elements of Mercury given by the DEA431 analytical series and by DE431 numerical ephemerides. Table 2. Maximum differences between orbital elements of Venus given by the DEA431 analytical series and by DE431 numerical ephemerides. Table 3. Maximum differences between orbital elements of the Earth–Moon barycentre given by the DEA431 analytical series and by DE431 numerical ephemerides. Table 4. Maximum differences between orbital elements of Mars given by the DEA431 analytical series and by DE431 numerical ephemerides. Table 5 gives the number of terms in the DEA431 analytical series, that is, the value of j i in equation ( 8 ) for the i th orbital element of every terrestrial planet. Table 5. Number of terms in the DEA431 analytical series. We also made a comparison of planetary heliocentric distances R , ecliptic longitudes L , and ecliptic latitudes B calculated using the DEA431 analytical series with those obtained from DE431 numerical ephemerides. Tables 6 – 9 present the maximum deviations between the two sets of spherical coordinates for every terrestrial planet over the total time interval of 30 000 yr covered by DE431. For comparison, some corresponding data available for the VSOP2013 theory from Simon et al. ( 2013 ) and a description of the theory's files on ftp 2 are given. In the case of VSOP2013, the numerical ephemerides INPOP10a (Fienga et al. 2011 ) extended to the time interval [−4 000, 8 000] were used as a reference. Table 6. Maximum differences between the spherical coordinates of Mercury given by the analytical series and by numerical ephemerides. Note. the coordinates R *, L *, B * are related to VSOP2013 planetary theory; the corresponding differences are taken from Simon et al. ( 2013 ). Table 7. Maximum differences between the spherical coordinates of Venus given by the analytical series and by numerical ephemerides. Note. the coordinates R *, L *, B * are related to VSOP2013 planetary theory; the corresponding differences are taken from Simon et al. ( 2013 ). Table 8. Maximum differences between spherical coordinates of the Earth–Moon barycentre given by the analytical series and by numerical ephemerides. Note. the coordinates R *, L *, B * are related to VSOP2013 planetary theory; the corresponding differences are taken from Simon et al. ( 2013 ). Table 9. Maximum differences between the spherical coordinates of Mars given by the analytical series and by numerical ephemerides. Note. the coordinates R *, L *, B * are related to VSOP2013 planetary theory; the corresponding differences are taken from Simon et al. ( 2013 ). Tables 6 – 9 show that the VSOP2013 theory better predicts the motion of terrestrial planets over 100 yr or so, but that the DEA431 solution performs better over long-term intervals such as thousands and tens of thousands of years. From Tables 1 – 4 and Tables 6 – 9 it can be seen that the DEA431 development has a rather stable accuracy over approximately 24 000 years centred at epoch J2000. In

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


value (see review in Zinnecker Yorke 2007 ). Increasing or decreasing the assumed excitation temperature by 5 K changes the column density by about 30%. If CS(1–0) is optically thick, as we assume for three sources in Section 4.2 below, then our calculation would be a lower limit. Given these assumptions we used the following relation (see Miettinen ( 2012 ) for a detailed discussion of the relations below): where Here 0 is the vacuum permittivity, μ el is the permanent electric dipole moment, S is the line strength, Z rot is the rotational partition function, ν is the frequency, g K is the K-level degeneracy, g I is the reduced nuclear spin degeneracy, E u is the energy of the upper-transition state, T ex is the excitation temperature and T bg is the background temperature. The dipole moment line strength ( S ) is taken from the JPL spectral line catalog (Pickett et al. 1998 ). The partition function ( Z rot ) is a linear fit to JPL data between T = 37 and 75 K. T bg was taken to be the cosmic microwave background temperature, 2.725 K. The uncertainty in the fit amplitudes and derived column densities is dominated by our flux-calibration uncertainty. Since the relationships are linear, we estimate the uncertainty in both as 20%. Figure 1. Detection of CS emission (gray) toward a candidate YSO (N92-3) on the rim of bubble N92. The Gaussian fit to the emission is shown in black. 3.2. CS Mapping The three mapped regions are shown as boxes overlaid on the GLIMPSE 8 μ m images in Figures 2 – 4 . The maps were cropped to just those regions showing emission above 3 σ . The regions containing emission were then exported to a FITS file using custom-built IDL tools and the FITS-format datacubes were analyzed using CASA. The standard moment maps (total intensity, average velocity and velocity width) are shown next to the corresponding 8 μ m emission images in Figures 2 – 4 . Figure 2. N56: (a) The region surrounding N56 mapped in CS (white outline) overlaid on a GLIMPSE 8 μ m survey image. (b) The integrated intensity map (c) the average velocity map (d) the velocity width map. Figure 3. N65: (a) The region surrounding N65 2 mapped in CS (white outline) overlaid on a GLIMPSE 8 μ m survey image (b) the integrated intensity map (c) the average velocity map (d) the velocity width map, (e) a spectrum centered on the primary peak marked in part (b), (f) a spectrum centered on the secondary peak and the shift in velocity, marked in part (c). Figure 4. N77: (a) The region surrounding N77 1 mapped in CS (white and black outline) overlaid on a GLIMPSE 8 μ m survey image (b) the integrated intensity map (c) the average velocity map (d) the velocity width map. The three maps of CS in N56, N65, and N77 (Figures 2 – 4 ) show evidence of supersonic gas motion in areas near each YSO. N56 has a weak peak in CS that corresponds to a velocity shift of ∼2 km s −1 in the gas immediately to the north. The CS emission in N65 has a clear, strong comma-like shape. The peak

O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Dataset I-Dataset O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-Survey B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O B-Software O O O B-Software I-Software O O O B-Software O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O B-Survey B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject I-CelestialObject O O O O O O O O B-Survey B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O B-Survey B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O


geometrically thin optically thick, radiatively efficient, and steady-state accretion disk (Shakura Sunyaev 1973 ), which is expected in AGNs. We assume the disk has viscosity parameter α = 0.1, and radiation efficiency , where and are the Eddington luminosity and Eddington accretion rate of an SMBH with mass M • . We further assume the accretion rate is constant during the lifetime τ agn of an AGN. These parameters determine the disk surface density Σ( r ), scale height H ( r ), isothermal sound speed c s ( r ), and midplane temperature T ( r ), which are dependent on the distance r from the SMBH. To derive these quantities we follow the model of Sirko Goodman ( 2003 ), assuming that viscosity in the disk is proportional to the total pressure: where T eff ( r ) is the blackbody temperature of the disk at r . , where r min is the inner radius of the disk; we set here, where . Ω is the Keplerian angular velocity of the disk, which is . τ is the optical depth at midplane. We assume the mean molecular mass . σ is the Stefan–Boltzmann constant, and k is Boltzmann constant. In order to solve these equations, we still need to provide the opacity κ . We approximate the opacity with (see Figure 1 in Thompson et al. 2005 ):where g −1 cm 2 K −2 and κ 1 = κ 0 (100 K) 2 = 2.4 g −1 cm 2 . κ m is the molecular opacity, which is about 0.1 Z , where Z indicates the mass fraction of heavy elements. We adopt Z = 0.01 in our simulations (Iglesias Rogers 1996 ). is the opacity due to the negative hydrogen ion:where and . The electron scattering opacity κ e can be written as X is the mass fraction of hydrogen. The Kramers opacity κ K is due to free–free, bound–free, and bound–bound electronic transitions:The equations above are valid where the Toomre Q -parameter Q = C s Ω/( πG Σ) ≳ 1. Beyond that region, out to 0.1 pc, we assume that the disk self-regulates, which we take into account by fixing the Q -parameter at 1 and replace Equation ( 1 ) with (see Equation (15) of Sirko Goodman 2003 ): 2.2. Orbital Variation Due to AGN Disk Crossing Consider a Cartesian coordinate system with the SMBH in its center, in which the AGN disk lies in the XY plane, and the major axis of the orbit of the stellar-mass BH orbiting the SMBH is in the YZ plane (see Figure 1 ). Let be the unit vector pointing in the direction of the major axis of the BH’s orbit in Cartesian coordinates, with θ being the angle is the orbit’s inclination angle compared to the AGN disk plane. Let a be the semimajor axis, e the eccentricity, and is the normal vector of the BH’s orbital plane. Let be the angle between and the YZ plane, where is the unit vector along the x axis. 5 5 Here we adopt the coordinates ( θ , ψ , φ ) that are the most suitable for the derivation of the results, even though these are not standard Keplerian orbital elements. They can be converted to Keplerian coordinates as follows. Longitude of the ascending mode: . Argument of periapsis: φ . Inclination: . We haveLet φ be the angle

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


given that the resulting cooling rate peaks in a temperature regime where the gas is expected to be fully ionized. To calculate an approximate cooling function, we solve equation ( C1 ) for a densities in the range 10 −2 –10 6 cm −3 and in the temperature range of 1–10 8 K. For each species, cooling rates are calculated at collisional excitation equilibrium. The net cooling is the sum of these terms: where the first sum is over species, n i ,s is the number density of an excited state for a given species, and ν ij is the frequency of the photon emitted by a transition from level i to level j . The two-dimensional table of cooling rates is averaged along the density axis to get an average cooling rate as a function of temperature. The average cooling function is show in Fig. C1 alongside the piecewise fit used in our models (equation 32 ). Figure C1. Average volumetric cooling rate (solid) and piecewise approximation (dashed) for collisionally excited line radiation. The shaded region indicates the 1σ error region, and the hatched region highlights the range of values averaged over to obtain the approximate cooling function used. APPENDIX D: DISC FINDING ALGORITHM We have developed an efficient parallelized disc finding routine that allows us to track the formation and evolution of formed discs in our models irrespective of scale or orientation. We implement this routine directly into athena for two reasons. First, this tool provides the advantage of ‘on-the-fly’ analysis which yields much greater time accuracy than is reasonably managed through post-processing. Secondly, the method is designed to take advantage of the parallelized structure of athena , which dramatically reduces the amount of time required to execute the necessary operations. The algorithm takes the following steps: (1) On each processor, or local grid, the total mass and angular momentum with respect to the origin are calculated within a search radius, R search . For simulations with refinement, overlapped cells are excluded from this total. The net angular momentum ( L net ) and mass ( M net ) are then calculated across all processors. (2) For each cell, the deviation angle is calculated as where L cell is the angular momentum vector of the cell. (3) The total mass for cells with θ dev θ thresh is then calculated. For our simulations, we found that setting θ thresh = 30° was a sufficiently strict condition for capturing the formation of a disc. (4) A mass fraction is then calculated as For our models, we consider f M > 0.75 to be indicative of a potential disc. (5) For disc candidates, we continue to a three-dimensional rotation of the mesh from the simulation reference frame into a reference frame in which the axis is parallel to L net , which we call the disc frame. We construct a new mesh in the disc frame with identical dimensions and hierarchical structure to the simulation frame of reference. We loop over all cells on the disc frame mesh and calculate the

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


gradient. The negative radial gradient of is expected because is associated with the surface mass density (e.g. Freeman Bland-Hawthorn 2002 ; Binney Tremaine 2008 ), which decreases with increasing Galactocentric radius (e.g. Xiang et al. 2018 ). The positive vertical gradient mainly reflects the increase of surface mass density enclosed by increasing height above the disc. The dispersion of J Z , however, has less dependence with R . Again, this is likely to be because of the nature of J Z , as discussed above. The skewness of V Z is close to 0 and uniformly distributes across the R – Z plane, except for the outermost region ( R > 11 kpc), whereas the skewness in J Z shows strong patterns. These characteristics are similar to the case of skewness for V R and J R , and they share similar physical reasons (see Section 3.2.1 ). 4 ORBITAL PARAMETERS AND STELLAR MIGRATION In this section, we present the orbital parameters of the RGB stars across the age–metallicity plane and the disc R – Z plane. We discuss the impact of non-circular orbital motion on the stellar distribution by comparing the current position of the stars with the guiding-centre radii of their orbits. We further estimate the birth radii of the stars based on their age and metallicity in the same way as Chen et al. ( 2019b ), initially proposed by Minchev et al. ( 2018 ), and we study the effect of stellar migration (churning). Note that here we refer to the stellar migration as the churning process, in which stars change their orbital radii through angular momentum transfer via scattering or perturbation (e.g. Sellwood Binney 2002 ; Minchev et al. 2013 ). 4.1 Orbital parameters As mentioned in Section 2 , the orbital parameters of our sample stars are derived using positions and proper motions from Gaia DR2, distances from BJ18 and radial velocities from LAMOST DR4. The orbit calculation is implemented with galpy , assuming the MWpotential2014 (Bovy 2015 ). Fig. 10 plots the median, dispersion and skewness of the stellar eccentricity distributions in the age–[Fe/H] plane and the disc R – Z plane. The figure illustrates that the young or old but metal-rich thin disc stars have mean orbital eccentricities smaller than 0.2, and have small (0.1) dispersion among stars in each mono-age and mono-[Fe/H] bin, which is consistent with previous work (e.g. Kordopatis et al. 2015 ). The old, metal-poor thick disc stars have larger eccentricities and broader distribution, and their mean eccentricities vary from 0.2 to larger than 0.5. For the thin disc stars, the eccentricity steadily increases with stellar age and exhibits little change with [Fe/H] at a fixed age, possibly suggesting that the steady disc heating processes have played a decisive role in determining the orbital evolution of these stars (e.g. Ting Rix 2019 ). At a given age, the [Fe/H] of a star is an indicator of its Galactocentric radius because of the existence of radial [Fe/H] gradient for mono-age populations (e.g. Boeche

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et al. 2013 ). Thus, the rather small source size of R = 4.8 × 10 15 cm obtained from the SSC fitting for the core of Cen A suggests that the emission region is located very close to the nucleus. To verify that a higher magnetic field can be observed closer to the core, we plotted the obtained magnetic fields from the SED fits for various FR I AGNs as functions of the distance from the core, as shown in Figure 3 (no error is included). It should be noted that for the sake of simplicity, the distance of the emission region from the nucleus is defined as , where R is the source radius obtained from previous studies (Chiaberge et al. 2003 ; de Jong et al. 2015 ; Fraija et al. 2017 , 2018 ; Baghmanyan et al. 2018 ; Tanada et al. 2018 ) and θ op is the opening angle of the jet. The VLBI observations revealed that the opening angle is collimated to the range of 2°–8° at a distance of 1 pc from the core; therefore, we set the angle as 5° (Junor et al. 1999 ; Horiuchi et al. 2006 ). For the distances of the AX2 and BX2 knots, we used the directly measured values, as shown in Table 1 . As can be seen in Figure 3 , the magnetic field decays as a power-law function , where the Schwarzschild radius of Cen A is r s = 2 G M BH / c 2 ≃ 1.6 × 10 13 cm. The obtained fitting parameters are the initial magnetic field of B 0 = (1.5 ± 0.2) × 10 4 G at r s and α = 0.88 ± 0.14. The conservation of particle number in a freely expanding conical jet requires the particle number density of N ( d ) ∝ d −2 , which implies B ( d ) ∝ d −1 from equipartition (Hirotani 2005 ) and it is consistent with our results. Furthermore, the core-shift analysis with the Very Long Baseline Array observations of blazars shows that the attenuation of the magnetic field at the vicinity of the core is described by B ( d ) ∝ d −1 appropriately (O’Sullivan Gabuzda 2009 ). According to the theory of the magnetically powered jet (Komissarov et al. 2007 ; O’Sullivan Gabuzda 2010 ), the initial magnetic field strength for a black hole launched jet can be calculated by G, where L rad ≃ 1.5 × 10 43 erg s −1 is the total jet luminosity of Cen A obtained from the SSC fitting, which is about one-fourth of the fitted value. Although there is a discrepancy between the theory and fitted value, which might be caused by the rough estimations such as the distance of the emission zones, these results support our suggestion that the SSC emission region of Cen A is located near the nucleus. Figure 3. Magnetic fields obtained from the SED fits for various FR I AGNs as functions of the distance from the core. The blue filled circle, square, and triangle represent the magnetic field strength of the Cen A core, AX2, and BX2, respectively. The blue open marks indicate the magnetic field strength of the core of the other FR I AGNs. The distances of AX2 and BX2 are directly measured using the Chandra analysis (Kraft et al. 2002 ). Except for these knots, the distance is calculated with the emission region size

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


× 28.9 arcsec beam), which has the characteristics of a rotating disc with a moderate rotation velocity and a signature of a warp in the SE. The black contours at the centre of this panel are the two highest H i column density contours from the medium resolution total intensity H i map. These H i contours coincide with the kinematic centre as well as the optical centre of the galaxy. The velocity field of UGC 2162 shows no significant sign of recent major interaction. The position velocity (PV) diagram for a slice taken along the H i major axis, position angle (PA) = 307 ° , is shown in the middle panel of Fig. 6 and confirms a rotating disc. A small mass of low density extra-planar H i is seen ∼15–20 arcsec (∼1 kpc) SE of the kinematic centre in the PV diagram. The lower panel of Fig. 6 shows the rotation curve extracted from bbarolo (Di Teodoro Fraternali 2015 ) fit to the medium-resolution cube. The figure shows a slowly rising rotation curve that is a common feature for dwarf galaxies. The bbarolo best fit gives an inclination of 55.5 ° , with V rot at the outermost fitted ring, r = 67.5 arcsec (4.05 kpc) ∼ 25 km s −1 . The M dyn enclosed within each ring fitted with bbarolo is used in Section 4.1 to estimate the galaxy’s virial mass ( M vir ). We also estimated the V rot using the H i W 20 (adjusted for the inclination from the bbarolo model fit) where V rot = 0.5 Δ V [km s −1 ]/sin(i) with Δ V = W 20 (50 km s −1 ) from the GBT spectrum and i = 55.5 ° /180. This second method gives V rot ∼ 30.8 km s −1 at the outer edge of the detected H i disc. These values are also used in the Section 4.1 analysis. Figure 6. UGC 2162 H i kinematics: Top: GMRT velocity field from the medium-resolution cube. The two highest column density contours from the medium-resolution H i integrated map are shown in black. The PV diagram (panel below) slice PA = 307 ° is shown with a grey line. Middle: PV diagram from a PA = 307 ° slice. Positive positional offsets are to the NW. The blue contours are from the data and the red are from the bbarolo best-fitting model. Bottom: Rotation curve derived from the bbarolo five ring model fit. In Fig. 7 , we compare the V rot from for UGC 2162 with the V rot derived from model fits to the VLA H i for the LITTLE THINGS dwarf galaxies by Oh et al. ( 2015 ). The plot shows that the UGC 2162 V rot = 30.8 km s −1 , derived from the GBT W 20 , falls within the range of V rot for LITTLE THINGS with similar M H i and below the LITTLE THINGS median V rot of 39.5 km s −1 . Fig. 8 shows the baryonic and stellar Tully-Fisher relations. Figure 7. V rot versus M gas : UGC 2162 (Black square) and LITTLE THINGS dwarf galaxies (red crosses) from Oh et al. ( 2015 ). The UGC 2162, V rot = 30.8 km s −1 is derived from the GBT W 20 in Section 3.2 . For both UGC 2162 and the LITTLE THINGs galaxies, M gas = 1.4 × M H i with the additional factor to account for molecular gas and He. Figure 8. UGC 2162 Tully–Fisher relations: The position of UGC

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Telescope O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O B-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O B-Telescope O O O O B-Survey I-Survey O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-CelestialObject I-CelestialObject B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Telescope O O O O O O O O O O O B-Survey I-Survey O O O O O O O O B-Survey I-Survey O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O B-CelestialObject I-CelestialObject O O O B-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Telescope O O O O O O O O B-CelestialObject I-CelestialObject O O B-Survey I-Survey O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-Model I-Model O O O B-CelestialObject


visible. 3. Physical and Observational Constraints of the Allowed Parameter Space Region that Enables the Production of FRBs The physical conditions in the region where the masing takes place can be constrained using the physics of the maser and constraints from observations, allowing us to place limits on the magnetic fields and number densities where the synchrotron maser can plausibly be the emission mechanism for FRBs. We consider a cataclysmic FRB progenitor. However, as the repeating burst FRB 121102 is the only one to have a known redshift, it is the only source that provides observational constraints for quantities such as the burst energy and the DM of the host galaxy. Therefore, we use the values it provides as representative limits for our calculations. The data enable us to obtain constraints linking the size and number density of the masing region to the magnetic field of the neutron star. These constraints are obtained from: (i) the energetics of the burst and size of the masing region, (ii) the efficiency of the maser mechanism, (iii) the dispersion measure of the burst, and (iv) the frequency of the signal. We consider that masing takes place in a spherical shell of thickness d , located a distance R from the central object. The maser will be activated by the formation of a population inversion in the shell. The magnetic fields and short timescale (≲ d/c Γ) required suggest that this object is a neutron star, though the timescale for maser emission is given by the duration of the maser itself (Ghisellini 2017 ). Assuming that the blast wave is relativistic, the shocked plasma has a width ∼ R /Γ (Blandford McKee 1976 ), where Γ is the Lorentz factor of the blast wave. This typical width provides the first constraint, on the thickness of the shell:The minimum thickness of the shell depends on the number of particles that contribute to the masing, , and their number density, n e . Here, E is the energy of the bursts (in the range 10 38 –10 40 erg for the repeater (Law et al. 2017 ; Tendulkar et al. 2017 )), η is the fraction of the electrons’ energy that contributes to the maser, and is the average energy of the masing electrons. These shocked electrons have a thermal energy of γ ≈ Γ. Constraint (ii) originates from the efficiency of the maser. For the maser to be a viable emission mechanism, the growth rate of the signal must be large enough to extract a fraction η of the particle energy. The maser will be quenched when the maser reaches saturation. The efficiency of the maser mechanism in simulations of relativistic shocks was shown to be η ≲ 10 −1 (e.g., Gallant et al. 1992 ; Sironi Spitkovsky 2009 ). In the case of AKR, the efficiency is in the range η ∼ 10 −1 –10 −3 (Wu 1985 ). The exact value, however, depends on the form of the particle distribution, which is uncertain. We therefore examine values of η in the range 10 −3 ≲ η ≲ 10 −1 in this Letter. Lu Kumar ( 2018 ) gave upper limits to the efficiency of η ≲ 10 −5 ,

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


decrease by more than 25% with respect to an older adjacent bin. The choice of just five coarse age bins is motivated by the low number of stars in the CMD, the large photometric errors, and the reached magnitude limit of just ∼1.5 mag below the RGB tip. Indeed, from the existence of an RGB we know that stars older than 1–2 Gyr are present in S1, but we are not able to infer in great detail the SFH from the age of the universe to ∼2 Gyr ago. The best-fit solution from SFERA is presented in Figure 9 ; in the top panels, we show the empirical and the model CMDs in the form of Hess diagrams, i.e., density plots, and the residuals in units of Poisson uncertainties ((data−model)/). As visible from the residuals, the model reproduces quite well the observed CMD above F814W ∼ 28. The TP-AGB star region at F814W ∼ 26, F606W−F814W ≳ 1.2 is better populated than in the previous SSP-fit case, although a shortage of counts is still present in the synthetic CMD. However, we do not consider this partial discrepancy as a problem here, as it is known that this evolutionary phase is very difficult to model and still lacks proper calibration over the whole range of metallicity regimes. Figure 9. Results of the SFERA procedure for the SFH recovery in S1. Top panels: Hess diagrams of the observed CMD, of the best-fit model, and of the residuals. Middle panel: derived metallicity behavior as a function of look-back time. Bottom panel: derived star formation rate as a function of look-back time. The total formed stellar mass is indicated. The middle and bottom panels of Figure 9 show the resulting metallicity and star formation behaviors as a function of look-back time. The solution recovered by SFERA implies that the vast majority of the SF activity occurred at epochs earlier than ∼500 Myr ago; the peak of the activity is predicted to have occurred in the 2–6 Gyr bin, while the adjacent 500 Myr–2 Gyr and 6–13.7 Gyr intervals were characterized by a lower SF. As already expected from the CMD appearance in Figure 7 , the SF rate in S1 was compatible with zero at epochs more recent than ∼0.5 Gyr ago. At earlier epochs we are not able to infer the details of the SFH with the data in our hands; however, the total formed stellar mass of (1.27 ± 0.41) × 10 6 M ⊙ can be considered a quite robust result and is in reasonable agreement with the estimates from both the SSP-fitting and the integrated light in Annibali et al. ( 2016 ). Concerning metallicity, the best-fit solution implies a continuous increase in Z starting from ∼0.0006 at the oldest look-back times and reaching up to Z ∼ 0.006 (∼40% solar) at recent epochs. This value is significantly higher than the Z ∼ 0.0004 (∼2%) solar metallicity measured in DDO 68's H ii regions, and also much higher than expected for a ∼10 6 M ⊙ stellar mass galaxy. Admittedly, given the strong age–metallicity degeneracy in the RGB phase combined with the relatively large photometric errors of our data, we can not claim a robust

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-EntityOfFutureInterest I-EntityOfFutureInterest O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


and along with the dust distribution, it sets the thermal profile of the disk. Likewise, UV and X-ray photons dominate the chemistry in the upper layers through molecular ionization and photodissociation. Given that photodissociating photons play a considerable role in the disk chemistry, we use a large number of photon packages for the X-rays and the UV frequencies to minimize the photon noise in the Monte Carlo simulation. We run the simulations using 5 × 10 6 photon packages for the optical and longer wavelengths, but we include a refinement factor of 10 3 for Ly α photons, 5 × 10 3 for UV photons, and 10 6 for X-rays. By using the refinement factor, we increase the number of photon packages in the respective spectral window by the corresponding refinement value. Even using a refinement for the UV photons, the dust opacity does not let the UV photons reach the midplane. Nevertheless, the UV flux reaching the midplane and its associated photodissociation rate are negligible compared to the optical irradiation and the gas-phase chemistry, so it does not change our final results. Further, Agúndez et al. ( 2018 ) show that FUV photodissociation rates for T Tauri disks are negligible for z / r 0.3 at 100 au. In our basic or benchmark model, the surface density, Σ basic , follows the self-similar solution from Lynden-Bell Pringle ( 1974 ), i.e.,where r e is the radius of the exponential cutoff, r 0 is the characteristic radius for the disk, and Σ 0 is the surface density of the disk at r = r 0 . In our basic model, the gas and dust components have the same profiles, where Σ 0 is the only changing value. For the disk with the dust substructure, we add the ring and the gap as modulations of the self-similar solution. Those modulations are further discussed in Section 2.1.1 . We aim to include how the gas and the distinct dust populations alter the chemistry and physical conditions in the disk. So, we set different vertical distributions for them, but with the same functional form. Both gas and dust have vertical Gaussian distributions (Armitage 2010 ):where and h is the scale height, which we vary for each population. Particularly for the gas, the aspect ratio is set to be h / r = 0.08 at r = 100 au. The scale height also has a radial dependence through a power law with a flaring index ψ :The flaring index is a common value for the gas and dust populations. We set it with a value of ψ = 1.22. Besides the gas, we include two main dust populations differentiated by the grain size they represent and an additional third component only within the inner 3.5 au. The two main populations are composed of a mixture of 80% silicates and 20% graphite. The optical opacities where taken from Draine Lee ( 1984 ), while the X-rays opacities come from the prescription in Bethell Bergin ( 2011 ). Both dust populations follow a grain size distribution with a standard power law, n ( a ) ∝ a −3.5 , which is taken from Mathis et al. ( 1977 ). We also set a

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


97% reliability threshold and matched to their input counterparts are plotted as blue circles. The red dashed line is the 1:1 relation, while the yellow contours indicate where most of the points lie. The Eddington bias flattening is clearly visible for input fluxes erg cm −2 s −1 . Bottom: ratio between the output and input flux as a function of the input flux for the simulations performed. The yellow filled circles are averages computed in bins of input flux, while the red horizontal line marks the 1:1 relation. 4. Source Detection We repeat the same procedure of source detection on the data mosaics. After deblending the list of potential sources, we detect 43 unique sources above the threshold of 99% reliability in at least one of the “canonical” F, S, and H bands. When considering the 97% reliability threshold, we detect 67 sources. We will refer to these catalogs as UDS99 and UDS97, respectively, and the detailed numbers for each band are reported in the last two rows of Table 5 . We note that the numbers of detections agree very well (within 1 σ of the distributions; see Figure 7 ) with the simulation expectations. To maximize the statistics, we will focus on UDS97 henceforth, keeping in mind that the spurious fraction of this catalog is 3%. 26 26 Details about this catalog can be found in the Appendix . For homogeneity with the other NuSTAR surveys, and given the few detections in the H1, H2, and VH bands, we will also consider only the 67 sources detected in the canonical bands (of which two are expected to be spurious), while discussing the results in the new bands elsewhere (A. Masini et al. 2018, in preparation). Table 6 shows how these 67 sources are distributed within the F, S, and H bands. Figure 7. Distribution of detections within the simulations. The six upper panels, in red, refer to the six bands at the 97% reliability threshold. The six lower panels, in green, refer to six bands at the 99% reliability threshold. From top left to bottom right, the bands are F, S, H, H1, H2, and VH. In each panel, the dashed lines mark the ±1 σ interval from the average, while the black solid line marks the position of the detections in the data mosaic, which are always within ±1 σ from the mean of the simulations, shown by the solid red or green line. Numerical values for the average number of detections in the simulations in each band and the number of detections in the data mosaics are explicitly addressed in the last four rows of Table 5 . Table 6 Summary of Detection, UDS97 Note. Capital letters are for sources above the threshold, while lowercase letters refer to sources detected but below the threshold. 4.1. Catalog Creation We create a catalog of the 67 sources detected in the three canonical bands of UDS97. Following the same strategy adopted for the simulations, we use dmextract to get total and background counts from circles of 20′′ radius, from the data and background mosaics, respectively. Similarly, we extract the exposures from

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


that in the inertial frame only 2l + 1 out of (2l + 1) 2 peaks dominate, consistent with the matrix remaining approximately diagonal even at substantial obliquities (see Fig. 9 ). Closer inspection of how the shape of this dominant sub-multiplet changes with obliquity reveals that while this may be asymmetric in general, it is still possible for this to appear nearly symmetric for some values of β and have a spacing close to the value expected of pure rotation (Fig. 12 , lower left-hand panel). However, the centroid of this sub-multiplet is offset compared to the pure rotational multiplet. At larger field strengths (see Fig. 14 ), there may be many peaks of significant amplitude in the inertial frame. Notably, due to the heavy mixing, it would be possible to find symmetric sub-multiplets among these (e.g. bottom row, middle panel). Figure 12. Splitting diagram for a p-dominated mode of Model C, with n = −16 and l = 1 (i.e. that used in Fig. 9 ), under the default rotation and field strength listed in Table 1 , for six different values of the obliquity angle. Short, thick, horizontal black lines joined by thin dotted lines illustrate the splitting process, from the unperturbed mode (leftmost line at ω 1 = 0) into a multiplet of 2l + 1 modes in the corotating frame (middle group), and then further into (2l + 1) 2 modes in the inertial frame (rightmost group). In the rightmost group, the various modes have been horizontally offset for clarity. The different grey-scale shades correspond to the absolute value of the associated a m coefficient, with black being 1 (maximum amplitude) and white being 0. For comparison, horizontal coloured dashed lines mark the position of the non-magnetic rotationally split multiplet in the inertial frame. Figure 13. As for Fig. 12 , but a mode with n = −28 and l = 2 (i.e. that used in Fig. 10 ). Figure 14. As for Fig. 12 , but with a field strength three times higher. Frequency splittings for all l = 1 modes of Models C and D are summarized in Fig. 15 , where points are coloured according to the associated value of | a m |. The increased m -mixing with larger obliquity and field strength can clearly be seen across all modes, both p- and g-dominated. The main difference between p- and g-dominated mixed modes appears to lie in the centroid frequency of the multiplet: this is located towards systematically higher values compared to the unperturbed value for g-dominated multiplets, a feature not present in the case of pure rotation (as seen in Figs 7 and 8 ), where the centroid offset is zero for all multiplets. This offset between p- and g-dominated multiplets increases towards lower frequencies, where magnetic effects become more significant. Figure 15. Inertial-frame frequency shifts versus unperturbed frequency, for all l = 1 modes of Models C (left) and D (right). Values of the field strength and obliquity are shown in the panel headers. Within each panel, the different colours correspond to the corotating-frame

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


source, no absorption occurs. An H i absorption line is centered at a frequency, blueshifted or redshifted, according to the H i cloud velocity along the line-of-sight (LOS or radial velocity, V r ). A more realistic scenario is depicted in Figure 1 with a number of H i clouds and continuum sources, represented as discrete clouds. The H i clouds in front of the object of interest (continuum source, red object) will absorb radiation. If the background LOS is adjacent, the distribution of H i and continuum clouds would be nearly the same. Figure 1. Geometry for the case of many H i clouds (yellow) and continuum sources (green) along the line-of-sight toward the object of interest (red). The equation of radiative transfer is , where is the specific intensity, ds is distance along the LOS, is the emission coefficient, and is the linear absorption coefficient. In local thermodynamic equilibrium, one has , the Planck function. For low frequency (), simplifies to the Rayleigh–Jeans formula where T B is the brightness temperature. The solution of the equation of radiative transfer in plane parallel geometry, ignoring scattering, is , with . This is converted to brightness temperature ( T B ) units using . The discrete form of this result using an arbitrary large set of optically thin H i clouds and continuum emission regions isHere the index m sums over the emission regions and the index n sums over the emitting and absorbing H i clouds. is the continuum brightness temperature of cloud m. Here, and are the total optical depth, from each continuum or H i region to the observer, is the optical depth of a single H i region along the LOS, and for all clouds in front of m . The “on” spectrum is the spectrum toward the source and the “off” spectrum is the spectrum toward the background. Subtracting the “off” spectrum from the “on” spectrum, the H i absorption spectrum is given as is the H i optical depth, and is plotted as the absorption spectrum. 3.2. Kinematic Distances and Resolving the Kinematic Distance Ambiguity The radial velocity of the object of interest, or limits on it, can be deduced from the H i absorption spectrum. Then the kinematic distance to the object of interest can be determined, assuming objects move with circular velocity V ( R ) in the Galaxy (Kwee et al. 1954 ). The radial velocity of the object with respect to the Sun is V r : or , where l is the Galactic longitude. Since the rotation curve provides V ( R ) for any R , , R can be found from this equation. The relation between distance, d , to the object from the Sun, in the direction of Galactic longitude, l , and the Galactocentric distance R is given by , which has the solution of . There is a unique solution for d if . For there are two possible distances. This is the kinematic distance ambiguity. For the purpose of this discussion we assume Galactic longitudes of 0°–90°, so that radial velocity is positive for objects inside the solar circle and negative for those

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O B-Model I-Model O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O


rows show the correlations and S/Ns after flipping the 185 MHz image about its vertical axis before the correlation calculation, giving an independent estimate of the noise. A significant correlation between 185 MHz and 4.5 μ m sources appears after masking infrared sources down to 1 mJy, which we show in Figure 5 cleanly removes the stars from the sample. The 185 MHz and 850 nm sources exhibit a marginal correlation after masking down to 1 mJy, though it does not appear significant in comparison to the noise correlation seen after flipping the radio image. We show in Figure 7 that this is due to increased stellar contamination at 850 nm compared to that at 4.5 μ m. The 185 MHz and 4.5 μ m catalogs exhibit a positive correlation peaking at 0.0332 ± 0.005 after masking infrared sources down to 10 −4 Jy (18.9 mag) and radio sources down to 1 Jy, and remain significant down to the completeness limits of these catalogs. There is no significant correlation detection after flipping the 185 MHz image, indicating that this detection is not an artifact of the analysis or of primary beam or vignetting effects. The 185 MHz and 850 nm catalogs exhibit a marginal correlation after masking infrared sources down to 10 −3 Jy (16.4 mag) and radio sources down to 0.3 Jy, though it does not appear significant in comparison to the level of correlation noise in the flipped image. To understand these findings, we begin by investigating which 4.5 μ m sources are responsible for this correlation. We select the subset of sources detected in the WISE 3.4 μ m, 4.5 μ m, and 12 μ m bands and plot them (Figure 5 , left panel) in the [4.6 μ m]–[12 μ m] versus [3.4 μ m]–[4.6 μ m] color–color space used by Wright et al. ( 2010 ) to illustrate the separation between different types of sources. Nikutta et al. ( 2014 ) study more quantitatively how sources separate in this space, finding that stars are isolated in the region , (1 σ ). In the right panel, we plot the faintest 90% of sources (fainter than 18.25 mag at 4.6 μ m) in the same color–color space and observe that this cut effectively cleanly excludes nearly all the stars. This explains the detection of a 185 MHz–4.5 μ m correlation only after masking the brightest 10% of sources. Figure 5. The ALLWISE sources are plotted in the color–color space of Wright et al. ( 2010 ) prior to any flux cuts (left panel), showing a cluster of stars near (0, 0). The color bar shows the number of sources in each cell. Then after cutting out the brightest 10% of sources (fainter than 18.25 AB mag), the stars are cleanly removed. We roughly split the remaining sources into AGNs () and SF galaxies (; Mingo et al. 2016 ), where [3.4 μ m]—[4.6 μ m]. To further probe which mid-infrared sources are responsible for this correlation, we make a rough cut to separate quasars and active galactic nuclei (AGNs; ) from star-forming (SF) galaxies (; Mingo et al. 2016 ). In Figure 6 we plot the power spectrum of 185 MHz sources (left panel), 4.5 μ m

O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O B-Wavelength O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O B-Telescope B-Wavelength I-Wavelength I-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength B-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength B-Wavelength I-Wavelength O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength B-Wavelength I-Wavelength O O O O O O O O O O O O B-Survey O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Wavelength I-Wavelength I-Wavelength B-Wavelength I-Wavelength O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Wavelength I-Wavelength O O O B-Wavelength I-Wavelength I-Wavelength


the outer and inner edges of the synthesized beams. When this ratio is greater than 1, the measured velocity becomes sub-Keplerian, and vice versa. Starting from the initial non-convolved image, this ratio can be perfectly known in the models and is shown in the bottom-center and bottom-right panels of Fig. 6 . As expected, the emission ratio I out ∕ I in is larger at 0.3′′ than at 0.1′′ and lead to stronger velocity biases. In real observations, the emission ratio can only be approached because the image received is already convolved by the observational beam. We show, however, in the bottom-left panel of Fig. 6 that the general tendency of the flux variations can be recovered, even if their magnitude is probably underestimated. In general, we find in our models that the flux ratio I out ∕ I in is proportional to the beam size, with ratios at 0.3′′ about three times larger than at 0.1′′, which is an expected result when the spatial variation of the line emission is relatively smooth compared to the synthesized beam size. Furthermore, biases are also sensitive to the velocity range Δ V probed by the synthesized beam, which is proportional to the beam size, and increases at smaller distances from the star dueto the steeper gradient of the Keplerian velocity. This leads to velocity artifacts generally larger at 0.3′′ compared to 0.1′′ and particularly important in the inner regions of the disk. Along the northern major axis, as shown in the top panel of Fig. 7 , the velocity deviations at 0.1′′ are about three times smaller thanat 0.3′′. Going to a higher spatial resolution will have then the double advantage of (i) reducing velocity biases and of (ii) allowing a better determination of the value and morphology of these biases, through a more precise knowledge of the disk structure. 4.3 Dust also affects the measured gas velocity The circumbinary disk around HD 142527 is not only made of gas but also contains a large concentration of dust particles at the north of the disk. The presence of this dust concentration, which is optically thick at 0.9 mm with τ Dust ~ 2–3 ( Muto et al. 2015 ; Boehler et al. 2017 ), has an important impact on the measured gas velocity at the north of the disk, as shown in the top panel of Fig. 7 . The four radial regions indicated in this figure are delimited by the radii at which the velocity shifts between a sub- and a super-Keplerian velocity at the spatial resolution of 0.3′′. Regions I and IV, which contain almost no millimeter dust emission, correspond to the two regions described previously in Sect. 4.2 . The 13 CO integrated emission uniformly increases for R ≤ 135 au and decreases for R ≥ 270 au, leading to the sub-Keplerian and to the super-Keplerian profiles. Fig. 7 Velocity deviations due to the presence of dust along the north of the major axis. Top : velocity deviations measured in the pure Keplerian model at two different spatial resolutions. Middle : 13 CO integrated emission and dust emission in

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


that we assume to be fully ionised and composed of hydrogen and helium with the equilibration timescales and where and stand for the electron and ion mass, respectively; is the ion number density with the proton mass and being the ion mean molecular weight; is the ion charge number; the electron charge; and is the Coulomb logarithm. Since , we neglect the term in the energy exchange rates. For a mixture of hydrogen and helium, the timescale of the temperature coupling between hydrogen and helium would be proportional to . Thus, the equilibration between hydrogen and helium operates times faster than the equilibration timescale between hydrogen and electrons or helium and electrons. For hydrogen or helium, the ratio of also simplifies to (not the case for heavier elements). Therefore, protons and HeIII can be considered as one single population sharing the same temperature. Writing , we see that The variation in energy is symmetric between the transfer of ion energy and electron energy, but this is not the case for the variation in temperature, which is . Therefore, the difference in temperature change between ions and electrons will be a factor , where is the “mean molecular weight of electrons” . For astrophysical plasmas essentially composed of hydrogen and helium, the change in temperature between ions and electrons is close to symmetric. To solve for the system of two non-linear coupled equations, we rewrite ( 16 ) as where . We introduce the vector , and rewrite the previous two equations as The goal is now to iterate on to find the value of and as required by equations ( 20 ). The derivatives and are obtained by differentiating equations ( 20 ) with respect to and , where the superscript replaces the superscript in ( 20 ). The values of are obtained with Cramer’s rule, and is added to the value of until the variation in . 3. Numerical tests If not indicated, we use a courant factor of with adaptive time-stepping between the different levels of refinement. Fig. 1 Top : diffusion of a pressure step function as a function of position at (black), (red), (blue) for the simulation (symbols) and the analytic solution (solid lines). The extra levels of refinement are indicated as dashed lines. Bottom : relative errors of the simulation with respect to the analytic solution. 3.1. Diffusion of a step function We performed a simple 1D test of the diffusion of a step function with a constant diffusion coefficient of without hydrodynamics. The initial setup is a pressure of between and outside with . We start with a minimum level of refinement of and refine up to level wherever the relative pressure variation is larger than . The time step is chosen to be (for level 7), where is the stability time step for diffusion (only useful for explicit schemes). The time evolution of the step function has the following analytical solution for a constant diffusion coefficient where . Figure 1 shows the result of the diffusion of the step function with the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


sources, while the HCO + line is either undetected or detected at a very weak level ( T mb 0.1 K) in disk-dominated Class II/Stage II sources. Another notable feature is an axisymmetric profile observed in the HCO + line, which may be indicative of an infalling envelope. Typically, a blue-dominated infall asymmetry has been seen in the HCO + profile of some low-mass protostars, while a red-dominated axisymmetry is usually associated with disk-only objects or the presence of an outflow (e.g., Thi et al. 2004 ; Evans et al. 1999 ; van Kempen et al. 2009 ). However, this is not always the case, since red-dominated asymmetry has been observed in embedded sources (D. Harsono et al. 2016, in preparation), an interpretation of which could be an expanding envelope that is due to an outflow. Among the bona fide detections in our present work, J182902.12, J182855.78, and J182956.67 have previously been categorized as Stage 0/I systems by Heiderman Evans ( 2015 ), based on the criteria of HCO + (3-2) integrated line intensity of ≥0.68 K km s −1 , while J182841.87 has been categorized as a Stage II system. We note that the Heiderman Evans ( 2015 ) criteria is solely based on a HCO + detection within 14′′ of the Spitzer c2d source position. As can be seen in Figure 6 , J182902.12 show a broad HCO + profile centered at the cloud velocity. The line model fit shows a typical Gaussian, which implies that there is no infalling envelope in this case. We have shown through SED modeling that J182902.12 is a Class Flat system and may possess a tenuous envelope. Using the “Stage” classification scheme also places J182902.12 in the Stage I-T category. Therefore, the Stage 0/I classification based on the HCO + detection is inconsistent with other schemes. For the case of J182855.78, results from line modeling indicate that the observed HCO + asymmetry can be explained by an infalling envelope model (Figure 6 ). The SED modeling results and the Stage classification also indicate that this is a genuine case of a Class 0/I or Stage 0/I system. For the case of J182956.67, the HCO + profile shows a hint of a blueshifted asymmetry and self-absorption at higher velocities (Figure 6 ). The results from line modeling for J182956.67 suggest that there may be blue-dominated infall asymmetry due to a weak envelope component with velocities between 0.5 and 1 km s −1 and a mass infall rate of 10 −7 M yr −1 . However, both the SED model fit and the Stage classification indicate that this is a Class II edge-on disk system or a Stage II source, rather than embedded Stage 0/I systems (Table 1 ; column 9). On the contrary, J182841.87 can be placed in the Stage I-T category and shows signatures of a tenuous envelope (Section 3.2 ), unlike the Stage II classification based on HCO + nondetection. The envelope mass in this system is too low to produce any detectable emission in a high-density tracer such as the HCO + (3-2) line. The submillimeter emission observed toward this source likely

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObject O B-CelestialObject O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Telescope B-Survey O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength O O O O O O


the intensity weighted first moment (i.e., the centroid) of the star. We measure the total flux using a 6 pixel (≈504) aperture radius. We weight pixels that lie partially within this circular aperture by the fraction of the pixel that lies within the ideal circular aperture. We also adopt an aperture correction to correct for the stellar flux that falls outside of our aperture. The typical size of an aperture correction is 0.04 magnitudes, but can vary from night to night depending on atmospheric conditions (predominantly seeing). In Table 2 , we provide the corrected photometry for LP 661-13 across all telescopes and both seasons of data. Table 2 MEarth Photometry for LP 661-13 (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. There is a marginal Roentgan Satellite ( ROSAT ) detection in the ROSAT faint source catalog (Voges et al. 2000 ). The potential source has a count rate of 3 ± 1 × 10 −2 counts per second. This corresponds to a flux density of approximately 1.70 ± 0.56 × 10 −13 erg cm −2 s −1 . While this may be interesting to assess the significance of the X-ray emission with stellar activity and radius inflation, further X-ray data to confirm this potential detection are needed. 3.3. Astrometry With two seasons of photometric data from the MEarth observatory, we are able to utilize the MEarth astrometric pipeline to measure the trigonometric distance to LP 661-13. The astrometric pipeline is described in detail in Dittmann et al. ( 2014 ), but we summarize it here. We eliminate nights for which the full width at half maximum for the image is greater than 5 pixels (approximately 3.5 arcseconds), or the average ellipticity of the target stars is greater than 0.5 (typically due to wind shake). We also eliminate images for which the pointing on the image is more than 15 pixels discrepant from the master MEarth image. We fit each image with a linear model in both the x and y coordinates, which allows for translation, rotation, shearing, and pixel scale variations. We also include a separate constant offset between data taken on opposite sides of the meridian, to accommodate the image flip that results from the telescopes’ German equatorial mount. We choose reference stars that lie within 600 pixels of the target star to avoid higher order effects at the edges of the CCD. We fit our astrometry iteratively, first fitting the linear plate model to each frame and then fitting for the proper motion and parallax of each star until convergence. Internal errors are estimated through a residual permutation. This provides the advantage that it preserves time-correlated noise in our error assessment but tends to be a weaker way to estimate errors for series with few data points. However, we have over 9000 data points and do not believe our astrometric error bars are underestimated. We perform

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O B-Survey O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope I-Telescope O B-Telescope O O O O B-Telescope O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O B-Observatory I-Observatory O O O O O O B-Software I-Software I-Software O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


takes place in TX Col (see Buckley Tuohy 1989 ; Norton et al. 1997 ; Wheatley 1999 ; Hellier 2002 ). Thus, a change in mass flow can be a reason for changing geometries between stream-fed dominance and disk-fed dominance. Further, since pure disk-fed accretion also takes place for some of the observational days, it implies that for those days, the contribution of mass accretion from the stream has been negligible in comparison to the disk. 2. They suggested that change in accretion mode is related to changes in the disk itself, which can be easily understood from our one-day analysis of TX Col. The sudden changes between the disk overflow to disk-fed or stream-fed cannot be only because of the change in the mass flow. The changes related to the disk, i.e., the change in the structure of the disk also takes place on such a timescale so that the combined effect of the two processes (i.e., the change in the mass accretion rate and the change in the structure of the disk) have shown such tremendous changes in the system. 4.2. Presence of QPOs We have also found periods of ∼5850–5950 s for days 46, 52, and 71, and these periods could be the QPOs. The flow of matter in IPs can take the form of diamagnetic “blobs” that orbit around the white dwarf (see King 1993 ; Wynn King 1995 for details). The accretion flow in TX Col can be explained as a combination of a stream and orbiting blobs (Hellier 2002 ). Keeping these ideas in mind, Mhlahlo et al. ( 2007 ) claimed that the QPO period results due to the beating of the Keplerian period of the orbiting “blobs” with the spin period. The theoretical calculation of QPO period demands the calculation of Keplerian period ( P kep = 2 π r d / v kep ), which we have calculated in the following way. The mass of the WD in the TX Col system was determined in the range of 0.48–0.82 M ⊙ (see Table 3 of Yuasa et al. 2010 ) in the different previous studies. Therefore, for our calculation, we have used the weighted mean of these mass values as 0.66 ± 0.03 M ⊙ , where the error on mass is the weighted standard deviation of different measurements. The WD radius, R WD , was calculated to be 0.012 ± 0.001 R ⊙ using the mass–radius relation of Nauenberg ( 1972 ). We have estimated the mass of the secondary as 0.61 ± 0.04 M ⊙ using the mass–period relation given by Smith Dhillon ( 1998 ). As a result, the mass ratio (q) was estimated to be 0.92 ± 0.07. The disk radius ( r d ) lies in the range of 2–3 times r r , where r r is the minimum outer radius of disk (Sulkanen et al. 1980 ). The estimation of r r involves binary separation (a), which was calculated using the relations as given in Warner ( 1995 ). The values of r r and “a” were found to be 1.08 × 10 10 cm and 12.17 × 10 10 cm, respectively. In this way, r d was estimated to be in the range of (2.2–3.2) × 10 10 cm. Using these parameters of TX Col, we obtained a range for Keplerian velocity () of 520–640 km s −1 . Further, we found that for the Keplerian velocity of ∼550

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


the sources. Prior information of such properties can be summarized in a joint density distribution P (). If the corresponding sub-hypothesis does not include a multi-messenger detection, then there may not be a requirement for a common source and the source parameters . In that case, if the detectors are independent from each other, the detection outcomes’ probabilities can be expanded as a product.where subscript i runs over different detectors and are the detection outcomes from the i th detector. Similarly, when there is a common source we can expand the detection outcomes’ probabilities for a fixed source as a product for different detectors.There we used the notation for representing a possible set of sources. There can be an additional level of complication related to combinatorics if the sub-hypothesis can be satisfied with different groupings of the detections. To illustrate this, consider the sub-hypothesis of having five detected particles, three of them from a source and all the rest being noise originated. In this case the sub-hypothesis can be satisfied with combinations. In such cases we expand the probabilities aswhere the sum is over all the combinations of detection outcomes satisfying the sub-hypothesis , the sets and are the detection outcomes of individual detections from different sources and is equal to the reciprocal of the total possible combinations for arising from detector i . For example, for a sub-hypothesis with a single source and w particles emitted from that source, and total of W detections; . are found by physics and empirical data. For memoryless detectors, the detections from different sources are independent soThe likelihoods are found via the detector characteristics and emission models. Now we look at our second term in Equation ( 3 ), . We transform it by using the Bayes’ rule.The denominator of Equation ( 8 ) cancel with the same term in Equation ( 2 ). P () is the joint density of source parameters being integrated over. The sub-hypothesis probabilities are found via the expected counts from the sources or the noise origin by assuming a source density and using the empirically known noise trigger rate. 4. Use Cases—Example: A Joint Gravitational Wave—High-energy Neutrino—Short Gamma-ray Burst Event Search The method for multi-messenger searches introduced above can be used in all scenarios. Specifically, in high-energy astrophysics, one can search for sources which emit more than one messenger. Those messengers can be in any wavelength in the electromagnetic spectrum, and can be neutrinos, cosmic rays, gravitational waves, or any other messenger. Joint emissions of gravitational waves, high-energy neutrinos and short gamma-rays from a binary neutron star or a neutron star black hole merger (Kimura et al. 2017 ; Berger 2014 ), or a binary black hole merger in a dense medium, such as an AGN disk, or surrounded by an accretion disk can be such examples (Ford et al. 2019a , 2019b ). In this section, we

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest O O B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O


for further studies of substructures and galaxy populations. The paper opens with a short description of the 15 constrained simulations used for the study proposed in this paper and of the unique Virgo candidate identified in each one. In a third section, the merger trees and in particular the main and second (second after the main) branches are studied in detail. Finally, we conclude that the constrained scheme proves to be efficient to some extent also at the merger tree level and provides some further indications regarding the mergers that formed the Virgo cluster within the last few gigayears. 2 Virgo halos 2.1 Constrained simulations Sorce et al. (2016b ) described in detail the scheme used to build the constrained initial conditions and to run the simulations. Furthermore the introduction of this paper summarises the various existing techniques. Thus, we summarise in this section only the main steps required to produce the simulations constrained with observational radial peculiar velocity catalogs we use here. We also give a brief description of their purpose with the latest references in the literature of the algorithms used: grouping (e.g., Tully 2015a , b ) of the radial peculiar velocity catalog to remove nonlinear virial motions that would affect the linear reconstruction obtained with the linear method (e.g., Sorce et al. 2017 ; Sorce Tempel 2017 ); minimizing the biases ( Sorce 2015 ) inherent to any observational radial peculiar velocity catalog; reconstructing the cosmic displacement field with the Wiener-Filter technique (linear minimum variance estimator, WF, Zaroubi et al. 1995 , 1999 ) applied to the peculiar velocity constraints; relocating constraints to the positions of their progenitors using the reverse Zel’dovich approximation and the reconstructed cosmic displacement field ( Doumler et al. 2013 ) and replacing noisy radial peculiar velocities by their WF 3D reconstructions ( Sorce et al. 2014 ). This ensures that, after evolving the structures with an N-body code from an early redshift until today, structures are at the same position (within the 2 h −1 Mpc limit of the linear-threshold) to that observed. Keeping constraints at their current position to build initial conditions would indeed result in a shift of ≥10 h −1 Mpc between observed and simulated structures after a complete evolution of the initial conditions until today; producing density fields constrained by the modified observational peculiar velocities combined with a random realization to restore statistically the missing structures using the constrained realization technique (CR, Hoffman Ribak 1991 , 1992 ); rescaling the density fields to build constrained initial conditions and increasing the resolution by adding small-scale features (e.g., GINNUNGAGAP code 1 ). Tully et al. (2013) supplied the observational catalog used as constraints. Since the Virgo cluster is the object of study here, Fig. 1 permits us to grasp the distribution of the constraints

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O B-Model I-Model O O O O B-Model B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Model B-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O