on pauses in X-ray bursts studied with RXTE , suggest that there is no link between the detection of double-peaked profiles and the detection of a pause during the rise. However, our interpretation would be incorrect if the pauses are short and weak manifestations of a peak in the X-ray burst profile, meaning that the X-ray burst in SAX J1808.4–3658 can be considered a triple-peaked X-ray burst, and B1 in MAXI J1807 can be considered a double-peaked burst. in’t Zand et al. ( 2003 ) reported on an unusual Type-I X-ray burst from GRS 1747-312 2 observed with RXTE whilst the system was apparently in quiescence. The 2–10 keV light curve profile exhibits a pause-like feature in the rise of the X-ray burst, while the full 2–60 keV light curve did not show a pause. The pause in the 2–10 keV light curve is considerably longer than those seen in S1 and B1 ( ∼20 s), and, judging from Fig. 2 a of in’t Zand et al. ( 2003 ), the pause occurs at roughly 70 per cent of the peak count rate. The X-ray burst also had an unusually strong PRE that was not at the start of the X-ray burst, as is expected. It is unclear whether this is a longer manifestation of the pause phenomena we see in B1 and S1, but it shows that the presence of pause-like features can depend on the energy band used. Independently of whether the pauses are related to the multipeaked phenomena discussed above, our results, together with those presented by Bult et al. ( 2019 ), show that short pauses in the rise of X-ray burst light curves occur in at least two sources, suggesting that these types of pauses might be present in the X-ray light curves of Type-I bursts from other NS-LMXBs. We only know of two sources (the accreting millisecond X-ray pulsar SAX J1808.4–3658 and MAXI J1807), each with one reported X-ray burst with a clear pause during the rise. This small sample size prevents us from a detailed study on the relation between pauses and X-ray burst characteristics/properties, and naturally it is unclear whether the pauses seen in B1 and S1 are caused by the same mechanism. However, under the assumption that we are seeing manifestations of the same phenomena in B1 and S1, even with a sample of two we can conclude that pauses in the X-ray bursts are possible in both pulsating and non-pulsating systems, suggesting that the strength of the magnetic field might not play an important role, unless the lack of pulsations in MAXI J1807 is due to an alignment between rotation and magnetic axes (Lamb et al. 2009a , b ). In addition, the occurrence of the pause does not appear to depend on the composition of the burning material either, or whether the X-ray burst reaches PRE. To our knowledge, standard X-ray burst models do not predict pauses in the X-ray burst rise. However, we note that Keek, Heger in’t Zand ( 2012 ) finds X-ray bursts with pauses in their rise when modelling X-ray bursts that occur shortly after a superburst. In the hot, He-rich atmospheres following a superburst, X-ray bursts

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


3.3 . 3.3 Expectation–maximization algorithm 3.3.1 Overview The measurements, , can be used to solve for τ c and τ s in , N c and N s in , and σ c and σ s in in equations ( 16 )–( 22 ) using any number of maximum-likelihood techniques. In this paper, we primarily discuss the expectation–maximization algorithm (Dempster et al. 1977 ; Shumway Stoffer 1982 ), which naturally extends to situations where observations are missing or Δ t is not constant (Shumway Stoffer 1982 ; Digalakis et al. 1993 ; Gibson Ninness 2005 ). The method cleanly separates process noise in the physical variables, characterized by , and measurement noise , characterized by . Two other common techniques include using Newton’s method to maximize the log-likelihood, log L , and the score vector method (Durbin Koopman 2012 ). The former method relies on numerical estimates of the inverse of the Hessian matrix, while the latter method calculates the gradient of log L with respect to the parameters and steps towards a maximum of log L (Durbin Koopman 2012 ). In this paper, we estimate the elements of , , and , as opposed to τ c , τ s , N c , N s , σ c , and σ s . It is possible to extend this method to directly estimating the parameters in equations ( 1 ) and ( 2 ) using ‘grey-box estimation’ (Ljung 1986 , 2013 ; Kristensen, Madsen Jørgensen 2004 ). 3.3.2 Algorithm and its logical basis The expectation–maximization algorithm is used to find stationary points of the posterior distribution in the absence of a direct observation of the states, . As in the previous section, we use as compact notation for the elements of , , and . We use the product rule to separate the posterior in terms of the states The first term on the right-hand side of equation ( 27 ) is a multivariate Gaussian with a straightforward interpretation, which we discuss in Appendix B . We will use this term to find stationary points of . To start, we treat as a random variable and take an expectation value of both sides, assuming some fixed value for the parameters, , leaving with The second term on the right-hand side of equation ( 28 ) is maximized at (Dempster et al. 1977 ; Gelman et al. 2013 ), while the first term on the right-hand side can be maximized analytically by a new choice of , which increases the marginal posterior distribution on the left-hand side of equation ( 28 ). This lends itself to the following iterative algorithm (where p now defines how many times we have iterated): Start with a guess of . Calculate (‘E-step’) using current estimate of the parameters . Maximize from step (ii) over (‘M-step’). Set to the maximum. Repeat steps (ii) and (iii) for a fixed number of iterations or until converges. In Appendix B , we discuss the application of this method to our specific case. We first present the full form of . We then discuss the E-step and M-step, along with the implementation used in carrying out the simulations discussed in Section 4 . In Appendix C , we show the detailed formulae used

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and disregarded signals below this cutoff as manifestations of polarization bias (see e.g. George, Stil Keller 2012 ). The FD and p values of our target sources were obtained by fitting a second-order polynomial to the highest peak in of each source (e.g. Heald et al. 2009 ; Mao et al. 2010 ; Betti et al. 2019 ). Specifically, we fitted to the seven data points nearest to the highest peak, with the FD uncertainty calculated by (e.g. Mao et al. 2010 ; Iacobelli, Haverkorn Katgert 2013 ). We did not correct for the Ricean polarization bias since we do not expect it to have significant effects for our case with S/N > 6 (Wardle Kronberg 1974 ; George et al. 2012 ). All these results from our new VLA observations (FD new and p new ), along with their counterparts in the original NVSS catalogue ( p NVSS ), the Taylor et al. ( 2009 ) catalogue (FD TSS09 and p TSS09 ), and Van Eck et al. ( 2011 ) observations (FD VE11 ) are listed in Tables 1 and 2 for on- and off-axis targets, respectively. We found that three on-axis and seven off-axis sources are unpolarized in our new observations (i.e. p new below our 6σ cut-off). The 1σ values in p new of these sources are reported within parentheses in the two tables. In total, we derived the FD values of 194 polarized EGSs, with 168 and 26 on- and off-axis targets, respectively. The sky distribution of our EGS FD is plotted in Fig. 1 (top panel). Figure 1. EGS FD values from (top) our new VLA observations and (bottom) Van Eck et al. ( 2011 ) observations, both plotted as colour dots. The background grey-scale map represents the WHAMSS Hα map (Haffner et al. 2003 , 2010 ). The typical uncertainty of our new FD is about , while that of the Van Eck et al. ( 2011 ) is about . The orange circle in the top panel outlines a prominent H ii structure, G26.5, and EGSs within this circle were not considered in our analysis (see Section 5.2 ). Table 1. RM-Synthesis results for on-axis targets. Notes: FD new and p new from this work; FD VE11 from Van Eck et al. ( 2011 ); FD TSS09 and p TSS09 from Taylor et al. ( 2009 ); and p NVSS from Condon et al. ( 1998 ). ⊙ Situated behind the prominent H ii structure G26.5. † Suffers from n π-ambiguity in Taylor et al. ( 2009 ). Table 2. RM-Synthesis results for off-axis targets. Notes: None of these sources are listed in Taylor et al. ( 2009 ) or Van Eck et al. ( 2011 ). FD new and p new from this work; and p NVSS from Condon et al. ( 1998 ). ⊙ Situated behind the prominent H ii structure G26.5. In order to assess the residual leakage signal level present in our data, we formed Faraday spectra of our unpolarized on-axis leakage calibrator (J1407+2827) using data from each of the seven observing blocks. We find that the highest peak out of the seven Faraday spectra is at a value of p = 0.024 ± 0.007 per cent, much lower than that of all of our polarized target sources. Note that the weak polarization signal we see from J1407+2827 here is likely due to polarization bias stemming

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of the maser emission, the spots could be bright at different times, and thus we consider the fit with nonzero positional offset to be more robust. Second, we performed an astrometric fit to the mean position of the shock front and derive a trigonometric parallax of π = 6.6 ± 0.6 mas, corresponding to a distance of pc. We show the parallax sinusoid from this fit in Fig. 6 , and give the resulting parameters in Table 5 . The values of the parallax derived from the fits to the three main spots are in good agreement with that obtained from the fit to the R.A. motion of the shock front. Table 5 Parameters obtained from the astrometric fits. The systematic errors that were added quadratically are also presented. The correlation coefficient matrices for all fits are shown in Appendix A . From these matrices we noticed a high (anti-) correlation between the parallax and the other fitted parameters in all the fits, as is expected given the short timescale covered by the observations. To check the reliability of our results, in view of these high correlations, we performed a series of Monte Carlo simulations which are presented in Appendix B . The result from these simulations is that, given the statistical and systematic errors in our measurements, the true astrometric parameters can be recovered from our fits even when the observations cover a period of only 3–7 months. The worst case is the individual fit to spot 1; this spot was only detected in four epochs and, consequently, the uncertainties on the fitted parameters are significantly larger. 4 Discussion 4.1 Distance to I16293 As we mentioned in the introduction, Imai et al. (2007) obtained a distance of 178 pc to I16293 by measuring the trigonometric parallax of water masers associated with this YSO. Imai et al. (2007) mentioned that the most severe factor causing their modest astrometric accuracy was the temporal variation in the brightness structure of the maser spots they detected. If water masers have short lifetimes of a few months, as is the case with water masers in I16293, it is possible to misidentify maser spots even from one epoch to the next. Taking into account this possibility, these authors performed several fits by using different combinations of observed positions. By excluding the oddest positions, they obtained an upper limit to the trigonometric parallax of 7.1 mas (corresponding to a lower limit in the distance of d = 141 pc). In the following, we discuss the differences between our analysis and that of Imai et al. (2007) . Fig. 6 Parallax fit to the mean positions of the shock front. Top: parallax sinusoid from the best fit to these data (blue line). The red squares are the mean positions of the shock front. In both cases, we have subtracted the proper motion and the RA position at JD 2453731.31 (the mean epoch from our observations) from the best fit. Bottom : residuals in right ascension. The first difference comes from their criterion for the selection of maser

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


The effect is best understood by examining Fig. 6 . The particular explosion energy for which incomplete silicon burning mostly ceases inside the oxygen-burning shell is shown by the dashed line at 5.75 B in Fig. 6 ( top ). For explosion energy 1 B − 2 B below this value, incomplete silicon burning occurs extensively against the inner edge of the boundary, and for greater explosion energy, incomplete silicon burning does not occur at all within the oxygen-burning shell. Examining the peak temperature profile in Fig. 6 ( bottom ), we see a 'bump' of increased temperature at the inner edge of the shell boundary (location shown by the dashed line), indicating that material in this region is subject to some source of extra heating. We understand the cause of the 'bump' in the temperature profile against the outer edge of the oxygen-burning shell as follows: as the initial supernova shock travels through the composition boundary, a reverse shock is launched as a consequence of the change in the sign of the gradient in ρ r 3 at this point. The quantity ρ r 3 is important for the shock velocity in supernovae. As is described by Kompaneets ( 1960 ); Klimishin Gnatyk ( 1982 ); Bethe ( 1990 ); and Matzner McKee ( 1999 ), the shock velocity, U , can be approximated by Here β is parameter that varies with the properties of the material that the shock is propagating through, typically, 0.2 ≲ β ≲ 0.5. Nevertheless, for the our purposes the important point is that the shock velocity is inversely proportional to ρ r 3 , so when ρ r 3 is increasing, the shock must decelerate, and when ρ r 3 is decreasing, the shock accelerates. The inner ρ r 3 profile is shown for the 15 M ⊙ model in Fig. 7(a) . We see that there is a reversal in the sign of the gradient of the ρ r 3 profile at the oxygen-burning/carbon-burning shell interface. Such a rapid change in ρ r 3 , from a negative to positive gradient, leads to a rapid slowing of the shock velocity. This abrupt deceleration results in an increased density at this point, as interior matter piles up against the slowed mass. This has a chain effect on the matter behind it, and so on, which propagates as an ingoing shock. Figure 6. In the upper and middle plots, we show chromium and iron mass fractions, respectively, as a function of mass coordinate and explosion energy for the 15 M ⊙ model, as in Fig. 1 . The lower plot shows the peak temperature profiles for four models with explosion energy at which explosive silicon burning is affected by the reverse shock. Figure 7. The ρ r 3 profile (black line) and electron mole fraction, Y e (grey line) for the inner 10 M ⊙ of each of the 15 M ⊙ –80 M ⊙ progenitor models. The piston location is indicated by the dashed line. Note, Y , where η is the neutron excess. The reverse shock provides additional heating to the material immediately interior to the shell boundary, as can be seen in the temperature profile at various time intervals (Fig. 8 ). This results in an approximately

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


values of according toEq. ( 8 ). We chose six values for the scaling size of the core, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9. All lines cross at point A which is the bulk density of a homogeneous Phobos. For core densities lower than the bulk density (left of point A shown in Fig. 3 ), the mantle is denser than the core, and vice versa, for core densities higher than the bulk density (right of point A), the mantle is less dense than the core. Figures 4 a and 4 b show the dependence of the mantle density on the size of the core for selected core densities. In Fig. 4 , only physical values (from 500 kg m −3 to 5000 kg m −3 ) for the densities of the mantle and the core are listed. For a given scaling factor of the core λ and the density of the core ρ c , 46 layering cases are considered in this paper. We removed the cases exceeding the allowable density range. Fig. 3 Range of densities for the Phobos two-layer model. The x -axis is the assumed density of the core, the y-axis is the resulting density of the mantle according to Eq. ( 8 ). Plots in Figs. 5 and 6 show that at the 3 σ level, the modeled and observed C 20 agree well when the mantle is denser than the core (see also Tables 3 and 4 ). To verify the effect of the core shape on the gravity field of the two-layer Phobos model, we used a triaxial ellipsoid core shape (homotheticto the best-fit triaxial ellipsoid of Phobos ( Willner et al. 2014 ), and recalculated the second-degree coefficients in a setting identical to the one used for Figs. 5 and 6 . The gravity potential of a homogeneous ellipsoid canbe computed using Ivory’s formula ( Garmier et al. 2002 ). Compared to the minimum absolute value of from Yang et al. (2019) , when the mantle is denser than the core, the cases listed in Table 3 (homothetic surface shapes) and Table 5 (homothetic ellipsoids) coincide. A test simulation for a core with a scaling factor of 0.6 and a density of 1400 kg m −3 produced a very small deviation of 0.097%, therefore, the deviation is not sensitive to the core shape, if the interface core-mantle is deeply buried. Only one test simulation is compatible with the data when the core is denser than the mantle. In this case, the core had a scaling factor of 0.1 and a density of 2200 kg m −3 , and the corresponding deviation was 0.002%.These results are summarized in Tables 5 and 6 . The disrupted and reaccreted body (DAR-F1) model (a rubble-pile internal structure), with a denser outer layer considered by Le Maistre et al. (2019) , is close to our two-layer model for a core with a scaling factor of 0.9 (the interface depth is about 1 km). Fig. 4 Mantle (a) and core (b) densities as a function of the core scaling factor of Phobos’ two-layer models. ( ρ c is the density of the core, λ is the scalingfactor of the core, and ρ m is the density of the mantle, units: kg m −3 ). All small bodies (radius smaller than 20 km) with known second order and degree gravity models ( Miller et al. 2000 ; Zuber et al. 2000 ;

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


of SFRs and molecular gas fraction. The initial density profile of the dark matter halo in a disc galaxy is modelled using the density distribution of the NFW halo (Navarro, Frenk White 1996 ) as suggested from cold dark matter (CDM) simulations. The c -parameter ( c r vir r s , where r vir and r s are the virial radii of a dark matter halo and the scale length, respectively) and r vir are chosen appropriately by using recent predictions from cosmological simulations (Neto, Gao Bett 2007 ). We adopt the Hernquist density profile for the bulge of a disc galaxy and the bulge mass fraction ( M b M s ) is a free parameter ranging from 0 to 1. The size ( R b ) and the mass of the bulge have the following scaling relation that can reproduce the properties of the Galactic bulge: kpc. The radial ( R ) and vertical ( Z ) density profiles of the stellar disc are assumed to be proportional to exp( R R 0 ) with a scale length R 0 0.2 R s and to sech 2 ( Z Z 0 ) with a scale length Z 0 0.04 R s , respectively. The gas disc with a size of R g has the radial and vertical scale lengths of 0.2 R g and 0.02 R g , respectively. Since we investigate disc models with M h similar to that of the Galaxy (i.e. M h 10 12 M ), the exponential disc has R s 17.5kpc for all models in the present model. The initial Toomres parameter Q is set to be 1.5 for gas and stars and the vertical velocity dispersion at a given radius is set to be 0.5times as large as the radial velocity dispersion at that point so that the dynamical equilibrium in the vertical direction can be achieved. A gas particle is assumed to be converted into a new star if the local H 2 density (, where and g are the H 2 mass function and gas density, respectively) exceeds a threshold density ( th ) and we investigate the models with th 10, 30 and 100cm 3 . These are H 2 -dependent SF models, and we also investigate H-dependent one in which g should be higher than th . We adopt these values because our previous numerical simulations of galaxy evolution with dust demonstrated that the models with the above values can explain the observed SFR g relation (see fig.3 in Yozin Bekki 2014 for the case of the Magellanic Clouds). We assume that SFR ( sf 1.5) in this study. Chemical enrichment processes, dust formation and evolution in ISM and SN feedback effects are included in the same way as B13 and B15 . We investigate disc models with corotating (referred to as standard rotation and labelled as S for convenience) and counter-rotating (C) gas discs for a given set of disc parameters. By comparing between the two models, we try to understand how counter-rotating gas can influence dynamics and star formation histories of disc galaxies. We show the results for only 24 models in this study and Table 1 describes the model parameters for each of the SC pairs. The total number of particles used for a simulation is 1033400 for a fiducial model (M1 and M2) and different initial gravitational softening lengths () are

O O O O 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 I-Citation I-Citation O O O O B-Model I-Model I-Model 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 B-Citation I-Citation I-Citation I-Citation 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 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 B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula 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 B-CelestialObject O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Model I-Model I-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 B-Citation I-Citation I-Citation 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 B-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O


galaxies, not the lower z ones in our low-mass sample (see XXL paper XXVII). We label these two as BG? in the table. Removing all seven likely or possible contaminants, 8 all of which have s > 5 arcsec, then leaves us with 20 likely genuine matched low-mass emission-line objects (though alternate matches may still exist for a few of them, cf. XXL paper XXVII). The X-ray sources matched to these 20 emission-line galaxies lie in the range L Xs = 10 39.2 to 10 41.1 erg s −1 . The highest X-ray luminosities appear to (just) fit in the expected range, which cuts off at ∼10 41 erg s −1 (e.g. Mineo et al. 2012 ) if the extension of HMXBs into ULXs is included. In total, then, a reasonable number, ≃20, likely XRBs have been found, probably mostly HMXBs in low-mass star-forming galaxies. For these, there is no correlation between X-ray luminosity and emission-line luminosity, which can easily be explained if the X-rays are from individual single HMXBs while the line emission reflects the ISM conditions and star formation in the galaxy. It is difficult to relate our present results to those in the literature on the prevalence of HMXBs in brighter/more massive star-forming galaxies, for a number of reasons. In particular, we are able to detect only the brightest X-ray sources and have a flux limited rather than volume limited sample. Our nearest and faintest source is at 10 39.2 erg s −1 while at the redshift of our most distant matched low-mass galaxy ( z ≃ 0.1) the detection limit is approaching 10 41 erg s −1 . This can be contrasted with, for example, the general study of local star-forming galaxies by Mineo et al. ( 2012 ) who found about one HMXB for every 0.3 yr −1 of star formation, but with a detection limit 10 38 erg s −1 , considerably below our accessible limits. Using a reasonable slope of the HMXB luminosity function (e.g. Grimm et al. 2003 ), we would need to correct the numbers detected in even nearby redshift bins (say at z ≃ 0.03, where the effective limit is around 10 39.5 erg s −1 ) by an order of magnitude, and in our more distant bins we are in the tail of the LF, near the upper cut-off, requiring very large (around two orders of magnitude) and very uncertain corrections. The best we can probably say is that, given our 20 actual detections, we would expect perhaps of order 500 detections if we had a similar X-ray limit to Mineo et al. In addition, the intercomparison of SFRs is problematic: translating H α luminosities to total SFRs (e.g. Kennicutt 1994 ) involves numerous uncertain steps (e.g. use of a particular initial mass function) and Mineo et al. ( 2012 ) use a completely different method based on infrared and ultraviolet luminosities. If, for a concrete example, we simply translate H α luminosity to an ‘indicative’ SFR via Kennicutt ( 1983 )’s relation SFR ≃ L H α /(10 41 erg s −1 ) yr −1 , ignoring any systematics in this translation, 9 then the total indicative SFR summed over our 21 low-mass host galaxies (we exclude

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in general (see e.g. Viti et al. 2001 ; and Hollenbach et al. 2009 , for a discussion of this specific aspect). Meaningful constraints on the oxygen chemistry could then be obtained through the observation of HO and O. Unfortunately, no velocity-resolved observations of water have been performed in Cep E by Herschel , and no instrument can allow them at the moment. O is best observed from space observatories (the transition at 487.249 GHz is close to an atmospheric absorption feature), but it has not been observed in Cep E, and its symmetry makes its detection difficult from the outset. Another possibility is that the shock that we observed is illuminated by the protostar that drives the outflow. In such irradiated shocks (e.g. Lesaffre et al. 2013 ; Melnick Kaufman 2015 ), both HO and CO can be photo-dissociated, increasing the gas-phase abundance of O i . This effect is not taken into account in the Paris-Durham code. However, the possibility that shocks at position BII could be affected by the distant protostar is relatively low: the BII and mm position are pc away. The Cep E-mm protostar has a luminosity of about 100 , and a T-Tauri star typically emits between 10 and 10 of its total luminosity in the UV range between 3200 and 5200 Å ( Gullbring et al. 1998 ). Conservatively assuming that Cep E-mm is radiating 10% of its total luminosity in this UV range and taking dilution into account, we found that the radiation field in BII is about 4.8 in units of Draine’s ( Draine 1978 ) or Mathis’s ( Mathis et al. 1983 ) interstellar radiation field (ISRF). This value has then to be corrected for the absorption by dust grains. We assumed an average local density in the outflow cavity of 10 cm, which translates into a column density of cm and into a visual extinction of mag ( Lefloch et al. 1996 provided local estimates for this value: 61 mag around the driving protostar, and 3.2–3.4 mag in the H knots along the flow). Using the exponential attenuation approximation , where , and is the effective attenuation cross section from Draine Bertoldi (1996) , we found that the radiation field really felt in BII is about in Draine’s or Mathis’s ISRF units. Another explanation is that several shock structures are present in our beam. In this case, the missing O i emission could be accounted for by the presence of another shock structure. This additional shock layer must present different physical characteristics, as simply combining shock layers similar to those we used would simultaneously increase the [OI] emission insufficiently and increase the CO emission too much, producing another type of discrepancy between observations and models. One such shock structure could be a stationary dissociative -type shock. In particular, a -type shock with cm, magnetic field parameter , and shock velocity km s ( Flower Pineau des Forêts 2015 ) could be combined with our current best-fit solution, providing a satisfying fit for the O i and CO emission. The additional

O O O O B-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 O O O O O O B-CelestialObject I-CelestialObject O B-Telescope 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 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 B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion 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 B-Wavelength O O B-Wavelength O B-Wavelength I-Wavelength O B-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O B-Wavelength O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O B-Citation O B-Citation I-Citation O O B-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 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 B-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObjectRegion O O O B-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 O O O O O O O O O O O O O O O O O O O O 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


internal energy per unit mass, g is the gravitational acceleration of the black hole located at ( r , θ) = ( z 0 , π), and f is the radiation force per unit mass (see Section 2.2 for detail). An adiabatic equation of state p /ρ = (γ − 1) e with γ = 5/3 is assumed in our calculations. In the last term of equation ( 5 ), is the approximate formula of the net heating rate introduced by Blondin ( 1994 ) where, n is the number density, G Compton is the heating rate via the Compton scattering G X is the difference between the heating rate by the X-ray photoionization and the cooling rate via the recombination and L b,l is the cooling rate by the bremsstrahlung and line emission In equation ( 9 ), the first term represents the effect of the bremsstrahlung and the last two terms show the line cooling in an optically thin atmosphere. The temperature T is obtained by T = μ m p p/k B ρ, where μ (=0.5) is the mean molecular weight, m p is the proton mass, and k B is the Boltzmann constant. The ionization parameter, ξ, is defined as where F X is the X-ray flux. In this study, the X-ray is assumed to come from the point source at the centre with luminosity L X = f X L D , where f X is a parameter. We also assume the temperature of the X-ray radiation to be . The X-ray flux is attenuated via the absorption of which the cross-section is set to be σ X = σ e for ξ > 10 5 , or 100σ e for ξ ≤ 10 5 , where σ e is the mass-scattering coefficient for free electrons. Such a treatment is employed in previous works (Proga et al. 2000 ; Nomura et al. 2016 , N17 ). We note that the scattered and reprocessed photons are ignored in our simulations (see also Section 4.2 ). 2.2 Line force The radiation force in equations ( 2 ) and ( 3 ) is described as where F D is the radiation flux of the disc emission integrated by the wavelength throughout the entire range, and F line is the line-driving flux (UV flux), which is the same as F D but integrated across the UV transition band of 200– Å. Both fluxes are calculated by integrating the radiation from the inner hot region of the disc, where the effective temperature is larger than . We divide the disc surface into 4096 grids for radial direction and 4096 grids for azimuthal direction, and numerically calculate and . The radial components of these fluxes are attenuated due to electron scattering, of which the optical depth is measured from the origin of the coordinate. On the other hand, θ-components of and are supposed not to be diluted. The force multiplier, M , is defined by Stevens Kallman ( 1990 ) Here, t is the local optical depth parameter written as Here, the thermal speed, v th , is set to that corresponds to the thermal speed of the hydrogen gas of the temperature of . The velocity gradient, d v /d s , depends on the direction of each light-ray from the disc surface to the point of interest. However, in order to reduce the computational cost, we evaluate d v /d s along the direction of the disc line-driving flux ( F

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


−3 are the critical densities of the S(0):2–0 and S(1):3–1 rotational lines. r op is the ortho-H 2 /para-H 2 ratio (set to 3 which is the equilibrium value). For the cooling rates due to the [C ii ]158 μm and [O i ]63 μm+146 μm fine-structure lines we adopt the parametrizations by Röllig et al. ( 2006 ). The C ii cooling rate () is where a carbon to hydrogen abundance ratio that scales with metallicity according to χ [C] = 1.4 × 10 −4 Z ′ is assumed. For the parametrization of the O i cooling rate () we refer to equations (A.5) and (A.6) in Röllig et al. ( 2006 ) and simply note that we adopt (in accordance with Röllig et al. 2006 ) an oxygen to hydrogen abundance ratio of χ [O] = 3 × 10 −4 Z ′. Λ CO is the cooling rate due to CO rotational transitions. We use the parametrization provided by Papadopoulos Thi ( 2013 ): where χ CO /χ [C] is the relative CO to neutral carbon abundance ratio, the value of which we determine by interpolation, assuming that χ CO /χ [C] = (0.97, 0.98, 0.99, 1.0) for cm −3 , respectively (Papadopoulos Thi 2013 ). Λ gas–dust is the cooling rate due to gas–dust interactions and is given by (Papadopoulos et al. 2011 ) where the dust temperature ( T dust ) is calculated using equation ( 15 ) (Section 3.4.3 ). APPENDIX C: GMC MODELS Each SPH particle is divided into several GMCs as described in Section 3.4.1 , and we derive the molecular gas density and temperature within each from three basic parameters which are SFR density, GMC mass, m GMC , and metallicity, Z ′/ Z ⊙ . Derived from these basic parameters are the far-UV and CR field strengths, the H 2 gas mass fraction of each SPH particles, as well as the GMC properties used to derive the CO excitation and emission; H 2 density and temperature. Histograms of the basic parameters are shown in Fig. C1 , while properties derived thereof can be found in Fig. C2 . Figure C1. Mass-weighted histograms of the basic parameters of the GMCs in G1 (red solid), G2 (green dashed), and G3 (blue dotted). From top left and clockwise: the local far-UV field () (and CR ionization rate since ), GMC mass ( m GMC ), metallicity ( Z ′) and external pressure ( P ext ). Black vertical lines indicate the , m GMC , Z ′, P ext -values for which curves were calculated (see Fig. C2 ) – a total of 630 GMCs which make up our grid GMC models. Each GMC in the galaxies is assigned the curve of the GMC model at the closest grid point. Figure C2. Kinetic temperature versus H 2 density curves for 80 out of the 630 grid model GMCs that span the full (, M GMC , Z ′) parameter space set by and marked on top of the distributions in Fig. C1 (see also Section 3.4.4 ) for a pressure of K cm −3 . The grid model most often assigned to GMCs in G1 is indicated by the red dashed curve in the highlighted panel and corresponds to ( s −1 ), , and Z ′ = 3.2. In general, higher metallicity (from top to bottom) leads to more cooling via emission lines of ions, atoms and molecules, and hence lower temperatures. On the other

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


redshift of the drag epoch, z d , can be calculated aswhere . Note that z d is well approximated analytically (Eisenstein Hu 1998 ) aswhere We use a fixed value of the baryon density from the Planck result (Planck Collaboration et al. 2016 ). 6 6 We have tested to marginalize over with a Gaussian prior derived from the Planck measurement and find that the result is largely unchanged. The baryon density can also be accurately determined in a CMB-independent way, e.g., using the primordial deuterium abundance in the Big Bang Nucleosynthesis (BBN) theory (Riemer-Sørensen Sem Jenssen 2017 ). Note that the quantities and can be estimated from anisotropic BAO measurements, while the quantity,is determined by isotropic BAO measurements. As shown above, the BAO distance measurements, or , are two-variable functions of and H 0 (once is known) in a flat ΛCDM cosmology, and therefore the Hubble constant can be, in principle, determined from the BAO distances with marginalized over. In what follows, we use isotropic or anisotropic BAO distance measurements to determine the Hubble constant with marginalized over, i.e., our parameter space is simply (assuming a flatness of the Universe)The sound horizon at the drag redshift r d is calculated using Equation ( 3 ). 7 7 Except for 6dF, the value of r d is rescaled by a factor , where value is calculated from CAMB (Lewis et al. 2000 , available at http://camb.info/ ) in the same fiducial cosmology (Bennett et al. 2014 ). We perform a Monte Carlo Markov Chain (MCMC) global fitting for parameter estimation using a modified version of CosmoMC (Lewis Bridle 2002 ). 8 8 http://cosmologist.info/cosmomc/ The BAO data sets used in this work include 1. the isotropic BAO measurements using the 6dFRS (6dF; Beutler et al. 2011 ) and SDSS main galaxy sample (MGS) (Ross et al. 2015 ) at effective redshifts and , respectively; 2. the BOSS DR12 anisotropic BAO measurements at three effective redshifts (BOSS 3 z bin) in Alam et al. ( 2017 ) or at nine effective redshifts (BOSS 9 z bin) in Wang et al. ( 2017 ) and Zhao et al. ( 2017b ); 3. the eBOSS DR14 isotropic BAO measurement at (Ata et al. 2017 ); and 4. a combination of 6dF + MGS + BOSS 3 z bin + eBOSS DR14 (All 3 z bin), or a combination of 6dF + MGS + BOSS 9 z bin + eBOSS DR14 (All 9 z bin). To check the consistency of the H 0 values determined from different data sets within the ΛCDM model, we compute the tension, T , based on the KL divergence (Kunz et al. 2006 ; Paykari Jaffe 2013 ; Amara Refregier 2014 ; Seehars et al. 2014 , 2016 ; Verde et al. 2014 ; Grandis et al. 2016 ; Raveri et al. 2016 ), which quantifies the distance between two probability density functions (PDFs), p 1, and p 2 . If both p 1 and p 2 are assumed to be Gaussian, the relative entropy in bits between the two PDFs can be evaluated aswhere is the best-fit parameter vector, is the corresponding covariance matrix, and d denotes the dimensions of the parameter space (e.g., d = 2 in our case where

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 B-Telescope O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Model I-Model O O O B-Telescope O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-Model I-Model I-Model B-Model O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Model O O O O O O O B-Model O O O O O B-Model O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O B-Model I-Model I-Model O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O B-Model O O O O O O O O O O O O B-Model O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Survey O O O O O O O O O O O O O O O O B-Software B-Citation I-Citation I-Citation I-Citation O O O B-URL O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Model I-Model I-Model I-Model B-Model O O O O O O O O O O B-Software B-Citation I-Citation I-Citation O O O B-URL O B-Model O O O O O O O O O O B-Model O O O B-Dataset B-Survey B-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset I-Dataset I-Dataset B-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Dataset I-Dataset O B-Model O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Dataset I-Dataset O B-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Survey O B-Dataset O B-Dataset I-Dataset I-Dataset I-Dataset O B-Dataset I-Dataset B-Dataset I-Dataset I-Dataset I-Dataset O O O O B-Survey O B-Dataset O B-Dataset I-Dataset I-Dataset I-Dataset O B-Dataset I-Dataset B-Dataset I-Dataset I-Dataset I-Dataset O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O B-Model I-Model 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 I-Citation I-Citation O B-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 O O O 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 B-Formula I-Formula I-Formula O O O O


super asymptotic giant branch (AGB) star. In an He star in the binary system, there is no H-rich envelope and, thus, AGB phase does not occur. For such a situation, the CO core growth depends on the efficiencies of the He shell burning and the mass loss of the He envelope. Because the mass-loss process in the binary system is not fully understood and long-time calculation of the CO-core growth through the He-shell burning is required, it is not easy to evaluate whether a CO core less than 1.36 M ⊙ in the binary system grows up to the critical mass and explodes as an ultra-stripped EC SN. The mass range of CO cores for EC SNe was recently discussed for the primary stars in binary system (Poelarends et al. 2017 ; Siess Lebreuilly 2018 ). However, the mass range would be still model dependent and, thus, we are not sure whether the binary evolution affects the mass range of CO cores of EC SN progenitors. The investigation of the growth of the ONe core with thin He layer is important to clarify the possibility of the evolutionary path to an EC SN in close-binary systems. How is low-mass NS formed from EC SN progenitors and ultra-stripped SN progenitors that have a low-mass Fe core? In the case of EC SNe, the electron fraction in the ONe core is ∼0.48 at the central Ne ignition (Takahashi et al. 2013 , 2018 ). The electron fraction in the ONe core scarcely decreases until the central Ne ignition. On the other hand, progenitors of ultra-stripped SNe cause the electron capture during their evolution and, as a result, the electron fraction of the Fe core decreases. The electron fraction at the centre is ∼0.43 when the central density becomes ∼10 10 g cm −3 in the collapsed models in this paper. Thus, the Chandrasekhar mass of EC SN progenitors is heavier than that of the progenitors that have a low-mass Fe core of ultra-stripped SNe. The hydrodynamical simulation in our previous study (Yoshida et al. 2017 ) showed that the baryon mass of the PNS of an EC SN is 1.32 M ⊙ . 4 This mass is higher than the baryonic mass range of the NSs formed from CO137–CO144 models. 6 SUMMARY In this paper, we investigated the minimum mass of NSs based on astrophysical scenarios, i.e. stellar collapse and supernova explosion. We calculated the evolution of CO cores with masses 1.35–1.45 M ⊙ . The stars with CO-core masses higher than 1.36 M ⊙ form an Fe core after off-centre Ne, O, and Si-burnings. We found that low-mass CO cores, which eventually form an Fe core and subsequently collapse, could result in an NS with mass ∼1.17 M ⊙ , which is comparable with the lowest NS mass precisely measured. According to our stellar evolution simulations, the minimum mass of CO cores that produce an Fe core is ∼1.37 M ⊙ and below this value an ONe core is formed, which would lead to EC SN instead. Because of its higher value of Y e , an EC SN would produce more massive NSs than a core-collapse SN from an Fe core. Therefore, the minimum mass of NSs is expected to be determined by the

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


Vidal et al. (2017) calculate a CS/HCS + ratio in better agreement with observations, especially at late times. The reason for the low abundance of HCS + in our gas-phase model may be related to the underproduction of H 2 S (see e.g. Agúndez Wakelam 2013 ), which is enhanced when including grain-surface chemistry, and which can increase the abundance of HCS + through reactions like H 2 S + C + and H 2 S + + C. While the calculated abundance of HC 2 S + is in agreement with the observed upper limit, the calculated C 2 S/HC 2 S + ratio is too low because the chemical model severely underestimates the abundance of C 2 S. This is because the chemical network kida.uva.2014 includes an efficient destruction channel for C 2 S through reaction with O atoms, based on estimations by Loison et al. (2012) . If this reaction is neglected, the calculated abundance of C 2 S is shifted up and becomes very close to the observed value, while the abundance of HC 2 S + also increases, remaining consistent with the observed upper limit (see dotted lines in Fig. 3 ). Accurate calculations of the rate constant of the O + C 2 S reaction at low temperatures are needed to shed light on the chemistry of C 2 S in cold dense clouds. 7. Conclusions We report the first identification in space of protonated C 3 S. Four harmonically related lines observed toward TMC-1 using the Yebes 40 m radiotelescope have been unambiguously assigned to this ion thanks to accurate ab initio quantum chemical calculations and laboratory measurements of the rotational spectrum of this species. The derived C 3 S/HC 3 S + ratio of 55 ± 20 is well reproduced by a gas-phase chemical model in which HC 3 S + is mostly formed through protonation of C 3 S and the reactions S + + C 3 H 2 and S + C 3 . Appendix A: Column densities under LTE for linear molecules The determination of column densities from the observed line paramaters of the rotational transitions of a molecule requires knowledge of the collisional rates of the molecular species under study. These values are not always available and rotational diagrams are used to derived rotational temperatures, T r , and column densities ( Goldsmith Langer 1999 ). Often only one line has been observed and therefore an assumption has to be made as to the rotational temperature. The line parameters of the species studied in this paper are given in Table A.1 . Table A.1. Observed line parameters in TMC-1 for CS, HCS + , C 2 S and C 3 S. Notes. (a) Predicted frequencies from the MADEX code ( Cernicharo et al. 2012 ). Numbers in parentheses are 1 σ uncertainties in units of the last digits. (b) Integrated line intensity in K km s −1 . (c) Line velocity with respect to the Local Standard of Rest in km s −1 . (d) Linewidth at half intensity derived by fitting a Gaussian function to the observed line profile (in km s −1 ). (e) Antenna temperature in K. In this section we analyse how the derived column density depends on the adopted T r for a range of

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


RWI is an analysis of a configuration with only one local pressure extremum). This analysis of the barotropic shear layer model gets simplified if we implement the strategy used in Umurhan ( 2010 ), wherein is assumed. In this limit, the continuous shear layer flow profile becomes an analytically tractable problem comprised of three piecewise linear shear profiles. The solution procedure follows the steps detailed in the Appendix relating to the same type of analysis done on the asymmetric jet (examined in the next section). 12 12 It ought to be noted that there are subtleties introduced because of the use of piecewise linear profiles. For instance, it is sometimes the case that piecewise linear profiles predict modes that do necessarily have counterparts in analysis of flow profiles that are infinitely differentiable. The normal modes generated herein, through the use of these piecewise linear representations, all have counterpart modes in their corresponding infinitely differentiable flow profiles. For the shear layer considered here it is a slight generalization of Rayleigh’s shear layer analysis (Rayleigh 1880 ) with the additional superposition of the background Keplerian shear profile. The resulting frequency response is given byInspection shows that this system becomes unstable when ω goes through zero. The physical condition for instability is the phase locking of two counter-propagating Rossby waves (Baines Mitsudera 1994 ; Heifetz et al. 1999 ). By setting we find the condition for marginal growth to be given by where is the shear layer amplitude corresponding to marginality. We do not consider the positive roots, , because these generally correspond to composite flows that are strongly cyclonic, i.e., flows in which while the negative root, on the other hand, often corresponds to anti-cyclonic shear profiles. 13 13 This is generally true on examination of the marginal condition in Equation ( 53 ). If we consider the setting examined in M13 , we suppose that the smallest azimuthal wavenumber is the one fitting the box considered therein. The corresponding azimuthal length scale in their box units is approximately . This corresponds to a fundamental azimuthal wavenumber . Similarly, the width of the shear line charge is approximately 0.28 (also in their code units). Translating this into our setup, the total width is , and thus we adopt a value of . In this case, the corresponding critical value of for the R-mode is given from Equation ( 53 ) by , with a corresponding critical Rossby number of . Values of should then be stable to the RWI. We find that the value of predicted in Equation ( 53 ) generally predicts more negative values of that are borne out in our models. The reason for this is that we adopt non-zero values of () for all the models reported in this section. The discrepency is generally 2%–5% of the predicted values. 14 14 We have separately checked (but have not included in this manuscript) and verified that as the model

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


the redshift of the clusters. 4.1. Ridge fitting Since we are trying to create a general method to use for many different sets of galaxies at different redshifts, we have decided to use a single set of parameters across all the bins. By doing so we retain a generally low dispersion in the fits as it makes sure that there are no systematic errors due to using different parameters. As can be seen in Fig. 4 , we mostly obtain good fits for both the ridges with a few outliers and even fewer ridges that we do not find at all. The main exceptions to this are at around redshift z 0.05 where the bins do not contain many galaxies and we end up with sparse ridges. For the redshift bins very close to us we cannot fit anything below redshift z = 0.021 as the ridges are too irregular and the method can no longer fit them to satisfying precision. For the star-forming ridge we see a large increase in dispersion at higher redshift. This is mainly due to how sparse the ridge becomes at this point, making it harder to detect. Fig. 4. Standard deviation of the fit of the ridges in each of the redshift bins. At redshifts of z 0.021, the method does not detect any ridges due to the number of galaxies in the bins. We also see increasing irregularity in the ridges below about redshift z 0.05. Besides this we the star-forming ridge having very high dispersion at redshifts above z 0.15 due to being very sparse at these redshifts. 4.2. Evolution of ridges in redshift By plotting each of our measured ridge SFRs, we can then see a relationship between the SFR and redshift (Fig. 5 ). It is obvious that the relationship is linear for the whole range of redshifts for the star-forming ridge and linear in about half the redshift range for the quenched ridge. It could be argued that a logarithmic fit would make a better fit for the quenched ridge but it seems more likely that there is another answer for the high redshift part of the quenched ridge, since the SFR values for the ridge stay almost constant in that part. This is likely due to a few different factors such as difficulty in observing the quenched ridge at this redshift or due to us using a shallower incline in the SFR– M ⋆ plane than appropriate or that the ridges are not linear in the SFR– M ⋆ plane but actually curve downwards at higher mass ranges as suggested by Schreiber et al. (2014) and Ciesla et al. (2017) . This would lead to higher estimates as we go to higher z since we only detect the high mass end of the ridge in that region. This fit does not seem to be widely accepted as people are still using linear ridges in some papers ( Renzini Peng 2015 and Davies et al. 2016 ). Fig. 5. Evolution of star-forming ridge and the quenched ridge SFR with redshift. For the star-forming ridge the best fit seems to be a linear one whereas for the quenched ridge it seems that above about redshift log( z + 1)=0.055 the measurements diverges from the linear fit. This could be due to a systematic error in the measurement of

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


contradicting the original hypothesis. A way to avoid this problem could be to look at the incidence of AGN activity as a function of environment, for example as was done by Arnold etal. ( 2009 ). With a sample of 10 groups and six clusters, they find that the fraction of AGN (with L 0.3-8keV 10 41 ergs 1 ) in groups is a factor of 2 higher than in clusters, although the result has marginal significance. Due to concerns about the differing morphological mix of galaxies in these different environments, they also looked at the AGN fraction in early-type galaxies, finding a similar result. Many other studies of the environment of X-ray selected AGN have used correlation analysis to assess the clustering bias on different scales as well as the typical halo mass. Recent examples include Gilli etal. ( 2009 ) who found a strong clustering signal for AGN with median z 0.98 and L 0.5-10keV 10 43.8 ergs 1 . The correlation length matched that of galaxies with stellar mass 10 10.5 M , and implied a typical halo mass of 10 12.4 M . Fanidakis etal. ( 2013 ) found that AGN (10 42 L 2-10keV [ergs 1 ] 10 44 ) typically reside in haloes of mass 10 13 M . DiPompeo etal. ( 2014 ) found similar halo masses for the infrared selected AGN they studied at z 1, with 10 13.3 M and 10 12.8 M depending on whether the AGN are obscured or not. These papers highlight that the luminosity, selection technique and redshift of an AGN population are also important factors in assessing environment. Indeed, Martini etal. ( 2013 ) show that at z 11.5 there may be a reversal in the incidence of AGN in clusters versus the field. More locally, using a large number of galaxies from the Sloan Digital Sky Survey (SDSS), with AGN identified via their [O iii ] line emission, Li etal. ( 2006 ) found little difference between the clustering of AGN and control galaxies on scales greater than 1Mpc, suggesting that the haloes of active and inactive galaxies have similar masses. Between 100kpc and 1Mpc AGN were more weakly clustered, a result these authors argued could be explained if AGN are preferentially located at the centres of their haloes. And at scales less than 70kpc, AGN appeared to be marginally more clustered. The importance of the local environment was also highlighted by Serber etal. ( 2006 ), who looked at quasars (identified by M i 22) in the SDSS, finding they are overdense on scales 100kpc, but have no difference compared to the general population at larger 1Mpc scales. This suggests that galaxygalaxy interactions might be important. However, from a snapshot survey of AGN (selected in the 0.33.5keV soft X-ray band, and identified as AGN using optical emission line widths and ratios), Schade, Boyle Letwasky ( 2000 ) emphasized the lack of evidence for strong interactions or merging activity. Indeed, Sabater, Best Heckman ( 2015 ) have argued that the availability of cold gas in the nuclear regions, rather than local galaxy density or galaxygalaxy interactions, is the key

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


note that the good agreement of the TSI with the red filters is expected to be valid only for the rotational variability, which is dominated by spots. In contrast, the solar irradiance variability on the activity cycle timescale is given by the delicate balance between facular and spot components, and consequently has a very sophisticated spectral profile ( Shapiro et al. 2016 ; Witzke et al. 2018 ). Thus, values of slopes from Table 1 cannot be extrapolated from rotational to activity cycle timescales (see Shapiro et al. 2016 , for the detailed discussion). Fig. 3. R 30 in different filter systems compared to the TSI (black solid line) for a time span of 1000 days over solar cycle 22. Top panels : Kepler , CoRoT, and TESS. Middle panels : three Gaia passbands. Bottom panels : three VIRGO/SPM channels. See the main text for the definition of R 30 . Table 1. Slopes of the linear regressions in Fig. 4 . Table 2 lists the cycle-averaged values of R 30 for all passbands in mmag. Together, Fig. 4 , and Tables 1 and 2 show that the TSI is a passable representative for the variability on the solar rotation timescale as it would be observed in the TESS, Kepler , CoRoT, Gaia G , Gaia G RP , and VIRGO-red filters, but it noticeably underestimates the variability in G BP , VIRGO-green, and VIRGO-blue. Fig. 4. Linear regression between the R 30 values calculated with the TSI and with the solar light curves as they would be recorded in different filter systems. The values of the slope, s , are given in the legend. The black lines have a slope equal to 1. Several studies (see e.g. Basri et al. 2010 ; Harrison et al. 2012 ) have assumed that the amplitude of the rotational solar variability as it would be measured by Kepler is very close to the amplitude calculated for the combined green and red VIRGO/SPM light curves (in the following VIRGO/g+r). Here we test this hypothesis. The variability R 30 for Kepler compared to VIRGO/g+r is shown in Fig. 5 , which is limited to the same time interval as Fig. 3 . The two curves are remarkably similar to one another. To test the similarity quantitatively, we show the linear regression of R 30 between Kepler and VIRGO/g+r for four solar cycles (21–24) in Fig. 6 . The Pearson correlation coefficient is very high (0.999) and the slope deviates by only +0.8% of unity averaged over four solar cycles. While these calculations are related to the amplitude of the rotational variability, R 30 , we additionally calculate regressions between Kepler and Virgo light curves in Sect. 4 . We also directly connect the TSI and VIRGO/g+r rotational variability. The linear regression between R 30 in TSI and VIRGO/g+r results in a slope of 0.88 (±0.002) and a Pearson correlation coefficient of 0.995. Fig. 5. R 30 over cycle 22 for VIRGO/g+r and Kepler . Fig. 6. Linear regression between the R 30 calculated for VIRGO/g+r and for Kepler light curves. The black solid line represents a linear regression with slope = 1. 3. Correction for the

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


model is able to replicate the spread in the CRIR. There are two main controlling factors for the CRIR in the clouds: the number of embedded sources and the cloud environment. Embedded sources create a natural dispersion in ζ for different molecular cloud masses and star formation efficiencies. Without internal sources, there is no spread in our modeled CRIR as a function of column density. In order to represent the observations, the external CRIR must be increased instead for different regions. Local sources of CRs, such as nearby OB associations or supernova, contribute significantly to the CR flux at the cloud boundary. As the external CR flux is increased, the impact of attenuation also increases due to the rapid reduction in low-energy CRs. Figure 11 shows that the impact of attenuation is different between the HNI and LNI models. For the LNI model, ζ changes by less than an order of magnitude across four orders of magnitude in A V . Conversely, the HNI model CRIR decreases by two orders of magnitude due to an overall reduction in MeV-scale CRs. The HNI model overpredicts the CRIRs measured in diffuse gas to CRIRs measured near high-mass protostars, excluding the galactic center sightlines. However, the spectrum is the maximal CR spectrum from Voyager -1 observations (Stone et al. 2013 ; Ivlev et al. 2015 ). The LNI and LNA models underpredict the observed CRIR for all but a few sightlines. Thus, Figure 11 demonstrates that it is essential to consider the cloud environment and properly treat the CR physics and cloud density distributions. Models without attenuation only represent the CRIR within narrow ranges of A V and not in the cloud interiors. Figure 11 also underscores that the low = energy CR spectrum, which, if often adopted in astrochemical modeling, is a poor fit to the majority of the observations. 4.3. Challenges for Deriving the CRIR from Chemical Diagnostics There have been numerous attempts to find chemical diagnostics that are strong tracers of the CRIR (Caselli et al. 1998 ; Indriolo et al. 2007 , 2015 ; Neufeld et al. 2010 ; Indriolo McCall 2012 ; Neufeld Wolfire 2017 ; Albertsson et al. 2018 ). Some of these, such as [DCO + ]/[HCO + ], cannot be modeled with the current 3 d-pdr version due to the lack of deuterium and isotopic chemistry. Most probes of the CRIR are based on the local abundance, which is difficult to directly ascertain from observations. The use of column density ratios typically assumes the line emission observed between species is cospatial. Figures 8 and 9 examine effects of CR physics on the , and column density ratio diagnostics. However, we find that none of these ratios are monotonically sensitive to the average CRIR, shown for the LDI model in Figure 5 . The [CO/C + ] column density ratio is anti-correlated with the density-averaged CRIR because the amount of CO declines while the amount of C + increases in large, massive clusters. Local abundance ratios are used for fitting observations with

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T / T c = 10 −2 . ( e ) GC3 WC, T / T c = 10 −4 . ( f ) GC3 WC, T / T c = 10 −6 . ( g ) GC4 LC, T / T c = 10 −2 . ( h ) GC4 LC, T / T c = 10 −4 . ( i ) GC4 LC, T / T c = 10 −6 . ( j ) GC4 WC, T / T c = 10 −2 . ( k ) GC4 WC, T / T c = 10 −4 . ( l ) GC4 WC, T / T c = 10 −6 . The orbital decay time for a wide range of parameters is shown in Fig. 8 for the two GCs inside the core radius of the Fornax dSph, GC3 and GC4, in the two density profiles considered. τ DF generally either attains a minimum value, τ DF, min , or approaches infinity. The minimum values in the region V > c − and δ ρ / ρ 0 1 are summarized in Table 3 , with τ DF in the range 67 Myr−197 Myr. These timescales are considerably smaller than the CDM result assuming the same density profiles, 515 Myr−1327 Myr, with the dynamical friction given by Binney Tremaine (2008) Table 3. Minimum orbital decay time τ DF, min of GC3 and GC4 in the LC and WC models for the Fornax dSph, found in the region V > c − and δ ρ / ρ 0 1 (for which perturbation theory is properly valid), with the CDM result given by Eq. ( 87 ) for comparison. Notes. The listed values are essentially constant for all temperatures for which the assumptions made are valid, T / T c ≲ 0.1. For V c − , perturbation theory instead predicts the dynamical friction to quickly vanish, causing τ DF to become infinite. The criterion on g / m 2 due to the hydrostatic core radius, Eq. ( 82 ), is also listed. The values for g / m 2 are given in units of eV −4 . where Λ ≈ r δ v 2 / G M , , δ v is the velocity dispersion of CDM particles, taken to be δ v ≈ V , and erf is the error function. The decay time remains small even if parameters used to model the Fornax dSph, the GCs, and the dynamical friction are varied, as illustrated in Fig. 6 . A notable exception is the position of the GC, for which τ DF is considerably shorter when closer to the halo center, and likewise longer when further away. This implies that the value for τ DF obtained from Eq. ( 79 ) overestimates the time it takes the GC to fully decay from its current position, but it also implies that the migration towards the halo center was slower in the past when the GCs were at larger radial distances. Indeed, estimates of τ DF, min for GC1, GC2, and GC5, all of which are located at r ≳ 1 kpc, give decay times in excess of 4 Gyr. In the CDM case, the decay times for these GCs are even longer: 17 Gyr and more. These estimates do not suggest a timing problem for the outer GCs, even if their decay times are considerably shorter for SIBEC-DM compared to CDM. However, we note that these GCs are near or outside the radius r s , where the density profile of the dSph falls sharply, and therefore we do not expect the result for the dynamical friction, nor Eq. ( 79 ), to necessarily provide a reasonable estimate of τ DF . Nonetheless, the present results show that for a large region of the relevant parameter space of the SIBEC-DM model considered here, GC3 and GC4 are currently

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