carbon. The luminosity was obtained by interpolating and fitting the optical ( B , V , i ′) and the NIR ( J – H – K s ) fluxes from Fransson et al. ( 2014 ) at several epochs, in addition to the five epochs (days 87, 254, 465, 621, 844) for which we have Spitzer data. Thereafter, the luminosity, L ( t ), was fitted using a power law of time. The exponents are found to be −0.55 and −4.0 with a switch at day 310, as shown in the figure. We also calculate the UVO luminosity in uniform temporal bins in order to reconstruct the UVO light curve. The evolution of L UVO was thereafter fitted using a power law, t − β , with β = 0.55 before day 310 and β = 4.0 thereafter. The best-fit scenario is presented in Figure 1 (top left). The nature of the luminosity, L ( t ), derived in this study compares well with the findings by Fransson et al. ( 2014 ), Jencson et al. ( 2016 ), and Gall et al. ( 2014 ). The rapid decline in luminosity at later times can be attributed to a change in the density profile at the shock front. A similar drop in luminosity is also reflected in the evolution of the X-rays along the line of sight (Chandra et al. 2015 ). 2.1. Possible Source of the IR Emission In this study, good fits to the IR spectra of SN 2010jl, shown in Figure 1 , were achieved using the contribution from any of the three sources (a blackbody, astronomical silicates, amorphous carbon) as the IR component. Therefore, the best-fit models do not have any strong bias toward any of the three sources being the most likely one. A blackbody fit is appropriate when the dusty shell is optically thick. Additionally, the blackbody radius ( R IR ) represents the minimum possible radius of the dusty shell from where the IR emission originates. The NIR fluxes can be fit well using the absorption coefficients of optically thin astronomical silicates. The upper limit on the 9.7 μ m flux at late times was obtained from the SOFIA observations (Williams Fox 2015 ). It indicates the absence of silicate features. Therefore, the presence of silicates is doubtful, unless the grains are large ( a ∼ 5 μ m) or the dusty shell is optically thick. With only photometric data in the 2.2–4.6 μ m region, many featureless dust species can fit the spectrum. We choose carbon as a likely dust type owing to its relatively large abundance in space compared to other metals. Moreover, the studies dealing with the IR spectra from the preexplosion era also hint at the presence of carbon dust in SN 2010jl (Dwek et al. 2017 ). However, the CSM of SN 2010jl is reported to have a N-rich (Fransson et al. 2014 ) C-depleted environment (N/C =25 ± 15, N/O = 0.85 ± 0.15). In such environments, O-rich dust species are known to be the primary dust components and all of the carbon mostly gets locked up in CO molecules (Sarangi Cherchneff 2013 ). To support this argument, the observation of a N-rich dusty shell around massive stars, such as the Homunculus nebula surrounding Eta Carinae, does not show any

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


Earth) for different ocean coverages. Panel B shows how pCO 2 varies with varied solar luminosity for different ocean coverages. Both Panel A and B assume an Earth-like outgassing rate of CO 2 . We did not plot solar luminosities greater than 1.2 S eff as the weathering rate dependence on CO 2 becomes negligible. Panel C shows how the surface temperature varies with varied CO 2 outgassing rates (relative to modern Earth) for different ocean coverages. And Panel D shows how pCO 2 varies with varied outgassing rates for different ocean coverages. Panel C and D assume a modern Earth solar constant. All panels assume a fixed planetary albedo of α = 0.3. Based on the results from Figure 1 , a planet’s buffering capability changes significantly when seafloor weathering becomes the dominant weathering sink, due to the different pCO 2 dependencies of seafloor versus continental weathering. We therefore determine the critical ocean fraction where weathering behavior switches from being dominated by seafloor weathering to being dominated by continental weathering by setting F cw = F sw , and solving for the ocean fraction: We solve for this critical ocean fraction for a range of solar luminosities, setting the rest of the variables equal to the median values that we used for Figure 1 , as a function of pCO 2 (see Figure 2 ). The solid curves in Figure 2 correspond to the at which the weathering fluxes from continental weathering and seafloor weathering are equal. While we calculated over a range of pCO 2 values, only certain pCO 2 pressures would be physical for a given solar luminosity using our weathering model. The dashed curves with the points are from our previous results, where we calculated the actual pCO 2 of a planet given its solar luminosity and ocean fraction (see Figure 1 ). Figure 2. The solid curve represents the ocean fraction at which seafloor weathering and continental weathering are equal, as a function of pCO 2 . We solved this critical ocean fraction curve for different solar luminosities and found that the difference was negligible (the black curve shown is for an S eff of 1.0). If a planet’s ocean coverage is above the curve, then seafloor weathering is the dominant sink of CO 2 , and if a planet’s ocean coverage is below the curve, then continental weathering is the dominant sink of CO 2 . The colored curves and points are data from Figure 1 , allowing us to constrain this parameter space by plotting the actual pCO 2 values that our model converged toward for a given ocean fraction and solar luminosity. We find that for solar luminosities lower than modern Earth’s, an ocean fraction of greater than 90% is needed for seafloor weathering to equal continental weathering. What we find from Figure 2 is that for relative solar luminosities less than or equal to S eff = 1.0, the planet requires an ocean fraction between 90% and 99% for the CO 2 flux from seafloor weathering to equal the CO 2 flux from continental weathering. This is

O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


thick disc was assumed to span from 9 to 11 Gyr. We also checked with an age span of 10–12 Gyr for the thick disc and found that it gives the same best-fitting parameters. The slight decrease in the mean metallicity of the thick disc compared to the estimate based on simply measuring the metallicity of stars within 5 R /kpc 11 and 1 | z |/kpc 2, is due to the inclusion of stars lying between 2 | z |/kpc 3 in the fitting process. This suggests that there is a small vertical gradient in the metallicity of the thick disc as found by others previously (Kordopatis et al. 2011 ; Duong et al. 2018 ). Note, stars in 2 | z |/kpc 3 were not included in the former scheme as they could be contaminated with stars from the metal-poor stellar halo and could suffer from volume incompleteness. However, they are included in the later scheme because it takes both these effects into account. In Fig. 12 we show the mean metallicity as function of Galactocentric radius R for different slices in height | z | for all stars with positive rotation about the Galaxy, which should eliminate star belonging to the halo. Results from APOGEE-DR14 and GALAH DR2c are shown separately. We also plot Galaxia predictions from the MP and MR models. The MP model clearly has a thick disc that is too metal poor to fit the observed data for | z | > 1 kpc. The new MR model reproduces the GALAH data very well. It also reproduces the APOGEE data very well, except for the slice closest to the mid-plane and the slice furthest from the mid-plane. Compared to GALAH, stars in APOGEE that are close to the mid-plane are metal rich, but the stars progressively become metal poor with increasing height above the mid-plane. These differences could be due to systematics in abundances between the two surveys, but could also be due to the different selection function of the surveys. The effect of selection function is clearly visible in Fig. 12(f) corresponding to the top most slice (2 | z |/kpc 3). Here, the MR model has a thick disc with a mean metallicity of −0.16. However, the mean metallicity of stars in this slice is −0.2. This is due to two reasons. First, because the metal poor stars are more luminous and are visible furthest in a magnitude-limited survey, we find that this effect decreases the metallicity of thick disc stars in this slice by 0.02 dex. Secondly, the top most slice has a larger contribution from the (metal poor) halo compared to other slices and this contamination decreases the metallicity of stars in this slice by −0.015 dex. Fig. 12 shows that close to the mid-plane there is a strong radial metallicity gradient. As we move away from the mid-plane the gradient diminishes progressively to zero. Close to the plane, a radial gradient of −0.07 (dashed line), as used in the Besançon (MP) and MR models, is roughly consistent with the observed data. Figure 12. Mean metallicity as function of Galactocentric radius R for different slices in height | z | for all stars with positive

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like a galaxy cluster creates a small-scale dipole pattern in the direction of the large-scale CMB temperature gradient. The QE (Hu et al. 2007 ) is based on extracting the correlation between the dipole and background gradient to estimate the lensing convergence . Specifically, the lensing convergence can be estimated from a weighted product of filtered versions of a gradient map and small-scale lensing map :Here, is the pointing unit vector, is angular multipole, and A l is a normalization factor. While designed to pull out the lensing-induced correlations between large-scale and small-scale CMB anisotropy, the QE is also sensitive to correlations due to foreground emission. Of particular concern is the cluster’s own SZ emission, which is typically an order of magnitude larger than lensing signal. As pointed out by Hu et al. ( 2007 ), the magnitude of the SZ bias can be lowered by decreasing the characteristic scale of the low-pass filter on the gradient map, which is set to 2000 in the current work. However, a stronger low-pass filter obviously reduces the number of modes used to measure the gradient, and thus decreases the signal-to-noise. The modified QE instead eliminates this bias by using an SZ-cleaned map for the large-scale gradient map. The large-scale gradient map is chosen because the CMB has much more power on large scales, so the noise penalty from SZ removal has minimal impact. Note that while multiple foregrounds can be removed in principle, in practice the focus has been on removing the SZ signal. This is simply because the SZ signal introduces the largest bias. With the SZ signal present in only one of the two maps, there is no SZ-induced correlation between the two maps and no net bias on the reconstruction of the lensing convergence. However the SZ emission in the small-scale lensing map does add noise to the lensing reconstruction. Since under self-similarity the SZ flux (Kaiser 1986 ), y , is expected to scale with cluster mass M as while the lensing signal is linear in mass (in 1-halo regime), the additional SZ variance will generally be more important for high-mass clusters. The SZ variance will also be more important in low-noise surveys, i.e., when it is larger than the instrumental noise in the convergence map. 3.2. Template Fitting to Reduce the SZ Variance An obvious way to eliminate this SZ variance is by using SZ-free maps for both the small-scale and gradient maps. Of course this would undo the advantages of the modified QE for the instrumental noise in the small-scale map. One can also reduce this extra variance by projecting out a model template for the SZ signal, and thereby reducing the total amount of SZ power in the small-scale map. Given the impossibility in creating a “perfect” SZ template, the template fitting will not completely eliminate the SZ signal in the small-scale map. However, template fitting can significantly reduce the SZ power in the small-scale map, and thus reduce the SZ noise penalty

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


SEDs from Abdo et al. ( 2010a ), a time-scale of one day due to the lack of variability data and γ min = 200 (Zhang et al. 2012 ). 4 RESULTS We used three types of EEDs to fit the SEDs of TeV BL Lacs. The SEDs for the maximum probability and the 1σ error band are shown in Fig. 1 . The one-dimensional (1D) probability distributions and the two-dimensional (2D) confidence contours of the model parameters are shown in Figs 2 – 5 . In the 1D distributions, the dotted lines show the maximum likelihood distributions and the solid lines show the marginalized distributions. In the 2D distributions, the inner contour denotes the 68 per cent CL, while the outer contour represents the 95 per cent CL. The model parameters are listed in Tables 1 – 3 . Figure 1. Left-hand panels: SEDs of the three objects fitted by three types of EEDs, where the solid black, blue dash-dot and magenta dash-dot-dot lines represent the SEDs given by the BPL, PLLP and PLC EEDs, respectively. Right-hand panels: SED contours of the three objects in γ-ray bands with the 1σ error bands, where the shaded areas in grey, blue and orange represent the SED contours given by the BPL, PLLP and PLC EEDs, respectively. Figure 2. 3C66A. Left-hand panels: Distributions of the parameters given by the three types of EEDs, where the dotted lines show the maximum likelihood distributions and the solid lines show the marginalized probability distributions. Right-hand panels: 2D confidence contours of the parameter correlations for the three types of EEDs, where the contours of the parameters with larger correlations are shown. The inner contours denote the 68 per cent CL, while the outer contours represent the 95 per cent CL. Figure 3. As Fig. 2 , but for PKS1424+240. Figure 4. As Fig. 2 , but for PKS1424+240 in the flare state. Figure 5. As Fig. 2 , but for PG1553+113. Table 1. Model parameters of the best fits and the marginalized 68 per cent confidence intervals (CI) for BPL EEDs. Table 2. The model parameters of the best fits and the marginalized 68 per cent confidence intervals (CI) for PLLP EEDs. Table 3. The model parameters of the best fits and the marginalized 68 per cent confidence intervals (CI) are listed for PLC EEDs. 4.1 3C66A The SED fitted by the PLC EED does not cover the Fermi /LAT bands, and χ 2 /dof is worse than that of the other two EEDs. It is found that the SED fitted by the PLC EED within the 68 per cent CL does not cover the Fermi band well compared with the other two EEDs. Hence, the PLC EED was ruled out in modelling the SEDs. We obtain that χ 2 /dof values are 1.22 and 1.20 for BPL and PLLP EEDs, and the parameters are well constrained except for α in both models. However, the flux in the 20–50 keV bands given by the PLLP EED is higher than that for the BPL EEDs, indicating that hard X-ray observations could distinguish between the two types of EED. In summary, the red shift of 3C66A is 0.16 and 0.24 from the BPL and PLLC models, respectively, and the red shift of the

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


and H i emission of several radio source populations. Assuming a luminosity dependence for the fractional polarization, O’Sullivan et al. ( 2008 ) realized a simulated polarized image based on the radio source catalogue of Wilman et al. ( 2008 ). Very recently a new simulated catalogue was produced (T-RECS; Bonaldi et al. 2019 ) based on cosmological dark matter simulations to reproduce the clustering of sources and it models the radio sky both in total intensity and polarization with updated information on radio sources. Our simulation, like the above simulations, aims at giving useful information for the advent of the SKA. Similarly, it is based on cosmological RLFs integrated over cosmological volumes but the models adopted to reproduce the characteristics of the radio sources and also the procedure are in general different. In addition, alternative to the previous works, we use observed high-quality images of extended radio sources to reproduce the morphology and the spectro-polarimetric properties of the simulated radio sources. This is especially important as these simulations will be used to study magnetic fields in galaxy clusters (Loi et al. in preparation). For each simulated radio source, our catalogue lists the following parameters: type , in principle we can classify our sources in several subclasses, radio-loud or radio-quiet AGN, SFG, quasar etc. Following Novak et al. ( 2017 ) and Smolčić et al. ( 2017a ) we consider two main families depending on the mechanism that triggers the radio emission: SFG and AGN; redshift , z; size , we used the relations adopted by Wilman et al. ( 2008 ) for radio-loud AGN and SFGs. The size model are redshift-dependent and in particular the SFG size depends also on luminosity; luminosity at 1.4 GHz , we extract this information from the RLFs of Novak et al. ( 2017 ) and Smolčić et al. ( 2017a ) for the SFGs and AGNs, respectively, based on the results of the VLA−COSMOS 3 GHz Large Project (Smolčić et al. 2017b ), extrapolated to 1.4 GHz assuming the spectral index derived in combination with the the VLA−COSMOS 1.4 GHz Large and Deep Projects (Schinnerer et al. 2010 , 2007 , 2004 ); coordinates , (x, y); morphology and spectro-polarimetry properties , we select a model of radio source from a dictionary depending on its luminosity and type. Each model of this dictionary consists of four 1.4 GHz images: the surface brightness I ν in total intensity; the spectral index distribution α determined by assuming that the flux density S ν at a frequency ν is S ν ∝ ν − α ; the fractional polarization defined as the ratio between the polarized intensity and the total intensity FPOL = P / I ; the intrinsic polarization angle which is defined with respect to the Q , and U Stokes parameters as: The images of the dictionary are real high-quality high-resolution images performed at high frequency. In particular, we used VLA images at C and X bands at arcsecond resolution so that the polarization properties

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


and GRMHD models, as well as from measurements of the ring diameter in the image domain. Our measurements remain consistent across methodologies, algorithms, data representations, and observed data sets. Motivated by the asymmetric emission ring structures seen in the reconstructed images (Section 5 ) and the similar emission structures seen in the images from GRMHD simulations (Section 6 ), we developed a family of geometric crescent models (see, e.g., Kamruddin Dexter 2013 ) to compare directly to the visibility data. We used two distinct Bayesian-inference algorithms and demonstrate that such crescent models are statistically preferred over other comparably complex geometric models that we have explored. We find that the crescent models provide fits to the data that are statistically comparable to those of the reconstructed images presented in Section 5 , allowing us to determine the basic parameters of the crescents. The best-fit models for the asymmetric emission ring have diameters of 43 ± 0.9 μ as and fractional widths relative to the diameter of 0.5. The emission drops sharply interior to the asymmetric emission ring with the central depression having a brightness 10% of the average brightness in the ring. The diameters of the geometric crescent models measure the characteristic sizes of the emitting regions that surround the shadows and not the sizes of the shadows themselves (see, e.g., Psaltis et al. 2015 ; Johannsen et al. 2016 ; Kuramochi et al. 2018 , for potential biases). We model the crescent angular diameter d in terms of the gravitational radius and distance, , as d = αθ g , where α is a function of spin, inclination, and R high ( α ≃ 9.6–10.4 corresponds to emission from the lensed photon ring only). We calibrate α by fitting the geometric crescent models to a large number of visibility data generated from the Image Library. We can also fit the model visibilities generated from the Image Library to the M87* data, which allows us to measure θ g directly. However, such a procedure is complicated by the stochastic nature of the emission in the accretion flow (see, e.g., Kim et al. 2016 ). To account for this turbulent structure, we developed a formalism and multiple algorithms that estimate the statistics of the stochastic components using ensembles of images from individual GRMHD simulations. We find that the visibility data are not inconsistent with being a realization of many of the GRMHD simulations. We conclude that the recovered model parameters are consistent across algorithms. Finally, we extract ring diameter, width, and shape directly from reconstructed images (see Section 5 ). The results are consistent with the parameter estimates from geometric crescent models. Following the same GRMHD calibration procedure, we infer values of θ g and α for regularized maximum likelihood and CLEAN reconstructed images. Combining results from all methods, we measure emission region diameters of 42 ± 3 μ as, angular sizes of

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


3 DM particles (Rasera et al. 2010 ; Alimi et al. 2012 ; Bouillot et al. 2015 ). For the flat ΛCDM model, the six key cosmological parameters were set at the central values determined by the Wilkinson Microwave Anisotropy Probe Seven Year data (WMAP7) (Spergel et al. 2007 ; Komatsu et al. 2011 ). For the other two DE models, the values of the seven cosmological parameters (the six key parameters + w ) were chosen to satisfy the same WMAP7 constraints. Table 1 lists the values of the matter density parameter, Ω m , linear power spectrum amplitude on the scale of , σ 8 , dimensionless Hubble parameter, h , dark energy equation of state, w ≡ w 0 + w a (1 − a ) with scale factor a ( t ) (Linder 2003 ), and mass of individual DM particles, m p , for the three DE models. For a more detailed description of the DEUS, we refer the readers to Alimi et al. ( 2012 ) as well as the DEUS webpage. 4 4 All data from the DEUS are publicly available at http://www.deus-consortium.org/ . Table 1 Initial Conditions and the Particle Mass Resolution A phantom DE ( w CDM) is characterized by its negative kinetic energy that exerts stronger repulsive force than Λ (Caldwell et al. 2003 ). Whereas, a quintessence model is characterized by the potential shape of a scalar field DE. For the DEUS, the Ratra–Peebles potential, whose shape is expressed as powers of linear combinations of the exponentials of the scalar field (Ratra Peebles 1988 ), was specifically chosen. Although the distant-field diagnostics based on the linear observables like the CMB temperature spectra and SN Ia luminosity–distance relations cannot distinguish among the three models (Bouillot et al. 2015 ), an efficient near-field probe based on the nonlinear observables might be able to pull it off since the formation and evolution of the nonlinear structures proceed in a different way among them (e.g., see Alimi et al. 2010 ). Table 2 Halo Spin Transition Zones The halo catalog from the DEUS comprises the gravitationally bound objects identified at various redshifts by the Friends-of-Friends (FoF) halo finder with a linkage parameter of 0.2 (Roy et al. 2014 ) and provides information not only on the center of mass position, , and virial mass, M h , of each object but also on the positions and velocities of its constituent DM particles, which allow us to compute its angular momentum vector, . The well-resolved FoF objects in the mass range of at z = 0 are selected for a sample of the galactic halos to be used for the current analysis. The low-mass FoF objects with are excluded from our analysis on the ground that the angular momentum vectors of the halos composed of less than 300 DM particles are likely to suffer from inaccuracy caused by the shot noise (Bett et al. 2007 ). The routine we go through for the investigation of the DE dependence of the halo spin transitions is similar to that described in Lee et al. ( 2020 ), the summary of which is provided in the following. 1. Divide the simulation volume

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


the Q 21,C ii ,C ii bispectrum should be identical to that of the Q 21, δ , δ bispectrum. That is, both of these cross-bispectra are expected to follow the form. Consequently, we study the statistics of Q 21, δ , δ here rather than Q 21,C ii ,C ii since the latter is more difficult to model. The main shortcoming of this approach is that it neglects the additional nonlinearities in the [C ii ] biasing relation, and these effects may reduce the agreement with the perturbative formula. In estimating the signal-to-noise ratio (S/N) at which may be measured in future surveys (Section 6 ), we assume a linear biasing model for [C ii ] emission. This allows us to predict the [C ii ] auto-spectrum and 21 cm-[C ii ] cross-spectrum (as required for the S/N calculation) from the statistics of the simulated density field. It may be possible to construct a subgrid model to include unresolved small halos with the correct statistical properties (e.g., McQuinn et al. 2007 ). However, the approach adopted in that work would need to be extended to capture the correct quadratic halo bias and is beyond the scope of this paper. 3.4. Estimating the Bispectrum In order to estimate the bispectrum from our 21cmFAST simulation outputs, we follow the method described in Smith et al. ( 2008 ). Throughout we consider two characterizations of the wavevector arguments. In both cases we calculate the average bispectrum in linearly spaced wavevector bins. Our first characterization is for illustration purposes only. Here we fix the magnitude of and while varying the angle θ 12 between the two wavevectors. The third side of the triangle, , is determined by these three parameters and the requirement of translation invariance (.) We consider spherical bins around each wavevector of thickness Δ k = k f , where k f = 2 π / L box = 0.00785 Mpc −1 is the fundamental mode of our simulation box. Our second characterization sets a minimum ( k min ) and a maximum ( k max ) magnitude for each wavevector argument. That is, we enforce for each wavevector , along with the usual constraint that . Owing to homogeneity and isotropy, all bispectra have three arguments: , , and . Our bins are cubes of side length 4 k f in Fourier space. This characterization captures nearly all the available modes. As discussed in Section 2 , we also enforce that k 1 ≥ k 2 ≥ k 3 . In estimating bispectra, we loop through all of the simulated Fourier modes that lie within this wavevector range. We determine B 21, δ , δ and P 21, δ from the same set of Fourier modes to reduce sample variance in our estimates of Q 21, δ , δ . In general, one would like to make k min and k max as small and large as possible, respectively. However, our estimates at small k are noisy, and the perturbative formula becomes less accurate at high k . We chose values of k min and k max throughout that give the best results, but we also show results for different values of k min and k max in Appendix A . 4. Results First, we consider

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


a slit orientation to better separate the components of the counterpart, but the seeing was not good enough to resolve them. In these observations, spectra of three objects are visible. Source 2 (see Fig. 1 ) corresponds to the elongated counterpart. All three spectra (sources 1 to 3) show a continuum with many absorption lines from neutral metals. The CO bandheads at 1.62 and 1.66 μm are also detected, proving that the objects are all K- or M-type stars (see Fig. 2 for the spectrum of source 2). We find no evidence for velocity broadening of the absorption lines. This is fully consistent with what we expect for RSGs, which have rotational velocities of at most a few km s −1 , undetectable at MOSFIRE's resolution of ∼80 km s −1 in the H band. Figure 2. The normalized spectra of NGC 925 J022721+333500 (source 2, from the Dec. 2013/Jan. 2014 observation, blue line) and a 3500 K model from Lançon et al. ( 2007 , red line, shifted downwards by 0.5 for clarity). The spectrum of J022721+333500 has been shifted by −506 km s −1 and has been interpolated over wavelength ranges that were strongly affected by noise from telluric emission lines (grey shaded areas). The ‘emission lines’ visible near some of these ranges are due to this noise. The dashed lines indicate the positions of absorption lines that are indicative of late-type stars – from short to long wavelength: Mg i triplet (1.574, 1.575, 1.577 μm), Si i (1.59 μm), CO bandhead (1.62 μm), CO bandhead (1.66 μm), Si i (1.67 μm), Al i triplet (1.672, 1.675, 1.676 μm), OH (1.69 μm) and Mg i (1.71 μm). From the cross-correlation with template spectra, we find a radial velocity of 495 ± 11 km s −1 for source 1. For source 2, we find a radial velocity of 506 ± 9 km s −1 for the Dec. 2013/Jan. 2014 spectrum and 500 ± 14 km s −1 for the Nov 2014 spectrum. For source 3, we find a radial velocity of 502 ± 8 km s −1 . This source is not centred on the slit, which causes an offset in the radial velocity. The sources are too faint to be visible in our through-slit alignment images. However, brighter stars selected to check the slit alignment were detected and we use the comparison of their locations on the through-slit image with that of the deep WHT/LIRIS image (Heida et al. 2014 ) to determine the actual location of the slit on the sky. From this, we derive a best estimate of the offset between the location of source 3 and the centre of the slit of 0.5 arcsec ± 0.05 arcsec (see Fig. 1 ). This corresponds to 2.8 ± 0.3 pixels on the MOSFIRE detector, or a velocity offset of 85 ± 10 km s −1 . Because the seeing was comparable to the slit width, this value is an upper limit on the velocity offset. If we correct for this, we find a radial velocity ≥420 ± 15 km s −1 for source 3. The radial velocities of sources 1 and 2 are in agreement with what is expected from the radial velocity field of NGC 925 (de Blok et al. 2008 ). This proves that the objects are located in NGC 925, confirming the absolute magnitude

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


is consistent with a solar value of log (Al∕H) = −5.55 ± 0.03 ( Asplund et al. 2009 ). However, Al can also be present in other refractory species, in which case our derived estimate is a lower limit on the true Al/H abundance in the atmosphere of WASP-33 b. In particular, the dominant Al-containing species in a solar-composition atmosphere in equilibrium, at temperatures near 3000 K, include Al and AlH ( Woitke et al. 2018 ). Under such conditions, our retrieved AlO abundance is ~ 10 3 × higher than that predicted in chemical equilibrium with solar abundances. Future studies may investigate the feasibility of our retrieved abundance of AlO in WASP-33 b, possibly due to chemical disequilibrium or other mechanisms. Future observations may also provide better constraints on the same. 4.2 The aluminum oxide feature The main spectral feature of the optical transmission spectrum reported in this work is an increase in the planet-star radius ratio blueward of about 560 nm. In general, an increase toward short wavelengths can be caused by third light of an additional stellar companion in the aperture of flux extraction. Moya et al. (2011) detected a stellar object 2 arcsec away from WASP-33, well included within our aperture. The authors estimated that the probable physical companion of WASP-33 should have an effective temperature of about ~3000 K. As carried out in von Essen et al. (2015) , within our wavelength region we estimated that the M dwarf should be at least 10 3 times dimmer than WASP-33. This would create a variability of planet-to-star radius ratio as large as 60 ppm. This upper value is well contained within our error bars given that the median precision of the dataset is 245 ppm. Thus, it is safe to neglect the contribution of WASP-33’s companion. Also, star spots might in principle cause a spectral slope in the transmission spectrum ( Sing et al. 2011 ; Oshagh et al. 2014 ). This possibility plays no role for WASP-33 either, since it has no or a too thin outer convection zone, making star spots unlikely to occur. In consequence, we have interpreted the spectral feature shortward of 560 nm as an increase in opacity in the planetary atmosphere of WASP-33 b. In Sect. 4.1 we retrieved AlO as the opacity source of the spectral feature between 450 and 550nm. Spectral absorption features by gaseous AlO have not been reported in the literature of exoplanetobservations, though atmospheric models predict its importance in hot Jupiters ( Gandhi Madhusudhan 2018b ). AlO was described as a potential condensate forming clouds or hazes in very hot exoplanet atmospheres (e.g., Lodders 2002 ; Wakeford et al. 2017 ; Pinhas Madhusudhan 2017 ). AlO is difficult to observe for M or L type stellar objects, first because the optical spectra of these low-mass stars are dominated by TiO and VO absorption features, and second because such spectral feature at ~ 500 nm is challenging to observe for very late type stars because of their intrinsic faintness at

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


measured in the various environments described in the Section 1 (Garrod et al. 2008 ; Barone et al. 2015 ; Codella et al. 2017 ; Quénard et al. 2017 ). Those models rely on different formamide formation routes (the gas-phase reaction NH 2 + H 2 CO in Barone et al. ( 2015 ), Codella et al. ( 2017 ), and a combination of irradiated surface chemistry and gas-phase chemistry in Garrod et al. 2008 and Quénard et al. 2017 ). On the contrary, the destruction routes are common in all models and are dominated by ion–molecule reactions involving formamide and HCO + , H, H 3 O + , He + , and C + . Moreover, some reactions involving neutral radicals have been invoked, such as the one involving the OH radical (Belloche et al. 2017 ), but no experimental or theoretical data are available on the rate coefficients for those reactions, which are, anyway, expected to be very slow based on what is known on the reaction NH 2 CHO + H (Haupa, Tarczay Lee 2019 ). The proton transfer reactions with HCO + and H dominate the formamide destruction in ‘standard’ conditions, while that with H 3 O + is a dominant formamide sink in the water-rich gas. Instead, the reactions involving He + and C + ions, which provide in general a smaller contribution to the formamide destruction, proceed via charge transfer and lead to a significant fragmentation of the ionized formamide, especially in the case of He + collisions because of the large amount of energy released in the process (see for instance, the article by Ascenzi et al. 2019 for the case of similar molecules). In summary, even though proton transfer processes can lead to fragmentation, NH 2 CHOH + is expected to be also an important by-product in the main destruction route of formamide. As already mentioned in the Section 1 , the most probable fate of protonated formamide will be its recombination with interstellar free electrons. The global electron–ion recombination rate coefficient assumed in the KIDA data base, and used in all the models cited above, is given by k ( T ) = 7.5 × 10 −7 ( T /300) −0.5 cm 3 s −1 , and very nicely compares with the value calculated in this work, which is larger by only 10 per cent, contrarily to other similar cases such as the one of protonated methanimine (CH 2 NH) where the guessed values substantially differ (Yuen et al. 2019 ). Our recommendation is to update the data bases for astrochemical modelling with the present value of 8.33 × 10 −7 ( T /300) −0.5 cm 3 s −1 , which is the result of rigorous calculations rather than an assumed value. Concerning the branching ratios, our method is not yet able to determine the yield in the different exit channels and, therefore, an experimental determination, such as those achieved by using the heavy ion storage ring CRYRING (Geppert Larsson 2008 ), is required. We note, however, that it is highly improbable that all the channels mentioned in the KIDA data base have the same yield. It can be instructive to compare the case of protonated formamide

O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Database I-Database I-Database O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Database I-Database I-Database O O O O O O O O O O O O O O O


v t ) = 0.03 km s −1 and a total error on [Fe/H] of 0.03 dex. This total error takes into account all the errors on the atmospheric parameters. The two methods agree well, and we adopted σ( T eff ) = 40 K, σ(log( g )) = 0.12, σ( v t ) = 0.04, and σ([Fe/H]) = 0.03 dex as our final errors on the atmospheric parameters. Then, we choose star no. 1367 as representative of the sample, varied its T eff , log( g ), [Fe/H], and v t according to the atmospheric errors we just obtained, and redetermining the abundances. Results are shown in Table 4 , including the error due to the noise of the spectra. This error was obtained for elements whose abundance was obtained by EQWs, as the errors on the mean given by MOOG, and for elements whose abundance was obtained by spectrum-synthesis, as the error given by the fitting procedure. Δ tot is the squared sum of the single errors, while Δ tot (obs) is the error as obtained from the four repeated stars. Δ tot (obs) is equal to or smaller than Δ tot . In Table 4 , for each element, we report the observed spread of the sample (RMS obs ) with its error and in the final column the significance (in units of σ) calculated as the absolute value of the difference between RMS obs and Δ tot divided by the error on RMS obs (we choose Δ tot instead of Δ tot (obs) to be conservative). This tells us if the observed dispersion RMS obs is intrinsic or due to observational errors. Values larger than 3σ imply an intrinsic dispersion in the species chemical abundance among the cluster stars. Table 4. Estimated errors on abundances due to errors on atmospheric parameters and to spectral noise for star no. 1367 (columns 2–6). Column 7 gives the total error calculated as the root squared of the sum of the squared of columns 2–6. This total error must be compared with the total error as obtained from the 4 repeated stars (column 8) and with the observed dispersion (RMS) of the data with its error (column 9). The last column gives the significance of the difference between the total error for star no. 1367 and the observed dispersion, in units of σ. 4 RESULTS 4.1 Iron-peak, α elements and heavy elements The mean iron content we obtained is with a dispersion of Reported errors are errors on the mean. Harris ( 1996 ) gives [Fe/H] = −1.32 and we can consider the agreement satisfactory. The measured iron dispersion in Table 4 well agrees with the dispersion due to measurement errors so we have no evidence for an intrinsic Fe abundance spread. As far as iron-peak elements are concerned, Cr and Ni are solar scaled, Mn and Cu are sub-solar, and V, Co and Zn are supersolar. The α elements Si, Ca, and Ti are overabundant compared to the Sun. This is a feature common to almost all Galactic GC and halo field stars as well as to very metal-poor stars ([Fe/H] −1.5) in outer galaxies. Based on these elements, we derive for the cluster a mean α element abundance of [α/Fe] = +0.34 ± 0.01 As far as heavy elements are concerned Y, Zr, Ba, and Eu are

B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O B-CelestialObject O O B-CelestialObjectRegion O O O O O O O O O B-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O


error margin at three different observatory sites. When dealing with PRV data that supplies the photon-weighted mid-point time and yet no flux curve, the simplest assumption is that the flux was uniformly distributed in time. 2 This allows us to add a second-order correction to the calculated barycentric correction, v (〈 t 〉). To obtain 〈 v 〉, we can either subtract the result of equation ( 10 ) or get the difference from a simple numerical simulation as in Section 4 . In fact, if we do not apply a second-order correction to v (〈 t 〉), we implicitly assume that all photons arrived in the same instant at the photon-weighted mid-point time. We argue that the assumption of a uniform flux distribution is much more realistic. Fig. 5 shows how the second-order errors of three different flux scenarios are all closer to the uniform scenario than zero. In Fig. 7 , we calculate the second-order error using real exposure meter data from 1315 observations with EXPRES (see Blackman, Ong Fischer 2019 ), combining all colour channels to one. The data was obtained during commissioning between March 2018 and February 2019. No exposures were longer than 20 minutes, and as a result the second-order errors are relatively small. We plot the real second-order errors against the errors estimated with equation ( 10 ) for a uniform flux curve. The plot shows that within 1 cm s −1 , the uniform flux approximation is indeed a realistic assumption, and much more realistic than setting the error to zero. There is no indication that longer exposure times would change this picture. Figure 7. For nearly one year of EXPRES observations, we compare the actual second-order error of each measurement with the uniform flux approximation. The multichannel exposure meter data have been binned to one channel. EXPRES mainly observes bright targets, so the exposure times are short (≤1200 s), resulting in errors of only a few cm s −1 . Nevertheless, the data show that the uniform flux curve is a good approximation if we for some reason do not have access to the exposure meter flux curve when we want to calculate the second-order error. In the rare case of dealing with data that completely lacks information about the temporal flux distribution, there is no way of knowing if the individual observations have large first-order barycentric errors. One can in principle use equation ( 8 ) to estimate the RV error bars, but it requires a qualified guess at the offset (〈 t 〉 − t 0 ). In our experience, a typical offset is of order 5 per cent of the exposure time, with some larger, negative offsets from when the star was lost on the fibre before the end of the exposure. The actual distribution depends on the instrument and the local sky conditions. 6 CONCLUSIONS This paper describes how to correctly apply a barycentric correction to an exposure using the exposure meter flux curve or an equivalent measure. Rather than following the common practice of computing a single value at the photon-weighted

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


− 2.4 per cent and a dispersion of ∼ 10 per cent (8.7 per cent when adjusted for random uncertainty), in the outer r / r max > 0.2 region of the haloes. Similar level of dispersion ( ∼ 7 per cent or ∼ 6 per cent when adjusted for random uncertainty) is also observed for the case of the DM haloes of BJ05 but with positive bias of ∼ 3 per cent. In perfect conditions, a spherical system in equilibrium, we can correctly reconstruct the mass profile and there is no bias and dispersion that can be accounted by random uncertainty on the estimated mass. When applied to data from simulations, we do see some bias and dispersion, meaning the bias and dispersion can be either due to system being aspherical and out-of-equilibrium or due to an unknown effect in simulations. We find little correlation of mass offsets with asphericity, so out-of-equilibrium effects seem to be the main cause behind the observed mass offsets. The out-of-equilibrium effects will shift the mass estimates at any given radius and this will lead to a non-zero dispersion. So the dispersion, in principle, sets a limit on the accuracy with which we can expect to measure mass at a given radius for any halo using the spherical Jeans equation. This limit is set due to the inherent nature of the ΛCDM haloes and is independent of the quality and the quantity of the observational data. Using galaxia (Sharma et al. 2011b ), a stellar population synthesis software utilizing Padova isochrones (Marigo et al. 2008 ), we estimate that, till the magnitude limit of 17 in V band, Gaia will have more than 5 times the number of tracers that we have investigated here. So, the error on the mass estimates of the Galaxy using stellar tracers provided by the Gaia using the Jeans analysis will be limited by the non-equilibrium effects as well as uncertainties in observed distances and tangential velocities rather than the sample size. The fact that the two different simulations (with DM particles as tracers) give similar level of dispersion (10 per cent for surfs and 7 per cent for BJ05 haloes) in estimated masses is reassuring for our estimate of the dispersion. When applied to BJ05 stellar haloes (stellar particles as tracers), we get a higher dispersion (14 per cent). This is the case that is of practical use, as we observe stars rather than dark matter particles. The higher dispersion here is also as expected, since the stellar haloes have more sub-structures than dark matter haloes. Dispersion could also be due to random uncertainty associated with our mass estimator (effect of Poisson noise). We have shown that the random uncertainty is a factor of 2 smaller than the measured dispersion for all analysed cases. The total variance being the sum of squares of random and intrinsic scatter, the random uncertainty should make very little contribution to the dispersion. Taking the random uncertainty into account, we estimate the intrinsic scatter due to non-equilibrium effects to be 12 per cent for the case

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


different dust sizes: Ohmic diffusivity (solid line), Hall diffusivity (dotted for positive and dash-dotted for negative), and ambipolar diffusivity (dashed line) vs. radius. The plots show at time kyr, when central temperature is K and central density cm. The asterix denotes that the mean dust cross section is minimal in model S5, and the mean dust radius is not a unique characteristic of dust properties. 4.1. Sign reversal of the Hall effect Owing to the total charge neutrality condition where is the charge of species, there are only three independent parameters that can be defined as , , and (see also Sect. 2 in Xu Bai 2016 ). In Fig. 3 we show the dimensionless parameters , , and . These parameters show how smaller dust size leads to larger inequality and the transition to the dust-ion regime and even dust-dust regimes, i.e. when main charge carriers are ions and (model S3), or and (model S5). Fig. 3 Various properties of charged species and magnetic diffusivities radial profiles shown for models S1 (black), S2 (red), S3 (blue), S4 (green), and S5 (lilac). The colours refer to the same models in all following figures. Plots show, from left to right , the density of ion vs. radius, the relation of electron density to the ion density, and the relative abundances of charged dust to the ion density ( bottom ). The Hall effect term, given in Eq. ( 2 ), becomes negative when the Hall conductivity is negative, . In case of single-charged dust and local charge neutrality, Eq. ( 4 ) can be rewritten as where we reduce the dust coupling parameter to because it is insensitive to the sign of the charge, i.e. . In our simulations, we observe that the coupling parameters everywhere, and and for radii AU (see also Fig. 4 ). Thus, we simplify the equation for further and derive the threshold value for , When , the Hall effect changes sign to the negative, which may lead to the interesting MHD effects during the collapse. The Hall effect becomes negative when the number of negative charge carriers, weighted over coupling parameter , dominates the number of -weighted positive carriers. This is likely to happen when both the negatively charged dust is abundant and the magnetic fields are strong enough for the strong coupling of ions (). Fig. 4 Top : relative number densities of positively and negatively charged dust, and , compared to the threshold value in Eq. ( 9 ). Middle : coupling parameter for electrons, ions, and dust (Eqs. ( 3 )–( 5 )). Bottom : magnetic diffusivities depending on the magnetic field strength with colours showing . From the previous equations we learn that both the amount of negatively charged dust and also the quality of coupling to the magnetic fields are important. It is now a natural question to ask what happens if we change the cloud magnetisation. We consider cases with . The various magnetisations are shown with colours in Fig. 4 for models S2 and S5. We observe that the radial position of the sign reversal in Hall effect shifts

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


merger fraction in all galaxies is low, leading to the misidentification of non-mergers as mergers. Assuming 1000 galaxies in which 10% are real mergers, even a 90% accuracy will cause 100 galaxies to be incorrectly identified. In an extreme situation where these 100 galaxies are all non-mergers misidentified as mergers, then the final CNN-identified merger sample will include 200 galaxies (100 real mergers plus 100 misidentified galaxies), causing half of the sample to be contaminated by non-mergers. This contamination exists in both the AGN sample and the non-AGN control sample. To assess the influence of this contamination and given the fact that it would be extremely time-consuming to visually inspect all the mergers, we only visually inspected the 184 and 213 mergers in the MIR AGNs of the SDSS sample and the GAMA sample respectively. For the controls we also visually inspected the 1200 mergers in the non-MIR AGNs of the SDSS sample and randomly selected 600 mergers from the entire 1560 mergers in the non-MIR AGNs of the GAMA sample. The visual inspection was done independently by three of the co-authors. By selecting all mergers that have more than one, two, and all three positive votes, in Fig. 14 we find an increase of the merger excess in MIR AGNs relative to non-AGN controls for the SDSS sample and the GAMA sample. Fig. 14. Merger excess in MIR AGNs relative to non-AGN controls for the SDSS sample and the GAMA sample. The x -axis indicates the number of positive votes based on visual inspection. There are advantages and disadvantages in terms of merger identification methods using CNN and visual inspection. Our CNN merger identification method is affected by non-merger contamination, but it can work efficiently on a large sample. Visual inspection, even though it is not always reliable, is less likely to be affected by contamination, but it would have a bias toward more obvious mergers, and is very time consuming. Despite the non-merger contamination, our results are in qualitative agreement compared to previous studies using visual inspecting to identify mergers. In addition, when applying a stricter merger threshold and visually inspecting a smaller sample of mergers, our results are more consistent. 5. Conclusion We selected the SDSS DR7 spectroscopic data at redshifts 0.005 z 0.1 and the GAMA spectroscopic data at redshifts 0 z 0.6. We took advantage of deep leaning convolutional neural networks to identify mergers based on SDSS and KiDS images. We adopted two methods to classify AGNs, a WISE MIR color cut and optical emission line diagnostics, totaling 799 MIR AGNs and 5420 OPT AGNs for the SDSS sample, and 543 MIR AGNs and 2750 OPT AGNs for the GAMA sample. We also selected LERGs to analyze the connection between mergers and low accretion rate AGNs. We built a strictly matched control sample for each subgroup to investigate the connection between mergers and AGNs. Our findings are as follows: (1) In terms of AGN fraction in

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Survey O O O B-Survey O O O O O O O O O O O O O O B-Wavelength O O O B-Survey O O O O O O O O O O O O O B-Wavelength O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O B-Survey O O O B-Survey O O O O O O B-Wavelength O O O O O O O B-Survey O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O B-Formula I-Formula I-Formula O O B-Survey O O O O B-Formula I-Formula I-Formula O O O O B-Model I-Model I-Model I-Model I-Model O O O O O B-Survey O B-Survey O O O O O O O O O B-Telescope B-Wavelength O O O O O O O O O B-Wavelength O O O B-Wavelength O O O B-Survey O O O B-Wavelength O O O B-Wavelength O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


−0.61. This is approximately Δα ≃ 0.15 flatter than the measured average spectral index in radio α R ∼ −0.765. The flattest indices are obtained at the locations of assumed active particle acceleration with high shock velocities, such as near the forward shock and the south-west quadrant of the remnant. These locations also coincide with the regions that are brightest in non-thermal X-ray emission. The flattening of the spectrum at high frequencies is in qualitative agreement with non-linear DSA theory. The spectral index is, however, still steeper than the predictions of the non-linear models without magnetic field amplification (see Malkov Drury 2001 , and references therein). We assess the possibility of a cooling break being present in the remnant. We place the break frequency lower limit at ν br ∼ 5 × 10 12 Hz in the likely most affected regions. Considering 100 yr as the age of the shocked plasma, this indicates allowed magnetic field strength up to B ∼ 2–2.5 mG in the south-east/north-east and B ∼ 1 mG in the south-west of the remnant. This is higher than the average magnetic field limits inferred for Cas A, but probes the magnetic field values suggested locally by Atoyan et al. ( 2000 ). Using the age of Cas A (∼340 yr) as the age of the plasma puts the upper limits close to the magnetic field estimates obtained with other methods. Further investigations are still necessary and south-east/north-east regions are the best candidate locations for further investigation of the cooling break. Doing this may put constraints on the history of electron acceleration in Cas A. APPENDIX A: EXAMPLE OF OBTAINING SPECTRAL INDICES PER PIXEL USING MONTE CARLO SIMULATIONS In order to illustrate our procedure for measuring a spectral index for a single pixel and to prove the overall robustness of our results given the statistical and systematic errors, we show here our procedure for a given pixel, and how it leads to a likelihood range in the spectral index. We chose a pixel in the south-east location at coordinates (RA 23h 23m 391, Dec. 58 47′ 238) as our example pixel to provide a more complete picture of our data-handling process. The location is ideal to showcase the effect of the strongest effect of background subtraction as this region provides the lowest mid-infrared flux. We use a Monte Carlo approach in order to propagate the uncertainties and to obtain the result. The basis of this method is to take a value and its uncertainty of a given parameter and to simulate a normal distribution around it. In our case, we produce one-dimensional arrays of 10 000 elements. These arrays are then used as input into equations as if it would be a single value. However, as we do it on 10 000 elements we are saving the information about parameter’s possible values and effectively we also propagate its uncertainty. A.1 Background subtraction The flux in the chosen pixel is 190.1 × 10 −7 Jy px −1 with an uncertainty 9.1 × 10 −7 Jy px −1 . The mid-infrared

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


of the WASP-South data for WASP-182 from three different years (left) along with folds of the data at the possible rotational periods (right). The folded data have been binned to 20 bins, each corresponding to 1.5 nights. The blue marks are at 30 and 15 d and the dotted horizontal line at 1 per cent false alarm probability (FAP). 4 SYSTEM PARAMETERS The full set of system parameters was modelled jointly using the discovery photometry, follow-up light curves, and RV data with the MCMC code described in detail in Collier Cameron et al. ( 2007 ) and Anderson et al. ( 2015 ). The analytic eclipse expressions derived by Mandel Agol ( 2002 ) are used with a four-parameter, non-linear limb darkening law of Claret ( 2000 , 2004 ). We have interpolated coefficients for stellar temperature and metallicity of each star, and in each photometric filter. The values used were perturbed during the MCMC via T L-D , the ‘limb-darkening temperature’, which has a mean and standard deviation corresponding to the spectroscopic T eff and its uncertainty. We ran the MCMC both with the eccentricity as a free parameter and fixed to zero, to check if the results are compatible with a circular orbit. We expect most giant planets in short-period orbits to have been circularized by tidal forces, and want to avoid overestimating the eccentricity in orbits that have none. Each circular model has six fitted parameters: orbital period, P , epoch, T C , transit depth in the absences of dark limb effects, ( R p / R s ) 2 , transit duration, T 14 , impact parameters, b , and stellar RV semi-amplitude, K 1 . The RV systemic velocity γ was fitted too, and in the case of WASP-171 along with a linear RV drift, . For WASP-182, where we have data from two different spectrographs, an offset between CORALIE and HARPS was modelled as well. For each target we ran 5000 MCMC steps as a ’burn in phase’ to initialize the main phase that was set to have 50 000 iterations. At each step the free parameters are perturbed and the models are refit. If the χ 2 of the fit is better than the previous step, the current parameters are accepted, if the fit is worse, the parameters are accepted with a probability proportional to exp (−Δχ 2 ). We used Gelman–Rubin statistics (Gelman et al. 2003 ; Ford 2006 ) to check how well the chains converge. In our case the Gelman–Rubin statistics indicated that all fitted parameters were well mixed. Continuing our practice from recent discovery papers (e.g. Hellier et al. 2019 ) we treat the stellar parameters through a two-step process; we first estimate the stellar density, ρ s , from the transit duration alone, independently of stellar models. Secondly, we obtain stellar masses by using ρ s , T eff , and [Fe/H] in the stellar evolution model bagemass , as explained in Section 3.2 . The resulting stellar mass estimate and its uncertainty are finally used as input in the MCMC to derive stellar radii. The stellar density for WASP-182 is poorly constrained by the

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


of UV radiation (e.g., Jonkheid et al. 2007 ). Because the observed HCO + gas gap at a radius of 30 au coincides with the dust gap, there is a deficit of dust grains in the HCO + gas gap (Okuzumi et al. 2016 ; Pinte et al. 2016 ). Thus, the gas-to-dust mass ratio is higher in the gap, if there is no actual deficit of H 2 gas. As demonstrated in chemical models, the HCO + abundance in a protoplanetary disk with a higher gas-to-dust mass ratio of 1000 could be a factor of ten lower than in a disk with a gas-to-dust mass ratio of 100 (Jonkheid et al. 2007 ). Therefore, the HCO + column density possibly decreases in these dust gaps, even if there is no deficit of H 2 gas in the dust gaps. Future observations to measure the column density of neutral molecules in the HL Tau disk, whose abundances are less sensitive to the gas-to-dust mass ratio and UV radiation, are needed to examine this possibility. 5.2. Origins of the Infalling Spiral An infalling gas stream toward HL Tau is detected in the HCO + (3–2) emission with our ALMA observations, and its rotational motion gradually becomes dominant over its infalling motion. We note that the HCO + (3–2) line is optically thick (Figure 13 ), and the observed line-of-sight velocity of the spiral structure is inconsistent with the Keplerian velocity at radii larger than 03 (Figure 16 ). In addition, no counterpart of this spiral is detected in the continuum emission, which traces the disk mid-plane. These results could suggest that the infalling spiral is not in the mid-plane but is located above the disk surface. The velocity of the infalling spiral matches with the Keplerian velocity at a radius of 03, and there is a bright peak at a radius close to 03 (Figure 1 ). Thus, the infalling spiral could merge with the HL Tau disk at a radius of 03. Future observations of shock tracers, such as SO, SO 2 , and CH 3 OH, are needed to examine this scenario and the presence of accretion shock (Ohashi et al. 2014 ; Sakai et al. 2014 ; Yen et al. 2014 ; Artur de la Villarmois et al. 2019 ). Spiral structures connected with central disks have been observed in millimeter continuum emission and infrared reflected light around several young stellar objects (Liu et al. 2016 , 2018 ). Infalling motion along spiral structures has also been detected around other protoplanetary disks (Tang et al. 2012 ; Yen et al. 2014 ). An infalling spiral could form when a gas clump falls freely toward the center with a conserved angular momentum (e.g., Ulrich 1976 ). Free-falling material with a conserved angular momentum follows a parabolic trajectory, which appears as a spiral structure after the projection on the plane of the sky. This freefall model has successfully explained the morphology and velocity of the infalling spirals observed in the protostellar source, L1489 IRS (Yen et al. 2014 ). In HL Tau, we found that the morphology of the outer part of the infalling spiral at a radius larger than 05 can indeed be described by a

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


are due to bending and breathing modes that might have been caused from the impact of Sagittarius dwarf falling into our Galactic disc (Gómez et al. 2013 ; D'Onghia et al. 2016 , Carrillo et al. 2019 ). Further Gómez et al. ( 2017 ) have shown that bending and breathing modes must be a common feature in most disc galaxies. 6.3 De-projection of corrugations seen in edge-on galaxies Simulations (Edelsohn Elmegreen 1997 ; Gómez et al. 2013 ) show that the bending waves generated by a Sagittarius-like perturber has both radial and azimuthal dependence. So far corrugations in other galaxies have all been detected in edge-on discs (all galaxies in Table 1 except Milky Way) where we see the combined effect of radial and azimuthal dependence. Looking for corrugations in less inclined discs is not an option, unless we look in the velocity space (Alfaro et al. 2001 ; Sánchez-Gil, Alfaro Pérez 2015 ). However, Kamphuis et al. ( 2013 ) combine the HALOGAS data with non-axisymmetric modelling (using a version of TIRIFIC) and deproject H i observations of edge-on galaxy NGC 5023. They find that their data are best explained by an edge-on disc showing radial corrugations (see their fig 10). Hence we stress the importance of non-axisymmetric modelling here as the need of the hour because, with the availability of high-resolution H i data on scores of nearby edge-on galaxies, a statistical study of corrugations is only a small step away. 7 CONCLUSION This study has focused mainly on characterizing corrugations in the dust lanes of a sample of five edge-on galaxies and understanding their physical state. Partial dust waves are seen in many galaxies but only a handful of galaxies show waves long enough to be characterized. This paper is one of the first attempts to give a quantitative estimate of their size and scale as well as address the issue of whether they are stable. In order to strengthen our analyses, we have added four more galaxies including the Milky Way where corrugations have been previously measured in stellar and gaseous discs. The fact that corrugations are seen in stars, gas and dust leads to the possibility of gravitational instability as the likely origin. In fact, five of the nine galaxies that we have tabulated are visibly perturbed (bar/tidal interaction). Our main conclusions from this work are the following: We present the first systematic study of dust corrugations in edge-on galaxies and characterize them by measuring their wavelengths and amplitudes. The amplitude of corrugation seems to depend on the host galaxy mass. The wavelength of corrugation does not show any dependence on host galaxy mass. It is roughly about the stellar disc scale length for most of our sample. By carrying out a stability analysis, we show that the corrugations with wavelengths larger than the scale length of the stellar disc, are stable. We find indications that corrugations can be as common as warps. We also find mounting evidence that corrugations play a

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