B-Person O O O O O O O O O O B-Person I-Person I-Person O O O O B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person I-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person O B-Person I-Person O O O O B-Person I-Person O O O O B-Database I-Database I-Database O B-Person I-Person 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 O O O B-Person O B-Person O O O O O B-Person O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Grant I-Grant O B-Grant I-Grant B-Person O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O O B-Fellowship I-Fellowship O O O O O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility B-Person O B-Person O O O O B-Organization B-Grant I-Grant I-Grant O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization O O B-Organization B-Grant O B-Grant O O O B-Survey O B-Survey O O O O O B-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization I-Organization O B-Location B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Location B-Organization O B-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization O B-Location O B-Survey O O O O B-Organization I-Organization I-Organization O O O O O B-Survey O O O B-URL O O B-Database I-Database I-Database O O O O O O O O O O O O O B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory

Y.T.L. dedicates this work to the loving memory of his mother, Ms. Chun-Chih Kang. We are grateful to Keiichi Umetsu, David Spergel, Jim Gunn, Cheng Li, Rita Tojeiro, Zheng Zheng, Frank van den Bosch, Masamune Oguri, Uros Seljak, Lihwai Lin, Chung-Pei Ma, and Ying Zu for helpful comments, to Gerard Lemson for help with the Millennium Simulation database, to Surhud More for providing the data used in Figure 4 in electronic form, and to Bau-Ching Hsieh for help with parallel computation. We thank the anonymous referee for a constructive report. Y.T.L. thanks I.H. for constant encouragement and support. Y.T.L. acknowledges support from the Ministry of Science and Technology grants MOST 102-2112-M-001-001-MY3 and MOST 104-2112-M-001-047. R.M. is supported by the Department of Energy Early Career Award Program and by a Sloan Fellowship. This work was completed in part using the computing resources provided by the University of Chicago Research Computing Center. B.D. and A.K. were supported by the NASA ATP grant NNH12ZDA001N and by the Kavli Institute for Cosmological Physics (KICP) at the University of Chicago through grants NSF PHY-0551142 and PHY-1125897. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The SDSS Web Site is http://www.sdss.org/ . The Millennium Simulation databases used in this paper were constructed as part of the activities of the German Astrophysical Virtual Observatory (GAVO).


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

stage. From our observations of SDSS 1137 we also see that the binary mostly emits in blue bands as occurs in other disc accreting AM CVns. The X-ray emission is an order of magnitude smaller than the UV one, and the X-ray spectrum is also soft (consistent with the one of SDSS 1411 within errors). This suggests that the X-ray component is less important in AM CVns with discs than the UV component. The main uncertainty in the light curve of Fig. 1 is the duration of the outburst. This is due to the poor photometric coverage at early epochs (between 2017 December 21 and 2018 March 25), as well as the instrumental fluctuations and relatively large photometric uncertainties of the ZTF data, especially in the g band. However, an upper limit can be established considering the observation date of the XMM – Newton data (2017 December 21) and the beginning of the first rebrightening, which then imply an outburst duration of 380 d. For comparison purposes we have plotted that value in the outburst duration versus orbital period relation for AM CVns (Fig. 4 ). This duration is comparable to that of the recently reported value for SDSS 0807 (Rivera Sandoval et al. 2020 ), for which the first detected superoutburst lasted 390 d. It is clear that these two systems do not follow expectations from the DIM. They exceed by several times their expected duration, suggesting that there are additional mechanisms at work causing the outbursts of (long-period) AM CVn binaries. Figure 4. Orbital period ( P orb ) versus outburst duration (τ dur ) for AM CVns. The orange line indicates the relation obtained from the DIM (Cannizzo Ramsay 2019 ) and the blue line the empirical relation (Levitan et al. 2015 ) derived for systems with P orb up to 37 min. Upper limits are marked in cyan colour. We superimpose the results for SDSS 1137 from this paper, for SDSS 0807 (Rivera Sandoval et al. 2020 ) and for SDSS 1411 (Rivera Sandoval Maccarone 2019 ). The first two systems clearly exceed the expected τ dur from the DIM by several times suggesting the presence of additional (or even different) mechanisms in the origin of the outbursts, while SDSS 1411 still fits the predictions from the model. Note that the duration of the outburst in SDSS 1137 is an upper limit, where the yellow circle denotes the duration between the first ZTF measurement and the chosen date for the end of the outburst (See Section 4 ). The dashed line indicates the AM CVn with the longest P orb identified so far (SDSS J1505+0659, P orb = 67.8 min) as reported by Green et al. ( 2020 ) and which up to date has not shown outbursts. Other data in the plot were taken from Cannizzo Ramsay ( 2019 ) and references therein. A error for the outburst duration for all systems that do not have an upper limit was considered, as done by Levitan et al. ( 2015 ). 4.1 Colour evolution during outbursts of AM CVns: evidence of enhanced mass-transfer In order to investigate the mechanism(s) that drive outbursts in long-period


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 B-Telescope O B-Person I-Person O B-Person I-Person O O O B-Person O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Person O O O O O B-Telescope O O B-Organization I-Organization I-Organization O O O O O O B-Organization O O O O O B-Organization O O O B-Archive I-Archive I-Archive I-Archive O O O B-Database O O O O B-Organization B-Location I-Location O O B-Database I-Database I-Database B-Database O O O O O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization O O O B-Organization

We are grateful to the referee for thoughtful comments that improved the content and clarity of the paper. We thank Andrew Siemion for assistance observing remotely with the GBT, and Carl Heiles and Leo Blitz for useful discussions. JM received support from a National Science Foundation Graduate Research Fellowship. KA is supported by funding through Herschel , a European Space Agency Cornerstone Mission with significant participation by NASA, through an award issued by JPL/Caltech. This research used NASA Astrophysics Data System Bibliographic Services, the SIMBAD data base, operated at CDS, Strasbourg, France, and the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.


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

Acknowledgments SS and MS acknowledge support from Higher Education Commission (HEC), Pakistan. MNS Qureshi acknowledges the Higher Education Commission (HEC), Pakistan grant 20-1886/RD10. PHY acknowledges NSF grant AGS1550566 to the University of Maryland, and the BK21 plus program from the National Research Foundation (NRF), Korea, to Kyung Hee University. He also acknowledges the Science Award Grant from the GFT Foundation to the University of Maryland.


O O O O O B-Telescope O O B-Organization O O O O O O O O O B-Organization O O O B-Organization O O O O O O B-Telescope O O O B-Location I-Location I-Location I-Location O O B-Organization I-Organization I-Organization O O B-Location B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory I-Observatory O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O B-Database I-Database I-Database O O O O O O O O O

Based on observations obtained with XMM–Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. Based on observations made with the WHT operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.


O B-Collaboration I-Collaboration O O O O B-Organization O O B-Grant I-Grant O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Telescope O O O O O O B-Person O O B-Person I-Person O B-Person I-Person O O O O O O

This Italian GAPS project acknowledges support from INAF through the “Progetti Premiali” funding scheme of the Italian Ministry of Education, University, and Research. We thank the TNG staff for help with the observations. K.B. also thanks V. D’Orazi and C. Sneden for fruitful discussions of heavy elements.


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

linearly by the axial mean magnetic field vector , then only the formulationis possible if pseudoscalars, such as or , are not available. Without global rotation, an effect in the formulation does not occur in the linear relation ( 16 ). The parameter describes the turbulence-induced eddy diffusivity in the fluid, while the vector gives a velocity which transports (“pumps”) the large-scale magnetic field downward for positive . If, however, terms that are nonlinear in are included in the heuristic expression ( 16 ), then the extra termsoccur within the second-order-correlation approximation (Kitchatinov et al. 1994 ). The first term of the right hand side represents the known magnetic influence on the eddy diffusivity, while the second one forms a magnetic-induced pumping effect. There is no term representing an effect with the large-scale current helicity of the background field as (pseudoscalar) coefficient. For nonrotating and nonhelically driven turbulence, the large-scale current helicity does not create an effect in this approximation. Indeed, we do not find an effect in the numerical simulations below. There is instead the – sometimes ignored – magnetic-induced new pumping term, which for positive advects the magnetic field towards the maximum magnetic field. This effect should appear in all numerical models with helical large-scale field and nonhelically driven flow (Yousef et al. 2003 ; Bhat et al. 2014 ). Equations ( 16 ) and ( 17 ) yieldThe nonlinear terms magnetically affect both the eddy diffusivity and the advection velocity, which are basically reduced if the factor is positive. The possesses the dimension of a time and can be estimated as in the high-conductivity limit. One easily finds for important cases. For , the coefficient is always positive for all spectral functions that do not grow with growing frequency. Moreover, for all the full expression ( 19 ) is positive for which represents long correlation times as well as for , i.e.very short correlation times (so-called white noise). If compared with the magnetic-diffusion time , the delta-like spectral line represents the low-conductivity limit, while the white-noise spectrum represents the high-conductivity limit. In both limits, the is positive and, therefore, both eddy diffusivity and diamagnetic pumping are magnetically quenched rather than amplified (“anti-quenched”). 3.1. Numerical results Fig. 5 Two horizontal components ( top : , bottom : ) of the turbulence-induced electromotive force for background fields without helicity. Left panels : , (only homogeneous magnetic field in direction); right panels : , (only homogeneous electric current in direction). The averages over all snapshots are noted in the plots. The two horizontal components of the electromotive force are calculated for two different cases. In the first simulation, the applied magnetic fields are assumed to be free of helicity, i.e. . The other possibility works with . If the results of both setups


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

overly optimistic. 9 In addition to uncertainties in the most representative noise model, even choosing an accurate number of pulsars is non-trivial. PTA continue to expand by adding new systems, which are a very important part of detection forecasts (e.g. Taylor et al. 2016b ), but have different frequency sensitivities and thus different leverages on SNR. Finally, the GWF detection statistics we use are suboptimal in at least two respects. First, the statistics in Rosado et al. ( 2015 ) do not account for eccentric effects, and we treat the GW energy from single sources that is split across multiple frequency bins as entirely independent. Secondly, the excess power statistics may not perform as well on data where noise processes are harder to distinguish from single GW sources. The presence of line-like noise, due to uncertainties in planetary ephemerides (and their harmonics) for example; or instrumental effects, could introduce significant interference. At the very least, it is likely that foreground sources with periods comparable to the observing duration will be hard to distinguish from red noise when using single pulsar searches. This may necessitate correlated searches with lower recovery efficiencies. 4.2 Conclusions Single GW sources resolvable above the background, which we refer to as GWF sources, tend to be most detectable at frequencies near ∼0.1–0.3 yr −1 . At higher frequencies there are fewer sources which also tend to be less loud. At lower frequencies the GWB from all other sources is more likely to drown out the singles. According to most of our models, we would expect there to be a foreground source with a characteristic strain of about 10 −15 (or timing residual of ∼30 ns) after 10 yr of observations. These values are roughly a factor of 2 higher than those predicted by Sesana et al. ( 2009 , e.g. Fig. 3 ) and Ravi et al. ( 2014 , e.g. Fig. 4 ). When taking eccentricity into account, the primary effect of non-circular evolution is to shift GW energy from lower to higher frequencies. Thus, increasing eccentricity decreases the occurrence rate of GWF sources, especially at lower frequencies, both by increasing the GWB amplitude and by diffusing the single-source strains themselves. Changes in the effectiveness of stellar scattering, and more generally the rate of environmental hardening, has a small effect on the properties and occurrence rates of the GWF. Measurements of the number of GWF sources can thus provide strong constraints on the eccentricity distribution of the underlying MBHB population. Even in the absence of foreground detections, measuring the number of sources contributing to the background could provide the same information. This could be done either by directly resolving numerous loud sources in particular frequency bins (i.e. in frequency space), or indirectly by inference from the degree of anisotropy of the GWB (i.e. in angular space). DPs are usually higher for the GWF than for the GWB – indicating


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

the observed acceleration g obs to that due to baryons g bar , where , , and g † is the only free parameter. This relation was established by analyzing the data points from 153 galaxies in a statistical sense. We wanted to test the constancy of G N , so we fit Eq. ( 2 ) to rotation curves fixing g † = 1.2 × 10 −10 m s −2 and varying G N from galaxy to galaxy. We also tried to vary g † and G N simultaneously, and obtained similar results. We show as an example the case that has the same number of fitting parameters as in Rodrigues et al. (2018a) . We set the likelihood function as with χ 2 given by where g RAR is the expected centripetal acceleration from Eq. ( 2 ) and is the uncertainty on the observed acceleration. We impose a flat prior on G N with G N > 0 as Rodrigues et al. (2018a) did for g † . Following the procedure of Rodrigues et al. (2018a) , we also impose flat priors on Υ ⋆ with a tolerance of a factor of two, though this is not consistent with stellar population synthesis models ( Schombert et al. 2019 ). Galaxy distance affects the fit quality as it is directly related to the contributions of the baryonic components. When the distance D is changed to D ′, the galaxy radius R and the contribution of each component will transform according to where k denotes gas, disk, or bulge. Rodrigues et al. (2018a) allow galaxy distance to vary freely within 20% of their observational values. This prior does not reflect actual observational errors. Distances measured through Cepheids or the tip magnitude of the red giant branch are highly accurate, so a 20% variation could be larger than 4 σ . The Hubble flow method is much less accurate; distances measured using this method could have errors up to 30%, so the imposed 20% free range is even within the 1 σ region. Rodrigues et al. (2018a) also ignored the uncertainty on disk inclination. Possible outer asymmetries and warps in the gas disk could mislead the determination of inclinations, as reflected in their uncertainties, which would in turn affect the observed rotation velocities. Despite the shortcomings of the methodology of Rodrigues et al. (2018a) , we choose to fit the same parameters (except G N instead of g † ) and impose the same priors to achieve the most direct comparison. We map the posterior distributions of the fitting parameters using the standard affine-invariant ensemble sampler in the Python package e m c e e ( Foreman-Mackey et al. 2013 ). The Markov chains are initialized with 200 random walkers and iterated for 2000 steps after 1000 burnt-in iterations. We record the best-fit parameters that maximize the probability function for every galaxy. Figure 1 shows the posterior probability distribution of G N . Though we analyze all the SPARC galaxies, we only include 101 of them in Fig. 1 based on the following quality cuts: we removed galaxies with quality flag Q = 3 (see Lelli et al. 2016 ), inclination smaller than 30°, or reduced . This is a quality cut similar to that


O O O O O O B-Person I-Person B-Person I-Person B-Person I-Person O O O O O O O O O O B-Person O O O O O B-Organization O O O B-Grant B-Grant B-Grant O B-Grant 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 B-Organization I-Organization O O O O B-Organization O B-Grant B-Person O O O O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship

The authors would like to thank Basu A., Heesen V., Calzetti D., and the anonymous referee for very helpful discussions and comments. LJT acknowledges the financial support from NASA through the grants NNX13AE87G, NNH14ZDA001N, NNX15AM93G, and NNX15AV24G. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The work at Ruhr-University Bochum has been supported by DFG through FOR1048. CJV acknowledges support for this work from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144468.


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

while gradually decreases with increasing , as can be seen from Equation ( 57 ). Figure 14. Normalized local effective sound speed () and normalized effective local vertical half-thickness () as functions of the normalized external pressure . The value of is maximum (=1.061) at (as shown by the dashed horizontal and vertical lines). Appendix E Notes on the Generalized Toomre Criterion In the limit of flux-freezing, the dispersion relation isTo minimize ω 2 , the criteria are d ( ω 2 )/ dk = 0 and d 2 ( ω 2 )/ dk 2 > 0 at , yieldingNow, to obtain the instability criterion, we set ω 2 0 at , which givesyieldingIn the dimensionless form, it becomeswhereSee Section 2.6 for further discussion. Appendix F Notes on the Stationary Field Limit In the limit of stationary magnetic fields, , the resulting dispersion relation can be obtained using Equations ( 42 )–( 45 ). It follows that where A 1 , A 2 , θ , and γ are explicitly written in Section 2.4 . Recall that θ and γ represent the case of AD and OD, respectively. Simplifying each individual term of Equation ( F2 ) yields For the case of only OD (set θ = 0), combining Equations ( F3 ), ( F5 ), and ( F6 ), we obtain the resulting dispersion relation and corresponding growth timescale as shown in Equations ( 91 ) and ( 92 ). For the case of only AD (set γ = 0), combining Equations ( F3 ), ( F4 ), and ( F6 ) we obtain the resulting dispersion relation and corresponding growth timescale as shown in Equations ( 95 ) and ( 96 ). Appendix G Additional Figures of the Normalized Growth Timescale versus Length Scale Figure 15 shows the curves of normalized growth timescale as a function of normalized length scale for different normalized rotation Q = 0, 0.1, 0.2, 0.3, 0.4. This figure represents our model of the protostellar disk for distinct normalized mass-to-flux ratios μ 0 = 0.5, 1, 1.1, 2, and 6, with and , corresponding to n n ,0 = 10 11 cm −3 . This is one of our fundamental results, which can be obtained by plotting the normalized form of the full dispersion relation as seen in Equation ( 63 ). Because of the combination of both nonideal MHD effects, the timescale versus length-scale curve attains a minimum at a smaller value of even for the subcritical case μ 0 = 0.5 as compared to the case when only one nonideal MHD effect is present. With the transition of μ 0 from subcritical to supercritical, these curves gradually approach that of the hydrodynamic case. See Sections 3.2 and 3.3 for a detailed discussion. Figure 15. Normalized growth time of the gravitationally unstable mode as a function of the normalized wavelength for models with a fixed normalized ohmic diffusivity and neutral–ion collision time for different normalized mass-to-flux ratios μ 0 = 0.5, 1, 1.1, 2, and 6. Each figure shows the timescale curves for models with different normalized rotation Q = 0 (black), 0.1 (red), 0.2 (blue), 0.3 (green), and 0.4 (magenta). Appendix H Monochromatic Perturbation Here we focus on the form of


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 B-Survey 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 O O O O O O O B-Survey O B-Citation I-Citation O O O O O O O O B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person 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

ACKNOWLEDGEMENTS We are very grateful to Alberto Bolatto and Tony Wong for providing us with a compilation of H i masses that they made for galaxies from the EDGE-CALIFA survey (see the references listed in item ii of Section 3 ), and to Ed Elson for providing us with measurements of mass and specific angular momentum that he made for galaxies from the WHISP survey (Elson 2017 ). In addition, we are very grateful to Oscar Agertz, Alberto Bolatto, Françoise Combes, John Forbes, Mark Krumholz, Claudia Lagos, Lorenzo Posti, and Florent Renaud for useful discussions. We are also very grateful to an anonymous referee for insightful comments and suggestions, for further constructive criticism, and for encouraging future work on the topic.


O O O O O O O B-Organization I-Organization I-Organization B-Grant I-Grant I-Grant B-Person O O O B-Organization O O B-Organization I-Organization O B-Person O O O B-Organization O O O B-Survey I-Survey I-Survey I-Survey I-Survey O O O O O B-Organization I-Organization I-Organization I-Organization O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O B-Survey O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O B-Organization I-Organization I-Organization O B-Survey O O B-URL O B-Survey O O O O B-Collaboration I-Collaboration I-Collaboration O O O O O O B-Collaboration I-Collaboration O O B-Collaboration I-Collaboration I-Collaboration O B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O B-Collaboration I-Collaboration I-Collaboration O B-Collaboration I-Collaboration I-Collaboration B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Organization I-Organization I-Organization B-Organization B-Location B-Organization I-Organization I-Organization I-Organization B-Location B-Organization I-Organization I-Organization I-Organization B-Organization B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Observatory I-Observatory B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Observatory I-Observatory I-Observatory B-Collaboration I-Collaboration I-Collaboration I-Collaboration B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization O B-Organization I-Organization O B-Observatory O B-Survey

This work was funded in part by National Science Foundation AST Award 1211005. M.P.H. acknowledges support from NSF/AST-1107390 and the Brinson Foundation, and R.A.F. acknowledges support from NSF/AST-0847430. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is www.sdss.org . SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the universe (IPMU)/University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. Facilities: Arecibo , Sloan.


O O O O O O O O O O O O O O O O O O O O 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Observatory 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 B-Wavelength 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Citation I-Citation I-Citation I-Citation 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 O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Telescope I-Telescope I-Telescope B-Instrument O O O O O B-Instrument 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 B-Citation I-Citation I-Citation I-Citation O O O O O O B-Wavelength O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength O O B-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength 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 B-Wavelength 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 B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O

distributions could be the following. For a given mass of gas and dust and assuming a homogeneous distribution of the material, it is more difficult to have CT lines-of-sight if the inner radius of the absorbing material is at larger distances from the X-ray source than that for local AGNs. The inner radius of the dust around hot DOGs can be calculated from the dust sublimation radius (e.g., Nenkova et al. 2008b ):where is the sublimation temperature of the dust grains. Considering silicate dust grains ( K) and the average bolometric luminosity of hot DOGs reported by Assef et al. ( 2015 ; ), we find that the inner radius of the dust is . This is much larger than the typical value expected for Swift /BAT AGNs (, considering ). It should be remarked that, while these distances might be systematically smaller by a factor of ∼3, as found by near-IR reverberation studies (Kishimoto et al. 2007 ), the ratio between the of local AGNs and hot DOGs would be the same. Therefore, if hot DOGs have a dust mass comparable to that of local AGNs and gas and dust are strongly coupled, a difference in the column density distribution is expected. While there might be a significant fraction of dust-free material within the sublimation radius, this material is also likely to be highly photo-ionized by the very luminous AGN. Moreover, as shown by Díaz-Santos et al. ( 2016 ) for WISE J2246−0526, it is possible that in hot DOGs, due to the strong radiation pressure, the gas is being blown away isotropically. Hot DOGs could be very different from local, less luminous AGNs, where the bulk of the gas and dust is believed to be distributed in a torus-like structure, and might represent a short-lived transition phase between heavily obscured and unobscured AGNs (Bridge et al. 2013 ), similar to red quasars (e.g., Urrutia et al. 2008 ; Banerji et al. 2012 ; LaMassa et al. 2016 ). The idea that this phase might happen following a major merger is supported by the recent results of Fan et al. ( 2016b ) based on Hubble Space Telescope /WFC3 images. In a study of SCUBA observations, Jones et al. ( 2015 ) showed that hot DOGs have an excess of SMG neighbors, which would increase the chances of a merger. These results are in agreement with the idea that the AGN unification model might not be valid at high luminosities, where galaxy mergers are more important than secular processes in triggering accretion onto the SMBH (e.g., Treister et al. 2012 ). 5.2. Are Hot DOGs Intrinsically X-Ray Weak? AGNs show a strong positive correlation between the 2–10 keV and the mid-IR luminosity (at 6 μ m or 12 μ m; e.g., Lutz et al. 2004 ; Ichikawa et al. 2012 ; Mateos et al. 2015 ), as confirmed by high angular resolution (∼0.35 arcsec) mid-IR studies (e.g., Gandhi et al. 2009 ; Levenson et al. 2009 ; Asmus et al. 2015 ) of AGNs in the Seyfert regime (). In the quasar regime (), Stern ( 2015 ) found evidence of a flattening of the relationship for , with most sources being fainter than


O B-Telescope O O B-Wavelength I-Wavelength 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 B-Wavelength O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-ObservationalTechniques I-ObservationalTechniques 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 B-Citation I-Citation I-Citation I-Citation O B-Model I-Model I-Model O O 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 O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-ObservationalTechniques I-ObservationalTechniques O O B-Software O O O O O O O O O O B-Software O O O O O O O O O O B-Instrument O B-Instrument O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission 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 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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O B-Instrument O B-Instrument O O O O O O O B-Model O O O O O O B-Mission O O 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 B-Mission O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O B-CelestialObject I-CelestialObject O O O B-Instrument O O O O O O O B-Mission 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 B-Model 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O 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

of Gaia’s astrometry. The V -band bolometric correction of − 0.847 ± 0.075 was derivedusing the polynomial equations from Flower (1996) combined with the coefficients and the solar bolometric magnitude from Torres (2010) . Assuming no reddening ( A v = 0 mag), we found a stellar radius of R ⋆ = 0.622± 0.051 R ⊙ , which agrees with the spectroscopic radius derived using SpecMatch-Emp (cf. Table 2 ). We finally converted T eff , R ⋆ , and [Fe/H] into stellar mass M ⋆ and surface gravity log g ⋆ using the empirical relations of Mann et al. (2015) and Monte Carlo simulations. We found that GJ 9827 has a mass of M ⋆ = 0.637 ± 0.051 M ⊙ and a surface gravity of log g ⋆ = 4.657 ± 0.042 (cgs), which agrees with the spectroscopic gravity derived using SME (cf. Table 2 ). According to the analysis performed with SpecMatch-Emp , the three stars 2 whose spectra best match the HARPS and HARPS-N spectra of GJ 9827 have masses between 0.62 and 0.64 M ⊙ , confirming our results. The derived stellar mass and radius are are given in Table 1 . 3.3 Stellar activity and rotation period The K2 light curve of GJ 9827 displays a quasi-periodic photometric variability with a peak-to-peak amplitude of about 0.4 % ( Fig. 2 , left panel). Given the late spectral type of the star (K6 V), the observed photometric variation is very likely caused by active regions (sun-like spots and plages) crossing the visible stellar hemisphere as the star rotates about its axis. This is corroborated by the detection of emission components in the cores of the Ca II H K lines ( Fig. 3 ), from which we measured an average S-index of 0.677 ± 0.034 and 0.739 ± 0.021 using the HARPS and HARPS-N spectra, respectively. Fig. 2 Left panel : Gaussian process regression model applied to the K2 light curve. The black points are the K2 light curve, and the purple curve and shaded band are the GP model predictive mean and 1-sigma uncertainty. Right panel : K2 light curve folded to the P rot ∕2 stellar rotational period. Fig. 3 Cores of the Ca II H and K lines of GJ 9827 as observed with HARPS. Applying the auto cross-correlation technique to the K2 light curve, Niraula et al. (2017) and Rodriguez et al. (2018) found that the rotation period P rot of the star is either ~17 or 30 days. A Gaussian process (GP) analysis of the K2 light curve (Sect. 5.2 ) yields a bimodal posterior distribution with rotational periods peaking at 15.1 ± 1.6 and 30.7 ± 1.4 days, and thus does not provide a conclusive result about the rotation period of the star. However, we note that the ratio between the two measurements is close to 2, suggesting that the first might be the harmonic of the second. A visual inspection of the K2 time series photometry reveals that there are two dips whose minima occur at BJD TBD -2454833 ≈ 2922 and 2971 days, with a duration of ~20 and 16 days, respectively ( Fig. 2 , left panel). If the observed dips are caused by active regions crossing the visible hemisphere of GJ 9827, the rotation period


O O O O O O O B-Organization I-Organization I-Organization O B-Grant O O O O O O O B-Organization I-Organization I-Organization O O B-Grant B-Grant B-Person O B-Grant B-Person O B-Organization I-Organization B-Person O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Person O B-Organization O B-Grant I-Grant B-Person O O B-Organization I-Organization I-Organization I-Organization B-Organization O O B-Grant B-Person O O O O B-Telescope O O O O B-Grant O O B-CelestialObject I-CelestialObject

ACKNOWLEDGEMENTS This work was supported by the Russian Science Foundation grant 19-12-00423. We also acknowledge the support from the Academy of Finland travel grants 317552, 324550 (SST), and 322779 (JP), the Väisälä Foundation (SST), the Netherlands Organization for Scientific Research Veni Fellowship (AAM), the DFG grant WE 1312/51-1 (VFS), and the German Academic Exchange Service (DAAD) travel grant 57405000 (VFS). We grateful to the NuSTAR team for approving the DDT observation of A 0535+262.


B-Person O O O O B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person O B-Person I-Person O O B-Person I-Person O O B-Instrument O O B-CelestialObject I-CelestialObject B-Person O O O O B-Organization B-Grant I-Grant I-Grant B-Organization O B-Grant O O B-Organization I-Organization I-Organization I-Organization O O O O B-Person I-Person O O O O O O O O O O O O O

BDM acknowledges helpful conversations with Damiano Caprioli, Koji Mukai, Jeno Sokoloski, Andrey Vlasov, and Bob Williams. We thank Fred Walter for providing optical data on nova Sco. BDM gratefully acknowledges support from NASA Fermi grant NNX14AQ68G, NSF grant AST-1410950, and the Alfred P. Sloan Foundation. We thank the reviewer, Guillaume Dubus, for helpful criticism and suggestions that led to an improvement of the manuscript.


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

the galaxy power spectrum in order to arrive at predicted BAO constraints. The previous Euclid forecasts ( Amendola et al. 2013 ) are currently being updated with the new HLFs and updated instrument parameters, and we anticipate that the public documents will be updated soon. The HLFs presented here have already been incorporated in the most recent WFIRST-AFTA science report ( Spergel et al. 2015b ). Appendix A: Cosmic variance This Appendix describes the treatment of cosmic variance in the Model 3 fits. In the linear regime, the cosmic variance error covariance between two luminosity function bins and coming from the matter density field is given by where is the number of independent fields, is the real-space matter power spectrum at redshift , is the growth rate (which boosts the cosmic variance in narrow-band surveys due to redshift-space distortions), and is the window function, the Fourier transform of the survey volume, normalized to . We have used this result here assuming a bias of (from a semianalytic model, Orsi et al. 2010 , although there is evidence that the bias of star-forming galaxies might be higher; see e.g. Geach et al. 2012 ). This reduces the cosmic variance matrix to (Here all the entries in the covariance are constant.) The HiZELS error bars do not incorporate a contribution from cosmic variance. However, we can estimate it from Eq. ( A.1 ) assuming the geometry of independent boxes of size deg each. The depth in the radial direction is given by the width of the narrow-band filter, and is , 0.030, 0.032, and 0.032 at , 0.84, 1.47, and 2.23 respectively. The faintest bins in HIZELS at come from the HAWK-I camera, and the survey volume is smaller in this case: it is a single field, with size deg, and width . For the 4 redshift bins and the luminosity function bins where the full field has been observed, the implied diagonal elements of the covariance are 0.100, 0.045, 0.032, and 0.025. For the HAWK-I data (faintest objects at ), we find a variance of 0.256. A more subtle issue is that the above procedure assumes that the bias is independent of . This assumption has been commonly used for the purpose of forecasting H survey performance and its dependence on survey design. However, in combination with Eq. ( A.1 ), it implies that the cosmic variance contributions in each bin are perfectly correlated. That means that a fit using Eq. ( A.1 ) will assume that the shape of the HLF has no cosmic variance: the cosmic variance term will instead allow only the normalization to float up and down with an uncertainty given by Eq. ( A.1 ). Since cosmic variance is the largest contributor to the errors in some luminosity ranges, the procedure above could lead to fit results that are artificially well-constrained, if the bias is in fact dependent on . There is no reason for to be exactly zero, although for star-forming galaxies it is not obvious which sign to expect. We have thus explored the possibility of averaging the covariance matrix


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

We thank Tony Pan for useful comments on X-ray emission from PPI SNe. This work has been partly support by the grants-in-aid for Scientific Research (24244028, 26400271, 26800100) from the MEXT of Japan and WPI Initiative, MEXT, Japan.


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

and fast (up to 0.1 local sound speed) accretion happening at disk surface, which is similar to the flow pattern in global ideal MHD simulations (Zhu Stone 2018 ). A similar trend can be seen in the radial mass flux ( ρv r ) plot in the lower middle panel. The Rφ and zφ components of total stress are plotted in the rightmost two panels. The Rφ stress at both radii are close to zero at midplane, as already shown by very low midplane α in Figure 4 . Almost zero Rφ stress at different radii also suggests that midplane accretion or decretion is not driven by the Rφ component. As calculated in Section 3 , the vertical ( z ) gradient of also contributes to . In the zφ stress panel, maximizes at midplane, which is in agreement with the ρv r spike in the lower middle panel. At R = 1.33 (13.3 au), the midplane decretion is driven by zφ stress’s negative z gradient. The surface accretion is related to the two positive z gradient parts around disk surface. The differences of at these two radii also suggest very different magnetic field structure. Figure 5. Various quantities along the vertical direction at R = 1 (blue) and R = 1.33 (orange, local diffusion maximum at the CO 2 snow line). The quantities are averaged over both the azimuthal direction and time (from t = 49 to t = 50 T 0 with Δ t = 0.1 T 0 ). Top row from left to right: gas density, gas radial velocity to local sound speed ratio, R – φ stress; bottom row: inverse of plasma β , radial mass flux, and z – φ stress. The origin of the differences in stress structure can be understood from the 2D plot on the z – R plane, showing how magnetic field is evolved in Figure 6 . At the beginning of the simulation, the magnetic field only has a vertical ( z ) component, so B φ , B R = 0 everywhere. Apart from the dominant Keplerian shear ( dv φ / dR ) at midplane, the disk is also differentially rotating vertically ( dv φ / dz ), as midplane is the fastest and the atmosphere portion, which is mostly pressure gradient supported, rotates much slower. After the simulation starts, the field lines closer to midplane are dragged toward the rotation direction, making the B φ positive below midplane and negative above midplane. The motion along the r direction is even more complicated. From t = 0.5, at the snow line, the velocity shear between wind and surface accretion forms a vortex-like flow pattern that drives gas infall to the midplane. This gas motion drags field lines toward midplane, making a kinked form of magnetic field. Accretion at midplane also drags field lines inward, amplifying horizontal field component B R , and the polarity of B R changes sign in the accretion stream. Close to midplane, Keplerian shear stretches horizontal field lines, just like the growth of azimuthal fields in the linear phase of MRI. The process makes the field kink at midplane even sharper. The vertical gradient of the B φ component tends to get steeper with AD (Brandenburg Zweibel 1994 ). The strength of AD is


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

property of the galaxy allows us to isolate its effects and explore its consequences. Here, we will discuss two criteria. The first is the metallicity threshold, already mentioned, that provides an upper mass limit on galaxies that will be able to host a GRB (see Equation ( 1 )). The second criterion we explore is that a given galaxy will be more likely to host a GRB if it has a high specific SFR. We implement this by assuming that to host a GRB, a galaxy will need to have a specific rate above a given threshold value that we further assume to be independent of redshift. Again, this provides an upper stellar-mass limit for galaxies hosting GRBs. Figure 2 shows examples of the critical masses as a function of redshift for three different values of S * thresholds, as well as three different metallicity thresholds. The critical mass increases more rapidly with increasing redshift for the S * thresholds than for the metallicity thresholds. In addition, at a fixed redshift, the critical mass increases with an increasing metallicity threshold, but decreases with an increasing specific rate threshold. Galaxies with masses above the critical mass values have, in this model, a very low probability of producing a GRB. Figure 2. Critical galaxy stellar mass (in solar masses) as a function of redshift, z , for three different values of S * (in Gyr −1 , solid curves) and three different metallicity thresholds (dashed curves). The critical mass increases faster with increasing redshift for the S * thresholds. For galaxies more massive than the critical mass, GRB production is suppressed. For comparison we also show the redshift evolution of M 0 and M 1 (dashed–dotted). It is clear that for both threshold types ( S * and metallicity), in fact, very few if any GRBs would be produced by galaxies in the mass range out to redshifts z ≲ 2. Indeed, galaxies of mass ∼10 11 M would not be hosting a GRB out to a redshift of z ∼ 3–4 in either scenario, as observed by Perley et al. ( 2016b ). Only in the case of relatively low values of S * and rather high values of the metallicity threshold does the critical mass reach the characteristic mass, M 1 , of Equation ( 2 ), at the highest redshifts shown. Massive galaxies, around M 1 or higher, are only likely to host a GRB at high redshift and thus contribute to the GRB-probed star formation if of high metallicity or of low specific SFR. 3. The SFR Fraction We now proceed as in Robertson Ellis ( 2012 ) and define the fraction of star formation occurring in galaxies with M M cr as:Here, P ( x , y ) = Γ( x , y )/Γ( x ), with Γ( x ) the gamma function,is the lower incomplete gamma function, and the second equality in Equation ( 5 ) holds for the functional forms adopted for and SFR( M , z ). Recall that ψ ( z ) is also the fraction of star formation producing a GRB (Robertson Ellis 2012 ). In Equation ( 5 ), t cr = M cr (1/ M 0 + 1/ M 1 ), and the redshift dependence of ψ ( z ) is only through the mass scales if the


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

gainer has an initial mass of 2.7 M ⊙ . With CF = 0.95 one has a shell with an initial mass of 0.135 M ⊙ . The solidly rotating core keeps its mass of 2.565 M ⊙ during the entire evolution. At the end of RLOF A, the mass of the gainer has grown up to 6.936 M ⊙ . Since the core maintains its initial mass, the mass of the shell raises up to 4.371 M ⊙ , containing 63% of the gainer’s mass (CF = 0.37). At the end of RLOF B, the mass of the gainer is again somewhat larger and the value of CF shrinks to 0.31. Fig. 4. Mass fraction of the core (CF) decreases with time during the evolution of the system shown in Figs. 1 – 3 . This is not so different from the evolution of a shell with, for example, an initial value of CF = 0.5 that ultimately evolves into a massive shell with CF = 0.2 (80% of the mass of the gainer being in the shell), giving rise to magnetic fields and magnetic braking that are very similar to those obtained with an initial value of CF = 0.95. This value was used in all our calculations. 3.4. Calculated cases We found 44 binaries for which the equatorial velocity of the gainer was published. These systems are shown in Tables 1 and 2 . Table 1. Calculated equatorial velocities of gainers that best fit observations. Notes. Masses are in M ⊙ , orbital periods in days and v eq in km s −1 . The masses mentioned by the observations of Glazunova Yushcenko (2008) are from Budding et al. (2004) . Table 2. Calculated equatorial velocities of gainers fitting observations less good than in Table 1 . Notes. Masses in M ⊙ and orbital periods in days. Masses for V356 Sgr are from Dominis et al. (2005) . Masses in Glazunova Yushcenko (2008) are from Budding et al. (2004) and Brancewicz Dworak (1980) . For every binary in these tables, we calculated the evolution of a number of possible progenitors. In the case of conservative evolution, the initial orbital period P 0 of every progenitor was determined from the presently observed period P 1 with relation ( 13 ), which obeys to the rule of conservation of angular momentum: The subscript d is for the donor and g for the gainer. Although we calculated the orbital period with relations ( 10 )–( 12 ) using our code, relation ( 13 ) yields correct initial periods in the case of conservative evolution. This demonstrates that neglecting the spin angular momenta of the individual stars, as done in relation ( 13 ), is of minor importance. In the case of liberal evolution initial periods are determined with relation ( 14 ). This relation is valid when all matter that leaves the system takes only the orbital angular momentum of the involved stars. Here, M is the total mass of the system, so that: The quantity Δ M SW , the amount of mass that leaves the system as stellar wind, is always small compared to the stellar masses. The quantity Δ M OUT is the amount of RLOF mass that is not accreted by the gainer. This amount can be large and is difficult to predict. In our sample of 44 binaries, only two of them (V356


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O

there is no over-exposure on the QS. Figure A.7 shows the CC constructed for the extreme exposure problems considered. The green, blue, and red curves display the imposed relation, the measured QS CLV, and the standardized linear relation, respectively. For all the data, our method allows calculating the NSB accurately enough so that the obtained CLV reproduces the imposed relation for almost the whole range of QS values. In particular, the maximum unsigned relative difference between the imposed CC and the one recovered by the method is less than 0.005. The accuracy drops slightly for the last point at the limb and only for the cases of severe under-exposure, rising to 0.017. Subset 4: Different disc sizes with linear CC without inhomogeneities in the image The photographic plates from different observatories have different sizes. Even among the images of the same archive there are variations due to changes of the spectroheliograph and of the photographic plates. The digitization affects the number of pixels covered by the disc as well. The disc size is also affected by the seasonal variation of the Sun–Earth distance. Therefore, this subset was created to test the efficiency of the proposed method when applied to datasets that have a different disc size. We selected 10 Rome/PSPT images covering the period 2000–2014 and re-sampled them, so that each was represented in the form of solar images with 10 different radii between 100 and 550 pixels. The range of radii was defined to include the sampling present in most available datasets, however the upper limit was dictated by the number of pixels covered by the disc on Rome/PSPT observations. Subset 4 consists of 100 images. Figure A.8 shows the relative differences between the images derived with our method and the original Rome/PSPT data. Panels a) and b) refer to values of NSB, while panels c) and d) refer to values of contrast images, each (coloured) box for a different image of subset 4. In each panel, rows (columns) show values derived from different observations (imposed radii). The relative difference between the retrieved NSB (SB) and imposed background decreases with increasing radius of the disc; the maximum unsigned difference measured is 0.03 (0.007) for the images with the smallest radius, average difference is , and maximum of rms difference is 0.004 (0.001). The relative difference between the contrast image derived with our method and the original data reaches up to 0.056; the average differences are ; the rms differences are . Fig. A.8 Colour-coded relative differences between the images retrieved by applying our method to images of subset 4 and the underlying original, undegraded Rome/PSPT images. Each box (with a given colour) corresponds to a different synthetic image; row (column) shows results derived from a given Rome/PSPT observation (imposed radius). Note the different scale of the colour code, as represented by the corresponding colour bars. a) Mean relative and b) rms


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

We wish to thank the referee, Dr. Thorsten Lisker, for the very thorough reading of the paper and the many insightful and useful suggestions. This work is supported in part by the Canadian Advanced Network for Astronomical Research (CANFAR), which has been made possible by funding from CANARIE under the Network-Enabled Platforms program. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. The authors further acknowledge use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and the HyperLeda database ( http://leda.univ-lyon1.fr ). This publication has made use of data products from the Sloan Digital Sky Survey (SDSS). Funding for SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. S.C. and J.T. acknowledge support from the Natural Science and Engineering Research Council of Canada through a Research Discovery Grant. E.W.P. acknowledges support from the National Natural Science Foundation of China under grant Nos. 11173003 and 11573002 and from the Strategic Priority Research Program, “The Emergence of Cosmological Structures,” of the Chinese Academy of Sciences, grant No. XDB09000105. This work is supported in part by the French Agence Nationale de la Recherche grant VIRAGE (ANR10-BLANC-0506-01). C.L. acknowledges the NSFC grant Nos. 11203017 and 11125313. P.G. and E.T. acknowledge support from NSF grants AST-1412504 and AST-1010039. Facility: Canada–France–Hawaii Telescope (CFHT).


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

as shown in the right panel of Figure 2 . A power-law index of −0.6 clearly agrees with the mass function slope for low-mass objects. This suggests that the differential mass functions are well fit by a single power law because there is a significantly larger number of planetesimals at small masses, thus weighting the fit toward the small-mass end. Therefore, while in reality the mass distribution will have a cutoff at some high-mass value (which depends on τ ), the differential distribution can be well fit by a single power law that is representative of masses away from the cutoff. Finally, we have also examined the evolution of p and the maximum and minimum planetesimal masses (in all three simulations) for times after which a significant number of planetesimals have formed; the results are shown in Figure 3 . Note that due to limitations with the clump finder algorithm (as described more fully in Simon et al. 2016 ), we smoothed the curves (with a boxcar average) to remove noise associated with this algorithm. As the figure shows, p and the minimum planetesimal mass are relatively constant in time, 7 7 We do observe a slight decrease in the minimum mass in some instances, a behavior that could arise from a combination of processes, including tidal stripping of smaller bodies by large planetesimals, fragmentation of smaller bodies, and/or the preferential production of smaller planetesimals as the reservoir of particles from which to produce planetesimals decreases in size (Johansen et al. 2015 ). Elucidating the nature of this behavior requires a more sophisticated clump-finding algorithm and will thus be reserved for future work. whereas the maximum planetesimal mass grows by a factor of order unity in each simulation, likely due to accretion of smaller particles and/or mergers with smaller planetesimals. Despite this growth, the mass function does not evolve appreciably once planetesimals have formed. Figure 3. Time evolution of the power-law index (solid black line), maximum planetesimal mass (dashed red line), and minimum planetesimal mass (dotted blue line) for the (left), (middle), and (right) simulations for a period of time after planetesimals have formed. Both the power-law index, p , and the minimum planetesimal mass are relatively constant in time, whereas the maximum planetesimal mass grows slowly, presumably due to continued accretion of small particles and/or mergers with smaller planetesimals. Our ability to directly measure any potential dependence of the mass function on particle properties is limited by the relatively small number of collapsed clumps, but across the range of τ considered, we can bound deviations at the level of . We can therefore exclude, with moderately high confidence, the possibility that the streaming instability might result in a steep mass function () with most of the mass in the smallest planetesimals. 4. Particle Clustering Prior to Collapse The similarity in the mass functions from the different


O O O O O O O O O O O O O O O O O O O O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O O O O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility O O O O B-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization O O O O O O O B-Observatory I-Observatory I-Observatory O O B-Organization O O O B-Location I-Location O O O O O O O O O O O O O O O B-Telescope O O O O B-Organization I-Organization B-Organization O O O O B-Organization I-Organization O O B-Organization I-Organization I-Organization O O O O O O B-Organization O O O B-Survey O O O B-Collaboration I-Collaboration O O O O O B-Organization I-Organization I-Organization O O B-URL O B-Telescope O

We thank the anonymous referee for providing comments that significantly improved the manuscript. The authors would like to acknowledge the contribution of an Australian Government Research Training Program Scholarship in supporting this research. This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the operation of the MWA is provided by the Australian Government (NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. We gratefully acknowledge the support of NASA and contributors of SkyView surveys. Funding for Breakthrough Listen research is provided by the Breakthrough Prize Foundation. 3 3 http://breakthroughprize.org Facility: MWA .


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

kpc which is almost the same as the scale length of the molecular disc which is ∼4.05 kpc. However, for our sample galaxies, the scale length of the molecular scale height is found to be approximately a factor of 2 greater than the scale length of the exponential surface density profile (see Fig. 6 ). This result strongly indicates that the coupling with the other disc components and with the dark matter halo significantly influence the molecular scale height. Figure 7. The behaviour of the molecular scale height in the absence of any external influence in NGC 3198. The solid red and the black dashed lines represent the molecular gas surface density profile (), and it’s fit to an exponential profile, respectively (marked by the left axis). The solid magenta and the dashed-dotted blue lines represent the estimated molecular scale height (HWHM) and its fit to a rising exponential function, respectively (marked by the right axis). The scale lengths of the exponential fits to the ( l CO ) and the HWHM profile ( l scl ) are found to be ∼4.05 and ∼4.06 kpc (as quoted in the top left), are almost the same. Next, we use the solutions of the hydrostatic equilibrium equation and the observed rotation curves to build a dynamical model of the molecular discs for our sample galaxies. We then incline these dynamical models to the observed inclinations and project it to the sky-plane to produce a column density map. These maps then convolved with the telescope beam (13 arcsec × 13 arcsec) to produce simulated maps that are equivalent to the observed ones. In Figs 8 and 9 we compare these simulated maps with the observed maps. In the left-hand columns of the figures, we plot the modelled column density maps in contour on top of the observed maps in colour scale. As we do not solve the hydrostatic equation for the central regions of our sample galaxies, the central region should be excluded for any comparison. Further, the convolution with the telescope beam at the edge of the central region would corrupt an area equivalent to the beam size. We hence exclude a region that we do not solve from the centre plus the beam size. The vertical dashed lines in each panel of the left-hand column represent this excluded region. As can be seen from the figure, the contours (modelled) match to the observed map reasonably well. However, for two galaxies NGC 925 and NGC 2976, the model seems to deviate from the observation. This is due to azimuthal asymmetry in their column density distribution that is one of our implicit assumptions. We would like to emphasize here that, the molecular surface density profiles of our sample galaxies are calculated through spectral stacking method (Schruba et al. 2011 ). This produces surface density profiles that are much more sensitive than what will be observed in the moment zero maps. Because of this, the surface density profiles of our sample galaxies extend to a much larger radius as given in Schruba et al. ( 2011 ) than what can be


B-Person O O B-Person I-Person 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 B-Person B-Person O B-Person O O O O B-Collaboration I-Collaboration I-Collaboration I-Collaboration I-Collaboration I-Collaboration I-Collaboration I-Collaboration B-Collaboration O O O B-Person I-Person O O B-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O B-Person O B-Person I-Person O O O O O B-Software B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Organization O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person 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 I-Organization O O O B-Grant O B-Grant O O O B-Grant I-Grant I-Grant O 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 B-Organization I-Organization O O B-Grant I-Grant I-Grant B-Grant O O O O O O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O B-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 I-Organization I-Organization O O O B-Grant O O O O B-ComputingFacility I-ComputingFacility O B-Organization I-Organization O O O O O O O O O O O O O B-Software B-Citation I-Citation O O O O O O O B-Archive I-Archive I-Archive I-Archive O O O O O B-Software O O O B-Software O O O B-Citation I-Citation I-Citation I-Citation I-Citation O

D.A.G. gratefully acknowledges Eric Linder for a careful reading of the manuscript and for thoughtful feedback. Greg Aldering and Alex Kim also deserve thanks for feedback that improved the paper. D.A.G., T.E.C., and P.E.N. are members of the Large Synoptic Survey Telescope Dark Energy Science Collaboration (LSST-DESC), and they acknowledge Phil Marshall and the Strong Lensing working group for bringing them together so that this work could be launched. D.A.G. thanks Kyle Barbary for leading the development of sncosmo (Barbary et al. 2016 ), which was essential to the analysis. The authors acknowledge support from the DOE under grant DE-AC02-05CH11231, Analytical Modeling for Extreme-Scale Computing Environments. D.N.K is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-05CH11231 and DE-SC0017616, and by a SciDAC award DE-SC0018297. This research was supported by the Gordon and Betty Moore Foundation through Grant GBMF5076. The authors are grateful to the Royal Society for an International Exchange Grant (IE170307), which has supported our collaboration on this work. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 as well as the savio cluster at UC Berkeley. Figures 2 , 9 , and 11 were rendered with code based on ChainConsumer (Hinton 2016 ). This research has made use of NASA’s Astrophysics Data System. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013 ).


O O O O O O O O O O O O O O O O B-Person O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Person O O O O O B-Fellowship I-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization I-Organization B-Person O B-Person I-Person O O O O O O O O O B-Software O O O O O B-ComputingFacility I-ComputingFacility O O B-Organization I-Organization I-Organization I-Organization

ACKNOWLEDGEMENTS We thank the referee for insightful comments, which were very valuable for improving the paper. GF is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy of Sciences and Humanities. GF also acknowledges support from an Arskin postdoctoral fellowship at the Hebrew University of Jerusalem. GF thanks Seppo Mikkola for helpful discussions on the use of the code archain. Simulations were run on the Astric cluster at the Hebrew University of Jerusalem.


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

is submitted to the electric field of a single proton. The calculation of this electric field requires knowing the distance between the electron and the proton, which can be taken as the mean distance between particles, which is the −1/3 power of their density, which is roughly the same for electrons and protons. This electric field derives from a potential, as does the gravitation field. Thus, there is also an escape velocity, as there is for gravitation. The escape velocity for freeing the electron from the corresponding single proton is then where q is the elementary charge, m e is the electron mass, and n the density, which is nearly the same for electrons and protons. In the solar interior conditions given by Allen (1973) , we then obtain that this escape velocity is nevertheless six times lower than the quadratic average thermal velocity for electrons, which then also escape from the protons, at least initially. As the escape velocity is obtained by equating the particle kinetic and potential energies, the total escape velocity from gravity and proton for an electron is and we obtain that for electrons it is 5.5 times lower than the quadratic average thermal velocity at 0.5 R ⊙ . The electrons then initially escape from gravity and protons. 6.4. Escape velocity from several protons As the electron cloud is free at the beginning, in a first approximation it begins to expand. The result is a decrease in the electron density, while the proton density remains unchanged. As a consequence, the number of electrons and protons is not yet the same in a given volume. As a result, an escaping electron is now submitted to the attraction of more than one mean proton, depending on the density ratio. If we denote as ρ this density ratio where n e and n p are the electron and proton density, respectively, the total escape velocity of the electron submitted to ρ > 1 protons and to gravity becomes because the mean distance between the electron and the other particles is . If we denote as x the ratio between the electron escape and thermal velocities we obtain from the preceding equations which permits us to compute the proton-to-electron density ratio ρ from the desired velocity ratio x . The electron escaping will certainly be suppressed when x = 3, which means that the escape velocity will be at least three standard deviations higher than the thermal velocity for each velocity component. In this case, ρ is found to be 6 × 10 3 at 0.5 R ⊙ in the Allen (1973) conditions. This corresponds to a mean distance between electrons 18 times larger than that between protons. These quantities are found to be quite constant along the solar radius. 6.5. Spreading time One important question is where exactly do the initially free electrons escape, and the answer is to the distance where the electron cloud density is divided by the ratio ρ defined above. We propose that the density scale height is the typical dimension of the cloud. We denote as h this density scale


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-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-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 B-Software 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 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-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O 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 B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O B-TextGarbage I-TextGarbage 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-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 B-Telescope O O O O O O

BYF40 HCN ( l , b , V ) data cube we chose as a test area for our code development, and has the advantage of being scriptable in a modern computing environment, as well as having a built-in tool for fitting the model across all spectra in a cube. Assuming LTE, it solves for the common parameters of all hyperfine components. In order to improve the fit quality, the original fits cube input for the package was binned by two spectral channels and convolved from an original resolution of 4060arcsec. Once a minimum signal-to-noise ratio (S/N) threshold is set for the hyperfine fitting, PySpecKit yields separate 2D maps of the excitation temperature T ex , opacity , mean velocity V LSR , and velocity dispersion V , as well as error maps for each parameter map. A sample fit is shown in Fig. 3 , while the parameter maps appear in AppendixA. 3.2 Column density calculation After obtaining the parameter maps and uncertainties as outputs from PySpecKit , the column density N was evaluated using a mix of C shell and miriad (Sault, Teuben Wright 1995 ) software. and T ex were measured in each voxel and then converted into N HCN using the equation Here, Q refers to the partition function, E up is the energy of the upper level above the ground state of the J upper J lower transition, h is the energy of the transition, and k is the Boltzmann constant. The derived N HCN map is shown in Fig. 2 , and has a flatter distribution compared to the overlayed I HCN contours. Figure 2. Maps of N HCN (a) and its uncertainty (b) across BYF40, plotted with contours at intervals of 10percent of the peak I HCN . Figure 3. Sample spectra from the original, model, and residual data cubes. Each was measured at the coordinate l = 284.002(3), b = 0.846(3). Under traditional methods, the X factor for 12 CO is often used to estimate column densities with a simple linear conversion. The accuracy of such conversions has frequently been debated, and recent work by our group (Barnes etal. 2015 , 2016 ) has shown that, in the case of CO at least, both the lines T ex and need to be properly accounted for, and must be self-consistently measured. In this paper, we calculate the uncertainty in N HCN by propagating all of the formal fit errors, and applying the constraints T ex 2.73K and 0 on the PySpecKit solutions. By using the error images and the output images together, relative errors were also determined for several intermediate quantities in the calculation, and for N HCN . The V LSR values fall in the range of 810kms 1 , V = 1.22.2kms 1 , T ex = 49K, and = 0.52.5. 3.3 Clump masses In Table 1 we give the observed and physical parameters of the BYF40 clumps used to derive these results. The values appear similar to table 4 from Paper I , due to the high correlation present between HCN and HCO in the clump. Additionally, Fig. 4 (b) shows that there is a high correlation between the integrated intensities of HCN and HCO . This can be seen in the Fig. 1 (a) Mopra map, where the blue of HCO