O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Wavelength O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O B-Survey O B-Survey O O O O B-Wavelength 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 B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

showing the distribution of distance values are shown in Fig. 12 . Figure 12. Distance distribution based on methods 1 and 2 described in Section 4.4 . We further analysed the differences in physical parameters computed using distances from methods 1 and 2. The left-hand panel of Fig. 13 shows a comparison of the mass distributions from method 1 (green hatched histogram) and method 2 (solid red histogram). As expected from the differences in the distance distribution, the mass distributions from the two methods are slightly different. However, the overall statistics are similar – while method 1 gives a mean and median mass of 1570 and 930 M ⊙ , the respective values obtained by distances from method 2 are 1785 and 925 M ⊙ . A t -test between the two mass distributions gives a t -value of 0.71 with a significance value of 0.48. Thus, there is no significant difference in the distribution of masses computed using distances from the two methods. Figure 13. Mass distribution using two methods (left-hand panel) and (b) MR plot obtained using the distances obtained with the approach presented by Reid et al. ( 2016 ) (right-hand panel).The dashed green line represents the power-law fit to the data. The upper and lower solid red line shows the surface densities of 1 and 0.05 g cm −2 , respectively. The right-hand panel of Fig. 13 shows the mass-radius relation computed using distances from method 2. With method 2, 287 out of 320 sources satisfy the Kauffmann criterion for potential to form massive stars. This is comparable to method 1 wherein 295 out of 320 sources satisfy the Kauffmann criterion. Thus, although the distances from method 1 are different from that of method 2, the overall statistical properties of sources associated with 6.7 GHz methanol masers are similar between the two methods. We are thus unable to make any distinction regarding accuracy of distances when considering the entire sample statistically. 5 CONCLUSION We have constructed SEDs from 870 to 70 μm for 320 6.7 GHz methanol masers using data from the ATLASGAL and Hi-GAL surveys. The SEDs from 870 to 160 μm were fitted with single-component grey body models. We observe a mean dust temperature of 22 K confirming the later evolutionary stage of the maser sources in comparison to infrared dark clouds, with some sources showing temperatures as high as 48 K. Almost 92 per cent of the methanol maser sources satisfy the Kauffmann criterion for potential to form massive stars. A comparison of the mass–luminosity diagram of the sample with simple evolutionary tracks from the turbulent core model suggest that most methanol masers are associated with massive young stellar objects with over 90 per cent in early evolutionary stages of accelerating accretion. However, there also appears to be a small population of sources that are likely to be associated with intermediate- or low-mass stars suggesting that the association between high-mass star formation and methanol maser emission is not


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ACKNOWLEDGEMENTS We thank Peder Norberg and Martin White for helpful discussions. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie grant agreement No 734374. JA and NP were supported by proyecto financiamiento BASAL CATA PFB-06 and Fondecyt 1150300. BL is supported by the European Research Council (ERC-StG-716532-PUNCA), and UK STFC Consolidated Grants ST/P000541/1, ST/L00075X/1. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility ( www.dirac.ac.uk ). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. YC and JAP are supported by the European Research Council, under grant no. 670193 (the COSFORM project).


O O O O B-Person I-Person O O O O O O O O O B-Person O O O O O O O B-Fellowship I-Fellowship O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O O O B-Fellowship I-Fellowship I-Fellowship B-Person O O O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization B-Grant O O O O O O O O B-Observatory O O O B-Observatory I-Observatory O B-Observatory I-Observatory O B-Observatory O O B-Proposal O O O O O B-Instrument O O O O B-Telescope O B-Organization O O B-Telescope I-Telescope B-Telescope O O O O O B-Organization I-Organization I-Organization B-Organization O B-Location O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O B-Location O O B-Organization I-Organization I-Organization O O O O O O O O O O O B-Software O O O B-Organization I-Organization I-Organization I-Organization O O O O B-Survey I-Survey I-Survey I-Survey O O O O B-Organization O B-Organization

Acknowledgements The authors thank Will Hartley for useful comments that have greatly improved this paper. PWH wishes to acknowledge support provided through an STFC studentship, and the Rector and Fellows of Lincoln College for support through the Graduate Research Fund. MJJ acknowledges support from the UK Science and Technology Facilities Council (ST/N000919/1). Based on data products from observations made with ESO Telescopes at the La Silla or Paranal Observatories under ESO programme ID 179.A-2006. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the CanadaFranceHawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de lUnivers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at Terapix available at the Canadian Astronomy Data Centre as part of the CanadaFranceHawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.


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

wavelengths, we can see that there appear simultaneous EUV flashes and enhanced 10830 Å absorption. Actually, the enhanced absorption at 21:31:00 UT has already appeared in Figure 4 . The EUV flashes over the sunspot certainly have a spatiotemporal association with the two valleys on the 10830 Å light curve, and at the same time, they are associated with the illumination of the two micro-flares. Though we need to carry out further detailed quantitative analysis with existing models, the nice correlation will serve as direct observational evidence supporting that the PRM mechanism really works. Here it is worth mentioning that Judge Pietarila ( 2004 ) reported no correlations between helium and He ii emission and overlying coronal emission. Their conclusions were made by analyzing quiet regions and coronal holes. Considering in the paper that the two micro-flares and the excitation site are over 5 Mm away, the excitation of helium atoms by EUV illumination should be a mean effect from overlying coronal emission as well as at least nearby coronal emission. Figure 6. Top and the third rows (panels S1–S8): snapshots for the micro-flare at ∼21:20:30 UT taken on the ground by NST and in space by AIA. AIA images are overlaid with contours of sunspots. The second and fourth rows (panels L1–L8): light curves for 10830 Å and EUV brightness from 21:00 UT to 21:40 UT obtained from the boxed areas shown in the upper corresponding images, with red curves from red boxes (over the sunspot) and blue curves from blue ones (over the sites of the two micro-flares). Two vertical dotted lines indicate the peak times of the two micro-flares. As mentioned in previous sections, the most notable phenomenon is that an oscillating absorption appears inside one flare ribbon when it sweeps into the sunspot. An overall description of this phenomenon is that a dark (absorption) strip rushes out into the sweeping forth sunspot ribbon and then returns back until the next several dark strips repeat the same behavior (Figure 1 , also please watch the online movie). From the space–time diagram, the oscillating behavior is revealed as a sawtooth-like pattern, the oscillating absorption strip actually starts when the ribbon reaches the inner penumbral region. For umbral flashes and even penumbral waves, the sawtooth behavior can be observed from spectrograms taken in certain chromospheric lines like Ca ii H K lines (Beckers Tallant 1969 ). It was also called a Z pattern with fast blueshift and slower redshift excursions, which were attributed to shock waves and post-shock falling back of the material contained in these shock waves. In this event, the observations were made in the He i 10830 Å blue wing −0.25 Å, which certainly highlights the blushift component. However, the 0.5 Å bandpass of the Lyot filter is comparable to the He i 10830 Å line width (∼0.75 Å) and the line is very weak, meaning slower redshift excursions will still be visible. So the features of blueshift and


B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-CelestialObject 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-Citation I-Citation I-Citation I-Citation 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 B-CelestialObject 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 B-CelestialObjectRegion 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-CelestialObject O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O B-CelestialObject 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O B-TextGarbage I-TextGarbage 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-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Citation I-Citation O O O O O O O O O B-CelestialObject O O O B-CelestialObjectRegion O O B-Citation I-Citation I-Citation I-Citation O O O O O O

Samsonov et al. (2020) that r 0 would be predicted more accurately by taking the effects of ρ SW and V SW individually into account rather than considering P d alone. However, Samsonov et al. (2020) also indicated that the changes in r 0 for a velocity change were greater than those for a density change for the same P d , especially for a southward IMF, which implied that the power-law index of V SW , M , was lower than that of ρ SW , N . We found a different result, however: A higher solar wind number density (or a lower velocity) results in a smaller r 0 of the Martian MPB for thesame P d ( M > N ). Moreover, these two results are also inconsistent with the results of Ramstad et al. (2017) , who reported that the Martian IMB was mainly pressure-dependent and that changing ρ SW or V SW individually had no significant effect for the equal dynamic pressure. The mechanism for the different effects of ρ SW and V SW on r 0 for the Earth can be explained by the solar wind electric field. It is generally greater for a higher solar wind velocity, which leads to the enhanced magnetopause reconnection rate and Region 1 current, especially for a southward IMF (e.g., Lee Lee 2020 ; Samsonov et al. 2020 ). As a result, the greater changes in r 0 occur for a velocity increase and not for a density increase for the same P d . For Mars, which is a weakly magnetized planet, the different results of ρ SW and V SW effects cannot be explained by the traditional concept of the solar wind electric field and the reconnection conditions at the Earth. On the other hand, the location of the magnetopause can also be affected and described by the pressure balance across it, as we mentioned above. To investigate the reason for the opposite effects of ρ SW and V SW at Mars as compared to Earth, we next studied the pressure balance throughout the MPB in more detail. Following Schield (1969) and Suvorova et al. (2010) , we introduced a factor k and f that describe the contributions of the different components, both thermal and magnetic, of pressure to the total pressure. We define the factor k as the ratio of the total pressure just outside the Martian MPB to the upstream solar wind dynamic pressure as follows: . Here , , and are the plasma thermal pressure, the magnetic pressure, and the solar wind dynamic pressure just outside the subsolar standoff point of the Martian MPB, respectively. is the solar wind dynamic pressure upstream of the bow shock. As the plasma thermal pressure and the magnetic pressure upstream of the bow shock are very low and can be neglected, the factor k can be used to describe how much the upstream solar wind dynamic pressure is converted into thermal and magnetic pressure just outside the MPB. The theoretical value of k in a gas-dynamic approximation assuming a normal shock is 0.881 ( Schield 1969 ). In addition, k also varies with the upstream solar wind and downstream magnetopause conditions (e.g., Suvorova et al. 2010 ). For a higher value of


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The authors thank the referee for valuable feedback. The authors thank Inga Kamp for sharing her ProDiMo model, Christian Rab for sharing prodimopy , Michiel Min for sharing OpacityTool , and Til Birnstiel for sharing twopoppy . We thank the above-mentioned and Rens Waters and Kaustubh Hakim for helpful discussions and Sebastiaan Krijt for his thoughtful comments on the manuscript. L.K. is supported by a grant from NOVA. C.W.O. acknowledges funding by the Netherlands Organization for Scientific Research (NWO; VIDI project 639.042.422). C.D. acknowledges funding by the NWO, project number 614.001.552.


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ACKNOWLEDGEMENTS We are grateful to an anonymous referee for valuable suggestions that have helped a lot to clarify our presentation. The study was supported by the Russian Foundation for Basic Research grant 17-02-00644. DS acknowledges support by the Deutsche Forschungsgemeinschaft through SPP 1833: "Building a Habitable Earth'' (SE 1962/6-1).


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Acknowledgements Support from the Centre National d'Etudes Spatiales (CNES, France) for this work, based on observations with CONSERT on board Rosetta , is acknowledged. The CONSERT instrument was designed built and operated by IPAG, LATMOS, and MPS and was financially supported by CNES, CNRS, UGA, DLR, and MPS. Rosetta is an ESA mission with contributions from its Member States and NASA. Rosetta 's Philae lander is provided by a consortium led by DLR, MPS, CNES, and ASI. The authors thank the teams of Rosetta (SGS and ESOC) and Philae (LCC and SONC) for making the CONSERT operations possible.


B-Person O B-Person O O O B-Organization O B-Organization O B-Grant O B-Grant O B-Person O O O O O B-Observatory B-Person O O O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization B-Location O O O O B-Organization I-Organization I-Organization O B-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O B-Grant O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Person O O O B-Fellowship I-Fellowship I-Fellowship O O B-Person I-Person O O B-Telescope O O O O O B-Person I-Person O O B-Observatory O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-Person I-Person O O B-Observatory O O O B-CelestialObject I-CelestialObject O O O O O O O O B-Person I-Person O O O O O O O O O O O O B-Observatory O B-Identifier B-Observatory O O O O B-Observatory O O O O B-Organization B-Location O B-Organization B-Location O O B-Organization B-Location B-Organization O B-Organization B-Location O B-Organization B-Location I-Location I-Location O O O O O O O B-Observatory I-Observatory I-Observatory I-Observatory O O O B-Observatory B-Organization O B-Observatory 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

M.C.A. and J.P.W, were supported by NSF and NASA grants AST-1208911 and NNX15AC92G, respectively. M.C.A. acknowledges student observing support from NRAO. N.M. is supported in part by the Beatrice W. Parrent Fellowship in Astronomy at the University of Hawaii. Leiden is supported by the European Union A-ERC grant 291141 CHEMPLAN, by the Netherlands Research School for Astronomy (NOVA), and by grant 614.001.352 from the Netherlands Organization for Scientific Research (NWO). C.F.M. gratefully acknowledges an ESA Research Fellowship. We thank J.M. Alcalá for providing VLT/X-Shooter spectra for deriving stellar parameters, Ilse Cleeves for providing ALMA continuum and integrated line fluxes for Sz 82 (IM Lup), and Scott Barenfeld for providing ALMA continuum fluxes for Upper Sco sources in advance of publication. We also thank Sean Andrews for his extremely useful comments. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00220.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC 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. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.


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

and 3C 273 we get signs of local nonstationarity that are not consistent among the various tests, for Mrk 421 we get multiple clues that may indicate local nonstationarity synchronous with large changes in its average flux. While we do not find any significant change in the PSD between multiple epochs for the same blazar we must note that a small temporal change in the PSD of red-noise variability may not always be detectable with the quality of data that is available. There may be an excess scatter in the binned PSD estimates obtained from light curves spanning hours to years leading to poor fits of a model PSD and subsequently higher rejection probabilities. Sampling window and Poisson noise give rise to distortions in the PSD, which are difficult to quantify, and cause uncertainties in the estimates of PSD slopes, which can eclipse any change in the slope of the PSD of the underlying process. In alternative methods of PSD modeling, such as CARMA, the difficulties in discerning a local nonstationary process persist. Kelly et al. ( 2014 ) showed, using an irregularly sampled simulated light curve stitched together from two separate stationary CARMA processes with differing variances, that a CARMA (5,2) model provided a good fit, as if the complete light curve was a single stationary process. It is as if the PSD of the combined process is a blend of the separate processes weighted according to their respective S/Ns. The local nonstationarity in the PSD of the X-ray variability of Seyfert galaxies such as IRAS 13224−3809 has been explained as being due to having an accretion rate close to the Eddington limit, or deformations in a time-evolving corona (Wilkins Gallo 2015 ; Alston et al. 2020 ). In the blazars the X-ray emission is mostly from the jet, thus if it exhibits local nonstationarity that will imply some important change in the emission process inside the jet. Alternatively, the jet variability may be driven by fluctuations in the accretion disk and in that case the nonstationarity may be due to specific kinds of changes in the disk. We find a linear flux–rms relation in the X-ray light curves of Mrk 421 separated by a period as long as two decades and over timescales of days to months. It is one of the few cases in which a flux–rms relation has been detected in a blazar across such long timescales. This, along with similar demonstrations in other blazars and other wave bands (Edelson et al. 2013 ; Kushwaha et al. 2017 ), implies that the flux–rms relation may be a ubiquitous feature of blazar variability, similar to other accreting black hole systems, e.g., BHXRBs. There is a significant difference in the best-fit slope of the flux–rms relation among the short-term and long-term observations of Mrk 421 across two decades. This may be the result of some systematic effect like error subtraction or it may be due to local nonstationarity in the source itself. For BHXRBs the slope and the flux intercept are known to depend on the


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The authors thank the anonymous referee for useful suggestions. W.P.L. acknowledges support from the National Key Program for Science and Technology Research and Development (2017YFB0203300). W.P.L. and X.K. acknowledge support by the NSFC projects (No. 11473053, 11233005, U1331201, 11333008) and the National Key Basic Research Program of China (No. 2015CB857001, 2013CB834900). X.K. is also supported by the foundation for Distinguished Young Scholars of Jiangsu Province (No. BK20140050). The simulations were run in the Shanghai Supercomputer Center and the data analysis was performed on the supercomputing platform of Shanghai Astronomical Observatory. W.C. is supported by the Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional (MINECO/FEDER, UE) in Spain through grant AYA2015-63810-P, as well as the Consolider-Ingenio 2010 Programme of the Spanish Ministerio de Ciencia e Innovación (MICINN) under grant MultiDark CSD2009-00064.


O O B-Person I-Person I-Person O O O O O O O O O O O O O B-Person O O O O O B-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Person I-Person O O O O B-Organization B-Grant O B-Location B-Person O O O O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O B-Grant I-Grant B-Person O O O O O B-Location B-Organization O B-Grant B-Person O O O O B-Person I-Person O B-Person I-Person B-Person O O O O B-Grant I-Grant O B-Location O O O O O O B-Observatory O B-Identifier I-Identifier B-Observatory O O O O B-Observatory O O O O B-Organization B-Location O B-Organization B-Location O O B-Organization B-Location B-Organization O B-Organization B-Location O B-Organization B-Location I-Location I-Location O O O O B-Location I-Location I-Location O O B-Observatory I-Observatory O O O B-Observatory B-Organization O B-Observatory O O O O O O O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O B-Observatory I-Observatory I-Observatory O B-Location O B-Observatory O

We thank Matthew S. Povich for his valuable constructive comments that have improved the presentation of the paper. S.O. is financially supported by a Research Fellowship from the Japan Society for the Promotion of Science for Young Scientists. H.V. C. acknowledges financial support from MOST 105-2119-M-007-022-MY3 grant, Taiwan. G.B. acknowledges the support of the Spanish Ministerio de Economia y Competitividad (MINECO) under the grant FPDI- 2013-18204. G.B. is also supported by the Spanish MINECO grant AYA2014-57369-C3-1-P. A.P. appreciates insightful discussions with Enrique Vázquez-Semadeni and Javier Ballesteros-Paredes. A.P. acknowledges financial support from UNAM-DGAPA-PAPIIT IA102815 grant, México. This paper makes use of the ALMA data: ADS/JAO.ALMA 2013.1.00312.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC 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. Data analysis was carried out on a common use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan. Facility: No:ALMA .


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

definition of the periphery changes with different norms. Therefore, it is possible to overcome this limitation by adopting a different norm. Consistent with Mitschang et al. ( 2013 ), we use the Manhattan distance, or L 1 norm, as the metric for implementing chemical tagging with DBSCAN. Figure 3 illustrates the two-dimensional projections of the chemical space for our sample and the recovered members according to the DBSCAN algorithm with = 0.4 and K = 6. We found that the recovery rate quickly diminishes when we deviate from these values. As compared to chemical tagging performed with K -means, DBSCAN recovers M13 at a similar rate (23.9%) and 18.4% of M5 members. However, the M13 members are recovered in four DBSCAN groups, compared to just one group for K -means. 3.3. Chemodynamical Clustering Previous sections showed that the process of identifying groups in chemical space can retrieve at most 30% of true members in known globular clusters. This result holds independently of the algorithm adopted to identify clusters in the chemical space. This is likely due to insufficient precisions of chemical abundances. We found that radial velocities of members from distinct globular clusters are almost indistinguishable in several cases. Thus, radial velocity alone is likely not sufficient to recover individual globular clusters either. As the next step, we add the kinematic information, i.e., radial velocity, to the chemical space and propose a new algorithm, SNN. We used AstroPy (Astropy Collaboration et al. 2013 ) and various SciPy packages (Jones et al. 2001 ), including Numpy (Walt et al. 2011 ) and Scikit-learn (Pedregosa et al. 2011 ), to build our algorithm. We applied this algorithm to our sample and identified star clusters in the chemistry–velocity space. All clustering techniques rely on assigning stars based on some distance metric. One problem with picking a distance metric is normalization when two spaces with different ranges are considered (e.g., the chemical space and radial velocity space). One of the main advantages of our algorithm is that it identifies clusters in the chemistry – velocity space by taking the intersection of the nearest neighbors in the chemical space and radial velocity space. This feature avoids normalization between chemical abundances and radial velocity and guarantees that the final nearest neighbors are neighbors in both the chemical and velocity space. The algorithm operates as follows. As suggested in Mitschang et al. ( 2013 ), for each set of elements , the distance in the chemical space between two stars, A and B , is computed as the Manhattan distance, defined below:where X i = [Fe/H], [K/Fe], [S/Fe], [Si/Fe], [Ca/Fe], [Mn/Fe]... The distance between the radial velocities of two stars is a simple Euclidean distance: For every star, a list of nearest neighbors in the chemical space and a list of nearest neighbors in the radial velocity space are respectively found. The neighbors for a particular


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

the zero-order solution: The last system of equations can be rewritten in the following form: Then, the solution of the last system has the form To obtain the first and next amendments, we use a form such as N i = C η −α Φ i ( x ), and when we insert N 0 into ( 11 ), we will obtain an equation for N 1 : If we expand the exponent in the last equation in a Taylor series, we obtain Here, the first condition ( 12 ) was applied, i.e. the integration was carried out from 0 to x . We will seek a solution of the last equation in the following form: After an expansion of the exponent in a series followed by integration, we obtain The last condition ( 12 ) for finding B was used, i.e. Φ 1 ( x 0 ) = 0, and then B has the following form: And, as a result, the first amendment is When we perform similar actions in order to find the second amendment, we will obtain An exact solution for this problem was initially obtained by Dolginov Toptygin ( 1967 ). In our notation, it has the following form: where F ( a, b ; x ) is the confluent hypergeometrical function (Bateman Erdelyi 1953 ). In particular, from the property of the confluent hypergeometrical function, it follows that for the simpler case when α = 3, the exact solution ( A11 ) has the form APPENDIX B: κ = <italic>k</italic>0η To obtain a solution for the problem, we introduce the following dimensionless variables: and . Then, taking into account ( 6 ) and ( 9 ), we obtain a system of questions for finding the zero-order solution: The last system can be rewritten in the following form: Then, the solution of the last system has the form After inserting N 0 into ( 11 ), we will obtain an equation for N 1 : After differentiating the left-hand side of the last equation, we obtain where Here, the limits of integration were selected as being from 0 to x in consequence of applying the first condition ( 12 ). Now, we consider the integral I separately. If we define , then the expression for I can be rewritten in the following form: Whereas N 0 has the form ( B3 ), then, taking into account the second condition ( 12 ), it is necessary to search for an N 1 of the following form: And, as a result, the first amendment takes the following form: It should be noted that N 1 is not defined for α = 0, 12. APPENDIX C: κ = <italic>k</italic>0βη To obtain a solution for the problem, we introduce the following dimensionless variables: and . Note that for such a problem, the diffusion coefficient κ can be rewritten in the following form: . This follows from the fact that and . Taking into account ( 6 ) and ( 9 ), we obtain a system of equations for finding the zero-order solution: The last system can be rewritten in the form Then, the solution of the last system has the form After inserting N 0 into ( 11 ), we will obtain an equation for N 1 : After introducing the variable: and simplifying the last equation, we obtain where . Note that the limits of integration in K, L, M were selected as being from 0


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

We thank the anonymous referee for useful comments on the paper. AT and RM acknowledge the support of NASA ROSES 12-EUCLID12-0004. We thank Rachel Bean, Jonathan Blazek, Nick Gnedin, Katrin Heitmann, Michael Schneider, and other member of the LSST-DESC collaboration for providing helpful feedback on this work. TDM has been funded by the National Science Foundation (NSF) PetaApps, OCI-0749212 and by NSF AST-1009781 and ACI-1036211. AK was supported in part by JPL, run under a contract by Caltech for NASA. AK was also supported in part by NASA ROSES 13-ATP13-0019. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231.


O O O O O O O B-Organization B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Grant I-Grant O O B-Organization 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-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Grant B-Organization B-Location O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization

We acknowledge partial financial support by contract PRIN/MIUR 2009: ‘Dynamics and Stellar Populations of Superdense Galaxies’ (Code: 2009L2J4MN) and by INAF/PRIN 2011: ‘Galaxy Evolution with the VLT Survey Telescope (VST)’. BV was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan, and by the Kakenhi Grant-in-Aid for Young Scientists (B)(26870140) from the Japan Society for the Promotion of Science (JSPS).


O O O O O O O B-Observatory I-Observatory I-Observatory B-Observatory O B-Location O O O O O B-Observatory B-Observatory O B-Observatory I-Observatory O O O O O O O O B-Observatory O B-Observatory O O O O B-Organization I-Organization I-Organization I-Organization B-Location O O O O O O O O B-Telescope O O O O O O B-Telescope O O O O O B-Telescope O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O B-Database I-Database I-Database B-Database O O O O O B-Organization I-Organization I-Organization 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 O O O O B-Survey I-Survey I-Survey I-Survey I-Survey O O O O O O O B-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization O O O B-Survey O B-Survey O O O O O B-Organization I-Organization I-Organization I-Organization O B-Collaboration I-Collaboration O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization O B-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Survey O O O B-URL O O B-Survey O O O O B-Organization I-Organization I-Organization O O B-Collaboration I-Collaboration O B-Collaboration I-Collaboration O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization I-Organization O B-Collaboration I-Collaboration I-Collaboration B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Collaboration I-Collaboration I-Collaboration O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Survey B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization B-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization B-Organization I-Organization O B-Observatory I-Observatory I-Observatory I-Observatory O O B-Organization I-Organization I-Organization O B-Telescope O B-Telescope O B-Telescope O B-Survey O B-Software B-Software B-Software B-Software B-Software B-Software B-Software

The optical observations were done at the Indian Optical Observatory (IAO) at Hanle. We thank the staff of IAO, Hanle, and CREST, Hosakote, that made these observations possible. The facilities at IAO and CREST are operated by the Indian Institute of Astrophysics, Bangalore. The H i observations were done using the GMRT. We thank the staff of the GMRT that made these observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. 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 publication makes use of data products from the Two Micron All Sky Survey, 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. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the higher Education Funding Council for England. The SDSS web site is http://www.sdss.org/ . The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of CH i cago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the CH i nese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, OH i o State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. Facilities: GMRT , HCT , GALEX , SDSS. Software: AIPS, IRAF, NEMO, MIRIAD, KARMA, PYTHON, SAOIMAGE.


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

Acknowledgements This work is based on observations obtained at the Bernard Lyot Telescope (TBL, Pic du Midi, France) of the Midi-Pyrénées Observatory, which is operated by the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France. Also based on observations obtained at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. O.K. acknowledges support by the Swedish Research Council (project 621-2014-5720), and the Swedish National Space Board (projects 185/14, 137/17). M.S. is supported by Natural Sciences and Engineering Research Council (NSERC) of Canada Postdoctoral Fellowship program.


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

bars stand for standard deviations of the mean correlation functions. As expected, dark matter particles within the closely bound regions of the haloes do not present any particular arrangement (red dashed line), they are homogeneously distributed around the centre of masses. When considering all the particles within a 6 h −1 Mpc radius sphere, the layout is not that much different, although the trend of a preferential direction begins to appear more clearly (black dotted line). Finally, selecting only infalling particles outside the closely bound regions but within 6 h −1 Mpc the signal (blue solid line) is quite strong: there are 5 ± 2 times more particles forming a small angle between themselves than when considering a random distribution and the particles, i.e. infalling particles are non uniformly distributed around the haloes. Knowing that particles falling on to the inner part of dark matter haloes are arranged along a preferred direction, the mean positions of the infalling particles in the different simulations can be derived. We find that the mean coordinates obtained for the different simulations are very similar in all the realizations. The resulting direction of ‘close’ (in the sense that only particles within 6 h −1 Mpc are considered) infall is, on average, along the axis formed between the mean position of Virgo haloes (−4.5 ± 0.6; 8.3 ± 0.4; 0.99 ± 0.8) h −1 Mpc in supergalactic coordinates and the point of supergalactic Cartesian coordinates (−4.8 ± 0.6; 9.3 ± 0.5; 0.26 ± 0.8) h −1 Mpc. To compare with the observed infall, it seems reasonable to look at farther particles. The resulting ‘far’ infall is illustrated by Aitoff projections in Fig. 6 where particles (yellow dots) are shown to fall along the filament linked to the observed Virgo cluster (thick red cross)/ simulated Virgo halo (orange cross). The particles come on average from the supergalactic Cartesian coordinates (−1.5,7.5,−0.05) h −1 Mpc, the supergalactic longitude and latitude (101.45,−0.337) in degrees and the galactic longitude and latitude (287.57,76.77) in degrees. The top projection is centred on the observer (blue cross) while the bottom projection is centred on the observed Virgo cluster. This is in agreement with West Blakeslee ( 2000 ) who suggest that infall of material along the Virgo-Abell 1367 filament has driven the formation of the Virgo cluster. The observational latitude and longitude coordinates, (235.08,73.02) in degrees, of the Abell 1367 cluster (violet triangle) are indeed very close to the direction of infall in the simulations. Figure 6. Aitoff projection in supergalactic (top) and galactic (bottom) coordinates of galaxies from the V8k redshift catalogue (black dots), of the observer (thick blue cross), of the observed Virgo cluster (thick red cross), of one Virgo halo (orange cross) and of the particles infalling on to this halo at redshift zero (yellow dots) The size of the dots in this latter case is proportional to the infall


O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O 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 B-Wavelength O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Wavelength O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O

in the selection of the most basic data sets, namely, the HJD dates of the midpoints of the X-ray flares and of the peak points of the IR flares that were taken as the events to be analyzed. This is why additional observations and analysis by different people are required in order to strengthen the evidence for the operation of the two pacemakers, or to refute this claim. The numbers S X = 0.1976 (Section 4.1 ) and (Section 6.2.3 ) should be compared with 0.2887, the StD of a rectangular distribution over the [−0.5, 0.5] interval. It seems that the grouping of the X-ray flares, and even more so of the NIR ones, around tick marks of the corresponding pacemakers is not very tight. The model presented in Section 8 suggests that the dispersion of the flare times around the pacemaker tick marks is inherent to the flare phenomenon itself. Rather than being well concentrated around some fixed phase of the pacemaker cycle, the flare events occur within a wide section of it. Therefore, if future observations are to strengthen the evidence for pacemakers in the Sgr A* system, it will not come in the form of diminishing the value of the S parameter as longer series of detected flare times are considered. Further evidence could come from finding in future new independent sets of observations the effect of two pacemakers of the same frequencies F X and F IR , but not necessarily with smaller dispersions. On the theoretical side, the model suggested here relies on post-Newtonian equations applied on a highly relativistic system. Such an approach could be too simplistic. For one thing, there is, so far, no observational hint regarding the rate of rotation of the BH at the center. If it is a fast rotator, the Schwarzschild metric itself is not an appropriate description of spacetime in its vicinity. But even within the context of the Schwarzschild metric it is yet to be proven that the Paczynsky–Witta potential function is an adequate approximation in the extreme conditions of existence of the alleged star. Furthermore, the equations in Table 4 are strictly consistent only for e ≈ 0, implying, according to Table 5 , that M ≅ 4.4. This value lies at the extreme end of the uncertainty in the value of M as estimated by other, more direct measurements. Also, as noted in Section 8.3.2 , P orb must be understood as referring to a mean value of the sidereal orbital cycle of the star, and the NIR pacemaker frequency F IR uncovered in the data is the mark of this mean value. In Equations (5) and (6), the relevant parameter x should represent an instantaneous distance r of the star from the BH that is varying between the extreme values and at the apicyclic frequency. For e > 0, the use of the constant a value in these equations may therefore be a poor approximation. Also for e > 0.02 the orbit leads the star to within the ISCO circle around the BH where the post-Newtonian approximation may break down. It may therefore be that in order to place the model on firmer


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

We are grateful for careful comments on the manuscript by Don Schneider and input from the SAMI, CALIFA, and ATLAS 3D teams. We also thank Adam Bolton, Stephane Courteau, Natascha Forster-Schreiber, Bruce Gillespie, Gary Hill, Arlette Pécontal, Rob Sharp, Ray Sharples, and Dennis Zaritsky for valuable feedback on the MaNGA instrumentation and survey design. Funding for SDSS-III and SDSS-IV has been provided by the Alfred P. Sloan Foundation and Participating Institutions. Additional funding for SDSS-III comes from the National Science Foundation and the U.S. Department of Energy Office of Science. Further information about both projects is available at www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions in both collaborations. In SDSS-III these include the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. The Participating Institutions in SDSS-IV are Carnegie Mellon University, Colorado University, Boulder, Harvard-Smithsonian Center for Astrophysics Participation Group, Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe, Max-Planck-Institut fuer Astrophysik (MPA Garching), Max-Planck-Institut fuer Extraterrestrische Physik (MPE), Max-Planck-Institut fuer Astronomie (MPIA Heidelberg), National Astronomical Observatories of China, New Mexico State University, New York University, The Ohio State University, Penn State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, University of Portsmouth, University of Utah, University of Wisconsin, and Yale University. This work was supported in part by a World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. J.H.K. acknowledges financial support to 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 no. PITN-GA-2011-289313. This work has also been supported by the Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences Grant No. XDB09000000 (S.M., C.L.) and by the National Natural Science Foundation of China (NSFC) under grant no. 11333003 (S.M.). M.C. acknowledges support from a Royal Society University Research Fellowship. C.C. acknowledges support from the Sloan and Packard Foundations. A.M.D. acknowledges support from The Grainger Foundation.


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

are required to find the true convergence time. Figure 6. Simulation of a theoretical 10 Myr old TWA-like association with four different initial velocity dispersions, σ init . The curves show the traceback age that is derived after randomly adding noise that simulates measurement uncertainties. The derived traceback age drops steeply with increasing measured uncertainties. The total measurement uncertainty in our data is dominated by uncertainties in the RV and parallax measurements. See Section 4.1 for more details. Typical expected velocity dispersions for open clusters are around 1–2 km s −1 (Adams 2000 ). Assuming an initial velocity dispersion for TWA of σ init = 2 km s −1 , we would need to measure the UVW velocities with a precision of 0.25 km s −1 to accurately measure the traceback time to within 1 Myr. Given our current mean measurement uncertainties of σ meas = 1.3 km s −1 , the initial velocity dispersion of TWA would have to be greater than 11 km s −1 for us to accurately measure the traceback age to within 1 Myr. The GAIA spacecraft will soon obtain parallaxes to a billion stars with unprecedented precision. Given the low uncertainties in GAIA parallaxes and proper motions, the RVs will become the dominate source of noise. From the calculations above, the RVs will need to have a precision of 0.25 km s −1 for a proper traceback analysis, a quarter of our current average RV uncertainty. We ran another traceback analysis, this time with only stars with RV uncertainties below 0.5 km s −1 (17 stars). This produces a traceback age of 2.9 ± 0.5 Myr. This is a slightly higher value than most of the subsamples tested, but still falls quite short of the expected value. Given the low traceback ages that we obtain for TWA, it is likely that its initial velocity dispersion is too low for an accurate determination of its age using a traceback analysis. 5. PRE-MAIN-SEQUENCE TRACK AGE 5.1. Calculating the Ages of Individual Stars To determine the ages of individual TWA stars, we compare the stars to the pre-main-sequence tracks of Baraffe et al. ( 2015 ). We use our parallax values to convert 2MASS apparent magnitudes to absolute magnitudes. We took optical spectral types from the literature and converted them to effective temperature using Table 6 of Pecaut Mamajek ( 2013 ). For stars M5–M7, we use the main-sequence values in Table 5 of Pecaut Mamajek ( 2013 ). It is interesting to note the disagreement between Pecaut Mamajek ( 2013 ) and Luhman ( 1999 ), which was used in Weinberger et al. ( 2013 ) to convert to effective temperature. The disagreement is typically about 100 K cooler for Pecaut Mamajek ( 2013 ) than for Luhman ( 1999 ), with particular disagreement around M5 with a difference of about 250 K. For stars with spectral type greater than M7, we used the polynomial relation in Table 19 of Faherty et al. ( 2016 ) for moving group members. Spectral types and temperatures for the entire sample are listed in Table 4 . Table 4


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

redshifts that is insensitive to assumptions about the cosmological parameters. In the next section, we derive relations between cosmological distances that are independent of cosmological parameters and allow us to construct combinations of that no longer carry a dependence on cosmology. 3. Distance relation in isotropic universes The two-dimensional separation vector between two infinitesimally close light rays follows the geodesic deviation equation where is the affine parameter along the light rays, and is the optical tidal matrix, which is determined by the Ricci and Weyl tensors of the spacetime metric (see, e.g., Chap.3 of Schneider et al. 1992 ; Seitz et al. 1994 ). In an isotropic universe, the tidal part of vanishes, so that , where is the two-dimensional unit matrix and is a scalar function, proportional to the local density. If the two light rays intersect at a vertex at under an angle , we can write , where satisfies the differential equation with initial conditions , , where the latter value depends on the choice of the affine parameter. By definition, is the angular-diameter distance of a source at , seen from an observer at . The second-order differential Eq. ( 7 ) has two independent solutions; we chose them to be and , that is, the angular-diameter distances as measured from observers located at and , respectively. A third solution, , must necessarily be a linear combination of the other two. If we write it in the form where is a constant, then we see that the first initial condition is satisfied, . For the second initial condition, we have where is the Wronskian From Eq.( 7 ), we directly see that , i.e., is constant. We can calculate this constant considering at , which yields yielding . Therefore, Thus, we have expressed the angular-diameter distance as seen from an observer at affine parameter by the corresponding expression for the angular-diameter distance for observers at and . The resulting expression contains the initial conditions at and . If we now specialize Eq.( 9 ) to the case of , with the corresponding angular-diameter distance denoted as , we obtain Combining Eqs.( 9 ) and ( 10 ), we then find The final equality no longer contains the initial conditions. It is valid for any spacetime that is isotropic around the observer through which the radial light bundle passes. Furthermore, this equation does not refer to the Einstein field equation and is therefore independent of the validity of General Relativity, as long a spacetime is characterized by a metric. A more graphical derivation of the result ( 11 ) is given in Schneider (2016) . If, in addition to isotropy, we assume a spatially homogeneous spacetime, which can then be expressed in the form of the Robertson–Walker metric, the initial conditions are , where is the redshift with which sources at are seen by the observer at (or redshift zero). For this, we chose a parametrization of the affine parameter such that it locally coincides with the comoving


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This work is funded by the UK Science and Technology Facilities Council, grant ST/I002871/1. The author thanks the CANARY team for allowing on-sky testing.


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for extracting information from the phase-space spiral. For example, Li Widrow (2021) infer the gravitational potential of the disk using the bulk stellar density in the ( z , w )-plane and then fit the spiral to the residual of the inferred steady state distribution. The fitted spiral constitutes a consistency check of the inferred potential and is seen as a straight line in the plane of angle and frequency. While our method is accurate in terms of one-dimensional simulations, the actual Milky Way is significantly more complex. To begin with, stars in the Galactic disk do not only oscillate in the vertical direction, but also move in the radial direction as they orbit the Galactic centre. Indeed, the phase-space spiral of the Milky Way has a different form depending on cuts in angular momentum or the galactocentric radius (see for example Fig. 14 in Laporte et al. 2019 , as well as Khanna et al. 2019 ). When applying our method to Gaia data for the first time, we plan to select stars of the solar neighbourhood that have approximately circular orbits, for which the assumptions of our model are most likely to be valid. It would be interesting, although probably quite complicated, to see what kind of information about the global structure of the Milky Way could be extracted from the shape of the spiral in different bins of Galactic angular momentum. Ultimately, we would want to test and refine our method using a high-resolution simulation of a galaxy in three spatial dimensions. Up until now, simulations of Milky Way-like systems (such as Laporte et al. 2018 and Bland-Hawthorn Tepper-García 2021 ) have not had sufficient resolution to produce well-resolved phase-space spirals in stellar density in the ( z , w )-plane. To do so requires billion-particle simulations, which have only recently become feasible ( Asano et al. 2020 , Hunt et al. in prep.). Furthermore, our modelling of the gravitational potential needs to be expanded somewhat; the Poisson equation is no longer one-dimensional but also has a smaller contribution from the radial direction (the so-called rotation curve term). A useful test in terms of systematic uncertainties would be to perform separate analyses of different segments of the phase-space spiral. The assumptions of our analytical model are probably only valid over a certain range in vertical energy; fitting the spiral over a large area in the z , w -plane would give a small statistical uncertainty, but potentially at the price of large systematic errors. It would be possible, at least to some extent, to quantify systematic errors by comparing the results of the smaller, separate spiral segments. Another test in a similar vein would be to abandon the form of the function as defined in Eq. ( 9 ), which might not be completely accurate due to effects of, for example, self-gravity; instead, we could simply assume that the spiral angle is a smooth function with respect to E z and model it as a Gaussian process. Observational


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above, that we assume the albedo distribution is not different for binaries and non-binaries. This assumption can be justified by the AD test result -value = 0.13, applied to samples from “TNOs are cool” program, which contains albedos for 108 single and 30 multiple objects. This fact indicates, that there is no occurred statistically significant difference between TNOs and TNBs albedo samples. The empirical albedo distributions are shown in Fig. 1 . Table 2 Result of the two-tailed AD test applied to diameter dataset using the whole MPC list. Notes. (a) Classification of Gladman et al. (2008) . CC denotes cold-classicals, HC denotes hot-classicals, Det. and SDO denote detached and scattered disc objects, Res. denotes resonant objects. (b) The -value is computed using the techniques developed by Scholz Stephens (1987) . and sd denote -values, calculated for the mean and mean standard deviation of the AD statistic, respectively. (c) The average number of objects. The number of objects with and/or varies, depending on Monte Carlo samples. The resulting -value for the average AD test statistic for diameter samples is summarised in Table 2 . The obtained result (-value , i.e. statistical significance >) indicates that the null hypothesis about identical diameter distributions is rejected. However, this result may be biased. To check for biases in statistical conclusions, we have repeated the AD test for different sub-samples. In a first step, the comparison is made for dynamically equivalent samples, namely using the classification of Gladman et al. ( 2008 ; see Table 2 ). The test statistic shows no significant evidence of the difference between TNBs and TNOs among detached objects and SDO (-value = 0.08). Thus, it appears that the occurred difference in diameter distributions can be caused by cold classicals, resonant objects, and possibly by cenataurs among which no binaries were observed. On the other hand, this conclusion can be not very robust from statistical point of view, since the small sizes of certain sub-samples for binary objects (e.g. less than 30 binary objects for HC, resonants, detached and SDOs) . It should be noticed, however, that the compared samples have significantly different size ranges. On the one side, the sample includes a very low binary fraction of small objects among plutinos, SDOs and centaurs. On the other side, the largest objects in the sample – dwarf planets – almost all have satellites. Consequently, the size distribution of TNBs may be biased towards the largest objects, whereas the sample of single TNOs may be biased by small objects. Thus, in a second step, we compare the size distribution over a restricted size range, namely, over the range from 140 km (the diameter of the smallest TNB 2001 RZ143 with known observed by Herschel ) to 1142 km (the largest TNO 2007 OR10 without satellites in the Herschel sample). In this case, the test indeed shows no significance about difference between two samples. This


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We thank Woong-tae Kim and Yonghwi Kim for helping us understand the Athena code. Hospitality at APCTP during the 7th Korean Astrophysics Workshop is also kindly acknowledged. The research presented here was funded by a grant from the Max-Planck Society under the cooperation agreement with the Chinese Academy of Sciences and was additionally supported by the 973 Program of China under grant no. 2014CB845700, by the National Natural Science Foundation of China under grant nos. 11333003, 11322326, and by the Strategic Priority Research Program “The Emergence of Cosmological Structures” (no. XDB09000000) of the Chinese Academy of Sciences. J.S. acknowledges support from an Newton Advanced Fellowship awarded by the Royal Society and the Newton Fund. This work made use of the facilities of the Center for High Performance Computing at Shanghai Astronomical Observatory.


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This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the NuSTAR Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA). G.Y. acknowledges support from NASA under NuSTAR Guest Observer cycle-2, proposal number 15-NUSTAR215-0024. L.C.H. was supported by the National Key RD Program of China (2016YFA0400702) and the National Science Foundation of China (11473002, 11721303). Y.T. is supported by JSPS Grants-in-Aid for Scientific Research 15H02070 and 16K05296. F.G.X. and F.Y. are supported in part by National Key RD Program of China (grants 2016YFA0400804 and 2016YFA0400704), the Natural Science Foundation of China (grants 11873074, 11573051, 11633006, 11650110427, 11661161012, 11303008, and 11473002), and the Key Research Program of Frontier Sciences of CAS (grants QYZDJSSW-SYS008 and QYZDB-SSW-SYS033). F.G.X. is also supported by the Youth Innovation Promotion Association of CAS (ID. 2016243) and the Natural Science Foundation of Shanghai (No. 17ZR1435800). We thank the referee for a concise and thorough reading of the manuscript that led to an improved version of the article.


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ACKNOWLEDGEMENTS We would like to thank the anonymous referee for thoughtful suggestions that improved the clarity and flow of our manuscript. We also thank Kelvin Au for many interesting discussions and for proofreading the manuscript. AR, AM, and BP wish to acknowledge the contribution of the Brandon University Research Committee and the Canada Summer Jobs Program. XE is supported by National Science Foundation of China (NSFC) Grant No. 11473032 and No. 11873006.


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ACKNOWLEDGEMENTS PAAL thanks the support of CNPq, grant 308969/2014-6; and CAPES, Programa Estágio Sênior no Exterior , process number 88881.120856/2016-01. ALBR thanks for the support of CNPq, grant 311932/2017-7. FD acknowledges long-term support from CNES. TFL acknowledges financial support from CNPq (grant 303278/2015-3). MT thanks the support from the PSL Research University. This research has made use of the SAO/NASA Astrophysics Data System, the NASA/IPAC Extragalactic Database (NED) and the ESA Sky tool (sky.esa.int/). Funding for the SDSS and SDSS-II was provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. A list of participating institutions can be obtained from the SDSS Web Site http://www.sdss.org/ . The 2dFGRS ( http://www.2dfgrs.net/ ) has been made possible by the dedicated efforts of the staff of the Anglo-Australian Observatory, both in creating the 2dF instrument and in supporting it on the telescope. We are also very thankful for the availability of the Final Release of 6dFGS ( http://www-wfau.roe.ac.uk/6dFGS/ ).


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

produced synthetic NIR light curves based on GRMHD models that incorporate a model for accelerating electrons from a thermal distribution into a power-law tail. A control model, which contains only thermal electrons, produces no significant NIR emission. A simple non-thermal model that places a constant fraction of the electrons into a non-thermal (κ distribution) component is able to produce the correct spectral slope and flaring flux density but not the variability. The next simplest model is based on the idea that electrons are accelerated due to resistive dissipation in reconnecting current sheets and, therefore, sets the fraction of electrons in the non-thermal distribution proportional to J 2 , where J is the 3-current density measured in the plasma frame. This model matches the spectral slope and flaring emission but also matches the flux density distribution fairly well. A final model fares similarly in matching observations but is both better motivated physically and more computationally expensive. It advects the non-thermal electrons with the fluid, accelerates electrons into the non-thermal component at a rate proportional to J 2 , and allows them to cool using a constant cooling time model. Because these last two models agree with each other relatively well, future work will focus on the simpler one. No model shows substantial NIR centroid wander or any correlation between the NIR flux density and image centroid. These models are the first to accurately produce the observed Sgr A* mm mean flux, mm image size NIR spectral slope, flare duration, and flux distribution at the same time. Future work will adapt these models to 3D and will include Compton scattering in order to examine X-ray flares. APPENDIX A: DISSIPATION IN RELATIVISTIC MHD Here, we prove that heating rate per unit volume due to resistive dissipation (the dissipation function) is J 2 /σ if we adopt the simplest covariant model for Ohm’s law. Maxwell’s equations are Project this into the space normal to the plasma four-velocity u μ using the projection tensor and define the projected current If we then write a simple model for Ohm’s law as where σ is the conductivity, it is easy to show that this reduces to the familiar form of Ohm’s law for a scalar conductivity. To evaluate the dissipation function, observe that since then For an ideal fluid stress-energy tensor, this implies where u is internal energy and p is pressure. This is just the internal energy equation as derived from the first law assuming the dissipation function ρ T d s /dτ = ρ Tu μ s ; μ = 0. Here, the scalars s ≡ entropy and T ≡ temperature. Now consider a magnetized fluid and break the stress-energy tensor into an ideal fluid and electromagnetic piece, . Then From MTW, Misner et al. ( 1973 ), exercise 22.10 (which uses a symmetry argument and Maxwell’s equations) Combining, Then Ohm’s law implies This is positive definite because J μ is spacelike by construction and, therefore, the second law is


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ACKNOWLEDGEMENTS This work has been supported by the National Natural Science Fund of China under grant nos. 11603005 and 11775038. We are grateful to Y. Sang for useful discussions.


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We thank the anonymous referee for valuable comments and suggestions that helped us to improve the paper. We thank Elvijs Matrozis for his helpful discussions. This work is supported by the Alexander von Humboldt Foundation. R.J.S. is the recipient of a Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation. B.W. acknowledges support from the 973 programme of China (No. 2014CB845700), the NSFC (Nos. 11322327, 11103072, 11033008, and 11390374), the Foundation of Yunnan province (No. 2013FB083).