stronger than the CF– L bol correlation. What causes this discrepancy? There are some possibilities, such as the difference in (i) redshift, (ii) sample selection (i.e., our sample is limited to type 1 quasars, while the sample in Ricci et al. 2017 is not), and (iii) definition of CF (i.e., is employed in this work, while a fraction of X-ray-obscured sources is employed in Ricci et al. 2017 ). It is hard to quantify, however, the influence of the difference in the sample selection and definition of the CF on the correlation strength, and it is beyond the scope of this work. We thus argue for the possibility that the difference in redshift could cause the discrepancy. Ricci et al. ( 2017 ) targeted nearby AGNs with a median redshift of z ∼ 0.037, whereas we focus on AGNs with much higher redshift up to z = 0.7. Figure 12 shows the redshift dependence of the CF– λ Edd relation. We find that the CF of AGNs with z 0.12 (which is similar to that of the sample in Ricci et al. 2017 ) tends to strongly depend on λ Edd , with a correlation coefficient of r = − 0.49 ± 0.32, while CF– λ Edd for AGNs at z > 0.2 shows weaker correlations than that at z 0.12. This indicates that the aforementioned discrepancy can be explained by the difference in redshift between the samples in Ricci et al. ( 2017 ) and this work. We also remind that our correlation analysis would be affected by the Malmquist bias, which could make the correlation strength weaker, as discussed in Section 4.2.2 . Figure 12. CF as a function of λ Edd . The blue, green, orange, and gray shaded regions represent quasars with z 0.12, 0.12 z 0.2, and z > 0.2 and the entire sample, respectively. This result also might suggest that the CF decreases with increasing redshift for at least z 0.7, a trend that is consistent with what is reported in Toba et al. ( 2014 ), although there is large uncertainty. On the other hand, the fraction of obscured X-ray AGNs increases with redshift for the same luminosity (e.g., Ueda et al. 2014 ). This discrepancy might be caused by the difference in sample selection and definition of CF. 4.6.2. BH Mass Dependence Recently, Kawakatu et al. ( 2020 ) constructed a model of a nuclear starburst disk supported by the turbulent pressure from Type II supernovae and investigated how the CF depends on AGN properties by taking account of anisotropic radiation pressure from AGNs. They found that the CF strongly depends on λ Edd for AGNs with M BH 10 8 M ⊙ , while the dependence of the CF on λ Edd is much weaker for AGNs with M BH > 10 9 −10 10 M ⊙ . Although they did not incorporate the AGN polar dust outflow, it is worth checking the dependence of the CF– λ Edd correlation on M BH . Figure 13 shows how the difference in M BH would affect the correlation strength of CF– λ Edd . The resultant r of CF– λ Edd for quasars with , , and are −0.56 ± 0.07, −0.50 ± 0.05, and −0.52 ± 0.22, respectively. This result indicates that the CF of quasars with 10 8 M ⊙ strongly depends on λ Edd .


magnetic field modulus of HD 50169: 5.6 kG, derived from consideration of the differential broadening of spectral lines having Zeeman patterns of different widths. We retrieved the Julian Date of this observation from Adelman (1973) , as the same spectrum was used for detailed abundance analysis. The representative point of Preston’s field determination appears as a salmon-coloured filled square in Fig. 2 . The uncertainty of this determination is difficult to assess. The most meaningful comparison between the field values derived by Preston (1971) through application of his Eq. (3) and the ⟨ B ⟩ determinations of Mathys (2017) , based on the splitting of the Fe II λ 6149.2 doublet, is for those stars for which the mean field modulus only shows small, almost insignificant variations, such as HD 2453, HD 137949, and HD 192678. For these stars, the estimates of Preston (1971) and the measurements of Mathys (2017) are in excellent agreement, within ∼100 G of each other. The comparison is less straightforward when the field modulus shows definite variability with phase, but at least for some stars (e.g. HD 12288 or HD 188041), the agreement between the values that were derived from line broadening by Preston (1971) and those that Mathys (2017) obtained from consideration of the wavelength separation of split line components, seems somewhat poorer. The error bar on ⟨ B ⟩ that we adopted for the data point from Preston (1971) in Fig. 2 , ±200 G, is only tentative, and it may well underestimate the actual uncertainty of the field values reported in this reference. Even with this “optimistic” accuracy, however, the magnetic field strength estimated nearly 50 years ago by Preston (1971) is consistent with the value of the rotation period that we derived above. With P rot = 10 600 d, HD 50169 becomes the longest-period Ap star for which magnetic field measurements have been obtained over more than a full rotation cycle. At 29.04 yr, its period is considerably longer than the next longest one, 21.75 yr for HD 9996. Furthermore, with respect to the latter, HD 50169 presents the advantage of showing resolved magnetically split lines throughout its whole rotation cycle, so that its mean magnetic field modulus can be exactly determined even around the phase of its minimum and, as a consequence, the structure of the field can be better constrained. 4. Magnetic field characterisation The observed variations of both the mean longitudinal magnetic field and the mean magnetic field modulus of HD 50169 can be well represented by the superposition of a cosine wave and of its first harmonic. The best least-squares fit solutions for P rot = 10 600 d are where the field strengths are expressed in Gauss, φ = (HJD − HJD 0 )/ P rot (mod 1) and the adopted value of HJD 0 = 2441600.0 corresponds to a maximum of the mean magnetic field modulus; ν is the number of degrees of freedom, and χ 2 / ν , the reduced χ 2 of the fit. The fitted curves are shown in Figs. 1 and 2 ; the


(e.g. Dalal et al. 2008 ; Desjacques Sheth 2010 ). Combination of the peaks approach plus N -body simulations led to further insights, including that haloes in dense regions that do not correspond to high peaks in the initial conditions accrete more slowly than haloes that do correspond to high peaks (Ludlow Porciani 2011 ). Faster supercomputers and better codes have led to improved N -body simulations, in which the haloes and subhaloes that host galaxies, groups and clusters are resolved. Such simulations have permitted more detailed analyses of the correlations of halo properties with each other and with the halo environments measured in various ways. One of the first papers to do this was Lemson Kauffmann ( 1999 ), which used Ω 0 = 1 τCDM and Ω 0 = 0.3 ΛCDM simulations and concluded that only the halo mass distribution varies as a function of environment, with more high-mass haloes in denser environments, in reasonable agreement with the analytic calculation by Mo White ( 1996 ) based on extended Press–Schechter theory and the spherical top-hat model. While subsequent N -body calculations confirmed this density effect on the halo mass function, they implied that environmental density also affects halo major merging rates (Gottlöber, Klypin Kravtsov 2001 ) and other halo properties. Bullock et al. ( 2001 ) found that haloes in dense environments tend to have higher concentrations than isolated haloes. Sheth Tormen ( 2004 ), using the same simulations as Lemson Kauffmann ( 1999 ), found evidence that low-mass haloes form somewhat earlier in dense environments. Avila-Reese et al. ( 1999 ) found that dark matter haloes that are isolated or in intermediate-density environments have outer density profiles r −β with β ≈ 2.8 ± 0.5 while haloes in denser regions have a wider range of β up to ∼5. Avila-Reese et al. ( 2005 ), again using ΛCDM simulations that were very small by modern standards, nevertheless found that haloes in dense regions had lower spin parameters and higher concentrations and were less prolate than haloes in lower density environments, with the differences arising mostly at low redshifts z 1 from phenomena such as tidal stripping in dense environments. In this paper, we confirm and expand on these results. Maulbetsch et al. ( 2007 ) used a 512 3 particle ΛCDM simulation in a (50 h −1 Mpc) 3 volume to study the mass accretion history of galaxy-mass haloes in different environments, and found that haloes of the same final mass accreted their mass earlier in denser environments, and also accreted a significantly higher fraction of their mass in major mergers. They suggested that this could help to explain the galaxy density–morphology relation and that early-type galaxies are more common in dense environments. They also found that ∼80 per cent of haloes in higher density environments are not accreting (i.e. have ), while this fraction is only ∼20 per cent in low-density environments. They defined the environmental density on a


indices of the thermal plasma/PUIs. The collisional thermal plasma stress tensor and PUI stress tensor in a 1D form may be expressed as , where η g is the thermal collisional viscosity coefficient and η I is the collisionless PUI viscosity coefficient associated with magnetic fluctuations (Jokipii Williams 1992 ; Zank et al. 2014 ). A 1D steady-state model in which all physical quantities depend on the x Cartesian coordinate is used and U = ( U x , 0, 0) and B = (0, 0, B z ). The 1D form of Equations ( 10 )–( 12 ) in normalized form is where is the thermal gas Mach number of the flow, , L ≡ K g / U 1 , is distance normalized to the diffusion length scale, and is the Schmidt number of the thermal plasma/PUI. The inverse compression ratio is y = U x / U x 1 and , , and . The inverse Alfvén Mach number squared of the flow far upstream is . Based on the thermal collisional term scales derived above from the expressions in Zank ( 2014 ), the dominant collisional terms in a collisional magnetized plasma can be estimated. The small mass of electrons compared to protons means they do not contribute significantly to either the thermal heat conduction or thermal viscosity. Proton–electron collisions are negligible since the mean free path is very large compared to that of proton–proton collisions. The dominant collisional terms derive from proton–proton collisions. The diffusion coefficient associated with proton–proton collisions, K pp , upstream of the observed VLISM shock is about 10 15 m 2 s −1 , and the thermal viscosity is 2.5 × 10 −8 kg m −1 s −1 , giving a thermal Schmidt number of about 0.25. Energetic PUIs outside the heliosphere are generated by secondary charge exchange, and their number density is very small in the VLISM ( n p ≈ 5 × 10 −5 cm −3 ; Zirnstein et al. 2014 ), therefore they contribute a very small pressure compared to the thermal plasma ( P I1 / P g 1 ≈ 0.05). A characteristic value for the PUI heat conduction, K I , in the VLISM is ∼10 14 m 2 s −1 . The pitch-angle scattering timescale in the VLISM is taken to be , giving 5.17 × 10 2 s, which gives a value for the PUI viscosity, η I , as ∼3.57 × 10 −13 kg m −1 s −1 , making the PUI Schmidt number about 10 −3 . The PUI Schmidt number is much smaller than that of the thermal gas (), thus PUIs and the dissipation associated with them cannot mediate the VLISM shock, as shown by Mostafavi et al. ( 2017a , 2017b ). Consequently, we can neglect PUI dissipation terms in favor of the corresponding thermal gas terms. Unlike the HTS, PUIs will therefore behave adiabatically, is constant through a broad VLISM shock, and the thermal gas will be heated preferentially. Recall that PUIs were predicted (Zank et al. 1996 ) and observed (Richardson et al. 2008 ) to be heated preferentially at the HTS, with the thermal plasma component remaining relatively unheated. The physical processes determining the structure of the HTS and the VLISM shocks are therefore different. We use the 1D form of the


Blandford 1990 ), and M 8 is the black hole mass in units of 10 8 M ⊙ . A was defined by Daly et al. ( 2018 ), A = α /( α + b ), a and b are the coefficients of the fundamental plane of black hole activity, where A = 0.43 (Merloni et al. 2003 ). L bol is the bolometric luminosity (Netzer 1990 ), L bol = 10 L BLR , and L Edd is the Eddington luminosity, L Edd = 1.3 × 10 38 ( M BH / M ⊙ ). The spin of the black hole can be estimated using the following formula (Daly 2019 ),where L j is the beam power, and g i = 0.1 and g bol = 1 are used (Daly 2019 ). The black hole spins obtained using this method can be compared with those obtained using other methods, such as the X-ray reflection method (Reynolds 2014 ). The spin values of GX 339-4 are 0.94 ± 0.02 (Miller et al. 2009 ) and (Garcĺa et al. 2009 ) using the X-ray reflection method, and the spin obtained using the method of Daly ( 2019 ) is 0.92 ± 0.06. The spin obtained using the method of Daly is consistent with those obtained with the X-ray reflection method. Because of the limitation of the observations, it is very difficult to directly measure the spin of large samples using the X-ray reflection method. Therefore, we use the method of Daly to measure the spin for our samples. The accretion disk magnetic field strength values of ∼10 4 G are also reported using other methods (Mikhailov et al. 2015 ). Thus, the accretion disk magnetic field strengths obtained here are similar to the values obtained using other methods. We use Equations ( 2 ) and ( 3 ) to estimate the magnetic field strength of the accretion disk and the spin of black hole, respectively. The values of the magnetic field strength and the spin of black hole are listed in columns (9) and (10) of Table 1 . 3.3. The Intrinsic <italic>γ</italic>-Ray Luminosity We estimate the intrinsic γ -ray luminosity using the formula (Nemmen et al. 2012 )where is the observational γ -ray luminosity. f b is the beaming factor, and f b is estimated as 1-cos(1/Γ), where Γ is the bulk Lorentz factor. The bulk Lorentz factor comes from Ghisellini et al. ( 2014 ). is the observed γ -ray luminosity from the catalog of LAT (Abdo et al. 2010 ; Ackermann et al. 2011 ). The values of and f b are listed in columns (7) and (11) of Table 1 , respectively. The values of the radiation jet power and the jet kinetic power come from Ghisellini et al. ( 2014 ). Ghisellini et al. ( 2014 ) used a simple one-zone leptonic model to get the radiation jet power and the jet kinetic power. They are listed in columns (12) and (13) of Table 1 , respectively. 4. Results and Discussions 4.1. The Distribution of Physical Parameters The density distributions of the spin of SMBHs and the magnetic field strength of the accretion disks are shown in Figure 1 . The average spins of SMBHs for FSRQs and BL Lacs are 〈 j FSRQs 〉 = 0.50 ± 0.23 and 〈 j BL Lacs 〉 = 0.44 ± 0.27, respectively. The average values of the magnetic field strength of the accretion disk for FSRQs and BL Lacs are and ,


found by Colombo et al. ( 2014 ). Their results, furthermore, impose a cautionary warning for the study of the mass–size relationship of our own Galaxy: if we are unable to distinguish whether an observed Galactic cloud is in one or another environment, we might be producing a mass–size relationship that arises from clouds in different physical environments. It is convenient to stress that, although Colombo et al. ( 2014 ) made an effort to avoid line blending in their cloud decomposition, the situation can be quite complex, since even at the smallest scales where coherent structures can be found (i.e. cores, fibres), substantial superposition along the line of sight can occur (Ballesteros-Paredes Mac Low 2002 ; Gammie et al. 2003 ; Shetty et al. 2010 ; Zamora-Avilés et al. 2017 ; Clarke et al. 2018 ). The situation for clouds in external galaxies might deserve a full further study, since the resolution is substantially different in external Galaxies (typically, 30–50 pc; see e.g.Sun et al. 2018 ; Utomo et al. 2018 ), with only one study (Faesi et al. 2018 ) achieving resolutions of ∼10 pc. A first puzzling challenge is to understand, for instance, the origin of mass–size relation with q 2 in the interarm region of M51, since q = 2 is the limit when clouds are defined via thresholds in column density. 4.3 Fractal dimension? Different studies have interpreted the exponent of the power-law mass–size relation as representative of the fractal dimension (e.g. Elmegreen Falgarone 1996 ; Roman-Duval et al. 2010 ; Beattie, Federrath Klessen 2019 ). This approach has two hypotheses: First, the mass can be a proxy for the area. Secondly, the linear size is related also to the area in a quadratic way, since in the observational studies the size is typically considered to be proportional to the square root of the area of the cloud. In principle, the first hypothesis seems plausible: observed molecular clouds exhibit rapidly decaying N -PDFs (Kainulainen et al. 2009 ). The same is found for MCs in numerical simulations (e.g. Vázquez-Semadeni García 2001 ; Tassis et al. 2010 ; Ballesteros-Paredes et al. 2011b ; Kritsuk, Norman Wagner 2011 ). This means that the filling factor of the dense structures is quite small, which in turn, implies that the mass should be proportional to the area. However, for a collection of Galactic clouds, the superposition effects we discussed are not compatible with invoking mass as a proxy for the area. As shown in the present work, the same fact that the exponent in the mass–size relation is larger than 2 means that the mass is not proportional to the area. In addition, invoking the square root of the area as a proxy for the linear size does not tell us what will be the linear size–area correlation, which is precisely what is intended to be measured with the fractal dimension. A more appropriate estimation of the fractal dimension will be that of comparing the actual area of an ensemble of clouds with an independent estimation


of a gas parcel being essentially independent of its collapse history – explaining why the asymptotic T − Δ relation holds so tightly despite the varied histories of cosmological fluid elements. (The tightness of this relation is absolutely striking in calculations that do not allow for shocking but allow for varied collapse histories. See the Zeldovich approximation calculations in Fig. 1 .) We further show that Compton cooling off of the CMB has no effect on the asymptotic slope of T (Δ) and that recombination cooling is just weak enough not to break the relation. This paper is organized as follows. We first enumerate the relevant heating and cooling processes in Section 2 . Next, in Section 3 , we derive an analytic expression for the asymptotic T − Δ relation in the limit of only photoheating and adiabatic heating/cooling. These are the processes considered in previous attempts to understand this relation analytically. Section 4 generalizes our solutions to include all relevant cooling processes. We briefly consider three applications of our results in Section 5 : (1) how quickly temperature fluctuations are erased, (2) the importance of shock heating, and (3) the thermal evolution of collapsing gas clouds. Lastly, Section 6 contrasts our results with previous attempts to understand the intergalactic T − Δ relation. Throughout we use the standard phenomenological parametrization for the temperature of the low-density IGM, where Δ is the baryonic density in units of its cosmic mean, T 0 is the gas kinetic temperature at Δ = 1, and γ − 1 is a power-law index. (Sometimes γ is called the ‘equation of state.’) When necessary, we assume a flat ΛCDM Universe with h = 0.7, Ω m = 0.3, Ω b = 0.045, and Y He = 0.25. Following Hui Gnedin ( 1997 ), we use the Zeldovich approximation to follow the density (necessary for computing the temperature) of fluid elements for certain calculations. Our Zeldovich approximation calculations draw matter elements randomly from the probability distribution of collapse eigenvalues, assuming that the gas traces the dark matter. This probability distribution only depends on the variance of the density field, which we set to 1.2. This choice is equal to the variance of a real space top hat filter containing a Lagrangian mass of 10 10 M ⊙ in our reference cosmology at z = 3 (and approximately the Jeans mass near the cosmic mean density at this redshift). 2 In addition, we also use a 20 Mpc, 512 3 gas and 512 3 dark matter particle simulation that was run with the gadget -3 smooth particle hydrodynamic code (Springel 2005 ). Reionization occurs instantaneously in this simulation at z = 9, heating all gas to 20 000 K, and the subsequent heating and cooling takes rates applicable for primordial gas if only the H i and He i are kept ionized with an ultraviolet background that is flat in specific intensity [erg s −1 Hz −1 cm −2 sr −1 ]. 2 RELEVANT HEATING AND COOLING PROCESSES The evolution of the temperature of a Lagrangian


and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. The similar line of C iv λ 1550.774 appears offset to higher velocity. Both features are formed mainly in the wind of the Be star, but shell variations are present on the high velocity side of the core. Figure 3.12. Al ii λ1670. The top panel shows the individual spectra of Al ii λ 1670.7867 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.13. Al iii λ1854. The top panel shows the individual spectra of Al iii λ 1854.7164 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.14. Al iii λ1862. The top panel shows the individual spectra of Al iii λ 1862.7895 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). Thespectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.15. Fe iii λ1895. The top panel shows the individual spectra of Fe iii λ 1895.46 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. The lines of Fe iii λ λ 1895.46, 1896.814 also appear in the figure. Figure 3.16. Mg ii λ2796. The top panel shows the individual spectra of Mg ii λ 2796.352 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in


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


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


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


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


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


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 ;


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


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


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


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


−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


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


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


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


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


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


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


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


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


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


but slightly lower mean ellipticity ⟨ε⟩ = 0.38 ± 0.05. Clampitt Jain ( 2016 ) used the technique to measure the quadrupole weak-lensing signal, and applied it to a sample of SDSS clusters. They corrected anisotropic PSF following Reyes et al. ( 2012 ) and Huff et al. ( 2014 ). They considered the noise from the intrinsic shape of galaxies and measurement on each background galaxy following Mandelbaum et al. ( 2013 ) and Sheldon et al. ( 2012 ). They obtained the best-fitting value of the mean ellipticity of ε = 0.19 with 1σ uncertainty of ∼0.05. They ascribed this smaller value to the misalignment between major axis of the BCG and that of cluster halo which is implicitly assumed to be aligned. Given the large uncertainty, their result is broadly consistent with our result. Shin et al. ( 2018 ) applied the quadrupole technique to SDSS clusters. They estimated anisotropic PSF, measurement noise, and noise from the intrinsic shape following Clampitt Jain ( 2016 ). The resulting mean ellipticity value with a prior of the satellite galaxydistribution is ⟨ε⟩ = 0.45 ± 0.09 after correcting for Poisson sampling. They also measured the ellipticity of satellite galaxy distribution as ⟨ε⟩ = 0.42 ± 0.04, and that derived from stacked weak lensing with a prior of the CG major axis as ⟨ε⟩ = 0.25 ± 0.06. By comparing these ellipticity values, they also estimated the rms misalignment angle of 30° between the CG and DM halo and 18° between satellite galaxies and DM halo. These misalignment values are in good agreement with our result (see also Section 6.2 ). van Uitert et al. ( 2017 ) used an estimator similar to Clampitt Jain ( 2016 ) and constrained the average ellipticity of galaxy groups obtained from Galaxy And Mass Assembly (GAMA) survey combined with the weak-lensing signal measured by Hildebrandt et al. ( 2017 ) from the Kilo Degree Survey (KiDS). They did not consider anisotropic PSF, but consider the intrinsic shape noise. They compared different priors for stacking analysis of weak-lensing signals at different scales. Their resulting values of the mean ellipticity are ε = 0.38 ± 0.12 (40 R 250 kpc) and ε = 0.05 ± 0.13 (250 kpc R 750 kpc) for the BCG prior, whereas ε = −0.04 ± 0.11 and 0.349 ± 0.13, respectively for the prior of the satellite galaxy distribution. They concluded that the BCG major axis (satellite galaxy distribution) is aligned (misaligned) with the DM halo orientation on small scales (250 kpc), whereas the BCG major axis (satellite galaxy distribution) is misaligned (aligned) with DM on large scales (>250 kpc). This result appears to be inconsistent with our result which indicates that the distribution of satellite galaxies is aligned better than the major axis of the CG at all scales, 100–1000 kpc. This discrepancies are partly because they use galaxy groups with M 200 ∼ 10 13 M ⊙ rather than galaxy clusters we considered in this paper, M 200 ∼ 10 14 M ⊙ . Nevertheless, further work is needed to explain this inconsistency, for


resonant stripes, where comets have experienced secular eccentricity pumping. This sometimes goes in the direction of circularizing the orbits, but very often leads into the sungrazing end state. This structure would be smeared out in the plot by Levison et al. ( 2006 ; their Fig. 1b) but, most likely, in reality it does not exist. The small nudges given to the comets in their integrations would inhibit the long-term action of these resonances in most cases (see Valsecchi et al. 1995 ), while we introduced the artefact of leaving the comets entirely in the hands of the giant planets. We find that deflected comets get dynamically ejected only if their starting semi-major axis is larger than 2.6 au. For smaller values, if Jupiter-crossing occurs, it is only for very high eccentricity, so encounters are rare and inefficient in changing the orbits. In these cases, collision with the Sun is very likely, and many of the survivors are “saved by the bell”, since our integrations stop after 100 000 revolutions. Looking at the statistics of end states for the deflected comets, we find a remarkably high probability (more than 12%) of collision with the Sun, and most of these belong to the above-mentioned category. We derived the steady-state number of comets in the Encke-like domain and made a comparison with the results of Levison et al. (2006) , which are quite different. In fact, using our first approximation to the transfer mechanism, we predict only of the number of Encke-like comets found by Levison et al. (2006) . Thus, our method is severely lacking in transfer efficiency, and several reasons for this are easily identifiable. First, consider the limitation caused by cutting the integrations to a total of 100 000 revolutions. Compare this to the example provided by Levison et al. (2006) of a comet meeting their requirement for an Encke-like comet. Before this happens, the comet spends a million years (roughly 250 000 orbits) in a decoupled state waiting for the completion of its transfer. If this situation is typical, we missed most of the transfers by stopping the integrations too early. Moreover, as in the case mentioned, most transfers may require several interventions by terrestrial planets – something that our procedure forbids. It is true that more than one encounter as close as those that we modelled is not likely to occur for the same object. However, significant perturbations may be caused by the more distant and more frequent encounters that our procedure neglects. The main reason for the inefficiency of our transfer model appears to be the premature cutting of the integrations and the limitation to the closest encounters, which are too rare to have a large effect. If we had integrated all the comets until ejection or collision, some more deflections would have occurred. However, the increase would not be large, as seen from the fact that more than 80% of the deflections that we followed up by further integrations happened before the 50


difference for the XIS detector normalization constants between both observations. We attribute this difference to the different data extraction modes and the lacking attitude correction for the first observation, as the simulation of the effective area of XIS, which is performed by the FTOOL xissimarfgen in the course of data reduction, is highly sensitive to the source position and extent. Possible deviations of the apparent source position due to thermal wobbling might therefore introduce an unknown systematic uncertainty to the flux measured by XIS. For this reason, we normalized the detector constants with respect to PIN. The fitted fluxes in the 15–50 keV band are about 10% higher than the corresponding NuSTAR observations, which agrees with flux calibration uncertainties ( Madsen et al. 2015 ). For future reference, we also determined the continuum flux for other energy bands than those given in Table 2 . For the first observation, the measured fluxes are and in the 2–10 keV and 3–60 keV band, respectively, and for the second observation and for the corresponding energy bands. Fig. 7 a) Unfolded phase-averaged spectra of Obs. I (red) and Obs. II (blue) and b) ratio of the phase-averaged spectra of Obs. I and Obs. II. For clarity, only XIS3, PIN and GSO data are shown in both panels. The ratios of the individual detectors have been corrected for the respective cross-calibration constants given in Table 2 . Fig. 8 Panels a) and b) : phase-averaged spectrum of XIS0 (blue), XIS1 (red), XIS3 (magenta), PIN (green), and GSO (brown) of Obs. I with best-fit model and residuals. Panels c) through g) : count rates of the respective phase-resolved spectra (1) to (5), divided by the count rate of the phase-averaged spectrum. Figure 7 shows the unfolded spectra of Obs. I and Obs. II in comparison and the photon ratio of both observations. We observe a slight change in spectral slope below 2.5 keV, while the ratio is mainly flat above this energy with slight differences in the Fe band. Above the cyclotron line energy the spectra are very similar in shape. Comparing the fit parameters of both Suzaku observations, we find a decrease of the photon index and folding energies toward higher flux. Fürst et al. (2014) observed the opposite trend regarding the first and second NuSTAR observation. There is, however, an artificial correlation between these two parameters, which is not taken into account here, so we cannot conclude that the Suzaku and NuSTAR results contradict each other. The excess of soft photons in Obs. I below 2.5 keV is at least partly explained by the increase of the blackbody temperature from Obs. I to Obs. II, while the relative normalization of the blackbody component is very similar in both observations. This behavior is consistent with NuSTAR . See Fürst et al. (2014) for a more detailed investigation of the parameter evolution over the outburst. The fitted value is comparable to the value of given by the Leiden/Argentine/Bonn


). CSM densities and radii of well-known Type Ia SNe and confirmed Type Ia SNRs studied in radio or X-rays, along with SN 1885A and G1.9+0.3 from this work (shown in red). The blue dashed and dotted lines indicate different CSM profiles of the form ( is mass-loss rate, u w is wind velocity, μ = 1.4) created by isotropic mass loss at constant rates and constant velocities given in the legend box. The horizontal gray bars show examples of cavity sizes created by accretion-driven outflows from the progenitor due to fast and slow winds (Badenes et al. 2007 ). The ruler on top shows the timescales on which these CSM profiles formed in the pre-supernova stage. The gray region shows the typical extent of the warm ISM phase (Ferrière 2001 ). The red dashed line indicates the average ISM density of the M31* circumnuclear region where SN 1885A is located (Li et al. 2009 ). Object references: SN 2011fe (Chomiuk et al. 2012 ), SN 2014J (Pérez-Torres et al. 2014 ), Tycho (Slane et al. 2014 ), SN 1006 (Raymond et al. 2007 ). Our upper limit on density of 0.04 ± 0.03 (sys) cm −3 derived in Section 5.1 suggests that, on average, SN 1885A has interacted with very little CSM material. A significantly denser CSM, if present, would have been captured in our measurements of n 0 in Figure 3 (a), and could have produced detectable radio emission. A low-density CSM (≲0.1 cm −1 ) around SN 1885A was also inferred from the non-detection of X-rays from SN 1885A (Perets et al. 2011 ). Our inferred density is lower than that of well-known Type Ia SNRs, and well-studied, recent nearby Type Ia SNe such as SN 2011fe and SN 2014J, although the density values are in the range of the warm ISM phase (Figure 5 ). The strong density limit rules out some portion of the SD parameter space for SN 1885A. For example, if the progenitor binary emitted a wind, the mass-loss rate, , would have been 10 −4 M ⊙ yr −1 (assuming wind velocity u w = 1000 km s −1 ) or 10 −6 M ⊙ yr −1 (assuming u w = 20 km s −1 , see Figure 5 ). Modeling the shock dynamics in such an r −2 profile can lead to even deeper constraints on , because the SNR would have evolved in much denser CSM in the past, and this would have left an observable effect on the radius and velocity of the SNR. However, as noted in Badenes et al. ( 2007 ), it is likely that the CSM structure on parsec scales resulting from an SD configuration is more complex than either a uniform or r −2 profile. Accounting for realistic CSM would require detailed modeling to predict the radius, velocity, and radio luminosity of the SNR, and is beyond the scope of this paper. In the future, it would be productive to revisit the observational constraints from this paper and Fesen et al. ( 2015 ) when a grid of models become available, predicting the CSM structure out to ∼10 pc for a variety of SN Ia progenitors (C. Badenes et al. 2019, in preparation). Such an analysis would allow us to make definitive statements about the progenitor of SN 1885A. As an


the interaction between the outgoing shock and the CSM. The X-ray data of SN 1978K prior to 2006 have been analysed; however, whether its flux has remained constant since the first detection by ROSAT or started to decline was still under debate (e.g. Lenz Schlegel 2007 ; Smith etal. 2007 ). It is worth noting that there are another two ULXs (X-1 and X-2) in NGC 1313. Thus, this region has been frequently visited by X-ray missions in order to explore the spectral evolution of these sources (e.g. Feng Kaaret 2006 ; Bachetti 2013 ; Weng, Zhang Zhao 2014 ). In the past decade, some additional long observations were carried out with XMMNewton (Table 1 ), which is a focusing X-ray telescope having a large effective area in the soft X-ray band. In this paper, we study the XMMNewton observations covering the period from 2000 to 2015 (spanning 5277d), aiming to explore the evolution in X-ray emission of SN 1978K at very late times. The XMMNewton data reduction is described in the next section, and the results are presented in Sections 3 . Discussion and conclusion follow in Section 4 . Table 1. XMMNewton observations of SN 1978K. Net exposure: clean exposure after background flares excluded. 2 DATA REDUCTION The detailed study presented here was on data from the main camera on board XMMNewton , the European Photon Imaging Camera (EPIC), which consists of two MOS arrays and one PN CCD array (Strder etal. 2001 ; Turner etal. 2001 ). We analyse all of the available XMMNewton observations of SN 1978K made from 2000 October to 2015 March; however, we discard those data that were affected by the strong background flares or when the source falls in the CCD gaps. The observational details of the data used in this paper are listed in Table 1 . The XMMNewton observations of SN 1978K are quite clean except two short data observed in 2003 December (ObsIDs = 0150280401, 0150280501), and only less than 12percent of data on average are excluded. The data are reduced with the Science Analysis System software ( sas ) version 14.0.0. We create the light curve above 10keV, and exclude background flares with a count-rate cut-off criterion. The data from good time intervals are filtered by setting FLAG = 0 and PATTERN 4 for the PN data, and PATTERN 12 for the MOS data to ensure the best quality spectrum (Snowden etal. 2016 ). The source photons are extracted from a circle aperture with a radius of 30arcsec, while the background is taken from the same CCD chips as the source and at a similar distance from the readout node. Since the source is relatively faint ( flux 10 12 erg cm 2 s 1 ), the pile-up effect can be neglected. We produce the response files with the sas tasks rmfgen and arfgen to facilitate subsequent spectral analysis. The task specgroup is carried out to rebin all spectra to have at least 20 counts per bin and not to oversample the instrument energy resolution by more than a factor of 3. We fit the spectra in the 0.510keV range using xspec with a 2percent