O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O B-Wavelength I-Wavelength O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-Organization O O O O O O O O O O O O O O O O O O O B-Organization O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Wavelength I-Wavelength O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Wavelength I-Wavelength O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Wavelength I-Wavelength

fit all transtions with K ′′ a up to 20 (excluding lines of K ′′ a = 18, which showed a clear trend in residuals between 197.48 and 221.45 GHz) for frequencies below 251 GHz. But in the range 310–506 GHz, the assignments appeared difficult in spite of larger intensities. As for other vibrational states of g - n -PrCN, the oblate pairing consisting of a -type and b -type transitions facilitated the assignments. However, the increasing separation of paired transitions with increase in quantum numbers made assignments more difficult combined with the blending and broadening in the high frequency range and the possible splits caused by internal rotations. Therefore very secure assignments with K ′′ a up to 4 or 5 (for each sub-branch) were first made. Even after the improvement of the parameters several lines at each K ′′ a could not be correctly fitted showing systematic residuals. All these lines are in the ranges of J ′′ from 68 to 73 and from 89 to 93. We attributed this as due coupling with other vibrational states. Finally 935 R -branch a -type transitions were identified in all four frequency ranges. The assignment of b -type transitions was not easy due to low intensities at low frequencies and blending and broadening at high frequencies. The confidently assigned b -type transitions at high frequency were all oblate paired with intense a -type transitions with K ′′ a less than 6. Totally, 487 b -type transitions were fitted. Q -branch b -type transitions were useful to correctly determine the parameter D K . We identified internal rotation causing splits (with the components separated by around 0.6 MHz) at frequencies between 89.25–126.75 GHz, in transitions with large J ′′. In this range, Q -branch b -type transitions have K ′′ a less than 11 and those that could be securely assigned have J ′′ less than 45. For Q -branch b -type transitions with K ′′ a between 13 and 18 in the range 171–251 GHz, only prolate pairs were assigned. These transitions have low J ′′ and hence are not affected by internal rotation. Finally 466 Q -branch b -type transitions were correctly fitted. Moreover, 110 hyperfine split transitions were fitted with fixed hyperfine parameters from v 30 = 1. These are R -branch a -type transitions with J ′′ approaching K ′′ a (less than 11), b -type transitions (both R -branch and Q -branch) with K ′′ a less than 4 at frequencies below 70 GHz. The derived parameters are shown in Table 6 . The fitted transitions are in Table 18 (available at the CDS), and those confidently assigned but not correctly fitted with systematic residuals are separated in Table 19 (available at the CDS). Using the newly derived molecular parameters, confident predictions of a -type transitions can be made up to J ′′ = 68 for K ′′ a ≤ 5 below 380 GHz, and J ′′ = 41 for K ′′ a ≤ 41 below 251 GHz (excluding lines of K ′′ a = 18 that show residuals that cannot be fitted). For b -type transitions, all R -branch predictions with K ′′ a ≤ 11 below 251 GHz


O O O O O O O O O O O B-Person I-Person O B-Person I-Person B-Instrument B-Observatory O O O O O O B-Person I-Person O B-Person I-Person O O O O O O O O O O O O O O O O O O O O O B-Person I-Person O O O O O O O O O O O B-Person I-Person O O O O O O O O O O O O O O O O O O O O O O O O B-Person O B-Person O O O O B-Grant O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Person O O O O O B-Collaboration I-Collaboration O O B-Grant I-Grant B-Grant I-Grant I-Grant O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O B-Organization O O O B-Grant B-Person O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O O B-Observatory O O O B-Observatory I-Observatory I-Observatory I-Observatory O O O B-Proposal

This work would have been impossible without the vigorous work of Fernando Selman and George Hau, MUSE Paranal instrument scientists. We are grateful to Francesca Iannuzzi and Francesca Fragkoudi for very useful input and discussions on bars in simulated galaxies, particularly on box/peanuts. It is also a pleasure to thank Eric Emsellem for useful discussions on the inner components of barred galaxies and Michael Vlasov for his permission to use a chart from his atlas. We are very grateful to the anonymous referee for a thoughtful and constructive report. M.K.S. and J.F.-B. acknowledge support from grant AYA2013-48226-C3-1-P from the Spanish Ministry of Economy and Competitiveness (MINECO). J.F.-B. also acknowledges support from the DAGAL network from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007–2013/ under REA grant agreement number PITN-GA-2011-289313. P.S.-B. acknowledges support from the Ramón y Cajal programme, grant ATA2010-21322-C03-02 from the Spanish Ministry of Economy and Competitiveness (MINECO). Based on observations made with ESO telescopes at the La Silla Paranal Observatory under programme ID 060.A-9313.


B-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O B-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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


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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.


O O O O O O O O O O O B-Person O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Person I-Person I-Person O B-Organization I-Organization B-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Fellowship O O O O O B-Person O O O O B-Organization I-Organization I-Organization I-Organization O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O O O B-Person I-Person O O B-Model I-Model O O O O O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O B-Organization I-Organization O B-Organization O O B-Grant I-Grant O O O O O O O B-Organization O O B-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization O B-Organization O B-Grant I-Grant O O O O O O O O O O O O B-Archive I-Archive I-Archive I-Archive I-Archive B-Archive O B-Telescope B-Instrument O B-Instrument O

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) .


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

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


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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


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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’.


O O O O O O O O O O O O O O B-Person O O O O O B-Organization I-Organization I-Organization O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person O O O O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O B-Grant O B-Telescope I-Telescope I-Telescope O O O O O O B-Location B-Location B-Location O B-Location I-Location B-Wavelength O O O O O O O O O O O O O O O B-Telescope O O B-Identifier O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization O B-Observatory I-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization O B-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization B-Organization O O B-Organization I-Organization O O O O B-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization O O O O O B-Grant I-Grant I-Grant I-Grant I-Grant O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Grant I-Grant I-Grant O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope O O O O B-Archive I-Archive I-Archive O O O O O O B-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Organization I-Organization I-Organization I-Organization B-Organization O O O O O O B-Archive I-Archive I-Archive I-Archive I-Archive I-Archive O O O O O O O B-Organization I-Organization I-Organization B-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O B-Software O O O B-Software O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Software O O O O O B-Software O B-Citation I-Citation I-Citation O

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 ).


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

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


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

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.


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

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


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

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


O O O O O O O O B-Wavelength O O B-Tag O O O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Wavelength O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Wavelength O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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


O O B-Person I-Person O B-Person I-Person O O O O O O O O O O B-Wavelength O O B-Telescope I-Telescope I-Telescope O O O O O B-Organization I-Organization I-Organization O O B-Grant O O O O O O O B-Organization B-Grant I-Grant I-Grant I-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization O B-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant I-Grant O O B-Organization I-Organization I-Organization O B-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization B-Person O O O O B-Organization I-Organization I-Organization B-Fellowship I-Fellowship B-Fellowship B-Person O O O O B-Organization I-Organization I-Organization I-Organization O O O B-Grant O O B-Location I-Location B-Organization I-Organization I-Organization O O O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope O O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O O O O B-Location B-Organization I-Organization I-Organization O B-Grant O O O O O B-ComputingFacility I-ComputingFacility O B-Organization I-Organization O O B-ComputingFacility I-ComputingFacility O B-ComputingFacility I-ComputingFacility O O O O O O O O O B-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O O O O B-Organization I-Organization I-Organization O B-Grant O B-Grant O O O O B-Location B-ComputingFacility I-ComputingFacility O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O O O O O O O B-ComputingFacility I-ComputingFacility O O O O O O O B-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Organization I-Organization I-Organization O O O O O O O O O O O O B-Software O O O O B-Software O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Software O O O B-URL B-Software B-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation I-Citation O B-Software B-Citation I-Citation O B-Software B-Citation I-Citation O B-Software O B-Software B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation O O B-Software O O O

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).


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

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


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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.


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

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


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

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


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

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


O B-Person O O O O B-Organization O O O O B-Person I-Person O O O O B-Database O O O O O O O B-Person I-Person O O O O O O O O B-Person O O O O O B-Person I-Person O O O O O O O O O O O O O O B-Organization I-Organization I-Organization B-Organization O B-Telescope O B-URL O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-URL O O O O B-Organization O O O O O O O O O O O O O B-Collaboration I-Collaboration I-Collaboration


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

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.


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

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


O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O O B-Citation I-Citation I-Citation O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Survey O O O O O O O O B-Survey O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObject O B-CelestialRegion I-CelestialRegion I-CelestialRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O

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


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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 (


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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


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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).