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

( 1963 ) with the origin located at the barycenter. There is an implicit perturbing force that induces the following accelerationwhich is implicit in the isotropic variation of the mass-losing body. This has been derived in several previous works (Hadjidemetriou 1963 , 1966a , 1966b ; Verhulst 1975 ). If the mass variation is anisotropic, i.e., either from nonconservative mass exchange and asymmetric ejecta, then the equation of motion will have an explicit perturbing force (e.g., Dosopoulou Kalogera 2016a , 2016b ). The orbital evolution can be described as an osculating orbit, which is the Keplerian orbit the binary would follow if the perturbation ceased immediately. The full set of equations for the time evolution of the osculating orbital elements from the isotropically varying mass are (e.g., Hadjidemetriou 1963 ) where the specific angular momentum is conserved, i.e., the quantity is a constant of the motion. Since there is no net force in the direction perpendicular to the orbital plane, , and the orbital motion thus remains coplanar. The mass variation causes the argument of periapsis to precess at a rate (see Equation 3 (c)). 3. Quadrupole Gravitational Radiation For eccentric Keplerian orbits, the orbital separation r between the binary masses obeysThe components of the mass quadrupole tensor I jk for an eccentric binary are where θ = ν + ω , which simplifies to θ = ν if the precession of the argument of periapsis is taken to be negligible. We can choose the initial θ to begin where ω = 0. For isotropic wind mass loss, the periapsis distance is constant, so . When including gravitational radiation reaction, the periapsis distance will indeed decrease, though one can take on timescales less than the mass-loss timescale when the GW coalescence timescale is much longer than the mass-loss timescale. The third time derivatives of the mass-quadrupole tensor components are If the periapsis precession is taken to be negligible, , then the first two terms of Equation 6 simplify to the canonical PM expressions. The additional terms in Equation 6 contain time derivatives of the masses of the binary, which may also be thought of as time derivatives of the binary’s chirp mass . Gravitational radiation carries away the orbital energy and angular momentum of the binary at the following rates where c is the speed of light and is the reduced (trace-free) quadrupole moment tensor . If we take the orbit to be circular, then the GW luminosity simplifies towhere the positive terms in Equation ( 8 ) correspond to GW energy loss from the orbit and the negative terms correspond to GW energy gain in the orbit. From the strain tensor in the transverse-traceless gauge,where is the luminosity distance to the source, the strain amplitude for a circular binary iswhere Ω is the angular frequency of the binary, . In the transverse-traceless gauge, the plus and cross polarizations of the strain are and . From Equation ( 10 ), we can define the following parameters


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

concept beyond one dimension (1D) or to unstructured/meshfree methods is less clear (e.g. Chiapolino, Saurel Nkonga 2017 ). A physical interpretation can be seen from our equation ( 6 ). For the drag term to provide a positive definite contribution to the entropy, and must have the same sign, such that d u /d t | drag is positive. Pairwise positivity is not strictly necessary so long as the sum over all neighbours is positive. We tried setting if the signs differ, but found this to be more dissipative than using slope limiters (see Fig. 3 ). We found the van Leer MC limiter to provide the best compromise between monotonicity and dissipation. 3 RESULTS We test our improved algorithm in 1D using the ndspmhd code (Price 2012 ) and in 3D using phantom (Price et al. 2018 ). We use explicit global time-stepping with a leapfrog integrator, the M 6 quintic kernel for the SPH terms with the double hump M 6 employed for the drag terms ( LP12a ). The results are not sensitive to the choice of kernel, provided a double hump kernel is used for the drag. The time-step was set to 0.9 times the minimum stopping time (we found that setting Δ t = t s exactly as in LP12a could result in instability with reconstruction). We use the van Leer MC limiter unless otherwise specified. 3.1 DUSTYWAVE Fig. 1 shows the results of the dustywave described in Laibe Price ( 2011 ), performed using 2 × n x particles with a fixed drag coefficient K = 1000, ρ g = ρ d = 1 and c s = 1 (giving t s = 5 × 10 −4 ) and a perturbation amplitude of 10 −6 . We use an adiabatic equation of state P = ( γ − 1) ρu with γ = 5/3 in the gas. In the absence of reconstruction, overdamping occurs when h ≳ c s t s , i.e. for n x ≲ 1024 (left-hand column), as found by LP12a . Adding reconstruction captures the true solution to within a few per cent for n x ≳ 64 (middle column), while the slope limiter does not visibly degrade it (right-hand column). Figure 1. Dust and gas velocities in the dustywave test after 10 wave periods, using K = 1000 with 2 × n x particles without reconstruction and with and without the slope limiter (see labels). Reconstruction avoids the need to resolve h ∼ t s c s (resolved at n x = 1024 particles). Exact solution shown in red. Fig. 2 shows the results in 3D using phantom . We follow the procedure used in Price et al. ( 2018 ), placing the particles using dense sphere packing and cropping the grid in the y and z directions at 12 particle spacings (for efficiency), giving 2 × 128 × 12 × 12 particles. The results in 3D are indistinguishable from those shown in Fig. 1 , showing our method also works in 3D. Figure 2. As in Fig. 1 , but in 3D with phantom using n x × 12 × 12 gas particles (solid) and n x × 12 × 12 dust particles (open) initially placed using dense sphere packing. Exact solution from Laibe Price ( 2011 ) shown in red. 3.1.1 Choice of slope limiter Fig. 3 shows the kinetic energy as a function of time in the 1D dustywave problem at a resolution of n x = 128. The


O O O O O O O O O O B-Grant I-Grant I-Grant I-Grant O O O O O O O O O O O O B-Organization I-Organization I-Organization I-Organization O O O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Grant

Acknowledgements This article was supported by the realization of the Project ITMS No. 26220120029, based on the supporting operational Research and development program financed by the European Regional Development Fund. The work was also supported in part by the Slovak Grant Agency for Science, VEGA, grant No. 2/0037/18, and by the Slovak Research and Development Agency under contract No. APVV-16-0148.


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Model I-Model I-Model I-Model B-Model O O O O O O O O B-Model B-Model O O O O O O O O O O O O O O O O O O O O O B-Model 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 B-Model O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Model O O O O O B-Citation I-Citation O O O O O O O O B-Model 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 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 B-Citation I-Citation O B-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation B-Person O B-Person O O O O O O O B-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 B-Citation I-Citation O O O O O O O O O O O O O O O O O O

et al. 2018 ). A common assumption involves the population of states, which, in the absence of knowledge to the contrary is assumed, for want of other choices , to be Maxwellian with density and temperature as parameters. This type of inversion is used to produce impressions of the temperature in plasmas far removed from the observer. A rather general argument is presented here that this mode of inverting photon information has a systematic bias by presuming the Maxwellian eVDF of the LTE when inferring the population of states involved in the line’s formation. A suggested alternative is presented based on this paper’s Steady Electron Runaway Model (SERM). Modeling astrophysical plasmas often involves solving the fluid-scale magnetohydrodynamical (MHD) equations. Within the derivation of the MHD fluid equations from the kinetic equations and their closure are further hidden assumptions of LTE (Hazeltine Walbroeck 2004 ) that are generally unwarranted on the same large scales of the modeling. These hidden assumptions also preempt consideration of non-LTE types of physics from occurring in such models (Scudder 1992 ; Meyer-Vernet et al. 1995 ). Barring improvements that do not make these small Knudsen number LTE approximations, this compromised modeling continues (Scudder 2019a ). To move beyond these (hidden) reliances on LTE requires the formulation of an alternative that challenges the likelihood that the Maxwellian eVDF is the most likely one to occur in the regions under study. To be sure, considering this possibility faces considerable resistance best redressed by studying plasmas with in situ experimental access, such as the solar wind as presented here. There finite is clearly common, and this paper argues that this parameter (or its dimensionless parallel electric field cousin) is the central index of the importance of non-Maxwellian eVDF’s that is explored in this paper. The present calculation and its comparison with solar wind measurements furthers the description of transport in plasmas with finite Knudsen number, , quantitatively defined bywhere is the local magnetic field-aligned unit vector. A fully defined Knudsen number associated with χ depends on the mean free path for scattering, , and the magnetic field-aligned scale, L , of χ . The pressure profile usually has the steepest gradient, determining the shortest transport scale L ; it thus determines the relevant for the mean free path over scale that enters as a small perturbative parameter in Chapman–Enskog closures (Chapman 1916 ; Enskog 1917 ), which are also discussed in Chapman Cowling ( 1939 ), Rossi Olbert ( 1970 ), Fitzpatrick ( 2004 ), and Zank ( 2014 ). Spitzer and Braginskii use slightly different versions of these expansions (Spitzer Harm 1953 ; Braginskii 1965 ). Between 0.3 and 5 au the observed solar wind plasma electrons possess . At 1 au (Scudder 2019b ). In all speed states of the wind “ubiquitously nonthermal” eVDF have been observed over the last 50


O B-Person I-Person O B-Person I-Person O O O O B-Person I-Person 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 B-Person I-Person O O O O B-Organization I-Organization I-Organization B-Organization B-Grant I-Grant I-Grant O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Person I-Person B-Person I-Person O B-Person I-Person O O O O B-Organization O B-Grant I-Grant I-Grant I-Grant B-Person I-Person O O O O B-Organization I-Organization I-Organization I-Organization B-Person I-Person O O O O B-Organization O B-Grant I-Grant I-Grant B-Person I-Person O O O O B-Grant I-Grant I-Grant B-Grant I-Grant I-Grant O O B-Organization I-Organization B-Person I-Person O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Organization O B-Grant I-Grant I-Grant B-Person I-Person O O O O B-Organization I-Organization O O O O O B-Grant I-Grant I-Grant I-Grant O O O O O O O O O O O B-Observatory O O O B-Observatory I-Observatory I-Observatory I-Observatory O B-Proposal I-Proposal I-Proposal B-Proposal O B-Proposal O O O O O O B-Survey I-Survey B-Organization B-Organization O O B-Survey O O O O O O B-Collaboration I-Collaboration B-Survey O O B-Survey O O O O O B-Organization O B-Organization O O O B-Organization O B-Organization O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Location B-Survey O O O B-Location O O O O O O O O B-Telescope I-Telescope O B-Dataset I-Dataset I-Dataset O O O O O O O B-Survey I-Survey I-Survey I-Survey O O B-Survey I-Survey I-Survey I-Survey I-Survey O O O O B-CelestialRegion I-CelestialRegion O O O O O O O O O O O B-Telescope B-Survey B-Telescope B-Survey B-Telescope B-Survey B-Telescope O B-Telescope B-Survey B-Telescope O B-Telescope O B-Wavelength O B-Wavelength O B-Survey O O O O B-Organization B-Location O B-Organization B-Location O B-Observatory O O O O O B-Survey O O B-URL O O O O O O O B-Software O B-URL O O O O B-Software O B-URL O B-Software O B-URL O O B-Software B-Citation I-Citation I-Citation O O O O O O B-Software 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 B-Person I-Person O B-Person I-Person O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

Acknowledgments M. Brouwer and M. Visser would like to thank Erik Verlinde for helpful clarifications and discussions regarding his EG theory. We also thank the anonymous referee for the useful comments that helped to improve this paper. The work of M. Visser was supported by the European Research Council (ERC) Advanced Grant 268088-EMERGRAV, and is part of the Delta Institute for Theoretical Physics (ITP) consortium, a program of the Netherlands Organisation for Scientific Research (NWO). M. Bilicki, H. Hoekstra and C. Sifon acknowledge support from the ERC under FP7 grant number 279396. K. Kuijken is supported by the Alexander von Humboldt Foundation. M. Bilicki acknowledges support from the NWO through grant number 614.001.103. H. Hildebrandt is supported by an Emmy Noether grant (No. Hi 1495/2-1) of the Deutsche Forschungsgemeinschaft. R. Nakajima acknowledges support from the German Federal Ministry for Economic Affairs and Energy (BMWi) provided via DLR under project no. 50QE1103. Dominik Klaes is supported by the Deutsche Forschungsgemeinschaft in the framework of the TR33 The Dark Universe. This research is based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017 and 177.A-3018, and on data products produced by Target OmegaCEN, INAF-OACN, INAF-OAPD and the KiDS production team, on behalf of the KiDS consortium. OmegaCEN and the KiDS production team acknowledge support by NOVA and NWO-M grants. Members of INAF-OAPD and INAF-OACN also acknowledge the support from the Department of Physics and Astronomy of the University of Padova, and of the Department of Physics of Univ. Federico II (Naples). GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KiDS, VISTA VIKING, WISE , Herschel -ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO and the participating institutions. The GAMA website is www.gama-survey.org . This work has made use of python ( www.python.org ), including the packages numpy ( www.numpy.org ), scipy ( www.scipy.org ) and ipython (Prez Granger 2007 ). Plots have been produced with matplotlib (Hunter 2007 ). All authors contributed to the development and writing of this paper. The authorship list is given in three groups: the lead authors (M. Brouwer and M. Visser), followed by two alphabetical groups. The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products. The second group covers those who have either made a significant contribution to the data products or to the scientific analysis.


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-Instrument 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 B-Wavelength I-Wavelength 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 B-Wavelength 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 O O O O O B-Instrument B-Wavelength I-Wavelength O O O O O O O B-Instrument B-Wavelength I-Wavelength 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 B-Wavelength 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Wavelength O B-Wavelength O B-Wavelength O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Telescope O O O O O O O B-Telescope O O O O O B-Wavelength O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject 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

could be adjusted, but should not be orders of magnitude lower or the jet would not be observed in the EUV given the TRACE 195 Å response and reasonable temperatures. For the closed loop, the thermal density is taken to be cm −3 . Values larger than this would violate three observational constraints: (1) 195 Å emission would be stronger than observed for all reasonable coronal (including flaring) temperatures; (2) SXR emission would be stronger than is observed; and (3) the microwave spectrum would be dominated by a free–free component, which would not match the observed microwave spectral shape. The density could be smaller but would require adjustment of the loop geometry; therefore, there is a range of thermal densities that could fit the observed data. For the likely value of 1 × 10 10 cm −3 for the jet density, a temperature of 2.05 MK fits the measured TRACE 195 Å emission from the jet location. As the TRACE 195 Å temperature response function (Handy et al. 1999 ) decreases for all higher temperatures, a higher density would be required for any jet temperatures higher than this (and would violate the density constraint from microwave data mentioned earlier). For the closed loop, a density of cm −3 can produce the observed 195 Å brightness at the jet base, where this loop is located, for a temperature between 7.3 and 10.6 MK. Since the plasma temperature is not constrained within this range, we choose a temperature of 7.3 MK. For the nonthermal electrons in the jet, the microwave spectrum deviates from observations if the upper cutoff to the electron energy E max is below ∼300 keV; it is not well constrained from above. Further decrease of this parameter could be compensated by an increase of the source area, but this would be in conflict with the microwave source size. For the closed loop, the microwave spectrum deviates from observations if . In the model we use . 4. Discussion To recap the observations, we find high time variability in HXRs and microwaves, with HXR spectral hardening strongly correlated with fast bursts of intense flux. HXR images are dominated by the flaring (thermal) loops at low energies and (nonthermal) footpoints at high energies, but at intermediate energies (∼18–30 keV), some RHESSI emission is elongated along the jet, where OVSA data also reveal an elongated microwave source at the location of the jet. We interpret the fast HXR spikes as bursts of particle acceleration, and the duration of these peaks gives the timescale over which electrons are accelerated. The very short time lag between the burst intensity and its hardening (less than 200 ms in the case of G1 and no lag in the case of G2) implies that electrons are accelerated quickly to ∼100 keV. Combining the time lag from Figure 4 with the peak width from Figure 5 , we conclude that the event exhibits electron energy increases on an average timescale of 0.2 s, with acceleration episodes lasting, on average, 1 s. Once accelerated, the electrons that have


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

has delayed or is preventing prodigious star formation in the Brick? Dynamical arguments, such as its highly turbulent nature (Kauffmann, Pillai Zhang 2013 ; Johnston et al. 2014 ; Rathborne et al. 2015 ) or its apparent location within a coherent ring-like stream of molecular clouds in orbit about Sgr A* (Molinari et al. 2011 ; Longmore et al. 2013 ; Kruijssen et al. 2015 ) have been cited as possible agents of star formation suppression. In order to set more stringent constraints on the level of star formation activity, it would seem worthwhile to examine other possible indicators, such as the presence of localized peaks in dust temperature that might signal the presence of protostars at early stages of evolution. With this in mind, we present here the results of a detailed study of the variation of dust temperature within the cloud. Studies of the temperature structure of the Brick show that the gas temperature is uniformly higher than the dust temperature and that those two components are heated by different mechanisms. The observed dust temperatures of T D ∼ 19–27 K (Longmore et al. 2012 ) are consistent with heating by the local interstellar radiation field (ISRF), assuming that the latter is ∼1000 times stronger than that in the solar vicinity (Clark et al. 2013 ). However, the gas (Güsten, Walmsley Pauls 1981 ; Mills Morris 2013 , T G ∼ 60–100 K) is most likely heated by a combination of cosmic rays (Clark et al. 2013 ) and turbulent compression (Ginsburg et al. 2016 ), with the former dominating at high densities. Our study of the dust temperature distribution makes use of the PPMAP procedure (Marsh, Whitworth Lomax 2015 ), whose purpose is to generate resolution-enhanced image cubes of differential column density as a function of dust temperature, T D , and sky position ( x , y ), given sets of observed images of dust continuum emission at multiple wavelengths. Differential column density, in this case, is defined as the column density of material (gas plus dust) per unit interval of dust temperature, and is expressed in units of hydrogen molecules per square centimetre per degree Kelvin. PPMAP is a Bayesian technique whose output is the expectation value of the differential column density in each cell of an image cube whose axes are ( x , y , T D ), given the input data. The a priori probability of cell occupancy is controlled by a ‘dilution’ parameter, η, such that the smaller the value of η, the more the algorithm tries to fit the data with as few source components as possible. The exact value of η is not critical, but the most appropriate value is one which results in a reduced chi-squared value of the order of unity, indicating that the data have been fit within the error bars using a minimum number of source components. The procedure also yields a corresponding image cube of uncertainty values. Errors are Poisson-like, and increase in regions of high total column density. A key underlying assumption is that the dust is


O O O O O O O B-Software O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-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-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula 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 O O O O B-Survey 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 B-CelestialObject I-CelestialObject 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 B-Survey 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 B-CelestialObject I-CelestialObject 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 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-CelestialRegion I-CelestialRegion O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject I-CelestialObject 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 B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-CelestialObject 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

elemental-abundance ratios ([X/Fe]) were derived with the BACCHUS code. Table 1 Atmospheric Parameters and Elemental Abundances of Stars in VVV CL001 Note. The errors were determined in the same manner as described in Fernández-Trincado et al. ( 2020a ) by varying the atmospheric parameters one at a time by the typical, albeit conservative, values of Δ T eff = ±100 K, g = ±100 cgs, and Δ ξ t = ±0.05 km s −1 . Thus, the reported uncertainties are defined as . Figure 2 shows the good quality of the APOGEE-2 spectrum compared to the best-fit synthetic spectra for selected atomic and molecule lines of the two stars in VVV CL001, from which the chemical-abundance ratios were determined. For each spectrum we were able to identify reliable lines for six chemical species (N, O, Mg, Al, Si, and Fe). The same figure also reveals the scarcely detectable Fe i lines. The resulting elemental-abundance ratios are listed in Table 1 . Figure 2. Detection of atomic and molecular lines. Spectral synthesis is shown for the determination of the [N/Fe], [O/Fe], [Mg/Fe], [Al/Fe], [Si/Fe], and [Fe/H] abundances for two stars in the innermost regions of VVV CL001. Each panel shows the best-fit syntheses (red line) from BACCHUS compared to the observed spectra (black squares) of selected lines (black arrows). The resulting chemical abundances are also displayed in Figure 3 , and compared to four GCs at similar metallicity taken from the APOGEE-2 GC sample of Mészáros et al. ( 2020 ). We find that the two program stars exhibit an iron abundance ratio of −2.47 to −2.44, suggesting that VVV CL001 has a mean metallicity [Fe/H] = −2.45, with an uncertainty due to systematics of 0.24 dex, which makes this cluster possibly the most metal-poor GC identified so far within the Sun’s Galactocentric distance, and with an extreme metallicity close to the apparent “floor” in the empirical metallicity distribution function for GCs in the MW and Local Universe (e.g., Geisler et al. 1995 ; Simpson 2018 ; Kruijssen 2019 ; Larsen et al. 2020 ; Wan et al. 2020 ). Note that VVV CL001 has the lowest Fe abundance among this sample, which includes the lowest metallicity GCs observed by APOGEE. With our limited sample, we do not find evidence of an intrinsic Fe-abundance spread. Figure 3. Elemental abundance of stars in VVV CL001 (black symbols) and four comparison metal-poor GCs. Each violin representation show the univariate kernel density estimate of the abundance ratios of each cluster as determined from Mészáros et al. ( 2020 ). Each GC has been slightly offset horizontally to distinguish them. Regarding the α -elements (O, Mg, and Si), VVV CL001 displays a modest α -element enhancement, a clear signature of the fast enrichment provided by SNe II events, and compatible with other Galactic metal-poor GCs at similar metallicity (see Figure 3 ). Furthermore, no Mg–Al anticorrelation is evident in our small sample; the two VVV CL001 stars exhibit an aluminum deficit, which places them within


O O O O B-Person I-Person O O O B-Dataset I-Dataset I-Dataset O O O O B-Person I-Person O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O B-Location I-Location O O O O O O O O O O O O O O O O B-Organization O O O O O O O O O O O O O O O B-Organization I-Organization I-Organization O O O B-Grant

We are grateful to Peter Behroozi, for updating the Bolshoi halo catalogue upon our request, to Hao-Yi Wu for sharing data from the Rhapsody simulation project, and to the anonymous referee for an insightful report that greatly helped to improve the presentation. We also thank the organizers of the programme ‘First Galaxies and Faint Dwarfs: Clues to the Small Scale Structure of Cold Dark Matter’ held at the Kavli Institute for Theoretical Physics (KITP) in Santa Barbara, for creating a stimulating, interactive environment that started the research presented in this paper, and the KITP staff for all their help and hospitality. This research was supported in part by the National Science Foundation under Grant No. PHY11-25915.


O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization B-Person O O O O B-Organization I-Organization I-Organization B-Organization O O B-Location I-Location B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O O O B-Grant

The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. T.B. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714769.


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

can be constrained to be From this equation, we can satisfy the observed mass scale () using reasonable θ and ε values. Although the individual parameters are still uncertain, the calculation shows that the gravitational contraction could release sufficient kinetic energy to account for the observed gas motion. Figure 4. A zoom-in of the PV plot along the three filaments. (a) The systemic velocity map showing the PV cuts. (b)–(d) The PV plot along the three directions. For each filament, the offset is measured from the MMS-7 centre. The solid line represents the v ( r ) function that closely goes through the spine of the PV emission region. The v ( r ) function is calculated from equation ( 5 ). The two dashed lines represent the v ( r ) curves with c 0 varying by a factor of 0.5 and 2, respectively. In (b) and (d), the red solid line denotes the central velocity variations along the PV sampling direction. 4.3 Velocity variations Despite the agreement in energy budget, the velocity distribution still has two features deviating from the modelled v ( r ) curves that should be further examined. First, the theoretical v ( r ) function will approach infinity towards the centre, while the observed velocity profiles are terminated at final velocities around the centre. For the NW filament (Fig. 4b ), the velocity profile reaches the maximum value in the area of r 4 arcsec around the centre. This area is comparable to the size of MMS-7 core ( r = 3.7 arcsec). As a reasonable explanation, this energy release of could be largely transferred to MMS-7, accelerating this core as an entire body to the final velocity. Second, although the three filaments have a similar velocity increase towards the centre, their final velocities are different. This could be attributed to IRAC-2405 and 2407. As shown in Fig. 4(d) , the north filament becomes weaker at IRAC-2407, but still connected to MMS-7, and the gas structure follows the v ( r ) curve well. This suggests that the gas motion is still towards the centre and would reach the same velocity with MMS-7. The south filament, according to its velocity gradient, could have a similar connection to MMS-7 except for a more significant interruption by IRAC-2406 (Core-2). The two YSOs could have dissipated the surrounding N 2 H + component but not significantly affected the global inward motion towards MMS-7. For the NW filament, its outer part ( r > 20 arcsec) could also be affected by the local dense cores. As seen in Fig. 4(b) , the emission region largely deviates from the v ( r ) curve in the spatial range between Core-3 and Core-4 with a value of Δ v ≃ 1.0 km s −1 , suggesting that the gas motion could be affected by these two cores. Although the filaments exhibit these systematic velocity variations, the N 2 H + (1 0 −0 1 ) line profile is usually single-peaked and has a relatively small linewidth of Δ v 0.65 km s −1 over the whole region. This suggests that the filaments should have no further internal


O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 B-Wavelength O O O O O O O O B-Wavelength O B-Wavelength 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 O O O O O O O O O O O O O O O O O O O

CO , CO and [OI] 63 m, respectively. However, the CO mm-line fluxes do not fully converge to the results obtained by the time-independent reference model, see arrows on the r.h.s. of Fig. C.1 , even for integration times as long as 10 Myr. This is consistent with the very long chemical relaxation timescales we find for those lines, namely 20 Myr, 50 Myr and 100 Myr in the line forming regions of CO , CO , and CO , respectively. The difference of the CO concentration between the Myr disk model and the time-independent reference model is shown in Fig. C.2 . All upper disk regions have practically indistinguishable CO concentrations, however, in the dark icy midplane (vertical visual extinction ), CO cannot freeze out directly where the dust temperature exceeds about 20 K (at radii AU in this model). Instead, the CO is very slowly dissociated in these regions by reactions with He created by cosmic rays. Most of the liberated oxygen atoms re-form CO, but a small part can form other molecules with higher adsorptions energies like water, and these will freeze out immediately. This mechanism provides a slowly ticking chemical clock in the dark midplane regions of protoplanetary disks, leading to gaseous carbon-to-oxygen ratios on Myr timescales between the water and the CO ice-lines, and to on even longer timescales, see ( Öberg et al. 2011 ; Helling et al. 2014 ). The (sub-)mm CO isotopologue line fluxes are somewhat affected by these differences, for example remainders of cold gaseous CO in the dark icy midplane can partly re-absorb the CO line photons emitted by the warm surface of the disk on the opposite side which cross the midplane. In this way, the CO isotopologue line fluxes of the time-independent model can be up to 30% larger than those of the time-dependent models. However, these effects are small compared to the impact of disk mass, size and shape, dust size distribution and settling, and the dust/gas ratio, and negligible for most optical and near-IR to far-IR lines. We conclude that the emission lines of chemically robust gas tracers, such as CO, H and the oxygen atom, are little affected by time-dependent chemical effects in our models. Disk evolution will rather have an impact on those lines via the changing stellar parameters, the changing shape of the disk, and physical gas and dust evolution. See e.g. Akimkin et al. (2013) for time-dependent disk models which include those effects. Appendix D: Properties derived from the model The mean gas temperature in the disk , the mean dust temperature , the near-IR excess, the m SED amplitude, the dust absorption mm and cm-slopes , and the millimetre and centimetre flux-slopes , are calculated as where is the gas mass density, the dust mass density, the local dust-to-gas mass ratio, the volume element, the distance, the spectral flux per wavelength interval, the flux from the naked star and the dust absorption opacity . Appendix E: Fluxes of optically thick emission lines It is noteworthy


B-Person O B-Person O O O O O O B-Location O O O B-Grant I-Grant I-Grant I-Grant I-Grant O O O O O O O O O O B-Person I-Person B-Person I-Person O B-Person I-Person O O O O O O O B-Person I-Person O O O O O O

G.L. and Ch.H. highlight the financial support of the European community under the FP7 ERC starting grant 257431. Our respective local computer support is highly acknowledged. We thank P. Woitke, C.R. Stark and P. Rimmer for helpful discussions and feedback. We thank L. Neary for insightful discussion on atmospheric mixing.


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

(in agreement with expectations about Gaia for this separation). But they do have very similar parallaxes (the average is 33.82 mas and the difference 0.04 mas) and radial velocities (Simbad gives − 4.6 ± 2.0km s −1 for HD 224636), so they are almost certainly a physical binary. If they were in a common plane perpendicular to the line-of-sight, their physical separation would be 73 au and their relative velocity 4.8 km s −1 . This compares to a relative orbital velocity of two solar-mass stars in a mutual circular orbit of this radius of 4.9 km s −1 (and their period would be 440 yr). As their proper motion difference of 35 mas yr −1 is about a quarter of their absolute proper motions, the velocity of their center-of-mass could differ significantly from that of either component. If we knew the masses of both components of the binary, we could solve for both the center-of-mass and the Keplerian orbital elements (6+6 parameters from the 3D position and 3D velocity for each star). The spectral type of this pair is given by Struve Franklin (1955) as F8+G1, with F8 presumably referring to GJ 4384 because this is slightly brighter and bluer (by 0.3 mag and 0.05 mag respectively in Gaia DR2). This agrees with their T eff and positions in the color-absolute magnitude diagram, as given by Gaia DR2. We adopt a mass of 1 M ⊙ for each component to compute the position and velocity of the center-of-mass. Integrating the center-of-mass’s motion back in time, we find the time, distance, and relative speed of its closest encounter with 2I/Borisov to be −1380 kyr, 3.64 pc, and 20.9 km s −1 respectively. This is considerably more distant than the encounter found when neglecting the binary motion (the second line of Table 2 ). We do not compute the covariance on these parameters due to an inadequately characterized correlation in the Gaia astrometric uncertainties for sources close to one another. The largest uncertainty is the radial velocity of s528. If we change this by ± 2 σ , the encounter distance remains between 3.3 and 4.0 pc. We conclude that the binary motion renders the GJ 4384–HD 224636 system a much more distant encounter than is shown in Table 2 . Although a binary system is an attractive option for ejecting 2I/Borisov, this encounter distance is far larger than the size of the system. We note that GJ 4384 is listed as the second closest encounter to 2I/Borisov as found by Hallatt Wiegert (2019) with similar encounter parameters to ours ( = 0.32 pc) when we neglect binarity. 3.3 Other candidates We note that several of our other close encounters are binary systems. One of these, HD 22781, has a planetary companion ( Díaz et al. 2012 ), but the encounter velocity of 36 km s −1 is rather high for this planet to be plausibly involved in ejecting 2I/Borisov. It is worth examining in addition those encounters beyond 1 pc (not listed in Table 2 ) that are particularly slow. The slowest encounter we find within 5 pc is with Gaia DR2


O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Wavelength O O O O B-Survey I-Survey I-Survey I-Survey I-Survey I-Survey I-Survey I-Survey I-Survey B-Survey B-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Observatory O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O B-Survey O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Wavelength B-ObservationalTechniques O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O B-ObservationalTechniques O B-ObservationalTechniques I-ObservationalTechniques O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Observatory B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-ObservationalTechniques I-ObservationalTechniques O O O O O O O B-Survey O B-Wavelength I-Wavelength 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-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O O O O O O 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 B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O B-Wavelength 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

the normalized results from Chapman et al. ( 2005 ; black, dashed), Wardlow et al. ( 2011 ; blue, solid), and Danielson et al. ( 2017 ; green, dotted). For the sample with robust and tentative counterpart identifications, the median redshifts are and , respectively. These results are in reasonable agreement with those from Chapman et al. ( 2005 ; median z = 2.2 ± 0.1 for a radio pre-selected sample); from the LABOCA Extended Chandra Deep Field South (ECDFS) submillimeter survey (LESS; Wardlow et al. 2011 ; median z = 2.2 ± 0.1); and also with the more recent ALMA surveys of Danielson et al. ( 2017 ; median z = 2.4 ± 0.1) and Brisbin et al. ( 2017 ; median z = 2.48 ± 0.05). In our photometric redshift distribution, there is a comparatively larger fraction of sources at z > 3, and the high-redshift tail extends farther out than that of the LESS distribution including one SMG with z phot ∼ 4.9. The existence of this tail is consistent with the millimeter spectroscopic confirmation of a growing number of SMGs at (e.g., Capak et al. 2008 ; Smolčić et al. 2012 ; Weiß et al. 2013 ; Smolčić et al. 2015 ; Strandet et al. 2016 ), and with the spectroscopic and photometric redshift distributions obtained by Danielson et al. ( 2017 ) and Brisbin et al. ( 2017 ) for samples of SMGs observed at the spatial resolution of ALMA. Su et al. ( 2017 ) also estimate photometric redshifts for nine strongly lensed DSFGs selected by ACT at 218 GHz (1.4 mm) and obtain a higher median redshift , which is expected since the highest-redshift sources remain bright at this lower frequency due to the negative K -correction. Our photometric redshift estimates support the existence of a SMG population, but it is unclear whether these sources’ number counts are consistent with the predictions of cosmological models. For example, semianalytic models that assume a top-heavy IMF and star formation driven by merger-triggered starbursts (Baugh et al. 2005 ) predict a redshift distribution centered at z ∼ 2, with very few sources at much higher redshifts. As a reference, the expected redshift distribution from Baugh et al. ( 2005 ) predicts that only ∼5% of SMGs with S 850 = 8 mJy ( S 870 ∼ 7.5 mJy) lie at z > 4, so for our subset of 25 SMGs with robust counterpart identifications, we would expect only one to two objects in this redshift range, in contrast to the ∼5 that we estimate (Table 5 ). Additional data are critical to confirm the existence and abundance of very high- z SMGs in our sample and confirm or refute our preliminary results; in particular, high-resolution submillimeter imaging is key to confirm counterpart identification for SMGs that are undetected at radio wavelengths. 6.3. Gravitational Magnification The detection of SMGs in LASCAR may be facilitated by the foreground galaxy clusters acting as gravitational lenses, but the resulting flux magnifications also affect the analysis of the intrinsic counts distribution and enhance the scatter in the contamination of


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

, z BOSS magnitudes. We have renormalized Maraston etal. ( 2013 ) stellar masses to a Chabrier IMF, by subtracting 0.05dex, and converted to our cosmology by subtracting 0.035dex, too. On average, our stellar masses are 0.08dex larger, this discrepancy is related to differences in data sample and template prescription adopted. However, the difference is also stronger at z 0.5, i.e. 0.12dex, and is inverted at 0.6 z 0.7, where our masses get smaller than BOSS estimates of 0.06dex. The trends of M dyn / M stay qualitatively the same. Thus, the results of this paper do not change if different stellar mass estimates are adopted, and conclusions are unaffected. Stellar mass uncertainties. The median uncertainty of the Chabrier IMF stellar masses determined in Section 2.3 in our massive galaxy sample is 0.13dex. We studied the impact of these uncertainties on our mass selection and all the trends discussed in this paper. We have randomly perturbed each mass, assuming a Gaussian with as mean the best-fitting mass and standard deviation the error of the best-fitting mass. Thus, we have selected the most massive galaxies with log M /M 11.2, using the new perturbed masses. The impact on the M dyn / M z trend is negligible. Stellar M / L gradients. We already shown how the gradients impact the results by comparing the g - and K -band results for the SPIDER sample in Section 3 . However, in general, at masses log M /M 11.2 the colour and stellar M / L gradients are weak, thus the correction would be small (Tortora etal. 2011 ). Later analysis would also take into account possible gradients in the IMF (Martn-Navarro etal. 2015 ). Mass profile slope. In addition to uncertainties in stellar mass estimates, the choice of the mass profile can be inappropriate for lower e (or M ) galaxies and cause an excess of negative M dyn / M values, mainly at high- z (e.g. Sonnenfeld etal. 2013 ). In particular, M dyn / M and the slope of total mass density are tightly correlated, with shallower density profiles corresponding to larger M dyn / M (Humphrey Buote 2010 ; Remus etal. 2013 ; Dutton Treu 2014 ; Tortora etal. 2014b ). To quantify the impact of a free-varying total mass density slope, , we have adopted a power-law mass density r , with slope steeper and shallower than isothermal. In Tortora etal. ( 2014c ) we have shown that using two extreme values for the density slope, i.e. 2.5 and 1.5, which bracket most of the results in the literature, then we find that the average M dyn / M get smaller (larger) of 0.1 (0.05)dex for 2.5 (1.5). However, in the realistic case with a varying slope with mass, smaller changes would be expected (Dutton Treu 2014 ; Tortora etal. 2014b ). If is constant with time, then, at fixed mass, these corrections would shift all the results of the same factor, leaving naturally unaffected the observed trend with redshift. On the other hand, if we assume that the slope is varying with redshift, then the only way to totally remove any M dyn / M


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

general for the other hydrogenated PAHs. It is also worth noting that the absence of bands at 3.51 and 3.56 μ m is characteristic of hydrogenated PAHs in which extra H atoms are located by pairs in different parts of the PAH (isomers a and e of Figure 5 and the lower spectrum in Figures 6 and 7 , which include the most stable hydrogenated PAHs predicted at the DFT level). Figure 6. Convoluted theoretical harmonic spectra of four stable isomers of C 54 H 18+12 . Isomers are schematically depicted as an inset. Bands located around 3.51 and 3.56 μ m are indicated by orange and green arrows, respectively. The corresponding H atoms that mostly contribute to those bands are identified with the same colors. Energy differences ( E rel ) correspond to the DFT values. Figure 7. Convoluted theoretical harmonic spectra of four stable isomers of C 96 H 24+10 . Isomers are schematically depicted as an inset. Bands located around 3.51 and 3.56 μ m are indicated by orange and green arrows, respectively. The corresponding H atoms that mostly contribute to those bands are identified with the same colors. Energy differences ( E rel ) correspond to the DFT values. 4. Astrophysical Implications Hydrogenated PAH molecules have been experimentally and theoretically studied in their neutral and cationic forms (Thrower et al. 2012 ; Klærke et al. 2013 ; Cazaux et al. 2016 , 2019 ; Jensen et al. 2019 ; Campisi et al. 2020 ). It has been demonstrated that specific isomeric forms are more abundant than others but that hydrogenation on the different positions (secondary and tertiary) occurs. For example, in the case of neutral coronene, the preferred isomeric forms of 2, 4, and 6 H atoms of the main and alternative sequences of hydrogenation (Jensen et al. 2019 ) are among the most stable isomers shown in Figure 3 . The structures with six additional H atoms previously found (isomers b and d) but also others found in this work (isomers c, f, and g) of Figure 3 are already hydrogenated in tertiary positions, therefore showing that even for low degrees of hydrogenation, tertiary positions are hydrogenated. The conditions in some regions of the ISM, for instance at the surface of PDRs, cause hydrogenated PAHs to become ionized, showing stable structures that are different from those of their neutral counterparts (Cazaux et al. 2019 ). In addition, it has been shown that H atoms attached to cationic PAHs can roam over the peripheral carbon atoms producing a “scrambling” of the different isomers upon absorption of interstellar FUV photons (Wiersma et al. 2020 ). Thus, although not all possible isomers are going to contribute equally to spectral features in the ISM, it is still under study which are the different isomers that contribute depending on the reactions that can take place under the particular conditions of the ISM. If “scrambling” of H atoms inside PAHs is a feasible process, it can be expected that the most stable isomers may have strong contributions to the


O O B-Person I-Person O O O O O O O O O O O O B-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization

We thank Uri Keshet for useful discussions. This research was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development.


O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model 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 B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

abundance used in our chemical reaction model. This table is from Garrod ( 2013 ). 4.2.2. The Physical Model The physical conditions used as input to the chemical model are based on the “fast warm-up model” from Garrod ( 2013 ). We have run four different models (A, B, C, and D) with different peak densities and/or temperatures as listed in Table 9 . The peak temperatures and densities were selected considering the recent observations toward hot cores (e.g., Hernández-Hernández et al. 2014 ). We will see how CH 2 NH formation processes depend on densities through the comparison of models A, B, and C, and on temperatures through the comparison of models A and D. The gas densities () were initially set to be 3 × 10 3 cm −3 , which gradually increased up to their peak densities along with the free-fall equation during the collapsing phase. While gas temperatures were fixed at 10 K in the collapsing phase, dust temperatures decreased from 16 to 8 K depending on the visual extinction, which was calculated from the gas density of the cloud. Once the peak densities were achieved, we fixed the densities, and the gas and dust temperatures were raised from ∼10 K to their peak temperatures in 7 × 10 4 years. Table 9 Peak Temperatures and Densities for the Models 4.2.3. Gas and Ice Phase Formation of CH2NH As a result of our model, CH 2 NH is formed both in the gas phase and on the dust grain surface. We summarize the main chemical reactions that form CH 2 NH on the surfaces in Table 10 . The dust surface reaction rates are proportional to both the probability of the reaction κ and the diffusion rates of reactant species (Hasegawa et al. 1992 ). κ is proportional to exp(- E ), where E A represents the value of the activation barrier. The diffusion rates depend on the exponential of the dust temperature. For the activation barriers of the hydrogenation processes of HCN and CH 2 NH, theoretically predicted values by Woon ( 2002 ) were employed. Table 10 Dust Surface Reactions Related to CH 2 NH Note. The dust surface reactions related to CH 2 NH are shown. E A represents the value of the activation barrier. Since radical species are so reactive, radical–radical reactions would have no activation barriers. The activation barriers for HCN and CH 2 NH were cited from the theoretical study by Woon ( 2002 ). The main reactions in the gas phase that form CH 2 NH are summarized in Table 11 . CH 2 NH is formed both by dissociative recombination processes and neutral–neutral reactions. The rate coefficients of gas-phase reactions are given by the Arrhenius equation, using the coefficients α , β , and γ :where T represents the gas temperature. As shown in Table 11 , the dissociative recombination processes have larger reaction rates. Table 11 Gas-phase Reactions Related to CH 2 NH Note. α , β , and γ represent the coefficients for the Arrhenius equation. 4.2.4. Modeling Results and Discussion The simulated gas-phase CH 2 NH abundances (relative to the


O O O O O O O B-Person I-Person I-Person O O O O O O O O O O O O B-Person I-Person O O O O O O O O O O O O O O O O O O O O O O O O O

Acknowledgements I sincerely thank my PhD advisor, A. R. Rao, for helpful discussions and suggestions during the entire course of the work; Pawan Kumar for his comments on the manuscript, thus improving its quality; and the referee for his critical comments which immensely improved the quality of the work.


O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Instrument O B-CelestialObject O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject 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-Survey O O O O O O O O O O O O O O O O

values of u 0 , are low. For rel , our prior is flat between 0 and 10masyr 1 (see Henderson 2015 ). The priors for and f , respectively, are flat from 0 to and 0 to 2rad. For inclination, the prior is flat in cos from 1 to 1. Note that while flat priors for some of our parameters are unrealistic, the assumptions are reasonable, given poor constraints and the resulting ease of computation. Since all of the priors are flat, the prior term can be pulled out of the integral in ( 3 ) as a factor of p V 1 , where V i ( i i ) is the volume of the priors. i is the upper boundary of the prior for parameter i and i is the lower boundary. Using the Fisher matrix approximation, the evidence is then We use Monte Carlo integration to compute the evidence integrals. 4 RESULTS The test in Bayesian statistics of support for one model over another is the Bayes factor, which is the ratio of the binary-lens evidence to the single-lens evidence: B E B / E S . We apply the scale from Jeffreys ( 1961 ) to determine support, where ln B 5 is strong evidence, 2.5 ln B 5 is moderate evidence, 1 ln B 2.5 is weak evidence and ln B 1 is inconclusive. In the limits that q 0 and s 0 0, the peak likelihood ratio 1, while ln B becomes negative, indicating that the single-lens model is favoured. We compute whether a BBH system with mass parameters similar to those of the first detected LIGO event, GW150914, would be detectable by microlensing in our own Galaxy. Our fiducial model is M 65M , q 0.8, rel 5.5masyr 1 and u 0 0.4 Einstein radii. We choose a lens distance D L 4kpc and a source star in the Galactic Bulge at D S 8kpc with apparent magnitude I 15, on the bright end of typical source magnitudes in current microlensing surveys. For this model, we find that BBH separations down to approximately s 0 0.0250 Einstein radii are detectable with strong evidence, given ideal observing conditions for one year around the peak of the event. For a dimmer source star with I 17, separations down to only s 0 0.0270 Einstein radii are detectable. We also calculate the 2 between the binary and single lenses, both for our fiducial separation s 0 0.01 Einstein radii and the minimum detectable separation s 0 0.0250 Einstein radii. For the fiducial case, ln B 7.65 corresponds to 2 1.36; for the minimum detectable s 0 0.0250 Einstein radii, ln B 5 and 2 53.4. The approximate merger time of BBHs is (see Peters Mathews 1963 ) as follows: where c is the speed of light and G is the gravitational constant. Therefore, BBHs that fit our fiducial model with a separation of s 0 0.0250 Einstein radii will merge in about 2.0810 3 Gyr, which is much greater than the 13.8Gyr age of the Universe. BBHs formed with these parameters in the early Universe can serve only as present-day GW sources if their initial separation was s 0 0.00716 Einstein radii, 3.5times smaller than the minimum separation detectable by KMTNet. The dependence of this minimum separation on the binary parameters is explored in Fig. 2 ,


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

only occur with a libration period, T , in years given by with and H θθ = − R θθ , which can be calculated numerically, and both H II , H θθ are evaluated at the stable equilibrium point. Fig. 1 Level curves of a fictitious Hamiltonian. O indicates a stable equilibrium point at ( θ s = 60°, I 0 = 2) and X an unstable equilibrium point at ( θ u = 240°, I 0 = 2). The arrows show the total resonance width, which is 2( I sep − I 0 ), where I sep = 2.44. To obtain the width of the resonance in terms of Δ I , we follow the same reasoning as in Gallardo (2020) . Figure 1 shows that the resonance half-width is equal to the difference between I 0 and I sep , where I sep is defined by the separatrix such that where θ s and θ u are the stable and unstable equilibrium points. When we assume that the domain of the resonance is symmetric with respect to the nominal position, I 0 , the total width is twice Δ I . In some cases, asymmetrieswith respect to I = I 0 are present and a precise examination of the Hamiltonian level curves is required to obtain the exact width. In general, however, because Δ H = H ( θ s , I sep ) − H ( θ s , I 0 ), we can approximate Evaluating the derivatives at the stable equilibrium point ( θ s , I 0 ) and using Eq. ( 14 ), we have Using Eq. ( 6 ), the left-hand side is while is given by Eq. ( 13 ), then We recall that Δ I is in fact Δ I 1 . Strength is commonly defined as SR = ⟨ R ⟩− R min ( Gallardo 2006 ), which can be associated with the coefficient multiplying cos ( σ ) when R is a well-behaved cosine-like function, but in general, R could be very different from a simple cosine. In these situations, SR therefore is a reliable estimate of the sensitivity of R with σ . The strength is a global property of the resonance, as are the libration center and period. It is an indicator of the dynamical relevance of the resonance, and it generates different Δ a on each planet. The width in terms of Δ a 1 is obtained from the definition of I 1 in Eq. ( 4 ) and Δ a 2 is obtained from I 2 = constant and is evaluated at the center of the resonance. We obtain the half-widths in au: The higher the planetary mass m i , the narrower the relative width Δ a i ∕ a i . The resonance is the same dynamical mechanism that affects the two planets, but stronger effects are present in the planet with the lower mass. It is important to stress that Δ a i are given interms of Poincaré elements, but for planetary systems, the computed widths Δ a i in Poincaré elements arevery similar to the widths in astrocentric elements. Moreover, the widths obtained by the formulae above are correct as long as Δ a ≪ a . If large widths are obtained using these formulae, it is convenient to recalculate the resonance limits ( I max , I min ) by a numerical procedure that considers the separatrix obtained from the Hamiltonian given by Eq. ( 6 ). In this circumstance, it is possible to observe asymmetries between the two limits of the resonance around the exact (nominal)


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Citation I-Citation I-Citation 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 B-Model I-Model I-Model 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 B-Model I-Model 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 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 B-Citation I-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 O O O O O O O O O O O O O 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 B-Model I-Model O O O O O O O O O O O O O O O O B-Tag I-Tag 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-Person O B-Person O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O B-Grant B-Person O O O O O B-Organization I-Organization B-Organization O O B-Grant I-Grant I-Grant I-Grant B-Person B-Person O B-Person O O O O B-Organization I-Organization O B-Grant O O B-Organization O B-Grant B-Person O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Grant O B-Grant O O O O B-Organization B-Grant B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization

We thank the anonymous referee for taking the time to review this manuscript. JRB and CAH were supported by the Science and Technology Facilities Council (STFC) under the grant ST/L000776/1. SVJ acknowledges research funding by the Deutsche Forschungsgemeinschaft (DFG) under grant SFB 963/1, project A16. HRAJ MT and FF are supported by the Leverhulme Trust grant, RPG-2014-281, and the STFC grant, ST/M001008/1. JSJ acknowledges funding by Fondo Nacional de Desarrollo Cientfico y Tecnolgico (FONDECYT) through grants 1161218 and 3110004, and partial support from CATA-Basal (PB06, Comisin Nacional de Investigacin Cientfica y Tecnolgica (CONICYT)), the GEMINI-CONICYT FUND and from the Comit Mixto ESO-GOBIERNO DE CHILE.


O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O 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 B-Citation O B-Citation I-Citation I-Citation O O O O O B-CelestialObject I-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-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject I-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 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 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 B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation O O O O O B-Wavelength O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation 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

(King 2008 , 2009 ). Moreover, we assumed L L X for the sake of simplicity, but in fact the bolometric correction of the inferred X-ray luminosity might vary from system to system. This might depend e.g. on the line of sight through the outflow, as hinted by the change in spectral hardness among the ULXs considered in Table 1 , as well as by the conjectured connection with ultraluminous supersoft sources (Middleton etal. 2011 , 2015 ; Pinto etal. 2017 ). According to this scenario, NGC 1313 X-1 is likely to be seen more face-on because of the harder spectrum, and therefore L L X is a reasonable assumption (but it is more likely to suffer from beaming though), while NGC 55 ULX and NGC 5408 X-1 have softer spectra that may come from the reprocessing of the harder radiation from the inner edge of the disc (and perhaps by a close hot corona) by the optically thick wind, for which L X L . This latter correction might partially compensate the effect of beaming, but note that L X as well is assumed isotropic and neglects beaming effects. None the less, even when reduced by a factor 23, our inferred masses are consistent with previous estimates of similar systems as well as with the aforementioned scenario of ULX formation. Our calculations also neglect general relativistic effects, such as spinning BHs, and magnetic fields. We explored the impact of a spinning BH by changing the inner boundary condition and the normalizing radiative efficiency . We find values of M up to 20percent lower than in the no spin cases, owing to larger values of and at similar accretion rates. Magnetic fields might also contribute to accelerate the outflow (Blandford Payne 1982 ), effectively lowering w and increasing L for the same accretion rate. This might also lower the value of the inferred central mass. Regardless of their masses, BHs are not the only possible accretors powering ULXs. Indeed, the light curves of three ULXs show sinusoidal pulses with a period 1s that are a distinctive signature of neutron stars (Bachetti etal. 2014 ; Frst etal. 2016 ; Israel etal. 2017 ). Recently, King Lasota ( 2016 ) and King, Lasota Kluzniak ( 2017 ) argued that neutron stars may actually power a fraction of ULXs larger than previously expected. According to their analysis, pulsations may only be observed during a rather short phase of the ULX evolution, preventing an easy detection. However, the variability of the optical spectrum has revealed the existence of a massive stellar BH, likely 2030M , in M101 ULX-1 (Liu etal. 2013 ) and perhaps also in X-ray binaries associated with WolfRayet stars (Prestwich etal. 2007 ; Crowther etal. 2010 ; but see also Laycock etal. 2015 ). Therefore, it appears evident that ULXs comprise a diverse variety of accreting objects, and whether the majority is represented by BHs or neutron stars must be scrutinized further. Clearly, our analysis can be applied only to the subset of ULXs with outflows that probably host an accreting BH; none the less,


O O O O O O O O O O B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory O O B-Telescope O O O B-Location I-Location I-Location I-Location O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Observatory O B-Proposal O B-Proposal O O O O O O B-Organization I-Organization I-Organization B-Organization O O B-Telescope O B-Telescope O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O B-Organization O O O O O O O O O O O O B-Collaboration I-Collaboration I-Collaboration B-Collaboration O B-Telescope O O O B-URL O O B-Archive I-Archive O O B-URL O O O O O O O O O B-Survey I-Survey I-Survey B-Observatory O B-Proposal O B-Person O O O O O O O B-Software I-Software I-Software I-Software O B-Organization B-Location O B-Organization B-Location O B-Software I-Software I-Software I-Software O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope O O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O B-Person I-Person O B-Person I-Person O O B-Model I-Model I-Model O O B-Person I-Person B-Person O O O O B-Fellowship I-Fellowship

ACKNOWLEDGEMENTS This article is based on observations made in the Observatorios de Canarias del IAC with the WHT operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Observatorio del los Muchachos. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programmes 096.C-0130 and 098.D-0222. This work presents results from the European Space Agency (ESA) space mission Gaia . Gaia data is being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia . The Gaia archive website is https://archives.esac.esa.int/gaia . Based on observations obtained as part of the VISTA Hemisphere Survey, ESO Progam, 179.A-2010 (PI: McMahon). Data processing has been contributed by the VISTA Data Flow System at CASU, Cambridge and WFAU, Edinburgh. The VISTA Data Flow System pipeline processing and science archive are described in Irwin et al. ( 2004 ), Hambly et al. ( 2008 ) and Cross et al. ( 2012 ). This publication makes use of data products from the Wide-field Infrared Survey Explorer , which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Synthetic spectra used in this paper are calculated by France Allard and Derek Homeier based on BT-Dusty model atmospheres developed by France Allard. ZHZ was supported by the PSL fellowship.


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

with distance estimates (Table A.1 ), as seen from the Galactic pole. Dynamically confirmed black holes are marked in orange circles, whereas BH candidates are indicated by yellow stars. Distance ranges are represented with bars while the systems with lower limits are indicated by arrows that follow the same colour index. Some systems are overlapped with other. Each faint white ring in the background image represents an additional 1.5 kpc from the Sun. Objects seem to concentrate along the direction towards the Galactic centre. The dynamical BHT in the upper right of the figure is BW Cir. Background image credit: NASA/JPL-Caltech/R. Hurt (SSC/Caltech). Fig. 3 Top : the distribution of dynamically confirmed BHs (circles) and BH candidates (stars) in the Galactic plane, using the Hammer projection. (Background image credit: Mellinger 2009 ). Some of the symbols are overlapped by others, especially in the Galactic centre region. Bottom : histogram of the distribution of BH transients in Galactic longitude ( left ) and latitude ( right ). A 10 bin size in longitude and 5 bin in latitude were used respectively. Here we revisit the problem of the size of the Galactic population of BHs, based on the currently observed sample, which is heavily affected by extinction. Duerbeck (1984) modelled the density distribution of several interacting binaries in the Galaxy. In a similar way, we want to obtain the density of Galactic BHTs and analyse the height distribution (; Col. 8 in Table A.1 ). Only 35 out of the 59 detected BHTs have estimated distances. Because of their transient nature, we have only detected those systems that went into outburst. Therefore, in a given period of time, we have only discovered a fraction of them (e.g. the brightest and/or closest BHTs – with the shortest outburst cycles). Thus, to derive the true distribution, we first need to find the maximum radial distance at which completeness of the sample is guaranteed, i.e. such that none of the X-ray outbursts are missed. The completeness of the sample can be obtained from the analysis of the density of objects as a function of the radial distance that is projected onto the Galactic plane (). The latter is derived from the Galactic latitude and the distance listed in Cols. 5 and 6 of Table A.1 . Here, we have used 31 out of the 35 BHTs with a reported distance (we exclude the four systems with lower or upper limits). In addition, we consider a conservative 50% uncertainty in systems with only a rough estimate of distance. To derive the density of objects () up to a radii , we count the number of BHTs lying in cylinders or radius centred at the Sun and infinite height (). However, given the large uncertainties with distance, in some cases an object could be placed in more than a single cylinder within errors. To account for this effect, for each object we randomly generate 10 000 values that assume a Gaussian distribution. Here, we naively assume the distance errors follow this sort of


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

detected oscillatory patterns. Double-peaked emitters 3C 390.3 and Arp 102B Our hybrid method made the distinction between these two objects based on the presence (absence) of underlying periodic variability, despite that the differences in the shapes of their spectral lines are subtle. 2D (auto)correlation maps of Arp 102B have a simple linear shape, but for 3C 390.3 a manifold of substructures is superimposed. Namely, our analysis has shown the fluctuations in the interactions between the oscillations in the continuum and the emission lines of 3C 390.3 due to substantially different levels of correlation polarity. The clusters of correlations are divided by gaps of very low (or no) correlations, and the islands of correlations switch their polarity (between positive and negative values). This can be seen as evidence that there is an activation process that reduces correlation coefficients (from positive values up to zero). Later, a different relationship is evoked between the periodicities in light curves (e.g. negative correlations). The break off of the correlation band (on diagonal) may be due to the change of variability in the light curves or due to the modulatory effects of some unknown background factors, which may have certain cycles. We applied the hybrid method to a series of artificially non-linear and non-stationary oscillatory signals simulated by equations ( 6 ) and ( 12 ) with unidirectional and bidirectional coupling. The degree of non-linearity/asymmetry is controlled by the ratio (differing from 1) of the periods (frequencies) of the sinusoids. To introduce the non-stationary mechanism, we assign a random value between 0 and 10 to the amplitude of signals, while they were fixed over trial realization, adding the red noise. The 2D correlation map of bidirectional coupling of three oscillatory processes (see Fig. 12 ) almost matches the map of the real light curves. Figure 12. Simulation of bidirectional coupled three-oscillator network for the case of 3C 390.3. Left: random realization of equation ( 12 ) from two time series (black is U a = OP 1 and red is U c = OP 2) of amplitudes A = 1.954, B = 1.729, C = 2.357, phase φ = φ 1 = 2.359 rad, coupling strengths cp a → b = 0.7, cp a → c = 0.5, cp b → a = 0.2, cp c → a = 0.3, frequencies f a = 1/1000, f b = 1/300, f c = 1/100, and time delay of 100 arbitrary chosen time units. Right: corresponding 2D correlation map, which clearly shows three clusters related to fundamental periods as well as clusters of negative correlation. We cannot produce a correlation image with negative ‘correlation islands’ if the time delays between processes are different. We note that time delays between the continuum and the Hα and Hβ lines are ∼120 and 95 d, respectively, as reported by Shapovalova et al. ( 2010a ). Based on this, one can see the importance of the value of the time lag for the expression of a negative link between oscillations, which implies that physical places of oscillation