we apply the formalism of Brogi Line ( 2019 ). We compute the log-likelihood function of each tested model by combining the data and model variances, and the cross-covariance of data and model according to their equation (9). We assume the same total observing time as for the data set presented in Benneke et al. ( 2019 ), which is 8.4 h. Since HRS does not necessarily need out-of-transit data, this integration time would allow us to observe four full transits of the exoplanet. Our results are shown in Fig. 5 . We can successfully and confidently (>5σ) recover the H 2 O signal for all instruments at the expected values of K p and V sys , as shown by the fact that the injection values (white crosses) are within or just outside the 1σ confidence interval. Given the relatively strong detection achieved, we conclude that the best-fitting atmospheric scenario for GJ 3470b would be easily detectable with current HRS with a relatively modest investment of telescope time. The size of the confidence intervals also suggests that GIANO and SPIRou are, respectively, the least and the most sensitive instruments based on our simulations. Note that CARMENES does not cover the K band unlike the other two spectrographs but does still show a strong H 2 O detection because of its high throughput. Figure 5. Detectability of the best-fitting atmospheric scenario for exoplanet GJ 3470b as derived by Benneke et al. ( 2019 ), based on 8 h (4 transits) of simulated observations with three near-infrared high-resolution spectrographs. Shown in the plot are confidence intervals on the rest-frame velocity and radial-velocity amplitude of the planet after the likelihood of Brogi Line ( 2019 ) has been applied to the simulated data. The white cross indicated the simulated values. 3.2 Constraining H2O and cloud decks – GJ 436b We now discuss how HRS may be able to constrain abundances and cloud decks on GJ 436b using the grid of model spectra discussed in Section 2.1.2 . We chose this planet as observations of the primary transit have revealed a largely featureless spectrum (Knutson et al. 2014a ). The only constraints on H 2 O thus far have been derived from low-resolution secondary eclipse observations (Stevenson et al. 2010 ). In Fig. 6 , we show three representative high-resolution transmission models with varying H 2 O and cloud top pressures but which resulted in similar muted HST WFC3 features. We studied their detectability with CARMENES, GIANO, and SPIRou, and the ability to discriminate cloudy scenarios from high-metallicity scenarios as shown in Fig. 2 . Figure 6. Model transmission spectra and binned HST WFC3 spectral points for GJ 436b. These are shown for three cases with varying cloud pressures and H 2 O abundances. The high-resolution spectra have been binned to the Knutson et al. ( 2014a ) observations, which are also shown with their associated error bars. We offset the models in the figure to highlight the spectral features. Fig. 6 shows the three


compensate for attenuation; a narrower attenuation region would imply a less steep radial dependence of the CO(3–2) effective emissivity. (vi) Values of the intensities integrated over the source size but not corrected for the loss of short spacing have been given for both CO(3–2) emission, 7.1 ± 0.4 Jy km s −1 , and continuum emission, 11.4 ± 0.5 mJy. Crude estimates of the short-spacing correction have been given at the respective levels of 1.6 Jy km s −1 and 5 mJy. The gas value is in good agreement with the value measured earlier by Dent et al. ( 2005 ) but the dust value is larger than that measured by Song et al. ( 2004 ) and close to the predictions of the disc models proposed by Hughes et al. ( 2008 ) and Roberge et al. ( 2013 ), according to which the CO mass of the disc is between ∼2 × 10 −4 and ∼10 −3 Earth masses and the 0.86 mm SED value is ∼19 mJy. The crudeness of our estimated short-spacing correction prevents a reliable resolution of this apparent disagreement, which will require new lower resolution observations. The use in this work of an effective emissivity to model the detected emission mixes the effects of temperature and density and the new results do not justify repeating the detailed analysis that Hughes et al. ( 2008 ) and Roberge et al. ( 2013 ) have made of the disc structure and chemistry for both gas and dust. (vii) Kinematics has been found to be Keplerian over a very broad range of distances from the star, typically 20–200 au, with a power index of and a rotation velocity of 5.1 ± 0.1 km s −1 at a distance of 60 au (1 arcsec) from the star, corresponding to a mass of 1.73 ± 0.10 solar masses for the central star. The observed line width was observed to be dominated by Keplerian shear over most of the disc volume and upper limits to the intrinsic line width (FWHM) have been evaluated, decreasing from 1.4 ± 0.3 km s −1 for | x | 1 arcsec to 0.8 ± 0.1 km s −1 for 2 | x | 3 arcsec. 6.2 Implications on the disc structure and chemistry (Hughes et al. <xref ref-type="bibr" rid="bib9">2008</xref>) The very detailed analysis of the chemistry that governs the morphology and temperature of the disc that was made by Hughes et al. ( 2008 ) remains essentially valid. However, some of the assumptions made by these authors need now to be modified; the inclination angle with respect to the line of sight is definitely non-zero; the dust extends to significantly larger distances from the star than the gas does, both radially and axially; the inner radius of the gas disc is nearly equal to the gravitational radius, ∼20 au, rather than twice this value; the lumpiness of the dust disc contrasts with the smoothness of the gas disc; the apparent warping of the gas disc suggested by the SMA data is not confirmed by the ALMA data; the central attenuation, for which the ALMA data gives strong evidence, can no longer be ignored. Yet, the analysis made by Hughes et al. ( 2008 ) of the factors governing the physics and chemistry of the gas


and φ is longitude. This is an important property for the analysis to follow. Because the eigenfunctions are a complete set (with zero Lagrangian-pressure perturbation on the outer boundary, e.g. Christensen–Dalsgaard 2003 ), a general wavefield may be written as where ω is temporal frequency, a q are coefficients representing the phase and amplitude of the q th mode and we write for the sake of notational convenience. The governing equation for driven, undamped oscillations is given by where is the unperturbed source, the zeroth-order wavefield and is the unperturbed operator (whose properties are derived from a basic solar model such as model S; Christensen-Dalsgaard et al. 1996 ) as defined in equation ( 2 ). Note that the MDI and HMI missions provide time series of spherical-harmonic coefficients of oscillations, i.e. a k (ω), measured at the solar photosphere. In practice, because we only observe the side of the Sun that faces us, it is not possible to precisely spatially decompose the measured oscillation signals into individual spherical harmonics. This partial visibility leads to leakage between modes across l, introducing systematical errors in eventual inferences (e.g. Larson Schou 2015 ; Woodard 2016 ). Further, at a given frequency (between ridges for instance), many different radial orders contribute, leading to diminished ability in distinguishing between them. Temporal gaps in observations, due to instrumental or other issues, lead to mixing across temporal frequencies (and hence radial orders), thereby exacerbating this issue. To mitigate the problem of radial-order leakage, we use measurements at frequencies within a linewidth of resonance, ensuring that the associated radial order contributes a dominant fraction of observed mode power. Nevertheless, leakage across radial orders and spherical-harmonic degrees will systematically influence the results of mode-coupling analyses and needs to be carefully considered. 2.1 Perturbation theory The effects of rotation, flows, magnetic fields or thermal asphericities are modelled by adding a perturbation to the operator to in equation ( 5 ), in turn leading to a perturbation to the wavefield, . Ignoring perturbations to the source , we obtain for the wavefield in a perturbed Sun. For time-varying perturbations, behaves as a convolution operator. To the first order, we have therefore where the convolution over frequency captures the time variability of the perturbation operator (equation D4 of Appendix D ). Perturbing equation ( 4 ) while assuming that the eigenfunctions are unmodified by the perturbation, we may express the change to the wavefield as a linear combination of the original set of eigenfunctions, Using equation ( 2 ), the left hand side of equation ( 6 ) may be simplified thus Solar modes are subject to a small amount of attenuation γ k ≪ ω k that perturb the eigenfunctions and eigenfrequencies (obtained from equation 1 ) by introducing small imaginary components. However,


determined by the resonant conditions ( 52 ) and ( 53 ) are of order unity (see ( 42 ). In the opposite short-length limit | k | ≫ 1, the eigenvalue relation ( 30 ) yields: , k, n ≠ 0, n = 1, 2, ..., i.e. the linear growthrates are also small in the radially short-length limit, , e.g. | k | ≫ 1 for n ∼ 1 and | q | ∼ 1. In the latter case the eigenfrequency is diminished together with the growthrate. Now, let us discuss the three-wave resonant interaction in the presence of the short-length modes. First, note that a resonant triad cannot be formed by three short-length modes because of the frequency resonant condition ( 53 ) that is degenerated with all three small frequencies which can be included in the mismatch value, δω. Second, the resonant condition ( 53 ) is also degenerated for two short-length modes with | k | ≫ 1 interacting with the single long-length mode with | k | ≪ 1, since all they have zero frequencies. Finally, evidently that the radial wavenumber resonant condition ( 52 ) cannot be satisfied by the single short-length term with | k | ≪ 1 and two long-length terms with | k | ≫ 1 (at axial wavenumber n ∼ 1). Thus, resonant triads cannot be formed by interacting short- and long-length modes. In particular, this means that energy transfer from very short- to very long-length modes and vice versa cannot be realized with three-wave resonant interaction. Rigorous analytical solutions of the reduced weakly non-linear VSI problem discovered in the present study can serve as a test for numerical modelling of the full non-linear hydrodynamic problem, since the intermediate non-linear stage of VSI should be inevitably crossed before saturation. Predictions of the reduced VSI model allow to identify and discover physical mechanisms that are driving the VSI and its transition to turbulence, as well as identify tuning parameters of the physical and numerical models of the full problem. Figure 1. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class I. The two-mode regime of the resonant triad evolution. n (1) = 1, n (2) = 1, n (3) = 2 (see Table 1 ); R = +1; S = +1; , φ 0 = π/4, δω = +1. The scaled resonant frequencies and radial wavenumbers are in Table 1 with and , , ( j = 1, 2). Dashed and solid lines are solutions of the linear and non-linear systems ( 71 )–( 74 ), respectively, for triad amplitudes. Figure 2. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class II. The two-mode regime of the resonant triad evolution as in Fig. 1 , but for S = −1. Figure 3. Triad amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class III. The three-mode regime of bounded oscillations of all interacting modes as in Fig. 1 , but for R = −1. Figure 4. Amplitudes α ( j ) versus slow time θ for the triad of class I as in Fig. 1 ( S = +1, R = +1), but at small initial amplitudes . Figure 5. Amplitudes α ( j ) versus slow time θ for the triad of class II as in Fig. 2 ( S = −1, R = +1), but


X-ray peak and a separation from continuum X-ray dominant region (DeLaney et al. 2004 ; Helder Vink 2008 ). On the other hand, we defined “south west,” “inner”, and “forward shock” regions as the “non-thermal dominant” region (light blue regions in Figure 2 ). Helder Vink ( 2008 ) and DeLaney et al. ( 2004 ) show the area dominated by a harder spectrum. As a matter of fact, these regions manifest themselves in hard X-rays. Our image also shows the inner region is bright with hard X-rays. In addition, the forward shock has a featureless non-thermal emission (e.g., Hughes et al. 2000 ; Bamba et al. 2005 ). In the forward shock, it was found that the average proper motion of Cassiopeia A is 030 yr −1 (Patnaude Fesen 2009 ), so we corrected the forward shock region of each year with this value (see the Appendix ). 3.2. Spectra With the regions defined in the Section 3.1 , we extracted spectra using a custom pipeline based on specextract in CIAO script. Here, the background spectra were extracted from outside of the whole SNR region defined in Figure 2 . Since X-ray emission from Cassiopeia A is very strong, the background contribution is almost negligible for a time variation estimation. The background fraction of the whole SNR is only ∼3% in the 4.2–7.3 keV band and this is almost constant value from 2002 to 2013. In 2000, this fraction shows a larger value (∼7%) presumably because of an increase in the charged particle flux experienced during this observation. After background subtraction, we fitted the 4.2–7.3 keV band spectra at each epoch with a power-law model and a Gaussian line with XSPEC version 12.8.2. The best-fit parameters are summarized in Table 2 . Table 2 Best-fit Parameters of the ACIS Spectra a Note. a The errors are at 1 σ confidence level. Figure 3 shows the spectra in the 4.2–7.3 keV band taken from the six regions. As described in Section 3.1 , the two bright non-thermal dominant regions south west and inner are remarkable in the hard X-ray continuum flux (Figures 1 and 3 ). Accordingly, the photon indexes of these regions are ∼2.6–3.0 while those of the thermal regions are 3.0–3.4. In the forward shock and whole SNR, photon indexes and evolutions are slightly different from the results in Patnaude et al. ( 2011 ). This is probably due to a difference in the energy band used in the spectral fitting. The fit residuals are larger for the thermal dominant regions than for the non-thermal regions. This is because weak thermal lines such as Cr–K (5.6 keV) appeared in the band. Figure 3. Time variation of the equivalent width of the Fe–K line (plot in the middle) and observed spectra in the 4.2–7.3 keV band (on both sides) at each region (whole SNR: WS, east: E, inner: I, north west: NW, south west: SW, forward shock: FS). In the central plot, the solid lines show best-fit linear models. Individual spectra were fitted with a model composed of a power law and a Gaussian. The error bars of all figures are 1 σ . As shown in Figure 3


by myriads of DM particles. This is the reason why two convergence criteria are applied to the CDM numerical simulations to declare their results reliable (e.g. Power et al. 2003 ). First, the finite mass of the DM particles used in the simulation artificially shortens the two-body relaxation time-scale; and this artificially-set time-scale is asked to be longer than the age of the Universe, t U . This requirement prevents the thermalization of the resulting DM haloes. Secondly, the artificial thermalization starts in the centre of the DM haloes so the simulated centres are discarded within the so-called convergence radius. However, when these two restrictions are relaxed then cores with polytropic shape appear in the DM haloes produced in the numerical simulation. In this paper, we analyse two types of numerical simulations showing such cores. If the DM is assumed to be self-interacting (SIDM) then the effective relaxation time-scale t SIDM is shortened by an amount that depends on the SIDM cross-section. As soon as , t SIDM ≲ t U and the simulations produce cores. We show in Section 4 how different SIDM from several authors produce cores which after normalization are very similar and equal to the core of a polytrope within error bars. The simulations comprise haloes that go all the way from dwarf galaxies to galaxy clusters. This agreement between simulated haloes and polytropes holds even when baryons are included in the simulation (Section 4.1 ). If standard CDM numerical simulations are analysed within the convergence radius, they also show cores (e.g. Wang et al. 2020 ). These cores are also indistinguishable from polytropes, a result which holds even with halo masses differing by as much as 10 orders of magnitude (Section 5 ). Outside the convergence radius, the profiles follow the canonical NFW or Einasto shape. Thus, because the profiles within the convergence radii are excluded from the standard analysis, it is easy to understand why the profiles resulting from these numerical simulations seem to be set by the initial conditions during the big bang (Cen 2014 ; Nipoti 2015 ; Brown et al. 2020 ). As a by-product of this work, we have shown how unresolved DM clump–clump collisions cannot thermalize the DM haloes within a time-scale comparable to the age of the Universe (Appendix A ). We conclude that the presence of cores in simulated or observed density profiles can used as evidence for systems in thermodynamic equilibrium. Cores are theoretically predicted and also generated in self-gravitating system known to be in thermodynamic equilibrium. APPENDIX A: TWO-BODY COLLISION CROSS-SECTION FOR A CONTINUOUS DISTRIBUTION OF UNRESOLVED DM CLUMPS Starting from the primordial density fluctuations, the gravitational collapse produces a full distribution of DM haloes with masses going from galaxy clusters to Earth-mass clumps and even smaller (e.g. Diemand, Moore Stadel 2005 ; Wang et al. 2020 ). However, numerical simulations cannot treat all


regression with an objective datapoint rejection, which models the outlier distribution. The method infers the parameters of the linear fit together with the mean and variance of the distribution of outliers as well as the number of outliers at the same time. The model is run through 50 000 MCMC realizations, taking the maximum of the posterior distribution and the standard deviation as the best values of the slope, intercept, and their uncertainties. We did not attempt to perform the fits when the number of stars with abundances is smaller than 15 or when the uncertainties in the abundances are greater than 0.05 dex. In Fig. 10 we plotted some examples of the differential element abundance pairs δ X 1 vs. δ X 2 . The full matrix of differential abundances for the three clusters are found in Figs. B.3 – B.5 . We also overplotted the obtained fits. For the Hyades, we obtain statistically significant correlations (larger than 3 σ ) of all pairs of abundances that we analyze. Some elements with larger intrinsic uncertainties (e.g., Ba, Nd, Cu) tend to present less significant correlations. For the other two clusters, we have a smaller sample of stars selected as most probable members and larger uncertainties in the abundance measures. The less significant fit is the one of Na I vs. Mn I in NGC 2632, which has a posterior with two peaks. For Ruprecht 147, two fits involving Si I have lower significances than 3 σ . In Fig. 10 we see no dependence of the differential abundance pairs with respect to T eff , showing that all types of stars correlate in the same way. Fig. 10. Left : δ Ti vs. δ Si. Right : δ Mg vs. δ Fe, rows correspond to the three clusters. Color represents effective temperature. Translucid lines are the resulting linear fit according to the obtained posterior distribution in the intercept and the slope. The slope and its uncertainty is indicated in each panel. For the Hyades, a large number of slopes were retrieved, so we computed the compatibility among all of them within the quoted uncertainties: (i) The fits involving Cu I and Ba II that have non-Gaussian posterior distributions tend to not be incompatible with the others. However, in these cases, uncertainties are not well represented by the standard deviation. (ii) In the case of Nd II , all the slopes are larger than 3 σ when compared to most of the other cases. We attribute this to the larger uncertainties of this element in comparison with the others, which makes the dispersion in its dimension increase, and thus increase the value of the slope. (iii) We find compatible slopes for all other pairs of elements. This indicates that the differences in star-by-star abundance are due to a zero-point difference in the overall metallicity among the stars in the cluster, with no variation depending on the chemical element. 6.3.2. Comparison with synthetic data To better interpret our results, we generated a set of simulated spectra of stars in two cases: a chemically homogeneous


is time independent. This is the case for GRB 160625B as presented in the right panel of Figure 8 . Moreover, typically is observed to increase over a pulse (Ryde Pe’er 2009 ); this is the case for most bursts in our sample, with two prominent cases of GRB 140329B and GRB 150330A. To derive the parameters in the rest frame, the selected bursts should have known redshift. A redshift measurement of 2.73 is found for GRB 140206B (Malesani et al. 2014 ), and 1.406 for GRB 160625B (Zhang et al. 2018 ). However, GRB 140329B and GRB 150330A are without known redshift. On the other hand, Iyyani et al. ( 2013 ) investigated how the different values of redshift affect the results and pointed out that, for different values of z , the estimated outflow parameters change within a factor, but the time evolution of the behavior of the parameters remains the same. We therefore assume an average value of GRBs (Bagoly et al. 2006 ), z = 2, for the calculation (see also Iyyani et al. 2013 ). 3.1.3. Lorentz Factor, Γ The coasting values of the Lorentz factor ( r ph > r s ) are derived by (see also Equation (4) in Pe’er et al. 2007 )where Y is the ratio of total fireball energy to the energy emitted in gamma-rays. As depicted in Figure 9 , temporal evolution of the Lorentz factor shows either an approximately constant (e.g., GRB 140206B) or a monotonic decay behavior (e.g., GRB 140329B exhibits a monotonic decline throughout the pulse), or both (initially approximately constant and then decaying faster, e.g., GRB 150330A, GRB 160625B). The best fitting gives power-law indices of 0.01 ± 0.07 for GRB 140206B and −0.51 ±0.03 for GRB 140329B. We fit GRB 150330A and GRB 160625B with a smoothly broken power-law function, which gives best-fitting power-law indices of 0.69 ± 0.52 (before the break) and −0.52 ± 1.44 (after the break) for GRB 150330A and 0.37 ± 0.27 (before the break) and −0.89 ± 0.33 (after the break) for GRB 160625B. Figure 9. Temporal evolution of the Lorentz factor, Γ. Color conventions are the same as in Figure 8 . GRB 140206B presents a constant behavior with best-fitting power-law index 0.01 ± 0.07, while GRB 140329B shows monotonic decay with index −0.51 ± 0.03. GRB 150330A and GRB 160625B exhibit a smoothly broken power-law decay feature, with power-law indices 0.69 ± 0.52 and −0.52 ± 1.44 before and after break for GRB 150330A and 0.37 ± 0.27 and −0.89 ± 0.33 before and after break for GRB 160625B. The global view on the evolution of the Lorentz factor decays with time from a few hundred Y 1/4 down to below 100 Y 1/4 (e.g., Fan Wei 2011 ; Peng et al. 2014 ). The decreasing Lorentz factor may have several implications (Iyyani et al. 2013 ). Since typically is observed to increase over a pulse, the decrease of the values of Γ = Γ( t ) is not surprising by realizing the fact that ; such a decrease must, therefore, be a common characteristic form over individual pulse structures in GRBs (Ryde Pe’er 2009 ). This is because, during the rising


of the two mini-BAL outflows in our sample. The Si iv trough is quite narrow and the S iv trough is unblended with other absorptions. In this case, the excellent template fit of the S iv trough is evident by the value f ( χ 2 ) = 0.4%. The S iv * absorption trough is heavily contaminated by a strong, likely Ly α , absorption trough from an intervening system. SDSS J0941+1331 has a multicomponent Si iv absorption trough. Following the procedure outlined in Section 4.2 , we use only the marked velocity interval as the template (see Figure 2 ). The Si iv template fits the S iv and S iv * absorption troughs well with f ( χ 2 ) = 6.8% in our survey. In order to account for possible systematic differences due to the Ly α forest, we compare the derived N ion from template fittings to the pure AOD extractions that includes the Ly α absorption features. In Figure 3 , we show the comparison of the observed S iv 1062.66 Å and S iv * 1072.97 Å absorption troughs. The AOD column densities integrated within the vertical dotted lines are N(S iv ) AOD = 1620 and N(S iv *) AOD = 740 (hereafter, all N ion are in units of 10 12 cm −2 ), while the template fitting results integrated over the same velocity range are N(S iv ) tmpl = 1450 and N(S iv *) tmpl = 540. Note that the AOD method overestimates the N ion because it includes the contributions from Ly α forest absorptions. These values yield a density sensitive ratio, N(S iv *)/N(S iv ), of 0.46 and 0.37 for the AOD method and the template fitting method, respectively. Using the same methodology described in Section 4.2 , the AOD method leads to a decrease of R by 25%. This difference is similar to our errors for the determined R value in Table 4 . Therefore, possible Ly α forest contaminations affect our derived R only moderately. Figure 3. Comparison of the normalized flux of S iv 1062.66 Å(blue) and S iv * 1072.97 Å(red) absorption troughs for SDSS J0941+1331 in velocity space. The vertical dotted lines are the range of Si iv template corresponding to Figure 2 . For SDSS J1135+1615, the Si iv absorption troughs are saturated and rather flat. We choose the two strongest local maxima to construct the Si iv template (see Figure 2 ). Fitting the S iv and S iv * absorption troughs with this template shows that the S iv * absorption trough is deeper than that of S iv . Including the many narrow Ly α troughs in the N ion extractions increase the N(S iv ) and N(S iv *) by 16% and 14%, respectively, and only marginally affects the derived R (6%). 5. Photoionization Analysis In order to derive the physical conditions of the outflow, we run the spectral synthesis code Cloudy (version c17.00, Ferland et al. 2017 ) to generate a grid of photoionization simulations. Cloudy self-consistently solves the photoionization and thermal equations necessary to model the physical conditions in the gas. The absorbers are modeled as plane-parallel, thin slabs of constant density gas with solar metallicity. They are irradiated by


12 . Panel (a) shows that current sheet disruption takes place earlier with a lower growth rate in Run H5, where the initial noise level is three orders of magnitude higher than that of Run S5. Because current sheet disruption marks the onset of fast reconnection, the onset also occurs earlier in Run H5. However, as can be seen from panel (b), the reconnection rates after nonlinear saturation are essentially the same for the two runs. Therefore, we can conclude that reconnection rate after saturation is nearly unaffected by the noise level and the condition at the onset. This conclusion is also in line with the earlier study by Huang Bhattacharjee ( 2010 ), where the dependence of the averaged reconnection rate on the external forcing amplitude is found to be fairly weak. Figure 12. Comparison of time histories of (a) the linear growth rate and (b) the reconnection rate for Run S5 (with an initial noise level ) and Run H5 (). The initial noise level affects the disruption time and the linear growth rate at disruption, but the reconnection rate after nonlinear saturation remains approximately the same. From an observational point of view, the period between current sheet disruption and nonlinear saturation is especially interesting. During this period, the outflow profile changes from a nearly monotonic Sweet–Parker-like profile to a highly fluctuating profile, as shown in Figure 13 (a). Accordingly, the histogram of the outflow profile changes from double-peak-shaped to triangle-shaped, as shown in Figure 13 (b). A triangle-shaped histogram is a natural consequence of the outflow profile being highly fluctuating. To understand this, we consider an outflow profile of the form , where and the first term mimics the Sweet–Parker-like outflow profile, and the second term mimics short-wavelength fluctuations. If the fluctuation is much smaller than the background, i.e., , the histogram is double-peak-shaped. Conversely, if the fluctuation is on the order of the background, i.e., , which is the case in the saturated phase, then the histogram is triangle-shaped. The transition from a double-peak-shaped histogram to triangle-shaped histogram can lead to observable effects through spectroscopy of emission spectral line profiles, because the observed spectral line profile is a convolution of the Doppler-shifted line profile along the line-of-sight direction. For reconnection events with outflows along the line-of-sight direction, the histogram of the outflow profile can be viewed as a proxy for the shape of the line profile. A triangle-shaped line profile has been reported in IRIS (Interface Region Imaging Spectrograph (De Pontieu et al. 2014 )) observations of the Si iv line during transition region explosive events (Innes et al. 2015 ). With the high-cadence mode of the IRIS spectrograph, it is possible to observe transitions from a double-peak-shaped line profile to a triangle-shaped line profile in transition region explosive events (L.-J. Guo


certain criteria. As an independent check the scatter in the DM was investigated among the models with a reduced χ 2 less than twice the minimum value. If there were five or more such models the dispersion (actually 1.48 ⋅ MAD) around the median was determined and compared with the formal error. This exercise revealed no systematic effects and the errors estimated in such a way are consistent with the formal errors. 6.2. Comparing dEBs and TRGB with Cepheid and RR Lyrae distances In Sect. 5.1 the TRGB distances are compared with the distances to 14 dEBs. One can also compare the TRGB distances with other independent distance estimates, as we did in Sects. 5.2 – 5.4 . We therefore took an identical approach as in Sects. 5.2 and 5.3 and determined the median DM and reddening value of CCs (in the LMC), and RRL (in the SMC) in the direction of the dEBs. The results are listed in Table 3 which first repeats the DM and reddening derived in the literature for the dEBs and the TRGB distance (based on the second-order derivative method) from Table 1 . Columns 5 and 6 give the DM and reddening values based on the CCs and RRL in those fields. Table 3. Comparison of distances for the dEBs. Notes. Columns 1–4 are taken from Table 1 . They indicate the name of the system, the DM and reddening based on the works listed in Col. 4 of Table 1 , and the TRGB distance using that reddening. Columns 5 and 6 list the DM and reddening of CCs (for the LMC objects) and RRL (for the SMC objects) in the direction of the EBs, and Col. 7 lists the TRGB distance using the reddening in Col. 6. Similarly, Columns 8 and 9 give the DM and reddening in Rubele et al. (2018) in the direction of the EBs, and Col. 10 lists the TRGB distance using the reddening in Col. 9. Finally, Col. 11 lists the reddening derived from Haschke et al. (2011) in the direction of the EBs, and Col. 12 lists the TRGB distance using that reddening. It is evident that the reddening estimates are smaller than adopted in the dEB analysis. In the SMC this is the case for all five objects. Although the differences are within the respective error bars it appears to be a systematic effect. In the LMC this is the case for eight out of nine objects but the differences appear to be smaller on average than for the SMC. To test the effect of reddening, the TRGB distance was derived using the E ( B − V ) from col. 6, and the results are listed in col. 7. It is clear that the effect on the DM is roughly inversely proportional to a change in E ( B − V ). Based on the definition of the sharpened magnitude, the absolute calibration equation (Eq. ( 3 )) and the reddening coefficients one expects a relation ΔDM/Δ E ( B − V )= − 1.1. The overall effect is noticeable however. The weighted mean DM of the nine LMC dEBs is shifted from 18.483 ± 0.006 mag to 18.523 ± 0.005 mag, and that of the five SMC binaries is shifted from 19.023 ± 0.007 mag to 19.051 ± 0.009 mag. In a similar way we used the data of Rubele et al. and took


than those within the UV/optical, and Figure 3 (top) indicates a diversity of X-ray/optical lags, with the X-rays leading the UV in three cases and lagging in the fourth. This poses a severe problem for the reprocessing model, for which no simple solution is currently apparent. Swift is the workhorse without which IDRM cannot (currently) be successfully performed on AGN, as it is the only fully operational observatory that provides coverage throughout the X-ray, UV, and optical regimes necessary for this experiment. However, it is also very helpful to expand coverage to longer wavelengths and higher spectral resolution than that afforded by the UVOT alone. We note that simultaneous ground-based optical monitoring has been performed on all four of these targets, and the first data set (on NGC 5548) has already been published (Fausnaugh et al. 2016 ). Reduction and analysis of optical photometry and spectroscopy on the other three is ongoing and will be published shortly. This should provide further constraints on the accretion disk fits (especially the overall size) and possibly on the effects of line and diffuse emission from the BLR. Note as well that HST can also play a crucial role, providing much higher spectral resolution that can allow direct detection and modeling of the BLR continuum component, and extending the UV coverage to wavelengths as short as ∼1150 Å. That has been used to resolve the excess lag around the Balmer jump in one of these targets (NGC 4593, Cackett et al. 2018 ), and it is expected to play a prominent role in future campaigns. These unprecedented data also reveal the need for improvements in our time-series analysis tools. The uncertainties on lags output by JAVELIN (Zu et al. 2011 ) are on average a factor of ∼2.5 smaller than the same quantity output by the ICCF FR/RSS technique (Peterson et al. 1998 ). The precise cause of this discrepancy has not been determined, but given the fact that the older ICCF FR/RSS technique is more conservative in its assumptions and results, it and not JAVELIN was utilized to derive the results reported above. More generally, there is no interband lag tool currently in use that allows separation of two distinct interband lag signals, as seems to be the case in the UV/optical, where short lags from the accretion disk and longer lags from the BLR appear to be present. Ultimately these techniques will need to be developed and directly compared so as to determine which are more reliable and suitable for studying AGN interband lags, but that is beyond the the scope of the current paper. These results also highlight the need for theoretical progress, especially in understanding the X-ray emission component(s) and relation to the disk and BLR that emit at lower energies. These observations present severe problems for the “lamp-post” reprocessing model. Can it be modified or must it be discarded? If it is the latter then what will take its place? The current set of reduced data are


Long Baseline Array (VLBA) images at 43 GHz (Agudo et al. 2011 ). A quasi-stationary feature within 2 mas of the core associated with a bent jet, was analyzed using Global mm-VLBI Array (GMVA) observations at 86 GHz (Hodgson et al. 2017 ). The spectral index showed different behavior between the core and stationary component at millimeter wavelengths. The spectral index of the core at 15–43 GHz and 43–86 GHz was inverted. When the core was flaring, the spectral index changed to become steep (Hodgson et al. 2017 ). In this paper, we report results from simultaneous multiple frequency VLBI observations of OJ 287 using the KVN at 22, 43, 86, and 129 GHz, from 2013 January until 2016 December. The paper is organized as follows. In Section 2 , we present the description of the observations for both the KVN data and all archival data used in this paper. Sections 3 and 4 describe results and various analyses of the flux density variability at multiple frequencies, respectively, and Section 5 includes a discussion on the results and analysis. Finally, we summarize this paper in Section 6 . We use luminosity distance D L = 1630 Mpc at a redshift of z = 0.306, assuming H =68 , , and = 0.302 (Spergel et al. 2015 ; Hodgson et al. 2016 ). Spectral index α is defined as . 2. Observations and Data Reduction 2.1. KVN Observations OJ 287 was observed by the KVN, which is a 500 km scale VLBI network consisting of three 21 m radio telescopes located in Seoul (KVN Yonsei), Ulsan (KVN Ulsan), and Jeju (KVN Tamna), Korea. The observations were conducted at 22, 43, 86, and 129 GHz bands simultaneously with a mean cadence of roughly a month since 2012 December except for maintenance seasons from June to August. The angular resolution of the KVN is 6 mas at 22 GHz, 3 mas at 43 GHz, 1.5 mas at 86 GHz, and 1 mas at 129 GHz. We used data obtained in 31 epochs in the period from 2013 January 16 (MJD 56308) until 2016 December 28 (MJD 57750). In each epoch, the source was observed with up to eight 5 minute scans per 24 hr. The observations were recorded in left circular polarization mode at a recording rate of 1 Gbps. The observing frequencies are 21.700–21.764, 43.400–43.464, 86.800–86.864, and 129.300–129.364 GHz, with a total bandwidth of 256 MHz (thus 64 MHz in each band). Detailed descriptions of the iMOGABA observations are reported in Lee et al. ( 2016b , 2017a ). 2.2. Data Calibration The obtained data were processed with the DiFX software correlator in Daejeon, Korea (Deller et al. 2007 , 2011 ), and the data was calibrated following the standard procedures (phase and amplitude calibration, fringe fitting, and bandpass correction) in the AIPS package from the National Radio Astronomy Observatory (NRAO). For the calibration, the KVN pipeline was used (Hodgson et al. 2016 ). The visibility phases in the 22 GHz band were transferred to those in the higher frequency bands (i.e., 43, 86, and 129 GHz bands) using the pipeline in order to increase the coherence time


of the CCH/HC 5 N ratio between the observed values and the model calculations (Taniguchi et al. 2019a ). The red and black curves show the modeled CCH/HC 5 N ratio and temperature, respectively, in (a) the slow warm-up period model with ζ = 1.3 × 10 −17 s −1 , (b) the fast warm-up period (5 × 10 4 yr) model with ζ = 1.3 × 10 −17 s −1 , and (c) the fast warm-up period model with ζ = 4.0 × 10 −14 s −1 . The orange line indicates the observed value in G28.28–0.36 and the yellow range covers the observed values in the three MYSOs including the standard deviation. The blue line in panel (a) indicates the observed value in L1527 and the blue range covers the error in L1527. Figure 6. Production fraction of formation pathways of HC 3 N during the warm-up period (Taniguchi et al. 2019a ). JHC 3 N denotes molecules on the dust surface. As discussed in Section 4.2 , the observed CCH/HC 5 N ratios in MYSOs agree with the model at a temperature of ∼85 K. This temperature corresponds to 1.08 × 10 6 yr, 16 16 This age does not mean a physical age of an MYSO, because we derive the age from the chemical network simulations and the age will change by the assumed warm-up timescales. as shown in panel (a) of Figure 5 . Just before this time ( t 1.07 × 10 6 yr), the main formation pathway of HC 3 N is the reaction between C 2 H 2 and CN (panel (a) of Figure 6 ). This is consistent with the reaction proposed based on the 13 C isotopic fractionation in G28.28–0.36. The electron recombination reaction of HC 3 NH + is dominant in the model with the high cosmic-ray ionization rate during the whole warm-up period (panel (b) of Figure 6 ). Thus, the 13 C isotopic fractionation of HC 3 N in G12.89+0.49 and G16.86–2.16 can be explained by this high cosmic-ray ionization rate model. These results suggest that the cosmic-ray ionization rates in G12.89+0.49 and G16.86–2.16 may be higher than that in G28.28–0.36 or physical structures in G12.89+0.49 and G16.86–2.16 allow cosmic rays to penetrate into regions where carbon-chain species are formed. The observed CCH/HC 5 N ratios in MYSOs agree with the age of (4.7–4.86)×10 5 yr as indicated in panel (c) of Figure 5 . In summary, the observed CCH/HC 5 N ratios around MYSOs can be reproduced in the chemical network simulations at a given temperature and time. Taking the 13 C isotopic fractionation of HC 3 N into consideration, G12.89+0.49 and G16.86–2.16 may prefer the model with the high cosmic-ray ionization rate ( ζ = 4 × 10 −14 s −1 ), while G28.28–0.36 agrees with the model with the standard cosmic-ray ionization rate ( ζ = 1.3 × 10 −17 s −1 ). These results imply that carbon-chain chemistry in G12.89+0.49 and G16.86–2.16 may resemble that in the OMC-2 FIR 4 young intermediate-mass protoclusters (Fontani et al. 2017 ; Favre et al. 2018 ), but that in G28.28–0.36 is different from the other MYSOs and OMC-2 FIR 4. The clear 13 C isotopic fractionation of HC 3 N in G28.28–0.36 supports its bottom-up formation during the


for each of the late-phase loops, the energy input in late-phase loops reaches ∼7.7 × 10 29 erg. For the main flaring loops (as mentioned in Sect. 3.1 ), both of the two-stage energy release processes contribute to the heating. Hence, the heating duration of the main flaring loops is double compared with the late-phase loops. Assuming that the amplitude of heating rate is equal to H 0 , the energy input in the main flaring loops is estimated to be (1.5−3.8) × 10 29 erg. Based on the EBTEL simulation, the energy input in the post-flare loops amounts to (9.2−11.5) × 10 29 erg. 3.6. Nonthermal energy in flare-accelerated electrons Under the thick-target assumption ( Fletcher et al. 2013 ), HXR emissions at the footpoints are produced by collisional bremsstrahlung during the passage of flare-accelerated nonthermal electrons through a dense target, where the electrons are stopped by Coulomb collisions and lose energy in the chromosphere. The localized plasma can be heated up to ∼10 MK accompanied by chromospheric evaporation and condensation ( Tian et al. 2014 ; Li et al. 2015 ; Zhang et al. 2016a , b ; Zhou et al. 2020 ). The distribution of injected nonthermal electrons above a low-energy cutoff ( E c ) is assumed to be in the form of single power law with a spectral index of δ , , where A 0 is a normalized parameter of the total electron flux (in units of 10 35 electrons s −1 ). In Fig. 6 , three characteristic HXR spectra made from detector 9 of RHESSI during the first stage of energy release are displayed. In each panel, the fitted thermal component is drawn with a red line, while the nonthermal component is drawn with a blue line. It is obvious that the isothermal temperatures from RHESSI are close to those derived from GOES. Fig. 6. Three characteristic HXR spectra with an integration time of 20 s (black lines) and the fitted thermal (red lines) and nonthermal (blue lines) components. T e and EM of the thermal component are labeled. The low-energy cutoff ( E c ), spectral index ( δ ) of nonthermal electrons, and of the residuals are also labeled. The delivered energy by nonthermal electrons above E c is calculated by integrating the injected power during 12:40−12:46 UT, where A nt is the electron injection area, E H is set to be 30 MeV ( Warmuth Mann 2016a ). The nonthermal energy in flare-accelerated electrons during the first stage is estimated to be ∼1.74 × 10 30 erg. Considering that the HXR observation from RHESSI was unavailable during the second stage of energy release around 12:47:30 UT, it is impossible to calculate the nonthermal energy of electrons from RHESSI during the second stage. In Fig. 2 b, the second peak of GOES derivative accounts for ∼1/4 of the first peak, implying that energy input during the second stage accounts for ∼1/4 of the first stage. In this way, the nonthermal energy in the flare-accelerated electrons during the second stage is crudely estimated to be 0.44 × 10 30 erg. Therefore, the total nonthermal energy


such as the Schmidt–Kennicutt relationship between the surface density and the star formation rate (Schmidt 1959 ; Kennicutt 1998 ) or the density threshold for star formation (Lada, Lombardi Alves 2010 ). In particular, the observed star formation rates for several of these clouds are an order of magnitude below those predicted by these relationships between the density and the rate (Longmore et al. 2013a ). A case in point is the cloud GCM-0.02-0.07 ( M ∼ 10 5 M ⊙ ; Johnston et al. 2014 ) nicknamed the Brick because of its high optical depth in the infrared and lack of internal infrared emission from star formation. Only 3–4 per cent of the mass of this cloud is in dense star-forming regions often called high-mass cores. The elliptical ring also includes the massive star-forming region Sgr B2 with some of the highest rates of star formation in the galaxy and the region Sgr C rich with H ii regions indicating earlier active star formation. Fig. 5 shows the locations of the clouds on their elliptical orbit. Figure 5. Schematic of the clouds in elliptical orbit in the Galactic Centre adapted from Kruijssen et al. ( 2014 ). The view is of the plane of the Galaxy from above. [Kruijssen, Dale Longmore ( 2015 ) suggest that the orbit is not exactly elliptical in that the path from Sgr C may not close back to the location of the Brick.] The clouds on the ring orbit in the direction from the Brick toward Sgr B2. The Brick and other clouds with low star formation rates are near the pericentre, 45 pc from the supermassive black hole Sgr A* (Johnston et al. 2014 ), while Sgr B2 is at the apocentre, twice as distant at 100 pc. Longmore et al. ( 2013b ) suggest that the Brick and nearby clouds are currently undergoing tidal compression that will result in intense star formation by the time the clouds arrive at the location of Sgr B2. The mini-starburst Sgr B2 would then be the result of compression in its earlier passage through the pericentre of the orbit. Observations indicate that the clouds on the orbit between the Brick and Sgr B2, the dust ridge clouds, may be just beginning to fragment (Walker et al. 2018 ). Alternatively, fragmentation and ultimately the star formation rate may be controlled by the turbulent-entropic instability. If orbital shear slows the effective turbulent dissipation rate by supplying fresh turbulent energy, then the difference in orbital shear along the elliptical orbit from pericentre to apocentre may prevent star formation near the pericentre and allow it at the apocentre. Following Kruijssen et al. ( 2014 ), we derive a rotation curve, v circ = 23.5 r 0.38 km s −1 from the mass distribution measured around the Galactic Centre by Launhardt, Zylka Mezger ( 2002 ). At the radial distance from the Brick to the Galactic Centre, 45 pc, the orbital shear across a 10 pc diameter cloud is Δ v = 8.4 km s −1 , while the one-dimensional turbulent velocity dispersion in the Brick is 4.3 km s −1 (Johnston et al. 2014 ). At the


from September 22. Additionally, in the image related to the observation from September 22, there is a radial polarization pattern across the Southern half of the core region. This suggests that the magnetic field B is roughly circular and centred on the brightness peak of the core, as one might expect from a helical field when one views it down the axis. This polarization pattern remained through 2016 March. The γ-ray LC from the Fermi -LAT and X-ray LC from the Swift -BAT do not show any obvious flux enhancement during the time of the radio flaring activity, although they show some activity (both BAT and LAT) about 40 d before the radio flare. During this time, there were no optical or VHE observations because of the Sun. Figure 4. A sequence of total (contours) and polarized (colour scale) intensity images of Mrk 421 at 43 GHz obtained with the VLBA. The image is convolved with an FWHM of 0.24 × 0.15 mas 2 along PA = −10°. The global total intensity peak is 329 mJy/beam, and the contours 0.35, 0.70, 1.4, etc. up to 89.6 per cent of the global intensity peak. The colour scale is the polarized intensity, and the black line segments within each image show the direction of the polarization electric vector, with the length of the segment being proportional to the polarized intensity values. The black vertical line indicates the position of the core. See Section 3.2 for details. The low polarization fraction at 43 GHz on 2015 August 1 implies that the magnetic field was very highly disordered in the core at this epoch. A radial polarization pattern, as measured at 43 GHz on 2015 September 22, can result from turbulent plasma flowing across a conical standing shock, as found in the simulations of Cawthorne, Jorstad Marscher ( 2013 ) and Marscher ( 2016 ). However, in such a scenario the linear polarization pattern is always present, since it is created by the partial ordering of the magnetic field by the shock front. Periods of polarization across the entire core should not be observed. An alternative picture ascribes the radial polarization pattern to the circular appearance of a magnetic field with a helical or toroidal geometry that is viewed within ∼0.2/Γ radians of the axis of the jet (Marscher et al. 2002 ), where Γ is the bulk Lorentz factor of the emitting plasma. In this case, the ratio of the observed polarization measured in the image, , to the value for a uniform magnetic field direction, , 6 implies that the helical field is superposed on a highly disordered field component that is ∼10 times stronger. If the helical field becomes disrupted by a current-driven kink instability, particle acceleration could cause a flare (Nalewajko 2017 ; Zhang et al. 2017 ; Alves, Zrake Fiuza 2018 ). The polarization pattern then becomes complex, with the possibility that the polarization becomes very low at some point (Dong, Zhang Giannios 2020 ). Such a flare would be expected to start at X-ray and VHE γ-ray energies upstream of the core, then


open circles joined with dotted line are for shock free accretion solutions. See the text for details. Until now, we have studied the various properties of accretion shock waves around black holes. These shocks are non-dissipative in nature as the specific energy remains conserved across the shock front (Chakrabarti 1989 ). Indeed, shocks of this kind are radiatively inefficient as well. Apparently, the realistic shock waves are likely to be dissipative, where a part of accreting energy is escaped from the shock location through the disc surfaces. This essentially causes the reduction of the overall specific energy profile in the PSC region (Singh Chakrabarti 2011 ). The plausible mechanism that perhaps regulates the energy dissipation at the shock is the thermal Comptonization process (Chakrabarti Titarchuk 1995 ) that eventually reduces the thermal energy in the PSC. Based on the above consideration, we assume that the loss of energy across the shock front is proportional to the temperature difference between the immediate pre-shock and post-shock flow and therefore, the energy loss across the shock front is calculated as (Das, Chakrabarti Mondal 2010 ) where a and a denote the immediate pre-shock and post-shock sound speed and f represents the fraction of the available thermal energy lost at shock. In this analysis, we treat f as a free parameter and chose its value as f 0.998 for representation. Evidently, this is the measure of energy dissipation across the shock (Das etal. 2010 ; Singh Chakrabarti 2011 ; Sarkar Das 2013 ; Kumar Chattopadhyay 2013 ). We then compute the maximum energy dissipation () by freely varying all the flow parameters and plot it as function of in for various values of accretion rates in Fig. 12 (a). Here, solid, dotted and dashed curves represent the results corresponding to , 0.075 and 0.1, respectively. We observe that for a given accretion rate (), initially shows gradual increment with in , which eventually approaches towards a saturation value in the gas pressure dominated domain. Meanwhile, it was shown by numerical simulation (Machida etal. 2006 ) that in the magnetically dominated state, accretion disc becomes cool as compared to a gas pressure dominated disc. This clearly indicates that the gas pressure dominated flow is expected to possess higher thermal energy content and accordingly, we obtain enhanced when in is larger. Moreover, as is increased, cooling becomes more effective in the flow resulting the lowering of thermal energy content. Therefore, for a given in , remains smaller for flows having higher . When in is large, the effect of synchrotron cooling due to the increase of accretion rate becomes practically insignificant. Hence, when in 10 4 , asymptotes to a saturation value irrespective to the flow accretion rate. Figure 12. Variation of maximum energy dissipation at shock () as a function of at inner sonic point ( in ) for three different values of accretion rates, , 0.075 and 0.1 denoted by


( 75 ) and ( 78 )) normalized by the total number of mergers. In these two examples, we set χ 1 = χ 2 . The top panel is for a 0 = 100 au (see Figure 19 ) and we include merging systems with the number of mergers is 673 for e out = 0, 790 for e out = 0.3, 1159 for e out = 0.6, and 2828 for e out = 0.9, so that N max = 5450. The lower panel is for a 0 = 20 au, and we include systems with the number of mergers is 146 ( e out = 0), 180 ( e out = 0.3), 227 ( e out = 0.6), and 411 ( e out = 0.9), so that N max = 964. The other parameters are the same as in Figure 19 . If the distributions of and are uncorrelated, as we may expect to be the case for , when the octupole effect is significant (see Figure 9 and the discussion in the last paragraph of Section 4.2 ), the distribution of χ eff can be derived directly from and , the distribution functions of and . Define where is the maximum possible value of χ eff for given m 1 χ 1 and m 2 χ 2 (this maximum is achieved at ). Then Equation ( 75 ) becomeswhere and . Note that and . Given P 1 ( μ 1 ) and P 2 ( μ 2 ), the distribution function of is In the special case when μ 1 and μ 2 are uniformly distributed, we have P 1 = P 2 = 1/2, and Equation ( 80 ) giveswhere we have assumed without loss of generality. Thus, even for uniform distributions of and (see the case of e out = 0.9 in Figure 19 ), the effective spin parameter χ eff is preferentially distributed around χ eff = 0 (see Figure 20 ). Note that the spin–orbit misalignment distribution and χ eff distribution obtained above refer to relatively wide BH binary systems ( a 0 ≳ 10–100 au) that experience a merger due to large LK eccentricity excitation. Such systems necessarily have . For BH binaries with smaller separations ( a 0 ≲ 1 au) and not much less than unity, the spin–orbit misalignment distribution can be quite different (see Liu Lai 2017 ). Antonini et al. ( 2018 ; see also Rodriguez Antonini 2018 ) have carried population studies of BH mergers in triple systems (based on DA secular equations) and have found a similar peak around χ eff = 0 in the χ eff distribution. Rodriguez Antonini ( 2018 ) also showed an example of the final spin–orbit misalignment distribution with a peak around 90°, in qualitative agreement with our result. They did not distinguish the difference in the spin–orbit misalignment distributions between largely quadrupole systems (small ε oct ) and strong octupole systems (large ε oct ). We do not agree with the reason(s) they gave for the peak in the χ eff distribution. In particular, it is important to recognize that, during the orbital decay, the adiabaticity parameter (Equation ( 61 )) transitions from ≪1 to ≫1, and this transition determines . 5. Summary and Discussion In this paper we have studied BH binary mergers in triple systems. A sufficiently inclined tertiary companion excites large eccentricity in the BH binary orbit through gravitational perturbations (the LK mechanism), significantly shortening its


conclude that the errors assigned by Reid et al. (2014) to and are conservative, and there do not appear to be any bias, given the available maser samples so far. For the rotation curve, the situation is somewhat different. Although the values found for are very close to the initial values adopted, the statistical spread is larger than expected. Reid et al. (2014) reported an error of 0.4 in the rotation curve which is optimistic compared with our findings. From our simulations, we would constrain the rotation curve value as given the BeSSeL data sample selection. The larger error possibly indicates that our assumed velocity distributions are too wide. Fig. 10 Marginalized posterior probability density distributions for correlated circular velocity parameters from 100 simulated galaxies mimicking the BeSSeL sample. Left panel : circular orbital speed of the Sun. Middle panel : orbital angular solar speed. Right panel : difference between the circular solar motion and average source peculiar motions. When more sources are added to the sample selection (sets A and B), Fig. 8 shows an initial improvement in the accuracy of the fundamental Galactic parameters. For set A, the errors in , , and can improve up to , , and , respectively, when more northern hemisphere sources are added. In contrast when southern hemisphere sources are also added (set B), the errors can decrease to , , and , respectively. Further improvements would require much better astrometry for weak sources, requiring for example much more sensitive observations. 4.2.2. Solar motion and average source peculiar motions Figure 7 shows the distribution obtained for solar velocity components (, , and ). The uncertainties derived were around 2 for all solar velocity components with respect to the initial parameters. For , the standard deviation found was 1.2 , which compares well to the results published by Reid et al. (2014) (i.e., ). In addition, the spread for and are narrower with values of 2.4 and 0.5 , respectively. Compared to the BeSSeL results (i.e., and ), we can affirm that the solar motion results published by Reid et al. (2014) have conservative estimates. For the radial and tangential average peculiar motions ( and ), Table 4 shows that indeed these peculiar velocities can be fitted with high accuracy using the simulations; however, the relative spreads are high compared with other parameters (see Fig. 7 ). Even so, compared to the BeSSeL results (i.e., and ), the errors in the simulated sample selection are still lower (i.e., and ). The reason for higher dispersions for and could be related to the number of parameters that must fit independently. The parameters and have largely the same effect on the observations for nearby sources and therefore are directly correlated (see Table 5 ). For , the correlation is high with two components (, ), which affects the fitting and hence the estimated accuracy. 4.2.3. Parameter correlations Pearson product-moment correlation


The real-time duration of the video is 9 s. (An animation of this figure is available.) The first column of Figure 2 presents the evolution of the photospheric continuum I c . We can observe that this expanding structure begins to form a central dark spot (Δ t = 252 s), which eventually grows in size while the EG splits into two smaller granules (Δ t = 504 s). Then, the southernmost descendant granule located at undergoes the same evolution (Δ t = 756 s), until it splits into four granules (Δ t = 1008 s). The second column of Figure 2 shows the integrated signal of V measured . The emerging flux region first appears at Δ t = 0 s, being located at [5′′, 4′′]. Magnetic field patches with apparent opposite polarity appear close to each other. These magnetic patches expand and occupy the EG area (Δ t = 252 s), showing a pattern of alternate polarities (salt-and-pepper or sea-serpent configuration). Some of these flux kernels intensify: notably, this occurs in the region where the first dark spot is forming (Δ t = 504 s). Later, this structure becomes an intergranular lane (Δ t = 756 s). Doppler velocity maps, displayed in Figure 2 (third column), highlight the plasma motions in the region. They indicate that the structure is generally characterized by conspicuous blueshifts, corresponding to upflows up to 1–1.5 km s −1 . In the central dark patch, downflows of ≃0.5 km s −1 are found (Δ t = 504 s and, partly, at Δ t = 1008 s), in agreement with previous observations. The same coherent behavior is also visible in the online movie. The maps of magnetic flux density (Figure 2 , fourth column) reveal that the distribution of the magnetic areas derived from V integrated is representative of the real magnetic flux patches. Indeed, a mixture of polarities is found where V integrated maps show an apparent salt-and-pepper configuration, suggesting once again that the quantity described in Equation ( 3 ) is a good qualitative proxy for the longitudinal component of the magnetic field, . The evolution of magnetic flux density qualitatively follows that of V integrated , as mentioned above. A polarimetric analysis of the dead-calm region (see Martínez González et al. 2012b ) in FoV QS , used as a control box for comparison, is presented in Figure 3 . This region of very quiet Sun entered the IMaX FoV after the first 200 s of the EG evolution. In Figure 3 (top panels) we present the maps of the integrated signal of V measured , at the same representative times shown for FoV EG in Figure 2 (second column). No new magnetic structures are seen to emerge during the analyzed time series. Figure 3 (bottom panel) shows a plot of the mean polarization signal averaged over FoV EG (black symbols) within the region of interest occupied by the EG inside FoV EG (red symbols), and over FoV QS (empty circles), along the sequence. The signal in FoV EG is on average 2.7 times stronger than in FoV QS . Moreover, the polarization signal in FoV QS remains almost constant during the


formation of multiple populations, as we are going to outline in the next. Figure 17. Left-hand panel: Difference between the mass-loss of 2Ge and 1G stars, , against the F 275 W − F 814 W colour extension of the HB. The groups of M3- and M13-like clusters are represented as orange and blue dots, as labelled. Right-hand panel: Present-day and initial cluster mass against the colour extension of the HB (top and bottom panels, respectively). 5.3 Extra-mass-loss in the 2Ge as a tracer of the multiple populations formation The discovery that 2Ge stars of all GCs lose more mass than the 1G (Fig. 11 , see also Tailo et al. 2015 , 2019b , a ) and, in particular, the correlations between the extra-mass-loss Δμ e and the extra-helium content of the 2Ge population δ Y max (Fig. 12 ) and the total (present-day and initial) mass of the clusters (Fig. 13 ) represent powerful constraints for the mechanisms of formation of multiple stellar populations. Since evolutionary effects alone cannot provide such increase (see Appendix D ) the need of an additional mechanism arises, allowing us to gain information on the environment where the distinct generations were born. Tailo et al. ( 2015 ) discuss, in the specific context of the blue hook morphology of ω Cen, how the formation environment of the different multiple populations can affect the stellar evolution during the RGB phase, mainly by affecting the initial stellar rotation. Here, we apply the same scenario to the less extreme environments encountered by the 2Ge of most clusters (see the schematic illustration in Fig. 18 ). Very young low-mass stars ( M /M ⊙ ∼ 0.1−1.0) behave as the T -Tauri stars in the Galactic field, pre-main sequence stars in the convective phase of contraction towards the main sequence. T -Tauri’s typically rotate with a period of 1–12 d, and their rotation rate is maintained constant in time by the magnetic coupling of the star with the accretion disc (magnetic disc-locking, see Armitage Livio 1996 ). Bouvier et al. ( 1997a ) and Bouvier, Forestini Allain ( 1997b ) developed a model to explain why main sequence stars in young clusters (such as α Persei) show a wide distribution of rotation velocities. They showed that, the earlier would pre-main sequence stars break the magnetic disc-locking, the faster would be their rotation when they reach the main sequence. Afterwards, during the main-sequence phase, the surface rotation rate (together with the rotation of the whole convective envelope) slows down thanks to the angular momentum loss associated with the stellar winds, but the core mostly preserves its fast rotation. Figure 18. Schematic interpretation of the different rotation of 1G and 2G stars. During the pre-main-sequence evolution, the rotation of the central contracting star is magnetically locked to the rotation of its residual accretion disc, having a diameter of 50–100 au. While the 1G formation occurs in an ambient of moderate density (yellow region), where the disc–disc


good for travel distances of about 8°–20° with measurement errors increasing with increasing travel distance. At very low travel distances (about Δ 6°), it can be seen that the forward-modeled travel times do not fit the measured ones. The reason for this effect lies in the fact that the simulated data only incorporates modes with harmonic degree l ≤ 170. This cut-off acts as an additional filter at a harmonic degree at which the filter with the lowest phase speed (L170) is centered. The modes selected by the filter thus do not form a proper wave packet, which may lead to artifacts in the cross-correlations (see, e.g., Zharkov et al. 2006 ). Additionally, it is noteworthy that the quality of the match of the power spectrum (compare matched versus unmatched) does not have a large effect on the travel times at higher travel distances (Δ > 8°). At smaller travel distances (Δ 8°), however, one can observe a substantial effect of the quality of the match of the power spectrum on the travel times. We note here that this conclusion may be different for other flow models. We also note that the absence of a proper averaging of multiple kernels in distance and latitude has a considerable effect for smaller travel distances (Δ 10°) but seems to vanish for the largest distances in the case of the flow model considered here. For comparison, forward travel times using ray kernels are also shown in Figures 5 and 6 . It can be seen that the magnitude of the ray kernel travel times is, in general, very similar to those obtained using Born kernels. However, Born kernels seem to better reproduce some features in the measured travel times, e.g., jumps in the magnitude of the travel times from one filter to another such as from Δ = 221 to Δ = 241 (see the middle panel in Figure 6 ). Because the given flow model varies on rather large length scales, a relatively good performance of the ray kernels is expected. This may be different for meridional flow models, which vary on smaller length scales, see Section 1 . We also note here that the measured travel times obtained using the filter with the highest phase speed (L035) encounter a spurious constant offset, see Figure 7 . This offset was corrected for by substracting the mean travel time at each distance. Such an offset is not present in real solar data (Kholikov et al. 2014 ; Jackiewicz et al. 2015 ). In Jackiewicz et al. ( 2015 ), this offset was not noticed in the simulated data, as a few distances had been dropped from the analysis, which resulted in an equivalent correction. The origin of this offset is not completely understood but may be connected to the relatively coarse spatial resolution of the simulated data (256 pixels on 180°). Figure 7. Measured travel times, without offset correction. Left and middle panels: same travel times as displayed in Figure 5 (top left and top middle panels). Right panel: measured GB04 and forward-modeled travel times from the Born kernels for all travel distances, at


al. (2011) . The percentages in Fig. 8 are therefore a minimum estimate only. Despite this, we can see in cases (ii) and (iii), a higher percentage of the flux remains at its original connection for the duration of the simulations. The change in connectivity of the field lines results in strong outflow velocities from the end of the current layers. White arrows showing the projected horizontal velocity components in the central region () in the mid-plane are displayed in Fig. 9 for case (i) at . These show large velocities from the narrow current layer along , where the maximum velocity is , where is the maximum Alfvén velocity in the mid-plane (at ), an indication of reconnection occurring. We note, that the magnetic field is 3D and reconnection is not confined to the mid-plane, but use the current contours and velocities in the mid-plane to illustrate the differences between the cases. Fig. 9 Contour of the magnitude of the current density in the mid-plane between at for case (i) with 2 sources. The arrows show the projected velocity in the plane. Fig. 10 Contours of the magnitude of the current density in the mid-plane between at a) and b) for case (ii) with 4 weak sources. The arrows show the projected velocity in the plane. Similarly, in case (ii), strong outflow velocities are seen from the current “wings” shown in Fig. 10 a at and very similar behaviour is observed for case (iii) (not shown). Later in cases (ii) and (iii), the current contours in Fig. 10 b show the “X” current formation to fragment and then (in addition to the outwards velocity towards the edge of the domain) there are inwards velocity outflows of reconnected field lines towards the centre of the mid-plane. This leads to further current being built up in the centre of the domain. The maximum velocity (in the mid-plane) for cases (ii) and (iii) are and , respectively, and occurs at , after this fragmentation of the “X” current formation. This is much smaller than in case (i) and suggests that the reconnection is weaker. The location of the sources in the case with two flux tubes appears highly idealised to produce the concentrated single current layer in the centre of the domain. In comparison, in the four-source case, as they are evenly distributed around the centre of rotation, a shear is first created between the neighbouring flux tubes, which due to the relative angles of the magnetic field lines will always be less than that induced in the two-source case and produces reduced current values and results in a smaller percentage of flux changing connectivity. 3.1.2. Energetics The energy evolution is shown by the volume integrated energy densities in Fig. 11 for the cases with two, four weak and four compact sources, displayed in pink, blue and black, respectively and can be described as follows: when the driving velocity acts on the field, the field lines are advected and the magnetic field becomes twisted, gradually increasing the magnetic energy in the domain (see


which was managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The fits had an average rms residual of 0.15 km s −1 and an average maximum residual of 0.3 km s −1 . Two observations have larger uncertainties. The signal-to-noise ratio (S/N) for the 814 cm −1 setting from 2007 was too low to use telluric lines in the GV Tau N spectrum. Instead, the dispersion function was fit to lines in the telluric standard (rms 0.06 km s −1 ), and this solution was applied to GV Tau N. Three measurable telluric lines in the GV Tau N spectrum were used to assign a zero-point (wavelength shift) uncertainty of 0.5 km s −1 . The 932 cm −1 setting from 2007 only contains three telluric lines; hence, an independent dispersion solution was not possible. In this case, the pipeline wavelength scale was retained, with an assumed uncertainty of 1 km s −1 . Model telluric transmission spectra were utilized for a few of the settings. Because the S/N for the 780 cm −1 setting was degraded as a result of insufficient S/N in the telluric calibrator, corrections for atmospheric lines were made using the software package TERRASPEC (Bender 2010 ). TERRASPEC was also used to improve the correction for telluric water lines in the 767 and 807 cm −1 settings. For all of the observations, additional continuum normalization was carried out for each order of interest by fitting a low-order polynomial to the continuum outside the intervals of absorption from known molecular transitions in GV Tau N. 3. Analysis In the TEXES spectra we detect numerous absorption lines of NH 3 , HCN, and C 2 H 2 ; a few lines of H 2 O in absorption; and H 2 in emission (Table 2 ). The H 2 emission from GV Tau was previously reported in Bitner et al. ( 2008 ) and is not discussed further here. Most of the HCN and C 2 H 2 lines were measured in 2006, with additional lines measured in 2007. The unexpected detection in 2006 of numerous NH 3 absorption lines was followed up in 2007 with grating settings that probed lower-energy NH 3 transitions. Weak absorption lines of water were also detected in the 2007 spectra. The entire set of pipeline-reduced spectra for GV Tau N is shown in Figure A1 . The positions and identifications of the detected lines are indicated. Table 2 All Lines Notes. a *: detection; x: nondetection; ...: not covered; t: telluric line; e: order edge/roll-off; u: high uncertainty; d: blended with other line. b ν 4 + ν 5 – ν 4 band, i.e., absorption out of the ν 4 level. c The 813.85 cm −1 ortho- and 813.83 cm −1 para-H 2 O lines were treated as a single line with combined g l . While the H 2 O lines are pure rotational transitions within the ground vibrational state, the NH 3 , HCN, and C 2 H 2 absorption lines are predominantly rovibrational transitions out of the ground vibrational state to a higher vibrational state; the C 2 H 2 ν 4 + ν 5 – ν 4 line is an exception (see Section 3.1 ). The HCN ν 2


of spectra for 10 K, 50 K, and 300 K, while series F[001]a lacks a 10 K spectrum. Figure 1 shows the measured transmission spectra ( T meas , solid and dashed lines) plotted over the wavenumber ( in cm −1 ); the corresponding wavelengths (30 μ m–500 μ m) can be inferred from the upper axes. Spectra for a-polarization are given in panel (a), b-polarization in (b), c-polarization in (c). The two incomplete series F[001]a and F[010]c are plotted as dashed lines. The former has been scaled by a factor of 1.8 in order to demonstrate the good agreement to series F[010]a after multiplicative scaling (see Sect. 3.2 ). The strong temperature dependence of the transmission of the forsterite crystal is easily recognizable in all three polarization directions. While measurable transmission is detected only below 50 cm −1 at room temperature, the crystal becomes transparent already below 250 cm −1 at 10 K. For each temperature, the transmission increases towards smaller wavenumbers, apart from absorption bands positioned at 202 cm −1 (particularly strong in a-polarization, but weaker also in c-polarization) and 145 cm −1 (in b-polarization, very weak in a- and c-polarization). These bands are the already mentioned 49 μ m and 69 μ m single-phonon bands of forsterite, respectively. They are excited exclusively by a- and b-polarized radiation, respectively, the weaker appearances in other polarization directions are due to the small error of the polarizer alignment and to the beam divergence of the FTIR spectrometer (±7.5 deg). Apart from these bands, a relatively weak minimum at ∼90 cm −1 also appears in several of the spectra for a-polarization. As there is no such long-wavelength infrared band predicted by theory ( Iishi 1978 ) and the band is not present at 10 K, we think that it is an artifact of the measurement setup. We note that the position of the band is one of the intensity minima created by multiple reflection in the inner cryostat windows. These minima occurring with a period of 18 cm −1 are sometimes not well compensated by the reference measurements and lead either to periodic structures in the spectrum (e.g., in the 10 K and 50 K spectra in Fig. 1 b), or coincide for unknown reasons with erratic peaks (mainly maxima) in the enstatite spectra (see Sect. 3.3 ). Fig. 1. Temperature-dependent transmission spectra of the forsterite crystal for orientation of the electric field parallel to the a-axis ( panel a ), b-axis ( panel b ), and c-axis ( panel c ). Complete temperature series (F[010]a, F[001]b, F[100]c) are plotted as solid lines, the two incomplete series as dashed lines. The F[001]a curves have been scaled by a factor of 1.8; they approximately match the F[010]a curves for the same temperatures. Dotted lines: calculated maximum transmission curves taking into account the intensity reflected by the crystal surfaces. 3.2. Maximum transmission curves and measurement errors The dotted lines in Fig. 1 indicate the theoretical maximum


limits were calculated from the misfit function when all solutions with a fit at least of 99percent of the best fit were accepted. Figure 2. Inversion for optimum cosmological parameters of (a) the CDM model and (b) the opaque universe model. The colour shows the mean of the absolute distance modulus residua between the predicted model and the SN Ia data as a function of (a) the Hubble constant H 0 and the dark energy , and (b) the Hubble constant H 0 and the B -band intergalactic opacity B . Only the most accurate 552 SNe Ia with z 1.4 and an error less than 0.50 mag are used. The optimum solutions marked by the white plus signs are defined by (a) and ( 0.122 and 2.466) and (b) and ( 0.088 and 1.920). The residual Hubble plots are shown in Fig. 3 . The optimum solutions have roughly comparable misfits for both models: 0.16 for the CDM model (Fig. 3a ) and 0.17 for the model of the opaque universe (Fig. 3b ). However, as illustrated in Figs 3(c) and (d), the CDM model performs slightly better than the opaque universe model, because of a better fit of the binned residua for redshifts less than 0.6. Nevertheless, the model of the opaque universe also accounts for the observed dimming of the SNe Ia within the confidence level of 95percent (Fig. 3d ). Figure 3. Residual Hubble plots for (a, b) the individual SN Ia data and (c, d) the binned SN Ia data. (a, c) The flat CDM model with and . (b, d) The opaque universe model with and . The error bars in (c, d) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Importantly, the optimum solutions are sensitive to the redshift interval of the SNe Ia used in the inversion. If the optimum model of the opaque universe is searched using the SNe Ia with redshifts z 0.6 only, and the SNe Ia are grouped in the following redshift bins: z [0, 0.025, 0.05, 0.1, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.5, 0.6], we get and (Fig. 4 ). We see in Fig. 4(b) that the model fits the SN Ia measurements at z 0.6 much better than in Fig. 3(d) , but a significant discrepancy appears at higher redshifts. A similar result was obtained by Goobar etal. ( 2002 ) and Riess etal. ( 2004 ), who concluded that the discrepancy at high redshifts excludes the SN Ia dimming to be produced by intergalactic dust. Figure 4. Residual Hubble plots for (a) the individual and (b) binned SN Ia data for the opaque universe model with and . The vertical dashed line in (b) denotes the upper redshift limit of the SN Ia data considered in the inversion. The error bars in (b) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Note that the best estimate of H 0 for the CDM model obtained by Riess etal. ( 2016 ) using the SN Ia data is . The precision of the distance scale was further improved by a factor of 2.5 by Riess etal. ( 2018 ). This value is within the uncertainty of the estimate of H 0 for the CDM model obtained in this paper and shown in Fig. 2(a) . The accuracy of the


larger intensity variations in the range and a patchy distribution peaked at the location of recent star formation events. The patchy ionization structure yields an escape fraction of , a value consistent with observations of lower redshift galaxies with similar brightness (e.g. Bouwens et al. 2016 ; Grazian et al. 2017 ), and averaged values of simulated galaxies with similar mass (Xu et al. 2016 ). We note, however, that large fluctuations are expected between different evolutionary stages (Trebitsch et al. 2017 ). The [C ii ] line luminosity of Freesia is . The emission extends on a few kpc scales, with peaks around Freesia-A, Freesia-B, and dense non-star-forming clumps found near the galaxy. The emission mostly comes from gas with and illuminated by an ISRF intensity . Instead, [O iii ] emission is concentrated around Freesia-A, the star-bursting component of the system, and its emission is dominated by gas with a wider range of properties, i.e. and U ≃ (6 ± 2) × 10 −3 . At the Freesia-A location, the oxygen line is very bright, . However, the smaller extent of the [O iii ] emitting region implies a lower galaxy-integrated emission, i.e. . With respect to [C ii ], the [O iii ] show both a spatial offset () and a spectral shift of (), reminiscent of similar evidence found in some systems at high z (Carniani et al. 2017 ). Freesia lies below the local [C ii ]–SFR relation (De Looze et al. 2014 , within ≃ 2σ), as it is in a starburst phase; that is, it sits above the KS relation (Schmidt 1959 ; Kennicutt 1998 , within ≃ 2σ). Spatial analysis reveals that patches of the galaxy that are above the resolved local – De Looze et al. ( 2014 ) relation are located below KS, and vice versa. In particular, due to the recent starburst, the dense centre () of Freesia lies above the KS () relation. At such high star formation surface densities the increases with less rapidly than expected from the local relation De Looze et al. ( 2014 ), which however only covers the low end of the range. Thus, the observed [C ii ]–SFR deficit can be primarily ascribed to negative stellar feedback during starburst phases, disrupting MCs around star formation sites. This interpretation was originally proposed in Vallini et al. ( 2015 ); our results fully endorse it. Metallicity effects have a weaker impact on the origin of [C ii ]–SFR deviations. Gas with extremely low ( Z ≲ 10 −2 Z ⊙ ) metallicities fails below both the local – and the KS relations, because of lack of C + ions and inefficient H 2 formation, respectively. However, gas with Z ≲ 10 −2 Z ⊙ is scarce in Freesia and its combined contribution is sub-dominant when galaxy integrated relations are considered. This situation is common to most of the galaxies in our sample. A metallicity effect might be relevant for galaxies that cannot retain their metal production (Xu et al. 2016 ; Jeon Bromm 2019 ), which are expected to be smaller ( M ⋆ ≲ 5 × 10 6 M ⊙ ), and with a lower star formation (), for which however


bias models that distinguish between centrals and satellites in upcoming work. 2.5 Incompleteness factor <italic>f</italic>ic The final parameter in our extended HOD model is the incompleteness factor (e.g. Leauthaud et al. 2016 ; Rodríguez-Torres et al. 2016 ; Guo, Yang Lu 2018 ). The inclusion of incompleteness is partially motivated by detection of incompleteness in the BOSS CMASS and LOWZ galaxy samples compared with theoretical stellar mass functions, but it is also needed to marginalize over uncertainties in the galaxy number densities since we do not model its full redshift dependence. We define f ic as a modification to in equation ( 1 ): where 0 f ic ≤ 1. Our implementation simply uniformly downsamples the mock galaxies by f ic to produce the desired galaxy number density. To summarize, our extended HOD model contains the five baseline parameters: M cut , M 1 , σ, α, and κ; six extended parameters: s, s p , s v , α c , A , and A e ; and one incompleteness factor f ic , for a total of 12 parameters. 3 DATA AND SIMULATIONS 3.1 BOSS CMASS galaxy sample The Baryon Oscillation Spectroscopic Survey (BOSS; Bolton et al. 2012 ; Dawson et al. 2013 ) is part of the SDSS-III programme (Eisenstein et al. 2011 ). BOSS Data Release 12 (DR12) provides redshifts for 1.5 million galaxies in an effective area of 9329 deg 2 divided into two samples: LOWZ and CMASS. The LOWZ galaxies are selected to be the brightest and reddest of the low-redshift galaxy population at z 0.4, whereas the CMASS sample is designed to isolate galaxies of approximately constant mass at higher redshift ( z > 0.4), most of them being also luminous red galaxies (LRGs; Reid et al. 2016 ; Rodríguez-Torres et al. 2016 ). The survey footprint is divided into chunks that are covered in overlapping plates of radius . Each plate can house up to 1000 fibres, but due to the finite size of the fibre housing, no two fibres can be placed closer than 62 arcsec, referred to as the fibre collision scale (Guo, Zehavi Zheng 2012 ). For this paper, we limit our measurements to the galaxy sample in the redshift range 0.46 z 0.6 in DR12. We choose this moderate redshift range for completeness and to minimize the systematics due to redshift evolution. Applying this redshift range to both the North and South Galactic Caps gives a total of approximately 600 000 galaxies in our sample. We showcase the number density variation over redshift in Fig. 1 . The average galaxy number density is given by n data = (3.01 ± 0.03) × 10 −4 h 3 Mpc −3 . Figure 1. The CMASS DR12 galaxy comoving number density distribution across our redshift range. The red dashed line corresponds to the North Galactic Cap, whereas the blue dotted line corresponds to the South Galactic Cap. The green solid line shows the combined number density. The two vertical lines mark z = 0.46 and z = 0.6, respectively. Fig. 2 shows the redshift-space 2PCF of the same BOSS sample and its corresponding correlation matrix,


since 2003. WIRCam is mounted at the prime focus of the 3.6 m CFH telescope. It is equipped with an image stabilization unit, which consists of a tip-tilt glass plate in front of the camera activated by the signal read-out from small 14 × 14 pixel regions centred on bright stars ( Puget et al. 2004 ). The camera cryostat includes the eight-lens field corrector and an entrance window. The detector focal plane consists of a mosaic of four 2 × 2 k HAWAII2-RG detectors separated by 7 mm. The sampling on sky is 0.′′3 per 18 μm pixel, providing a total field of view of 20′ × 20′ with a 2′ wide central cross between the four detectors. The transmission curve of the Y -band filter used in this work is shown on Fig. 1 , together with the MegaCam filter transmission curves. Fig. 1 Instrument and telescope total efficiency (optics and detector) in the MegaCam u *, g ’, r ’, i ’, z ’ filters and WIRCam Y -band. The survey observations consisted of four dithered 75 s Y -band images of the same field. They were split into two visits, with dithering within a visit and between visits of 2′′. The two visits were separated by at least 20 days in order to discard slow moving objects and strongly variable objects. The time difference between visits of the same field varied strongly between the four CFHTLS fields. For W1 and W4, the observations were performed very rapidly at the beginning of the survey, while for W2 and W3 (winter and spring fields), they suffered from poorer weather conditions and were executed over much longer periods of time. The interval between visits of the same fields were in the range of 20–40 days for 50% (W1) and 90% (W4) of the observations, whereas for the W2 and W3 fields, most of the visits were separated by approximately 1 yr. The mapping of the fields was organized so as to match the CFHTLS one-square-degree tiles, with nine WIRCam pointings per CFHTLS tile. The observations were defined so as to prevent the execution of the second-visit observations in the same sequence as for the first visits, which could have generated similar persistence patterns in the data. A total of approximately 150 h of observations were conducted during four CFHT semesters, 2010B, 2011B, 2012A, and 2012B. The observations were carefully checked by our team in the days following the observations using the Elixir-IR pre-processing package (see next section), and observations that were executed under high background conditions and/or poor seeing conditions were not validated and were subsequently re-executed. 3 Data processing 3.1 Needfor a custom pipeline: Elixir-IR The CFHT offers to all its WIRCam users a data detrending and calibration service, the “I”iwi pipeline ( Thanjavur et al. 2011 ), which aims to remove the instrumental signature and delivers an absolute photometric calibration within 5%. This pipeline relies on generic recipes, such as acquiring flat fields over a whole observing run in order to create master detrending frames that are applied


(), by using equation ( 36 ) for a clear atmosphere The red curve in Fig. 4 depicts our approximation, which follows the solution of Lecavelier des Etangs etal. ( 2008 ) when s , and is zero otherwise. Fig. 5 shows the amount by which our approximation underestimates the effective atmospheric thickness. It is at most E 1 (e ) 0.492 when s 0.561. We can apply the same line of reasoning to quickly determine which CIA features rise above the surface continuum. We use instead of equation ( 43 ), where k is the modified mean collision-induced cross-section with its minimum set at the surface collision-induced cross-section ( k s ), given by We then determine the spectral modulation between any two spectral regions with equation ( 34 ). The surface decreases the spectral modulation only if s or k s k in one of the two spectral regions. Figure 5. Amount by which the effective thickness of the atmosphere (in units of scaleheight) is underestimated for q 1 (see equations 11 and 12 ) with the approximation outlined in Section 3.4 as a function of the surface slant optical depth. Can we use this approximation to quickly determine the location of a surface from observations? If the observed spectral modulation is due to a single species and is less than that expected in a clear atmosphere, we can relate the observed spectral modulation, ( h / H ) obs , with respect to a local maximum in its cross-section ( i ) by Unfortunately, the surface location cannot be unambiguously determined because the mole fraction ( f i ) of the species is unknown. Indeed, equation ( 46 ) shows that the spectral modulation of a species increases with abundance in the presence of a surface, as demonstrated numerically by Btrmieux ( 2016 ) for a refractive boundary. The only way to break this degeneracy are two-fold: either the spectral signature is that of a major species (i.e. f i 1) and we can determine the location of the surface, or we can predict the location of the surface and then determine the abundance of the minor species. Given that refractive boundaries are dependent on the bulk properties of the exoplanetary atmosphere as well as the planetstar geometry, quantities that are retrieved during the analysis of transit data, location of refractive boundaries can be predicted in principle and used to place a lower limit on the abundance of detected minor species (Btrmieux 2016 ). 3.5 Transmission spectrum versus formalism To illustrate the impact of the choice of formalism on a computed transmission spectrum, we apply the following formalisms on simulations by Btrmieux ( 2016 ): the formalism of Lecavelier des Etangs etal. ( 2008 ), the exact solution embodied by equation ( 35 ) with q 1, and the approximation discussed in Section 3.4 . The chosen atmospheric model is that of a Jovian-like planet with an H 2 He atmosphere identical to Jupiters, a 600K isothermal profile, and a constant H 2 O abundance with altitude with a mole fraction of 10 8 . We choose the


indeed become an important ionization source in T Tauri disks. However, our model should be seen as a first attempt at a comprehensive modelling of SP ionization in protoplanetary disks. In the following we discuss further important aspects like variability, non-stellar origin of SPs, the importance of magnetic fields and future prospects for SP modelling in protoplanetary disks. 4.3.1. Flares and variability In our models we assumed continuous (i.e. time-averaged) particle fluxes and X-ray luminosities. Although this is a reasonable assumption (see Sect. 2.2 ) it is likely that the disk is also hit by singular powerful X-ray and/or SP flares. From the X-ray COUP survey of the ONC cloud, Feigelson et al. (2002) and Wolk et al. (2005) derived a median X-ray flare luminosity of , comparable to the X-ray luminosity in our high X-ray models, and peak flare luminosities up to . The duration of such flares can last from hours up to three days with a typical frequency of roughly one powerful flare per week. Ilgner Nelson (2006) argue that for such flare properties the disk ion chemistry responds to time-averaged ionization rates. However, in case the duration between two flares is longer than the recombination time scale in the disk, such singular and strong flares would produce asymmetric features in molecular ion emission (i.e. a singular flare only affects a certain fraction of the disk). The spatially resolved SMA observations of TW Hya show indeed such an asymmetric structure ( Cleeves et al. 2015 ). However, as discussed by Cleeves et al. (2015) these features could also have a different origin like spiral arms or an hidden planet locally heating the disk. If such asymmetric structures are caused by stellar flares they would provide complementary constraints on the X-ray/SP activity of the star. Multi-epoch data of spatially resolved molecular ion emission is required to prove the flare scenario (i.e. the features should disappear quickly). The current (sub)mm interferometers like ALMA, NOEMA and SMA provide the required spatial resolution and might even allow for monitoring of disks in molecular ion lines on a daily/weekly basis in the future. Modelling of such observations does not necessarily require 3D chemical models. Assuming that the disk physical structure is not affected by flares, radial cuts through the disk can be modelled with 2D (time-dependent) thermo-chemical models as presented here. 4.3.2. Magnetic fields In our model we neglect the impact of magnetic fields on the SP transport. Stellar and disk magnetic fields are in particular relevant for the question if particles actually hit the disk (see Feigelson et al. 2002 , for a discussion). Magnetic fields can either drag the particles away from the disk (e.g. like in a wind) but could also funnel the particles and concentrate the ionizing flux in particular regions of the disk. In the first scenario SPs can still have an impact on the upper layers of the disk but certainly not on


have a constant absolute magnitude of −7 mag. The investigation on decline times for all novae in the catalogue indicates that the novae with D- and J- type light-curve shapes may not have linear decline. If so, only considering the decline times ( t 2 or t 3 ) may not characterize the absolute magnitude at outburst maximum and the decline well. All of these arise uncertainties when using of MMRD relation if the type of nova is not clarified. Furthermore, the luminosities of novae may differ depending on their stellar populations or their evolutionary states, i.e. on the stellar age and metallicity. For the scatters found in other distance indicators, e.g. Cepheids, it has been suggested that a different metallicity of the host galaxy may significantly affect the zero-point of the relation (Freedman et al. 2001 ). In this case, there is a possibility on any environmental dependence on MMRD relations as well. Since our and the other studies in literature depend on disc novae in the Galaxy, the contribution of other populations (bulge, halo, even thick disc) is not well known. Our study does not dispel the concerns regarding the validity the MMRD relation, but has raised open questions. To understand the MMRD relation better, the novae in galaxies, which have reliable distances (and extinctions), should be studied with high quality photometric and spectroscopic data. A larger sample of them would also be very welcome as well as corresponding parameters and well-determined light curves that allow us to classify their light-curve shapes. Our catalogue may also help to investigate the nature of MMRD diagrams in the long-term. Studying the MMRD relation is important to understand the origin of the nova eruption itself, besides using it as a tool to estimate absolute magnitudes. In this study, we did not find any signs that the spectral classification represents the stellar population of the Galactic novae, as suggested by Della Valle Livio ( 1998 ). However, our decline time histogram implies that the He/N+Hybrid novae are fast systems with a peak time at t 3 = 13 d, while the slow Fe ii novae have a peak decline time at t 3 = 42 d. In contrast to the suggestion of Della Valle Livio ( 1998 ), we found that both Fe ii and He/N novae or fast and slow novae are located in the whole range of vertical distances, but mainly concentrated throughout the Galactic disc. The differences between the results in these two studies may arise from the samples used in the analyses. The sample in Della Valle Livio ( 1998 ) contains a smaller number of novae with spectral classification, which reach only to a vertical distance of ∼2 kpc, whereas our sample have much more systems reaching up to two times farther in vertical direction from the Galactic plane. A relatively large number of slow novae located close to the Galactic plane and fast novae farther from Galactic plane must have been undetected in their analyses. The approach proposed by Della Valle Livio ( 1998