infrared yielding a water production rate in the range 3.2–3.0 × 10 28 molec. s −1 for the 7–11 September period, just before our IRAM observations. Faggi et al. (2019) , also from infrared observations of H 2 O, found water production rate in the range 2.6–2.8 × 10 28 molec. s −1 for the 8 September–7 October 2018 period. We thus assumed molec. s −1 at the time of IRAM observations ( Table 3 ). Comet 41P outgassing was monitored by several observatories during its 2017 approach, including the Solar Wind ANistropies (SWAN) instrument on the SOHO spacecraft (31 March to 20 April, Combi et al. 2019 ), TRAPPIST (11 March to 24 May for OH, Moulane et al. 2018 ) and with the Nançay radio telescope (4 March–5 June). The comet was not observed at Nançay at the time of the IRAM observations because of poor OH inversion. OH production rates of ≈ 1.1 × 10 28 and 0.6 × 10 28 molec. s −1 were obtained for the 28 March–3 April and 18–30 May, respectively. Most of the observations showed a plateau of activity ±20 days around perihelion with 4, 3 and 8 × 10 27 molec. s −1 respectively. We adopted a median value of molec. s −1 for the time of our observations. For comets 64P and 38P, around the time of our observations, water production rates were estimated from ground-based spectro-photometric observations of OH. D.G. Schleicher (priv. comm.) provided molec. s −1 for 64P on 16 December 2018. For comet 38P his values were molec. s −1 on 3 and 16 December 2018. We adopted water production rates of molec. s −1 for 64P and molec. s −1 for 38P, both to compute collision rates and determine molecular abundances. 64P and 38P were only observed in September and October at Nançay. 6 Production rates and abundances Line intensities have been converted to production rates using our time-dependent non-LTE code taking into account collisions with neutrals and electrons as well as solar radiative pumping via vibrational bands for HCN, CH 3 OH, CS, H 2 CO, CO, and CH 3 CN, as used in previous papers ( Biver et al. 2000 , 2006 , 2012 , 2014 , 2015 , 2016 ). Photolysis constants are also from previous papers and detailed in Biver et al. (2021) . Production rates or upper limits are provided in Table 4 . For species with multiple lines observed at the same time, the weighted average is provided. 6.1 Relative abundances Abundances relative to water were computed using the average water production rates provided in Table 3 . For 41P we assumed a lower water production rate (0.3 × 10 28 molec. s −1 ) for the second day (1.0 May 2017) because HCN and CH 3 OH showed a significant decrease in production rate and Moulane et al. (2018) also found a decrease in the water production rate between 26.1 April and the 28 April–3 May period. For 38P, abundances relative to water assume molec. s −1 . Although no line is detected with a signal-to-noise larger than 2, the combined signal of the 13 strongest methanol lines at 165–170 GHz and 242 GHz yields a formal signal-to-noise of 3.5 and a


at PA = 259o as the distance to the source increases. Table 8. Parameters of the continuum and CH 3 CN line best-fitting model, which consists of an Ulrich envelope and a flared disc. Parameters in bold under the last column are the 10 free parameters that can be constrained by the observations. The inner radius depends on the stellar luminosity. a Defined at a height z = 10 4 au. b 2 models had the same best overall ranking value. The parameters of the best-fitting model during the visual inspection step are listed in Table 8 and the ranking and reduced χ 2 values for each observation are listed in Table 9 . The best-fitting model has a type A density distribution (Ulrich 1976 envelope and a flared disc). The envelope radius was well-fitted during this step and has a value roughly a factor of 1.4 smaller than the initial visual inspection model and has a similar size to the model with disc of Johnston et al. ( 2013 ). The disc scale height is smaller by roughly one order of magnitude than those in Johnston et al. ( 2013 ), but this parameter cannot be constrained by our observations, thus it was less explored. The shape of the cavity changed from the initial exponent of 1.5 to 2.0, and its reference density decreased an ∼15 per cent from its initial value and its density exponent did not change. Table 9. Overall ranking and reduced χ 2 values for the continuum best-fitting models of each type. a Average weighted ranking as described in Appendix A1.2 (online). Similarly, in Table 8 and 9 , we list the results for the overall best-fitting model after the grid-fitting step. The disc mass and radius increased with respect to the visual inspection best-fitting model. The former is roughly a third of the disc mass in the control model of Johnston et al. ( 2013 ), but the disc radii are similar. This difference is explained by the difference in the disc scale height. The cavity opening angle is equal to the one from the visual inspection best-fitting model, and the inclination angles differ by 15o. The luminosities are the same between the best-fitting models at each step, whilst the envelope infall rate is ∼1.8 times higher in the overall best-fitting model. The model SED agrees reasonably well with the observations as shown in Fig. 10 . The total mass of the model, M = 2.1 × 10 3 M ⊙ , is enough to fit the 10- μ m silicate feature relatively well at an inclination of 25o and closely fit the submm points. This is 900 M ⊙ more massive than the model with disc of Johnston et al. ( 2013 ), but 600 M ⊙ less massive than their envelope only model. Models with WD01 dust can also fit in the submm with lower masses, however, they underestimate the fluxes between 3 and 8 μ m by a factor of ∼10. The near-IR maps in Fig. 7(a)–(c) show that the near-IR band images are well reproduced. However, Fig. 8 shows that the small scales closer to the star position probed by the speckle observation are not well reproduced. In general, most models with a cavity


tens of solar masses (Abbott et al. 2016a ). Based solely on the gravitational-wave signals recorded by two LIGO detectors, the current hypothesis is that GW150914 was the result of a merger of two black holes with initial masses of and at a luminosity distance of Mpc. The Fermi Gamma-ray Burst Monitor ( Fermi -GBM) reported a possible weak gamma-ray transient source above 50 keV at 0.4 s after the GW150914 trigger (Connaughton et al. 2016 ). However, the upper limit provided by the INTEGRAL Advanced Camera for Survey instrument in a gamma-ray energy band similar to the Fermi -GBM energy band is not consistent with a possible gamma-ray counterpart of GW150914 suggested by the Fermi -GBM (Savchenko et al. 2016 ). No electromagnetic counterpart of GW150914 was found in radio, optical, near-infrared, X-ray, or high-energy gamma-rays (Abbott et al. 2016b ). GW151226 (LIGO-Virgo trigger ID: G211117) is the second gravitational-wave candidate identified by both LIGO Hanford Observatory and LIGO Livingston Observatory with a high significance (the false-alarm rate of less than one per 1000 years by the on-line search) at 3:38:53.647 UT on 2015 December 26 (Abbott et al. 2016c ). According to a Bayesian parameter estimation analysis, the event is very likely a binary black hole merger with initial black hole masses of and and a final black hole mass of (Abbott et al. 2016d ). The luminosity distance of the source is estimated as Mpc, which corresponds to a redshift of . As far as the electromagnetic counterpart search of GW151226 in the gamma-ray regime is concerned, Fermi -GBM (Racusin et al. 2016 ), Fermi Large Area Telescope (LAT; Racusin et al. 2016 ), High-Altitude Water Cherenkov Observatory (Wood the HAWC Collaboration 2016 ), and Astrosat -CZTI (Bhalerao et al. 2016 ) reported no detections around the GW trigger time. According to Racusin et al. ( 2016 ), the flux upper limit of Fermi -GBM is from 4.5 × 10 −7 to 9 × 10 −7 erg cm −2 s −1 in the 10–1000 keV band. The Fermi -LAT flux upper limit using the first orbit data after the LIGO trigger is from 2.6 × 10 −10 to 7.8 × 10 −9 erg cm −2 s −1 in the 0.1–1 GeV band. The CALorimetric Electron Telescope ( CALET ; Asaoka the CALET Collaboration 2015 ; Torii et al. 2015 ) mission, which was successfully launched and emplaced on the Japanese Experiment Module—Exposed Facility of the International Space Station (ISS) in 2015 August, was fully operational at the time of GW151226. CALET consists of two scientific instruments. The Calorimeter (CAL) is the main instrument, which is capable of observing high-energy electrons from ∼1 GeV to ∼20 TeV, protons, helium, and heavy nuclei from ∼10 GeV to 1000 TeV and gamma-rays from ∼1 GeV to ∼10 TeV. The field of view (FOV) of CAL is ∼45° from the zenith direction. Another instrument, CALET Gamma-ray Burst Monitor (CGBM; Yamaoka et al. 2013 ), is a gamma-ray burst (GRB) monitor using two different kind of scintillators (LaBr 3 (Ce) and BGO) to achieve a broad


except when the wind speed at the orbital separation is so low compared to the orbital speed that the effective cross section given by the BHL formula is larger than the orbital separation and looses any meaning: relying on the Roche lobe instead makes much more sense in this regime. Fig. 3. Comparison of the mass-transfer rates we measure as a function of η (solid lines) to the BHL mass-accretion-rate formula (dashed lines) for a filling factor of 90% and mass ratios of 2 ( upper panel ) and 15 ( lower panel ). The BHL prescription leads to mass-accretion rates much lower than the mass-transfer rates we derive, sometimes by more than an order of magnitude. 3.2. Accretion luminosity Another insight revealed by this analysis is the identification of the configurations where ULXs cannot be the product of mass transfer via wind-RLOF and can be explained only if the star fills its Roche lobe. In Fig. 2 , we represented in black the regions where μ is inferior to 0.2%, the minimum fraction of stellar wind captured under which mass transfer without RLOF is not able to produce a stationary X-ray luminosity of 10 39 erg s −1 , even for a stellar mass-loss rate as high as 10 −4 M ⊙ yr −1 : Given the maximal μ we found at ∼20%, this analysis suggests that mass transfer via wind-RLOF in a ULX is only possible for stellar mass-loss rates above ∼10 −6 M ⊙ yr −1 . We note here that matching a mass-transfer rate Ṁ high enough to reach ULX levels is a necessary but not sufficient condition: the final X-ray luminosity is determined by the physics at stake within the Roche lobe of the accretor, where the flow geometry can be disk-like ( El Mellah et al. 2019 ) or spherical ( Shakura et al. 2012 ) and where, for an accreting NS, the magnetosphere eventually dominates the dynamics ( Bozzo et al. 2008 ). For accreting BHs, powerful disk outflows come into play in the vicinity of the compact object, lowering the final mass-accretion rate ( Pinto et al. 2016 ). Rather than covering the large scope of regimes possible, let us focus on three donor stars in one SgXBs and two ULXs. In the SgXBs Vela X-1, Gimenez-Garcia et al. (2016) reported a stellar mass-loss rate Ṁ ⋆ ∼ 6.3 × 10 −7 M ⊙ yr −1 for the B0.5 Ib supergiant donor star. In M101, Liu et al. (2013) inferred a mass-loss rate of 2 × 10 −5 M ⊙ yr −1 for the Wolf-Rayet donor. In P13, we think the stellar mass-loss rate of the B9 Ia supergiant donor star could have been underestimated: in Motch et al. (2014) , it is assumed that the donor star fills its Roche lobe on the basis that the wind mass-loss rate derived by Kudritzki et al. (1999) would be inferior to 10 −6 M ⊙ yr −1 . However, Kudritzki et al. (1999) did not include late-type B supergiants in their study because of the difficulty to measure their terminal wind speed ( Howarth et al. 1997 ). Vink et al. (2000 , 2001) provided fits for the mass-loss rate and the terminal wind speed which extend below the second bi-stability jump (i.e., for effective


a comparison needs to be made statistically using a large sample of stars. Unfortunately measurements are available for very few A and B stars in Kepler and K2 . A comparison using 30 stars with T eff in the range 8300–12 000 K supports the identification of the photometric period with the rotation period (Balona 2017 ). A similar study using TESS observations of B stars comes to the same conclusion (Balona et al. 2019 ). Both these studies involve small numbers of stars. TESS observations have revealed the rotation frequencies of many more A and B stars for which measurements are available. Table 2 shows that there are 274 stars with both rotation frequencies and . In Fig. 1 a comparison between and the photometric v is made for 193 stars with T eff > 8000 K. The temperature limit was chosen to minimize contamination by stars in which surface convection may still be present. Since cannot exceed v , the points are expected to lie below the line in the figure. Figure 1. The projected rotation velocity, , as a function of the equatorial rotation velocity, v , estimated from the photometric period for stars with effective temperatures T eff > 8000 K. The straight line is . The 1σ error bars are shown on the top left. Table 2. The number of stars with known photometric periods, n , and the total number of stars, N , in the given effective temperature range, T eff , is given. The number of stars with both photometric periods and measurements is . The number of stars used in the frequency distribution is . The increase in numbers of stars above the line at low rotation rates is to be expected. Measurements of are mostly obtained by measuring the widths of selected spectral lines and applying a calibration to convert to . The calibration is obtained using high-dispersion spectra and modelling line profiles. The error in is approximately constant at all values of . Analysis of the rotational velocity catalogue of Glebocki Gnacinski ( 2005 ) gives a standard deviation of of about 30 km s −1 . This means that at low values of , the error is comparable to, or larger than, itself and many points will lie above the line. This is further compounded by the fact that is constrained to be greater than or equal to zero, although measurements of line width may lead to negative values of on applying the calibration. As the rotation rate tends to zero, measurements of ever increasing precision are required to ensure that stays below the line. This is clearly not possible to attain. Another consideration is that low photometric frequencies are more difficult to measure because a longer time span is required to resolve the frequency peak in the periodogram. Thus instrumental drift becomes important. This is further compounded by the fact that the periodogram noise increases sharply towards zero frequency. Thus the uncertainty in v increases as v approaches zero, contributing to moving the point above the line. The standard deviation for v is about 40 km s −1 as


values of polynomial fits at each pixel and plot the resulting model over the original spectrum. In the lower panel of Fig. 3 , we show the residuals between the model and the input spectrum, with horizontal lines marking the median uncertainty about zero for this spectrum. With just eight principal components, we can model this particular spectrum to within the typical measurement uncertainty. In Fig. 4 , we show the distribution of model residuals for all spectra in our example slice of the red clump for models using zero, four, and eight principal components. As we expect, increasing the number of principal components used in the model causes the residual histogram to become more peaked towards a value of 0, with shallower wings. Given this continual improvement in modelling, it is necessary to define a cut-off to determine when improvements to the model explain variations due to noise, not intrinsic stellar differences. Figure 4. Histograms of the residual normalized flux after subtracting from each spectrum in the example slice a model using 0, 4, and 8 PCs. As more PCs are used, the histogram becomes more sharply peaked in the centre, with shrinking wings. 3.3.2 Assessing principal components We now define some useful statistics that can be computed from the results of EMPCA, which allow us to assess the results of the algorithm. We define V data mean as the mean-based variance of , We define a similar quantity V data MAD as the median absolute deviation (MAD) based variance of where is the p th row of , the operation ‘med’ is taking the median of the data, and the factor of 1.4826 scales the MAD to the standard deviation for normally distributed data, i.e. V data mean = V data MAD for data drawn from a Gaussian distribution. Equations ( 11 ) and ( 13 ) represent the total variance in the data. We wish to compare these to the variance after removing the principal component model produced by EMPCA. To do this, we define two additional quantities that compute the variance in the data after the contribution from the first n principal components has been subtracted out: Here, is the original data set after the model using the first n principal components has been subtracted (equation 10 ), and is the p th row of . Note that in the case where n = 0, V model mean (0) = V data mean and V model MAD (0) = V data MAD . Also note that when the number of components is maximized ( n = min ( N, P ) = n limit ), V model mean ( n limit ) = 0 and V model MAD ( n limit ) = 0, because in that case the model can perfectly represent all of the data. In what follows, we use V data and V model without superscripts when either the mean-based or MAD-based versions could be used. We compare these variances using R 2 ( n ), defined as The R 2 ( n ) curve should increase in height with the number of iterations on the model components. The maximal value for R 2 ( n ) is 1, which indicates the data can be perfectly reproduced by a linear combination of n components


by the WIND spacecraft at 1 au on 2011 June 24, with n i = 2.4 cm −3 , V SW = 599.0 km s −1 , and . (a)–(c) Time variations of the magnetic field vector in geocentric solar ecliptic coordinates. (d) Time variation of field magnitude. (e) Power spectrum of the magnetic field obtained from the FFT method with 11-point centered smoothing. The spectrum with spacecraft frequency f 10 −3 Hz is not shown here, since there are not enough points at this domain when smoothing the spectrum. The two red solid lines show the linear fits to the spectrum in the low-frequency and high-frequency ranges, respectively. Red dots indicate the upper and lower frequency limits used to perform the fits. The obtained spectral indices for the low-frequency ( α L ) and high-frequency ( α H ) ranges and their uncertainties are shown, along with the values of f b , f di (blue arrow), f ρi (green arrow), f ri (black arrow), and β . The vertical dashed line denotes the frequency of the spectral break. Panel (e) of Figure 1 shows the power spectrum of the magnetic field fluctuations. The spectrum is computed by applying the Fast Fourier Transform (FFT) on the three components of B . The trace of the spectral matrix gives the total power spectrum (hereafter power spectrum for simplicity). The spectral break frequency is determined as follows. First, we estimate the location of the break point by eye, which is roughly at 0.3 Hz. The break point divides the power spectrum into two parts: a shallower low-frequency part and a steeper high-frequency part. We then perform linear least-squares fit to the spectrum on the two sides of the roughly estimated break point, respectively (see the two red solid lines in Figure 1 (e)). The upper and lower frequency limits (the red dots) used in the fitting procedure are determined by the frequency range that is best fitted by power law based on chi-square minimization. For the low-frequency part, the band used to perform the fit is between 0.008 and 0.1 Hz. The spectral slope as well as its 1 σ uncertainty is estimated as α L = − 1.694 ± 0.025, which is close to the typical Kolmogorov scaling f −5/3 in the inertial range of the solar wind turbulence (Matthaeus et al. 1982 ; Goldstein et al. 1995 ; Tu Marsch 1995 ; Podesta et al. 2006 ). The high-frequency part of the spectrum beyond the break point is also fitted by a red solid line in the same way, but at the frequency range between 0.4 and 1.2 Hz. The intersection of the two red solid lines associated with the power-law fits thus gives the exact frequency of the break point f b = 0.288 Hz (marked by the red vertical dashed line). This method about determining the break frequency has already been used in several previous studies (e.g., Leamon et al. 1998 ; Smith et al. 2006 ; Hamilton et al. 2008 ; Markovskii et al. 2008 ; Bruno Trenchi 2014 ; Wang et al. 2018 ). In addition, we notice that at the frequencies higher than 1.5 Hz, the power spectrum becomes shallower again, which


resolution. In order to compare clumps in our galaxies with those in high-redshift galaxies, we identified clumps by visual inspection of the z = 2 images in the F435W band. Measurements were made on the original non-redshifted images since the simulated redshifted images were Gauss-blurred and re-pixelated with artificial noise added, so flux would have been difficult to determine accurately. Instead, contour plots were made on the local images and compared with the z = 2 images where the clumps were distinct. In practice, the contours at 10× sky σ on the local images matched the clumps in the high -z images, so that is where the boxes were defined for photometry. The IRAF task imstat was used to determine total pixels and mean counts per pixel within the boxes. Magnitudes were then determined for each clump, using the zero-points from the WFC3 and ACS handbooks. There was no background subtraction, because the photometric fitting described below included an underlying component plus the star-forming component, described further below. This is the same procedure as we used in measuring clumps in high-redshift galaxies (Elmegreen et al. 2009b ), so our results for clumps in our galaxies can be directly compared to the high-redshift results. For the galaxies in this paper, this means the clump boundaries sometimes contain more than one large clump, since some get blended at high z . The measured clumps are indicated as boxes in Figure 8 . For Haro 11, contour plots are also shown as an example of how clumps were identified. The right-hand contour plot is on the simulated z = 2 F435W image, where the three clumps are distinct. The middle contour plot on the original image has an arrow indicating the ∼10× sky σ contour where the boxes were then drawn. A total of 51 clumps were identified in the seven galaxies (the number in each galaxy is listed in Table 1 ). The average absolute B- band magnitude of the clumps for the seven galaxies in this sample is −15.4 ± 2.6, while the average ( B − V ) color is 0.4 ± 0.5. The clump diameters are approximated by the square root of the area of the boxes defined for photometry, and range from 170 pc to 2.9 kpc, with an average of 1.0 ± 0.7 kpc. Figure 8. The seven galaxies studied here are shown in the F555W filter. Boxes identify the clumps that were measured; see Figures 1 – 4 for physical scales. For Haro 11, contour plots are also shown to indicate how clumps were identified. Measurements were done on the non-redshifted images for the boundaries determined from the redshifted images. In practice, these boundaries were at a surface brightness ∼10× sky σ in the F435W image, as indicated in the middle image. The right-hand contour plot is for the simulated z = 2 F435W image, where the three clumps are distinct. 4.2. Clump Ages and Masses The photometric measurements of the clumps were used to determine ages, masses, and extinctions through spectral energy distribution (SED) fitting, as detailed in Elmegreen


is undetected in one scenario, an upper limit on its abundance can suffice to disfavour biological production for certain molecules. Fig. 7 shows the posterior distributions for the mixing ratio of each molecule from our retrievals. A 10 Prism transit campaign suffices to constrain the CO 2 and CH 4 abundances of the modern Earth to within an order of magnitude (0.78 dex and 0.85 dex, respectively); for the prebiotic Earth only CO 2 is constrained (0.62 dex). An increase to 20 Prism transits offers two new results: (i) an order of magnitude constraint on the H 2 O abundance for the prebiotic Earth (0.70 dex); and (ii) a upper limit on the prebiotic Earth CH 4 abundance in tension with the confidence interval of the modern Earth CH 4 abundance. The latter is a diagnostic difference between the modern and prebiotic scenarios accessible with only 20 transits. A further increase to 100 Prism transits offers dramatic improvements: constraints on the CO 2 , H 2 O, and CH 4 abundances to ∼ a factor of 2 (0.3 dex), and an order of magnitude constraint on the modern Earth N 2 O abundance (0.78 dex). Constraints on the O 3 abundance of the modern Earth can be obtained, to within an order of magnitude, with either 100 Prism + 100 MIRI transits (0.63 dex), or 200 Prism transits (0.59 dex). However, 200 Prism transits result in significantly better constrained abundances for the other molecules. As above, we recommend additional Prism transits instead of dividing time between the Prism and MIRI. Figure 7. Retrieved atmospheric composition from simulated JWST observations of TRAPPIST-1e. The predicted mixing ratio constraints for CO 2 , H 2 O, CH 4 , N 2 O, and O 3 for the 1 bar modern Earth (green) and prebiotic Earth (orange) are shown by the marginalized posterior histograms. Each row corresponds to a different JWST observing campaign, ranging from 10 NIRSpec Prism transits (top) to 200 Prism transits (bottom). Various combinations of NIRSpec Prism + MIRI LRS transits are also included. Predicted detection significances for each molecule are annotated (green text for the modern Earth; orange text for the prebiotic Earth). Abundance constraints (error bars) are overlaid, with the interval annotated where constraints 1 dex are possible. For non-detections, abundance upper limits are shown (vertical lines with arrows). The prebiotic and Modern scenarios can first be distinguished with 20 Prism transits via the CH 4 abundance (the prebiotic upper limit precludes the modern abundance). Similarly, the N 2 O abundance can be differentiated with 200 Prism transits. Even 200 Prism transits (or 100 Prism + 100 MIRI transits) are insufficient to distinguish the O 3 abundances. O 2 is unconstrained in all cases. Adding MIRI data slightly increases the detection significances of O 3 , but adding the same quantity of Prism transits produces tighter constraints on all other atmospheric parameters. Our analysis demonstrates that the most effective way to


is evaluated. The θ = 0 plane corresponds to the equatorial plane of the system. Bottom : angular ejection distribution after the most energetic interaction. Other elements in the plots are the same as in the top three plots. In contrast, we find that Pops. II and III are more isotropically distributed. These are the stars that come from outer regions on centrophilic orbits, and therefore the effect of random motion of the black holes does not play a role, like it does in the case of Pop. I. We notice that the third and especially second population show a prevalence of ejections along the SMBH binary orbital plane. This is in agreement with previous studies which find that HVSs are primarily ejected near the SMBH binary orbital plane when their SMBH orbital plane is aligned with the rotational place of the system (e.g., Sesana et al. 2006 ; Wang et al. 2014 ; Zhong et al. 2014 ; Lezhnin Vasiliev 2019 ; Rasskazov et al. 2019 ). Our findings show that not all of our three populations are uniformly distributed in the azimuthal direction. While there is some disagreement in the literature over the existence of a preferential ejection direction in eccentric binaries ( Rasskazov et al. 2019 ), in the case of a circular SMBH binary like ours, previous studies report a uniform distribution of ejected stars in the orbital plane ( Rasskazov et al. 2019 ; Lezhnin Vasiliev 2019 ; Sesana et al. 2006 ). We argue that the motion of black holes around the center can result in anisotropic ejections for the central population of stars (Pop. I), while stars originating from outside of the SMBH binary influence sphere are uniformly distributed (Pop. II and III). From the apocenter distribution in Fig. 11 , we conclude that Populations II and III correspond to stars on centrophilic orbits that are responsible for repopulating the loss cone. Depending on the shape and degree of nonsphericity of the nuclei (e.g., axisymmetric or triaxial), different orbital families may fulfil this role ( Merritt 2013b ). We can distinguish between the different orbital families if we look at their angular momentum at a time significantly before the interaction. Namely, because of the symmetries of different geometries, we can distinguish between spherical, axisymmetric, and triaxial orbits. Spherical orbits have conserved angular momentum, and in order to satisfy Eq. ( 12 ) are necessarily contained at all times within the bottom dashed square region of the top panel of Fig. 12 . Axisymmetric orbits, such as the saucer and tube orbits, in turn have conservation of the z -component of angular momentum and are therefore necessarily contained in the | L z / L crit | 1 region of phase space, denoted by vertical dashed lines in the same figures. Finally, triaxial orbits (pyramids, boxes or chaotic orbits) can have arbitrarily small or large values in the ( L , L z ) plane as a result of a lack of an integral of motion in that plane. This simple classification scheme is inspired


This holds true in particular for CO line emission which is optically thick. A second parameter to explore is , the power-law index of the surface density distribution. Models with different values are presented separately in the lower panels of Fig. A.1 . The line luminosities produced by models with and are almost indistinguishable, while differences can be found comparing these with models with . In particular in the intermediate disk mass regime, models for show three different populations that reflect models with different values. This effect is less clear for models with . Finally, it is interesting to investigate how much the central star influences the CO isotopologues emission in the disk. Models with the same physical disk parameters, but irradiated by a T Tauri-like and a Herbig-like star are presented in Fig. 2 (respectively in the left and in the right panel) and in Fig. 3 . For the Herbig disks, the derived line intensities are higher for more massive disks and lower for less massive disks compared with those obtained for the T Tauri disks. Since Herbig disks are warmer, less material is frozen-out onto grains in the higher mass regime and can contribute to the total emission enhancing the line luminosity. Also, the higher temperatures enhance the moderately optically thick CO lines. On the other hand, the strong UV flux coming from the Herbig star is very efficient in photodissociating CO isotopologue molecules for lower mass disks (, ) where the column densities are not high enough for shielding the photons. This combination of high UV field and low column densities reduces the line luminosities, thus they are lower than those from T Tauri disks. 3.2.4. Dependence on gas-to-dust ratio To test the sensitivity of the results to the gas-to-dust ratio for a fixed gas mass, we ran two T Tauri disk models ( au, , , , , ) with reduced gas-to-dust ratios of ten and one. The ray-traced line intensities for CO isotopologues are presented in Fig. A.3 as functions of the gas-to-dust ratio, normalized to the value at gas-to-dust =100. For the disk the optically thin CO intensities increase by a factor of 3.5 if the gas-to-dust ratio is lowered by a factor of 100. This can be understood by the fact that if the dust mass is enhanced for a fixed gas mass, the FUV flux will be more efficiently attenuated. Accordingly CO and its isotopologues survive more easily and their lines are brighter for low gas-to-dust ratios, especially if their emission is optically thin. In addition, if the dust opacity dominates over the line mutual shielding, isotope-selective processes are minimized. The CO and CO emission on the other hand stay the same within 30% because they are more optically thick. There are also small temperature differences of order 20% between the models in the emitting layer which can affect their line intensities. For the disk the trend is opposite: line intensities decrease if the gas-to-dust ratio is lowered for all three


et al. 2017 ). This implies that low- and intermediate-mass stars in such a late evolutionary phase are often old and hence metal-poor on average. The distributions of [Fe/H] and [S, Zn/H] are very similar in the LMC and in our Galaxy (see Fig. 7 ), indicating that depletion and hence dust-gas segregation and subsequent gas accretion, is very similar in both environments. On average the LMC has a lower metallicity than the Galaxy but it does not affect the efficiency of the depletion process. Additionally, in none of the systems, there is an evidence that the abundances of the s-process elements, systematically deviate from the abundance of elements with similar dust-condensation temperature. A good tracer of the depletion which is independent of the initial metallicity is the [Zn/Ti] abundance ratio. Zn and Ti are both Fe-peak elements and have the same nucleosynthetic history. If only nucleosynthesis determines the abundances ratios, (i.e. no selective depletion of elements), then [Zn/Ti] will remain zero. Since Ti is likely to be more strongly depleted than Zn, [Zn/Ti] is an ideal indicator of depletion. Therefore, we define the efficiency of depletion as a function of [Zn/Ti]. In principle [Zn/Fe] can also be used as a proxy for depletion; however, Ti has a higher dust condensation temperature and hence more easily depleted than Fe. So we define the efficiency of depletion as a function of [Zn/Ti]. Strongly depleted sources are defined such that [Zn/Ti] ≳ 1.5. Mildly depleted sources are defined as those with 0.5 ≲ [Zn/Ti] ≲ 1.5 while sources with [Zn/Ti] ≲ 0.5 are considered to show little or no depletion. In Fig. 8 we show the spread in [Zn/Ti] abundance ratios as a function of T eff for the LMC post-AGB sources (left-hand panel) and the depleted Galactic sources (right-hand panel). We find that the LMC as well as Galactic post-AGB stars show a large spread in depletion patterns, ranging from strongly depleted to stars with mild and no depletion. The plots also show that there is no clear trend in the efficiency of depletion with respect to T eff . The advantage of the LMC sample is their accurately derived luminosities. The derived photospheric luminosities of the LMC sample of depleted objects show that chemical depletion is active over a range of luminosities (see Section 5 and Fig. 8 , left-hand panel). Hence we can conclude that there is also no clear trend in the efficiency of depletion with respect to luminosity ( L ph ) of the star. Figure 8. Left: The [Zn/Ti] abundance as a function of T eff for the depleted sources in the LMC. Right: The [Zn/Ti] abundance as a function of T eff for the depleted post-AGB stars in the LMC (black filled circles) and their Galactic analogues (magenta filled squares). The dashed line represents the limit of what we call ‘strongly depleted’ while the dotted line marks our definition of ‘mild depletion’. In the case of the Galactic sources, majority of these objects show the presence of a stable


not correspond to the velocity expected from the kinematical model presented in this work. We thus conclude that the clumps of the EHVO seen in Fig. 7 correspond to material that was swept up by a different outflow, which is faster and likely older than the HVO. Additionally, there seems to be traces of entrained material of the EHVO outflow that appear as “horns” of the HVO and confined within the region traced by the CO( J = 1 → 0) emission. Therefore, it is likely that the extension of the region traced by the CO( J = 1 → 0) emission determines the radius of CO photo-dissociation by interstellar UV radiation. 4.3 Kinetic energy and scalar momentum of molecular outflow Since our model is fundamentally different from the one presented by Sahai et al. (2017) , here we re-calculate the scalar momentum and kinematic energy of the molecular outflow. To obtain the kinetic energy and scalar momentum, in principle, it is possible to use the mass spectrum shown in Fig. 6 . However, the velocity offset in this plot is simply the component of the expansion velocity on the line-of-sight. The 3D expansion velocity is necessary for the calculations. To circumvent this problem, one can take advantage of the morphology adopted for the emitting region, which has an elongated, narrow shape. In such a case, it is possible to de-project the velocity vectors by multiplying the velocity offsets by a factor of 1/sin θ inc , with θ inc = 45°. This is not strictly true for all the velocity vectors, since the gas is not really moving along one single line. Particularly, for the gas located close to the central source, which might be expanding along the line-of-sight, the expansion velocity will be overestimated by a factor of 1/sin θ inc . For such gas, this leads to an overestimation of the scalar momentum by a factor of and for the kinetic energy by a factor of 2. In addition, there is an uncertainty due to the optical depth of the CO( J = 3 → 2) emission since it was assumed to be optically thin to obtain the mass spectrum of Fig. 6 . If the emission is optically thick the estimated value of the mass simply represents a lower limit. Having this in mind, we proceeded to calculate the kinetic energy and scalar momentum in the manner introduced by Bujarrabal et al. (2001) and described inthe following. Firstly, we multiplied the mass in each channel by the square of the corresponding velocity to obtain the kinetic energy per channel. By adding the kinetic energy of all the channels we obtain K LVC = 8 × 10 44 erg and K HVC = 1 × 10 45 erg for the LVC and HVC, respectively. Similarly, the derived scalar momentum is 1.6 × 10 38 gm cm s −1 and 6 × 10 37 gm cm s −1 for the LVC and HVC, respectively. The total kinetic energy and scalar momentum in the outflow are of the same order of magnitude as the values calculated by Sahai et al. (2017) . Subsequently, the scalar momentum in the molecular outflow transferred by the jet, Δ P , is obtained by subtracting the scalar


the maximal error is roughly 6 times smaller than this upper limit. Table 2. In this table we compare nonlinear correction spectra computed with EuclidEmulator (EE) on the one hand and with an emulator based on Takahashi’s HALOFIT (THF) on the other hand to nonlinear correction spectra of full N -body simulations for 36 cosmologies outside the ED over the entire k and range. The numbers in the table correspond to the relative errors in per cent between the emulated and simulated nonlinear corrections. We find that the errors predicted with the THF emulator are broadly consistent with the EE errors. This implies that the configuration of the EE based on a HALOFIT emulator is actually valid and that there is no need to make a huge investment to run N -body simulations just for the configuration study of an emulator. This result is particularly important for finding the size of the ED (see Fig. 5 ). 4.2 Sensitivity analysis Sobol’ indices (Sobol’ 1993 , 2001 ) measure how sensitive the eigenvalues λ i (ω b , ω m , n s , h , 0 , σ 8 ) (introduced in equation 9 ) are to each single input parameter as well as to any of their interactions. For an introduction into Sobol’ sensitivity analysis, we refer to the previous references or to Marelli et al. ( 2017 ) and Le Gratiet, Marelli Sudret ( 2016 ). Sobol’ indices are based on the Hoeffding-Sobol’ decomposition, which states that any square-integrable function over a hypercube input parameter space can be cast as a sum of a constant, a set of univariate functions of each input parameter, another set of bivariate functions, etc. This decomposition is unique and the various terms are orthogonal with each other (with respect to the uniform probability measure over the hypercube). The variance of the output can then be apportioned to each input parameter, each pair, triplets, etc.: these contributions are called Sobol’ sensitivity indices S . Although their classical estimation relies on costly Monte Carlo simulations, Sobol’ indices can be computed analytically from a PCE as in equation ( 9 ), see Sudret ( 2008 ). In our case, we get one such Sobol’ expansion for each principal component. Each Sobol’ index corresponds to the fraction of the total variance of the respective eigenvalue that is caused by the parameter(s) under consideration. The bigger this number, the more λ i depends on the considered set of input parameters. Fig. 7 shows the first-order (no interactions) Sobol’ index plots for the first and fifth principal components (see Fig. A1 in Appendix A ). It is interesting to notice that the input parameter ω b has no leading-order effect on the output nonlinear correction. The nonlinear correction starts to show non-vanishing sensitivity to this parameter only at fifth order and higher. This is explained by the fact that baryons are treated as a background quantity that only come into the simulation data via the transfer function. As they are not directly evolved themselves in PKDGRAV3 , their


disc. Fig. 7 confirms that the time lag increases with increasing R acc and with increasing ξ, and does not change significantly with τ r and τ w . Observations show time lags of about 50–250 Myr which are reproduced by most of our simulations, but a few of our simulations produce time lags that have values of up to 550 Myr. Such large time lags have not been directly observed yet. However, Wild et al. ( 2010 ) only measure the average time lag of a sample of about 400 galaxies with a result of 250 Myr, thus some of the galaxies involved will have time lags that exceed this average value. Fig. 8 shows SFR, BHAR and AGNAR versus time for different levels of resolution. We increase the number of particles by a factor of 2 and a factor of 4, respectively. For the former resolution level the gravitational softening length for the halo particles is 0.64 kpc and 0.4 kpc for all other particles, for the latter resolution level the gravitational softening length for the halo particles is 0.5 kpc and 0.32 kpc for all other particles. Figure 7. Time lag between starburst and AGN activity with the values of R acc , ξ, τ r and τ w from our parameter study (Table 2 ) and our reference model. Figure 8. SFR, AGNAR and BHAR as functions of time for different resolution levels. Left: particle number increased by a factor of 2, right: particle number increased by a factor of 4. The vertical lines mark the peak of the SFR and the peak of the BHAR, respectively. Both resolution levels show the same features as our reference model: the simulations produce a starburst and a peak in the BHAR, and a time lag between these phenomena of about 200 Myr. However, there are still small differences between these simulations, to further reduce these deviations additional adjustments to the respective resolution levels would be necessary. For instance, Hopkins, Narayanan Murray ( 2013 ) argue that the critical density for the onset of star formation ρ crit has to be adjusted for resolution. But as the above mentioned key features are already reproduced we refrain from such a fine tuning of our simulation parameters. 5 SUMMARY With only one model setup we were able to reproduce three observational findings that have been identified in galaxies. The observed time lag between starburst and AGN activity is, in our work, principally caused by a viscous time lag the gas needs to flow through the AGN's accretion disc until it reaches the central black hole. Our results match the observed M BH -σ correlation, but additionally include the aforementioned time lags. As, e.g. Di Matteo et al. ( 2005 ) or Debuhr et al. ( 2011 ) have already shown, AGN feedback is responsible for this relation. The large scatter of the M BH -σ correlation is, in our work, caused by the continuing evolution of the black hole mass after the merging event. The time lag does not only occur between starburst and AGN activity, but also between the peak in the AGNAR and the peak in the BHAR. Thus it is


in the range 0.4%–0.5%. For the dwells with mHz detections, the symbol size is proportional to the fractional amplitude (rms). 3. Properties of mHz Oscillations Since the power spectra (Figure 3 ) from the two 2017 September dwells with strong mHz oscillations are consistent with rather coherent oscillations during these intervals, we computed phases for all events in each of these dwells separately using the frequencies corresponding to the peak Fourier power obtained from each dwell. To combine the two dwells we first found a constant phase offset for the second dwell that maximized the signal power of the sum of both dwells (Buccheri et al. 1983 ). The resulting phase-folded pulse profile in the 0.4–7.5 keV band is shown in Figure 8 . We fit a model of the form and find an amplitude B / A = 1.12 ± 0.05%. The fit is good with a minimum χ 2 = 11.0 for 13 degrees of freedom, and the best-fitting model is also plotted in Figure 8 (red curve). To explore the energy dependence we also computed profiles for different energy bands and then fit them with the same model as above. Figure 9 shows the derived amplitudes, B / A , and phases, φ 0 , versus energy. The amplitude shows a modest increase from about 0.7% at 1 keV to ≈2% at 7 keV. There is no strong evidence for a variation in φ 0 with energy. Figure 8. Average, phase-folded mHz oscillation profile from the two NICER dwells shown in Figure 2 . The profile includes all events in the 0.4–7.5 keV band. The best-fitting model, , is also shown (red curve). Phase ranges used to extract spectra are indicated by the vertical lines. See Section 3 for additional discussion. Figure 9. Fractional pulsed amplitude (top) and reference phase, φ 0 (bottom), obtained from fits to the phase-folded mHz oscillation profile are shown as a function of energy. The dashed line in the bottom panel marks the mean reference phase value. This constant value provides an acceptable fit to the distribution of values, with a χ 2 = 6.9 for 5 degrees of freedom. To explore the pulse-phase spectrum we extracted two spectra, from regions around the pulse minimum and maximum. The phase ranges used for these extractions are indicated by the dashed and dashed–dotted vertical lines in Figure 8 , respectively. In each case, we used a range of pulse phase equal to 0.25 cycles, so that each spectrum has an exposure of 0.25 × (2(2200)) = 1100 s. We used XSPEC version 12.9.1 with version 1.02 of the NICER instrument response, and we fit the spectra in the range 0.6–9 keV. We use a model comprising thermal comptonization ( comptt in XSPEC) for the persistent accretion-driven emission and a blackbody ( bbodyrad in XSPEC) to capture the neutron star thermal emission (Thompson et al. 2008 ; Chenevez et al. 2016 ). To model Galactic absorption we employ the tbabs model with abundances from Wilms et al. ( 2000 ), and we fix (freeze) the value of n H = 0.4 × 10 22 cm −2 based on prior measurements, since the source is in the same (soft) state


low-density conditions ( Westermann et al. 2001 ). It allows operation at 10 −10 –10 −6 mbar and can be used in a static background mode in molecular-beam mode to simulate the outflow of cometary gases ( Graf et al. 2004 ). To calibrate aliphatic and aromatic hydrocarbons in the DFMS laboratory model, the compounds were transferred from liquid into gas phase by heating the sample tube to 323 K at low pressure in the range of 10 −6 mbar. The gas was introduced through a leak valve into CASYMIR/DFMS where measurements were performed. This procedure was repeated at three different pressure levels p = 3 × 10 −9 mbar, 2 × 10 −8 mbar, and 1 × 10 −7 mbar to cover a maximum density range. In addition, background measurements were performed and subtracted from the actual measurements. Calibration of DFMS on aliphatic and aromatic hydrocarbons was necessary for several reasons: (I) in addition to charging neutral molecules positively, the electron impact ionization used in mass spectrometry also breaks molecular chains and thus leads to a detection of the fragments and the parent molecule in the mass spectra. This process is called fragmentation and depends on the electron energy used for ionization. It results in a fragmentation pattern with species-dependent abundances, which can be used to identify or exclude species in mass spectra. However, the pattern also depends on the molecular structure of the ionized species and thus can be used to distinguish isomers such as n-butane and isobutane. Compounds such as pentane, heptane, and octane with higher masses are significantly affected, leading to complex fragmentation patterns. (II) The DFMS sensitivity on a compound is calculated from the detected number of ions. Therefore pressure conditions must be stable over the measurement period. This can be especially challenging for volatile organic compounds. Thus, a thermal valve-control system was used to stabilize the partial pressure of the component in CASYMIR during the calibration cycle. The DFMS calibration campaign included the saturated aliphatic compounds up to octane (integer mass/charge 114 u/e). The calibration campaign on butane, pentane, hexane, heptane, and octane was made with unbranched molecule version (e.g., n-pentane), while for methane, ethane, and propane, no differentiation in terms of structural isomerism appears. For benzene, the DFMS calibration was not performed because the compound is highly toxic in the gas phase. In addition to the fragmentation, the DFMS calibration campaigns were performed to derive the sensitivity of DFMS for measured compounds. Hereby the ratio of the ion current to the molecular density was derived for each pressure level. The slope between ion current and density for each of the three pressure levels is the sensitivity factor. As an alternative, when no calibration could be obtained, the sensitivity was calculated based on the ionization cross-section of the molecule. The sensitivities used in


higher post-outburst luminosities lead to shorter durations (∼hours to days) when DNB contributes significantly to the total luminosity and surface temperature. The total amount of energy available from burning hydrogen is set by the initial hydrogen column, and corresponds to the integral of the elevated luminosity cooling curves, which is constant in all cases with the same initial hydrogen column. For the initial hydrogen column of y H ∼ 1.2 × 10 6 g cm −2 , the total amount of energy that is released from hydrogen burning is 10 38 erg (i.e. ∼7 MeV nucleon −1 ). Without residual accretion or a large helium buffer that may slow down the depletion of hydrogen, the times when DNB significantly increases the observed surface temperature for an H–C envelope are ∼0.1–80 d from the end of the accretion outburst, depending on the initial hydrogen column and cooling luminosity. For larger initial cooling luminosities (and thus higher surface temperatures), the initial change in surface temperatures increases, while the relevant duration decreases, as the available hydrogen is consumed more rapidly. 4.2 Effect on cooling curves due to changing composition In addition to releasing energy through nuclear reactions, DNB can alter the observed cooling after accretion outbursts by altering the light element column in the envelope over time. In Section 2.1 , we show the effect on the T s – T b temperature relations for varying hydrogen column sizes in H–C envelopes (see Fig. 1 ). The decreasing hydrogen column due to DNB will only affect the cooling curve when the hydrogen column is varied in the transition region (as shown in Fig. 2 ). At the relevant temperatures for NSs in quiescence (with initial post-outburst surface temperatures T s > 8 × 10 5 K), this means that the same boundary temperature (at the bottom of the envelope) corresponds to different surface temperatures when the hydrogen column varies in the range 10 4 ≲ y H ≲ 10 7 g cm −2 . When the initial hydrogen column after the accretion outburst is smaller than y H ∼ 10 4 g cm −2 , a further decreasing hydrogen column will not affect the cooling curve. When the initial hydrogen column is larger than y H ∼ 10 4 g cm −2 , the cooling curve transitions from relatively hydrogen-rich envelope temperature relations towards those for a hydrogen-poor envelope. We show this effect in Fig. 8 , where we ignore the effect of heat released by DNB and only show the effect of a changing envelope composition. The dashed cooling curves in Fig. 8 correspond to fixed hydrogen columns, while the solid cooling curves correspond to the scenario where the hydrogen column decreases over time due to DNB. When DNB is taken into account, all the plotted cooling curves (irrespective of initial column density) correspond to that of a hydrogen-poor envelope after ∼50 d. Note that for larger post-outburst luminosities, it can take less time (∼1 d) to decrease the hydrogen column below y H ∼ 10 4 g cm −2 (see e.g. the


(Barnes et al. 2018 ). 5 THE LINK TO AGN EFFICIENCY The inclusion of ATC leads to an increased CC fraction, increased central cumulative gas fraction, and flattens the CC fraction redshift evolution relative to simulations that neglect it. The temperature profiles and the gas fraction profiles both suggest that the central black hole injects less energy into the ICM when thermal conduction is included. Therefore, we now examine the energy injected by the central black hole and how this energy couples to the ICM. We define the central black hole of each cluster as the most massive black hole within of the potential minimum at z = 0. Its properties are then traced as a function of redshift. During the formation of the ICM ( z ≤ 2) the low accretion rate, kinetic mode feedback dominates (Weinberger et al. 2018 ). In Fig. 9 , we plot the median cumulative energy injected by the central black hole in the kinetic mode feedback for simulations with and without ATC. At z = 0, the inclusion of ATC results in a 24 per cent reduction in the energy injected by the low accretion rate kinetic feedback mode, which agrees with the results from the temperature and gas fraction plots. Figure 9. Median cumulative energy injected in the kinetic mode (solid lines) as a function of redshift for the central black hole compared to the energy lost via radiative cooling within r 500 (dash–dot-dotted lines) for simulations with (red) and without (blue). For simulations with ATC, we plot the conductive heating (short dashed line) within as a function of redshift. The right scale shows the median stellar mass (long-dashed) and black hole mass (dash-dotted) with (dark red) and without (dark blue) ATC. When ATC is included the central black hole injects less energy, but the stellar and black hole masses are unchanged. Conductive heating only contributes a small fraction of the energy difference. The right axis of Fig. 9 shows the median stellar mass of the brightest cluster galaxy, defined as the mass within twice the stellar half-mass radius for the most massive subhalo, and the median central black hole mass as a function of redshift. Despite the reduction in energy produced by the inclusion of ATC, the masses of the galaxy and black hole have a negligible difference for simulations with and without ATC, especially for z ≤ 1. Therefore, the central black hole is still able to self-regulate and control its growth and the growth of the galaxy. One explanation of the difference in injected energy with and without ATC is that there is significant heat transfer from outer radii into the core of the cluster, providing the additional energy required to offset radiative losses. To investigate this we define the conductive heating with a sphere by its conductive luminosity. This is defined as the heat flux integrated over the surface of the sphere where we chose a sphere of radius to account for the collapse and growth of the cluster with cosmic time. Only gas that is


Fig. 9 favour periodic models more strongly when the light curve is sinusoidal and favour QPO models more strongly when the light curves are sawtooth like. We introduced the higher order p 4 QPO models with periods locked in a 2:1 ratio and a common coherence time to explore if this difference was driven by the assumption of sinusoidal waveforms in the one-component QPO model. We show in Fig. 10 that using the two-component QPO models halves the number of RRab and RRd variables that prefer a QPO compared to the one-component model. The increase in periodic preference going from a one-component to the two-component QPO models for the more sinusoidal RRc and RRe sources is marginal. An even larger shift is seen for the fundamental mode Cepheids. These results are in agreement with the behaviour we found in Section 3 for the experiments comparing artificial light curves with sawtooth or sinusoid waveforms. It is not surprising, therefore, that a higher order QPO model results in a greater preference for periodicity. The two-component QPO is a better approximation to the sawtooth-like light curves of RRab, RRd and fundamental mode Cepheids, so the coherence time can act more like the desired physical parameter, and less as a proxy for light-curve shape. Figure 10. Distribution of the RRab+RRd, RRc+RRe, fundamental Cepheids and first overtone Cepheids in the AIC likelihood difference for the QPO and periodic models when using either the standard QPO model (dotted) or the higher order model (solid). The changes are small for the more sinusoidal RRc+RRe and first overtone Cepheids, but shift strongly towards the periodic case for the more sawtooth-like RRab+RRd and fundamental mode Cepheids. In addition to light-curve shape, we also noted in Section 2 that mischaracterized errors can bias the results, irrespective of light-curve shape. If some of the noise level assumed in the model is less than the true noise, the extra variance will be interpreted as signal. The natural way to model this extra variance is to move the model away from periodic to model the mischaracterized noise as stochastic variability. If we take the best-fitting model and re-estimate the errors so that it has 2 /dof1, then the fraction of Cepheids and RRLyrae favouring the periodic model increase significantly, from 70 and 10percent to 90 and 65percent in the case of first overtone and fundamental Cepheids, respectively, and increasing from 80 and 15percent among RRc/RRe and RRab/RRd to 95 and 90percent, respectively. 4.3 Variable distributions in QPO parameters The lesson from Section 3 is that the QPO parameter cannot simply be interpreted as a measure of coherence because it is significantly affected by light-curve shape. This is, however, no barrier to using the QPO parameters to classify variables. Fig. 11 shows the distribution of all the sources in the P , P and QPO parameter spaces. We see that the P plane maps physically distinct variable types to nearly unique spaces.


of dust extinction. Generally, the single-band SFRs are in agreement or are slightly lower than those derived from SED modeling with MAGPHYS. If we are unable to use either of the first two methods and we have at least one detection in another filter, we scale the SEDs from the DLA counterparts that were fit with the first method to match the detected host galaxy flux. We then use the scaled SEDs to estimate the rest-frame UV flux and use the median and standard deviation of the scaled SEDs to calculate the rest-frame UV SFR. Finally, if there are no detections in any band but there are upper limits in the rest-frame UV band, we calculate SFR upper limits using the second method. Photometric measurement were made using the aperture photometry technique, using the HST point-spread function (PSF) for GRB-DLA counterparts with HST data and the DCT 12 PSF for the ground-based data (corresponding to ∼2 and ∼17 kpc diameter apertures respectively). The large difference in apertures comes from the fact that HST is able to resolve the host galaxy. We assume that the light from unresolved sources is solely from the host galaxy and background sky. All SFRs are calculated from dust-corrected observations unless otherwise stated. The host extinction, A V , is taken either from SED models or from GRB afterglow measurements using a Small Magellanic Cloud (SMC)-like extinction law which has been shown to best depict the GRB explosion environment (e.g., Schady et al. 2012 ). We assume the host extinction is the same as the GRB line-of-sight extinction, which Perley et al. ( 2013 ) has shown is fairly consistent within a factor of two to three. If the host extinction is an upper limit, we use that value in all dust-corrected calculations and report SFR upper limits. In Table 1 , we report these GRB-DLAs with SFR error estimates but treat these as SFRs upper limits in all plots using dust-corrected SFRs. Our host extinction is, in general, higher than the reported for DLAs in the SDSS survey for our sample’s column densities (Murphy Bernet 2016 ). This may likely be because GRB-DLAs are found at a smaller impact parameter of (Blanchard et al. 2016 ) than the general DLA population of (Fumagalli et al. 2015 ) or more simply because our DLA sample traces, in general, metal-rich, and likely dust rich, systems (for example, see the correlation between and metal line equivalent widths in Murphy Bernet 2016 ). 3.1.1. SED Fitting Star Formation Rate We use MAGPHYS with the HIGHZ extension (da Cunha et al. 2008 , 2015 ), to model the host galaxy SEDs from photometry. MAGPHYS models templates to the data and returns an SED with fitted parameters which include SFR, stellar mass ( M ), dust mass ( M ), and A V . This particular package is well suited for galaxies and takes into account bursty star formation which is appropriate for GRB host galaxies as suggested by Hunt et al. ( 2014 ). MAGPHYS uses a continuous model of star formation with superimposed random bursts that


image for illustration purposes and to help explain the caveats of the algorithm (compare it to the segmentation obtained for COR2B). Figure 7. CME event observed on 2008 December 12 on COR2A (upper panels) and COR2B (bottom panels) FOVs. The red line indicates the CME segmentation obtained by CORSET. Earlier stages of the CME development in the FOV of the COR2 instruments are not shown in Figure 7 . At these earlier time instances, CORSET was able to segment the CME; however, these results had to be ignored because they led to NARD equal to zero on the application of CORSET3D. The radial speeds derived by CORSET3D as a function of latitude and longitude are shown in Figure 8 . The CORSET3D reconstruction shows that this CME event propagates in the inner portion of northwest quadrant of the solar corona (as seen from Earth, see Figure 8 ) up to at least a heliocentric distance of 16 solar radii (de-projected distance by the time the event reached the end of the time period selected, i.e., 13:37 UT). Figure 8 indicates that the radial speed is lowest at the northernmost portion of the reconstructed CME front. Further analysis is necessary to assess whether the speed differences along the front are artifacts resulting from the segmentation or a true effect. Byrne et al. ( 2010 ) also studied this event using a technique based on triangulation and found a speed of about 400 km s −1 between 10 and 15 solar radii. Using the GCS model, Möstl et al. ( 2014 ) found that the initial speed of this CME was about 497 km s −1 . Figure 8. Radial speeds of the front of the CME event observed on 2008 December 12 as a function of latitude and longitude (in the Stonyhurst coordinate system), as derived by CORSET3D. If we assume that this event continues propagating radially after 13:37 UT, it would be observed at Earth with its center in the negative side of the y axis of the Geocentric Solar Ecliptic (GSE) coordinate system. This assumption seems to be in agreement with the magnetic flux estimated by Howard DeForest ( 2014 ). According to these authors, the corresponding magnetic structure reached Earth first in the negative portion of the y axis, i.e., westward of the Sun–Earth line. A previous study from Liu et al. ( 2010 ) used triangulation in both STEREO /COR2 and STEREO /Heliospheric Imagers’ FOVs to track this CME event along its evolution to Earth (see, e.g., Figure 9 in Liu et al. 2010 ). They identified two structures in the time-elongation maps constructed from running difference images along the ecliptic plane. They called these two structures “feature 1” and “feature 2,” although they seem to be part of one complex CME structure, which is the one CME event considered in this work. Liu et al. ( 2010 ) found that the speed of both features ranged between 300 and 600 km s −1 , developing westward during their entire evolution from the Sun to the Earth. Lugaz ( 2010 ) computed the direction of this complex event with respect to the Sun–Earth line


for the freedom in the low-order gravity field, because the likelihood function used to obtain our posterior distribution of interior models assumed an inflated variance on J 2 to accord with the numerical precision of our theory of figures implementation (see Section 2.1 ). Because the resulting distribution included a diversity of heavy-element and helium distributions, envelope transition locations, and core masses, the seismology suggests a tension with the Voyager rotation rate commonly assumed for Saturn’s interior that different three-layer interior models seem unlikely to resolve. This conclusion based on the ring seismology adds support to the notion that periodicities in Saturn’s magnetospheric emission (e.g., Desch Kaiser 1981 ; Gurnett et al. 2005 ; Giampieri et al. 2006 ) may not be consistently coupled to the rotation of Saturn’s interior (e.g., Gurnett et al. 2007 ; Read et al. 2009 ). The present model is potentially oversimplified in two major ways. First, the model is not suited to address the close multiplets of waves observed to have the same azimuthal wavenumber m , namely, the multiplets of waves in the C ring with m = 2 and m = 3. The bulk of these seem naturally explained by the model of Fuller ( 2014 ), wherein avoided crossings between the f -modes and deep g -modes of higher angular degree give rise to a number of strong perturbations with the same m value. However, in the wealth of new OLR and OVR wave patterns that have been measured from increasingly low signal-to-noise VIMS data since Hedman Nicholson ( 2014 ), it seems that only two waves add to the mixed-mode picture, both with m = 2: the close-in W76.44 wave, and the Maxwell ringlet wave whose frequency and m number were predicted by Fuller ( 2014 ). The f -modes of higher angular degree have less amplitude in the deep interior and so are less likely to undergo degenerate mixing with any deep g -modes strongly. Indeed, there is not yet any direct evidence for f -modes with l > 3 undergoing avoided crossings with deep g -modes, although the outlying (5, 4) OVR warrants closer scrutiny in the mixed-mode context. The second major simplification of the present model is the assumption that Saturn rotates rigidly. While upper limits can be established for the depth of shear in Jupiter or Saturn’s envelopes on magnetohydrodynamic grounds (Liu et al. 2008 ; Cao Stevenson 2017 ), evidence gathered from spacecraft indicates that zonal wind patterns do penetrate to significant depths (Smith et al. 1982 ; Kaspi et al. 2018 ). It has been proposed that the insulating molecular regions of these planets may be rotating differentially on concentric cylinders (Busse 1976 ; Ingersoll Pollard 1982 ; Ingersoll Miller 1986 ), the zonal winds being the surface manifestation of these cylinders of constant angular velocity.The mode identifications made in Section 5 and Table 1 reveal that the seismological data set now samples a variety of radial (via the angular degree l ) and


of the burst spectrum (see Lewin et al. 1993 ; Strohmayer et al. 2006 , for a review). We scrutinised the NICER, XMM-Newton EPIC-MOS/PN, Swift -XRT, and NuSTAR data for the presence of thermonuclear bursts at different time scales (1, 10, and 100 s), but we found none. Next we focused on INTEGRAL JEM-X (3–35 keV) data and identified one thermonuclear burst candidate. We describe the analysis of the JEM-X data covering the event below. 6.1. Properties of the type-I X-ray burst We searched INTEGRAL/JEM-X data for bursts within revolutions 1986–2009, following the same approach as Mereminskiy et al. (2017) . We focussed on one of the two JEM-X cameras, JEM-X1; JEM-X2 was operating simultaneously, but this instrument seems to be experiencing reduced sensitivity in recent years. We identified one candidate thermonuclear (type-I) burst from IGR J17591−2342, in Scw-500010 of INTEGRAL revolution 2001 on Sept. 19, 2018; 23:07:33 (MJD 58380.96358 TT). The key properties of the burst are listed in Table 3 . Based on its intensity, this burst should have also been detectable by ISGRI; however, due to the loss of sensitivity over time in the lower ISGRI energy band, 18–27 keV, we did not detect this event in those data. Table 3. Parameters of the type-I X-ray burst observed by INTEGRAL JEM-X during the outburst of IGR J17591−2342 in 2018. Notes. (a) 58380.96358 MJD. (b) The τ fit has been measured after a plateau of ∼10 s. (c) Burst peak flux in 0.1–40 keV energy band. (d) Burst peak temperature. (e) Pre-burst unabsorbed flux in 0.1–250 keV energy range. (f) Burst fluence in 0.1–40 keV energy band. The type-I X-ray burst occurred ∼60 d after the outburst commenced, just after the source was at the highest X-ray luminosity (see Fig. 2 ). We defined the burst start time as the time at which the X-ray intensity of the source first exceeded 10% of the burst peak flux (above the persistent intensity level). The event showed a fast (∼1–2 s) rise time, followed by a ∼10 s long plateau at its peak (Fig. 8 ). Therefore, we conclude that most likely a photospheric radius expansion took place (see e.g. Falanga et al. 2007 ; Galloway et al. 2008 ; Bozzo et al. 2010 ). After the plateau, the burst decay profile could be well fitted with an exponential function and the correspondingly derived e-folding time is τ fit ∼ 9 s. The total duration of the burst was ∼40 s. Fig. 8. Light curve of the type-I X-ray burst detected by JEM-X and reported for the first time in this paper. The burst start time was 58380.96358 MJD. The JEM-X light curve was extracted in the 5–25 keV energy range with a time bin of 2 s. For the burst spectral analysis, we first determined the persistent flux before the burst onset by converting the ISGRI counts from Rev-2001 to the total broadband spectrum obtained from Revs. 1994–1995 (see Sect. 5 ). We note that during the outburst, the ISGRI data in the 20–150 keV band alone were fitted by a power-law model, and the spectral parameters were


in star formation: both in short bursts (during close passages of the satellite) as well as sustained high star formation rates lasting several Gyrs. In Starkenburg et al. (2016) we show that the general properties of our simulated dwarf systems compare very well to a large sample of dwarf irregular galaxies and blue compact dwarfs from the literature. In this paper we provide a quantitative characterization of the morphological and kinematic properties of the dwarf systems during the minor merger events and thereby facilitate a more detailed comparison to observations. For the analysis we used morphological descriptions that have been applied to characterise where galaxies lie along the Hubble sequence, to disentangle interacting from isolated systems, and to describe the stellar distributions in major mergers of or larger spiral galaxies, such as the CAS (concentration, asymmetry and smoothness) and GM (Gini coefficient and M20) indicators (see e.g. Conselice et al. 2000 ; Conselice 2003 ; Abraham et al. 2003 ; Lotz et al. 2004 , and references therein). These have also been applied to describe the stellar components of isolated irregular dwarf galaxy samples ( Conselice 2003 ; Lotz et al. 2004 ), to characterise the gas distribution in starbursting dwarf galaxies ( Lelli et al. 2014 ) and in simulations of major mergers ( Holwerda et al. 2011a ), as well as a variety of observational samples ( Holwerda et al. 2011b , c , d , e , 2012 , 2013 , 2014 ). This paper is organised as follows. The hydrodynamical simulations are described concisely in Sect. 2 , while in Sect. 3 we focus on one specific simulated system and highlight key tidal features as the merger takes place. In Sect. 4 we introduce the morphological and kinematic indicators used and compare the results to some observational samples. We present a brief comparison to two dwarf galaxies with peculiar properties, namely IC10 and NGC 6822 in Sect. 5 . The summary and conclusions are given in Sect. 6 . 2. Method We analyse the structural properties of dwarf galaxies during minor mergers with dark satellites for a suite of hydrodynamical simulations recently presented in Starkenburg et al. (2016) . The simulations span a range of initial conditions for the dwarf galaxy, its satellite and a variety of orbital configurations for the interaction. They were performed using the OWLS ( Schaye et al. 2010 ) version of Gadget-3 (based on Springel et al. 2001 ; Springel 2005 ) with implementations for star formation and feedback as described in Schaye Dalla Vecchia (2008), Dalla Vecchia Schaye (2008) . Table 2 Parameters for the orbits. Notes. (a) The apo-to-peri ratio is an average for the first pericentric passage for different main and satellite masses and inclinations. Fig. 1 Evolution of the model-A dwarf galaxy with , merging with a 20% mass ratio dark satellite with on a co-planar relatively radial orbit (the intermediate orbit in Table 2 ). All figures show an inclined view of the


and in quiescence, along with their rates of change. We have also refined the supercycle length (and possible changes in superoutburst occurrence rate). The structure of this paper is the following: Sect. 2 presents details about observations, data reduction, and photometry analysis. Section 3 contains information about the photometric behaviour of the system and a thorough study of positive and negative superhumps. We discuss this information in Sect. 4 and present a summary of our campaign dedicated to MN Dra in Sect. 5 . 2. Observations and data reduction The observational data of MN Dra presented here were collected throughout three observation campaigns. The first data were obtained during 6 nights from 2009 October 12 to October 22, in the Skinakas Observatory, Greece. At that time, the star was in a quiescent state. The second campaign was conducted during 15 nights from 2013 July 09 to September 10. On that occasion, data were gathered in Poland, at the Borowiec station of the Poznań Astronomical Observatory, at J. Kochanowski University in Kielce, and at the Ostrowik station of the Warsaw University Observatory. During the 2013 campaign we detected two superoutbursts of MN Dra. The longest campaign was organized in 2015, from June 4 to December 18. Data covering 84 nights of observations were collected at the Borowiec Station, in Pisa in Italy, at Antelope Hills Observatory and at the MDM Observatory in the USA. Once again, two superoutbursts in MN Dra were observed. In Table 1 we present the journal of CCD observations of MN Dra. In total, the star was observed 374.37 h, and 8717 useful exposures were obtained. To summarize, eight telescopes with diameters ranging from 0.25 to 1.30 m were used to collect data during the 2009, 2013, and 2015 campaigns. All data of MN Dra were gathered in a clear filter (“white light”). This approach allowed us to obtain reliable photometry around 18–20 mag when the object was in quiescence. Bias, dark, and flat-field correction of raw files was made in a standard way. For the data reduction we used the IRAF package. The profile (point-spread function) photometry was obtained with the DAOPHOTII package ( Stetson 1987 ). Relative unfiltered magnitudes of MN Dra were derived from the difference between the magnitude of the star and the mean magnitude of two comparison stars marked as C1 and C2 in Fig. 1 , where the map of the observed region is shown. The equatorial coordinates of comparison stars C1 (, , mag in filter) and C2 (, , mag in filter) were taken from the 2MASS All-Sky Catalog ( Cutri et al. 2003 ). Instrumental magnitudes were transformed into the passband, which is very close to observations made in white light. Fig. 1 Finding chart with the position of MN Dra marked as V1 and two comparison stars C1 and C2. The field of view is about . 3. Light curves 3.1. Global photometric behaviour Fig. 2 Global photometric behaviour of MN Dra. For convenience, we use only a day number HJD-2 450 000 [d]


so the shell velocity equals the ratio of the swept-up momentum and the swept-up mass M sh ( t ) ≈ f Ω M ej + M w,sh ( t ), where the swept-up mass of the pre-dynamical wind is given by and t end = t run ( v w / v sh ) is the time required for shock to reach the outer edge of the pre-dynamical outflow. Although the time evolution of v sh is complicated by the implicit dependence of M w,sh on v sh , we find that for fiducial parameters its value varies by less than a factor of 2 from its asymptotic value of where in the final line we have taken M w ≲ M ej , f Ω = 0.3 and . In our analytic estimates that follow we assume a temporarily constant value of , where ξ ≈ 0.5–1. Emission escapes on the photon diffusion time-scale through the fast ejecta, which at the shock radius R sh ≈ v sh t is given by where we have used equation ( 3 ) for δ o = 5 and δ i = 2. If the opacity remains roughly constant in time, then equating this to the expansion time t gives the time-scale for the shock-powered emission to rise to its peak (Arnett 1982 ) where κ 0.3 ≡ κ/(0.3 cm 2 g −1 ) is normalized to a typical value (Fig. 4 ). In the second line, we have normalized the ejecta mass to a fixed fraction of the total stellar mass; the fast ejecta velocity to the binary escape speed; and the semimajor axis of the binary to 10 times the main-sequence stellar radius R ms ≈ R ⊙ ( M / M ⊙ ) 0.8 . All else being equal, equation ( 21 ) shows that mergers between more massive stars, or evolved stars with radii much larger than their main-sequence values, will result in later peaking light curves, consistent with our full numerical solutions (Fig. 5 ). However, overall the dependence on binary parameter is relatively weak, consistent with the factor of ≲5 range of observed values t pk ≈ 30–150 d for a ∼ 2–100 R ms and M ∼ 1.5–30 M ⊙ (Fig. 1 , Table 1 ). Also note that, by assuming a fixed, relatively high value of the opacity in deriving equation ( 21 ), we cannot account for cases when the shock heating is insufficient to keep the ejecta ionized (e.g. the 2 M ⊙ model in Fig. 5 ). Table 1. Properties of LRN and their stellar progenitors. References: (1) Mason et al. ( 2010 ), (2) Munari et al. ( 2002 ), (3) Tylenda et al. ( 2005 ), Afşar Bond ( 2007 ) but see Munari et al. ( 2005 ), (4) Blagorodnova et al. ( 2017 ), (5) Smith et al. ( 2016 ), (6) Mauerhan et al. ( 2015 ), (7) Mauerhan et al. ( 2017 ), (8) MacLeod et al. ( 2017a ). a Ejecta velocity inferred from optical spectra (note that this may not always represent the velocity of the bulk of the ejecta, but instead the fastest components). b Approximate peak luminosity of second peak. c Assuming the second peak was bolometrically about 1 mag fainter than the first. If the forward shock dominates the shock power (equation 7 ), then the peak luminosity of the shock-powered second light-curve peak is given by (in the limit v sh ≫ v w ) where we have used equations ( 1 ) and ( 2 ). Normalizing again to the


Rincon, Ogilvie Proctor ( 2007 ), where a subcritical dynamo generated by the interaction between (fluctuating) non-axisymmetric and (mean) axisymmetric toroidal magnetic field leads to ultimate saturation in a bounded domain with zero initial magnetic field. Indeed, in our theory the domain-integrated helicities of the flow and the magnetic field are both zero, while the domain-integrated cross-helicity vanishes because and are out of phase. The theory also allows us to investigate the astrophysically relevant weakly dissipative regime. We do this by showing that the inviscid limit of the theory, Re ≫ 1, is well behaved, and then examine the further limit of nearly ideal MHD by considering large values of Re m . Although the zonal magnetic field fluctuation becomes very large in this regime the angular momentum flux remains well behaved for all wavenumbers and decreases as | m | −1 (equation 34 ). A simple prediction for the magnitude of this flux follows from the assumption that the dominant wavenumber m corresponds to the optimum wavenumber from linear theory that yields the fastest MRI growth rate. The resulting expression for the angular momentum flux, , resembles the prescription of the α-model of Shakura Sunyaev ( 1973 ) but with the sound speed replaced by the Alfvén speed. A similar dependence of the momentum flux on the square of the magnetic field was observed in the simulations of a compressible but magnetic field dominated disc by Hawley et al. ( 1995 ), although subsequent computations by Sano et al. ( 2004 ) and Pessah, Chan Psaltis ( 2007 ) suggest a weaker dependence on v A . We emphasize that throughout this paper the symbols Re and Re m refer to scaled values of the Reynolds number and the magnetic Reynolds number, i.e. the condition Re m ≫ 1, for example, refers to Re m * ≫ ε −1 while the condition Re m ≪ 1 refers to Re m * ≪ ε −1 . The latter condition can therefore be satisfied for the large Re m * expected of astrophysical systems provided only that these are small compared to ε −1 , a quantity defined here in terms of the shear rate. Our approach has several attractive features, one of which is the fact that angular momentum is inevitably transported outward. In this connection we mention that Lesur Longaretti ( 2007 ) find that the transport coefficient F v /|σ| in a turbulent disc increases with the magnetic Prandtl number Re m / Re . In our (non-turbulent but fully non-linear) study we find a similar tendency when m is fixed (cf. Fig. 7 a): in the supercritical regime, above the marginal stability curve, F v decreases as Re increases for fixed Re m ; when Re is fixed, F v increases as Re m increases. Possible concerns include neglecting the possibility that saturation is accompanied by net outflow in the vertical direction, as measured by the quantity although this is a consistent assumption that follows from the relation obtained from equation ( 1a ). This relation is identically satisfied by . A further concern is


that the mass point C is under the action of an arbitrary additional force F (per unit mass) that is not gravitational and acts on the gas component of the supercloud. In our case, F will represent an hydrodynamic force due to the friction between the gas supercloud and the surrounding interstellar medium or due to the pass of a shock wave that affects the gas supercloud. Then, the motion equations in the reference system ( x, y ) are given by Figure 1. Illustration of the stationary coordinate system ( x, y ) centred at the Galactic centre and the rotating coordinate system (η, ξ) of origin O moving on a circular orbit of radius R 0 . The figure plane represents the Galactic plane and the rotation sense is clockwise. The angle θ, between the y -axis and ξ-axis, increases in the sense of the Galactic rotation. where F x and F y are the components of F and subscript c refers to the coordinates of the point C. Because the force K ( R c ) is equal to the centripetal force at R c , we will write K ( R c ) = R c ω( R c ) 2 , where ω( R c ) refers to the Galactic angular velocity at R c and for short will be written as ω c . Since we are interested in referring the motions to the rotating system, we should convert equations ( 1 ) into ones depending on (η, ξ). The formulae that govern this transformation are which are obtained from simple geometrical considerations of Fig. 1 . Besides, we should have into account that F x = F ξ sin θ + F η cos θ and F y = F ξ cos θ − F η sin θ, and that θ is a function of time t given by θ = θ 0 + ω 0 t . We adopt θ 0 = 0, i.e. the y -axis and ξ-axis agree at t = 0. Differentiating twice equations ( 2 ) with respect to the time t, we obtain where and . From the second members of equations ( 1 ), with the corresponding replacements, we get that where and . By equating equations ( 3 ) to the corresponding equations ( 4 ), we have The system of equations ( 5 ) is equivalent to the system a 1 = c 1 and a 2 = c 2 . Thus, the exact epicyclic equations of motion for the mass point C are If we adopt a flat rotation curve for the Galaxy, with a rotation velocity V 0 = 220 km s −1 , in approximate agreement with the measurements in a large range of Galactocentric distances (e.g. Sofue, Honma Omodaka 2009 ; Bovy et al. 2012 ), and . Since (see Fig. 1 ), we have For the study of the orbit of the local supercloud, the origin of the system (η, ξ) should be located in the solar neighbourhood at the Galactocentric distance of the Sun (i.e. R 0 ≈ 8 kpc). Replacing equation ( 7 ) into equations ( 6 ), this equation system can be solved numerically. The initial conditions needed to solve the differential equation system ( 6 ), i.e. η c (0), ξ c (0), , , will be calculated from the LSR velocity and position of the supercloud's gravitational centre at t = 0. With this purpose, we use the relations ( 2 ) to obtain η c = x c cos θ − y c sin θ and ξ c = x c sin θ + y c cos θ − R 0 . Hence, and . Since θ = ω 0 t = 0 at t = 0, the former


Figure 8. The dependence of dimensionless critical accretion rate on IBHs dimensionless mass m M /(10M ). Solid, dashed, and dotted curves correspond to JWST detection limit, that of MM without the confusion effect being taken into account, and that of MM with this effect being accounted, respectively. Figure 9. The dependence of the parameter on m M /(10M ). Curves of different styles describe the same cases as in Fig. 8 . Figure 10. Same as Fig. 9 , but now the dependence of parameter eff is shown. Figure 11. The minimal number of black holes within CMZ needed for observations with JWST and MM, for fixed values of the efficiency parameter , as a function of mass. Solid, dashed, and dotted curves corresponding to observations with JWST and MM without and with the confusion limit being taken into account. Curves with progressively larger values of N bh for a given value of m correspond to progressively smaller values of . The accretion rate given by expression ( 1 ) should be smaller than its critical value for 1. This requirement sets a limit of the possibility of observations of IBHs by a given instrument and for a given value of the density of the interstellar medium, IBH mass, and velocity. In order to compare ( 1 ) with , we express the accretion rate given by equation ( 1 ) in terms of our characteristic unit: , set c s 0 in equation ( 1 ), substitute there our chosen value of density of interstellar medium 3.34 10 22 gcm 3 and assume that a typical IBH velocity 225kms 1 . We obtain where . To begin with, let us consider a black hole travelling at the typical velocity and set in equation( 17 ). In this case from the requirement that for 1, we get lower limits on black hole masses, M crit , available for observations. For larger black hole masses, the condition gives a lower limit on the efficiency parameter , which is shown in Fig. 9 . One can see from this figure that JWST and MM without taking into account the confusion effect could be able to observe IBHs with masses larger than M crit 30M , while allowing for the confusion effect MM is able to detect IBHs only when M 90M , respectively. The range of available values of is rather limited to 0.1 even for the largest considered black hole mass 100M when the confusion effect is neglected and 0.8 in the opposite case. That should be relatively large may be a problem for observations since some works suggest significantly smaller values, see e.g. Igumenshchev Narayan ( 2002 ). On the other hand, assuming that high accretion efficiency is sustained and taking into account that the available volume of CMZ 10 6 pc 3 , it could be possible to observe IBHs with number density as small as 10 6 pc 3 . Available values of and M crit can be smaller in a situation when there are many IBHs within CMZ, with their number N bh 1 and, accordingly, their number density 10 6 pc 3 . In this case, assuming the Maxwellian statistics of distribution of IBHs over velocities we can estimate the number of IBHs


circles mark the mass coordinate . The last few profiles show the envelope oscillating due to mass transfer rates falling below the steady He-burning regime. On the low mass donor end, however, carbon may not ignite at all. The model with initial donor mass stays at lower accretion rates and falls far enough below the steady helum burning range that the mild helium oscillations become powerful enough to blow off mass. The blue dashed line shows how the maximum temperature in the shell will oscillate between the flashing helium burning shell and the hot, compressed C/O ashes below the burning layer. At these accretion rates, however, the accretion efficiency stays above 70%, allowing the WD to gain enough mass to trigger carbon ignition in the core. We compare our accreted mass retention efficiency calculated in the final stages of this model using super-Eddington winds to the analytical estimates of Kato Hachisu ( 2004 ) and Wang et al. ( 2015 ) in Figure 7 . Wang et al. ( 2015 ) used MESA (r3661) for their study, but used the default super-Eddington wind settings, whereas we triggered super-Eddington winds while the WD is still in a compact configuration in order to speed the computation. Systems with donor mass , therefore, will fall to lower accretion rates and efficiencies, and ultimately lose contact before a core ignition occurs. The components, now both C/O WDs, will spiral in due to GWR and contribute to the double-degenerate channel for SNe Ia, as discussed in Section 6 . Figure 7. The top panel shows the accreted mass retention efficiency during the last stages of the system with the donor stars. The black line shows what we calculated by taking the ratio of the growth rate of the accretor and the mass loss rate of the donor. The red short-dashed lines show the retention efficiency given analytically by Kato Hachisu ( 2004 ) for 1.3 and cores. The blue long-dashed lines show the retention efficiency given analytically by Wang et al. ( 2015 ) for 1.3 and cores. The bottom panel shows the mass transfer rate being donated to the WD. Between the high and low mass donor limits is the region where core carbon ignitions occur. This is where the donor mass is high enough that the accretion rates do not fall too far below the stability boundary before is reached, but also low enough that it avoids a carbon shell ignition from long sustained high accretion rates. A clear example of this is shown in Figure 8 , with the trajectories of the core and envelope shown with profiles in space for a system with . In Figure 9 we show the composition and temperature profile when this model has grown to , the same profile as that in Figure 8 . As the C/O envelope is , this shows that the base is at a temperature minimum, and the temperature maximum in the envelope is only deep. The core ignition in these models indicates that they are likely SN Ia progenitors. Figure 8. Profiles of of the WD in the binary system with . Markings have the same meaning as


unobscured sources, to dex for faint and highly obscured sources. The average value of the error is dex, with standard deviation . We show the average errors with a black cross in the left panel of Fig. 4 , while the specific value for each source is used in the following analysis. In order to look for intrinsic correlations between these two quantities, one possibility is to compute the partial Spearman rank correlation between two variables in the presence of a third, and to assess the statistical significance of such correlation (e.g., Macklin 1982 ). To derive the correlation coefficient between and , conditioned by the distance, (, , ), we evaluate the Spearman coefficient related to each pair of parameters and then combine them according to the expression (Conover 1980), which returns the partial correlation between and , corrected for the dependency on . The resulting is 0.15, and the associated confidence level, in terms of standard deviations, that the first two variables are correlated, independently of the influence of the third, is , following Eq. (6) of Macklin ( 1982 ). Therefore, the two quantities appear to be significantly correlated, after the effect of redshift on both of them is taken into account. 3.2. Redshift bins The second approach, often used in the literature, is to define redshift bins that are as narrow as possible to minimize the distance effect, and to look for correlations between the two quantities. Thanks to the large sample collected in this work, we can divide the sample into five redshift bins. For every redshift bin, a large distribution in both luminosities can be observed, with the typical luminosity increasing with redshift (Fig. 4 ). Most of the observational works mentioned in Sect. 1 looked for the distribution of average in bins, or average in bins (but see Gruppioni et al. 2016 ). Both hydrodynamical simulations (e.g., Volonteri et al. 2015 a,b) and semi-analytic models (e.g., Neistein Netzer 2014 ) show that, in the - plane, there may be the superimposition of a weak correlation for the bulk of the population, and a strong correlation only for the most extreme merger phases, corresponding to the highest and . If the underlying distribution shows such a complex shape, the results of the two approaches (average in bins or average in bins) may be very different. In Fig. 5 (left) we show the result of plotting average in bin of (both in log scale) in five redshift bins. As can be seen, there is no correlation at all and, as expected, there are no sources below the relation computed for a pure AGN template in Mullaney et al. ( 2011 ), similar to the one used in Netzer et al. (2009). Following this approach, we are therefore able to reproduce the results of Shao et al. ( 2010 ), Rosario et al. ( 2012 ), and others that claim no correlation between AGN activity and SF over several orders of magnitude in luminosity. On the other hand, computing average in bins (in log scale) from the same bivariate


by a Gaussian probability density function (PDF) with mean and standard deviation . For the flux limits, we used the 1 σ upper limits as as well as σ for the PDFs. All PDFs are set to zero for negative fluxes. Each of these Gaussian PDF representations of the measurements are corrected by the lens magnifications listed in Table 2 , using the intrinsic flux as simply the observed flux divided by the magnification and that the variance on the intrinsic flux is given by:Here is the estimated lensing magnification for each image. The uncertainties on the magnification factors quoted in Table 2 are not symmetric, and we therefore, conservatively, use the largest absolute value of the upper/lower magnification ranges as . The product of the individual magnification-corrected PDFs for each image gives us a representation of the intrinsic flux of the background source including information from both the detections and the upper limits. If any of the detections and/or upper flux limits were in disagreement such a product would be driven to 0. Within 3 σ, all of our measurements are in agreement with each other. Sampling the distribution resulting from taking the product of the individual PDFs, we obtain the combined intrinsic magnification-corrected line fluxes or line flux limits. In Table 3 , we quote the median value and 68% (∼1 σ ) confidence intervals for the final distribution. Comparing these values with the measurements performed directly on the stack also given in Table 3 scaled by the average HFF lens magnification of images A–E (), the de-magnified Ly α and C iv fluxes agree within the errors (especially if the large uncertainties on the magnification are folded into the error budget). The intrinsic flux limits appear to be roughly a factor of three tighter from the stack than from the Gaussian PDF if the uncertainties on the lens magnification are ignored. The tentative detection of the O iii ]-doublet in the stacks agree with the magnification-corrected flux limit set by the Gaussian PDF combination. 5.2. Interpreting UV Line Fluxes and EWs As mentioned in Section 1 , EWs and flux ratios of rest-frame UV lines are strongly correlated with the ionization strength and the gas-phase metallicity of the stellar populations dominating the SEDs of the emitters (e.g., Stark et al. 2015b , 2017 ; Feltre et al. 2016 ; Gutkin et al. 2016 ; Jaskot Ravindranath 2016 ; Mainali et al. 2017 ; Pérez-Montero Amorín 2017 ). Hence, the obtained flux ratios (limits) from the GLASS spectra can be used as an indicator of the physical properties of the multiply imaged source behind RXC J2248.7–4431 as described in the following. The C iii ] EW and the flux ratio between C iv and C iii ] are sensitive to the gas-phase metallicity and the ionization parameter of the stellar populations (Feltre et al. 2016 ; Gutkin et al. 2016 ; Jaskot Ravindranath 2016 ; Pérez-Montero Amorín 2017 ). From the GLASS spectra, we get rest-frame (1 σ ). As shown by Rigby et al. (