et al. 2013 ). However, a long-term evolution of a super-Earth’s atmosphere after the formation stage (≳10 Myr) is beyond the scope of this paper. The paper is structured as follows: We present detailed descriptions of models in Section 2 . Results of gas accretion rates onto planetary cores and their orbital evolution are shown in Section 3 . The radial dependence of the occurrence rate of gas giants and the validity of our model are discussed in Section 4 . We summarize this study in the last section. 2. Model 2.1. Disk Evolution For the evolution of a protoplanetary disk including effects of magnetically driven disk winds, we adopt the model of Suzuki et al. ( 2016 ), in which the diffusion equation is given bywhere Σ g , Ω, c s , and H are the gas surface density, the Keplerian angular velocity, the sound speed, and the disk scale height, respectively. The first two terms on the right-hand side induce radial mass accretion


atmospheres with the classical 1D approach we can calculate abundance corrections and abundance errors. Given a measured LTE abundance, we define the abundance correction as the abundance difference compared to LTE, such that the non-LTE equivalent width matches the LTE equivalent width. 3 Results 3.1 Departure coefficients In order to understand the behaviour of the abundance corrections and , we can look at the variation of the departure coefficients, b ( τ R ) = n NLTE ∕ n LTE , for the 1D hydrostatic models and 2D models. For the first, b 1D values were calculated for the lower 3s 5 S and upper 3p 5 P levels of the O I 777 nm triplet in the 4D parameter space τ R , T eff , log g eff , ξ t . For each combination a cubic interpolation that splits the multi-dimensional case into a sequence of 1D interpolations was performed on the grid of 1D models to obtain departures from LTE as functions of the Rosseland optical depth. In the 2D models – due to the horizontal inhomogeneities produced by convection – departures from LTE b ( τ R ) are different for in- and outflows (e.g. Amarsi et al. 2016 ). In order to compare departures from LTE for 2D models with the 1D case, b 2D values for each individual 2D snapshot were horizontally averaged over surfaces of constant optical depth. Both 2D models predict a similar dependence of the abundance corrections on effective temperature. Since they are similar we focused on the nine-day model, and calculated median 2D departures from LTE. We plot the departure coefficients in Fig. 3 . Namely, the median 2D departures from LTE as a function of τ R and photometricphase for the lower and upper levels are shown by solid blue and red lines, respectively. One-dimensional interpolated b 1D departures from LTE are shown by dashed lines using the same colours for the levels. Qualitatively, the departures from LTE of 1D and 2D models show a similar dependence on the photometric phase. The lower and upper levels of the O I 777 nm triplet, namely 3s 5 S and 3p 5 P, are almost always overpopulated during the pulsating cycle. At photometric phases φ ph ≈ 0.3...0.8 the lower and upper levels are less underpopulated compared to the rest of the phases, due to the minimum on the light curve and effective temperature. Quantitatively, for this range of phases 1D and 2D models predict very similar optical depth dependences of the departures from LTE. As a result, the abundance correction is small. Moreover, the 2D non-LTE versus 1D LTE abundance correction turns out to be smallest, as shown in Sect. 3.5 . 3.2 O I 777 nm triplet profiles We illustrate the temporal evolution (over a pulsation cycle) of the normalised emergent LTE and non-LTE O I 777 nm triplet flux profiles in Fig. 4 for M3 and M9. The line profiles are shown without adding instrumental broadening or noise. This allows us to identify the occasional multi-component structure of the line profiles, for example at photometric phases φ ph = 0.299 and 0.318 for M3 and


8 of Savino et al. 2019 ) is about 0.5–0.6 × 10 −4 M ⊙ yr −1 , in good agreement with the SFR of about 0.4 × 10 −4 M ⊙ yr −1 measured in our work. The both measurements show the comparable total stellar mass 3.13 × 10 6 M ⊙ (4.2 × 10 6 M ⊙ ). The bottom panel of Fig. 4 shows the stellar mass formed in the last 2 Gyr versus the galaxy absolute magnitudes. The percentage of the mass formed is small for all the objects. This plot could demonstrate how intensive is the residual SF in dSph galaxies depending on the degree of isolation of the object. Highly isolated galaxies clearly show recent residual SF (>10 per cent of the total stellar mass), while within the virial radius it is practically zero. And XVIII and Tucana are situated beyond the virial radius (but within the LG zero-velocity sphere). Following the logic of the above plot, they should be located between those two extreme cases. None the less, Tucana is again out of the trend, located among the galaxies inside the virial radius. It also could be an argument in favour that in the past this dwarf was among the closer satellites of M 31. Simpson et al. ( 2018 ) predicted quenching of SF in satellite galaxies of the LG from a suite of 30 cosmological zoom simulations of MW-like host galaxies. The authors found that systems within host galaxy R 200 are comprised of two populations: first infall and ‘backsplash’ galaxies that have had a much more extended interaction with the host. Backsplash galaxies that do not returned to the host by z = 0, exhibit quenching properties similar to galaxies within R 200 and are distinct from other external systems. The last scenario are well match with the SF properties of Tucana dSph. It should be noted that the results are affected by the well-known problem of age–metallicity degeneration in the process of SFH measurement, and some dependence of the results on the isochrone set used (Williams et al. 2017 ). Besides, the obtained CMDs are somewhat not deep enough, in the sense that the photometric limit is brighter than MSTO. At the same time, uniform data set and its homogeneous processing should reduce the influence of these factors on the results of our comparative analysis to a minimum. When analysing SFHs of dwarf satellite galaxies in the nearby galaxy groups (first of all, the LG), one should take into account, that even in a particular mass range and fixed environment, there is a large degree of scatter in the SFHs, frequently noted in various researches (see e.g. Garrison-Kimmel et al. 2019 ). The mentioned study has used a set of about 500 dwarf galaxies taken from FIRE-2 (The Feedback In Realistic Environments 2) simulations to explore how their SFHs vary with environment. They conclude, that the satellites ( d ≤ 300 kpc) tend to form their stars earlier than equivalent 300 d 1000 kpc dwarfs, consistent with the picture, where interactions with MW-mass hosts inhibit SF. A galaxy could loss its gas due to ram pressure effects and tidal


of SNe Iax, which are in the range 12+log(O/H)=∼8.16–8.87 (Magee et al. 2017 ; Lyman et al. 2018 ), somehow lower than the solar abundance (Caffau et al. 2008 ). This indicates that the site of SNe Iax may have relatively lower metallicity compared with other subtypes of SNe Ia, but comparable to that of SNe II or SNe Ibc. Note that Lyman et al. ( 2018 ) recently carried out an analysis of metallicity for the host galaxies and explosion sites of a large sample of SNe Iax through both integral-field and long-slit spectroscopy and they also found that the SN Iax explosion site metallicity distribution is similar to that of core-collapse SNe. We also measure the fractional flux of the SN site, which is the fraction for which the total host light in a pixel is fainter than or equal to the light in the pixel at the location of the SN. This parameter can be used to trace the difference/similarity between luminosity at SN locations and the light of the hosts (Fruchter et al. 2006 ). As the fractional flux is independent of galaxy morphology and size, it is thus a better parameter to quantify the correlation between SN progenitor environments and the light of their hosts (i.e. Wang et al. 2013 ). Such an analysis can be performed for 21 SNe Iax of our sample by using the host-galaxy images from SDSS DR12 (Alam et al. 2015 ) and Panstarrs (Chambers et al. 2017 ); the results are also tabulated in columns (9)–(11) of Table 5 . The distribution of the fractional fluxes of our SN Iax sample in the ugr bands is shown in Fig. 14 . The fractional flux distributions of SNe Ia, SNe Ibc and SNe II are overplotted for comparison and the measurements of these samples are taken from Wang et al. ( 2013 ). One can see that the progenitor populations of SNe Iax belong to the brighter regions of the host galaxies, similar to the case for SNe Ibc (especially in the g and r bands), suggesting that SNe Iax may arise in larger star-forming regions that produce more massive stars. This is also evidenced by the Kolmogorov–Smirnov (KS) test results, as tabulated in Table 6 . One can see that SNe Iax and SNe Ia trace the host-galaxy light differently, with a very low probability (i.e. P = 0.04 in r ) that they come from the same stellar populations. SNe Iax also show some differences from SNe II in the light distribution of progenitor populations, with KS test results of P = 0.13 in g and P = 0.17 in r , respectively. In the u band, however, we notice that SNe Iax seem to show large differences from all kinds of SNe in the fractional flux distribution. This is likely related to the relatively larger uncertainty in the SDSS u -band photometry. As for the comparison between SNe Iax and SNe Ibc, P values of 0.33 obtained in the g band and 0.37 in the r band indicate that the hypothesis that they share a similar fractional flux (and hence environment) distribution cannot be rejected. Figure 14. Histogram distribution of the fractional flux of the host-galaxy light at the location


× 5 pixel 2 raster pixd (after neighbour subtraction) centred on the target stars, and PSF( i, j ) is the value of the local PSF-model expected in the pixel ( i, j ). Introduced by Bedin et al. ( 2008 , see also Bedin et al. 2009 , 2010 ; Simioni et al. 2018 ), the parameter RADXS is a shape parameter that allows us to distinguish sources that deviate from a PSF shape. It is a comparison between the measured flux of the source outside the core (in an annulus 1.0 r 2.5 pixels) and the flux expected from the PSF-model. For RADXS >0, the source is broader than that expected from the model (i.e. galaxies), while for negative values of RADXS the source is sharper than the PSF (i.e. cosmic rays and artefacts). Finally, the KS2 routine gives the number of images in which a star is found ( N ) and the number of consistent measurements of the star used to compute its average position and flux ( N ). 3 COLOUR–MAGNITUDE DIAGRAMS In Fig. 2 , we show an example of selection of well-measured stars for the case of NGC 6752 and the photometric method 1. Similar plots can be made for the methods 2 and 3. The top and middle panels of Fig. 2 show the selection of the stars based on the distribution of the photometric errors ( RMS , left-hand panels), the quality of fit ( QFIT , centre panels), and the shape of the sources ( RADXS , right-hand panels), in the case of F275W (top panels) and F336W (middle panels) filters. The selection based on RMS and QFIT are performed as done by Milone et al. ( 2012 ): we divided the distributions in 12 magnitude bins and, in each bin, we calculated the 3.5σ-clipped average of the magnitude and of the parameter, where σ is the standard deviation associated with the average value in the given bin. We added to the mean parameter of each bin 3.5 × σ, and we linearly interpolated these points (green line). We excluded all the points above (in the case of the RMS ) or below (in the case of the QFIT ) the green line (azure points). For the sharp parameter, we selected all the stars that satisfy the condition: −0.15 RADXS +0.15. Bottom panels show the m F275W versus ( m F275W − m F336W ) colour-magnitude diagram (CMD) for the stars that pass the selection criteria in both filters (left-hand panel) and for the stars that were rejected in at least one filter (right-hand panel). In red, the stars that are saturated in at least one filter and that have been recovered from the ‘first-pass’ photometry. From the CMDs, it is clear that many stars with poor photometric quality are rejected. Figure 2. Procedure adopted for selecting well-measured stars in the cluster NGC 6752. Top and middle panels show the selection of the stars based on RMS (left-hand panels), QFIT (central panels), and RADXS (right-hand panels) parameters: in azure the stars that are rejected. Bottom panels show the m F275W versus ( m F275W − m F336W ) CMD cleaned by rejected stars in both filters (left-hand panel) and the CMD of the stars rejected in at least one filter. Red


solves the MHD process of an isolated, twisted magnetic flux tube, initially placed in the convection zone, buoyantly rising into the atmosphere. The full set of basic resistive, adiabatic MHD equations in conservative form are solved by a finite difference scheme with the spatial derivatives by the fourth-order central differences and the temporal derivatives by the four-step Runge–Kutta scheme. Periodic boundary conditions are applied for the x -direction and symmetric boundaries for both the y - and z -directions. The initial background atmosphere of the simulation is gravitationally stratified, which consists of an adiabatically temperature-gradient convection zone (, where km is the length unit of the FES model), the cool isothermal photosphere/chromosphere (), and the hot isothermal corona (). The flux tube is initially placed at with the form of and , where r is the radial distance from the tube’s axis, the radius, kG the axial field strength, and the twist intensity (the negative sign indicates a left-handed twist). The middle of the tube is made buoyant by a small density deficiency and the tube started to emerge due to its own buoyancy. As the time-dependent bottom boundary data for the data-driven model, we extracted the time sequence of 2D data at fixed heights from the FES model. To test the behavior of the data-driven model with different inputs, data sets at three different slices from the FES model are used, including (referred to Z00 , representing the photosphere), ( Z10 , i.e., the chromospheric layer), and ( Z20 , i.e., the coronal base). The slices were sampled at every (where s is the time unit of the FES model) from 0 to 500. The slices spanned over , resolved by the uniform grid spacing of 200 × 200, thus the resolution is . The data sets include three components of magnetic and velocity field, i.e., , and v z . As we have mentioned in Section 1 , the Lorentz force in the boundary data is an important factor affecting the output of the data-driven model. Thus we quantitatively assess the Lorentz force as well as torque in the data of different layers. The sum of Lorentz force (and torque ) in a volume can be expressed by surface integral (Aly 1989 ; Sakurai 1989 ), and by neglecting contribution from the side and top boundaries, they can be written as integrals of the 2D data sets:andFollowing Duan et al. ( 2020 ), to assess the data with respect to the force-free condition, the forces can be normalized by the integrated magnetic pressure force F p that is given byand the normalized forces are ratios defined as , , and , respectively. Similarly, the normalized torques are defined as , , and , where T p is magnitude of the net torque induced by only the magnetic pressure force, whereFor a field being close to force-free, it must have ratios much less than unity, i.e., and . Metcalf et al. ( 1995 ) suggested that the magnetic field can be considered as force-free if the normalized forces are all less than or equal to


consistent with previous studies that show an anticorrelated relationship between the magnetic field strength and the ionopause altitude (Elphic et al. 1980 ; Russell Vaisberg 1983 ; Phillips et al. 1985 ; Angsmann et al. 2011 ). The ionospheric thermal pressure balances the magnetic pressure difference at the ionopause. To sustain stronger magnetic pressure, it needs higher thermal pressure, and thus higher plasma density in the ionosphere. Therefore, the ionopause would be pushed down to a lower altitude with higher plasma density. However, it is important to point out that this process is based on an assumption that the ionization of the ionosphere is not significantly changed. Figure 4. Left to right: the magnetic field strength near the periapsis prior to, during, and after the passage of the ICME on November 4, November 5, and November 6, respectively. The IMA measurements of heavy ion density are overplotted (red dots). The heavy ion density on November 6 is not available. Nevertheless, the altitude of the ionopause during the strong ICME (Figure 4 (b)) did not decrease at all considering that the magnetic field strength increased a lot. The magnetic pressure differences () across the ionopause on November 4 and 6 are 0.7, and 1.1 × 10 −8 dyn cm −2 , respectively. But on November 5, the magnetic pressure difference reaches 2 × 10 −7 dyn cm −2 ( from 250 to 116 nT) or 9.0 × 10 −8 dyn cm −2 even if we use 200 nT as the magnetic strength in the ionosphere. To balance such atypically higher magnetic pressure, the plasma density of the ionosphere must be correspondingly much greater. The heavy ion densities measured by the IMA instrument are also shown in Figure 4 . Although the resolution of IMA data is low, we can still easily see that the ionospheric plasma density was over 1000 cm −3 during the strong ICME with respect to that less than 400 cm −3 during quiet solar wind. The much higher ionospheric plasma shows that the ionosphere had been greatly excited and perhaps entirely expanded under the impact of the strong ICME. The strengthened ionosphere during the passage of ICME has been evidenced by the fact that the magnetized ionosphere can be demagnetized by some ICMEs during solar minimum (Zhang et al. 2008a ). Due to the limitation of measurements, we have no ideas about the reason why the ionosphere was greatly enhanced by the ICME. However, previous study of the Venus and Mars ionosphere has determined that the solar energetic particles (SEPs) could cause extra ionization of the neutral particles in exosphere through impact ionization and charge-exchange reactions (Futaana et al. 2008 ). Simulations also show that the SEPs can effectively increase the ionization rate of the Venusian atmosphere (Plainaki et al. 2016 ). 4. Discussions and Conclusions We have presented very detailed observations of the induced magnetosphere and ionosphere during the passage of the extremely strong ICME on 2011 November 5. From our observations, a clear


must come from combinations of the harmonics contained in . Recall that we defined containing both synodic terms with harmonics λ 1 − λ 2 and λ 2 − λ 3 , which we wrote in equation ( 16 ) in terms of the new variables . Therefore, the harmonics that are included in are combinations of these synodic harmonics; more specifically, they come from the products of their cosines. 6 Using the standard trigonometric identity , the resulting harmonics are so the harmonic with the lowest fraction of is . Specializing now to the case of a chain with the same resonance index k (1) = k (2) = k , this simply gives With the aid of an algebraic manipulator, one can compute the full expression of and select the desired harmonic term (we used the software package wolfram mathematica ), thus obtaining its coefficient (actually, one can see that this term emerges solely from the term ). We avoid writing here the full expression, which is rather cumbersome; moreover, as in ( 35 ), we evaluate it at the reference values of the actions so the term multiplying the cosine becomes a numerical coefficient, and we write this term as Since we want to compare the frequency of to that of , we need to consider the resonant Hamiltonian in the variables, expand the ‘barred’ variables around the equilibrium point characterized by the equilibrium actions and the equilibrium angles (equation 20 ) as in Section 4.3 , and then introduce the transformation to the action-angle variables , which transforms into the sum of decoupled harmonic oscillators plus higher order terms (equation 22 ). It is also useful to translate the value of around its initial reference value introducing . Therefore, we write in terms of the variables . The Hamiltonian will now contain harmonic terms of type where is an integer combination with coefficients of the angles φ 1 , ..., φ 4 , which we can calculate explicitly. Therefore, whenever , a secondary resonance is crossed. We can rewrite this expression as . Since the Hamiltonian has d’Alembert characteristics in each pair ( I l , φ l ), and the values of the actions are initially (that is, before their excitation) small, the strongest secondary resonances will come from lowest integer combinations , that is, where most h l are zero. We also note that since and the frequencies ω l are all negative, a secondary resonance term can only appear when , which, together with the requirement that be small, is tantamount to requiring that all non-zero integers h l are positive. Since we calculated ω ( m pl ) in Section 4.3 , we can calculate for each the relative frequency as a function of m pl , and check if any of these vanish for some value of m pl , which corresponds to crossing a secondary resonance. We carried out the calculation with the aid of the mathematica software in the reference case k = 3 and a 1 ≃ 0.1, which corresponds to the evolution shown in Fig. 1(b) (and also Figs 2 and 3 ). We found that contains, among many others, the following


which could make the analyses more complicated. We analyzed the PILS data for lines of CH3SD similarly to what Drozdovskaya et al. (2018) did for CH3SH. We adopted an excitation temperature T ex = 125 K, a source size of 0.5′′, and a full line width at half maximum (FWHM) of 1 km s −1 . We did not detect CH3SD, but we derived an upper limit on its column density of 8.8 × 10 14 cm −2 (±0.88). The 12 most constraining lines are shown in Fig. 3 . Drozdovskaya et al. (2018) determined a column density of 4.8 × 10 15 cm −2 (±0.48) for CH3SH 2 . Our upper limit corresponds thus to a D/H ratio of less than 0.18 ± 0.025; the uncertainty was estimated as previously ( Drozdovskaya et al. 2018 ). This value is probably not particularly constraining compared with the deuteration of H2CS per H atom of ∼0.05 ± 0.007 ( Drozdovskaya et al. 2018 ). Deeper interferometric observations are required in order to conclusively verify whether the level of deuteration of H2CS is lower, higher, or preserved at the next level of chemical complexity into CH3SH. Fig. 3. Sections of the Protostellar Interferometric Line Survey (PILS) toward a position slightly off-set from source B displaying the 12 most constraining lines of CH3SD. 5. Conclusion and outlook In the present work, the rotational spectrum of CH3SD has been investigated in the frequency range 150−510 GHz in order to provide accurate predictions for astronomical searches. Extensive assignments have been made for the ground, first, and second excited torsional states up to high J and K a quantum numbers (see Table 1 ). The detailed modeling of the absorption spectra of the CH3SD has been performed in a global fit of a dataset of 4905 rotational transitions to the RAM Hamiltonian containing 78 parameters. The overall weighted standard deviation of the global fit is 0.9 in the range 150–510 GHz, indicating that our set of parameters reproduces the assigned lines within experimental uncertainties. Transition frequencies calculated from these parameters should be reliable for astronomical observations. A first attempt to search for CH3SD in the PILS data turned out to be negative. The upper limit, though not unreasonable, is not very constraining either. Future, even more sensitive observations may provide insight into the deuteration of CH3SH. A search for the potentially more abundant CH2DSH was hampered by insufficient laboratory data reported for this isotopolog. Calculations of the rotational spectrum of CH3SD will be available in the catalog section of the Cologne Database for Molecular Spectroscopy, CDMS, ( Endres et al. 2016 ). The input and output files of the fit as well as auxiliary files are available in the data section of the CDMS 3 . Appendix A: Additional tables Table A.1. Spectroscopic parameters of CH3SD (cm −1 ). Notes. (a) n = t + r , where n is the total order of the operator, t is the order of the torsional part and r is the order of the rotational part, respectively. The ordering scheme of


not trivial, the main obstacle being the lack of robust estimates for key parameters (e.g., distance, Doppler factor), as well as the lack of contemporaneous multi-wavelength radio observations. In most cases, the range of available frequencies was confined in the lower frequency range (43 GHz). The observed flux density at higher frequencies was the result of extrapolation by assuming that the spectral index derived from the highest available frequencies of each microquasar remains constant up to 142.33 GHz (which is the highest available frequency for the supermassive BH sample). To estimate the relativistic boosting and revert to the rest frame, we gathered for each source all available estimates in the literature for the velocity of the jet, distance, and viewing angle (Table 1 ). For each source, we created a parameter space set by the extrema of the aforementioned quantities. We then uniformly drew a random value for each of these parameters and calculated . We repeated this process 10 4 times and calculated the mean and spread (minimum, maximum) of for each source. The y -values for the microquasars in Figure 3 are the mean, and the y -axis errorbars the spread for each of the sources that resulted from the random sampling. We follow a similar procedure for the , by gathering all the available estimates in the literature and calculating the mean and spread. The x -values for the microquasars in Figure 3 are the mean, and the x -axis errorbars are the spread for each of the sources. The microquasars, parameter ranges, and the references to them are the following. For SS 433. we used radio data from Trushkin et al. ( 2003 ), and parameter values from Seward et al. ( 1980 ), Fender ( 2001 ), Trushkin et al. ( 2003 ), Bowler ( 2010 ), Panferov ( 2010 ), Cherepashchuk et al. ( 2013 ), and references therein. For GRS 1915+105, we used radio data from Rodriguez et al. ( 1995 ), and parameter values from Mirabel Rodríguez ( 1994 ), Fender ( 2001 , 2003 ), Fender Belloni ( 2004 ), Miller-Jones et al. ( 2005 ), Punsly ( 2011 ), Reid et al. ( 2014 ), Zdziarski ( 2014 ), Punsly Rodriguez ( 2016 ), and references therein. For Cyg X-3, we used radio data from Koljonen et al. ( 2010 ), and parameter values from Fender et al. ( 1997 ), Ling et al. ( 2009 ), Hjalmarsdotter et al. ( 2009 ), Vilhu et al. ( 2009 ), Dubus et al. ( 2010 ), Vilhu Hannikainen ( 2013 ), Zdziarski et al. ( 2012 , 2013 , 2016 ), and references therein. For GRO J1655-40, we used radio data from Hannikainen et al. ( 2000 ), and parameter values from Hannikainen et al. ( 2000 ), Trushkin ( 2000 ), Mirabel et al. ( 2002 ), Fender ( 2003 ), Fender et al. ( 2004 , 2010 ), Narayan McClintock ( 2005 ), Motta et al. ( 2014 ), Stuchlík Kološ ( 2016 ), and references therein. For V* V404 Cyg, we used radio data from Gallo et al. ( 2005 ), and parameter values from Cherepashchuk et al. 2004 , Heinz Merloni ( 2004 ), Miller-Jones et al. ( 2009 ), Khargharia et al. ( 2010 ), Xie et al. ( 2014


for small values of the BH spin and θ v , those effects should have a minor weight on the results. For the fitting procedure, we used the sources NGC 5548 and MRK 509 (see Appendix C for details) also studied by KD 18. We found that, as for KERRBB, the modeling with AGNSED of the optical-UV-X SED is degenerate: by changing the model parameters (i.e., BH mass, spin, Eddington ratio, corona size) appropriately, it is possible to reproduce the same set of data (for some reference values, see Fig. C.1 ). For NGC 5548, results from both models are compatible while for MRK 509, KERRBB BH masses are larger than the AGNSED ones by a factor ≲0.2 dex (Fig. C.2 , right panel). We argue that for NGC 5548, the absence of relativistic effects in AGNSED is balanced out by modeling a large corona above the disk, leading to the same BH mass. Instead, for MRK 509, the smaller X-ray corona leads to different results because it does not compensate the differences between the two models. In general, KERRBB observed disk luminosity is dimmer than the intrinsic one because of a compact corona located in the inner region of the disk, the presence of which modifies the AD emission and leads to an overestimated BH mass. By correcting the KERRBB disk lumminosity by a factor ≲0.2 dex, results from the two models related to MRK 509 show better compatibility (see Appendix C for more details about this correction). The partial compatibility between KERRBB and AGNSED shows that BH masses estimated though the SED fitting procedure clearly depend on the adopted model and their physical background. Despite these findings, KERRBB results for both sources are still compatible with SE and RM ones within uncertainties (Fig. 4 ). Moreover, results presented by KD 18 are compatible with ours for what concerns the BH spin: taking their finding for the BH masses as reference values, both their results and ours favor low values of a . 6. Discussion and conclusions We used the relativistic thin AD model KERRBB ( Li et al. 2005 ) to infer the BH masses of 28 sources from the AGN Black Hole Mass Database ( Bentz Katz 2015 ). These sources have a BH mass estimate from the RM studies and show clear evidence in the UV band of the so-called Big Blue Bump produced by the radiation coming from an AD around a SMBH. Since we did not have information about the viewing angle for the majority of them, we assumed that the Type 1 QSOs were observed with θ v ≤ 45° in order to avoid the absorption from the dusty torus (assumed to have an average opening angle of ∼45°). Our results can be summarized as follows: The majority of the sources of our sample show a good match between data and the modeling with KERRBB. The modeling led to a relatively good estimate of the spectrum peak position (i.e., peak frequency and luminosity; see Appendix B ) with a small uncertainty (∼0.05 dex on average). The total uncertainty related to the AD BH mass estimates is ∼0.45 dex, which is connected to the unknown BH spin


9799, the total population of GCs out to (the limit of the ACS field of view, as well as the limit of the halo-light photometry) is brighter than I = 27.0 from integration of the radial profile. This limit is 0.12 mag fainter than the GCLF peak (Paper I) and thus should include 54% of the total over all luminosities assuming the GCLF is Gaussian in number per unit magnitude, and using the parameters from Paper I. We therefore obtain for (110 kpc). Integrating the light profile (Seigar et al. 2007 ) to the same radius gives a total integrated magnitude , which corresponds to assuming a mean color for giant early-type galaxies. The specific frequency of the GC system is then . If we were to choose to integrate the entire GC system profile all the way to the WFC3 field at R = 300 kpc, the result would be clusters, but this estimate is much more uncertain. For UGC 10143, if again we restrict the calculation conservatively to the ACS field of view (), the total number of GCs with is , which translates into over all magnitudes. Integration of the halo light (Donzelli et al. 2011 ) into the same radius gives , . Finally then, to that radius. The more risky extrapolation of the GC profile out to the WFC3 field limit at (270 kpc) would give . The UGC 9799 GC system is therefore relatively richer than for UGC 10143, but the specific frequencies of both are in the range observed for other BCGs (e.g., Harris et al. 2013 ). For NGC 4889, the profile integration gives clusters brighter than (); this limiting magnitude is very near the GCLF peak point. We therefore double that number to , which when combined with gives a specific frequency of . This galaxy is very luminous, but relative to its size, it does not have an exceptionally populous cluster system in the more typical BCG range. The story is somewhat different and more complex for the central Coma cD NGC 4874: Peng et al. ( 2011 ) estimate GCs excluding the more extended IGC population. When we use , as did Harris et al. ( 2009 ), we obtain . If instead N tot is normalized to the entire luminosity profile out to a much larger radius of 520 kpc, then S N decreases markedly to 3.7 ± 0.1 (Peng et al. 2011 ); but since the IGC clusters outnumber the “intrinsic” GCs that are more clearly associated with the galaxy itself, this lower specific frequency is perhaps more of a statement about S N (IGC). In this respect, it is worth noting that Durrell et al. ( 2014 ) find for the Virgo cluster GCs in their entirety, of which a large fraction are IGCs, and less than a quarter are from M87 itself. 8. DISCUSSION For extremely broad and nearly featureless MDFs such as we find in NGC 6166 (Paper II), UGC 9799, and UGC 10143, the imposition of a bimodal Gaussian numerical model, or any simplistic multimodal fitting process, begins to look increasingly arbitrary. Transformation of their CDFs back into the underlying metallicity distribution by way of Equation ( 1 ) yields a similarly broad unimodal MDF. We stress


C.1 shows the losses at these three wavelengths, and for the weak sources with fluxes 400 mJy these losses clearly become more significant at the short wavelength range (closer to m), while the losses are moderate above the m. We used this information to correct the average correlated fluxes of the sources for this work. For the source with high correlation losses we took the average correlated flux obtained after the data reduction and multiplied it by the decorrelation correction factor obtained from our dilution experiment. Appendix D: Dust sublimation Fig. D.1 Temperatures for the three smallest grain sizes of each species in a shell with a radial optical thickness of ( top ) and ( bottom ). The colors indicate the different species of grains: blue – graphites , green – graphites , and red – silicates. Fig. D.2 Convergence of the temperature for the three smallest grains. Each line indicate the relative difference after each iteration. The darker the color, the higher the number of iterations. The radiative transfer code RADMC-3D does not include an internal computation to account for dust sublimation. To include dust sublimation in our models, we slightly modified the code. Each time a photon package enters a cell, it increases the energy of the cell and thus increases the temperature of the dust of this cell. When the dust in the cells exceeds the dust sublimation temperature, the dust inside the cell is completely removed. To accurately estimate the temperature of the system using dust sublimation, we performed an iterative process for which the code several times computed the temperatures and removed the corresponding cells. When the differences between different iterations were below a certain tolerance (we took a value of 5%), we performed a last computation without removing any dust in the cells to ensure the conservation of energy in the system. To show how our implementation of the dust sublimation works in different situations we show examples of an optically thick and optically thin case. In Fig. D.1 we show the temperature of the three smallest grains of each species for a spherical shell with and , respectively. Both shells have an initial inner radius of 0.3 pc and an outer radius of 5 pc. We can observe clearly that in the optically thick case, the temperature rises more quickly and the sublimation radius for the three species are closer than in the optically thin case. We made sure that we had enough cells close to the inner radius to obtain an accurate solution. In Fig. D.2 , we show for the optically thick case, the relative differences between each iteration and our final computation. After a few iterations, the temperature in every cell reaches a relative difference below our tolerance value. Appendix E: Acceptance levels For our method we used linear transformations computed for each model to remove the mean values, normalize the variance, and remove correlations. We expect that if our models agree with the


μ) and Q 0 (0, μ) in the case of ( 1 7 ) are simple: In Table 1 we give the angular distribution J (μ) = I 0 (0, μ)/ I 0 (0, 0) and the degree of polarization p (μ) = − Q 0 (0, μ)/ I 0 (0, μ) per cent at and 1 for the incident non-polarized radiation perpendicular to the surface. Note that negative Q 0 (0, μ) corresponds to the wave electric-field oscillations perpendicular to the plane ( nN ). This holds in our case. Table 1 shows that the chaotic Faraday rotations strongly decrease the polarization and weakly affect the angular distribution. Table 1. The angular distribution J (μ) = I 0 (0, μ)/ I 0 (0, 0) and the degree of polarization p (μ) = − Q 0 (0, μ)/ I 0 (0, μ) per cent at and 1 for the incident non-polarized radiation normal to the surface. Recall that we consider the atmosphere without pure absorption (a conservative atmosphere). It follows from equation ( 13 ) that the zero moments of functions β(μ) and γ(μ) are equal to and , respectively. These moments are μ-integrals over the interval (0,1) (see Silant’ev et al. 2019 ). Taking the integral μ I 0 (0, μ) dμ over the same interval and using equation ( 13 ), we obtain that the flux of outgoing radiation is equal to the flux F I of the incident radiation. For a conservative atmosphere this is the evident result. In Fig. 1 the polarization degree p (μ) of the reflected radiation for different parameters h is given. The angular dependence J (μ) practically does not depend on h . Therefore, we give J (μ) for and 30. The numbers over the curves denote the values of h . Figure 1. The angular distribution J (μ) = I 0 (0, μ)/ I 0 (0, 0) and polarization degree p (μ) = − Q 0 (0, μ)/ I 0 (0, μ) per cent of reflected radiation. The incident radiation is perpendicular to the surface. The numbers denote the values of parameter h . It is of interest to compare the multiple scattered outgoing intensity I 0 and parameter Q 0 to the results for single scattered radiation: The flux of outgoing single scattered radiation is equal to F single = 0.168 F I , i.e. only ≃ 17 per cent of the total flux. The other 83 per cent of the emerging radiation is concerned with two and more scatterings in the semi-infinite atmosphere. In our case, the considered normal incident radiation penetrates deeper into the atmosphere than in other inclinations. It means that F single = 0.168 F I is the minimum value of the single scattered radiation fluxes. The corresponding angular distribution and the degree of polarization have the form The polarization p single (μ) at μ = 0 is equal to 100/(1 + h ) per cent. This result clearly demonstrates the decrease of polarization with increasing parameter h . This decrease holds along the outflow of single scattered non-polarized radiation. Recall that single scattering of non-polarized radiation gives rise to 100 per cent polarization of the scattered one if the scattering angle (the angle between the incident radiation and the scattered one) is equal to 90° (see, for example,


mass with a ratio coefficient f : and the accretion rate will be where f may be of the order of 0.5, as argued by van den Heuvel Bitzaraki ( 1995 ). Because the mass statistics of NSs and MSPs illustrate that the accreted mass at least is about for MSP formation (Zhang et al. 2011 ; Kiziltan et al. 2013 ; Ozel Freire 2016 ; Antoniadis et al. 2016 ), we prefer f to be about 0.1−0.2. A NS/LMXB with m = 1.4 and R 6 = 1 for possesses an accretion rate higher than . However, it cannot accrete at a rate over the Eddington limit, namely . With the limit of the accretion rate , it is possible to derive an upper limit companion mass in the main sequence according to equation ( 11 ): With 0.1 f 0.2, will be about to . Hence a possible mass distribution of the companion in the main sequence in a LMXB is , corresponding to an accretion rate of . In such a LXMB, the NS will evolve to be a MSP with a magnetic field of . For , however, the accretion time might be too short for the binary system to produce a MSP. Table 2. The 16 radio MSPs with a low magnetic fields of (data from the ATNF pulsar catalogue). 2.4 On the minimum magnetic fields of AMXPs AMXPs exhibit an X-ray outburst during the accreting phase, with a luminosity a few magnitudes higher than that in the quiescent phase (Wijnands van der Klis 1998 ; van der Klis 2000 , 2006 ; Mukherjee et al. 2015 ). The variation of the luminosity leads to a change in the accretion rate, and thus the magnetic field of AMXPs can be derived by equating the magnetosphere radius with the co-rotation radius R co of the NS (Burderi et al. 1996 ; Zhang Kojima 2006 ): where P −3 is the spin period P of AMXPs in units of milliseconds. When R co equals the NS radius, the AMXP will have the minimum magnetic field. For 19 AMXPs with spin periods in the range ~1.7–6.1 ms (Patruno, Haskell Andersson 2017b ), their R co are in the range of ~24–56 km, which is larger than the radii of NSs of ~10–15 km. Thus, it can be concluded that the magnetic fields of 19 AMXPs have not achieved their lowest values. Another method for judging the minimum magnetic field of AMXPs is comparing their spin frequencies to the kilohertz quasi-periodic oscillation (kHz QPO) frequencies. The upper-limit kHz QPO frequency of a AMXP is believed to be the Keplerian frequency with orbital accreting matter (van der Klis 2000 ; Zhang 2004 ): where G is the gravitational constant. For a NS with m = 1.4 and R 6 = 1.5, the orbital frequency ν k is . Some AMXPs have an upper kHz QPO frequency over (van der Klis 2000 , 2006 ; Wang et al. 2017 ), which is comparable to ν k . However, the spin frequency of 19 AMXPs ranges from 164 to (Patruno et al. 2017a ), which is much lower than the upper kHz QPO frequency or ν k , and hence we conclude that the minimum magnetic fields of 19 AMXPs have not yet been achieved. If the co-rotation radius approaches the stellar radius, AMXPs will obtain the minimum magnetic field through accretion. 2.5 The minimum magnetic


which describes the adopted stellar mass ( M ), polar radius ( R p ), polar effective temperature ( T eff,p ), equatorial rotation velocity ( v e ) and inclination angle ( i ). For example, ‘m40r40t150v300i30’ is the case of M = 4.0 M ⊙ , R p = 4.0 R ⊙ , T eff,p = 15 000 K, v e = 300 km s −1 and i = 30°. (2) Polar radius (in R ⊙ ). (3) Equatorial radius (in R ⊙ ). (4) Polar effective temperature (in K). (5) Equatorial effective temperature (in K). (6) Mean effective temperature in K (which was derived by averaging the local effective temperature over the observed stellar disc while weighting according to the local continuum intensity). (7) Product of v e and sin i (in km s −1 ). (8) First-zero frequency of He 4471 (in Å − 1 ). (9) Ratio of to , where is the first-zero frequency of classical rotational broadening function (see, e.g. Gray 2005 ), which is related to v e sin i as (λ: wavelength, c : velocity of light). (10) First-zero frequency of Mg 4481 (in Å − 1 ). (11) Ratio of to . The wavelengths of the reference pseudo-continuum, necessary for reducing the normalized profile, were defined in almost the same way as in Section 4.1: Having found the critical wavelength λ* corresponding to the maximum level between these two lines, we set [, ] = [4471 − (λ* − 4471), λ*] and [, ] = [λ*, 4481 + (4481 − λ*)]. The Fourier transforms were then calculated for the resulting R (λ) profiles of He 4471 and Mg 4481 lines. Fig. 6 illustrates the behaviours of | f (σ)| as function of σ for the same selected cases as in Fig. 5 . The first-zero frequencies (σ 1 ) measured from the computed transforms are presented in Table 2 . Figure 6. Fourier transform amplitudes [| f (σ)|] plotted against σ, for the theoretical profiles of He 4471 (left-hand panels) and Mg 4481 (right-hand panels) computed for six representative models (same as those shown in Fig. 5 ). The upper panels (a) and (b) show the results for T eff,p = 12 000 K, while the lower panels (c) and (d) are for T eff,p = 15 000 K. 5 DISCUSSION 5.1 Characteristics of first-zero frequencies We first discuss the trend of theoretical σ 1 modelled in Section 4.2 . Since this quantity is expressed as σ 1 = 0.660/(λ v e sin i / c ) in the classical approximation, the projected rotational velocity (or the linewidth for the case of rotation-dominated broadening) plays the most significant role for determining σ 1 , though its precise value is affected also by the profile shape. The resulting values of σ 1 corresponding to the theoretical line profiles simulated on the gravity-darkened models are plotted against v e sin i in Fig. 7 , where the classical σ 1 versus v e sin i relation is depicted by a slanted solid line and the observed σ 1 values for six stars are indicated by horizontal dashed lines. Figure 7. The values of theoretical σ 1 , measured from the Fourier transforms of simulated spectra for gravity-darkened models, are plotted against v e sin i . The results for He 4471 (left-hand panels)


rate coefficients. Notes. TW18 and MP17 refer to models with nominal values of the oxygen-bearing influx (see Sect. 4 ). ( a ) CO 2 best-fit mixing ratio at 0.15 mbar from photochemical modelling in Orton et al. (2014b ) is 8.0 × 10 −11 . ( b ) Ice grain ablation influx giving rise to 1.2 × 10 5 and 4.5 × 10 3 cm −2 s −1 for H 2 O and CO 2 , respectively. The outcome of this sensitivity test indicates that a CO relative abundance of 7.0 × 10 −9 at 0.4 mbar allows an optimal match of the line core and wings. Table 4 lists different cases where ( D , t impact ) have been tuned to obtain q CO = (7.0 ± 0.1) × 10 −9 at the ~0.4 mbar region, the delivered CO mass (in g), and the CO mixing ratio obtained in 200–100 mbar. The resulting profiles in the 200–100 mbar region reflect that larger impactors give rise to larger CO abundances although below the upper limits in Teanby Irwin (2013) . Table 4 shows that the solution is degenerate; in other words, the same stratospheric CO mixing ratio abundance can be obtanied with pairs of ( D , t impact ) that represent larger impacts having taken place longer ago, or smaller impactors occurred more recently. For instance, the temporal evolution of the material delivered by a cometary impact indicates that the CO currently observed in the Uranus stratosphere could be due to a comet nucleus of 3.5 km diameter that hit the planet 822 yr ago. Figure 4 shows the comparison of the measured spectra and the modelled radiative transfer as a function of the different CO vertical distribution resulting from steady ice grain ablation and cases 3 and 4 for cometary impact. Any other of the cases in Table 4 will give rise to a CO mixing ratio of ~ 7.0 × 10 −9 at the 0.4 mbar pressure level that will give good match to the Herschel data. Stratospheric CO mixing ratio profiles with an abundance larger than the value in the pressure range probed by Herschel (i.e. between 3 and 0.4 mbar) can in principle be discarded as they will overestimate the observations. Similarly, the tropospheric CO mixing ratio profile has to be lower than the upper limit 2.1 × 10 −9 established by Teanby Irwin (2013) . Equations (14) and (15) in Zahnle et al. (2003) , adapted to Uranus where the impact frequency is ~ 4 times smaller than on Jupiter, allow us to plot the frequency of impact versus the impactor size. Figure 5 displays two different scenarios depending on the origin of the impactors: the size-number distribution of impactors is like that inferred at Jupiter (ecliptic comets), and those impactors more representative of nearly isotropic comets (as long-period or Halley-type comets). Zahnle et al. (2003) conclude that known craters on the Saturnian and Uranian satellites are consistent with either case; that is, both kinds of impactors are in the Uranian system and they could have entered the Uranus atmosphere. Also, Zahnle et al. (2003) assume that the impact rate on Uranus is ~ 0.00125 comets per year (i.e. one impact every ~800 yr)


This observational evidence makes HL Tau an example of a young (∼1 Myr; Briceño et al. 2002 ), embedded, disc-hosting star. HL Tau has therefore become a prime target for studies aiming to characterize young discs (Greaves et al. 2008 ; Carrasco-González et al. 2009 ). In particular, HL Tau was the first circumstellar disc observed with the long baselines of Atacama Large Millimeter/submillimeter Array (ALMA). This revealed an ordered series of concentric rings and gaps in the disc across multiple millimetre wavelengths (2.9, 1.3, and 0.9 mm; ALMA Partnership et al. 2015 ). Many theories have been put forward to explain the origin of these structures (e.g. Zhang, Blake Bergin 2015 ; Okuzumi et al. 2016 ), but one of the most persistent involves the growth of planets within the disc (Dipierro et al. 2015b ). While efforts to detect thermal emission from young planets in the gaps have been unsucessful, the corresponding upper limits on their masses still lie at the higher end of the giant-planet regime (10–15 M Jup ; Testi et al. 2015 ). The combination of the potential for massive planets at large radii, coupled with evidence of a high disc mass (∼0.1–0.13 M ⊙ ; Guilloteau et al. 2011 ; Kwon, Looney Mundy 2011 ), has seen gravitational instability (GI; Boss 1997 ) being invoked to explain both the observed dust structures in HL Tau (Akiyama et al. 2016 ; Takahashi Inutsuka 2016 ), along with the origin of any planetary companions (Nero Bjorkman 2009 ). However, recent measurements of the gas mass in the HL Tau disc using have shown appears to be too low for GI to occur ( M gas = 2–40 × 10 −3 M ⊙ ; Wu et al. 2018 ). In this Letter, we present a detection of the J = 3–2 transition towards the disc around HL Tau. We have demonstrated this line to be a robust tracer of disc gas mass (Booth et al. 2019 ). We use these data to derive a lower limit to the total gas disc mass and discuss how this impacts the gravitational stability of the HL Tau star–disc system. 2 OBSERVATIONS HL Tau was observed by ALMA in Band 7 on 2017 November 24 for 1.46 h with 49 antennae in configuration C43-8 under project code 2017.1.01178.S (PI: E. Humphreys). Baseline lengths ranged from 92 to 8547 m, and the spectral window (SPW) covering the J = 3–2 transition (321.852 GHz) had a native spectral resolution of 0.908 km s −1 (which we note does not resolve the J = 3–2 hyperfine structure). Data (self-)calibration and imaging were performed with casa version 5.1.1 (McMullin et al. 2007 ). Continuum subtraction was performed in the uv -plane using a constant baseline (fitorder 0). Line imaging was performed with the clean algorithm, using a Keplerian mask with the measured position and inclination angles of the disc (e.g. PA = 138° and i = 46°; ALMA Partnership et al. 2015 ). The native resolution of the data resulted in a natural beam size of 0.14 × 0.11 arcsec 2 (−13°). We used a 0.2-arcsec uv -taper, which was found to maximize the signal-to-noise ratio (S/N) of the line


above even a high percentile point necessarily belong to a real filament. The histograms in Fig. 4 g are plotted for the intersection of two sets of pixels, those that are above the 95% percentile of significance (given by the and values) and those that coincide with a real filament (see Fig. 4 d). The RHT parameters are this time optimised manually, only keeping = 10 fixed. This leads to values = 5 and = 0.65. By visual inspection of the results, TM is slightly more successful in highlighting the location of filaments (Fig. 4 c) and estimating their position angles (Fig. 4 g). Fig. 4 Comparison of RHT and TM results in the case of non-linear filamentary structures on a fluctuating background. For the description of individual frames see Fig. 2 . Figure 5 shows how the results change as a function of the parameter values. For TM, the values of and are varied in 30% steps. For RHT, the parameter is kept at a value of 0.65, which was found to give the best results in Fig. 4 , but and are also change in 30% steps around the best parameter combination. The results of TM are less sensitive to these changes in the input parameters. Fig. 5 Change in RHT and TM results as a function of the method parameters. RHT parameters and are varied and parameter is kept at a value of 0.65. 3.2. Reliability and completeness We address questions of TM reliability estimation in the case of the simulation shown in Fig. 4 . equals 2.8 pixels and any two estimates that are separated by approximately five pixels can be considered independent. We first analyse maps similar to those in Fig. 4 but without filaments and with different realisations of noise and of large-scale fluctuations. According to this Monte Carlo exercise, 84%, 97.7%, and 99.9% confidence limits correspond to values of 0.0457, 0.0638, and 0.0850, respectively. This can be compared to the values of the same percentiles that are obtained directly from the input map of Fig. 4 . Those values are 0.057, 0.23, and 0.43 and they would thus result in lower estimates of the statistical significance of a given value. For example, , the apparent value corresponding to the 84% confidence limit, is in reality closer to the 97.7% confidence limit. The values are higher because of the presence of real elongated structures. The more real structures the map has, the more conservative are the limits derived from the statistics of the image itself. For a 360 360 pixel image, one would expect 2981 spurious pixels above the given threshold of . We count as spurious all pixel whose distance to real filaments exceeds three pixels. The number of such pixels is 3174, by 6.5% higher than expected. However, there are still some pixels with that are associated to filaments but are at distances exceeding three pixels. This suggests that the threshold obtained from the Monte Carlo simulations gives a rather accurate estimate of the reliability. For the 97.7% confidence limit, we expect the map to contain 68 instances where two


we find that the median total number of stars that are enriched per event is ∼13, which is comparable to the maximum number of G dwarfs (stars with similar mass to the Sun, 0.8–1.2 M ⊙ ) that were enriched in the more populous clusters studied in Parker et al. ( 2014a ) as are the fraction of enriched G-dwarf stars that were unperturbed (∼0.5 per cent). In an attempt to increase the fraction of enriched stars, we explore a host of initial conditions for the star clusters. However, we find that no initial conditions result in a significantly higher (more than twice) fraction of enriched stars. The principal reason for this is the effects of two-body relaxation. In order to enrich more than half of the stars in a low-mass cluster, the required stellar density at the time of the SN(e) explosion(s) corresponds to a radius of only ∼0.4 pc, which is impossible to achieve after 5 Myr of evolution. Two-body relaxation results from a stellar system attempting to reach dynamical equilibrium and even a moderately dense system (100 stars pc −3 ) such as our default model expands significantly in the first 10 Myr, as shown in Fig. 7 . Here, the coloured lines represent the median local stellar density with time in each of the 20 realizations of these initial conditions. The solid black line is the mean central density across all 20 simulations (where we define the central density as the density within the half-mass radius). Clearly, the clusters have expanded significantly before the SN explosions (shown by the vertical dashed lines). Figure 7. Evolution of the median local volume density in each simulation as a function of time. Coloured lines indicate each of the 20 simulations and the black line indicates the average central density. Vertical dashed grey lines indicate times of the SNe. Irrespective of the initial conditions, the clusters never attain significant enrichment because the initial density is either too high (and two-body relaxation causes significant expansion), or the initial density is so low that despite the initial subvirial motion of the stars, significant clustering of the stars never occurs. The fraction of enriched unperturbed G-dwarf stars across all initial conditions range between ∼0.07 and 0.55 per cent. 3.4 Low-mass clusters containing one massive star The second median cluster chosen from our Monte Carlo sampling contained 113 stars, of which 1 was massive (24 M ⊙ ), resulting in an SN explosion at ∼7 Myr. The same set of initial conditions are focused on (α vir = 0.3, r F = 1.0 pc and D = 1.6) before summarizing the others. An average of six stars were enriched by the SN, a similar number to the median cluster containing two massive stars. The dynamical interaction and cluster density histories were also similar, resulting in ∼0.1–0.4 per cent of G dwarfs being enriched and unperturbed. With almost identical enrichment fractions and dynamical histories, the difference between clusters containing one or two massive stars is


Strassmeier 2012 ) observed in total 11 transits in five observing seasons using the filters Johnson B and Sloan r ′. The first seven transits were observed with both filters alternating. These r ′ band light curves were already published by Seeliger et al. ( 2014 ). The four light curves from 2014 to 2016 were taken only in Johnson B . WiFSIP holds a back-illuminated 4k × 4k 15 μm pixel CCD and offers a field of view (FoV) of 22 arcmin ×22 arcmin. To minimize the read-out time, a windowing of the CCD was used reducing the FoV to about 15 arcmin × 15 arcmin. A small defocus was applied to spread the point spread function to an artificial full width at half-maximum (FWHM) of about 3 arcsec. One transit was observed with the NOT as a Fast Track program using ALFOSC in imaging mode. ALFOSC contains a 2k × 2k E2V CCD providing an FoV of 6.4 arcmin × 6.4 arcmin. One transit was observed with BUSCA, the four-channel imager at the Calar Alto Observatory 2.2 m telescope (Reif et al. 1999 ). The instrument performs simultaneous photometry in four different bandpasses with an FoV of 11 arcmin × 11 arcmin. For the bandpass of shortest wavelength, we used a white glass filter and the beam splitter defined the limit of λ 430 nm. For the other bandpasses from blue to red, we used a Thuan–Gunn g , Thuan–Gunn r , and Bessel I filter. Unfortunately, the observation of a pre-ingress baseline was lost due to weather, and the observing conditions remained to be non-photometric. We discarded the light curve of shortest wavelength because it exhibited significantly larger correlated noise than the other three light curves. One transit was observed with the IAC80 telescope, owned and operated by the Instituto de Astrofísica de Canarias (IAC), using its wide field imager CAMELOT. It is equipped with a CCD-E2V detector of 2k × 2k, with a pixel scale of about 0.3 arcsec pixel −1 providing an FoV of about 10.5 arcmin × 10.5 arcmin. Two transits were observed with the 70 cm telescope of the AIP, located in the city of Potsdam at the Babelsberg Observatory. The telescope is equipped with a cryogenic cooled 1k × 1k TEK-CCD providing an FoV of 8 arcmin × 8 arcmin. The first transit was observed in Johnson B and Johnson V quasi-simultaneously with alternating filters, the second one in Johnson V only. We applied a 3 × 3 pixel binning to reduce the detector read-out time. Data of an additional transit were taken with the triple beam, frame-transfer CCD camera ULTRACAM (Dhillon et al. 2007 ) mounted at the WHT. The instrument optics allow for the simultaneous photometry in three different bandpasses. We chose the filters Sloan u ′, Sloan g ′, and a filter centred on the sodium doublet at 591.2 nm with a width (FWHM) of 31.2 nm. To avoid saturation of the brightest stars, the exposure time was extremely short with 0.3 s. However, the overheads are negligible owing to the frame-transfer technique. The detectors were read out in windowed mode. For the u ′ channel, 10


results of both the binary pipeline ( Fig. 2 ) and the main GALAH pipeline ( Fig. B.1 ), confirming their close agreement. We observe some trends and biases in these figures, such as the inconsistency for the lowest metallicities in the bottom panel of Fig. 2 , which are a consequence of both the flexibility of the The Cannon model and the adequacy of the training set,where it is difficult to ensure equal representation of spectra in all bins of the modelled parameter space (see Sect. 3.3.1 in Buder et al. 2018 ). 4.2 Photometric validation In this validation check and in Sect. 4.3 , we used the benchmark validation sample, which is a subset of GALAH objects withexternal information about the temperature from the infrared flux method (IRFM; Casagrande et al. 2019 ) and radii (asteroseismology; Stello et al. 2017 ; Sharma et al. 2019 ). This sample of 620 giant stars is therefore suitable for benchmark testing, and is dominated by stars from K2 campaigns C1 and C6 ( Sharma et al. 2019 , see also Fig. 13 ). Compared to the previous spectroscopic validation, this sample has almost no overlap with the training set and hence no SME parameters to compare to. We took the results from the binary pipeline, using only the photometric fit (Eq. ( 7 )), and compared them to the IRFM temperature values and reddening values from SFD (Schlegel, Finkbeiner, David) reddening maps ( Schlegel et al. 1998 ) and 3D reddening maps based on Pan-STARRS 1 photometry ( Green et al. 2018 ; hereafter the Bayestar maps). This comparison is shown in Fig. 3 , where we display results with and without the Gaia magnitudes in the photometric fit. We distinguish the Gaia magnitudes from other photometric passbands used in this work because their uncertainties are approximately an order of magnitude lower. They therefore affect the fit more strongly. Fig. 2 Comparison of SME and The Cannon labels for The Cannon validation sample. The Cannon values produced by the binary pipeline are subtracted from the SME values. Fig. 3 Comparison of T eff (orange) and E ( B − V ) (green) parameters as derived by the binary pipeline (using only the photometric fit) to the literature values (indicated on the horizontal axis). The literature values are subtracted from those given by the binary pipeline. The lighter shaded distributions correspond to results from the binary pipeline when the Gaia magnitudes are excluded from the fit. Figure 3 shows some important problems. The most striking is a bias towards higher temperatures as compared to both the GALAH DR2 values and those from the IRFM. An additional test that we performed using a larger validation sample (without the need for external information about the radii) also shows strong grid effects, which diminish considerably, however, when we exclude Gaia magnitudes. The grid effect has a spacing of about 250 K, and it might originate from the SED model in combinationwith small uncertainties of Gaia data. Excluding Gaia magnitudes from the fit


membership probabilities for the 7312 sample stars. Figure 8 shows that the stars with high GMM membership probabilities ( P GMM ≥ 0.8) generally have high RF membership probabilities, and the stars with low GMM membership probabilities ( P GMM 0.2) have low RF membership probabilities in general. However, some stars with near zero GMM probabilities still have a certain probability of being RF memberships, indicating that the color and magnitude parameters play non-negligible roles in the RF-based membership determination. In our previous work, we have confirmed that color and magnitude parameters cannot be ignored in the RF-based membership determination (Gao 2018a ). As a model-dependent clustering technique, the GMM method cannot directly handle the color and magnitude parameters since they do not follow Gaussian distributions. Fortunately, the RF method does not require any mathematical model, only a reliable training set is needed. This allows us to easily identify most likely cluster members of M67 in the 11-dimensional parameter space, even if the color and magnitude parameters are taken into account. The 1502 likely cluster members ( P RF ≥ 0.6) are listed in Table 3 , this table is published in its entirety in the electronic edition. It should be noted that Gaia Collaboration et al. ( 2018a ) have investigated M67 using Gaia -DR2 data and obtained 1520 cluster members, but they did not calculate membership probabilities for these stars. Among the 1520 cluster members in Gaia Collaboration et al. ( 2018a ), 1240 (∼82%) and 1201 (∼79%) stars are determined to be likely ( P RF ≥ 0.6) and high-probability ( P RF ≥ 0.8) cluster members by the RF method, respectively. This once again indicates the high effectiveness of the RF method for identifying likely cluster members in high-dimensional parameter space. Figure 8. A comparison between the GMM and RF membership probabilities of the 7312 sample stars. Table 3 A Catalog of Fundamental Information for 1502 Likely Cluster Members with P RF ≥ 0.6 Note. This table contains Gaia -DR2 identifier, positions and RF membership probabilities for the 1502 likely cluster members (≥0.6). Only a portion of this table is shown here to demonstrate its form and content, a machine-readable version of the full table is available. ID: Gaia -DR2 identifier; R.A., decl.: right ascension and declination; P RF : RF membership probability. (This table is available in its entirety in machine-readable form.) Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available. 3. Distance, Proper Motion, and Radial Velocity We have obtained 1361 high-probability cluster members ( P RF ≥ 0.8), 1260 of which have quite high membership probabilities ( P RF ≥ 0.9). We use the 1260 cluster members to calculate the weighted mean parallax (distance) and proper motion of the cluster, but also taking into account the observational errors. Using 687


surface term for stellar models is the method proposed by Kjeldsen et al. ( 2008 ). They note that the offset between observed and best model frequencies turns out to be closely fitted by a power law. However, there are issues with many of the surface-term corrections, even for main-sequence stars; and most models perform very badly in the subgiant and red-giant region. Schmitt Basu ( 2015 ) found that the two-term model proposed by Ball Gizon ( 2014 ) works better than other models across a large portion of the HR diagram, and consequently we adopt that for this work. 4. Results and Discussions We show the evolutionary track of our models in Figure 3 . The figure also shows the two error boxes. T eff = 4766 ± 200 K and have been used for the large error box while T eff = 4974 ± 161 K and for the smaller error box. We find that many models fall into the two error boxes. In order to obtain the best-fit model for KIC 8263801 from among these models, we calculated the likelihood for each model and determined the model with the highest likelihood. The likelihood function is defined aswherewith the quantity indicating the observed T eff , [Fe/H], and frequency ν n , l ; while the corresponds to these values of the model. The quantity σ i represents the observational error of . In the study, we use the function as follows:where In the above expression, are the surface-term corrected frequencies of the models. The values of likelihood function for model of KIC 8263801 as a function of mass, metallicity, age, and radius are shown in Figure 4 . We choose the model that maximizes as a candidate for the best-fit model. The model parameters are shown in Table 5 . To clearly compare all of the theoretical frequencies of the best-fit model with observed frequencies, we show the échelle diagram of the best-fit model in Figure 5 . In the figure, open symbols are the model frequencies corrected for the surface term, the filled symbols refer to the observable frequencies. Circles are used for l = 0 modes and squares for l = 2 modes. Figure 4. Likelihood values for models of KIC 8263801 as a function of mass, metallicity, age, and radius, corresponding to panels (a)–(d) respectively. Figure 5. Échelle diagrams for the best-fit model. Open symbols are the surface-term corrected frequencies of the best-fit model; filled symbols refer to the observed frequencies. Circles are used for l = 0 modes, squares for l = 2 modes. Table 5 The Parameters of the Best-fit Model Finally, we derive the parameters of star as the likelihood weighted mean and standard deviation from the models and obtain a mass of , age of t = 1.48 ±0.21 Gyr, and radius of R / R ⊙ = 10.53 ± 0.28. Our best-fit model has an age of 1.596 Gyr, the age as determined from the likelihood weighted average is not too different, and thus the question arises as to whether this star can be a member of NGC 6866. There are a number of results about the age of NGC 6866. For example, Loktin et al. ( 1994 ) derived


well-studied extragalactic radio source Cygnus A to demonstrate the applicability of our code. This is one of the nearest Fanaroff–Riley type I (FR II) galaxies and has a well-defined jet and counterjet with a jet inclination angle thought to be between 55 ○ and 85 ○ (Bartel et al. 1995 ), with a redshift of z = 0.0562 ± 0.000067 (Carilli Barthel 1996 ). There is observational morphological evidence to suggest that the jets are precessing ( K19 ). We visually identified 48 jet points (with an astrometric error of 0.4 arcsec from VLA data) from the 5-GHz map of Perley, Dreher Cowan ( 1984 ), as shown in Fig. 5 . Jet points are taken to be any discrete, distinct feature that appears to be part of the jet; this includes both compact knots and sub-regions of more extended bright parts of the jet. For longer sub-regions of the jet, we placed multiple points, spaced by at least the resolution of the image, in order to constrain the curvature of the jet in these regions. Figure 5. Locations of visible jet knots within Cygnus A used in the model. Note the curvature in both the jet and counterjet paths. The final points in both the jet and counterjet lie above the primary hotspots of the two lobes, as discussed in the text. The inner, straighter points of the jet used in Fig. 7 are identified as being within the indicated brackets. One important constraint on the application of the MCMC model to real-world data is the requirement for the precessing jet to stay within the lobes. Since the model does not take lobe structure into account beyond the terminal hotspot, the constraint was implemented by a joint prior on position angle and precession cone opening angle. This dual parameter calculates the two extreme edges of the cone for any given position angle, and keeps them within the lobes, which are approximately 25 ○ across at their widest point. Without this additional prior, we found that unphysical jet paths were permitted as part of the fit, broadening the credible interval on precession period. As with the tests described in earlier sections, the priors were flat within the permitted ranges. We initially fitted the model to all identified points (see Fig. 6 ), including the terminal hotspot, which in both lobes is taken to be the more compact or ‘primary’ hotspot; we do not include the larger, brighter secondary hotspot since the position of this is almost certainly dominated by post-shock hydrodynamics (Cox, Gull Scheuer 1991 ). Then, with respect to the potential for hydrodynamic influences to dominate the jet path within the lobes, we fitted the model only to the straight portion of the jet, terminating just inside the lobes (Fig. 7 ). The full number of regions produced a peak in the posterior distribution corresponding to a precession period of 1 Myr. Figure 6. Main: ‘Corner plot’ of one-dimensional and two-dimensional marginalized posterior probability distributions for the model parameters, given the observed jet path of Cygnus A, including