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no notable difference in the spatial distribution of variable stars compared to the unclassified (nonvariable) sources according to Gaia DR2. 5.6. P-L Relations Accurately determined periods have been the means to distinguish variable stars, particularly pulsating variables (LPV, Cepheids, RR Lyrae, RV Tauri, etc.) within Gaia (Gaia Collaboration et al. 2019 ). Figure 13 displays the distribution of periods available from Gaia for sources in the local sample. It can be noted that the whole sample contains LPV stars with a wide range of periods, but that those within the solar neighborhood are restricted to stars with periods of ≲1250 days, presumably Mira or SR variables, as classified by Gaia (Molnár et al. 2018 ; Mowlavi et al. 2018 ). Figure 13. Period distributions obtained for variable stars in Gaia DR2 for the Gaia–BAaDE and local samples (see Figure 5 and Section 5.6 ). It has been established that P-L relations are a very powerful tool to distinguish AGB stars of different natures (Wood et al. 1999 ; Ita et al. 2004 ; Lebzelter et al. 2019 ). By recognizing that Mira variables pulsate dominantly in the fundamental mode, they can be promising candidates for distance determinations of remote galaxies, using empirical relations based on the LMC (Whitelock et al. 2008 ). With 2MASS K magnitudes, Gaia DR2 parallaxes, extinction maps, and periods for a subsample of the local sample ( period sample), we are able to make a comparison of the BAaDE targets with previously studied variable stars. In Figure 14 , we present the P-L relation for those BAaDE stars in the local sample with measured Gaia periods ( period sample), where there is a spread in the magnitude, resulting from uncertainties in the apparent K magnitude, extinction, and distance (indicated by the error bars) and IR variability (not indicated). A comparison is made of the P-L distributions with known variability sequences, associated with distinct pulsation modes, that have been derived from Gaia DR2 data for LMC populations as discussed by Lebzelter et al. ( 2019 ). These sequences have been transformed to , using the LMC distance modulus in that work (18.49 mag). Moreover, the established P-L relations for Miras from Whitelock et al. ( 2008 ) and Matsunaga Team ( 2006 ) in the LMC are added. Figure 14. Period–luminosity relations found for the variable stars within the local sample. The sequences marked as C , C ′, and D represent different known variability sequences, associated with distinct pulsation modes derived for the LMC based on Gaia data (Lebzelter et al. 2019 ). Note that the sequences were corrected for a distance modulus of 18.49 mag. Period–luminosity relations reported for LMC using other surveys are also shown (Matsunaga Team 2006 ; Whitelock et al. 2008 ). The local sample appears to be much flatter than the empirical P-L relations for the LMC, and it is clear that most of the stars in the local sample fall below the LMC P-L relations (Matsunaga Team 2006 ;


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at the redshifts of interest) and that the seeing was good (0.65 arcsec), slit losses are likely ≲30 per cent. Moreover, because we are primarily concerned with redshifts rather than line luminosities, aperture corrections to the line fluxes are unnecessary. The spectra were obtained using the standard two-position ‘mask nod’ in which the telescope position was dithered ±1.5 arcsec along the slit direction, with individual integration times of 120 s ( H band) or 180 s ( K band). Total integration times of 2880 s were obtained, with 24 and 16 individual exposures in the H and K bands, respectively. The MOSFIRE data were reduced using the publicly available data reduction pipeline developed by the instrument team; 12 see Steidel et al. ( 2014 ) for a detailed description of the procedure. One-dimensional spectra were extracted, flux-calibrated, and analysed using the mospec package (Strom et al. 2017 ). 2.1.2 GNIRS observations Near-infrared spectra of resolved components of two multicomponent single-dish submm sources (UDS292 and UDS306) were obtained using the cross-dispersed mode of the Gemini Near-Infrared Spectrograph (GNIRS 13 ) on the Gemini North 8.1 m telescope. This configuration provides continuous spectral coverage from 0.84 to 2.48 m at a spectral resolution of R ≈ 1500 with a spatial scale of 0.15 arcsec pix −1 . The slit dimensions were 1.0 arcsec × 7.0 arcsec. The observations used an ABBA pattern, nodding along the slit to keep the galaxy on slit at all times. Eight individual on-source integrations of 240 s each were performed for each source. Observations of standard stars were obtained before and after each set of SMG observations. These were used to correct the spectra for telluric absorption. The spectral reduction, extraction, and wavelength and flux calibration procedures were performed using the Gemini iraf package and pyraf . 14 Briefly, the processing consists of removing cosmic ray-like features, dividing by flat-fields taken with infrared and quartz halogen lamps, subtracting sky emission using exposures taken at a different point in the dither pattern, and finally rectifying the tilted, curved spectra using pinhole flats. Wavelength calibration is performed using Argon arc spectra, and then a spectrum of each order is extracted, divided by the standard star observation to cancel telluric absorption lines, and roughly flux-calibrated using the telluric standard star spectrum. The different spectral orders for each extraction window are merged into a single 1D spectrum from 0.84 to 2.48 m. In all cases, the agreement in flux between the overlapping regions of two consecutive orders was very good, and scaling factors of only 3 per cent or less were necessary. 2.1.3 XSHOOTER observations One target was observed on 2015 March 4 with the XSHOOTER echelon spectrograph (Vernet et al. 2011 ) on the ESO VLT-UT2 (Kueyen) as part of programme 094.A-0811(A), providing near-continuous spectroscopy from 0.3 to 2.48 m with a 1.2


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of the initial rotational energy. Other energies increase the irreducible mass of the BH. For a • (0) = 0.5, M • = 0.98 M • (0), or 2% of the initial mass can be used to power a GRB. 10 10 Atteia et al. ( 2017 ) found the maximum isotropic energy of GRBs when they studied the GRB energy distribution within redshifts z = 1–5. Jet break measurements are needed to derive the beaming-corrected energy, which can be compared with our model predictions. As the magnetic field on the BH is supported by the surrounding disk, there are some relations between B • and . In a hyperaccreting flow in a GRB, it is possible that a magnetic flux is accumulated near the BH horizon. Considering the balance between the magnetic pressure on the horizon and the ram pressure of the innermost part of the accretion flow (e.g., Moderski et al. 1997 ), one can estimate the magnetic field strength threading the BH horizon , where is the radius of the BH horizon. One thus hasInserting it into Equation ( 31 ), we obtain the magnetic power and torque as a function of the mass accretion rate and BH spin, i.e., where . It is found that X (0) = 1/6 and X (1) = π − 2. Both neutrino-annihilation and magnetic powers depend on the disk mass accretion. In Figure 3 , we present the BZ power as a function of accretion rate for different BH spins and compare it with the neutrino-annihilation power. We find that (1) the magnetic power is much greater than the neutrino-annihilation power for a moderate to high-spin BH, (2) the neutrino-annihilation power dominates over the BZ power for BHs with a very small spin at high accretion rates, and (3) compared with the magnetic power, is much more sensitive to the mass accretion rate . Therefore, if the disk accretion rate is variable, the jet driven by the neutrino-annihilation process should be highly variable. However, the MHD jet is usually subject to instabilities, such as kink instability (Wang et al. 2006 ) and magnetic reconnection (e.g., the Internal-Collision induced Magnetic Reconnection and Turbulence, ICMART, in Zhang Yan 2011 ). These MHD processes will add complexity to the GRB light curves. Figure 3. Magnetic power as a function of accretion rate for different BH spins, a • = 0.01 (thick red solid line) and 0.99 (thick blue solid lines). The dashed lines show the neutrino-annihilation power calculated with Equation ( 22 ) for a • = 0.01 (red dashed line) and 0.99 (blue dashed line). The thin blue line is produced with the analytic expression of Tchekhovskoy et al. ( 2011 ) for a • = 0.99, where the average magnetic flux and κ = 0.044 are taken based on the numerical simulation Model A0.99f. In the magnetic model, baryons from the disk will be suppressed by the strong magnetic field lines. For , the baryon-loading rate for the BZ-driven jet is (Paper I )For , the dependence of on will be . In Equation ( 39 ), f p is the fraction of protons and r z is the distance from the BH in the jet direction, which is normalized to 10 11 cm.


O O B-Person I-Person B-Organization B-Person I-Person B-Organization B-Person I-Person B-Organization B-Person I-Person B-Organization O B-Person I-Person B-Observatory I-Observatory O B-Software I-Software B-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation O O O O O O B-Archive I-Archive I-Archive I-Archive I-Archive I-Archive O O O O O O O B-Location I-Location B-Grant I-Grant O B-Organization O O B-Grant B-Person O B-Organization I-Organization O O B-Organization I-Organization I-Organization O O O O O B-Person O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Fellowship I-Fellowship O O B-Survey O O O O O B-Observatory I-Observatory I-Observatory O B-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization B-Grant I-Grant O B-Grant I-Grant O B-Organization I-Organization I-Organization B-Grant I-Grant I-Grant O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization O O B-Location B-Grant O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Location O B-Organization I-Organization I-Organization B-Grant I-Grant I-Grant I-Grant I-Grant O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization O O B-Location B-Organization B-Observatory I-Observatory O O B-Organization I-Organization I-Organization B-Organization O B-Organization O B-Survey O O O B-URL O

We thank Borja Anguiano (Macquarie), Jonathan Bird (Vanderbilt), Sven Buder (MPIA), Sofia Randich (INAF), and Kevin Schlaufman (Carnegie Observatories). Software: The Cannon (Casey et al. 2016b ), Astropy (Astropy Collaboration et al. 2013 ), TOPCAT (Taylor 2005 ). This research made use of NASA’s Astrophysics Data System Bibliographic Services. This work was partly supported by the European Union FP7 programme through ERC grant number 320360. A.R.C. thanks King’s College of the University of Cambridge for their support. Funding for K.H. has been provided through the Simons Foundation Society of Fellows and the Marshall Scholarship. Funding for RAVE has been provided by the Australian Astronomical Observatory; the Leibniz-Institut fuer Astrophysik Potsdam (AIP); the Australian National University; the Australian Research Council; the French National Research Agency; the German Research Foundation (SPP 1177 and SFB 881); the European Research Council (ERC-StG 240271 Galactica); the Istituto Nazionale di Astrofisica at Padova; The Johns Hopkins University; the National Science Foundation of the USA (AST-0908326); the W. M. Keck foundation; the Macquarie University; the Netherlands Research School for Astronomy; the Natural Sciences and Engineering Research Council of Canada; the Slovenian Research Agency (research core funding No. P1-0188); the Swiss National Science Foundation; the Science Technology Facilities Council of the UK; Opticon; Strasbourg Observatory; and the Universities of Groningen, Heidelberg and Sydney. The RAVE web site is https://www.rave-survey.org .


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We thank the Juno science team, Eli Tziperman, and Adam Showman for useful discussions and comments. This research has been supported by the Israeli ministry of Science under grants 3-11481 and 45-851-641, and the Helen Kimmel Center for Planetary Science at the Weizmann Institute of Science.


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We acknowledge the support from the following agencies: US National Science Foundation–Office of Polar Programs, US National Science Foundation–Physics Division, University of Wisconsin Alumni Research Foundation, the Grid Laboratory Of Wisconsin (GLOW) grid infrastructure at the University of Wisconsin–Madison, the Open Science Grid (OSG) grid infrastructure; US Department of Energy, and National Energy Research Scientific Computing Center, the Louisiana Optical Network Initiative (LONI) grid computing resources; Natural Sciences and Engineering Research Council of Canada, WestGrid, and Compute/Calcul Canada; Swedish Research Council, Swedish Polar Research Secretariat, Swedish National Infrastructure for Computing (SNIC), and Knut and Alice Wallenberg Foundation, Sweden; German Ministry for Education and Research (BMBF), Deutsche Forschungsgemeinschaft (DFG), Helmholtz Alliance for Astroparticle Physics (HAP), Research Department of Plasmas with Complex Interactions (Bochum), Germany; Fund for Scientific Research (FNRS–FWO), FWO Odysseus program, Flanders Institute to encourage scientific and technological research in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford, United Kingdom; Marsden Fund, New Zealand; Australian Research Council; Japan Society for Promotion of Science (JSPS); the Swiss National Science Foundation (SNSF), Switzerland; National Research Foundation of Korea (NRF); Villum Fonden, Danish National Research Foundation (DNRF), Denmark.


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ACKNOWLEDGEMENTS The authors would like to thank the anonymous referee for extremely valuable comments and suggestions. A special thank goes to Kristoffer Stensbo–Smidt for all the work and the precious suggestions he put in reviewing and commenting this work. MB acknowledges the INAF PRIN-SKA 2017 program 1.05.01.88.04 and the funding from MIUR Premiale 2016: MITIC . GL and MDV acknowledge the UE funded Marie Curie ITN SUNDIAL that partially supported this work. SC acknowledges support from the project ‘ Quasars at high redshift: physics and Cosmology ’ financed by the ASI/INAF agreement 2017-14-H.0. Topcat has been used for this work (Taylor 2005 ). C 3 has been used for catalogue cross-matching (Riccio et al. 2017 ). DAMEWARE has been used for ML experiments (Brescia et al. 2014a ).


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on the neutron star. All MeerKAT observations were processed in a homogeneous way using PSRCHIVE. The first few days of Parkes and MeerKAT observations (see Figure 1 (b)) were used to improve the rotational ephemeris thereafter used to fold all MeerKAT observations. Radio frequency interference (RFI) was excised in a multi-step process: first, 10% of the recorded band was removed due to bandpass roll-off, yielding the nominal 900–1670 MHz MeerKAT L band; next, a mask with known persistent RFI signals (e.g., GSM, GPS) was applied to the data; finally, PSRCHIVE tools and clean.py, an RFI-excision script provided by the CoastGuard data analysis pipeline (Lazarus et al. 2016 ), were used to remove remaining RFI. More than 400 MHz of clean band was typically retained after this flagging. We summed all frequency channels, integrations, and polarizations for each observation to obtain total-intensity profiles. We then obtained times-of-arrival (TOAs) for each observation (i.e., two or three per day) by cross-correlating individual profiles with a standard template based on a very high signal-to-noise ratio observation. The same procedure was applied to MeerKAT observations of PSR J1644−4559. Using the TEMPO software we confirmed that the timing solution derived for PSR J1644−4559 was consistent with published parameters (see Manchester et al. 2005 ). Like many magnetars, PSR J1622–4950 displays unsteady rotation (e.g., Dib Kaspi 2014 ). Nevertheless, we were able to obtain a simple phase-connected timing solution for the three week period spanning the first two sets of paired Chandra and NuSTAR observations (Table 1 ), fitting the TOAs with TEMPO to a model containing only pulse phase, rotation frequency ( ν ), and frequency derivative (). This solution is presented in Table 2 . Over the five month span of all the radio timing observations presented here, the for PSR J1622–4950 changed by a factor of 7, and changed sign. In order to probe the evolution of the spin-down torque (), we fit short-term overlapping timing models where a fit for ν and proves adequate, i.e., with featureless timing residuals. Each of these short-term timing solutions spanned approximately one week. Figure 3 shows the run of from these solutions spanning five months. Figure 3. Time evolution of properties of PSR J1622–4950. Top panel: frequency derivative measured from short-term timing solutions (spanning horizontal error bars) obtained from MeerKAT data (see Section 3.3 ). Bottom panel: 0.3–10 keV absorbed X-ray flux as measured from the best-fit blackbody model (see Table 3 ). Swift observations are shown as open squares and combined Chandra and NuSTAR observations are represented simply by their error bars. The X-ray flux appears to be decaying exponentially with a time constant of 111 ± 19 days (see Section 4.4 ). All error bars represent 1 σ confidence levels. The gray band in both panels represents the timespan during which the X-ray outburst and radio revival most


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

it may not be the case for () point statistics. For instance, peak-peak correlation functions on small scales are not very well reproduced by this approximation even for large contrasts because the contribution from the other critical points actually dominates at small distance; there is at least one saddle point between two peaks. However, in the context of this work, we impose the peak constraint in one location only and therefore the rare peak approximation is expected to be accurate for . As an illustration, Fig. 2 shows the Gaussian mean number density of minima, saddle points, and peaks ( Longuet-Higgins 1957 ; Adler 1981 , and later generalised to weakly non-Gaussian fields by Pogosyan et al. 2011 ) and compares the latter to the high- approximation (related to the genus) which can be easily computed The relative error between the number density of peaks and its high- approximation is shown on the right-hand panel of Fig. 2 . Fig. 4 Zero lag annulus correlation function in units of for a central peak with height and eigenvalues and in for a Gaussian random field with power spectrum smoothed with a Gaussian filter (similar to Fig. 3 when the two points on the annulus are the same). The separation is given in units of the smoothing length. 2.4. Density correlations on the circle surrounding a central peak with given geometry The expected product of projected density in two locations of space and such that and and given a peak in of height and curvatures along the first and second coordinates can be analytically computed. For instance for a power-law power spectrum with (and ), we get where is the trace of the density Hessian at the location of the peak, is the ellipticity of the peak and the unconstrained correlation function is To start with, Fig. 4 shows the zero lag contribution to the annulus correlation function (i.e. when the two points are at the same location on the annulus, ) in the frame of the central peak. As expected the amplitude of fluctuations around the peak have an ellipsoidal shape that is more elongated along the smallest curvature . This zero-lag annulus correlation is dominated by the square of the mean density profile at small separations and by the fluctuations at larger separations. Figure 3 then shows the full constrained correlation function on the annulus. We use different orientations of the pair () with regard to the axis of smaller curvature (corresponding to ) of the central peak. The orientation of is described by the angle which is taken to be 0, and from the left-hand to the right-hand panel. On each plot, the angle between and , namely , vary between 0 and and the separation to the central peak is described by the value . We show the result for two different peak heights: the most likely value (top panels) and a rarer case (bottom panels), which is more relevant to our study. In each case respectively, we fix the peak curvatures to their most likely values and , respectively (see Appendix A for a


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

of ∼30 km s −1 . The same grid-based Monte Carlo code “STaRS 2 2 http://github.com/csj607/STaRS ” is used to investigate the distribution and kinematics of the neutral hydrogen regions in NGC 6886 and NGC 6881. Figure 2 shows a schematic illustration of the scattering geometry adopted in this work. The scattering region is a partial spherical shell surrounding the central He ii region. The covering factor CF is defined as , where θ is the opening angle of the partial spherical shell. The H i region is assumed to be of uniform density expanding with a single speed . Figure 2. Schematic illustration of the simulation. The neutral region is assumed to be a partial spherical shell with the inner and outer radii R in and R out , respectively, and the opening angle θ . The scattering region is divided into a large number of small cubical grids to trace each photon until escape. Far-UV He ii λ 1025 photons generated in the central He ii region are incident on the H i region. The neutral region is optically thick to Rayleigh-scattered UV photons, and optically thin to Raman-scattered optical photons. Each far-UV photon is traced until escape from the neutral region through multiple Rayleigh scattering, or after Raman scattering. The incident far-UV He ii λ 1025 is assumed to be described by a single Gaussian function determined in Section 3.2 . A simulation model is set with the following free parameters: the covering factor CF of the scattering region, H i column density N HI , and expanding speed . The incident He ii λ 1025 flux is deduced from the observed He ii λ 6560 line flux and using case B recombination theory (Storey Hummer 1995 ). We refer to Hyung et al. ( 1995 ) and Pottasch Surendiranath ( 2005 ) for NGC 6886 and also Kaler et al. ( 1987 ) for NGC 6881 for the values of T e = 1.25 × 10 4 K and n e = 10 4 cm −3 . In Figure 3 , we present our best-fit results for NGC 6886 and NGC 6881 in the middle panels. The gray lines represent the observational data and the black lines show our best-fit simulation results. Despite severe blending of the Raman-scattered He ii features with strong [N ii ] λ 6548 lines, the broad features with a red-tail structure are conspicuous in both PNe. These red-tailed features are attributed to the expansion of H i region. Figure 3. Comparison of the observational data (gray) and our simulation data (black) for NGC 6886 (left) and NGC 6881 (right). The best-fit models are found in the middle panels. In the middle-left panel, the model parameters are CF = 0.3, N HI = 5 × 10 20 cm −2 , and (CF30-N5E20-V25). We also present models with a lower expansion speed of (CF30-N5E20-V20, top-left) and a higher value of 30 km s −1 (CF30-N520-V30, bottom-left) for comparison. In the middle-right panel, the model parameters are CF = 0.6, N HI = 2 × 10 20 cm −2 , and (CF60-N3E20-V30). The top-right and bottom-right panels show the models with (CF60-N3E20-V20) and 40 km s −1 (CF60-N3E20-V40), respectively. In the case of NGC


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

. Figure 2.40. SDSSJ2324+0105. The observational data, B-spline-subtracted image, predicted lensed image, final residual, and background source model are shown from left to right, respectively. Images are orientated such that north is up and East is to the left. For each system, the results of the two source models are split into two rows with the parameterized source model on the top and the pixelized source model on the bottom. The white lines in the last panels are the caustics of the lens model. The source plane panels are magnified by factors from 2 to 32 relative to the image plane panel as indicated in each panel. The color bars indicate the intensity levels in units of electrons per second per pixel 2 . Table 2 Strong-lens Model Parameters of the 40 S4TM Grade-A Lenses Note. Column 1 is the SDSS system name. Columns 2–4 are the Einstein radius, minor-to-major axis ratio, and major-axis position angle of the SIE component with respect to the north. Column 5 indicates the number of Sérsic components used. Column 6 is the average magnification. Column 7 is the total projected mass within the Einstein radius from the best-fit lens model. Column 8 is the estimated stellar mass assuming a Chabrier IMF from Shu et al. ( 2015 ). Column 9 is the inferred dark-matter fraction within half of the half-light radius. Column 10 provides the χ 2 value and the dof. 5. Discussion The S4TM survey is optimized to select strong-lens systems with relatively lower-mass lens galaxies as a complementary sample to the SLACS sample. In Figure 3 , we compare the stellar velocity dispersion, Einstein radius, total projected mass within the Einstein radius, and stellar mass between the two lens samples. The SLACS sample refers to the 63 grade-A lenses with lens models in Bolton et al. ( 2008a ). Figure 3. Distributions of the stellar velocity dispersion, Einstein radius, total enclosed mass within the Einstein radius, and stellar mass for the S4TM (solid histograms) and the SLACS (dashed histograms) lens samples. The stellar mass is derived from HST F814W-band photometry assuming a Chabrier IMF as explained in Shu et al. ( 2015 ). The distribution of the stellar velocity dispersions σ SDSS of the S4TM sample has a strong peak at about 200 km s −1 and declines rapidly on both ends. On the other hand, the distribution of σ SDSS of the SLACS sample is almost flat from 200 to 320 km s −1 . As a comparison, the median σ SDSS of the S4TM sample is 203 km s −1 , while it is 243 km s −1 for the SLACS sample. The median Einstein radius of the S4TM sample is 100, almost 15% smaller than that of the SLACS sample (117). The distributions of the Einstein radii for the two samples further suggest that the S4TM sample is more abundant in systems with Einstein radii smaller than 08 (12/40 versus 3/63) and lack systems with Einstein radii larger than 12 (8/40 versus 30/63). We basically expect the distributions of M Ein to be similar to those of b SIE because Σ crit , which is


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M.P. is a member of the working team of projects AYA2013-40979-P and AYA2013-48226-C3-2-P, funded by MINECO. P. M. acknowledges the support from the European Research Council (grant CAMAP-259276), the government of Spain (MINECO grant AYA2013-40979-P), and the regional government of Valencia (grant PROMETEO-II-2014-069).


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

where we expected to be incomplete, and fewer at bright magnitudes, where we expected to be very complete. Flux uncertainties and missing VISTA fluxes were assigned in the same way as during the contamination test in Section 3.3 . Mock galaxies were not assigned photometric flags. This likely results in an incompleteness of only a couple percent, as the fraction of all sources affected by flags is small. All sources with an S/N of greater than 3.5 in any one band are flagged in another band less than 2% of the time on average (never exceeding 4%). In addition, all images have ∼0.5% of all pixels flagged on average (never exceeding 2%). The photometric catalog of 600,000 mock galaxies was then run through EAZY to generate z phot PDFs, and our high- z galaxy selection was applied. The completeness was defined as the number of mock galaxies recovered divided by the number of input mock galaxies, as a function of input absolute magnitude and redshift. Figure 7 shows the results of our simulation. We define the 50%-completeness limit as the absolute magnitude where the area under the curve falls to less than 50% of the areas under the average of the M UV, i ′ = −25 to M UV, i ′ = −28 curves, which we find to be at M UV, i ′ = −22 ( m = 24 for z = 4). 4. Results 4.1. The Rest-frame UV <italic>z</italic> = 4 Luminosity Function We utilize the effective volume method to correct for incompleteness in deriving our luminosity function. The effective volume ( V eff ) can be estimated aswhere is the comoving volume element, which depends on the adopted cosmology, and C ( M UV, i ′ , z ) is the completeness as calculated in Section 3.4 . The integral was evaluated over z = 3–5. To calculate the luminosity function, we convert the apparent i -band AB magnitudes ( m i ′ ) to the absolute magnitude at rest-frame 1500 Å ( M UV, i ′ ) using the following formula:where d L is the luminosity distance in pc. The second and third terms on the right-hand side are the distance modulus. Our UV luminosity function is shown as red diamonds in Figure 8 . We note that we do not include the luminosity function data points in bins M UV, i ′ ≥ −22 in our analysis, as these bins are below our 50%-completeness limit as discussed in Section 3.4 , where the completeness corrections are unreliable owing to the low S/N of our data. This is confirmed by comparing to the HST CANDELS results in these same magnitude bins, which are more reliable owing to their higher S/N. In Figure 8 we also include the UV luminosity function of z = 4 UV-selected galaxies from deeper HST imaging by Finkelstein et al. ( 2015 ) as green squares. In the bin where we overlap with this data set ( M UV, i ′ = −22.5), both results are consistent; however, if the luminosity function derived from HST imaging is extrapolated to brighter magnitudes, it would fall off more steeply than our luminosity function. While our luminosity function declines from M UV, i ′ = −22 to M UV, i ′ = −24, it flattens out at


O O O O O O O O O O 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 O O O O O O O O O O B-Organization O O O B-Person I-Person B-Person I-Person O B-Person I-Person O O O O O O B-Person O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization O B-Organization B-Person O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Person O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Grant I-Grant I-Grant B-Person O B-Person O O O B-Organization O O O B-Grant O B-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization O B-Organization O B-Grant I-Grant B-Person O O O O O B-Organization I-Organization O O O O O O O O O B-Person I-Person O O O B-Organization I-Organization I-Organization 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 O O O O O O O B-Database O O O O O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Survey O O O O O O O O B-Location B-Organization I-Organization I-Organization

ACKNOWLEDGEMENTS We thank the anonymous referee for the comments, especially for the suggestion to separate our original manuscript into two standalone publications, which have improved the clarity of the papers. We also thank Rainer Beck for his careful reading and valuable suggestions and comments as the MPIfR internal referee, and Aristeidis Noutsos, Shane O’Sullivan, and Dominic Schnitzeler for insightful discussions about this project. YKM was supported for this research by the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. YKM acknowledges partial support through the Bonn-Cologne Graduate School of Physics and Astronomy. AB acknowledges financial support by the German Federal Ministry of Education and Research (BMBF) under grant 05A17PB1 (Verbundprojekt D-MeerKAT). ASH and SKB acknowledge support by NASA through grant number HST-AR-14297 to Haverford College from Space Telescope Science Institute, which is operated by AURA, Inc. under NASA contract NAS 5-26555. ASH is partially supported by the Dunlap Institute, which is funded through an endowment established by the David Dunlap family and the University of Toronto. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the NED which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The WHAMSS have been funded primarily through awards from the U.S. National Science Foundation.


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

a function of time (symbols). We use Bayesian analysis (Sect. 2.4 ) to calculate how our oscillation model describes the observational data. Our Bayesian analysis returns the posterior PDF for each of the model parameters. These may be plotted as histograms, such as those for and in Fig. 6 , or summarised by quoting the median and 95% credible intervals as in Table 2 . The red shaded region represents the 99% credible interval for the loop position described by our model, including the estimated noise . The parameters (e.g., vertical dotted line denoting ) or model components (e.g., background trend shown by blue line in top left panel) plotted in figures correspond to the median values of the relevant parameters. For comparison, the results for analysing Loop #1 with a model including the fundamental kink mode only is shown in Fig. 16 . The top right panel shows the wavelet analysis of the signal. The colour contours represent the spectral amplitude (square root of the spectral power). The dashed lines show the time-dependent periods of oscillation , which demonstrate a gradual increase during the course of the oscillation of approximately 20%. The middle right panel shows a 2D histogram of the seismologically determined transverse density profile parameters based on our MCMC model samplings (with intensity normalised to unity). The red bars correspond to the median values and 95% credible intervals based on the individual 1D histograms (bottom panels) and quoted in Table 2 . They give a convenient summmary of the localisation of the parameters by our model fit, though we note that they do not reflect the strong inverse relationship between and in determing the damping rate (see Fig. 2 ). Consequently, the 2D contour (likely density profile parameters) is significantly more localised than considering the posterior credible intervals independently would suggest. It is necessary to appreciate this inverse relationship when considering the quality of the constraint on density profile indicated by the data. For example, is generally less well constrained for lower density contrasts than for larger density contrasts, as discussed in Sect. 2 . We note that the 95th percentiles quoted in this paper correspond to a confidence interval, in contrast to Paper I where the errors were estimated values of . For example, Paper I gives the density contrast for Loop #1 as , and here (Table 2 ) it is , which corresponds to a that is approximately five times smaller. On the other hand, the for the here is comparable to that in Paper I. The solid curves in the histogram plots are fits to the data using the exponentially modified Gaussian function of the form where is the complementary error function, is a constant determining the amplitude, and are the mean and standard deviation of the Gaussian component, respectively, and is the rate of the exponential component. Equation ( 18 ) is found to describe the histogram profiles well, in particular when the


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation 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 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 B-CelestialObject O B-Mission 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 B-Organization O O O O B-Software I-Software B-Software O O O O O B-Software O O O O O B-CelestialObject O O B-Mission O O O O O O O O O O B-Organization O O O O O O O B-Mission O B-Software O O O B-CelestialObject O O O O O O O O O O O O O B-Mission O O O O O O O B-Software O O O O B-Organization O O O B-Organization O O O O O B-Mission 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 B-Organization 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

to the nucleus, e n directed along the orbital angular momentum perpendicular to the orbital plane, and e t being orthogonal to both e r and e n . By comparing 14 cometary orbits, Marsden et al. (1973) derived the following model for the non-gravitational acceleration: with the constant parameters A 1 , A 2 , and A 3 and the empirical activity function The solar distance r ′ = r ( t − Δ t ) includes a time-shift Δ t introduced by Yeomans Chodas (1989) to account for activity asymmetries with respect to perihelion, as studied by Sekanina (1988) . As we discuss later, g ( r ) is not directly proportional to the sublimation flux, as originally stated by Marsden et al. (1973) , Eq. ( 4 ). The Marsden parameters provide an excellent albeit empirical description of the non-gravitational acceleration. The integration of a Marsden-type orbit proceeds by solving Eq. ( 1 ) with the non-gravitational acceleration Eq. ( 2 ). 2.2 Orbit determination based on spacecraft data Moving beyond the Marsden model requires more detailed data from spacecraft missions or radar observations about the magnitude and direction of the observed non-gravitational acceleration in space. For 67P, the Rosetta mission provides a two-year data set for the rotation-axis orientation (see Kramer et al. 2019 ) and the orbital evolution, but an iterative process was required to obtain the three-dimensional non-gravitational acceleration with high precision. We started with the multi-arc orbital solution from the flight dynamics team at ESOC, which is available as SPICE kernel (CORB_DV_257_03___T19_00345.BSP) and is returned from the Horizons system as the position for 67P during the Rosetta mission. In the following we refer to it as r ESOC ( t ). Before and after the Rosetta mission, Horizons returns the solution 67P/K154/2 based on Marsden parameters with a discontinuous jump by 2000 km into the Rosetta period. In addition to this discontinuity in Horizons, further discontinuities in the ESOC position vector r ESOC ( t ) during the Rosetta mission exist, which do not allow us to obtain the acceleration by a second derivative of the position vector (see the discussion by Attree et al. 2019 ). Table 1 Representative initial conditions for the orbit integration at JDB 2 456 897.71970 (≈ −350 days from perihelion), mean equator, and equinox of the terrestrial J2000 frame. Notes. An additional 30 nearby initial conditions have been identified that differ on average by less than 22 km over the (−400, 400) day interval from the partly nonphysical ESOC orbit. Table 2 Non-gravitational acceleration parameters used by Horizons in conjunction with the standard parameters α = 0.1112620426, k = 4.6142, m = 2.15, n = 5.093, and r 0 = 2.808 for g ( r ) in Eq. ( 3 ). An accurate estimate of the non-gravitational acceleration is tied to finding the best possible initial condition during a time with negligible cometary activity. We iteratively refined the orbit to identify the initial conditions


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ACKNOWLEDGEMENTS We gratefully thank N. M. Frster Schreiber and the KMOS and SINS/zC-SINF team for sharing the reduced KMOS and SINFONI data analysed in this work. We thank P. G. van Dokkum for the GNIRS spectrum and P. Rosati who provided the reduced MUSE data. We thank also P. Martini for having kindly made available the reduced data cube of M82. AC, MT and FL acknowledge the support from the grants PRIN-MIUR 2015, PRIN 2017, and ASI n.2018-23-HH.0.


O O O O O O O O O O O O O B-Person I-Person O O O O O O O B-Person I-Person O O O O O O B-Software O B-Instrument O O B-Person I-Person O O O O O O O B-Telescope 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 B-Software I-Software B-Citation I-Citation I-Citation I-Citation O O O O O O B-Organization O B-Grant O B-Grant O B-Person O B-Organization O B-Grant O B-Grant I-Grant O B-Person B-Instrument O O B-Telescope I-Telescope I-Telescope I-Telescope O O O O B-Collaboration I-Collaboration I-Collaboration I-Collaboration O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O O B-Telescope I-Telescope I-Telescope O O O O O B-Organization I-Organization I-Organization O B-Grant I-Grant I-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O O B-Archive I-Archive I-Archive I-Archive I-Archive B-Archive B-Organization O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization O B-Grant O O B-Archive O O B-Telescope O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant O O O O O O O B-Telescope I-Telescope O O O O B-Organization I-Organization I-Organization O B-Organization O B-Grant O B-Grant O O B-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization I-Organization I-Organization O O O O O O B-Database O B-Database I-Database I-Database O O B-Organization B-Location I-Location O B-URL O O O B-Archive I-Archive I-Archive I-Archive O O B-Software I-Software I-Software I-Software I-Software O B-URL O O O B-Organization O B-Organization O B-Telescope B-Telescope I-Telescope B-Telescope

Thanks to the anonymous referee for their fast and generous comments. Thanks to Keivan Stassun for his comments on activity diagnostics, to Michael Cushing for providing a pre-release version of SpexTools for uSpeX, and to Coryn Bailer-Jones for providing the spectral response profiles for Gaia . Additional thanks to Adam Kraus and Aaron Rizzuto for useful discussions on the content of this manuscript. PPD Figures were made with the aid of the triangle.py package (Foreman-Mackey et al. 2014 ). This research was supported by NASA grants NNX10AQ36G and NNX11AC33G to E.G. and NASA grants ADAP12-0172 and 14-XRP14 2-0147 to T.B. SNIFS on the UH 2.2 m telescope is part of the Nearby Supernova Factory project, a scientific collaboration among the Centre de Recherche Astronomique de Lyon, Institut de Physique Nuclaire de Lyon, Laboratoire de Physique Nuclaire et des Hautes Energies, Lawrence Berkeley National Laboratory, Yale University, University of Bonn, Max Planck Institute for Astrophysics, Tsinghua Center for Astrophysics, and the Centre de Physique des Particules de Marseille. Based on data from the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement no. NNX-08AE38A with the National Aeronautics and Space Administration, Science Mission Directorate, Planetary Astronomy Program. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non- HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. The CHARA Array is funded by the National Science Foundation through NSF grants AST-0606958 and AST-0908253 and by Georgia State University through the College of Arts and Sciences, as well as the W. M. Keck Foundation. This research made use of the SIMBAD and VIZIER Astronomical Databases, operated at CDS, Strasbourg, France ( http://cdsweb.u-strasbg.fr ), and of NASA Astrophysics Data System, of the Jean-Marie Mariotti Center SearchCal service ( http://www.jmmc.fr/searchcal ), co-developed by FIZEAU and LAOG/IPAG. Facilities: IRTF, UH:2.2 m, CHARA


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

measured ratios range between 1.04 ± 0.05 for the meteorite Malakal and 4.56 ± 0.38 for the meteorite Jilin. The ratio for Jilin is not representative because this meteorite experienced a complex exposure history and 26 Al is not in saturation, making the 22 Na/ 26 Al ratio too high (Begemann et al. 1985 ). Ignoring Jilin for the discussion, the upper limit is reduced to 2.48 for the meteorite Dhajala. Again, excluding Jilin, the average value for the 22 Na/ 26 Al ratio for the studied meteorites is 1.46 ± 0.27. Both the range and the average value are in agreement with the model predictions. In addition, the 22 Na/ 26 Al ratios for a variety of ordinary chondrites studied by Bhandari et al. ( 1994 ) range between ∼1 and ∼2.3, again in agreement with the model predictions. For the meteorite Chelyabinsk, Alexeev et al. ( 2019 ) give 22 Na/ 26 Al ratios of 1.4 ± 1.2 for a depth larger than 100 cm, 2.42 ± 0.17 at a depth of 10–50 cm, and of 2.53 ± 0.24 for depths of 50–100 cm. For the Košice meteorite, the same authors measured 1.45 ± 0.20 for depths below 10 cm and 1.52 ± 0.21 for depths between 10 and 50 cm. All values are well within the range of the predicted 22 Na/ 26 Al ratios we modeled for an L-chondrite with a radius of 45 cm (see Figure 6 ). The data for meteorites indicate that the 22 Na/ 26 Al production rate ratios vary enough to be useful for studies on the temporal variability of the solar modulation function. For the following, more qualitative discussion, we use the reconstruction of the solar modulation function given by Usoskin et al. ( 2017 ) shown in Figure 2 . Because the data are for 1 au, we adjusted them to the meteoroid orbits at ∼3 au by simply subtracting ΔΦ = 60 MeV, which we determined above as the best value for the long-term average gradient in the inner solar system (see also Section 2.3 ). Note that this procedure is surely not entirely correct because the radial gradient in the inner solar system is expected to vary on short timescales. However, for our qualitative discussion, this rough assumption is sufficient. Considering now a meteorite that fell in 1970, the 22 Na production rate is determined by the solar modulation function from ∼1965 to ∼1970, which was 〈Φ〉 ∼ 480 MeV. According to Equation ( 18 ), this corresponds to a 22 Na/ 26 Al (1965–1970) production rate ratio of ∼2.08. For a meteorite that fell in 1980, the relevant parameters are 〈Φ〉 ∼ 420 MeV and 22 Na/ 26 Al (1975–1980) ∼2.25. Finally, a meteorite that fell in 1985 experienced 〈Φ〉 ∼ 670 MeV, corresponding to 22 Na/ 26 Al (1980–1985) ∼1.7. Consequently, the variability in 22 Na/ 26 Al production rate ratios for the three cases is more than 30%, sufficient for detailed studies of the temporal variability of the solar modulation function using freshly fallen meteorites. 5.2. Can We Put Constraints on the Meteoroid Orbits? For the qualitative discussion above, we assumed that the meteorites spent their entire lifetime in the same orbit at ∼3 au and


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

to bounce against the surface to gain velocity in the direction perpendicular to the surface. The simplest way to treat this process is to assume an elastic collision, Here is the fraction of kinetic energy retained by the product colliding with the surface, which has an effective mass . For the specific case of hyperthermal scattering of O on graphite surface, Hayes et al. (2012) have proposed an equivalent mass of 130 a.m.u. The effective mass is much higher than one single C atom or even a carbon ring because the rigidity of the surface induces a collective behavior of the atom and a higher effective mass. The chemical energy is certainly split in many different ro-vibronic states, not at all in thermal equilibrium. We assumed that an equal share of energy is spread throughout all the degrees of freedom, so that the total chemical energy available for the kinetic energy perpendicular to the surface is . We assumed that the only velocity perpendicular to the surface corresponds to a distribution of speed and temperature. In this sense, . The probability for the product to have an energy (or liberation velocity) higher than the binding energy becomes In Fig. 3 we compare the results obtained experimentally (blue triangles) to our predictions. We first used a constant of 0.4, which implies that we assumed that 40% of the total energy is retained by the product (red hexagons). In this case, our predictions do not reproduce the experimental results well. In a second step, we considered a fraction that depends on the mass of the products, as described in Eq. (1). These predictions are represented as green squares and do match the chemical desorption efficiencies observed experimentally very well. This clearly indicates that the chemical desorption process is sensitive to the binding energy, the energy distributed in the products (depending on their mass), and the degrees of freedom. For some of the reactions such as OH+H, O+H, our predictions slightly underestimate the experimental results. Table 2 Binding energies on bare and icy surfaces. Notes. For HCO, CHO, and CHOH we assume . (a) Cazaux Tielens (2004) ; (b) Bergeron et al. (2008) ; (c) Al-Halabi van Dishoeck (2007) , Amiaud et al. (2007) ; (d) Pirronello et al. (1997) ; (e) Amiaud et al. (2006) for low coverage; (f) Minissale et al. (2015) ; (g) Dulieu et al. (2013) ; (h) Speedy et al. (1996) , Fraser et al. (2001) energy derived of 5800 K with pre-factor of 10 s here corrected to have a pre-factor of 10 s; (i) Noble et al. (2012a) ; (j) Borget et al. (2001) , Minissale (2014) ; (k) Karssemeijer Cuppen (2014) ; (l) Garrod Herbst (2006) ; (m) Noble et al. (2012b) ; (n) Collings et al. (2004) ; (o) Fuchs et al. (2006) ; (p) extrapolated from Kimmel et al. (2001) . The formula (Eq. (2)) can be translated to the case of silicates surfaces. The efficiency is slightly lower, and thus reducing the effective mass from 130 to 110 matches the most constrained reaction (O+O) very well. Taking the


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ACKNOWLEDGEMENTS I thank B. Hansen and B. Zuckerman for useful discussions.


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We thank the anonymous referee for providing constructive comments that have improved the contents of this paper. We thank Ronin Wu for sharing PACS observations. STO2 is a multi-institutional effort funded by the National Aeronautics and Space Administration (NASA) through the ROSES-2012 program under grant NNX14AD58G. This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Y.S. acknowledges support from the NASA postdoctoral program. M.G.W. was supported in part by NSF grant AST-1411827. C.K. was supported through NSF grant AST-1410896. Kristina Davis is currently supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1801983.


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

measurements in -band for the two prototypes are presented in Fig. 3 . Fig. 3 Azimuthal average of the amplitude transmission in band for the chromium oxide (green) and aluminum (blue) SLLC prototypes as a function of the normalized distance from the apodizer center (). The increase in transmission close to zero corresponds to the pattern introduced in the design to facilitate the opto-mechanical centering of the apodizer. The design specifications are represented by a gray envelope. The transmission of the microdots map designed by Aktiwave is also plotted in orange. This profile appears noisier because the microdots map is binarized. The top plot shows the normalized transmission measurement, and the bottom plot shows the error with respect to the nominal transmission profile. The vertical dotted lines correspond to the radii where we plot the transmission as a function of wavelength in the left panel of Fig. 4 . The transmission map of the aluminum prototype in band is shown as an inset in the top left corner of the plot. Fig. 4 Chromaticity of the amplitude transmission of the aluminum prototype. Left : normalized amplitude transmission as a function of wavelength at different radii in the pupil (dotted vertical lines in Fig. 3 ), for which the value is indicated on the right side of each curve. Right : measured amplitude transmission at the four wavelengths as a function of the original specification. The dashed and dash-dotted lines show the 1 decade/decade and 0.5 decade/decade trends. From the very first measurements it appeared that the chromium oxide prototype did not meet the specifications in the low-transmission part (20%) of the apodizer. Because only one free position was available in the SPHERE apodizer wheel and because limited time was available for extensive tests during the reintegration period, we chose to keep the aluminum prototype inside the instrument, which is closer to the specifications. For this reason, we do not discuss the results from the chromium oxide prototype in more detail, and we refer to Vigan et al. (2014) for some additional information and measurements. From now on, we refer only to the aluminum prototype. For the aluminum prototype, the transmission is closer to the specifications, but there are still some problems in the low-transmission parts of the apodizer. The minimum transmission obtained in the central part and at the edge of the pupil is 2.5%. This measurement illustrates the difficulty of manufacturing apodizers for the near-IR with very high optical densities, as was already identified by previous studies in the context of the GPI instrument ( Sivaramakrishnan et al. 2009 ). A microdots apodizer relies on the absorption and diffraction of light by small (20 m in our case) metallic dots deposited on a substrate whose distribution is such that the density of dots produces the required transmission profile. As detailed by Dorrer Zuegel (2007) , the spatial filtering produced by the


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O

. This correction therefore also accounts for the average offset induced from the PSF residual ellipticities which is of similar amplitude to the c -terms 15 . We note that we choose not to implement an additional empirical PSF-dependent correction for the PSF contamination, as our α measurements are noisy. Furthermore, if α is not a constant, and is instead correlated with galaxy properties, atmospheric seeing, or image depth, for example, applying an average correction would artificially imprint a PSF-correlation across our full data set that could potentially be more problematic than the low-level average bias that we currently detect. 3.5.2. Accuracy requirements and validation of the first-order systematics model In Fig. 7 we compare two measured star-galaxy cross-correlation functions with the amplitudes predicted by the linear systematics model from Eq. ( 12 ). The standard star-galaxy cross-correlation function, , is related to the linear systematic model parameters as 16 Fig. 7. Comparison of two star-galaxy cross-correlation functions with their corresponding linear systematics model predictions (Eqs. ( 15 ) and ( 16 )) for the five tomographic bins. The star-galaxy cross correlation function (dark blue) is in reasonable agreement with the model (blue bar), where the width of the model reflects the 2 σ uncertainty. The residual star-galaxy cross correlation function (shown pink), and corresponding model (grey), are scaled by a factor of ten in order to display the measurements on the same scale. We also scale the correlation functions by to aid visualisation of the large-scale signal. We can also construct a residual star-galaxy cross correlation function , where As with previous sections, we focus on the ξ + ( θ ) term only, as the ξ − ( θ ) term is consistent with zero for the OmegaCAM PSF. Figure 7 shows reasonable agreement between the measured star-galaxy correlation function (shown dark blue), with the model prediction (blue band, Eq. ( 15 )) which we calculate by taking α from Fig. 6 , β = −1, δ ε PSF from the detector level model in Sect. 3.4 , and the other terms measured directly from KiDS-1000. We note that the band encompasses the 2 σ uncertainty from the measurement of α . All other error terms are sub-dominant. In contrast we find little agreement between the measured residual star-galaxy correlation function (shown pink), with the model prediction (grey dotted line, Eq. ( 16 )). This suggests that our faint magnitude linear extrapolation of the residual PSF ellipticity, δ ε PSF , is not representative of the detector level bias that the galaxies have experienced. We find that the average residual star-galaxy correlation is significantly lower than the expectation from the extrapolated chip-dependent residual PSF ellipticity model. We therefore conclude that whilst we find significant flux dependence in the PSF residual ellipticity for 3/32 OmegaCAM chips (see Fig. 4 ), there is no evidence in the shear catalogue


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

0.7 dex), which seems unlikely to be consistent with our assumed scatter of , given the small size of that sample. Figure 3. Halo-to-stellar mass ratio (the inverse of the SHMR) as a function of stellar mass. The red and blue dots denote the measurements for the low- and high-redshift samples, respectively. The red solid curve and cyan-shaded area are the best-fit SHMR relation and its 1 σ uncertainty derived by fitting our low-redshift sample combined with the results of H15. The blue solid curve is the best-fit SHMR relation of our high-redshift sample combined with H15. The red and blue dashed curves are the relations fitted by Leauthaud et al. ( 2012 ), using COSMOS measurements. We also compare our measurements with other galaxy–galaxy lensing measurements, including SDSS (Mandelbaum et al. 2006 ), CFHTLenS (H15), COSMOS X-ray clusters (Leauthaud et al. 2010 , 2012 ), and GAMA (Han et al. 2015 ). This scatter for individual systems suggests that selection effects may still be a concern, in our and other published determinations of the SHMR. For instance, Leauthaud et al. ( 2015 ) found that the CMASS sample is significantly affected by mass incompleteness, and is ∼80% complete at only in the narrow redshift range z = [0.51, 0.61]. The LOWZ sample, on the other hand, is ∼80% complete at for z = [0.15, 0.43]. If completeness is correlated with high or low stellar mass for the primary galaxy, it could mean that the true scatter is larger than what we have assumed. Follow-up studies of the individual objects in our samples, as well as data from larger forthcoming lensing surveys, will be required to determine the exact effect of these biases on the mean relation. A Chabrier ( 2003 ) IMF was assumed by the measurements of H15 , Leauthaud et al. ( 2012 ), and Han et al. ( 2015 ). Ilbert et al. ( 2010 ) investigated the possible sources of uncertainty and bias by comparing stellar mass estimates between methods. The expected difference between a Kroupa IMF and a Chabrier IMF is not significant. 5. Conclusions Combining samples of massive galaxies and clusters from the redMaPPer and LOWZ/CMASS catalogs, respectively, with the shear catalog of the CS82 survey, we have measured the projected density profiles around central galaxies at two different redshifts. We fit the lensing shear signal around these galaxies as a sum of contributions from the main DM halo, the central galaxy, centering errors and physical offsets between the galaxy and the halo, and neighboring halos. From the resulting fits, we determine the mean halo concentration as a function of mass, and also the mean relation between (central) stellar mass and halo mass (the SHMR). Our main results are as follows. 1. We find that halo concentration decreases weakly with mass, c = A ( M / M 0 ) B , with the amplitude A ∼ 5–6 and slope B ∼ −0.14. These parameter values are in good agreement with the predictions of simulations, e.g., Duffy et al. 2008 or Klypin et al. 2016 , given a


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The authors thank the anonymous referee for comments and suggestions that improved the paper. The authors also thank J. Martin-Pintado for allowing us to utilize his 2001 7 mm observations of Sgr B2 North. C.D.P. and M.M.M.L. were supported by NSF grant AST-1211460. M.M.M.L. was additionally supported by NSF grant AST-1109395 and the Humboldt Foundation. T.P. acknowledges financial support through a Forschungskredit of the University of Zürich, grant No. FK-13-112, and from the DFG priority program 1537 “Physics of the interstellar medium.” R.S.K. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) via the SFB 881 The Milky Way System (sub-projects B1, B2, B5, and B8), and the SPP (priority program) 1573, Physics of the ISM. He also thanks the European Research Council for significant funding under the European Community’s Seventh Framework Programme (FP7/2007-2013) via the ERC Advanced Grant STARLIGHT (project number 339177). R.G.M. acknowledges support from program UNAM-DGAPA-PAPIIT IA101715.


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lines is explained by (i) the very similar upper level energy and critical densities ( E up = 33 K, and 35 K, and n cr ∼ 7−5 × 10 5 cm −3 , and 17−9.5 × 10 5 cm −3 , at 20−100 K, for the H 2 CO and the CS line, respectively, Shirley 2015 ) and (ii) the similar distribution of the H 2 CO and CS abundance in the disk suggested by thermo-chemical models (e.g., Fedele Favre 2020 ). Both molecules are released from grains and/or formed in the gas phase in the molecular layer, following the photodissociation of CO, which makes available C for the formation of small hydrocarbons, together with atomic O. The CS is formed from reactions of small hydrocarbons with either ionized or atomic S (e.g., Le Gal et al. 2019 ), while the main formation route of H 2 CO is via the reaction CH 3 +O (e.g., Loomis et al. 2015 ). Additionally, CO 2−1 and CS 5−4 emissions strongly decrease in the outer disk-midplane, where molecules freeze-out onto dust grains (freeze-out layer). By contrast, H 2 CO is only a factor of 2 less intense, which indicates that the H 2 CO molecules trapped on the icy grains in the cold midplane are partially released by nonthermal processes (UV, X-ray, cosmic-rays-induced, and/or reactive desorption). We tentatively detected CH 3 OH emission at 3 σ in the disk region where H 2 CO and CS emissions are bright, that is to say the warm molecular layer where molecules should be efficiently released from grains, while it remains undetected in the outer disk midplane. In the right panel of Fig. 3 , the [CH 3 OH]/[H 2 CO] abundance ratio inferred for the Class I disk of IRAS 04302 is compared with the estimates obtained in hot corinos around Class 0 protostars, in the other Class I disks observed by ALMA-DOT, in the Class II disks of TW Hya and HD 163296, and in comets. The abundance ratio in the disk of IRAS 04302 (∼0.5−0.6 for T ex ∼ 20−100 K) is consistent with the upper limits derived for the other Class I and early Class II disks observed by ALMA-DOT (between 0.7 and 3.2, Podio et al. 2019 ; Garufi et al. 2020 , and in prep.). On the other hand, it is lower by a factor of ∼2.5 than that inferred for the Class II disk of TW Hya (∼1.27−1.73 for T ex ∼ 25−75 K, Walsh et al. 2016 ; Carney et al. 2019 ), while it is larger than the upper limit obtained for HD 163296 ( 0.24, Carney et al. 2019 ). As methanol forms on the dust grains due to CO freeze-out and subsequent hydrogenation, the lower [CH 3 OH]/[H 2 CO] in HD 163296 may be due to a smaller degree of CO ice chemistry in the warmer disks around Herbig stars (e.g., Pegues et al. 2020 ). The abundance ratios in Class I and Class II disks are from a factor of 2 to 2 orders of magnitude lower than the values inferred in the hot corinos around the Class 0 protostars IRAS 16293-2422 A and B in Taurus ( Jørgensen et al. 2018 ; Persson et al. 2018 ; Manigand et al. 2020 ), and NGC 1333-IRAS 4A and IRAS 2A in Perseus ( Taquet et al. 2015 ). This difference may be due to an evolution of the methanol and


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Acknowledgements I am very grateful to the anonymous referee for his/her helpful comments and suggestions that have led to a significant improvement in the presentation of this manuscript. I am also grateful to Amherst College for its support through a John Woodruff Simpson Lectureship, and to Purple Mountain Observatory in Nanjing, China, for its hospitality while part of this work was being carried out.