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


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


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


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


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


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


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


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


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


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


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


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


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


wavelength. The blue line is a fit to the data. Extinction according to Cardelli et al. (1989) is shown for A v / E ( B − V ) values of 1.5 (green triangles, dashed line) and 3.1 (red diamonds, continuous line), respectively. Dirsch et al. (2005) estimated the maximum and the mean B -band attenuation in a 4′′ diameter area in the patch to be A ′ B ∼10 and 04, respectively. Their maximum attenuation is somewhat higher than the value A ′ B 005 obtained from the GMOS spectrum. This difference is possibly due to the positioning of the GMOS spectrum slit that does not pass exactly through the densest part of the patch, and also to the selected off position. Had the off position been chosen northeast of the patch, the attenuation would be 10% lower, but the wavelength dependence of the attenuation would not change. Our A ′ B /( A ′ B − A ′ R ) ratio (1.6 ± 0.11) is the same as (1.6 ± 0.25) reported in Dirsch et al. (2005) and it is, again, substantially lower than the corresponding value for the mean Milky Way extinction curve for point sources, A B /( A B − A R ) = 2.3. The background surface brightness attenuation in a clumpy foreground screen has been modelled by Natta Panagia (1984) . Clumping causes the reddening curve to flatten, meaning R ′ V and A ′ B /( A ′ B − A ′ R ) increase as compared to a homogeneous screen. Thus, clumping cannot explain the steeper wavelength dependence as observed for the patch. In several cases, dust lanes or patches have been observed in early-type galaxies with R ′ V values as low as 1.9 ( IC 4320 , Warren-Smith Berry 1983 ) and 2.15 ( NGC 5626 , Goudfrooij et al. 1994 ). Such cases have been interpreted as evidence for dust-grain size distributions with mean and maximum grain sizes smaller than those for the mean Milky Way extinction curve (with R V = 3.1). The dust mass for lanes and patches in galaxies has frequently been estimated using the following formula: where ⟨ A ′ B ⟩ is the mean attenuation as measured over the object’s area of Σ ( Sadler Gerhard 1985 ; van Dokkum Franx 1995 ). For the Milky Way’s mean dust properties, the value of the mass absorption coefficient Γ B is Γ = 4 × 10 4 mag cm 2 g −1 ∼ 8 × 10 −6 mag kpc 2 M ⊙ −1 . Adopting as the mean attenuation over the area of Σ = 4′′ × 4′′ of the patch, we obtain the following for its dust mass: M d = 4.8 × 10 4 ( d /50.7 Mpc) 2 M ⊙ . Goudfrooij et al. (1994) presented a method to estimate the value of Γ B for extinction (attenuation) curves with R ′ V values other than 3.1. For NGC 5626 and IC 4320, which have the smallest values of R ′ V in their sample (2.15 and 2.08), they found that the dust masses as obtained from the above formula have to be multiplied by a factor of ∼0.66. In the case of the patch with R ′ V = 1.5, we estimate that a correction factor of ∼0.5 could be appropriate. Thus the dust mass estimate for the patch becomes M d ∼ 2.6 × 10 4 ( d /50.7 Mpc) 2 M ⊙ , which corresponds to a molecular-gas-to-dust ratio of ∼55. If at the


= 3.5 − 750 are displayed in Figs 14 (a, b). From Fig. 14(a) , it looks, as if, the acoustic modes are divided into two groups as follows. The first group of waves, which looks dispersive in nature, is localized in the domain, k = 10 −4 − 6 (). The phase velocity of this group increases radially outwards and acquires supersonic value with sub-sonic transition beyond about k = 6. The second group of waves, which is almost non-dispersive in nature within a wide-range of the ξ-space, is localized in a wide-range of k -domain [Fig. 14(a) and Fig. 15(a) ]. The propensity of weak dispersion is reflected near the SSB. These waves are found to travel within sub-sonic range of flow speed. From Fig. 14(b) , it is clear that the group velocity is almost constant in the k -space. Moreover, no distinction is noted between the two groups. However, the group velocity increases radially outwards with the asymptotic sonic speed value at 1 au. In Fig. 14(c) , we depict the time period plot in the k - and ξ-spaces. The categorization of the two distinct groups of waves is clearly visible. The time periods of the dispersive mode group are found to be larger than those of the non-dispersive wave group, but at different ξ. It also reveals that discreteness in the low k -value of the fluctuations results in discreteness in the oscillation period peaks. In the range of relatively high k -value and beyond, such discrete peaks in the oscillation period are not observed. In Figs 15 (a, b, c), we depict the phase velocity, group velocity and time period in the extended k -space. Also, the time period decreases towards shorter wavelength. Figs 16 (a, b) depict that the electrostatic potential fluctuation level, being higher, dominates over the SWP flow fluctuations. One more thing, to say, is that the amplitude profile of the flow fluctuations is the same as that of the SIP. Finally, Figs 17 (a, b) show the SWP fluctuation parameters [ and ] in the extended k -space. The electrostatic potential fluctuations increase towards the shorter wavelength regime. The flow fluctuations are not affected much in the shorter wavelengths. This is in conformity with the flow-pattern phenomena as observed in the SIP as well. Some specific features in addition, are enlisted as below. The phase velocity of the non-dispersive p-modes, as identified and discussed above in Figs 14(a) and 15(a) , varies in the sub-sonic range only. But, that of the dispersive branch of the p-modes, varies from sub-sonic to supersonic regime with maximum v p ∼ 200 km s −1 at 1 au. The phase velocity plot in the extended k -domain [Fig. 15(a) ] exhibits a similar type of behaviour of sub-sonic and supersonic speeds. The corresponding group velocity [Fig. 14(b) and Fig. 15(b) ] acquires sonic speed, v g ≈ 100 km s −1 , at 1 au in all the k -domains. The time period plots [Fig. 14(c) and Fig. 15(c) ] exhibit strong radial dependence with two distinct peaks, τ ≈ 20 min each, lying at ξ ≈ 30 ( away from the SSB) and


estimate the average period of the rotational motion, which turns out to be 569 s (with an error of 91 s). Taking the average width 38 Mm of the jet into account, the average line speed of the rotational motion during the off-limb stage is then estimated to be about 209.8 ± 33.6 km s −1 . As the upward Lorentz force acting on the jet after reconnection resulted from the untwisting motion (Shibata Uchida 1985 ), the nearly invariable rotating period is consistent with the above result that the downward acceleration in the late ascending phase and in the descending phase were almost the same. 3.2. Kinetics of the Twin Jets A similar method reveals almost the same inclination of the twins’ axes as the preceding jet related to the plane of the sky. To probe the axial motions of the twin jets, we place two slices along their axes labeled as “C1” and “C2” in Figure 1 (e) for the left and right twin jet respectively. Their corresponding running-difference time–distance plots are shown in Figures 3 (C1) and (C2) with the y -axis representing the distance along the slices after correcting the projection effects. Figure 3. (C1) and (C2): running-difference time–distance plots for slices “C1” and “C2” in Figure 1 (e) along the axis of the left and right twin jet respectively. The blue stars represent trajectories of materials within the twins, with the blue curves representing their parabolic fitting results. (D1)–(D4): running-difference time–distance plots for slices “D1” to “D4” perpendicular to the twins’ axes as shown in Figure 1 (e). Only very weak rotational motion of the twins could be found at their early off-limb stage from these four time–distance plots. As in Figure 3 (C1), the left twin jet reached a maximum distance along the slice of about 330 Mm at around 01:09 UT. The downward acceleration during its ascending phase and descending phase turned out to be similar. Parabolic fitting (blue dashed curve in Figure 3 (C1)) on one of the trajectories (blue stars in Figure 3 (C1)) gives us an average deceleration of about −228.9 ± 8.1 m s −2 . It is hard to determine the maximum distance of the right twin jet (Figure 3 (C2))—it seemed to reach a distance that was beyond the FOV of SDO observations or faint enough to be invisible. However, the dynamics seem to be similar to the left one with the average downward acceleration of about −234.5 ± 10.7 m s −2 (one of the trajectories is shown as the blue stars with the parabolic fitting result shown by the blue dashed curve in Figure 3 (C2)). All of the above values resemble the average local gravity (−220 m s −2 ), which indicates that there was almost no extra driving force acting on the twins. Meanwhile, we place four parallel slices “D1” to “D4” at different heights perpendicular to the twins’ axes to investigate the rotational motion of the twins if there was any. Corresponding running-difference time–distance plots are shown in Figures 3 (D1) to (D4) with the y -axis representing the distance


excited normal modes, since we only need the normal-mode amplitudes. We then obtainwhere S p ( ω ) is the spectral density of effective pressure fluctuations acting on the Moon’s surface isotropically owing to a steady flux of meteoroids, which also depends on the formation of ejecta, elasticity properties of the impact, and the transfer efficiency of impact energy to the seismic field (typically a small fraction of the kinetic energy of a meteoroid; Lognonné et al. 2009 ). The last equation has the on-resonance formThe term depends on the Moon’s internal structure and is of the order of 0.2/ M for lower-order quadrupole modes, where M = 7.3 × 10 22 kg is the mass of the Moon, so that we can approximate the last equation asThe spectral density of pressure fluctuations is related to pressure rms by below 1 Hz (Theodorsen et al. 2017 ), where τ is the average duration of meteoroid impacts henceforth assumed to be τ = 0.1 s neglecting complications like the formation of ejecta. The effective pressure 〈 p 〉 produced by meteoroids with flux F and momentum mv is given bywhere α is the efficiency of momentum transfer to the seismic field. For our order-of-magnitude estimate, we go with the approximation , where we use F 0 = 10 −5 m −2 s −1 for the flux and m 0 v 0 = 10 −3 kg m s −1 for the average meteoroid momentum (Grün et al. 2011 ), and α = 0.01, which seems to be a generous overestimation of the impact-energy-to-seismic conversion efficiency (Lognonné et al. 2009 ; Quillen et al. 2019 ). Note that this expression is independent of the dynamical timescale τ of the impact since we are interested in the average force or pressure, which means that the relevant timescale Δ t of momentum transfer Δ P is given by the average time between impacts, which is contained in the meteoroid flux. It is possible that the total (undirected) momentum transfer onto the Moon is dominated by rare impacts of larger meteoroids, where rare means less frequent than once per Q n / f n , which is about once per day for the lower-order quadrupole modes. With 4 π R 2 × 1 day = 3 × 10 18 m 2 s, we estimate that impacts of meteoroids with mass greater than 1 kg occur less than once per day (Grün et al. 2011 ). These events need to be treated individually and cannot be considered part of the stationary background noise anymore. As for any other larger seismic disturbance, one needs to exclude the stretch of data containing the large impact event from GW analyses, or attempt a subtraction of the seismic signal from the seismometer data. Inserting the expression of the surface pressure from meteoroid impacts into Equation ( A6 ), we obtainConfronting this equation with the on-resonance normal-mode response to GWs, we can say that the effective strain noise in mode n from meteoroid impacts iswhere the numerical value depends weakly on the order of normal modes below about 10 mHz according to our simplified response model. The value lies well below the LGWA strain-noise target of 10


axis due to the boundary conditions. The minimum value of v θ is also located around θ ∼ 43°. Moreover, the bottom right panel of Figure 1 presents the Mach numbers. We plot this figure to check the existence of the sonic point in the high-latitude region. As we can see, no sonic points exist in this region. The Mach number decreases from the equatorial plane to about 65°, where the radial velocity is zero. Then, it increases until θ = 30° and again decreases rapidly to the rotation axis. This behavior can be explained by the isothermal sound speed, c s = p / ρ . The profiles of the density and the total pressure in Figure 1 show that the density decreases rapidly from the equator to the pole while the pressure is almost constant in the range 0 θ 30°. Therefore, the Mach number declines in this range. Also, has a maximum value at θ = 43° and becomes null at both axes due to the boundary conditions there, i.e., v θ = 0. For both lines, the Mach numbers are less than unity, which clearly shows that there is no critical point at high latitudes. Figure 1. Angular profile of the physical variables at r = 10 r s . Top left panel: radial velocity in units of the light speed, c . The dashed line shows the location of v r = 0, which is about 65°. Top middle panel: latitudinal velocity in units of the Keplerian velocity, v 0 . Top right panel: angular velocity in units of the Keplerian angular velocity, Ω 0 . Bottom left panel: density in units of the density of the midplane at r 0 , i.e., ρ 0 . Bottom middle panel: radiation pressure in units of . Bottom right panel: Mach numbers. In the left panel of Figure 2 , we plot the two-dimensional distribution of the density overlaid with the streamlines of the flow. The results are shown for and the poloidal velocity is normalized by 0.1 c . It can be seen that the density tends to be larger around the midplane than around the polar axis. Moreover, the streamlines are directed toward the black hole at low latitudes with small magnitudes. At high latitudes, they are pointed outward and become outflow. For a high accretion rate, as in our present study, the outflow can be driven by the radiation pressure produced at inner radii. This strong radiation interacts with the gas particles and can push them away as a high-velocity outflow. This figure clearly shows the region of a high velocity field at high latitudes at r = 10 r s . As shown in the top left and top middle panels of Figure 1 , the poloidal velocity of the outflow reaches about ∼0.25 c . These results are fully consistent with the results obtained in Yang et al. ( 2014 , see their Figure 1) and Ohsuga et al. ( 2005 ), where the inflow is presented at low latitudes around the equatorial plane of the disk while the outflow is presented in the high-latitude regions. Figure 2. Two-dimensional distribution of the density (left) and the temperature (right) based on the self-similar solutions. Both panels are overlaid with the poloidal velocity. In the left


mode i is given by where P i is the discrete version of the power spectrum P ( k ). In the fixed-power approach, the PDF for mode i is instead given by where δ D indicates the Dirac delta-function. One can sample from Pr f straightforwardly by setting with θ i drawn with uniform probability between 0 and 2π, and θ − i = −θ i . The second of the pair of simulations is then generated by transforming θ i → π + θ i (Pontzen et al. 2016 ). Sampling from Pr f results in an ensemble that is not equivalent to sampling from Pr g . However, Pr f can stand in place of Pr g for many practical calculations (the analytic justification is discussed in Section 3 ). We verified that, despite the fixed amplitudes, the one-point input overdensity PDF in real space, , is still a Gaussian deviate owing to the central limit theorem. Furthermore, Fig. 1 shows the distribution of overdensities in the initial conditions at z = 9, δ IC , averaged over spheres of 8 h −1 Mpc radius for a subset of traditional simulations (grey lines) and the two paired-and-fixed (orange lines; these overlap almost perfectly). The corresponding PDF for the combined volume of the two paired-and-fixed simulations is shown by the red dots. There is excellent agreement between this characterization of the density fields of traditional and fixed simulations, with both following a near-Gaussian distribution. The mild skewness (which also agrees between the cases) arises from the 2LPT particle displacements. Figure 1. The probability density function of non-linear overdensities in spheres of radius 8 h −1 Mpc, as measured in the fixed simulations (orange lines) and in an ensemble of Gaussian simulations (grey lines), at the starting redshift, z = 9. The blue line and red circles show the mean of the respective cases, as indicated by the legend. 2.2 Results Fig. 2 shows the dark matter power spectrum measured from the z = 1 outputs. In the top panel, the results of the fixed pair (red circles) are indistinguishable from the traditional ensemble mean (blue line) over all the scales plotted, confirming that the approach correctly predicts the ensemble average power spectrum in linear and in non-linear regimes. Figure 2. The power spectrum of the dark matter at z = 1. In the top panel, measurements from the ensemble of 300 traditional simulations are shown as grey lines, with the mean shown by a blue line. The solid red circles show the average of the two simulations in the paired-and-fixed set. Finally, the horizontal dotted line marks the shot noise limit. In the bottom panel, we show the differences with respect to the average ensemble measurement, in units of the standard deviation in the ensemble. As in the top panel, red symbols show the final estimate from the pair of fixed simulations. We additionally show residuals in each of the two individual fixed simulations by the orange lines. The envelopes bounded by dashed lines mark a 1 per cent (left) and 0.1 per cent (right) uncertainty in


a fair fraction of these lower luminosity candidates are attributed to gas motions in the narrow-line region due to other effects such as outflows (e.g., Rosario et al. 2011 ; Comerford et al. 2012 ; Fu et al. 2012 ; Müller-Sánchez et al. 2015 ). Only a modest number of more luminous dual quasars have been identified at close separation, i.e., during the final stages in the merger at ≲20 kpc (Komossa et al. 2003 ; Hennawi et al. 2006 ; Inada et al. 2012 ; Shields et al. 2012 ; More et al. 2016 ; Eftekharzadeh et al. 2017 ; Goulding et al. 2019 ), due to limitations with instrumental spatial resolution. Current and future wide-field deep imaging surveys have the potential to significantly improve upon the number of confirmed dual quasars, particularly at higher luminosity and closer separations. For instance, the wide-area coverage, depth, and superb seeing of the Subaru Strategic Program (SSP; Aihara et al. 2018 , 2019 ) with Hyper Suprime-Cam (HSC; Miyazaki et al. 2018 ) can be used to search for rare and key phases of black-hole growth, including dual quasars as signposts of major galaxy mergers. Simulations show that quasar accretion is episodic (e.g., Capelo et al. 2017 ), thus the SMBHs in a major merger will both shine as a quasar only a small fraction of the merger time. The areal coverage of HSC is now sufficient to detect these rare events in significant numbers. Just as importantly, the median seeing of the HSC imaging is in the i -band, meaning that dual quasars can be identified with projected separations as small as ∼3 kpc at z ∼ 0.3. For those at z 1, the underlying host galaxies can be robustly detected with HSC including those with faint structures such as tidal debris from an ongoing merger (Goulding et al. 2018 ). To exploit the capabilities of HSC on this topic, we are carrying out a program to search for Sloan Digital Sky Survey (SDSS) quasars, falling within the HSC survey footprint, that are dual quasar candidates based on an automated 2D image detection algorithm to identify multiple compact emission on the scale of the HSC point-spread function (PSF). These objects were initially identified as single SDSS quasars based on optical photometry and spectroscopy, given the lower spatial resolution of the SDSS data. Dual quasars with separations >2′′ are also missed by SDSS, as the survey could not obtain spectra of close pairs of objects due to a limitation on the minimum distance between two optical fibers on a single spectroscopic plate. Even so, subsequent dedicated spectroscopic follow-up programs of dual quasar candidates have confirmed 47 cases with most at wider separations ( Eftekharzadeh et al. 2017 ) with essentially no overlap with the HSC SSP program. Using the Subaru HSC SSP imaging, we report here on the identification of dual quasar candidates having projected separations between 3 and 30 kpc. An optical spectroscopic campaign is underway using Keck I/Low Resolution Imaging Spectrometer (LRIS), Subaru/FOCAS


method applied to the observed data as described in Section 3 . For each template the process was repeated 100 times and the distribution of recovered metallicities is shown in Fig. B1 . It can be seen that although there is a systematic offset to lower metallicities for all but the lowest metallicity template (for which it is impossible to measure a lower metallicity), the effect is relatively small (≤10 per cent). We therefore conclude that the low metallicities derived for our VANDELS sample are robust against the effect of redshift uncertainties. Figure B1. The distribution of recovered metallicity for each of the five default Starburst99 WM-Basic template spectra ( Z ∗ = 0.001, 0.002, 0.008, 0.014, and 0.040) after mimicking the effect of redshift uncertainties when creating composite spectra (see text for details). Each panel shows one of the five default templates with the true metallicity indicated by the dashed vertical red line. The distribution of recovered metallicities is shown in black. The offset between the median recovered metallicity and the true metallicity is quoted in the top left-hand corner of each panel. APPENDIX C: BPASSv2.1 STELLAR POPULATION SYNTHESIS MODELS In our main analysis, we focused on using the SB99 WM-Basic stellar population synthesis models to fit the VANDELS spectra for the reasons outlined in Section 3 . However, we have also performed the same analysis using the BPASSv2.1 stellar population synthesis models described in Eldridge et al. ( 2017 ). It is well known that using different models will cause systematic offsets in derived parameters (e.g. Conroy 2013 ; Cid Fernandes et al. 2014 ) and it is useful to have an estimate of how significant this offset might be in the case of FUV-derived metallicities. The unique feature of the BPASSv2.1 models is the inclusion of massive binary star evolution, which can have a strong effect on the predicted UV spectrum, particularly at low metallicities. For example, at Z ∗ /Z ⊙ 0.35, a phenomenon known as ‘quasi-homogeneous evolution’ can occur (QHE; e.g. Eldridge Stanway 2012 ), which results in stars living longer on the main sequence and becoming hotter than single stars at similar mass and metallicity. QHE can have a profound effect on the estimated ages and ionizing photon output of stellar populations. Stellar metallicities should be less affected, particularly since the O-star models in BPASSv2.1 are also generated with WM-Basic code, albeit with a different parameter set to SB99. Nevertheless, it worth testing this assumption. We considered a 100 Myr constant star formation BPASSv2.1 model that included binary star evolution and had an IMF cut-off of 100 M ⊙ , with IMF index of −1.3 between 0.1–0.5 M ⊙ and −2.35 above 0.5 M ⊙ . The model is referred to as BPASSv2.1-100bin. We considered the following set of metallicity values Z ∗ = (0.001, 0.002, 0.003, 0.004, 0.006, 0.008, 0.010, 0.014, 0.020, 0.030, 0.040), interpolating between the models in the same


both approaches, was carried out by restricting the EGMRT antenna locations to lie within the central square. Further, we included only the 14 GMRT central square antennas as ‘fixed’ antennas in the optimization. In the tomographic projection method, beginning with the above fixed GMRT antennas, we add N new antennas to the array, starting with N = 1, and increasing the number of new antennas by one at each iteration. For each value of N , we carried out the tomographic projection optimization for 100 random initial conditions for the new antenna locations, and evaluated the final Residual RMS for each of the 100 realizations. We then chose the realization with the lowest Residual RMS as the optimal configuration for the N new antennas. This minimum Residual RMS was saved, along with the locations of the new antennas, and the process repeated for ( N + 1) new antennas. The Residual RMS initially declines steeply with each new added antenna, as the UV coverage approaches a 2D Gaussian UV distribution. However, beyond some number of new antennas, there is no significant decrease in the Residual RMS, i.e. improvement in the UV coverage, on adding more antennas. We fixed the number of new antennas to the number above which the Residual RMS does not change by more than ≈10 per cent with the addition of further new antennas. In the random-sampling method, we again add N new antennas to the array, starting with N = 1, and increasing the number of new antennas by one at each iteration. However, here we randomly assign the new antennas to available cells in the GMRT central square, keeping the locations of the existing GMRT antennas fixed. We then evaluate the Residual RMS for the configuration, comparing the UV coverage of the configuration with the ‘ideal’ 2D Gaussian of FWHM = 0.5 km. We carry out this random assignment process 10 4 times for each value of N , and determine the minimum Residual RMS for the 10 4 evaluated configurations. This minimum Residual RMS is then stored, again along with the locations of the new antennas, and the process is repeated, by adding ( N + 1) antennas. We find a pattern similar to that seen in the tomographic projection approach, i.e. that the Residual RMS initially declines steeply with each added antenna, but, beyond some number of new antennas, does not decrease significantly on adding more antennas. Again, the number of new antennas is then fixed to the number above which the Residual RMS does not change by more than ≈10 per cent with the addition of further new antennas. As noted earlier, the above process was carried out independently for four declinations, δ = −30°, +30°, and +60°. Fig. 3(A) shows the Residual RMS plotted against the number of antennas for the four different declinations, with the results obtained via the tomographic projection method. The dashed vertical line indicates eight added antennas, beyond which the Residual RMS improves only slowly with additional antennas (for all four


and rest-frame UV spectra for each of the ALPINE galaxies are shown in Appendix B . Throughout the paper, we assume the ΛCDM cosmology with , Ω Λ = 0.70, and Ω m = 0.30. All magnitudes are given in the AB system (Oke 1974 ) and stellar masses and SFRs are normalized to a Chabrier ( 2003 ) initial mass function (IMF). 2. Spectroscopic Data and Selection 2.1. Spectroscopic Selection of ALPINE Galaxies The ALPINE survey is only possible due to a spectroscopic pre-selection of galaxies from large spectroscopic surveys on COSMOS and ECDFS. This is because the ALMA frequency bands are narrow (), and in order to observe [] emission, the redshift has to be known within a precision of . The galaxy selection is refined to optimize the efficiency of the ALMA observations by creating groups of galaxies in spectral dimensions. Our sample also includes seven galaxies that were previously observed with ALMA by Riechers et al. ( 2014 ) and Capak et al. ( 2015 ). These are HZ1 , HZ2 , HZ3 , HZ4 , HZ5 , HZ6 / LBG-1 , and HZ8 , which correspond to the ALPINE galaxies DC _ 536534 , DC _ 417567 , DC _ 683613 , DC _ 494057 , DC _ 845652 , DC _ 848185 , and DC _ 873321 , respectively. Furthermore, four galaxies from the VUDS survey ( vc _ 5101288969 , vc _ 5100822662 , and vc _ 510786441 in COSMOS and ve _ 530029038 in ECDFS) are observed twice (resulting in a total number of 122 observations). The duplicate observations are used for quality assessment. Bethermin et al. ( 2020 ) describes the combination of these observations. The rest-frame UV spectroscopic data from which the ALPINE sample is selected combine various large surveys on the COSMOS and ECDFS fields. Out of the 105 ALPINE galaxies on the COSMOS field, 84 are obtained by the large DEIMOS spectroscopic survey (Capak et al. 2004 ; Mallery et al. 2012 ; Hasinger et al. 2018 ) at the Keck telescope in Hawaii. The remaining spectra on the COSMOS field are obtained from the VIMOS Ultra Deep Survey (VUDS; Le Fèvre et al. 2015 ; Tasca et al. 2017 ) at the VLT in Chile. In total, six of the VUDS spectra are independently also observed as part of the Keck/DEIMOS survey ( vc _ 5100559223 , vc _ 5100822662 , vc _ 5101218326 , vc _ 5101244930 , vc _ 5101288969 , vc _ 510786441 ). The redshifts are consistent within , and we do not find any systematic offsets between the two observations (see also Section 2.4.1 ). Out of the 13 galaxies in the ECDFS field, 11 are obtained from spectroscopic observations with VIMOS (9) and FORS2 (2 29 29 One of these galaxies, ve _ 530029038 , has also been observed by the VUDS survey. ) at the VLT (Vanzella et al. 2007 , 2008 ; Balestra et al. 2010 ), and two come from the HST grism survey GRAPES (Malhotra et al. 2005 ; Rhoads et al. 2009 ). The spectral resolution of the different data set varies between R ∼ 100 (ECDFS/ GRAPES grism), R ∼ 180 (ECDFS/VIMOS), R ∼ 230 (COSMOS/VUDS), R ∼ 660 (ECDFS/FORS2), and R ∼ 2500 (COSMOS/DEIMOS). Biases toward dust-poor star-forming galaxies


the presence of perturbations is at (for details about that, see Pereira et al. 2018 ). From Equations ( 7 ) and ( 12 ), one has thatwhere use has been made ofas well as Equation ( 4 ) and . The perturbed surface tension , with the help of Equations ( 8 ) and ( 11 ), can be written asTherefore, likewise to the slow phase conversion case, no thin shell surface degrees of freedom are induced in perfect fluids when the phase conversion is rapid. Now we investigate solid hybrid stars. In this case, as commented in the previous section, . Besides, now changes with respect to its perfect fluid counterpart, Equation ( 17 ), and it is given by (a mere replacement of by )where summarizes the contribution to the radial shear stresses in the presence of perturbations (see Penner et al. 2011 and Krüger et al. 2015 for realizations of them). For slow phase conversions in solid hybrid stars, due to Equations ( 11 ) and ( 13 ), it still follows that and . However, nontrivial results hold for solid hybrid stars under rapid phase conversions. Indeed, for , we haveBefore calculating the thin shell surface tension for this case, let us expand Equation ( 13 ), because . From our assumption to elastic aspects in hybrid stars, one can generically write . Thus, after some simple calculations it follows from Equations ( 4 ), ( 13 ), and ( 23 ) that (recall that )which turns out to be exactly the same as that for slow reactions in solid stars (because there and ). From the above equation, one can clearly see that indeed in general. (Equation ( 24 ) also makes it clear that the Lagrangian change of the pressure is null for perfect fluids, as already known from other methods; Pereira Rueda 2015 ; Pereira et al. 2018 .) Now, for the thin shell surface tension induced by rapid reactions in solid hybrid stars, from Equations ( 12 ), ( 22 ), and ( 24 ), one obtains that The fact that rapid conversions in solid stars induce thin shell surface degrees of freedom could have interesting consequences; for instance, the induction of a surface mass (general relativistic energy) on a phase-splitting surface. Since the induced thin shell surface energy density is a local quantity, so is the shell’s proper mass (note that for first-order quantities, such as those associated with surface degrees of freedom, it is irrelevant to use R or R 0 for functions multiplying them)Total mass measurements at infinity will have to take into account the correction factor (related to the gravitational redshift), as we shall show in Equation ( 31 ). From Equations ( 7 ), ( 26 ), and ( 5 ), it follows thatGiven that , when only linear terms in ξ are kept, the above equation can be simplified to . Thus, from Equations ( 26 ) and ( 27 ),where it has been assumed that , m 0 being the background mass function ( because the background has been assumed not to have surface degrees of freedom—see the Appendix ) and m 1 being its first-order correction in the presence of perturbations. In summary, among other


lines to fade from TPSB to QPSB. Figure 3. Stacked spectra for SF (top), TPSB (middle), and QPSB (bottom). All the spectra are normalized at 4100 Å. The shadow marks the region of H δ . 2.4. SFR, Gas-phase Metallicity, and ΣH <sc>i</sc> The emission lines may be contaminated by active galactic nuclei (AGNs) and shocks, which would lead to the overestimation of the star formation rate (SFR) when we use H α as tracer of star formation. To this end, we investigate the excitation mechanisms for SF and TPSB regions according to the traditional BPT diagram (Baldwin et al. 1981 , left panel of Figure 4 ). In this diagram, we impose the S/N of these four emission lines (i.e., H α , [N ii ], H β, and [O iii ]) to be greater than 3. For the spaxels with H α and [N ii ] S/N larger than 3, we use the relation between the EW (H α ) and [N ii ]/H α (right panel of Figure 4 ) as the diagnostic diagram of excitation mechanisms (Cid Fernandes et al. 2010 ). We find no spaxels in the Seyfert region, which indicates that PGC 26218 does not host an AGN. Although some spaxels locate at the composite and LINER regions, the effect from shock is ignorable considering that most of the spaxels locate at the SF region and the contribution from shocks to emission lines could be low in the SF region. Taking into account that all of the SF and TPSB spaxels have EW(H α ) −3 Å and the percent light contribution (at λ = 5100 Å) of young populations in those spaxels is larger than 4%, we believe that most of the contribution is dominated by ionization due to young stars and only a tiny fraction by hot low-mass evolved stars/post AGB stars (González Delgado et al. 2016 ; Lacerda et al. 2018 ). At any rate, our SFR value is an upper limit. Figure 4. Left panel: the traditional BPT diagnostic diagram for SF and TPSB regions. The blue squares and red triangles represent the spaxels with S/N of four emission lines larger than 3 for SF and TPSB, respectively. The dashed and solid lines are from Kauffmann et al. ( 2003a ) and Kewley et al. ( 2001 ), respectively. Right panel: an alternative diagnostic diagram of excitation mechanisms for SF and TPSB regions. The hollow squares and triangles represent the spaxels with S/N of H β or [O iii ] lower than 3 for SF and TPSB, respectively. The blue solid squares and red triangles are the same as in the left panel. The horizontal dashed line is from Kewley et al. ( 2006 ) and the vertical dashed line is from Kewley et al. ( 2001 ). In order to estimate the spatially resolved SFR from extinction-corrected H α emission lines, we calculate the extinction according the Balmer decrement assuming a Cardelli et al. ( 1989 ) extinction curve with case B condition for each Voronoi bin. The extinction-corrected SFR is calculated using the formula given by Kennicutt ( 1998a ):where L (H α ) is the extinction-corrected H α luminosity. For the total extinction-corrected SFR, we construct a spectrum by stacking all spaxels in Voronoi bins


ratios compared to the theoretical values expected from Doppler boost model. In three of the four candidates (J0247, J0249, and J0254), the observed values show deviations from the expected values at the ≳2σ level in at least two band pairs with p-values of 0.189, 0.013, and 0.113. In J0246, the observed values are generally consistent with those expected from Doppler boost. Figure 4. Observed frequency-dependent variability amplitude ratios compared with the expected values from the Doppler boost model. Different colours represent different combinations of band pairs. The black lines represent the one-to-one relation. The error bars denote 1σ uncertainties. See Section 5.2 for details. Fig. 5 shows the allowed parameter space for the Doppler boost model to explain the observed multiband variability amplitudes (Table 4 ). The parameters considered are the total black hole mass M BH , binary orbital inclination angle i , the fraction of the total emission from the secondary black hole f 2 , and the mass ratio q . Our other fiducial parameters are the rest frame orbital period P orb = P GLS /(1 + z ) and α ν in griz bands (Table 4 ). Figure 5. Doppler boost model parameter space estimates for the four candidate periodic quasars. The four sub-panels for each quasar represent griz bands. The dashed (shaded) contours denote f 2 = 1.0 ( f 2 = 0.8), where f 2 is defined as the fraction of the total emission from the secondary black hole. Different colours represents different mass ratios with q = 0.0, 0.05, 0.1, and 0.2 for blue, green, orange, and red, respectively. The black vertical line with the grey-shaded region shows the virial mass estimated from SDSS spectra (Fig. 3 ) and its 1σ statistical error. The orbital inclination angle i = 90° for an edge-on view and i = 0° for a face-on view. For J0246, while the observed frequency-dependent variability amplitudes are generally consistent with the expected values under the Doppler boost hypothesis (Fig. 4 ), there is almost no allowed parameter space unless the following conditions are all met: 1. The virial black mass is significantly underestimated, even after considering a 0.4-dex systematic error (e.g. Shen 2013 ), 2. A high fraction (e.g. ) of the total emission is from the secondary black hole fueled by its mini accretion disc, and 3. The binary system is close to being an edge-on view. For the other three quasars, the observed multiband variability amplitudes are generally consistent with the expected values under the Doppler boost hypothesis, although there is a stringent requirement on the allowed parameter space. Liao et al. ( 2020 ) have tested the Doppler boost hypothesis for J0252. They suggest that the Doppler boost model is disfavoured for J0252 based on both the frequency-dependent variability amplitude ratios and the multiband variability amplitudes. In summary, there is evidence against the Doppler boost hypothesis in all five periodic candidates in our parent sample based on the


The results we obtain span 7 10 7 2 10 9 M . 3.3 X-ray luminosities and Eddington fractions We returned to the Swift XRT X-ray spectra we presented in S11 in light of the new redshift measurements, and fitted each spectrum with an absorbed power-law model in xspec , making use of the most up-to-date Galactic column densities (Willingale etal. 2013 ) and adopting Cash statistics. Swift XRT observations additional to our fill-in programme were available for one source, XMMSL1J002202.9254004. We included these in a combined X-ray spectrum totalling 12ks of exposure, created using the method of Evans etal. ( 2009 ) and after establishing no significant changes in spectral hardness had occurred. From the XRT best fits, we obtained the observed 210keV X-ray luminosities of our AGN. We estimated the Eddington luminosities, L Edd , using our mass estimates in the equation ergs 1 . To obtain the Eddington ratio, the bolometric luminosity is required. Our measured X-ray luminosities significantly underestimate the total output, by an approximate factor of 10 (Elvis etal. 1994 ). Marconi etal. ( 2004 ) derived a bolometric correction to convert from 210keV X-ray luminosity to L Bol (their equation 21), which we adopt here. The resulting bolometric luminosities and Eddington ratios are listed in Table 4 . Table 4. Estimates of black hole mass and Eddington ratio. Optical photometry from USNOB1.0 ( R , I ) or NOMAD ( V ) is given for the observed band closest to rest frame 5100 . Black hole mass estimates use the virial method described in Shen etal. ( 2011 ) and Vestergaard Peterson ( 2006 ) (H linewidth taken from SDSS for this source). Bolometric luminosity estimates, L Bol , use the corrections given in equation 21 of Marconi etal. ( 2004 ). Most of these AGN have X-ray luminosities that imply a bolometric luminosity around a tenth of a per cent Eddington (mean value 0.2). The spread is large, however; our least luminous AGN, XMMSL1J131651.2084915 with L 210keV 510 42 ergs 1 , is emitting at a few hundredths of the Eddington rate, whilst our most distant source, XMMSL1J095336.4161231 at z 0.873, may be emitting at or close to the Eddington rate. 3.4 -ray and radio searches At the hard X-ray/-ray energies of 15150keV, none of our sources are detected in the Swift BAT transient monitoring programme spanning 20052016 May (Krimm, private communication). This supersedes the low-significance detections we tentatively reported in S11 . All of our sources, listed in Table 1 , have coverage in either the NRAO VLA Sky Survey (Condon etal. 1998 ) or Sydney University Molonglo Sky Survey (Mauch etal. 2003 ), three are also covered by the VLA Faint Images of the Radio Sky at Twenty centimeters survey (White etal. 1997 ) and one by the Westerbork Northern Sky Survey (WENSS; Rengelink etal. 1997 ). A search in radio catalogues at these positions results in no spatially coincident radio sources (within 30arcsec radius). However, XMMSL1J164859.4800507 (for which


of one year of on-bulge time, but spread over several years of the mission. Table 5 lists the areal coverage and survey efficiency for a number of current and proposed ground and space microlensing surveys. For the space missions (Euclid and WFIRST) the efficiency in Table 5 is simply given by the likely total on-bulge observing time of the proposed programme. For the ground-based surveys the efficiency represents the fraction of the yearly rate likely to be achievable within a given observing season. The numbers in Table 5 can be used to scale the rate yields in Table 3 to approximate expected seasonal rates for a given survey in the limit of perfect detection capability across all event timescales. Figure 9 shows the maximum event rate detectable as a function of survey area for ground and space surveys listed in Table 5 . The actual area of the high-cadence regions of each survey is indicated on each line. We note that these maximum rates are obtained by rank sorting regions of the rate maps by the highest yielding regions and therefore a given total areal coverage may correspond to non-contiguous regions which do not correspond to the actual area being surveyed by the surveys. They instead represent the largest event yield which can be obtained for a survey with the same total areal coverage, wavelength and sensitivity. The advantage of a multi-site ground-based survey like KMTNet over single-site surveys is apparent from Fig. 9 . However a high-cadence near-IR survey with VISTA covering a similar area could prove even more effective owing to the increased sensitivity of near-IR observations. Figure 9 shows how space-based surveys are particularly effective for detecting low-mass exoplanets. Even though Euclid and WFIRST will cover areas of only around a couple of square degrees, their detection yield for Earth-mass FFPs is expected to be 12 orders of magnitude higher than ground-based survey seasonal rates which span an area of sky 10 times larger. For Jupiter-mass planets the space mission yields are 1030 times higher, and so are still likely to exceed yields from the ground, with the possible exception of a high-cadence VISTA survey operating over at least a decade. Overall, a short space-based campaign over a modest area can significantly outperform a gtound-based campaign and can provide important statistics on the FFP population over a range of FFP mass. In our simulation, the population of one FFP per star is assumed, but as Sumi et al. (2011) mentioned, the population of planets may be larger than stars. The detection probability of FFP is therefore expected to rise from our event rate values. Moreover, Varvoglis et al. (2012) and Wang et al. (2015) calculated the motion of FFPs in a cluster and resulted they tend to travel within the cluster unless any outside effect such as a close encounter with other clusters happens. The capture and release of FFPs ( Varvoglis et al. 2012 ) may also temporarily and slightly affect the


time is dominated by mid latitudes where R ∼ r , so magnetic energy can only be dissipated at the pole as fast as it migrates from mid latitudes. Assuming loops of azimuthal field are totally dissipated near the pole, their effective damping rate is equal to their migration rate γ mig . Because the loop migration is essentially a random walk process, the migration time-scale is where N step is the migration length divided by a step length, and t step is the time it takes to complete each step. The number of steps is N step ∼ r /ξ ⊥ . The time of each step is a magnetic diffusion time across a radial wavelength, . In order for the the instability to grow, this diffusion rate must be smaller than the growth rate . So we have . Then using equations ( B3 ) and ( B4 ), we have This maximum migration rate will be realized when ω A ∼ ω c , with the critical field strength ω c defined by equation ( 1 ). When ω A ∼ ω c , field loops can reconnect with loops of opposite oscillation phase (i.e. those separated by radial distance ∼1/ k r ) after ∼1 oscillation cycle, such that they can migrate by a distance ∼ξ ⊥ each oscillation cycle. When ω A > ω c , reconnection requires many oscillation cycles, the migration rate will depend on the magnetic diffusivity, and it will be smaller than equation ( B5 ). Why do we still think the instability damping rate is δ v A / r ? Let’s consider whether migration of magnetic loops toward the pole can destroy them at faster rates. Assuming a loop reconnects with its neighbour after being displaced horizontally by ξ ⊥ at a rate , the two loops have moved apart from one another at a speed . Then the maximum rate at which the instability can be damped due to loops migrating to the pole is But since ω A Ω, γ diss is smaller than the damping rate δ v A / r . The actual destruction rate is likely much slower due to the random walk process discussed above, and is given by equation ( B5 ). This means that unstable perturbations will damp faster by weak turbulence than they will by migrating toward the pole. So the instability is still limited by weak turbulence, and setting the growth rate equal to the damping rate still implies The migration rate of equation ( B5 ) implies that the background field, whose energy density is , is destroyed at this rate. The assosiated energy damping rate per unit volume is The turbulent energy damping discussed in the paper will operate regardless of the loop migration, and will be more important when the reconnection time-scale is longer than an oscillation times scale. Hence, while damping from loop migration may be relevant when ω A ∼ ω c , we do not expect any of our scaling arguments or results to be altered. APPENDIX C: EFFECTIVE STRATIFICATION As discussed in Spruit ( 2002 ), the effective Brunt–Väisälä frequency N eff depends on the thermal diffusion time-scale across the Tayler instability length-scale. This thermal diffusion time-scale in turn depends on N eff . Spruit ( 2002 )


of the accretion disk (White Swank 1982 ). The spectral evolution during dips depends on the ionization state of the absorbing material (Diaz Trigo Boirin 2016 ). In the case of XMM0042, we do not detect any significant spectral variation possibly due to the low statistics. Van Paradijs McClintock ( 1994 ) study the optical emission from LMXB disks; if X-rays from the central source are reprocessed by the accretion disk, this implies that the optical luminosity in the V-band M V scales with the X-ray luminosity and size of the accretion disk. We considered the relation from Van Paradijs McClintock ( 1994 ) between L X , the orbital period, and M V , taking into account erg s −1 (following Van Paradijs McClintock 1994 ), our orbital period of 4.01 hr, and our mean luminosity erg s −1 :obtaining (1 σ error) for the optical counterpart. Among the 17 positionally consistent HST candidate optical/UV counterparts reported in Yukita et al. ( 2017 ), all but one are very similar to the surrounding stellar populations. The remaining source shows an excess in the blue band, making it an interesting potential counterpart of the accretion disk of an LMXB. From the blue apparent magnitude of the counterpart m B = 24.78, standard extinction toward M31, and Equation (4) from Barnard et al. ( 2012 ), we obtain an absolute visual magnitude . This is consistent (within 2 σ ) with the one derived from the X-ray luminosity and period of XMM0042, thus making it the likely optical counterpart. The X-ray spectrum of XMM0042 (as seen by Yukita et al. 2017 and in this work) is quite hard, with and . This is consistent with what is seen in dipping LMXBs in our Galaxy, where a wide range of spectral properties is displayed, with photon indexes varying from 0.4 to 2 and cutoff energies from 3.5 to 80 keV (Diaz Trigo et al. 2006 ). The most luminous Galactic dipper, X 1624–490, has an X-ray luminosity in the 1–30 keV energy band of erg s −1 (in the 0.5–50 keV energy range—Galactic dippers reach persistent luminosities only ∼3 times lower than XMM0042; Balucinska-Church et al. 2000 ; Iaria et al. 2007 ) and a much softer spectrum. The period, optical counterpart, and the spectrum of XMM0042 are reminiscent of a dipping LMXB system. The mean luminosity we found through our XMM-Newton analysis, ranging from 0.8 to erg s −1 , makes XMM0042 the most luminous dipper to date. Trudolyubov et al. ( 2002 ) found the first dipping source of M31, located in Bo 158. The period of the dipping behavior was 2.78 hr with a flux modulation of ∼83%. A study of the other XMM-Newton , Chandra , and ROSAT observations (Trudolyubov Priedhorsky 2004 ) revealed the amplitude of the dips to be anti-correlated with the source luminosity ( erg s −1 ), disappearing at high luminosities. The source also showed hour-timescale and month-timescale luminosity variations. This source has a rather hard spectrum, but it has never been observed by NuSTAR , so we have no information about the cutoff energy or


d for the longitudinal field, veiling, and radial velocity (RV), respectively. Modulation is also obvious in Ca ii IRT and He i emission (with periods of 5.82 ± 0.16 d and 5.65 ± 0.08 d, respectively) and longitudinal fields from Ca ii IRT emission (with period of 5.50 ± 0.15 d). Since all periods we derive are quite consistent (within 1.5σ) with the rotation period used to phase our data (5.70 d, see Table 1 ), we report no evidence for differential rotation at the surface of LkCa 15. 4 TOMOGRAPHIC MODELLING OF LkCa 15 We use tomographic imaging, and more specifically Zeeman–Doppler Imaging (ZDI), to simultaneously reconstruct the large-scale magnetic field at the surface of LkCa 15, and the distributions of photospheric brightness and accretion-induced excess emission in Ca ii IRT lines. To achieve this we proceed as in previous studies (e.g. Donati et al. 2011 , 2012 ) where the imaging method was outlined. We thus only recall the main steps in this paper. ZDI iteratively looks for a set of images with lowest information content that fits the data at a given reduced chi-square level, starting from blank images containing no information. Whereas photospheric brightness (with cool spots only) and accretion-induced excess emission (with bright features only) are directly described through their distributions at the surface of the star, the large-scale magnetic field, decomposed into its poloidal and toroidal components, is expressed as a set of spherical harmonics (SH, Donati et al. 2006 ). We assume the field is dominated by odd SH modes (as in, e.g. Donati et al. 2011 ) to ensure that accretion occurs mostly towards the polar regions. The local Stokes I and V profiles of both photospheric lines and Ca ii emission are computed using Unno-Rachkovsky’s analytical solution to the polarized radiative transfer equations, taking into account the local values of the modelled distributions (i.e. the brightness map for photospheric lines, the accretion map for Ca ii emission, and the magnetic map for both sets of lines). Local profiles are then integrated over the visible hemisphere to obtain the synthetic profiles of the rotating star at each observed phase. Our model is able to convincingly reproduce most observed profile distortions and Zeeman signatures, as shown in Fig. 2 . We note that the sharp increase and decrease of the longitudinal field of Ca ii IRT lines are better matched for an inclination angle i of 70° than of 50° (the angle measured for the outer accretion disc, Thalmann et al. 2014 , with no reliable inclination estimate for the inner disc). We also obtain that the data are best fit for km s −1 and for an average radial velocity of LkCa 15 with respect to the Sun of km s −1 . Figure 2. Observed (thick black line) and modelled (thin red line) LSD Stokes I and V profiles of the photospheric lines (lef-handt panels) and of the emission core of IRT lines (right-hand panels) of LkCa 15. Rotation cycles and 3σ error bars (for Stokes V


a simple threshold of 450 km s −1 . It is clear that in general, inverted HMF is much more similar to slow wind than fast wind. Indeed, inverted HMF is distinct from fast uninverted HMF in all parameters except α :p. The CDFs of Fe:O for inverted and fast uninverted HMF are clearly separated in the upper part of the distribution, despite the fairly modest β ′ value. Inverted HMF is nevertheless distinct from slow wind in terms of the HMF properties, which is weaker and increasingly inclined the radial direction. 4. Discussion and Conclusions Inverted HMF occurs in short bursts and preferentially in slow, dense solar wind. It generally exhibits lower HMF intensity and increased inclination to the radial direction compared to the solar wind as a whole. We have further shown that inverted HMF is similar to slow solar wind in terms of elemental composition and oxygen, carbon, and iron ion charge states, suggesting common coronal source conditions. Inverted HMF is strongly differentiated from fast uninverted HMF in all parameters except the elemental composition measures and iron ion charge-states. Specifically, inverted HMF is weakly differentiated from fast uninverted HMF in terms of Fe:O abundance ratios, but there is little difference in the α :p abundance ratio, as is often the case with the slow wind in general (Fu et al. 2017 ). It is possible that gradients in elemental abundance are produced in the corona, but that differential streaming of heavy ions relative to the bulk (proton) solar wind (Berger et al. 2011 ; Alterman et al. 2018 ) is able to traverse the short inverted HMF intervals and remove this signature by 1 au (though this should equally wash out the carbon and oxygen charge-state signatures). It may be possible to identify any such compositional signatures using observations from close to the Sun, discussed below, or focusing only on the longer-lived HMF inversions, though this prohibits statistical analysis. Interpretation of ion charge states in terms of coronal temperature is not straightforward, with coronal electron density, temperature, and plasma velocities all contributing to the observed in situ values (Zhao et al. 2014 ). But in general, elevated oxygen and carbon charge states are produced by increased temperatures in the lower corona, whereas elevated iron charge states are more likely to result from heating through the extended corona (Song et al. 2016 ). Thus inverted HMF is associated with increased heating in the lower corona, which suggests that the material was released from hot coronal loops. This is consistent with previous results that show an association between inverted HMF and closed coronal loops in dipolar and unipolar coronal streamers (Owens et al. 2013 ). Hot coronal material on closed coronal loops can be released by interchange reconnection (Crooker Owens 2011 ) with an open magnetic flux tube. This could be initiated by either random foot-point motions or more large-scale convection of open


12 CO(2–1) transition at 230.53 GHz with a high spectral resolution of 0.485 MHz (or 0.63 km s −1 ). Each of our nine sources was observed for ~1.4 min in total. The system temperature has a mean value of 91 K. The data was pipeline-calibrated using standard procedures of CASA 4.5.2. The continuum was imaged concatenating together the 128 channels of the spectral window dedicated to the continuum emission, which is a valid approach for pre-stellar structures for which no complex organic molecule emission lines are expected ( Motte et al. 2018a ). The 12 CO(2–1) emission line was slightly smoothed to a spectral resolution of 0.7 km s −1 . Both the continuum and 12 CO line emissions were imaged with the Clark method using the weighting Briggs with a robust parameter of 0.5 1 . The mean achieved root mean square (rms) noise level is of ~0.34 mJy/beam in continuum and of 16.6 mJy beam −1 per channel in 12 CO(2–1), at a resolution of 0.7 km s −1 . Table A.1 details the rms for each source. Fig. 1 Herschel -SPIRE-250 μ m emission of the NGC 6334 region. The 9 MDCs of the present work are located by the blue stars together with their number, following the nomenclature by Tigé et al. (2017) . Red arrows and Roman numerals (I, V) denote bright far-infrared sources from Kraemer Jackson (1999) . Credits: ESA, Herschel Science Archive. 2.2. MOPRA observations These observations were performed in September 2008 with the ATNF MOPRA 22-m telescope with the 3 mm receiver and the MOPRA spectrometer (MOPS) in the “zoom mode” that allows to observe simultaneously up to 16 different frequencies, among them the HCO + (1–0) at 89.18 GHz, the H 13 CO + (1–0) at 86.75 GHz, and the N 2 H + (1–0) at 93.17 GHz emission lines. Small maps (between 3′ × 3′ and 5′ × 5′) were observed in the direction of the 40 dense cores of Russeil et al. (2010) . These maps cover seven of our nine MDCs. These maps were done in the OTF mode and the data reduction was performed with Livedata and Gridzilla tools 2 producing data-cubes with a spectral and spatial resolution of 0.11 km s −1 and 30′′, respectively. The MOPRA dataset is complemented with observations from the MALT90 survey ( Jackson et al. 2013 ) to cover one additional MDC. To sum up, we gather a coherent sample of spectral lines from the MOPRA telescope for all our nine MDCs but the MDC #44. 3. Analysis 3.1. Large-scale view of the MDCs Figure 1 shows the Herschel 250 μ m emission of NGC 6334, overshowing the column density map plotted with the positions of the nine MDCs studied here. Figures 2 – 4 show the column density map over each MDC (left columns) and the molecular line emissions in the MDC (right columns). All observed MDCs are detected in HCO + . Five MDCs are detected in H 13 CO + , and seven in N 2 H + (see Table 1 ). Only the MDC #11 was undetected in both N 2 H + and H 13 CO + . We fitted the lines in CLASS 3 using the hfs procedure to fit the multiplet N 2 H + molecular emission line and Gaussian functions to fit