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

Whilst a reasonable harmonic fit to the ESPaDOnS data can be achieved using this period, it does not produce an acceptable phasing of all available 〈 B z 〉 measurements. Figure 1. Photometric ( top ) and magnetic 〈 B z 〉 ( bottom ) measurements, phased with periods determined from (left to right) K2 photometry, all 〈 B z 〉 measurements, and all photometric measurements. 〈 B z 〉 measurements were obtained from ESPaDOnS by Shultz et al. ( 2018 ) and photopolarimetric data by Borra et al. ( 1983 , BLT83) and Bohlender et al. ( 1993 , BLT93). The solid and dashed curves show, respectively, the best second-order harmonic fit to the ESPaDOnS 〈 B z 〉 data and the 1σ fit uncertainty. Photometric data are from K2, SMEI, Hipparcos ( H p ), and Strömgren y photometry published by Sterken et al. ( 1995 , S95), Manfroid et al. ( 1995 , M95), and Pedersen Thomsen ( 1977 , P77). For clarity the K2 and SMEI data have been binned by phase, using bin sizes of 0.01 cycles. The mean magnitude of each data set was subtracted for display purposes. All of the periods obtained since the year 2000 are approximately consistent with one another; the two periods determined using photometric data obtained prior to the year 2000 are also consistent with one another. At the same time, the difference between the Bowman et al. ( 2018 ) and Catalano Leone ( 1996 ) periods, 0.00015 d, is almost eight times larger than the formal uncertainty in the Bowman et al. period (the less precise of the two). This suggests that the period may have changed. 3.2 Period analysis We now turn to a re-examination of the period analysis of the individual data sets; the results are summarized in Table 2 . Period analysis was performed using standard Lomb–Scargle statistics (Lomb 1976 ; Scargle 1982 ), utilizing both the idl program periodogram 4 and the period04 package (Lenz Breger 2005 ). The uncertainties were determined in the same manner as by Shultz et al. ( 2018 ), i.e. according to the analytic method described by Bloomfield ( 1976 ). Examining the ESPaDOnS data in isolation yields P rot = 0.97887(6) d, consistent within uncertainty with the K2 period (Bowman et al. 2018 ). The period determined from all available magnetic data (i.e. including the previously overlooked 〈 B z 〉 measurements of Borra et al. 1983 ) is 0.979855(5) d. The photometric and magnetic data are shown phased with this period in the middle panel of Fig. 1 . Phasing 〈 B z 〉 with this period and fitting the ESPaDOnS data with a second-order sinusoid yields a reduced χ 2 = 5, i.e. a poor fit. This period also does not provide a coherent phasing of the majority of the photometric data set. Notably, the variation in the phase-binned SMEI data almost completely disappears, likely because observations are being binned as though they were at the same phase when, in fact, they are not. We can also combine all available photometric data (shifted to the mean magnitude of each data set, as in Fig. 1 ), which yields P rot =


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comparison once the angular positions of the satellites are randomized around this point. From this tensor we compute its eigenvalues, λ 1 > λ 2 > λ 3 , and corresponding eigenvectors, , , . We define the size of the three ellipsoidal axis as a = λ 1 , b = λ 2 , and c = λ 3 . We also define as the vector perpendicular to the planar satellite distribution. We also define the RMS plane width w as the standard deviation of the satellite distances to the plane defined by the vector . To summarize we characterize the satellite distributionwith the following quantities obtained from the inertia tensor: RMS plane width, w . c / a axis ratio. b / a axis ratio. 3.3 Characterizing asphericity with normalized scalars We compare each satellite distribution against its own spherically randomized distribution. We keep fixed the radial position of every satellite with respect to the central galaxy and then randomize its angular position. We repeat this procedure 10 4 times for each satellite distribution and proceed to measure the quantities mentioned in the previous section: w , c / a , and b / a . This allows us to build a normalized version of all quantities of interest by subtracting the mean and dividing by the standard deviation of the values from randomized samples. We use the normalized quantities to build the analytic probability distributions for the scalars describing asphericity. 3.4 Building an analytic asphericity probability distribution After building the normalized variables for the simulated data we perform a Kolmogorov–Smirnov test with the null hypothesis of belonging to a normal distribution with mean and standard deviation computed from the mean and standard deviation estimated from the data. Although the physical quantities of interest are bound, we find that the distributions for the normalized w , c / a , and b / a are indeed consistent with Gaussian distributions. Based on this result we build a multivariate normal distribution for the joint distributions of the normalized w , c / a , and b / a where X = [ w , c / a , b / a ] T is a vector variable with the normalized quantities, μ is the vector mean, and the Σ is the covariance matrix. We compute the preferred covariance matrix and the mean distribution values with a jackknife technique. That is, out of the n pairs in each simulation, we perform n different covariance and mean value measurements using only n − 1 pairs. The reported covariance and mean values correspond to the average of all measurements, the corresponding standard deviation also helps us to estimate the uncertainty on every reported coefficient. This compact description allows us to generate samples of size N that are consistent by construction with their parent simulation. Finally, we use the generated samples to estimate how common are the deviations from sphericity that we measure in the observational data. We use a double-tailed test around the mean in this comparison, meaning that we count the fraction of


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ACKNOWLEDGEMENTS The authors thank an anonymous reviewer for comments and suggestions that strengthened our manuscript. NWCL gratefully acknowledges support from the American Museum of Natural History and the Richard Guilder Graduate School, specifically the Kalbfleisch Fellowship Program, as well as support from a National Science Foundation Award No. AST 11-09395.


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BDM gratefully acknowledges support from NASA Fermi grant NNX14AQ68G, NSF grant AST-1410950, and the Alfred P. Sloan Foundation. We thank the anonymous reviewer for several helpful suggestions and for emphasizing the applicability of our results to Type IIn supernovae.


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I thank Duncan Galloway, Zac Johnston, and Hendrik Schatz for useful discussions, and acknowledge the instructive tutorials from Ed Brown, Andrew Cumming, and Rob Farmer located on the MESA Marketplace ( http://cococubed.asu.edu/mesa_market/ ) that helped at the beginning stages of this work. Relevant MESA inputs for this work are available on the MESA Marketplace. This work was supported in part by the U.S. Department of Energy under grant No. DE-FG02-88ER40387. This material is based on work supported by the National Science Foundation under grant No. PHY-1430152 (Joint Institute for Nuclear Astrophysics–Center for the Evolution of the Elements).


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

Lasota (2007) have updated this work, for a sample of ten CVs. The importance of this test was revived by Schreiber Gänsicke (2002) , who pointed out that the HST FGS parallax distance of SS Cyg, the archetypal dwarf nova, led to a revised mass transfer rate that placed this outbursting system in the stable region of the ( R out , Ṁ ) plane. This led to much hand-wringing ( Schreiber Lasota 2007 ; Smak 2010 ) until a radio VLBI parallax distance ( Miller-Jones et al. 2013 ), confirmed with Gaia ( Ramsay et al. 2017 ), firmly placed the system in the unstable region (as anticipated by Lasota 2008 ). Although there are dozens of CVs with relatively well-known binary parameters, a thorough investigation of this test has yet to be performed. Ramsay et al. (2017) proposed to test the DIM by taking the absolute magnitude at a specific stage of a dwarf nova outbursts (at the end of long outbursts or in stand-stills) to compare the derived mass accretion rate to the critical rate for the system, under the assumption that both should be nearly equal. Good agreement was found for the sample of a dozen CVs with distances in the Gaia first data release. Its main advantage is that the required optical magnitude is straightforward to extract from the light curves. However, a major drawback of this approach is that it relies on assumptions on how matter is transported in the disk because one needs to predict when the mass accretion rate happens to be equal to the critical rate during the outburst cycle. Hence, it is a much less fundamental and robust test of the basic premises of the DIM. Indeed, an important difficulty in using the fundamental test is that it requires extracting the average mass transfer rate from CV optical light curves, which in many cases are unevenly, inhomogeneously, and incompletely sampled. The situation is somewhat easier for X-ray binaries, for which the DIM also applies because X-ray monitors provide continuous coverage and X-rays provide a closer measure of the bolometric luminosity in those systems than optical magnitudes do for CVs, which in outburst emit primarily UV radiation. Coriat et al. (2012) were able to derive average mass accretion rates for ~50 X-ray binaries and showed that the DIM, modified to include X-ray irradiation ( van Paradijs 1996 ), separates consistently transients from steady systems. We carry out, for the first time on CVs, a similar analysis to that performed on X-ray binaries by Coriat et al. (2012) to test the fundamental prediction of the DIM: the separation of CVs into stable and unstable systems is determined by the orbital period (a proxy for disk size) of the binary and mass transfer rate from the companion. We use a subsample of the CV light curves gathered by Otulakowska-Hypka et al. (2016) . We consider all types of dwarf nova outbursts, including so-called super-outbursts that are not supposed to be described by the standard version of the DIM (see, e.g., Lasota 2001 , and references


O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Telescope O O O O O O B-CelestialObject O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-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

an antenna with cos 2 beam would have a maximum temperature at the LST when the Galactic center is at zenith. This also corresponds to the spectrum having the largest rms residual among the various mock spectra considered here. For various large-area samplings of the sky, we have thus demonstrated that the foreground component in a beam-averaged spectrum is adequately describable by an MS function. Thus for observations with a frequency-independent beam, MS functions can be used to separate foregrounds from the global EoR signal. We next discuss mathematical statistical detection based on MS-form modeling of foregrounds. 5. On the Detection of the Global Reionization Signal A mock observation of the radio sky as observed by a cos 2 beam is shown in Figures 14 . A GMOSS model for the sky has been adopted and CMB with a spectral distortion corresponding to generic global EoR has been added. We fit this mock observation of the sky spectrum with a function given by the sum of an MS function to model the foreground and a Planck spectrum to account for the undistorted CMB, as described by Equation ( 9 ). We use the downhill simplex (Nelder Mead 1965 ) optimization algorithm to iteratively fit this model to the synthetic spectrum, adopting a successive approximation strategy. The MS polynomial is expressed as a polynomial expansion about a frequency given by the parameter . The first iteration fits a function that has four parameters: the CMB temperature , the pivot frequency , and two coefficients that describe a single power law for the foreground. We successively include more terms using the optimized parameters from the previous iteration as initial guess and zero for the new coefficient introduced. The residuals on subtracting a model with a Planckian function and MS functions of degree 2, 5, 7, 8, 10, and 15 are also shown in Figure 14 . This may be compared to the residuals obtained above on fitting unconstrained polynomials to the same mock observation containing the EoR signal, which was shown in Figure 6 . As noted earlier, for the case where the foregrounds were modeled as unconstrained polynomials, the residual amplitude gets progressively lower with increasing polynomial order and there are additional turning points introduced. On the other hand, while using MS functions, the residual remains unchanged once the MS function encounters a shape that cannot be described as smooth and thenceforth increasing the order of the MS function no longer changes the residual. This would continue to be the case on fitting the spectrum with MS functions of arbitrarily higher order. The residual recovered on fitting the mock spectrum with such components that may accurately model the undistorted CMB and foreground appears qualitatively similar to the adopted global EoR signal, which was included in the simulation. We now proceed to quantify the confidence in signal detection and the likelihood of false positives. Figure 14. (a) Mock spectrum generated


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

origin. Our findings from this Section for the radial metallicity distribution of red and blue GCs in massive ETGs are summarized in a schematic way in Fig. 2 . This representative diagram shows the same radial metallicity gradient for the red (metal-rich) GCs and the blue (metal-poor) GCs, only offset by 1 dex in metallicity. We have argued that the stellar metallicity gradient is similar to that of the red GCs. At some large effective radius (8 R e in the diagram), the blue GC metallicity gradient gives way to an outer halo gradient that is flat, i.e. constant metallicity. We refer the reader to fig. 1 of Forbes et al. ( 2011 ) and fig. 10 of Faifer et al. ( 2011 ) for real-world examples of our schematic plot, which show not only the mean gradients but also the individual GC data points. Figure 2. Schematic representation of globular cluster metallicity gradients as described in Section 3 . The diagram shows the metallicity gradient of the metal-rich GCs in red. This gradient is similar to that of the stellar metallicity gradient. Approximately 1 dex lower in metallicity, the metallicity gradients for the metal-poor GCs are shown in blue. At several effective radii ( R e ), this radial gradient may give way to a constant metallicity for metal-poor GCs to large radius. 4 COMPARISON WITH MAGNETICUM SIMULATIONS Although a number of simulations have modelled the formation of GCs in a hierarchical cosmology (e.g. Bekki et al 2008 ; Muratov Gnedin 2010 ; Li Gnedin 2014 ; Kruijssen 2015 ; Choksi et al. 2018 ) with reasonable success in reproducing bimodal metallicity distributions and some global scaling relations, none have predicted the radial colour or metallicity profiles for GC subpopulations (although see Pfeffer et al. 2018 , for a first step in modelling the spatial distribution of GC systems in Milky Way-like galaxies). However, simulations of the stars in massive ETGs are available and, as noted above, GC metallicity gradients tend to be similar to those of their host galaxy stellar metallicity profiles. Thus, it is useful to investigate simulated stellar gradients as a proxy for GC gradients. The Magneticum hydrodynamical cosmological simulation suite (Dolag et al., in prep.) produces in its highest resolution ( and as every gas particle can spawn up to four stellar particles) massive ETGs with properties that successfully reproduce observations (e.g. Teklu et al. 2015 , for stellar-mass–angular-momentum properties; Remus et al. 2017 , for mass–size relations at different redshifts and dark matter properties; Teklu et al. 2017 , for satellite fractions and the morphology–density relation; Schulze et al. 2018 , for kinematic properties). The simulation adopts a WMAP7 (Komatsu et al. 2011 ) ΛCDM cosmology with σ 8 = 0.809, h = 0.704, , Ω M = 0.272, and Ω B = 0.0451, and an initial slope for the power spectrum of n s = 0.963 (see Hirschmann et al. 2014 ; Teklu et al. 2015 , for more simulation details; Dolag, Mevius Remus 2017 , for


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

of the CSM density profile and the forward shock evolution together determine the temporal behavior of the optically thin radio flux density. For s = 0 and , the flux density rises with time as once , the flux density evolution steepens to (visible as a kink in the model light curve of Figure 1 with cm −3 around 100 days after explosion; such a kink was also noted by Pérez-Torres et al. 2014 ). Meanwhile for s = 2, the optically thin scaling is when later, when bottoms out at unity, then . Therefore, the temporal evolution of the optically thin flux density can be used to estimate the density profile of the CSM. 4.5. Caveats and Complexities As discussed by Horesh et al. ( 2012 , 2013 ), the most significant uncertainty in converting radio luminosity to a density of the CSM is the efficiency of magnetic field acceleration, . In the case of uniform density CSM and optically thin synchrotron emission, the density derived from a given flux density will depend on as when , and when . For s = 2 and , we find , and when converges to unity then . In addition, as seen by comparing Figures 3 with 4 , and Figures 5 with 6 , the synchrotron emission becomes optically thin faster for lower values of . Figure 3. Each panel shows a set of radio light curve models at a distinct frequency (marked top left), assuming , , , , and s = 2. The model grids span a range of values () with spacing of 1 dex, with purple-blue models corresponding to low values and red models to high . Observed 3 σ upper limits at each frequency are overplotted as black triangles. Figure 4. Each panel shows a set of radio light curve models at a distinct frequency (marked top left), assuming a more conservative (compared to in Figure 3 ) and retaining , , , and s = 2. The model grid and color scheme represent the same range of values as in Figure 3 . Observed 3 σ upper limits at each frequency are overplotted as black triangles. Figure 5. Each panel shows a set of radio light curve models at a distinct frequency (marked top left), assuming , , , , and s = 0. The model grids span a range of n 0 values ( cm −3 ) with resolution of 1 dex, with purple-blue models corresponding to low n 0 values and red models to high n 0 . Observed 3 σ upper limits at each frequency are overplotted as black triangles. Figure 6. Each panel shows a set of radio light curve models at a distinct frequency (marked top left), assuming a more conservative (compared to in Figure 5 ) and retaining , , , and s = 0. The model grid and color scheme represent the same range of n 0 values as in Figure 5 . Observed 3 σ upper limits at each frequency are overplotted as black triangles. Radio light curve models are also affected by the assumed values of SN ejecta mass and explosion energy. If we take conservatively low values of and , we find forward shock velocities of and at t = 10 days since explosion for and , respectively. These blast wave velocities are high enough that synchrotron self absorption still dominates over


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We first thank M.S. Ruderman for helpful comments. A.H. thanks the School of Mathematics and Statistics, University of Sheffield (UK) for the support received. R.E. is thankful to the NSF, Hungary (OTKA, Ref. No. K83133) and is grateful to STFC (UK) for the awarded Consolidated Grant, and, The Royal Society for the support received in a number of mobility grants.


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

where and are two cylindrical coordinates; is the axial ratio, is a structure parameter, and is a function of , such that becomes the density at radius , which defines a volume containing half of the total mass of the component (for details, see Tamm et al. 2012 ). The gravitational potential derivative of a component, necessary for calculating epicyclic frequencies, can in this case be expressed in a convenient form (see e.g. Tenjes et al. 2001 ) where is eccentricity and . The formula is general, but the epicyclic frequencies were calculated within the galactic plane only. While the mass distribution parameters of the spheroidal components and the main stellar disc can be used as presented in Tamm et al. (2012) , the flat young disc needs to be split further into main subsystems. At this stage we do not yet know the relative contributions of different flat components to the overall galaxy stability, thus we have to analyse each one of them. First we separated gas and stars by taking the mass of young stars to be and the gas mass to be ( Tamm et al. 2012 ). Thus we now had two stellar discs, one containing old and intermediate-age stars, and the other containing young stars. The parameters of these flat stellar components are given in Table 1 . Several subcomponents can also be distinguished in the gas disc. Although four to five different phases can be distinguished in the interstellar medium ( McKee Ostriker 1977 ; Chemin et al. 2009 ), here we have considered the neutral gas (including both the atomic and the molecular component) to consist of two main phases, the cold neutral medium (CNM) and the warm neutral medium (WNM; see Brinks Burton 1984 ; Wolfire et al. 2003 ; Kalberla Kerp 2009 ). For M 31 Dickey Brinks (1993) estimated that the mass of WNM is about 60 per cent and the mass of CNM is 40 per cent of the total neutral gas mass. These proportions have been found to also be valid in the Milky Way: according to Heiles Troland (2003) , the mass of WNM is 60 per cent of the total neutral gas mass. Thus we took the masses and . Unfortunately, direct measurements of the thickness of these two gas components in M 31 do not exist. Brinks Burton (1984) estimated from their gas velocity dispersion measurements that CNM has . In addition, taking into account the broad similarity between M 31 and the Milky Way, we have applied here the estimate by Kalberla (2003) that at the solar distance, the WNM disc of the Milky Way is about three times thicker than the CNM disc. Thus we take and . However, to consider the impact of uncertainties, we also performed calculations for some other flatnesses. All the other structure parameters and effective radii were kept unchanged. The parameters of the two gas discs are given in Table 1 . To complete the set of input parameters for stability calculations, we also needed to know velocity dispersions and the speed of sound in the disc components. Following Dorman et al. (2015) , we took velocity dispersions of


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version of the full table is available. Table 3 The Spectral Parameters Associated with the H β Emission Line Note. Column information is as follows: Column [1]: 4FGL name; Column [2]: redshift; Column [3]: log-scale continuum luminosity at 5100 Å; Column [4]: FWHM of H β line, in km s −1 ; and Column [5]: log-scale H β line luminosity, in erg s −1 . (This table is available in its entirety in machine-readable form.) Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available. Table 4 The Spectral Parameters Associated with the Mg ii Emission Line Note. Column information is as follows: Column [1]: 4FGL name; Column [2]: redshift; Column [3]: log-scale continuum luminosity at 3000 Å; Column [4]: FWHM of Mg ii line, in km s −1 ; and Column [5]: log-scale Mg ii line luminosity, in erg s −1 . (This table is available in its entirety in machine-readable form.) Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available. Table 5 The Spectral Parameters Associated with the C iv Emission Line Note. Column information is as follows: Column [1]: 4FGL name; Column [2]: redshift; Column [3]: log-scale continuum luminosity at 1350 Å; Column [4]: FWHM of C iv line, in km s −1 ; and Column [5]: log-scale C iv line luminosity, in erg s −1 . (This table is available in its entirety in machine-readable form.) Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available. 3.2. Absorption-line Spectrum The optical spectra of 346 γ -ray blazars show prominent absorption lines, e.g., the Ca HK doublet, primarily originating from the stellar population in the host galaxy. It has been shown in various works that the mass of the central black hole significantly correlates with the stellar velocity dispersion ( σ * , see Ferrarese Merritt 2000 ; Gültekin et al. 2009 ; Kormendy Ho 2013 ). Therefore, we have used the penalized PiXel Fitting tool ( pPXF ; Cappellari Emsellem 2004 ) to derive σ * for blazars present in the sample. This software works in pixel space and uses a maximum penalized likelihood approach to calculate the line-of-sight velocity distribution (LOSVD) from kinematic data (Merritt 1997 ). A large set of stellar population synthesis models (adopted from Vazdekis et al. 2010 ) with a spectral resolution of FWHM = 2.5 Å and the wavelength coverage of [3525, 7500] Å (Sánchez-Blázquez et al. 2006 ) was used in this work. The pPXF code first creates a template galaxy spectrum by convolving the stellar population models with the parameterized LOSVD. To mimic the nonthermal power-law contribution from the central nucleus, we further added a fourth-order Legendre polynomial to the model. The regions of bright emission lines were masked prior to the fitting. The model was then fitted on the rest-frame galaxy spectrum, and σ * and the


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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 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 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 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 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 B-Telescope I-Telescope I-Telescope O B-Telescope I-Telescope I-Telescope B-Telescope O O O B-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Observatory O O O O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Telescope I-Telescope I-Telescope I-Telescope O O O O O B-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Survey I-Survey I-Survey I-Survey I-Survey B-Citation I-Citation I-Citation I-Citation O O O O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Telescope O B-Telescope I-Telescope I-Telescope O O O O B-CelestialRegion I-CelestialRegion O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Observatory O O O B-CelestialObject I-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject I-CelestialObject 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission O O O B-CelestialObject O O O O O O B-CelestialObject I-CelestialObject O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O

identified toward these positions. Column densities of this species are presented in Table 3 . Figure 5. Detected transitions of CH 3 NHCHO in the spectrum toward position MM1-nmf. The observed spectrum is plotted in black, with the synthetic spectrum overplotted in blue and the rest-frequency center of the transition is indicated by the red dotted line. The upper state energy of each transition is given in the top-left corner. At a number of positions a handful of spectral features are identified that can be assigned to NH 2 C(O)NH 2 . The most and clearest lines are identified toward position MM1-v, see Figure 6 , with at least three unblended lines. While these lines are not enough for a secure detection, a tentative identification can be claimed. The tentative and upper-limit column densities of NH 2 C(O)NH 2 are presented in Table 3 . Figure 6. Overview of NH 2 C(O)NH 2 spectral features toward position MM1-v. The observed spectrum is plotted in black, with the synthetic spectrum overplotted in blue and the rest-frequency center of the transition is indicated by the red dotted line. The upper state energy of each transition is given in the top-left corner. 4. Discussion 4.1. Chemical Comparisons To get a better understanding of the chemical processes leading to amides, it is useful to compare molecular ratios of the positions within NGC 6334I, but also with other inter- and circumstellar sources. In the following subsections molecular ratios with respect to NH 2 CHO are discussed. Comparisons are made with the following works: IRAM 30 m and Green Bank Telescope (GBT) single-dish observations of G + 0.693-0.027 (hereafter G + 0.693), a quiescent giant molecular cloud (Zeng et al. 2018 ; Jiménez-Serra et al. 2020 ); ALMA observations of the incipient hot corino Barnard B1b-S (Marcelino et al. 2018 ); Northern Extended Millimeter Array observations of the low-mass protostars SVS13A, NGC 1333-IRAS4A2, NGC 1333-IRAS2A(1) (Coutens et al. 2018 ; Belloche et al. 2020 ); ALMA Protostellar Interferometric Line Survey (Jørgensen et al. 2016 ) observations of the low-mass protostar IRAS 16293B (Coutens et al. 2016 , 2018 ; Ligterink et al. 2018a ); GBT and IRAM 30 m single-dish observations of the Galactic center source Sgr B2 (Hollis et al. 2006 ; Belloche et al. 2013 ); ALMA observations of the HMSFR Orion KL (Cernicharo et al. 2016 ); the high-mass star AFGL 4176 (Bøgelund et al. 2019a ); and of Sgr B2 (Belloche et al. 2017 , 2019 ). One has to be aware that precise knowledge of the source size or emitting area of a certain molecule is essential for ratio comparisons. In particular, single-dish telescopes do not have the spatial resolution to determine the source size and these are often assumed or estimated, potentially resulting in large column density errors. Finally, mass spectrometric measurements of the Rosetta mission to comet 67P/C-G are used to compare with a solar system object (Goesmann et al. 2015 ; Altwegg et al. 2017 ). The


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Archive I-Archive I-Archive O O O O B-CelestialObject 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

core. Figure 12 shows that such a quiescent gas environment barely affects the line ratios, as the difference in the line ratios calculated at the turbulence 0 and 150 km s −1 is very small. Figure 12. Line intensity ratio relative to w vs. log of hydrogen column density. Solid: no turbulence. Dashed: turbulence 150 km s −1 . Dotted: turbulence 500 km s −1 . We also plot the line ratios at the turbulence of 500 km s −1 . Such a high value of the turbulence is not applicable in our region of interest, but may apply for gas in other environments such as merging galaxy clusters (Cassano Brunetti 2005 ). Optical depth can be expressed as a product of column density ( N ), and absorption cross section ( α ν ):where α ν is inversely proportional to the total Doppler velocity.where is the downward transition probability, is the oscillator strength and other symbols have their usual meaning (Mihalas 1970 ). If the temperature is held constant, an increase in the turbulent velocity decreases α ν , which decreases the overall optical depth. Ideally, to achieve the same optical depth at zero and higher turbulent velocity, a higher column density should be required for the latter case. However, this outcome gets reversed due to line interlocking of x, y, and z with other ions (e.g., Fe 23+ , Cr 22+ , and Fe 22+ ; see Paper I ). The higher the turbulent velocity is, the higher the Doppler width of a line and the more pronounced the line interlocking effects will be. Total line-center optical depth will therefore increase with turbulent velocity rather than decrease, as Equation ( 3 ) suggests. 6.4. Interplay between Column Density and Metallicity Figure 13 shows the line ratios relative to w as contour plots by varying the hydrogen column density and metallicity. In the low-column-density limit, the line ratios are minimally affected by the change in metallicity. Figure 6 shows the effects of metallicity on the line ratios at the reported value of hydrogen column density by Hitomi Collaboration et al. ( 2018b ) (). The uncertainties in temperature and collision strengths from the sources mentioned previously contribute to the uncertainties in the line ratios (shown with the solid enclosed lines in Figure 6 ). At the reported hydrogen column density, x, y, and z are optically thin (Paper I ). Therefore, the line ratios are minimally affected with change in metallicity as these lines are still optically thin for the metallicity range (0 ≤ Z (solar) ≤ 5) shown in the figure. Figure 13. Left, middle, and right panels show x/w, y/w, and z/w as a function of hydrogen column density and metallicity, respectively. Hitomi Collaboration et al. ( 2018a ) use deprojected heavy element abundances relative to solar in the inner 150 kpc region from Chandra data archive for a spherically symmetric Perseus model. At a distance of 30–60 kpc from the central AGN of the Perseus core, the reported range in metallicity is ∼0.5–0.75 of solar (see Figure 7 in their paper). We


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This work is based on observations made with the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 526555. These observations are associated with program #13364. R.C. acknowledges support from NSF grant 1517819. M.C. and M.T. acknowledge support from the INAF PRIN-SKA 2017 program 1.05.01.88.04. We thank Francesca Annibali for useful discussions. We also thank the referee for many useful comments that improved the paper. Facility: HST .


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We are grateful to Francisco Prada, Risa Wechsler, Chiaki Hikage, and Surhud More for useful discussions. We acknowledge Yu Lu and Andrew Benson for useful discussions related to SAMs. This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. Numerical computations were partly carried out on Cray XC30 at Center for Computational Astrophysics, National Astronomical Observatory of Japan. SS is supported by a Grant-in-Aid for Young Scientists (Start-up) from the Japan Society for the Promotion of Science (JSPS) (No. 25887012). The MultiDark Database used in this paper and the web application providing online access to it were constructed as part of the activities of the German Astrophysical Virtual Observatory as result of a collaboration between the Leibniz-Institute for Astrophysics Potsdam (AIP) and the Spanish MultiDark Consolider Project CSD2009-00064. The Bolshoi and MultiDark simulations were run on the NASA's Pleiades supercomputer at the NASA Ames Research Center. The MultiDark-Planck (MDPL) and the BigMD simulation suite have been performed in the Supermuc supercomputer at LRZ using time granted by PRACE. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/ . SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including 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.


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

We are indebted to Jimmy Irwin, Matthieu Roman, Jean-Baptiste Melin, Adam Mantz, Nabila Aghanim, Maxim Markevich, Erwin Lau and Daisuke Nagai for valuable comments. AM gratefully acknowledges the hospitality of the Harvard-Smithsonian Center for Astrophysics and of the NASA's Goddard Space Flight Center. We acknowledge support from Chandra grants GO2-13160A, GO2-13102A, GO4-15115X and NASA grant NNX14AI29G. This work was made possible in part by a grant of high-performance computing resources and technical support from the Alabama Supercomputer Authority. We thank the anonymous referee for the careful reading of the manuscript and suggestions, which improved the presentation of our work.


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This work was supported by INFN and ASI under ASI-INFN Agreement No.2014-037-R.0. M.S.P. and E.E.V. acknowledge partial financial support from the the South African National Research Foundation (NRF) under their Research Cooperation Programme. E.E.V. thanks the Space Science Division of the South African Space Agency (SANSA) for partial financial support during his PhD studies. R.M. acknowledges partial financial support from The Italian Space Agency (ASI) under the program “Programma PAMELA—attivita” scientifica di analisi dati in fase E.” The authors would like to thank Dr. M. Boezio for the useful discussion and helpful suggestions. Part of this work is based on archival data provided by the ASI Science Data Center (ASDC).


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

shells, with secondary typical scales as small as (0.5–1) × 10 9 cm. The slope of the VSH power spectra stays consistent over time regardless of the presence or absence of rotation. In all cases we find nearly circular shapes for the characteristic convective elements. In terms of the 2D convective flow properties, our results are in agreement with those of Meakin Arnett ( 2007 ; see also Meakin et al. 2015 ; Cristini et al. 2016 ). In particular, we find a highly intermittent flow and mixing due to turbulent entrainment. In terms of the effects of rotation on the strength and properties of convection alone we find our 2D simulations suggest minimal impact and generally lead to an overall small reduction of the total convective power stored in solenoidal motions (“no-rot” versus “rot-2” models) regardless of the nature of convective burning (C, O, or Si shell). We suggest this is due to the effects of the centrifugal forces mildly expanding the star, and thus changing the locations of the convective shells over time, subjecting the shell material to lower temperatures that, in turn, trigger lower nuclear burning rates and weaker convection. That alone may lead to initial velocity perturbations of smaller amplitude than in the case of zero rotation which may have a small impact on the susceptibility to a successful SN explosion following iron core collapse (Couch Ott 2013 ). We emphasize that, to answer the question of whether the inclusion of rotation significantly changes the ICs to the core-collapse SN mechanism, self-consistent evolution with the effects of rotation included is the proper approach to take. In our analysis we have followed the evolution of a 20 M ⊙ SN progenitor with both the effects of rotation and magnetic fields included in the transport of angular momentum and chemical mixing using the MESA code (“rot-ST” models). We find differences in the sign of the effect depending on the nature of convective shell-burning: during C- O-shell burning there is more power stored in the solenoidal components than in the case of no rotation while during Si-shell burning the effects are very small, with hints of even reduction of the solenoidal mode power by the end of the simulation. We attribute this effect to differences in the initial MESA models for the two distinct shell-burning stages. A careful look in the upper panel of Figure 2 shows that the “rot-ST” nuc profile during C- O-shell burning has a secondary peak (due to O-burning) that is nearly an order of magnitude greater than the corresponding one for the “no-rot” model. On the contrary, in Figure 3 we see that during Si-shell burning the peaks in the nuc profile for the “no-rot” model in the region (4–6) × 10 8 cm are greater than the corresponding ones for “rot-ST.” This is due to the enhanced chemical mixing by the ST mechanism during the C- O-shell burning phase that effectively recycles fresh fuel from outer layers to deeper and hotter regions enabling faster specific


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

drops (to 30%) in the lowest redshift bin. This decline may not be reliable for two reasons. First, the size of the star-forming sample at 0.5 z 1.0 is very small (only 10 objects), and therefore the sample proportions are not statistically significant. Second, the H -band images for the low- z objects are missing the bluer parts of the galaxy flux (see also Elmegreen Elmegreen 2014 ). Figure 9. Examples of galaxies with apparent spiral structures at different redshifts. The residual images, after removal of the bulge+disk model from the original galaxy image, allow for more effective detection of fine substructure. Table 5 Estimated Fraction of Massive Galaxies with a Prominent Stellar Disk Using Different Diagnostics Note. Column (1) Massive galaxies with bulge-to-total ratios less than 0.5. Column (2) Massive galaxies with visually detectable spiral structures. Column (3) Edge-on massive galaxies with e > 0.6. The conditions necessary for the formation of spiral arms are complex (see Dobbs Baba 2014 for a review), but one requirement is clear—the existence of a disk. Thus, from the point of view of one of the main themes of this paper, the clear detection of spiral features in high- z massive galaxies constitutes arguably the strongest, most model-independent evidence for the presence of a substantial disk component in these systems. We see the spiral features only in the star-forming galaxies and not in quiescent systems, but, by analogy with local S0 and spiral galaxies, this is not surprising. An edge-on view of a galaxy can reveal another indisputable signature of a prominent stellar disk. Figure 10 gives several examples of highly flattened ( e > 0.6) galaxies in our sample that are consistent with disk structures seen edge-on. Interestingly, most of them are relatively thick. The fraction of galaxies with a stellar disk can be inferred from the frequency of detected edge-on galaxies. For this estimation, we assume a uniform distribution of ellipticity with 0 e 0.8 for a population of bulge+disk systems. The inferred fraction of galaxies with a disk hovers around 40%–50% at 1.0 z 2.5, among both star-forming and quiescent galaxies, but the incidence of edge-on quiescent galaxies below z = 1 drops to zero (Table 5 ). Highly flattened systems (especially at high z ) can be hallmarks of a merger. However, considering the fact that the majority of these galaxies are compact, the chance of this degeneracy is low. Figure 10. Examples of massive galaxies with an edge-on disk at different redshifts. 5. Implications for Galaxy Evolution The discovery of red nuggets has captured much attention in recent years because it requires a new paradigm for the formation and evolution of massive elliptical galaxies. The observed compactness of red nuggets at z ≈ 2 initially raised the question of whether the massive red galaxies have indeed increased their sizes by a factor of roughly ∼3–5 while maintaining their passive state, or whether the size


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This material is based upon work supported by the National Science Foundation under grant No. AST-1109683. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation Based on observations (GS-2011A-Q-91) obtained at the Gemini Observatory, processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnologa e Innovacin Productiva (Argentina), and Ministrio da Cincia, Tecnologia e Inovao (Brazil). This paper uses observations made at the South African Astronomical Observatory (SAAO).


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

from stellar activity). The likelihood function is given by where i indexes the observations acquired with the I th instrument, is the nominal uncertainty of each RV measurement, and s I are the zero-point and extra noise parameters (also called jitter) of each instrument, and the Doppler signal from a companion on the star is encoded in the model , which is a function of time t and the Keplerian parameters . The Keplerian parameters of the p th companion in the system are: the orbital period P p (in days), the semi-amplitude K p (in m s −1 ), the mean anomaly at the reference epoch t 0 (in degrees), the eccentricity e p , and the argument of periastron . The second and third terms in Equation ( 3 ) account for the possible presence of a long-period candidate whose orbit is only detected as a trend (acceleration, ) plus some curvature (jerk, ). These two terms are especially important for the analysis that follows. When performing the Bayesian MCMC analysis, one needs to specify some prior distributions for these parameters. In this paper, we use uniform distributions for the angles and ω , as any value would be equally likely a priori. Given that the objects involved are rather massive and the signals are large, we also allow for a uniform eccentricity distribution between [0,0.95). For K p , , s I , , and , we assume unbound non-normalized uniform priors. While this can cause issues when normalizing the posterior, we are only using the MCMC analysis to sample the shape of the posterior, so precise values of the bounds and the normalization factors are unnecessary. Furthermore, because we will later try to constrain the signal with a period much longer than the span of the observations, the possible values of these three quantities will be correlated, so bound priors might eliminate many long period solutions that would otherwise be highly probable. For example, a large K in general requires a large γ unless the companion is precisely crossing the plane of the sky at t 0 (which is highly unlikely). Regarding the prior on the period, in this work we assume that the prior is uniform in (equivalent to uniform in frequency). This is motivated by the following. When analyzing time series, the local solutions to periodic signals are approximately equally spaced in frequency. For example, if one produces a Lomb–Scargle periodogram of a time series and plots period versus power, one will quickly appreciate that the peaks become much broader with increasing period (Scargle 1982 ). However, when making the same plot in frequency, the peaks appear uniformly distributed over the possible frequencies. As discussed in Tuomi and Anglada-Escudé ( 2013 ), in Bayesian statistics the choice of the parameter automatically imposes implicit priors on all other alternative parameterizations. In the case of the period, a uniform prior at very long periods can outweigh the information content on the likelihood, producing a biased result. While this issue


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-CelestialObject I-CelestialObject 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 B-Wavelength B-Tag O O O O O O O O O O O B-Wavelength O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O 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 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 B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength O O 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-CelestialObject I-CelestialObject O O O O O O O O O B-CelestialObject I-CelestialObject O O O B-CelestialObject I-CelestialObject O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


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

conditions requires equilibrium total pressure balance across each interface. This demands that the total (gas plus magnetic) pressure is constant:where μ is the permeability of free space. The sound speed in each region can be expressed as for . Here, γ is the adiabatic index, which is uniform across every domain, under the assumption that the plasma composition in the entire volume is the same. Another characteristic speed in the plasma, i.e., the Alfvén speed, is denoted by for . Equation ( 2 ), which describes the equilibrium pressure balance, yields the following relationship between the density ratios and characteristic speeds for any two regions:Disturbances within the slab, as well as in its magnetic environment, are governed by the ideal MHD equations: where , , p , and ρ are the velocity, magnetic field strength, pressure, and density, respectively. Equation ( 4 ) expresses mass conservation, Equations ( 5 ) and ( 6 ) are the momentum and induction equations, respectively, and Equation ( 7 ) is the energy equation for adiabatic processes. We linearize these equations about a static basic state and seek plane-wave solutions propagating in the direction of the formwhere k is the z component of the wavenumber vector defined as , and ω is the angular frequency. The linearized governing equations then reduce to a single ordinary differential equation for each region, which governs perturbations in that region, given bywhereare wavenumber coefficients andare the tube speeds for each domain for . The solutions to Equation ( 9 ) are a linear combination of the hyperbolic functions. Physically realistic solutions prescribe that the slab perturbations have no effect on plasma far away from the system; that is, the waves are evanescent outside the slab. This imposes the restriction on that as , , and so and . The general solution in this case takes the formHere, A , B , C, and D are arbitrary real constants. Physical solutions further require that two boundary conditions be fulfilled: the perturbations in velocity have to be continuous, and total pressure balance has to be upheld across the slab boundaries at . Applying these boundary conditions gives us four homogeneous algebraic equations in terms of the constants A , B , C , and D . Nontrivial solutions only exist when the determinant of the coefficients of this system of equations is zero. This gives us the dispersion relation aswhere By substituting for into Equation ( 13 ), we can express the dispersion relation in terms of wavenumbers, frequencies, and characteristic speeds as 2.1. Validation with Earlier Studies There are at least two earlier models that the current results should remain consistent with. If we set , thus , then the wavenumbers m 1 and m 2 revert back toFurthermore, the dispersion relation (Equation ( 16 )) loses the factors containing the external Alfvén speeds and simplifies towhich corroborates with the result for the magnetic slab embedded in a nonmagnetic asymmetric


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

Swift Observatory short GRBs suggests that of those with an X-ray afterglow, >30% have evidence for additional energy input at later times, most commonly in the form of a long-lived plateau, persisting at bright levels for hundreds to thousands of seconds (with the majority at the shorter end), or in the form of X-ray flares. Such properties are also common in long GRBs and are widely ascribed to additional energy injection. The origin of the prompt emission in GRB 170817A is difficult to explain with an off-axis-viewed typical short GRB (Kasliwal et al. 2017 ; Lamb Kobayashi 2018 ); however, some component of the observed GRB could be due to emission from the midpoint of a structured jet (Ioka Nakamura 2019 ). The required core energy and Lorentz factor are at the upper limits for the short GRB population but consistent with those from afterglow modeling assuming a Gaussian-structured jet (e.g., Gill Granot 2018 ; Lamb et al. 2019a ; Troja et al. 2019b ). For the refreshed shock models shown here, the top-hat jet structure, the initial on-axis isotropic equivalent kinetic energy ∼10 51 erg, and the separation of ι / θ j ∼ 3 for both models suggest 10 10 If we assume a typical γ -ray efficiency η γ ∼ 0.15, where , and following the method in Ioka Nakamura ( 2019 ) and including the opacity for high-energy photons as in Lamb Kobayashi ( 2016 ), we find an isotropic equivalent γ -ray energy for an observer at ι = 0.28(0.31) rad of erg and ∼2.5 × 10 38 erg for Models 1 and 2 respectively, with spectral peaks at E p ∼ 0.55 (0.23) keV, and well below the erg and E p ∼ 185 ± 62 keV for GRB 170817A. that the prompt emission of GRB 170817A is better explained as the shock breakout from a cocoon of material that is inflated by the successful, ultrarelativistic jet (Bromberg et al. 2018 ; Duffell et al. 2018 ; Gottlieb et al. 2018b ; Pozanenko et al. 2018 ). Then, GRB 170817A is more a “burst of γ -rays” than a “classical” GRB (Kasliwal et al. 2017 ). An interesting result of the posterior distribution from Model 1 for the Lorentz factor of the initial outflow is the preference of the model for “lower” values, see Figure 5 . Given the energy of the initial outflow, erg, any dissipation within an outflow with Γ 0,1 ≲ 25 is likely to be into an optically thick medium, resulting in a failed GRB (Dermer et al. 2000 ; Huang et al. 2002 ). This implies that GRB 170817A may be associated with a population of failed GRBs that would result in on-axis orphan afterglows for all but the nearest events where the burst of gamma rays from the shock breakout may be detected. Such a population could dominate the neutron star merger population; however, at cosmological distances the electromagnetic counterparts, even for an on-axis observer, would be difficult to detect (Lamb Kobayashi 2016 ). However, if we restricted the prior range for the initial Lorentz factor to Γ 0,1 ≳ 25, then higher Lorentz factor solutions to the fit exist, with some at Γ ≳ 100 (see the top


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

We thank Ralf-Dieter Scholz for constructive conversations during the preparation stage and the anonymous referee for indicating revisions that have improved our analysis and results. We also thank the Cambridge Astronomy Survey Unit (CASU) and the Wide Field Astronomy Unit (WFAU) in Edinburgh for providing calibrated data products under the support of the Science and Technology Facility Council (STFC) in the UK. M.R. acknowledges support from the German Academic Exchange Service. R.d.G. acknowledges research support from the National Natural Science Foundation of China (grant 11373010). This work was supported by STFC grant numbers ST/J001333/1, ST/M001008/1. 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.


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

would either require a larger sample of observed dwarf galaxies with smaller error bars and systematic errors of (a difficult prospect) or obtaining spatially resolved rotation curves for 400 pc (within the half-light radii) in small dwarfs, where Fig. 9 shows that CDM/WDM and SIDM/SIWDM could be differentiated. While the differences between DM models in Fig. 10 are marginal, the diversity in the observed population does appear greater than what we find in the suite of simulated dwarfs. Two field dwarfs, Cetus and IC 1613, have derived V 1/2 values that are lower than the 68 confidence region of the analytic fit. While the low V 1/2 values would be more consistent with a lower halo mass, the large sizes of these dwarfs are not expected for a lower halo mass given our simulations’ M halo – r 1/2 relation. Read et al. ( 2016b ) have noted that the gas morphology of IC 1613 implies a state of disequilibrium, which may explain why it is not consistent with our expected relation. A systematic difference in V circ (e.g. from the observed V circ underestimating the true V circ ; Verbeke et al. 2017 ; Oman et al. 2019 ) would explain the difference. We note, however, that we are plotting observationally determined V circ values that are derived from the galaxy kinematics, not from the underlying gravitational potential, for the simulations as well. Moreover, there are other FIRE -2 CDM simulations that agree fairly well with the two points with low V 1/2 values. Chan et al. ( 2018 )’s sample of ultra-diffuse dwarfs () occupy the same space as Cetus and IC 1613, with r 1/2 ∼ 2 kpc and V 1/2 ∼ 20 km s −1 , and the suite of ultra-faint dwarfs in Wheeler et al. (2019 ) contains a number of dwarfs that are directly in line with the analytic fits for haloes with . The two ‘compact’ dwarfs with high V 1/2 values and small half-light radii, Tucana and NGC 6822, are also difficult to explain with the results of our suite. Both reside much further outside the confidence region than the dwarfs with low V 1/2 values. Though our suite of simulations is not part of a Local-Group-like environment, FIRE -2 ΛCDM simulations that simulate LG analogues have found similar difficulty matching high V 1/2 values for isolated dwarf galaxies: Garrison-Kimmel et al. ( 2018 ) found no counterparts in the ELVIS on FIRE suite to observed ‘compact’ dwarfs with high V 1/2 values and small r 1/2 values (e.g.Tucana, NGC 6822, and even satellites such as NGC 205, NGC 147, and IC 10). It has been argued that Tucana has had a previous passage through the MW or M31 disc (Teyssier, Johnston Kuhlen 2012 ), which could invalidate the assumptions of Wolf et al. ( 2010 ) used to derive its V 1/2 . Finally, it is worth noting that the role of ‘chaos’ – more precisely, sensitivity of high-level results to very small numerical changes – could be significant when studying the combined effects of dark matter physics and hydrodynamics. This issue has not been studied extensively, but recent work by


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all but one observation the exposure time was particularly low (i.e. 0.5–1 ks). We estimated the Bayesian 95 per cent upper limit (Kraft et al. ( 1991 ) to the source count rate from the observation with the longer exposure (4.2 ks, n. 4, obs ID 00087498001). It was equal to 0.0025 count s −1 . Using WebPIMMS, it converts into an observed (unabsorbed) 0.3–10 keV flux of ∼4 × 10 −13 erg cm −2 s −1 (∼7.6 × 10 −13 erg cm −2 s −1 ). We assumed the same spectral model as from the Swift / XRT observation n. 3. 3.4.3 Investigation at lower frequencies with archival data Halpern et al. ( 2007 ), using the precise Chandra localization of XTE J1829–098, individuated its probable infrared counterpart through images in the K and H bands as obtained with the MDM 2.4 m telescope on 2005 June 17. The measured magnitudes were H ∼ 13.9 and K ∼ 12.7, no measurement was obtained in the J band. Back in 2005, no additional infrared measurements were available from other infrared surveys, e.g. the UKIDSS survey started on 2005. Currently, we note that the infrared counterpart proposed by Halpern et al. ( 2007 ) has been also detected during the UKIDSS GPS data release 6 with magnitudes J = 16.147 ± 0.009, H = 14.064 ± 0.003, and K = 12.591 ± 0.002. If we use the reddening-free NIR diagnostic Q (Negueruela Schurch 2007 ) with the extinction law of Fitzpatrick ( 1999 ), we note that it has a Q value of –0.7 which is very typical of early-type OB stars. In particular, the negative value strongly indicates an infrared excess. Adopting the UKIDSS infrared magnitudes of such proposed counterpart, we can obtain a rough idea about its distance: if we assume a supergiant B0.5Ib star (with M V = −6.8 mag, V − K = −0.7 and V − J = −0.54), the source should be located at a distance d ∼ 18 kpc (with A V ∼ 21 mag). This implies a V magnitude fainter than V ∼ 30–31 mag. A main-sequence B0 star (assuming M V = −4 mag, V − K = −0.9 and V − J = −0.7), would imply a distance d ∼ 4.5 kpc (with A V ∼ 22 mag), while a giant B0 star (assuming M V = −5 mag, V − K = −0.8 and V − J = −0.7), would be located at about 8 kpc. 3.5 Search for periodicities We performed a timing analysis using the Lomb–Scargle method (Lomb 1976 ; Scargle 1982 ) of the long term INTEGRAL -IBIS and Swift -BAT light curves of the sources under study, in order to search for periodicities which could eventually be ascribed to their orbital periods. For all four sources, the long term IBIS/ISGRI ScW light curves (18–60 keV) cover the observational period from 2003 March to 2015 January. We applied standard optimum filtering in order to exclude poor quality data points which could eventually disrupt the periodic signal (see Goossens et al. 2013 and Sguera et al. 2007 for details). Periodicities were searched for in the range 1–100 d, but none were found. The public access Swift /BAT light curves (15–50 keV) on daily time-scale of all sources but IGR J12341–6143 were also searched for periodic signals in the range 2–250