et al. 2007 ), disk formation in the central parsec followed by a migration process (Levin 2007 ; Madigan et al. 2009 ), and rejuvenation of older stars by mergers or severe atmosphere alteration by envelope-stripping (Lee 1987 ; Genzel et al. 2003 ; Davies King 2005 ). A successful scenario should explain the young ages and randomly inclined orbits of the S-stars, via an efficient transport/relaxation mechanism. External formation followed by binary disruption is among the favorite scenarios, as it also explains some of the hyper-velocity stars that escape the Galaxy (e.g., Brown et al. 2009 ). However, the field binary disruption scenario requires some minimum time span to explain the thermal distribution of eccentricities in the S-star cusp (Perets 2009 ). This may be incompatible with the shorter main-sequence lifetime of some S-stars. While it is plausible to assume that S-stars were initially formed in an outer stellar disk in the central parsec, it has proved difficult to find an explanation of how their original angular momenta are strongly perturbed and later redistributed. Finally, many of the scenarios that suggested environmental mechanisms to significantly alter the photosphere of in fact old S-stars (Lee 1987 ; Alexander Morris 2003 ; Hansen milosavljević 2003 ; Davies King 2005 ) either have serious problems with explaining the S-stars’ stellar properties or are not elaborated in detail. For example, it is not clear how quickly after a major atmospheric perturbation a star will relax, and whether such products experience significant mixing (e.g., Gaburov et al. 2008 ). With all the aforementioned observational constraints and theoretical complexities the question of the origin and distribution of young stars in the GC has become one of the most remarkable issues in this field. Here, we perform quantitative spectroscopy of S-stars and obtain their individual stellar parameters. This enable us to locate these stars for the first time in a Hertzsprung–Russell (HR)-like diagram and provide new insights on their evolutionary stage and their puzzling origin. However, such an analysis demands a high signal-to-noise (), comparable to that achieved in spectra of nearby Galactic OB-stars. In this study, we co-add more than 100 hr of adaptive-optics (AO) assisted spectroscopic observations of the GC over twelve years, to construct ultra-deep spectra of eight S-stars. The paper is organized as follows: in Section 2 we list the observations used in this analysis and describe the reduction of the data to obtain deep spectra on a number of stars in the GC. Section 3 describes the stellar atmosphere models used to match the data and our fitting strategy . In the results section (Section 4 ), we present the spectral and luminosity (gravity) classification of the S-stars as well as parameters of individual stars based on the results of atmosphere model fits. In the final paragraph of the results section we compare the inferred stellar


Acknowledgements We want to thank Antonio J. Cuesta, Benjamin Alan Weaver, and Jeremy Tinker of SDSS-III BOSS DR12 for the help on BAO fitting and weight assignment of QPM mocks. We also want to thank Yuting Wang, Gong-Bo Zhao and Bin Hu for helpful discussions. S-JW wants to thank Shuangpeng Sun, Shihong Liao, Junyi Jia, Zhen Jiang and Chunxiang Wang for the help on idl coding, and Wei-Ming Dai and Li-Wei Ji for the help on fortran coding during this work. S-JW also wants to thank the hospitality of Misao Sasaki when the work was finished during the visit at the Center for Gravitational Physics of Yukawa Institute for Theoretical Physics of Kyoto University. We acknowledge the use of the cluster PANGU and NOVA of NAOC. QG is supported by a Newton Advanced Fellowship, NSFC grant (No. 11573033, 11622325), the Strategic Priority Research Program The Emergence of Cosmological Structure of the Chinese Academy of Sciences (No.XDB09000000) and the Recruitment Program of Global Youth Expertsof China, the NAOC grant (Y434011V01). R-GC is supported in part by the Strategic Priority Research Program of the Chinese Academy of Science (CAS), Grant No.XDB09000000 and by a key project of CAS, Grant No.QYZDJ-SSW-SYS006 and by the National Natural Science Foundation of China under Grants No.11375247 and No.11435006.


We thank the referee for a critical appraisal of this paper. R.N. was supported by Yale Astronomy's Dorrit Hoffleit Undergraduate Research Scholarship, Alice Peter Tan, and Yale-NUS College's Summer Independent Research Program (SIRP) while working on this research. P.O. acknowledges support by the Swiss National Science Foundation through the SNSF Professorship grant 157567 “Galaxy Build-up at Cosmic Dawn.” We are grateful to Akio Inoue for providing Monte Carlo realizations of the IGM transmission at high redshift. The primary data for this work were obtained with the Hubble Space Telescope operated by AURA, Inc., for NASA under contract NAS 5-26555. Support for this work was provided by NASA through grant HST -GO-13871 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). Facilities: HST (ACS , WFC3) .


the UV magnitude and flux were measured by selecting a 5-arcsec source region and a larger background region avoiding any visible sources. Flux measurements were made using the uvotmaghist software. Flux measurements from individual observations are plotted in Fig. 2 (top panel), where orange circles indicate uvw1 (2253–2946 Å) and blue diamonds are uvw2 (1599–2256 Å). They show that in both the uvw1 and uvw2 filters the first observation measured a higher flux. The following four observations all measured a slightly lower flux by about 1–2 × 10 −13 and there was no significant variability between any of the four observations from 2019 July 13 to 2019 August 10. Figure 2. Top panel: Swift UVOT flux (×10 −13 erg cm −2 s −1 ) for each observation taken over 5 weeks. Observations were made using two filters. The uvw1 filter (orange circles) and uvw2 (blue diamonds). Bottom panel: the X-ray count rate over the five observations from 2019 July 7 to 2019 August 10 in the 0.3–10 keV energy range. There is no significant change in the X-ray count rate in contrast to the decline in the UV flux. X-ray count rates were extracted from each observation using the automatic XRT data builder (Evans et al. 2007 ). The X-ray light curve in Fig. 2 , bottom panel, shows no significant variability across all the observations, including the first one where there is a higher flux in the UV. Since there is very little variation in the X-ray light curve and hardness ratio, it appears the source of the X-rays is consistent across all the observations. Therefore it is possible to combine the X-ray data to produce a single spectrum that represents the source as shown in Fig. 3 . Figure 3. The combined X-ray spectrum with a total exposure time of 10 ks. The spectrum is fitted with a two-temperature apec model without absorption. Model parameters are given in Table 3 , column 4. 3 X-RAY SPECTRAL ANALYSIS Spectral files including ancillary response file ( arf ) and response matrix file ( rmf ) were generated using the XRT automatic analysis tools (Evans et al. 2009 ) and fitting was carried out using xspec . The spectrum has relatively few photons but shows a peak at ∼1 keV and very little flux above 2 keV that are characteristics of a stellar coronal spectrum (Güdel Nazé 2009 ). We therefore fit the spectrum with an apec plasma model (Smith et al. 2001 ) as shown by the red line in Fig. 3 . The low number of counts in the spectrum requires the use of the C-statistic when fitting, rather than standard χ 2 . Single and two-temperature models were compared, with results shown in Table 3 . Fitting with a two-temperature model without absorption gave an improved fit with a reduced χ 2 of 1.27 compared to the one-temperature model’s 1.42. The temperature of the cool plasma component is 0.31 keV that is consistent with the values found for the similar K5-type T Tauri star LkCa 15 (Skinner Güdel 2013 ). However the higher temperature plasma component is almost a factor of 2


Finally, we computed the cumulative number of sources, N (> S ), brighter than a given flux ( S ) in each X-ray band. To this aim, we computed the sky coverage (i.e., the sky area Ω covered as a function of the flux limit) of the J1030 field to correct the incompleteness of our catalog. We computed our sky-coverage by dividing the number of output sources flagged as “good” in our simulations by the number of input sources as a function of input flux and then multiplying for the total geometric area of the J1030 field covered by Chandra . The sky-coverage values were then fitted with a spline to obtain a smooth monotonically increasing function. The sky coverage in the full, soft, and hard bands are plotted in Fig. 11 . The derived flux limits over the central ∼1 arcmin 2 region are ∼3 × 10 −16 , 6 × 10 −17 , and 2 × 10 −16 erg cm −2 s −1 in the full, soft, and hard bands, respectively, making the deep Chandra survey in the SDSS J1030+0524 field the fifth deepest X-ray survey field achieved so far (see Fig. 1 for area-flux curve comparison with other surveys). Once the sky coverage is known, the cumulative source number was computed using the equation: Fig. 11. Sky coverage (i.e., sky area vs. flux limit relation) in the full (black solid line), soft (red dashed line), and hard (blue dash-dotted line) bands derived from our simulations. The horizontal cyan dotted line represents the total geometric area of our J1030 field (335 arcmin 2 ). where N S is the total number of detected sources in the field with fluxes higher than S , and Ω i is the sky coverage associated with the flux of the i th source. The three log( N )–log( S ) relations in the three X-ray bands are reported in Fig. 12 . The red points represent the cumulative number of sources of our J1030 field, while the log( N )–log( S ) of our mock catalog are shown as the blue dot-dashed line. For comparison, we also plotted the log( N )–log( S ) relations found in the 7 Ms (magenta line) Chandra Deep Field-South by Luo et al. (2017) , and the log( N )–log( S ) relation (green solid line) found in the COSMOS field by Civano et al. (2016) . From Fig. 12 we conclude that the log( N )–log( S ) relations derived in the J1030 field are in general agreement with those from the literature. Besides cosmic variance, we caution that some of the differences among the log( N )–log( S ) seen in Fig. 12 could be produced by systematic uncertainties in the different methods used to derive the sky coverage and the individual source fluxes in the various surveys. Fig. 12. Cumulative number counts (number of sources brighter than a given flux) for the main source catalog (red dots) in the full ( top ), soft ( middle ), and hard ( bottom ) bands. The blue dot-dashed line represents the cumulative number of sources from our mock catalog. For comparison, we plot the log( N )–log( S ) relations found in the 7 Ms Chandra Deep Field-South by Luo et al. (2017 , magenta line), and in the COSMOS field by Civano et


GMA and BV acknowledge financial support from the ERASMUS–ASTROMUNDUS programme hosted by the University of Innsbruck, where this article was conceived in 2013. We thank an anonymous referee for useful comments that have greatly improved the quality of the article. BV, MMC and MM acknowledge financial support from the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project #ON176021 ‘Visible and invisible matter in nearby galaxies: theory and observations’.


We thank the anonymous journal referee for their insightful comments, which improved the manuscript. RS gratefully acknowledges support from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Advanced Grant RADIOLIFE-320745. EKM acknowledges support from the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. The European VLBI Network is a joint facility of independent European, African, Asian, and North American radio astronomy institutes. Scientific results from data presented in this publication are derived from the following EVN project code: GN002. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Long Baseline Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Arecibo Observatory is a facility of the National Science Foundation (NSF) operated by SRI International in alliance with the Universities Space Research Association (USRA) and UMET under a cooperative agreement. The Arecibo Observatory Planetary Radar Program is funded through the National Aeronautics and Space Administration (NASA) Near-Earth Objects Observations program. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA). This research has made use of NASA’s Astrophysics Data System Bibliographic Services. This research has made 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. This research made use of Astropy, a community-developed core Python package for Astronomy ( Astropy Collaboration et al. 2013 ). This research made use of APLpy, an open-source plotting package for Python ( Robitaille Bressert 2012 ).


λ , to dimensionalize our expressions. It is thus, in Equation ( 26 ), that we use to formally eliminate a . We assume that the horizontal motion of the open flux has deflected it by some small angle, say 10°, relative to the stationary and undeflected arcade flux. This angular difference, ±5°, produces the tangential discontinuity that is the secondary CS. The field strength in the vicinity of the flare ribbon region was estimated by BLQ to be 150 G. The single-sided 10° deflection will therefore produce a reconnecting component of 150 sin (5°) = 13 G. Assuming a typical electron density of gives the value v a = 500 km s −1 , which was used in our nondimensionalization. The sawtooth pattern is observed to propagate with phase velocity perpendicular to the IRIS slit of km s −1 . This was found in the plane of the sky (primed coordinates) but applied to horizontal motion some 40° from disk center. Foreshortening effects will reduce the velocity V in the horizontal x- direction to an apparent velocitywhere is the horizontal distance from disk center. The observed phase velocity of 15 km s −1 thus corresponds to a real horizontal velocity of km s −1 . Scaling this to the reconnection-component Alfvén speed, v a = 500 km s −1 , yields . 3.3. The Saturated State We use these dimensions to construct a synthetic observation from the velocity structure of our model. We do so by first using the velocity structure given by the linear eigenmode, , discussed above. We construct the stream function by superposing this using the amplitude, , in Equations ( 24 ) and ( 25 ). Figure 6 shows the magnetic field lines (black) and stream lines (red) of the resulting construction. This specific eigenfunction results from and , which has a growth rate of in agreement with Equation ( 26 ). The green contour shows the trajectory of a fluid element that begins at . Figure 6. Contours of TM/KH eigenfunctions for , and with . The green contour shows the trajectory of a particle that started at . Because the growing linear eigenmode has been used in the nonlinear regime, the stream lines (red) cross the field lines (black) in this construction. The magnetic field evolution implied by continuing island growth requires this cross-field flow in the regions outside the diffusion region. This cross-field flow will naturally cease when the linear phase ends and the island growth ends. Fully nonlinear treatments of the TM, in which island saturation occurs self-consistently, have found magnetic structures that are very similar to those produced by the linear modes with (Ofman et al. 1993 ; Chen et al. 1997 ; Li Ma 2010 ; Zhang et al. 2011 ; Doss et al. 2015 ). However, these treatments find a velocity field very different from the linear eigenmode. In particular, the velocity must align with the magnetic field outside the diffusion region, which is not the case with the unsaturated linear eigenmode, as seen in Figure 6 . Nonlinear studies of island saturation in the presence


We thank N. Featherstone for helping to formulate the idea of creating a rotationally unconstrained near-surface fluid layer in simulations by increasing density contrast, as well as for his work as the primary developer of the Rayleigh code. We thank K. Julien for helpful conversations on the subject of rotational constraint. We thank M. Miesch for illuminating discussion with regard to the high-latitude interaction between differential rotation and meridional circulation. NASA grants NNX13AG18G and NNX16AC92G provided the primary support for this research. Additional support was provided through NASA grants NNX14AG05G, NNX14AC05G, and NNX17AM01G. Computational resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. We thank the Computational Infrastructure for Geodynamics ( http://geodynamics.org ), which is funded by the NSF under awards EAR-0949446 and EAR-1550901, for supporting the development of the Rayleigh code. L.I.M. was supported during this work by a University of Colorado Boulder Chancellor Fellowship and a George Ellery Hale Graduate Fellowship.


strong enough, it will be difficult to detect them in low resolution spectra especially when the data is noisy. Unfortunately, the Ly β spectral region in most of our spectra has low SNR. This limits our ability to accurately measure the H i column density in individual systems. We, therefore, try to stack these spectra to increase the SNR. The higher SNR of the stacked spectrum would in principle allow us to determine the H i column density by fitting the damping wings of the Ly β absorption. If the majority of the systems have high H i column density, the damping wings of the Ly β absorption should be clearly and prominently visible in the stacked spectrum. In order to increase the chances of clearly detecting the damping wings of the Ly β absorption in the stacked spectrum, we carefully choose the quasar spectra that are used in the construction of the stacked spectrum using the following procedure. We first create a synthetic Ly β absorption line with H i column density of N (H i ) = 10 21 cm −2 and b -value of 100 km s −1 . We then exclude quasar spectra whose Ly β absorption are narrower than this synthetic absorption line. For example, in the lower panel (upper panel) of Figure 8 , we show two quasar spectra with their Ly β absorption being narrower (broader) than the synthetic one. This conservative approach ensures that only spectra of the absorbers with relatively higher H i column density and stronger damping wings contribute in the construction of the stacked spectrum. Applying this constraint on the S β sample leaves us with 21 spectra. We then median-stack these spectra following the method described in Paper I . Figure 8. Ly β spectral regions of four ghostly absorbers. In each panel, the red curve shows a synthetic Ly β absorption line with log N (H i ) = 21.0 and b -value of 100 km s −1 . The upper panels (lower panels) show Ly β absorption that are broader (resp. narrower) than the synthetic spectrum. Figure 9 shows the resulting stacked spectrum around the Ly α , Ly β , and Ly γ spectral regions. The Voigt profile fit for log N (H i ) = 21.5 and b -value of 50 km s −1 is overplotted on the observed spectrum as the red curve. This column density is 0.5 dex higher than the H i column density derived from the stacked spectrum of Paper I . As shown in Figure 9 , the damping wings (especially the red wing) of the Ly β absorption is clearly visible in the spectrum. This implies that the majority of the ghostly absorbers whose spectra are used in the stacked spectrum, are DLAs. Follow-up observation with higher resolution and SNR are needed to precisely measure the H i column density in the individual systems, and confirm which of them are indeed DLAs. Figure 9. Voigt profile fit (red curves), with log N (H i ) = 21.5 and b -value of 50 km s −1 , on the Ly β and Ly γ absorption lines in the stacked spectrum of the ghostly absorbers with strong Ly β absorption lines (see the text). The H i column density is constrained by the strong


three regimes: the stable regime, the intermediate regime, and the unstable regime. In the stable regime, the Laplace–Lagrange rapid angular momentum exchange between the two inner planets dominates over the gravitational perturbations of the outer orbit. The left column of Fig. 3 depicts this evolution; the system in this column resides in the bottom left of the parameter space in Fig. 2 . The system remains stable for 10 Gyr of evolution and the orbits never come close to one another. The two inner planets’ inclinations oscillate together around the initial z -axis, due to the precession of the nodes (see below). In the intermediate region, systems might appear to be stable for a long time but the outer orbit perturbations become too dominant, causing instability of the inner system. We show the dynamics of this type of evolution in the middle panel of Fig. 3 . This system resides very close to the stability criterion plotted in Fig. 2 , inside the transition zone between e 3,c ( f LL,min ) and e 3,c ( f LL,max ). In the unstable regime, gravitational perturbations from the third planet cause high amplitude eccentricity excitation in the inner planets, causing orbits to cross. We show this behaviour in the right column 3 . This system lies in the top right of the parameter space depicted in Fig. 2 . Figure 3. Here are the three prominent types of dynamics from each region. From left to right, stable, intermediate, and unstable, for our nominal system. Shown in the top panels are the apocentres and the pericentres of the innermost planets ( m 1 in blue and m 2 in red). The middle panels show the inclinations of the inner planets ( i 1 in blue and i 2 in red). Note that this inclination is not with respect to the total angular momentum, but rather with respect to the initial angular momentum of the two inner planets . The bottom panels show, coloured in black, the difference between the longitude of ascending nodes of the two inner planets as cos (Ω 1 − Ω 2 ), and, in yellow, the difference between the longitude of the periapsis as cos (π 1 − π 2 ). These two parameters are present in equation ( 11 ). Recall that the nominal system has the following parameters: M = 1 M ⊙ , m 1 = m 2 = 1 M ⊕ , m 3 = 1 M J , a 1 = 1 au, a 3 = 100 au, and we set initially ω 1 = ω 2 = ω 3 = Ω 1 = Ω 2 = Ω 3 = 0, e 1 = e 2 = 0.001, i 1 = i 2 = 0.001 and i 3 = 45°. The only difference between each integrated system is the second planet’s separation and the third planet’s eccentricity. For the stable system we chose a 2 = 3 au and e 3 = 0.8. For the intermediate system we had a 2 = 3 au and e 3 = 0.9 (which placed it on the stability curve), and finally for the unstable system we had a 2 = 5.9 au and e 3 = 0.551. In the system’s reference frame (see Fig. 1 ), the inner planet’s inclinations are the angles between their respective angular momenta and the normal to the initial orbits. Thus, the inclination modulation shown in the stable system (the left column in Fig. 3


this figure; both parameters are prior dominated. 5.1 CMB lensing with <italic>A</italic>s prior and supernova Ωm constraint With a conservative external A s prior, the L eq – A s degeneracy can be broken, leading to a tight constraint on L eq . Since L eq depends only on Ω m and H 0 , given an additional Ω m constraint, H 0 can then be determined. As described in Section 4 , our conservative A s prior (10 9 A s = 2.19 ± 0.09) is motivated by current constraints from the primary CMB, while our Ω m constraint (Ω m = 0.3 ± 0.012) is forecast for future supernovae measurements. Both the A s and Ω m constraints are independent of assumptions about r s . In Fig. 3 , we show (blue curves) the results of imposing the CMB-motivated A s prior on the CMB lensing constraints from Fig. 2 . Imposing this prior results in tighter constraints on L eq than in Fig. 2 , as expected. The results of additionally imposing the supernova prior on Ω m are shown with the red curves in Fig. 3 . We find that an uncertainty on H 0 of 3.5 km s −1 Mpc −1 is achieved. We will refer to this as our pessimistic forecast, since it assumes a conservative A s prior and a modestly improved Ω m from future surveys. Figure 3. Forecast constraints for a CMB-S4 measurement of the CMB lensing power spectrum upon including external A s and Ω m information. By imposing a conservative, CMB-motivated (but r s independent) prior on A s we break the A s – L eq degeneracy, leading to a constraint on L eq ≡ k eq χ * (blue curves; compare to Fig. 2 ). Since , the resultant constraints on Ω m and H 0 are degenerate (the black dashed curve illustrates the expected degeneracy). By including an additional Ω m constraint from supernovae we break the Ω m – H 0 degeneracy, and obtain a constraint on H 0 (red curve, with marginalized parameter uncertainties reported along the diagonal). As an alternative to a supernova constraint on Ω m , we also consider adding information from an LSST-like measurement of cosmic shear (orange curve); in combination with CMB lensing, galaxy lensing also provides Ω m information that serves to break the H 0 –Ω m degeneracy. Because of large-scale information in the CMB lensing power spectrum, our constraints are not very sensitive to the A s prior. If this prior is completely removed, we still obtain a constraint on H 0 (grey dashed curve). We note that the constraints on H 0 are not very sensitive to the A s prior. As can be seen in Fig. 2 , even in the absence of any A s prior, we obtain some constraint on L eq . This is likely because low- L information in the lensing power spectrum breaks the degeneracy between A s and L eq . In the absence of the A s prior, the combination of CMB lensing and the Ω m constraint yields an H 0 uncertainty of roughly 6 km s −1 Mpc −1 (grey dashed curves in Fig. 3 ). It is also interesting to consider how the constraints on H 0 could improve with more optimistic constraints on A s , Ω m , and Ω b . For


We thank Hao-Yi Wu and Olivier Doré for the theoretical prediction for the cross-correlation between CIB and CMB lensing. The South Pole Telescope program is supported by the National Science Foundation through grant PLR-1248097. Partial support is also provided by the NSF Physics Frontier Center grant PHY-0114422 to the Kavli Institute of Cosmological Physics at the University of Chicago, the Kavli Foundation, and the Gordon and Betty Moore Foundation through Grant GBMF #947 to the University of Chicago. The McGill authors acknowledge funding from the Natural Sciences and Engineering Research Council of Canada, Canadian Institute for Advanced Research, and Canada Research Chairs program. C.R. acknowledges support from a Australian Research Council Future Fellowship (FT150100074). B.B. is supported by the Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy. This publication makes use of data products from the Wide-field Infrared Survey Explorer , which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Argonne National Laboratorys work was supported under U.S. Department of Energy contract DE-AC02-06CH11357. Computations were made on the supercomputer Guillimin from McGill University, managed by Calcul Québec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), the ministère de l’Économie, de la science et de l’innovation du Québec (MESI) and the Fonds de recherche du Québec—Nature et technologies (FRQ-NT). This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. This work used resources made available on the Jupiter cluster, a joint data-intensive computing project between the High Energy Physics Division and the Computing, Environment, and Life Sciences (CELS) Directorate at Argonne National Laboratory. The results in this paper have been derived using the following packages: Astropy , a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013 ), CAMB (Lewis et al. 2000 ; Howlett et al. 2012 ), cmocean , 49 49 http://matplotlib.org/cmocean/ HEALPix (Górski et al. 2005 ), IPython (Pérez Granger 2007 ), LensPix (Lewis 2005 ), Matplotlib (Hunter 2007 ), NumPy and SciPy (van der Walt et al. 2011 ), PolSpice (Chon et al. 2004 ), and quicklens (see footnote 39).


based as closely as possible on standard physical principles. It includes winds and heating from young massive stars as well as feedback from SN explosions, the latter following the known physical processes that govern the different phases of SN remnants. We use the smoothed particle hydrodynamics (SPH) code SPHGal (Hu et al. 2014 ), an implementation that incorporates several improvements into the Gadget code (Springel 2005 ) to address some known SPH problems. Our approach is most similar to the stellar feedback in Agertz et al. ( 2013 ) and Agertz Kravtsov ( 2016 ), as well as the Feedback in Realistic Environment ( Fire ) project, presented in Hopkins et al. ( 2014 ) and used by Hopkins et al. ( 2012 , 2013 ), Muratov et al. ( 2015 ), and Su et al. ( 2016 ). We apply our stellar feedback to an isolated Milky Way-like galaxy within an isolated dark halo. The tools developed here are in use by the cosmological code in Choi et al. ( 2016 ). We break our approach into three parts: feedback from young, pre-SN stars, feedback from dying low-mass stars, and feedback from SN events. (1) We include the effects of stellar winds from young stars by transferring momentum to neighboring gas particles for the first few Myr. We also heat gas particles within the Strömgren sphere of young star particles, mimicking the effects of ionizing radiation from massive stars (see e.g., Renaud et al. 2013 ). (2) We transfer momentum from old, low-mass stars to the surrounding gas throughout several Gyr after a type II SN has occurred to imitate the slow winds from asymptotic giant branch (AGB) stars. (3) In our SN feedback model, we transfer SN energy to the closest gas particles to the SN event, each one receiving mass and energy according to a computed high-resolution evolution of SN remnants (SNR). Detailed very high resolution hydrodynamic simulations of SNR evolution (see Li et al. 2015 ), for details and a review) show that they pass through three successive phases: free expansion (FE), energy-conserving Sedov–Taylor (ST), and momentum-conserving snowplow (SP) (see also Kim Ostriker 2015 ; Martizzi et al. 2015 ; Walch Naab 2015 ). In our model, which we call the SP model, each neighboring gas particle receives energy following the physics governing the SNR phase in which the gas particles lie. If the gas particle lies within the radius of the ejecta-dominated FE phase, SN energy is transferred by conserving the ejecta momentum. If the gas particle lies outside the FE radius but within the radius of the adiabatic blast wave ST phase, SN energy is transferred 30% as kinetic energy and 70% as thermal energy. Finally, if the gas particle is beyond the ST radius and into the realm of the pressure-driven and momentum-conserving SP phase (Chevalier 1974 ), then part of the original SN energy is dissipated and only a fraction is transferred to the gas particle, with the thermal portion dissipating more quickly than the kinetic portion, following a prescription


We thank all of the members of the Chandra team for their enormous efforts, specifically D.P. Huenemoerder and M. Nowak for easing data processing and fitting procedures.


all models and for both metallicity evolution scenarios. This indicates that BH–BH mergers are expected to be dominant GW sources within the framework of the evolution adopted in models V1, V2, and V4. If no detections are made after a modest increase of LIGO/Virgo sensitivity, these specific evolutionary models will be eliminated in the context of BH–BH formation. Our calculations incorporate the evolution of average metallicity with redshift into the cosmological model of the universe; a broad spectrum of metallicities is employed in our population synthesis rate predictions. Changing the details of the adopted metallicity evolution model, within the uncertainties, produces at best modest effects on the rate predictions. Specifically, the low metallicity models produce somewhat higher merger rate densities for heavy mergers (BH–BH) and somewhat lower merger rate density for low-mass mergers (NS–NS). These trends are understood in terms of the effects of metallicity on the BH–BH formation (low metallicity boosts the formation) and on the NS–NS formation (low metallicity suppresses the formation; see Dominik et al. 2012 ). For example, the standard model (V2) merger rate density changes from for low metallicity evolution (see Table 3 and Figure 2 ) to for high metallicity evolution (see Table 4 and Figure 3 ) for BH–BH mergers with total mass 25–37 . The decrease of merger rate density from low to high metallicity evolution is rather moderate: only a factor of ∼2. Note that metallicity itself plays a crucial role in the formation process of BH–BH binaries (Belczynski et al. 2007 , 2010a ). For example, a group of stars with solar metallicity will form ∼10–100 times fewer BH–BH mergers than the same group with 10% solar metallicity (de Mink Belczynski 2015 ). However, once we incorporate this very strong dependence into cosmological models of the universe and employ a broad spectrum of metallicities (as in our methods II and III), the merger rate density predictions are not as strongly affected. Both metallicity evolution models generate (alas with a different efficiency) low metallicity stars (typical progenitors of BH–BH mergers). A broad BH–BH merger delay time distribution (Dominik et al. 2013 , 2015 ) allows for both models to produce BH–BH binaries that will merge at low redshifts. 5.2. Comparison of Models with Forecasted Advanced LIGO/Virgo Upper Limits In Figures 5 and 6 and Tables 5 and 6 we present evolutionary model predictions for double compact object merger rate densities. The predictions are constructed with method II (see Section 2.2 ) and contrasted with the forecasted advanced LIGO/Virgo upper limits (see Section 4.2 ). Our models are described in Section 3 . Figure 4. Merger rate density for BH–NS systems for our low metallicity evolution scenario predicted (with method II) for advanced LIGO/Virgo. Note that only the delayed SN explosion model (V4) and the optimistic model (V1) make detections likely, while the other models


eigenvalues . The lengths of the semimajor and semiminor axes are and with the axis ratio b ′/ a ′. The details of the iterative method for measuring axis ratios can be found in Tenneti et al. ( 2015b ). In the first iteration, we start with the half-lengths of the principal axes all equal to 1 and determine the eigenvalues and eigenvectors of the ellipsoid. After each iteration, the lengths of the principal axes of ellipsoids are rescaled such that the enclosed volume is constant and particles outside the ellipsoidal volume are discarded. This process is repeated until convergence is reached such that the fractional change in axis ratios is below 1%. In addition to the distribution of the axis ratios, b / a and c / a of the stellar components of subhalos, we are also interested in the orientation of the major axis of the stellar shape with the shape of dark matter in subhalos. So, we compute the probability distribution of the misalignment anglewhere and are the major axes of the shapes defined by the dark matter and stellar matter components respectively. 3.2. Two-point Statistics In this paper we quantify the intrinsic alignments of galaxies with the large-scale density field using the ellipticity–direction (ED) and the projected shape–density () correlation functions. The ED correlation function cross-correlates the orientation of the major axes of the 3D shapes of dark matter or stellar component of galaxies with the large-scale density field. Consider a subhalo centered at position x with the major axis direction . Let the unit vector in the direction of a tracer of the matter density field at a distance r be . Based on the notation in Lee et al. ( 2008 ), the ED correlation function is given bywhich is zero for randomly oriented galaxies in a uniform distribution. In the simulations the matter density field is traced using the positions of dark matter particles. The projected shape correlation function, is directly related to the correlation function measured in observations. Following the notation of Mandelbaum et al. ( 2006 ), we define the the matter–intrinsic shear correlation function and the corresponding projected two-point statistic . In this paper, r p is the comoving transverse separation of a pair of galaxies in the XY plane and Π is their separation along the Z direction. The components of the projected ellipticities of a galaxy are given bywhere b ′/ a ′ is the axis ratio of the projected shape of the stellar component of a galaxy, and φ is the position angle of the major axis with respect to the reference direction (position of the dark matter particle). Here, e + refers to the radial component and e × is the component rotated at 45°. The matter–intrinsic shear correlation function is given by,where S + represents the “shape sample,” selected on the basis of a binning in subhalo mass and the “density sample” labeled by D consists of the dark matter particles used to trace the matter density field. S + D is given by the


into components which contribute significantly to the total density above z − , labelled Σ b 1 , and those which do not, e.g. are concentrated close to the disc plane below z − , labelled Σ b 0 . For this latter component, above z − the density is close enough to zero for the surface density to be considered constant. Hence, for z > z − , the total surface density is Σ( z ) = Σ b 0 + Σ b 1 ( z ) + 2ρ dm z , and equation ( 9 ) then becomes Note that the baryonic density shape inside z does not enter in the calculation of σ z ( z ); hence, there is no need to model the shapes of the baryonic components inside the z region where we have tracer data. Also note the term 2ρ dm ( h + z ) of equation ( 10 ) can be expressed as Σ dm ( z ) + Σ dm ( h ). Despite having a constant DM density to an infinite height, the contribution of this component to is finite. The term Σ dm ( h ) is a manifestation of the tracer density decaying exponentially with a scaleheight of h . Thus, the impact of the assumption of constant DM density is controlled by the scaleheight of the tracer population, with hotter populations reaching higher above the disc being affected more. 3.3 Tilt term At the midplane ( z = 0), a potential symmetric in z is separable up to second order (Binney Tremaine 2008 ), with the radial and vertical motions decoupling. Thus, at z = 0 the tilt term vanishes, though it may increase rapidly as z increases. Intuitively, the motion of the tracer stars can be viewed as a combination of oscillation around the disc, and orbital motion around the Galactic Centre; the tilt term can then be seen as a consequence of this orbital motion around the Galactic Centre. A star in an elliptic orbit around the Galactic Centre will on the part of its orbit where it is moving towards the Galactic Centre (i.e. has v R 0) typically also move towards the disc plane; hence, this star will for z > 0 have , and for z 0. Similarly, in the part of the orbit where the star moves away from the Galactic Centre (i.e. has v R > 0), it will typically also move away from the disc plane, again resulting in having the same sign as z . Hence, as , we arrive at the result that σ Rz carries the same sign as z . See Silverwood ( 2016 ) for further motivation on this result on the sign of σ Rz . The σ Rz data given in B15 has a sign error, as it is negative for positive z , and so as done in Silverwood et al. ( 2016 ) we apply a sign correction. We again assume symmetry above and below the disc plane, and for z > 0 we model the vertical profile of σ Rz using the functional form σ Rz = Az n , where A and n are fitted to the σ Rz data. For A ≥ 0 and n > 0, this model naturally encompasses the requirements that σ Rz ≥ 0 for z ≥ 0 and σ Rz = 0 for z = 0. The modelling σ Rz = Az n also fits the σ Rz data well, see Fig. 8 and Silverwood et al. ( 2016 ). The tilt term includes a radial derivative of ν and σ Rz , and so we must also model their radial behaviours. From Bovy et al. ( 2016 ), we see


This work is supported by JSPS KAKENHI grant number 24740190 and 24105007. HY is supported by the Advanced Leading graduate school for Photon Science (ALPS). ACF acknowledges support from ERC Advanced Grant Feedback, 340442.


to interval E4, as well as the full spectrum irrespective of the angle. The spectral break between the 1∕ f and f −3∕2 ranges for the unconditioned spectrum (black line) is found approximately (by inspection) at f b ≈ 2 × 10 −3 Hz, indicated by the vertical dotted line. Right : spatial energy spectrum of magnetic fluctuations E ( k ⊥ ) reconstructed from the frequency spectra P sc ( ω , θ VB ) for each sampling angle as well as by using the standard TH (black line). 4.2 Reconstructing field-perpendicular spectrum E(<italic>k</italic><sub>⊥</sub>) In the following, we proceed to reconstruct the reduced energy spectrum E ( k ⊥ ) using data from within each angle bin above 30° and for which we find a large statistical sample. For illustration purposes, we concentrate on a 24 h interval just before the fourth perihelion (E4), which was also investigated using standard TH analysis by Chen et al. (2021) . Table 3 shows the values of ε , V ⊥ , and the number of statistical samples (counts) associated with this interval for each ten-degree angle bin around 30°, 40°, 50°, and 60°. In this statistical sample, ε ≤ 0.2, in which case the BP19 methodology in the TH limit ( ε → 0) is applicable, that is, combining Eqs. ( 2 ) and ( 3 ) leads to The most common form of TH follows from this expression by performing the integral in a Cartesian coordinate system with one axis along the streamwise direction so that Here E 1 D ( k s ) is the one-dimensional energy spectrum with respect to the streamwise (sampling) direction. This expression resembles Eq. ( 6 ) with Λ α , ε = 1 and leads to the familiar ω = − k s V expression to relate spacecraft frequencies with streamwise wavenumber k s . Table 3 Relevant parameters for each angle bin considered in our analysis with a sufficiently large number of samples. Notes. The parameter ε remains below 0.15, which allows for the application of the TH in the framework of BP19. For reference, at 1 s resolution, the total number of counts in any given 24 h interval is 86 400 samples. It is also important to note that values for Λ α , ε were calculated for α = 3∕2 and using Eq. ( 18 ) Λ TH ≃ 0.7628. However, because we are interested in reconstructing the angle-integrated field-perpendicular spectrum E ( k ⊥ ), Eq. ( 16 ) must be integrated in cylindrical coordinates. Using the change of variables x = ω ∕ k ⊥ V ⊥ , we obtained 3 where Again, Eq. ( 18 ) also resembles Eq. ( 6 ) with Λ α , ε replaced by Λ TH . In fact, it can be shown that , which is consistent with the BP19 model in the limit when ε → 0. The scaling factor Λ TH is due to the integration with respect to the angle φ in the dot product k ⊥ ⋅ V = k ⊥ V ⊥ cos φ ( Bourouaine et al. 2012 ; Bourouaine Chandran 2013 ; Martinović et al. 2019 ). Using Eqs. ( 11 ) and ( 12 ) within each statistical sample described in Table 3 , P sc ( ω , θ V B ) was calculated for magnetic field measurements b ( t ) according to Eq. ( 12 ). The left panel of Fig. 3 shows frequency spectra P


scale rad is equal to the radial distance separating adjacent arms near the Sun and the scale azim is the angular distance between them. To reiterate, this method does not afford a reliable estimate of the angle p and the number m but only of rad for a given m . The best-fitting values of , , , , u 0 , v 0 , p and 0 for a given m are found by fitting that minimizes 2 defined as In equation ( 2 ), i counts the N stars in the sample, q is the number of parameters to be derived, v obs, i and v mod, i are the observed and modelled velocities and v obs, i is the uncertainty of v obs, i . The model depends linearly on the set of six parameters (), which can be determined by the standard least-squares fitting for specified 0 , m and p . The quantities and are periodic with respect to 0 with period , while the quantities and are periodic with period 2. For a trailing pattern and the potential , out of the two Suns phase values 0 and 0 that minimize 2 , the values that satisfy the condition are chosen (Mishurov etal. 1997 ; Griv, Ngeow Jiang 2013 , 2014 ; cf. Mishurov etal. 1979 , p. 149). The values of () will completely determine the Galactic differential rotation and the values of ( m , p ) (or rad ) and 0 will determine the nearby spiral pattern. 3 DATA The data of stars were collected from the APOGEE-RC DR13 catalogue of 19291 objects, which is constructed using the method described in Bovy etal. ( 2014 ). This catalogue of intermediate-old giant stars in the near-infrared on scales between 25pc and 10kpc has the estimated uncertainties in the distances and the line-of-sight velocities of 5percent and 0.1kms 1 . Ages of 705 local APOGEE giants with the mean age t age 310 9 yr were estimated by Feuillet etal. ( 2016 ). Details on the APOGEE project in the Northern hemisphere and the dual-hemisphere APOGEE-2 project can be found in Bovy etal. ( 2015 ) and Majewski etal. ( 2016 ), respectively. 1 Fig. 1 shows space locations of all 19291 stars with known d projected on to the Galactic X Y , X Z and Y Z planes. As is seen, the stars are limited mainly within 5kpc from the Sun. In the object distribution, there are no obvious arm-like structures in the X Y plane (Fig. 1 , left-hand panel). A scaleheight of the distribution is h 200300pc (Fig. 1 , middle and right-hand panels). Such a scaleheight h 250pc is typical to intermediate-old population of the local Galactic disc (Kharadze etal. 1989 ; Rix Bovy 2013 ; Camargo, Bica Bonatto 2015 ). Figure 1. The space distribution of all 19296 stars with known photometric distances from the Sun projected on to the Galactic X Y , X Z and Y Z planes. Cartesian X , Y and Z coordinates are shown, the X -axis points to the direction of the Galactic rotation, the Y -axis points to the Galactic anticentre, the Z -axis points to the North Galactic pole, and the axis of the Galactic rotation is taken oriented along the Z -axis, with the Sun at (0, 8.3, 0). The solar circle ( r 0 8.3kpc) and the Z 0 plane are indicated by


Megeath et al. 2012 ) and included in the Herschel Orion protostar survey (HOPS; Furlan et al. 2016 ) as source 383. The submillimeter to bolometric luminosity ratio of HOPS 383 , 1.4% ( Safron et al. 2015 ) well surpasses the (0.5%) threshold for bona fide Class 0 designation ( André et al. 1993 , 2000 ). Notably, HOPS 383 is the first Class 0 protostar known to have undergone a mass-accretion-driven eruption ( Safron et al. 2015 ), which peaked by 2008 and ended by 2017 September (Appendix B ). 2. Observations We observed HOPS 383 three times with the Chandra X-ray Observatory ( Weisskopf et al. 2002 ) from 2017 December 13 to 14 with simultaneous near-infrared imaging on 2017 December 14 using the 4.1 m Southern Astrophysical Research (SOAR) telescope ( Krabbendam et al. 2004 ) in Chile. We used Chandra ’s Advanced CCD Imaging Spectrometer (ACIS-I) in the very faint mode with a frame time of 3.141 s (Appendix C.1 ). Chandra ACIS-I has on-axis a full-width half-maximum (FWHM) angular resolution of and a FWHM spectral resolution of 261 eV at 6.4 keV. Data reduction is described in Appendix C.2 . The SOAR Spartan infrared camera ( Loh et al. 2012 ) is composed of 4 CCDs of 2048 × 2048 pixels. We selected the wide-field mode with a single-detector field-of-view of (i.e., edge-to-edge) and a pixel scale of with the K -band and H 2 narrow-band filters. We used a five-point dithering pattern where, in the first exposure, HOPS 383 was put near the center of the northeastern detector (det3), which was then moved by 30′′ from its initial position sequentially toward the south, north, west, and east in the four other exposures. This sequence was repeated until the requested exposure was achieved (Appendix D.1 ). Data reduction is described in Appendix D.2 . 3. Near-infrared results The near-infrared nebulosity that was prominent during the outburst ( Safron et al. 2015 ) is not detected in our SOAR K -band image (Fig. 1 a), indeed, it vanished by 2015 December 30 ( Fischer Hillenbrand 2017 ). The H 2 -narrow filter image shows an emission knot of shocked molecular hydrogen at ∼15′′ southeast from the mid-infrared location of HOPS 383. We identify this H 2 source with the H 2 emission knot SMZ 1-2B ( Stanke et al. 2002 ) that was observed on 1997 September 13, yielding a proper motion of southeast in 20.25 yr (Fig. 1 b). The corresponding velocity is ∼(95 ± 53)×(sin i /sin69.5°) −1 × ( d /420 pc) km s −1 , where d is the distance to HOPS 383 and i is the inclination angle (90° corresponds to an edge-on view; Table 1 of Furlan et al. 2016 ). This is typical of the velocity values observed in the outflows from Class 0 protostars ( Bontemps et al. 1996 ; Reipurth Bally 2001 ). Assuming a constant velocity, the angular distance of ∼16′′ between HOPS 383 and this emission knot corresponds to a kinematical timescale of ∼180 ± 100 yr. The proper motion direction is consistent with the orientation of the outburst nebula, the 12 CO blueshifted outflow (


by the electrons in the converging flow on to a compact star and they called such Comptonization BMC, i.e. Bulk-Motion Comptonization. The BMC model has been adopted to interpret the PL component in the broad-band X-ray spectrum of BHXBs well (Shrader Titarchuk 1998 ; Titarchuk Seifina 2009 ; Titarchuk Shaposhnikov 2010 ). The BMC is one of the models of xspec and it is formulated as This expression consists of two subcomponents: one is the injected blackbody-like component ( BB ) to provide the seed photons for Comptonization, and the other is BB * G to describe the Comptonization emission, where G is the response of the Compton medium. In xspec , this model includes four parameters, i.e. kT bb , α, log ( A ), and C N , which are the seed photon temperature, spectral index (photon index Γ = α + 1), logarithm of weighting factor, and normalization, respectively. The observed fluxes of BMC are comprised of a direct emission flux and a Comptonized emission flux. The fraction of the direct emission is calculated with the factor 1/(1 + A ), whereas the fraction of Comptonized emission is inferred by a Comptonization factor f ( f = A /(1 + A )). In two cases, the BMC model is reduced to BB: one is with A ≪ 1, indicating that the value of f is very small, so the Comptonized emission is negligible; the other is with α ≫ 1, meaning that the energy for the seed photons to get from the high-energy electrons is very small, so the Comptonization is inefficient. The theoretical study of Titarchuk et al. ( 1997 ) shows that the BMC process taking place around the compact star can only result in the PL component with Γ > 1. Therefore, theoretically, the BMC model can only account for the relatively soft hard X-ray PL tail in the X-ray spectrum of XRBs. In xspec , in order to ensure Γ > 1, the spectral index α is set to be a positive parameter, i.e. α > 0. In this work, the obtained photon index of the PL hard tail spans a range of ∼(1.1–2.3) and ∼(0.7–2.1) on the HB and on the NB, respectively. Considering the error of the lower limit of NB photon index, the photon index of HB or NB satisfies the condition of Γ > 1. Assuming the hard X-ray tail in Sco X-1 to be originated from the BMC process around the NS, we form model to fit the broad-band spectra of this source. Assuming the NS to provide the seed photons for the dynamic Comptonization in the converging accretion flow on to the compact star, the BB emission of the NS as well as the outgoing PL component is included in the BMC model. Adopting that there exists an ADC in Sco X-1, the disc emission is described by a CPL (Church et al. 2012 ). Taking into consideration the iron line and the additive BB in the PCA spectrum, finally, we form a multicomponent model of BMC+LINE+CPL+BB to fit the PCA+HEXTE spectra of HB and NB in which hard X-ray PL tails are detected. The fitting results are listed in Table 4 and four unfolded spectra are shown in Fig. 9 . As listed in Table 4 , the values of the reduced


Acknowledgments The author is grateful to an anonymous referee for a careful reading of the manuscript and several suggestions that helped him to substantially improve the present work. AFL wishes to thank Drs. A. Maggio, S. Sciortino, I. Pillitteri, I. Pagano, and S. Desidera for interesting discussions on magnetic star–planet interactions. This work was partially supported by the project WOW, one of the Progetti Premiali of the Italian Ministry of Education, University, and Research funded to the Italian National Institute for Astrophysics (INAF).


We thank the staffs of the ESA Herschel Science Centre and NASA Herschel Science Center, and the HIFI, Instrument Control Centre (ICC) for their help with the data reduction routines. In addition, we owe special thanks to David Teyssier for clarifications regarding the hebCorrection tool. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.


ACKNOWLEDGEMENTS This work is partly supported by Ministry of Economy and Competitiveness (MINECO) (Spain), FIS2016-76363-P, by COST Action PHAROS (CA16214), by project 2017 SGR247 Agency for Management of University and Research Grants (AGAUR, Catalonia) (SDO). AVA thanks the Program 5–100 Immanuel Kant Baltic Federal University (IKBFU, Russia).


= 5 in the top panel of Fig. 5 . The result using the Tiamat halo merger trees is shown for comparison and to indicate the convergence of the quasar luminosity function at z ∼ 5. The model is in agreement with observations at the bright end (Shen Kelly 2012 ; McGreer et al. 2013 ). We indicate the quasar luminosities of the three analogues in the top panel of Fig. 5 and find that the progenitors of ZF-1, ZF-2 and ZF-3 are bright quasars in the range that optical surveys such as SDSS are able to detect. This suggests that ZF-COSMOS-20115 is likely the descendent of a high-redshift quasar and illustrates the possible connection between MQGs and high-redshift quasars (Maiolino et al. 2012 ). Figure 5. Top panel: quasar UV 1450 Å luminosity function at z ∼ 5. The results using the Tiamat-125-HR and Tiamat halo merger trees are shown with solid and dashed lines, respectively. The shaded regions represent the 95 per cent confidence intervals around the mean using 100 000 bootstrap re-samples for the Tiamat-125-HR result. The observational data is shown with different symbols: Shen Kelly ( 2012 ) using SDSS DR7 data at z ∼ 4.75 () and McGreer et al. ( 2013 ) using SDSS, UKIDSS and MMT at z ∼ 4.7–5.1 (▵). From left to right, the vertical lines indicate the quasar UV magnitudes of ZF-1, ZF-2 and ZF-3, respectively. Bottom panel: correlation between high-redshift black hole mass ( z = 5) and low-redshift SFR ( z = 3.7). The size of the empty circle represents different stellar masses and ZF-3 is indicated with the filled circle. In order to better demonstrate this, for each massive galaxy selected at z = 5.3, we present the correlation between black hole mass and subsequent SFR at z = 3.7 in the bottom panel of Fig. 5 . ZF-3 is indicated with a filled circle. We see that galaxies with less massive black holes ( M BH 10 7.5 M ⊙ ) can have a range of subsequent star formation levels. This is due to self-regulated stellar feedback being the dominant mechanism in these galaxies. However, due to persistent AGN feedback, galaxies with more massive black holes do not have intense subsequent star formation. This indicates that a high-redshift bright quasar will likely become a MQG at later times, when its accretion disc has been completely consumed. 5.3 The reionization history The three ZF-COSMOS-20115 analogues are hosted by massive haloes, which provide deep gravitational potentials that efficiently accrete baryons. The high baryon accretion efficiency induces intense star formation at high redshift, providing a significant number of ionizing photons that reionize surrounding hydrogen at a very early time. In this section, we discuss the reionization history of ZF-COSMOS-20115 . In Fig. 6 , we show the evolution of the average neutral hydrogen fraction within boxes with different sizes around the three analogues. The neutral hydrogen fraction is calculated using 21 cmFAST (Mesinger, Furlanetto Cen 2011 ) with the galaxy catalogue provided by Meraxes (see


from (numerical) frequency runs. In that case, reorganization energies (e.g. ) computed from single-point calculations were utilized to get the adiabatic values. APPENDIX A2: ADDITIONAL DETAILS ON THE CALCULATION OF ROVIBRATIONAL PROPERTIES Deceptive as it may sound for a theoretical chemist, like in the case of other carbon chains (e.g. Bâldea 2020b ), the DFT approach appears to perform better in quantifying rovibrational spectra for MgC 4 H species than much more computationally demanding ab initio methods: For MgC 4 H 0 neutral, our very tight optimization at B3LYP/aug-cc-pVTZ level yielded MHz. The relative deviation from the experimental value MHz (Cernicharo et al. 2019 ) is per cent, which is significantly better than both 0.3 per cent at CCSD(T)-F12/cc-pCVTZ-F12 (Cernicharo et al. 2019 ) and 1.7 per cent at CCSD(T)/aug-cc-pVTZ (Cernicharo et al. 2019 ). Likewise, our value D DFT = 0.0641 kHz obtained at the B3LYP/aug-cc-pVTZ level is only 13 per cent smaller than the experimental value D exp = 0.074 ± 0.002 kHz (Cernicharo et al. 2019 ). This is better than 17 per cent, the deviation of CCSD(T)-value (Cernicharo et al. 2019 , and Table A2 ), which, in turn, is somewhat better than 18 per cent, which is the deviation of the CCSD(T)-F12-based value (Cernicharo et al. 2019 ). Noteworthily, the best estimate D Kratzer = 0.0741 kHz (cf. Table 4 and A3 ) is that based on the Kratzer formula (equation 1 ) used in conjunction with the B3LYP/aug-cc-pVTZ B and ω MgC . This Kratzer-based value cannot be distinguished from the experimental value D exp = 0.074 ± 0.002 kHz (Cernicharo et al. 2019 ). Table A3. Calculated centrifugal distortion constants D (kHz) presented along with estimates deduced via Kratzer approximation (equation 1 ) using harmonic frequencies ω MgC of the MgC stretching mode. a CCSD(T)/cc-pVTZ. b CCSD(T)/aug-cc-pVTZ. c B3LYP/6-311++G(3DF, 3PD). d B3LYP/aug-cc-pVTZ. e CCSD(T)/cc-pCVTZ. Table A4. Adiabatic and vertical values of electron attachment (EA) and ionization (IP) energies computed for MgC 4 H linear chains at several levels of theory. Their most reliable estimates are depicted in italics and were used to deduce the chemical hardness η, chemical softness S , chemical potential μ, Mulliken electronegativity χ, and electrophilicity index ω. Energy unit is eV. a Δ-CCSD(T)/cc-pVTZ. b EOM-CCSD/cc-pVTZ. c Δ-CCSD(T)/cc-pCVTZ. d EOM-CCSD/cc-pCVTZ. e Δ-CCSD(T)/aug-cc-pVTZ. f .EOM-CCSD/aug-cc-pVTZ. g Δ-B3LYP/6-311++G(3DF, 3PD). h CBS-QB3. For sure, neither a DFT/B3LYP/aug-cc-pVTZ calculation like that found performing the best above nor whatever other much more sophisticated theoretical approach currently available is precise enough to enable detection in space. In view of this state-of-the-art of theory, aiming at guiding subsequent laboratory measurements and interpretation of not yet available rovibrational spectra (as the case of ), it is current practice in astrophysics (e.g. Pauzat et al. 1991 ) to estimate rotational


This work was supported by NSF SHINE grant AGS1622413 to the University of New Hampshire (UNH). C.W.S. was also supported by Caltech subcontract 44A-1062037 to UNH in support of the ACE/MAG instrument. J.E.S. is supported by STFC(UK) grant ST/N000692/1.


origin ( t = 61 minutes) up to its decay phase ( t = 70.7 minutes) in the vertical experiment for the three LOSs. (An animation of this figure is available.) To extend the analysis, it is also of interest to consider the total emitted intensity for each vertical column, i.e.,which, following Equation ( 1 ), is equal to the column density of the emitting species along the LOS (except for a constant factor). Equation ( 5 ) has been calculated separately for Si iv and O iv and with the emissivities obtained assuming either NEQ or SE ionization, to better gauge the importance of disregarding the NEQ effects. The results are shown in the middle and bottom panels of Figure 10 for Si iv and O iv , respectively. The top panel of the image contains the 2D map of the emissivity for Si iv for context identification. Combining Figures 9 and 10 , we are able to discern and describe characteristic features of the spectral profiles: 1. The most prominent feature in the synthetic profiles of Figure 9 is the brightening associated with the location of the internal footpoint of the surge. In the corresponding movie, we can see how that footpoint is formed at around t = 64 minutes, as the surge detaches from the emerged dome. During those instants, the associated synthetic profiles are characterized by large intermittent intensities and bidirectional behavior with velocities of tens of kilometers per second, as apparent, e.g., in the (b0) and (c0) panels at x ∈ [15, 16] Mm. In Si iv and O iv (panels (b) and (c)), we find that the internal footpoint is usually the brightest region, although there are some instants in which the brightest points can be located in the crests or the external footpoint. This is a potentially important result from the observational point of view because it can help us to unravel the spatial geometry of the surge in future observational studies: if strong brightenings are detected in Si iv and also in O iv within the surge, it would be reasonable to think that they correspond to the internal footpoint of the surge. In this region, the intensity ratio between Si iv and O iv ranges between 2 and 7, approximately. Note that, in general, the intensity ratio values vary depending on the observed region and features (Martínez-Sykora et al. 2016 ). In Section 5.2 we will see that LOS effects play a major role in causing the large brightness of the internal footpoint (and other bright features) compared to the rest of the surge. Here we consider the parallel question of the role of NEQ: what would be obtained for the intensity of the internal footpoint if SE were assumed? Comparing the values for Si iv in Figure 10 (middle panel, 13.8 ≤ x ≤ 16.1), there is roughly a factor of two, on average, between the NEQ and SE intensities; for O iv (bottom panel), there is no major difference in the intensity between both calculations. One could conclude that while NEQ is important for the Si iv diagnostics, SE could be applied for the O iv case;


The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE). This publication makes use of data products from the 2MASS, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The computations in this paper were run on the Odyssey cluster supported by the FAS Science Division Research Computing Group at Harvard University. G.G. and D.F. are supported by NSF grant AST-1312891. E.S. acknowledges funding by Sonderforschungsbereich SFB 881 “The Milky Way System” (subproject A3) of the German Research Foundation (DFG). N.M. gratefully acknowledges the CNRS for support through PICS project PICS06183.


ACKNOWLEDGEMENTS We thank the anonymous referee for helpful comments that improved this paper. We gratefully acknowledge support from the Science and Technology Facilities Council (HPE through grant ST/K501979/1, SR through ST/N50404X/1, TPR through ST/P000541/1). HPE acknowledges support under NASA contract NNG08FD60C. This research made use of data obtained from the 3XMM XMM–Newton Serendipitous Source Catalogue, based on archival observations with XMM–Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We also acknowledge the use of public data obtained from the Chandra Data Archive and the Swift Data Archive.


which may not be particularly informative, as Fig. 9 demonstrates. To this end, we do not interpret the total flux from the nebula in terms of any particular spectral model; rather, we present a total integrated flux of the nebula emission to interpret the overall source strength. The resulting spectral flux obtained for the nebula from the summation of the 38 independent 0.26°× 0.26° spectral boxes is shown in Fig. 10 . For illustration, the HESS J1825–137 spectrum obtained from Fermi -LAT ( Grondin et al. 2011 ) is also shown, although we recall that the region from which this spectrum is obtained, based on a morphological symmetric Gaussian fit template to the Fermi data, does not fully correspond to the same spatial region. The 2011 analysis and the 3FGL (the third LAT Catalogue, 20 MeV to 300 GeV, Acero et al. 2015 ) both use a Gaussian with σ = 0.56° as the spatial template, whereas the 3FHL (the third LAT High-Energy Catalogue, 10 GeV to 2 TeV, Ajello et al. 2017 ) uses a Gaussian with σ = 0.75°, which is consistent with the Fermi Galactic Extended Source catalogue best-fit Gaussian with σ = 0.79 ± 0.04 ( Ackermann et al. 2017 ). Whilst these templates are only approximations to the source morphology, in Fig. 10 the IC peak may be inferred to occur at around 200 GeV. Fig. 10 Spectral energy distribution of HESS J1825–137 showing the total combined H.E.S.S. energy flux with statistical error bars, as determined with the method outlined in Sect. A . The Fermi -LAT spectral fit from Grondin et al. (2011) is also shown, together with spectra from the 3FGL and 3FHL catalogues ( Acero et al. 2015 ; Ajello et al. 2017 ), both of which were best fit by a log-parabola spectral model. A plausible IC emission model with multiple contributing radiation fields is shown, using the parameterisation of the Popescu et al. (2017) radiation model and the Naima modelling package ( Zabalza 2015 ). The emission is mostly dominated by the FIR field; the CMB becomes dominant at a few tens of TeV and below ~ 10 GeV. The total flux above 300 GeV was found to be (6.07 ± 0.13 stat ± 0.07 sys ) × 10 −11 cm −2 s −1 , corresponding to ~54% of the flux of the Crab nebula. Above 1 TeV, a flux of (1.12 ± 0.03 stat ± 0.25 sys ) × 10 −11 cm −2 s −1 was obtained, such that the proportion relative to the Crab nebula increases slightly to ~ 64% of the Crab nebula flux. The source remains strong at ~61% of the Crab nebula flux above 10 TeV. It is impressive that the γ -ray flux is so strong given that the distance to HESS J1825–137 is approximately twice that of the Crab nebula (at 2 kpc), despite (or perhaps because of) the significantly older age of the system. Interestingly, as part of the second HAWC catalogue, it was found that the confused source associated with the combined emission from the region, 2HWC J1825–134, is brighter than the Crab nebula at 7 TeV ( Abeysekara et al. 2017 ). This point is included in Fig. 10 and is compatible within the HAWC systematic


MJ-H is supported in this work by a Marsden Grant administered by the Royal Society of New Zealand. SS was supported by a Victoria Doctoral Scholarship. We would like to thank Dr Phil Edwards for allocating Director's time for the H214 ARDES observations. Additionally, SS deeply thanks Dr Robin Wark for her assistance with remote-observing, and Dr Jamie Stevens and Dr Mark Wieringa for their invaluable comments on CABB data reduction and imaging in MIRIAD . Finally, we thank Dr Siamak Dehghan for assistance with the generation of the multipanel figures. The ATCA is part of the Australia Telescope National Facility which is funded by the Australian Government for operation as a National Facility managed by CSIRO. The Digitized Sky Surveys were produced at the Space Telescope Science Institute under US Government grant NAG W-2166.


are the amplitude index (Belokurov et al. 2017 ) and the Gaia DR2 variability classification (Mowlavi et al. 2018 ), in the left and right panels, respectively. The local sample (without YSOs and MS/RGB stars) is also shown for comparison as a black unfilled histograms. We found that the amplitude index method seems to find variability in objects that were not classified as variables in the Gaia DR2. 5.2. Absolute Magnitudes for the Foreground Mira Population Several studies have been carried out to estimate IR absolute magnitudes of the AGB populations in the LMC, where the distance to the stellar system is known, and therefore, the distance modulus (and presumably also the IR extinction) can be assumed to be the same for each object (see e.g., Whitelock et al. 2008 ). From Figure 6 , we can determine an average absolute K magnitude of mag for the local sample. Although the magnitude values found roughly correspond to those found in the LMC (i.e., between −8.0 and −6.0 mag; see, e.g., Whitelock et al. 2008 ), one should keep in mind that we established a fainter limit of . Nevertheless, with the aim of analyzing the absolute magnitude distribution using the variability classification defined by Gaia (Mowlavi et al. 2018 ) and the amplitude index (Belokurov et al. 2017 ), we made Figure 9 , where the distribution of absolute magnitude for the local sample without considering the filtering of YSOs and MS/RGB (made in Section 4.3 ) is shown. Figure 9 shows that by including these younger sources, the distributions are much broader in terms of absolute magnitude range. This can be partly explained by the current sample being mostly based on a single 2MASS observation and including the effects of large amplitude variability. However, when we filter the sample by a variability qualification (as shown in both panels of Figure 9 ), the low-luminosity tail (in ) is cut out. In this sense, the Gaia classification as Mira or SR variable seems to narrow the distribution more. Following a similar discussion in Mowlavi et al. ( 2018 ), we argue that the low-luminosity tail in Figure 9 and also Figure 7 is due to contamination with YSOs and MS/RGB stars that can also peak in the IR but do not show the same variability (Lewis et al. 2020b ; Y. M. Pihlström et al. 2020, in preparation). 5.3. Bolometric Magnitudes for the Foreground Mira Population The bolometric luminosity is a fundamental property useful for classifying stellar populations and evolutionary stages (Srinivasan et al. 2009 ), as it measures the intrinsic stellar power. Although its definition is straightforwardly formulated as the total integrated power over all frequencies, in practice, complete photometric measurements that allow a direct bolometric luminosity estimate are hardly ever available. Therefore, under various assumptions, a limited set of photometric measurements, preferably near the peak of the SED, can be used to apply a bolometric correction (BC) in order to determine the


paper, as explained below. 3.1 Peebles–Hauser The Peebles–Hauser (PH) estimator is based on the correlation function proposed by Peebles Hauser ( 1974 ) and can be expressed as where DD (ρ) is defined as the pair of observed quasars counts within a separation radius ρ normalized by the the total number of pairs, N obs ( N obs − 1)/2 and RR (ρ) follows the same definition for the random catalogues. 3.2 Landy–Szalay The Landy–Szalay (LS) estimator is based on the correlation function proposed by Landy Szalay ( 1993 ). In addition to the quantities DD (ρ) and RR (ρ) defined above, DR (ρ) denotes the pair of quasar-random counts normalised by the available number of pairs. Hence, the LS estimator is given by For the PH and LS estimators, the quantity D 2 ( r ) is calculated by replacing equations ( 4 ) and ( 5 ) into equation ( 3 ), respectively. 4 THE HOMOGENEITY SCALE In order to estimate the homogeneity scale from the SDSS-IV DR14 quasars (henceforth ), we first obtain the values of D 2 ( r ) for each redshift slice using both estimators above. Then, a polynomial fitting is used to obtain the corresponding scale where the transition to homogeneity is identified (we refer the reader to Gonçalves et al. 2018 for more details). Following previous analyses (Scrimgeour et al. 2012 ; Laurent et al. 2016 ; Ntelis 2017 ), we define the homogeneity scale as the characteristic scale where the Universe can be considered homogeneous within 1 per cent of the expected fractal dimension for a homogeneous distribution, D 2 = 3 – thus, the scale where the spatial distribution of QSOs reaches D 2 = 2.97. Although arbitrary, the 1 per cent criterion is widely used in the literature, and is justified given some observational issues, such as the survey geometry and incompleteness, as well as the sample noise. Since we use 20 random catalogues to obtain D 2 , we also perform a bootstrap analysis over the 20 values of , quoting its mean and standard deviation as our measurement of the homogeneity scale and its corresponding uncertainty, respectively. 1 Our first measurements of are presented in the second column of Tables 2 and 3 . For the sake of example, Fig. 1 shows the results for both and D 2 obtained at the redshift using the PH estimator (similar results are also obtained for the LS estimator). The red points stand for the observational data with the respective errors. In the right-hand panel of Fig. 1 , the horizontal lines correspond to the D 2 = 2.97 and D 2 = 3 thresholds. From our analysis, we find that the PH estimator provides smaller error bars than the LS estimator. A possible explanation may be attributed to the fact that our approach does not use mock catalogues. Instead, we relied on an accurate bootstrap analysis of the 20 random catalogues we have produced. A full assessment of the homogeneity scale using mock catalogues that reflects the matter power spectrum, redshift distribution, QSO clustering, and angular selection function of the


We thank the referee for constructive and insightful comments. This paper makes use of ALMA data ADS/JAO.ALMA#2011.0.00294.S; #2012.0.00307.S; #2013.1.00118.S; #2012.1.00090.S; #2015.1.01528.S; #2016.1.00434.S; and #2017.1.01492.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST , which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. P.L., D.L., and E.S., acknowledge support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 694343). I.R.S. and A.M.S. acknowledge support from STFC (ST/P000541/1). F.B., B.M., E.V., K.H. and EFJA acknowledge support of the Collaborative Research Center 956, subproject A1, funded by the Deutsche Forschungsgemeinschaft (DFG). S.L. acknowledges funding from Deutsche Forschungsgemeinschaft (DFG) Grant BE 1837/13-1 r. E.A.C. acknowledges support from the ERC Advanced Investigator Grant DUSTYGAL (321334) and STFC (ST/P000541/1). M.J.M. acknowledges the support of the National Science Centre, Poland, through the POLONEZ grant 2015/19/P/ST9/04010; this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778. S.T. acknowledges support from the ERC Consolidator Grant funding scheme (project ConTExt, grant No. 648179). The Cosmic Dawn Center is funded by the Danish National Research Foundation. E.V. acknowledges funding from the DFG grant BE 1837/13-1. J.L.W. acknowledges support from an STFC Ernest Rutherford Fellowship (ST/P004784/2), and additional support from STFC (ST/P000541/1).


considering equation ( 4 ) and measured P values of ∼1–10 h for AMXPs (Salvo Sanna 2020 ). Figure 2. Numerically computed neutron star spin frequency ( ν ) evolution curves due to transient accretion (see Sections 3.1 and 3.2 ). The dotted grey horizontal line shows a constant long-term average accretion rate () of 5 × 10 15 g s −1 . The solid grey curve shows an evolving with a fast decay (see Section 3.1 ) in the RLDP on the right of the dotted vertical line. Various ν evolution curves are for g s −1 , and based on if EM spin-down and evolution are included not, as mentioned on the plot. This figure shows that ν can increase in the RLDP (see Section 3.2 ). 3.2 Spin evolution of transient sources First, we study how ν evolves if the neutron star accretes transiently throughout the LMXB phase (see Fig. 2 ). For an evolution duration of two billion years and for the examples given in this figure, ν remains much lower than ν eq,eff (not shown in the figure) for g s −1 . Note that an –10 16 g s −1 is common for many neutron star LMXBs (Lamb Yu 2005 ), although could be significantly higher in the initial stage of the LMXB phase (e.g. see Tauris 2018 ; Chen, Liu Wang 2020 ). The aim here is to find out how ν evolves in the RLDP. Fig. 2 shows that ν can increase in the RLDP, even if decreases drastically, and regardless of the inclusion of an additional spin-down (e.g. due to the EM torque (see Bhattacharyya Chakrabarty 2017 ) in the quiescent phase; see Section 1 ). This is not unexpected, as ν neither approaches nor tracks ν eq,per during a transient accretion (see Section 2 ). However, we note that ν either approaches or tracks ν eq,eff for such an accretion. Hence, ν either increases or decreases due to transient accretion, both in the RLDP and before the RLDP, depending on whether increases or decreases (see Section 2 and equation 5 ). Therefore, ν can decrease in the RLDP, if decreases. But, since is expected to depend on the binary orbital period P (with an estimated relation of , Lasota 2001 ), and the change of P is moderate (e.g. Bhattacharya van den Heuvel 1991 ), ν should not considerably decrease in the RLDP. 3.3 Spin evolution of persistent sources How does ν evolve for a persistent source? The neutron star first spins up relatively quickly towards the spin equilibrium frequency ( ν eq,per ), and then tracks it (see Introduction and Section 2 , Fig. 3 ). In the RLDP, if decreases rapidly, ν decreases for a persistent source, but cannot track ν eq,per anymore, which was shown by Tauris ( 2012 ), and is confirmed in Fig. 3 . But in reality, a persistent source becomes a transient for , and consequently, ν increases, as it does not track the lower spin equilibrium frequency ν eq,per anymore, and now approaches a higher effective spin equilibrium value ν eq,eff (see Fig. 3 , and also Section 2 ). Therefore, since even an initially persistent source should become a transient at least in the RLDP due to the decrease of , ν should increase in


directions, uncorrected for Faraday rotation. The FWHM dimensions and orientation of the elliptical Gaussian restoring beam are indicated by a cross in the lower left corner of the map. Figure 4.4421. Source 1954+513 at epoch 2011-Aug-26. Each panel contains two contour maps of the radio source, the first consisting of I contours in successive integer powers of two times the lowest contour level, with linear fractional polarization overlaid according to the color wedge. A single negative I contour equal to the base contour level is also plotted. The second map includes the lowest positive I contour from the first map, and linearly polarized intensity contours, also in increasing powers of two. The sticks indicate the electric polarization vector directions, uncorrected for Faraday rotation. The FWHM dimensions and orientation of the elliptical Gaussian restoring beam are indicated by a cross in the lower left corner of the map. Figure 4.4422. Source 1954+513 at epoch 2015-May-18. Each panel contains two contour maps of the radio source, the first consisting of I contours in successive integer powers of two times the lowest contour level, with linear fractional polarization overlaid according to the color wedge. A single negative I contour equal to the base contour level is also plotted. The second map includes the lowest positive I contour from the first map, and linearly polarized intensity contours, also in increasing powers of two. The sticks indicate the electric polarization vector directions, uncorrected for Faraday rotation. The FWHM dimensions and orientation of the elliptical Gaussian restoring beam are indicated by a cross in the lower left corner of the map. Figure 4.4423. Source 1954+513 at epoch 2015-Nov-25. Each panel contains two contour maps of the radio source, the first consisting of I contours in successive integer powers of two times the lowest contour level, with linear fractional polarization overlaid according to the color wedge. A single negative I contour equal to the base contour level is also plotted. The second map includes the lowest positive I contour from the first map, and linearly polarized intensity contours, also in increasing powers of two. The sticks indicate the electric polarization vector directions, uncorrected for Faraday rotation. The FWHM dimensions and orientation of the elliptical Gaussian restoring beam are indicated by a cross in the lower left corner of the map. Figure 4.4424. Source 1954+513 at epoch 2016-Mar-05. Each panel contains two contour maps of the radio source, the first consisting of I contours in successive integer powers of two times the lowest contour level, with linear fractional polarization overlaid according to the color wedge. A single negative I contour equal to the base contour level is also plotted. The second map includes the lowest positive I contour from the first map, and linearly polarized intensity contours, also in increasing powers of two. The sticks indicate the


ACKNOWLEDGEMENTS This study has benefited greatly from discussions with K. Doi. We also thank the reviewer for many useful comments on this paper. The present research used the computational resources of the HPCI system provided by Cyber Sciencecenter, Tohoku University; Cybermedia Center, Osaka University, Earth Simulator, JAMSTEC through the HPCI System Research Project (Project ID:hp160079, hp170047, hp180001, hp190035). This study was supported by JSPS KAKENHI Grant Numbers JP17K05387, JP17H02869, JP17H01101, and JP17H06360. Simulations reported in this paper were also performed by 2017 Koubo Kadai on Earth Simulator (NEC SX-ACE) at JAMSTEC. This work was partly achieved through the use of supercomputer system SX-ACE at the Cybermedia Center, Osaka University. Simulations were also performed by 2018 Koubo Kadai on Earth Simulator (NEC SX-ACE) at JAMSTEC.


The authors thank the staff at the Subaru Telescope for their excellent support in the observations. The authors thank Takaya Nozawa for his constructive comments on the manuscript. The work by K.M. is supported by the JSPS Grant-in-Aid for Scientific Research (No. 26800100) and by the WPI Initiative, MEXT, Japan. Support for H.K. is provided by the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. H.K. acknowledges support by CONICYT through FONDECYT grant 3140563. The work by M.T. is supported by JSPS Grant-in-Aid for Scientific Research (15H02075, 15H00788). The work by K.N. is supported by the JSPS Grant-in-aid for Scientific Research (No. 23224004 and 26400222).


Hazra et al. 2014 ; Passos et al. 2014 ). Indeed, LC17 observed that when their model enters into an extended grand minimum of the weaker magnetic field, the generation of poloidal field stops due to a lack of BMRs, and the model never recovers from that quiescent phase. Their model recovers only when the magnetic field does not fall below a certain level. 2. Model In this Letter, we explore the Babcock–Leighton dynamo mechanism during grand minima, focusing in particular on how the dynamo might recover from such episodes through the Babcock–Leighton process alone. To do so, we first produce grand minima. In our study, we build on our recent work, KM17 , which is an updated version of the original model (Miesch Dikpati 2014 ; Miesch Teweldebirhan 2016 ). In this model, BMRs are produced near the surface based on the toroidal flux at the BCZ and most of the statistical properties of BMRs are based on solar observations. We refer readers to Section 2 of KM17 for the details of this model. From KM17 , we consider Runs B10–11, in which the diffusivity in the bulk of the CZ is in the order of 10 12 cm 2 s −1 . The BMR flux distribution is fixed at the observed value and the time delay between two successive BMR emergences is obtained from the observed log-normal distribution, which changes in response to the toroidal field at the base of the CZ to allow more frequent BMRs when the toroidal field is strong and vice versa. The BMR tilt has a Gaussian scatter around Joy’s law with standard deviation ( σ δ ) of 30° (15°) for Run B11 (B10). Another key ingredient of the model is the downward magnetic pumping which has the same form as in KM17 and it is given bywhere γ CZ = 2 m s −1 and γ S = 20 m s −1 . Thus, the pumping has a value of γ S only near the surface, while in the rest of the CZ it is γ CZ . Pumping is a process in which the magnetic flux can be transported in a stratified convective medium due to the topological asymmetry in the convective flow. A variety of theoretical and numerical models suggest that it is operating in the solar CZ, particularly near the surface where both the velocity amplitude and the density stratification are largest (e.g., Drobyshevski Yuferev 1974 ; Krause Rädler 1980 ; Petrovay Szakaly 1993 ; Tobias et al. 1998 ). Although at present, we do not have a stringent constraint on the strength of the pumping, previous studies (Käpylä et al. 2006 ; Karak et al. 2014b ) suggest its value to be at least a few tenths of the convective velocity. Keeping in mind that the upper layer of the Sun is highly convective with the observed surface convection speed of a few km s −1 , a value of 20 m s −1 pumping speed is realistic (Spruit 1997 ; Nordlund et al. 2009 ). This value is also large enough to make the magnetic field near the surface approximately vertical, which improves the agreement between Babcock–Leighton dynamo models and Surface Flux Transport (SFT) models. 4 4 Cameron et al. ( 2012 ) showed that the inclusion of


layer is still unknown. It may be either an ice-rock layer between a rocky core and a water-rich outer mantle, or an outer rocky core composed of hydrous silicates (Lunine et al. 2010 ; Fortes 2012 ). If a mixed ice-rock layer indeed exists in the interior of Titan, then a newly discovered high-pressure solid solution in the H 2 O–CH 4 binary system called MH-III (also CH 4 filled-ice Ih) may hinder the outer transport of internal CH 4 into the atmosphere. The formation of MH-III was first reported by Loveday et al. ( 2001a ). Contrary to what was previously assumed for the H 2 O–CH 4 system, upon increasing the pressure above about 2 GPa, at room temperature, the classical structure H cage clathrate transforms into MH-III, rather than experiencing phase separation into water ice VII and solid CH 4 . In this way the solubility of H 2 O and CH 4 is kept throughout the entire pressure range inside Titan. This is also likely true for the much higher pressures spanning water ice mantles of water-rich exoplanets, given that MH-III was found to be stable up to about 100 GPa (Hirai et al. 2006 ), and temperatures higher than 1000 K (ichi Machida et al. 2006 ). We refer the reader to Loveday et al. ( 2001b ) for a detailed description of the crystallographic nature of MH-III. If CH 4 is indeed locked in grains of MH-III, scattered within a deep ice-rock layer above the rocky core of Titan, then how much of it may be transported outward depends on the buoyancy of these grains with respect to melt pockets and the solubility of CH 4 under such conditions. Such an analysis requires thermophysical data for MH-III at the appropriate conditions. In the context of Titan-like exoplanets, previous work on the transport of CH 4 across the interior pinpointed the necessary thermophysical data and the lack thereof (Levi et al. 2013 , 2014 ). Quantifying the needed thermophysical data is the prime object of this work. In Section 2 we probe the pressure regime in a possible undifferentiated ice-rock layer inside Titan, to better pinpoint our molecular simulations and the possible role of MH-III. In Section 3 we explain our computational methods. In Section 4 we derive the thermal equation of state (EOS) and thermal expansivity. Section 5 is dedicated to the heat capacity. In Section 6 we estimate the thickness of an MH-III enriched layer in Titan’s interior, and its carbon content capacity. In Section 7 we use a 1D thermal evolution model to asses the stability of such a layer. Section 8 is a discussion, and Section 9 is a summary. 2. The Pressure Regime in the Undifferentiated Ice-rock Layer In this section we estimate the pressure at the boundary between a possible internal ice-rock layer and Titan’s inner rocky core. We consider three basic constraints for Titan, its radius, mass and normalized MOI. We adopt a layered model for the internal structure (see Figure 1 ). The outermost condensed layer is a shell composed of ice Ih with a mass density of


We thank the referee for a thorough, probing report that has improved the manuscript. We also thank Holly Capelo, Alan W. Harris, and Anders Johansen for useful discussions. BTG and DV benefited by support by the European Union through ERC grant number 320964. SE would like to acknowledge the support of the NEOShield project funded by the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement no. 282703 (NEOShield) as well as Paris Observatory's ESTERS (Environnement Spatial de la Terre : Recherche Surveillance) travel grants.


however, is insufficient for the pressure gradient to change sign. The top panel of Figure 6 shows the mdot1e-7 case with parameters as in Table 2 , but N 2 = 3 and N 3 = 170. It shows that the active zone inside the rounded-off rim is in its final size. After the active zone has reached a certain size small temperature deviations start to occur and travel through it. These small deviations manifest in the depiction as walls where the dust is present. Since the simulation does not include latent heat it is not possible to decide whether these small deviations have to occur or not, but they were consistently found for every parameter set that produced a dust-free zone. The surface density in Figure 7 mirrors the cavity that is also present in Figure 6 . The dust density falls to g cm −2 in the dust-free zone. To further analyze the equilibrium in the dust-free zone an alternative sublimation formula is in progress. Figure 7. Comparison of gas (solid line) and dust (dotted line) surface density for model mdot1e-7 . The optical depth at constant radiusfor near-infrared radiation in Figure 8 suggests that the dust-free zone forms after a large enough depth in the θ direction is reached because diffusion of radiation energy is no longer sufficient to cool the disk. Figure 8. Vertical optical depth in NIR for model mdot1e-7 . 5. Discussion This work represents an early stage in developing a self-consistent model of the inner rim of protoplanetary disks. In this section some important limitations of the model will be discussed. The simplifications introduced in Flock et al. ( 2016 ), such as a uniform dust-to-gas ratio in the shadowed region and a non-frequency-dependent gas opacity are also applied in this work. A more elaborate sublimation formula is needed in the future, especially with respect to the dust-free cavity inside the disk. A detailed implementation of gas line radiation transfer would go beyond the scope of this work. Further, the value of α was chosen to be constant throughout the box. However, the strength of the turbulence is expected to be influenced by the temperature and could be lower in MRI inactive regions (Lesur et al. 2014 ; Turner et al. 2014 ; Simon et al. 2015 ). A temperature-dependent α will be explored in further work. The temperature deviations and dust walls described at the end of Section 4.3 will have to be further scrutinized in combination with the modeling of the dust-to-gas ratio. Small differences in the temperature can lead to significant changes in the dust-to-gas ratio around the sublimation temperature. This is therefore a delicate problem that needs to be explored tentatively and might necessitate the inclusion of further effects like dust diffusion, latent heat, and temperature-dependent viscosity. 6. Conclusions and Outlook This paper presents a 2D model for protoplanetary disks that consistently includes viscous heating and thermal conduction. The model expands previous works (Flock et al. 2016 )


compared to the unambiguously identified [Si ii ] 34.81 and [S iii ] 33.48 μ m makes the [O iv ] identification much more plausible. If [Fe II] 25.98 μ m is not detected, the putative identifications of the intrinsically weaker [Fe II] 35.35 and [Fe II] 17.94 μ m (Smith et al. 2009 ) in the PR equilibrium ejecta are also doubtful. Isensee et al. ( 2012 ) clearly saw the 17.94 μ m line, and we interpret this as coming from the reverse shock. Gerardy Fesen ( 2001 ) saw even higher excitation [Fe II] lines with wavelengths in the range ∼1–2 μ m that are clearly from fast-moving knots or quasi-stationary flocculi and therefore must result from a shock–clump interaction. Koo et al. ( 2016 ) and Lee et al. ( 2017 ) presented mid- and near-IR observations of [Fe II] from ejecta and CSM knots in Cas A. Koo et al. ( 2016 ) demonstrated that in radiative shock models of this emission, most of the [Fe II] radiation comes from the postshock cooling zone, similar to our Figure 5 . Lee et al. ( 2017 ) found ejecta knots with abundances commensurate with O burning, with elements like Si and S present, as well as a sample of ejecta knots showing strong [Fe II] lines but no Si or S. They interpreted these knots as products of complete Si burning leaving almost “pure” Fe, as found in X-rays by Hwang Laming ( 2009 , 2012 ), or in some cases the products of α -rich freeze-out. It is unclear why such ejecta should be in knots, as opposed to diffuse clouds inflated by radioactivity, but compared to SN 1987A, Cas A was a smaller progenitor at explosion, with a presumably faster expansion and lower opacities leading to a reduced effect of this sort. Koo et al. ( 2018 ) estimated a mass of [Fe II]-emitting shocked dense ejecta of ∼3 × 10 −5 M ⊙ . Extrapolating this clumping to the whole inner ejecta is difficult to do with any accuracy, but assuming that the Fe clump density is the same throughout the inner ejecta as it is at the reverse shock, the total Fe mass should be 3 × 10 −5 M ⊙ multiplied by the ratio of the volumes represented by the inner ejecta (6 × 10 56 cm 3 ) and the reverse shock emission. This last piece is ambiguous, but taking it to be cm 3 , where R r = 1.7 pc = 5 × 10 18 cm is the reverse shock radius and the knot diameter 2 r knot ∼ 2.5 × 10 17 cm gives a ratio of volumes of 7.5 and an implied Fe mass in knots of 2 × 10 −4 M ⊙ . This could be uncertain by at least an order of magnitude either way but is likely to be smaller than the Fe mass in the diffuse Fe–Co–Ni bubbles, estimated below. Features coinciding with [Fe V] 20.85 and 36.34 μ m, which would imply Fe masses of order ∼10 −3 M ⊙ , are also visible in the inner ejecta region. These only appear at marginal significance and must wait for better data for confirmation. The [O iv ] 25.89 and [Si ii ] 34.81 μ m morphologies (Smith et al. 2009 ) and the mass estimate of Arias et al. ( 2018 ) suggest that most of the inner ejecta mass expected by Hwang Laming ( 2012 ) are clumped into a volume