radius from the particle data of the N -body simulation. We also determine the circular frequency, Ω, by averaging the radial accelerations of particles in an annulus with a width of 50 pc at each radius. In Fig. 8 , we show Ω + κ/ m ( m =2, 3, 4, and 5) and Ω as functions of R . The horizontal line in Fig. 8 indicates the pattern speed of the bar. The radii at which the line intersects with the frequency curves correspond to the guiding-centre radii for the 2:1, 3:1, 4:1, 5:1, and the corotation resonances. In our model of the MW, the Sun, at a distance of ∼8 kpc from the Galactic Centre, is close to the 3:1 OLR. It is then not surprising that this resonance is dominant in the local stellar population. Figure 8. The orbital frequencies as functions of the galactic radius, R . Ω was determined from the velocity curve and Ω + κ/ m ( m =2, 3, 4, and 5) as expected within the epicycle approximation are plotted. The vertical axis is normalized by Ω 8kpc . The horizontal line indicates the bar’s pattern speed and the vertical lines represent the guiding radii for the respective resonant orbits. 4.2 The MW’s bar pattern speed When projecting the Gaia DR2 data into the R – v φ plane, a series of ridges appear over a wide range of R (Antoja et al. 2018 ; Kawata et al. 2018 ; Ramos et al. 2018 ). Although the origin of these structures is still in debate, previous studies (e.g. Monari et al. 2017b , 2019a ; Fragkoudi et al. 2019 ; Barros et al. 2020 ) have discussed that these ridges may originate from resonances, while other studies have suggested perturbations by a satellite galaxy such as the Sagittarius dwarf (Khanna et al. 2019 ; Laporte et al. 2019 ) or by winding transient spiral structure (Hunt et al. 2018 , 2019 ). In this section, we study the relation between the observed ridges in Gaia DR2 and resonances in our simulated MW model: by projecting our simulated and classified stars into the R – v φ plane, we can directly compare to the same projection of the Gaia DR2 data. If the stars classified by resonance type align with the observed ridges, then this provides compelling evidence that these ridges are indeed caused by resonantly trapped stars. In Fig. 9 , we show the distributions of stars in the R – v φ plane as it arises in our simulated MW model (left-hand column) and in the Gaia DR2 (right-hand column). The top panels show 2D histograms of particles within 2 kpc from the Sun [, in our model]. This panel reveals a selection effect caused by Gaia ’s measurement uncertainties: due to us having selected stars with relative parallax error below 10 per cent the right-hand panel displays systematically fewer stars in approximately concentric rings around the Sun’s position at Visually, this causes the smaller extend of the red area in the right-hand panel. We expected, however, that our analysis is robust with respect to this selection effect: it implies that fewer stars make it into our DR2 projection than into the projection of our


J.H. greatly appreciates the CNES post-doctoral fellowship program. J.H. and M.D. were supported by the project under the contract 11-BS56-008 (SHOCKS) of the French Agence National de la Recherche (ANR), JD by grant GACR 15-04816S of the Czech Science Foundation, DO by the grant NCN 2012/S/ST9/00022 of Polish National Science Center, and A. Marciniak by grant 2014/13/D/ST9/01818 of Polish National Science Center. We thank the referee, Mikko Kaasalainen, for his thorough review of our manuscript and his constructive comments and suggestions that led to a significant improvement of the text. The computations have been carried out on the “Mesocentre” computers, hosted by the Observatoire de la Côte d’Azur, and on the computational cluster Tiger at the Astronomical Institute of Charles University in Prague ( http://sirrah.troja.mff.cuni.cz/tiger ). Data from Pic du Midi Observatory were partly obtained with the 0.6 m telescope, a facility operated by observatoire Midi-Pyrénées and Association T60, an amateur association. The Joan Oró Telescope (TJO) of the Montsec Astronomical Observatory (OAdM) is owned by the Catalan Government and operated by the Institute for Space Studies of Catalonia (IEEC). We thank Franck Pino (INO-AZ) and Lech Mankiewicz (EU-HOU/Comenius) for the remote access to Ironwood North.


supported. The key features of each multiphase flow are summarized as follows. As rotation dominates over turbulent motions (Ta t v rot / v 1) in the massive galaxy, the condensation occurs along a conical helix, spinning up the gas which finally settles on to the equatorial region in a multiphase disc (Section 3 ). The disc is cylindrically stratified: the ionized (UV) gas envelops the neutral (optical/IR) layer which englobes the very thin molecular (radio) structure, dominating 95percent of the condensed mass. There is no major fragmentation observed, with the warm phase experiencing mild perturbations via pressure torques induced by condensation. The BHAR is suppressed by almost 2 dex compared with the cooling rate as most gas has found circularization. Accretion proceeds along a narrow polar funnel (10) or after reaching the inner equator through gas which condensed out of the plasma with low angular momentum. After 50Myr the disc has reached 1kpc extension. Extended molecular and warm discs with similar surface density have been observed in 1/3 of observed massive, typically quiescent, galaxies (Caon etal. 2000 ; Ocaa Flaquer etal. 2010 ; Young etal. 2011 ; Werner etal. 2014 ; Hamer etal. 2016 ). When rotation and turbulence are both inhibited, the hot plasma condenses radially, forming a compact multiphase cloud within the inner 100pc (Section 4 ). The inner multiphase density profiles are steep, following a free-fall slope ( r 3/2 within the SMBH influence region). The BHAR is almost identical to the cooling rate. Adiabatic, compressional heating inhibits the formation of the third molecular phase. The warm 8000K (optical/IR) gas dominates the condensed medium, albeit highly concentrated, with a very thin ionized UV layer. Similarly, a significant number of quiescent massive galaxies display a spherically symmetric warm H core (Caon etal. 2000 ; Werner etal. 2012 , 2014 ) and tend to lack cold gas, in particular when no AGN feedback or other major perturber are detected. In the more frequent, intermediate state the hot halo experiences both rotation and subsonic turbulent motions (160kms 1 ) leading to CCA (Section 5 ). The multiphase condensation occurs in the high density peaks via local runaway cooling. AGN heating prevents a global cooling inflow, facilitating the growth of multiphase structures. Initially, warm filaments condense out of the low-entropy plasma. The denser, perturbed regions further condense into molecular clouds, following a top-down rainfall. The CCA angular momentum transport is driven by pressure torques on 1kpc scale and dominated by chaotic inelastic collisions in the core (ram-pressure drag is secondary). The accretion rate can be modelled via quasi-spherical viscous accretion (Section 5.2 ), with an effective clump collisional viscosity c v c ; the mean free path is c 100pc with large scatter. The core cloud volume filling is a few per cent, rapidly declining beyond 1kpc. As ensemble, the multiphase clouds


ACKNOWLEDGEMENTS This research has been supported by Arizona State University. We would like to thank Soma De, Tanmay Vachaspati, Daniel C. Jacobs, Adam Beardsley and Samuel Gordon for their insightful conversations. Some of the results in this paper have been derived using the healpix (Górski et al. 2005 ) package. This research made use of astropy , a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013 ).


z LAEs with high-resolution HST narrow-band imaging (Bond et al. 2010 ; Finkelstein et al. 2011 ). Bond et al. ( 2010 ) suggested that Ly α sizes are compact and similar to UV emission, but Finkelstein et al. ( 2011 ) posited that the half light radius of Ly α appears ∼1.6 times larger than the half light radius of UV continuum. These narrow-band HST images of high- z LAEs are very hard to obtain and have low S/Ns. Thus, the Ly α and UV sizes of low- z LAEs are valuable. Because Green Peas are analogs of high- z LAEs, our results suggest that most high- z LAEs likely have larger Ly α than UV sizes. The extended Ly α emission probably indicates gas outflows around galaxies illuminated by Ly α light. One interesting question regards the redshift evolution of Ly α sizes of LAEs. Recently, Wisotzki et al. ( 2016 ) measured Ly α radial profiles of a sample of LAEs at z = 3–6 from VLT/MUSE data. They found that, in 12 LAEs with both Ly α and UV continuum sizes, the Ly α light is considerably more extended than the UV continuum light. Here, we compare the sizes of Green Peas and the MUSE LAEs. Using the Ly α radial profiles in Wisotzki et al. ( 2016 ), we measured the deconvolved Ly α scale radius r e , assuming intrinsic exponential profiles, so that the methods are same when measuring the r e of MUSE LAES and Green Peas. As shown in Figure 5 , the Ly α -to-UV size ratios of Green Peas and MUSE LAEs are very similar. Notice that some MUSE LAEs have extended Ly α halos far beyond the scale radius. For Green Peas, however, we do not have robust data to characterize the Ly α emission beyond a few Kpcs. Figure 5. Left: Comparison of the Ly α- to-UV scale length of Green Peas (green squares) and MUSE z = 3–6 LAEs sample (red dots) (Wisotzki et al. 2016 ). The Ly α scale lengths of MUSE LAEs are measured from the radial profile in Wisotzki et al. ( 2016 ) using the same method as Green Peas. Right: Comparison of the Ly α -to-UV Petrosian 20% (P20) radius of LARS ∼ 0 galaxies and MUSE LAEs. Notice that the Petrosian 20% radius of MUSE sample is measured from the best-fit model of radial profile. The dashed cyan lines show constant ratios of 1:1, 2:1, 5:1, and 10:1. In the right panel of Figure 5 , we compare the Petrosian 20% radius ( R P20 ) of MUSE LAEs (Table 2 in Wisotzki et al. ( 2016 )) to that of the LARS sample (Table 1 in Hayes et al. 2014 ). Compared to the five strong Ly α emitters in LARS sample (marked by stars, LARS01, 02, 05, 07, and 14), the MUSE LAEs only have about two times larger ratios of R P20 (Ly α ) to R P20 (UV). One caveat of the comparison is that the Petrosian radius of MUSE sample is measured from the best-fit model of radial profile, instead of the observed data, which is different from the method used in the LARS sample. This might be the reason that the R P20 (UV) of MUSE LAEs are about 2–4 kpc, approximately a factor of three larger than the R P20 (UV) of the five LARS Ly α emitters. Based on our rough comparison of Green


this correlation is presently unclear. Figure 8. Periastron distance (top panel; ) and the tidal force acting on the primary star at periastron divided by the star’s surface gravity (bottom panel; see Equation ( 4 )), both as a function of the full amplitude of the photometric HB signal at periastron. In both panels the markers’ radius is linear in the primary star’s (See Table 1 ). 4.5. Higher Multiplicity Systems Many of the HB stars examined here may be members of triple or higher multiplicity systems. Several works (e.g., Duquennoy Mayor 1991 ; Meibom Mathieu 2005 ; Tokovinin et al. 2006 ; Raghavan et al. 2010 ) have found that the fraction of higher multiplicity systems among short period and highly eccentric binaries is very high, exceeding 90% for binaries with days (Tokovinin et al. 2006 ). Indeed, tertiary bodies may excite the orbital eccentricity of HB progenitors via Kozai–Lidov oscillations, producing close periastron passages and allowing them to be detected as HB systems. Quadruple systems composed of two binaries may also be common among HB systems, and these types of systems have been predicted by Pejcha et al. ( 2013 ), who found that HB systems formation is greatly enhanced in quadruple systems relative to triple and binary systems. Despite the probable multiplicity of many HB systems, we find it unlikely that many of our Keplerian RV solutions have been greatly affected by orbital motion induced by a third body. A putative third body exciting the orbital eccentricity would likely have an orbital period, P 3 , more than several times longer than the HB period (depending on its mass, inclination, and eccentricity) in order to allow long term dynamical stability (Kiseleva et al. 1994 ). If a typical HB progenitor system was born with an orbital period days before having its orbital eccentricity excited, we expect for most third bodies, with much larger values of P 3 in most cases. Although the orbital RV amplitude induced by such a third body can be several km s −1 , we expect only a small RV change over the ∼100 days baseline of our observations, and therefore this motion can be safely included as a constant RV offset. We are currently pursuing a follow-up survey on many of these HB systems that will reveal whether any of them have tertiary companions with orbital periods . 4.6. Future Prospects The orbits measured here comprise a relatively large sample obtained with a single instrument, and they can facilitate several follow-up scientific studies in addition to that of tidal circularization and the shape of the e – P distribution (Section 4.3 ). Other examples include: 1. Stars in binary systems with highly eccentric orbits are expected to rotate near a “pseudo-synchronous” rotation period, where the stellar rotation is synchronized with the orbital motion close to periastron which is faster than the mean orbital motion along the entire orbit. The pseudo-synchronous rotation rate depends on eccentricity and orbital period


tidal forces from the Milky Way, or any other physical process 9 9 For example, relaxation cannot be responsible for the spatial distribution of the different stellar populations in Carina since the relaxation times in dwarf galaxies are much larger than the age of the universe (e.g., Freeman 2008 ). , were not strong enough to move a significant fraction of these stars to the outer region. Otherwise, a large number of intermediate-age stars would be present in the outskirts of Carina, and thus we would not observe such a strong break in the SFH and MDF of Carina between the intermediate and outer regions. Since the radial trends in Carina do not seem to be governed by the tidal influence of the Milky Way, we suggest that this hints that the SFH is not dominated by tidal forces either. In this line, the fact that we find two episodes of star formation in Carina instead of three, as has often been claimed (Hurley-Keller et al. 1998 ; Rizzi et al. 2003 ; Pilkington Gibson 2012 ; McMonigal et al. 2014 ), makes it harder to explain the star-formation episodes in Carina as being the result of periodic close passages to the Milky Way (as done in Pasetto et al. 2011 ), even more so since proper motions suggest that the orbital period of Carina is close to 2 Gyr (Piatek et al. 2003 ; Pasetto et al. 2011 ). Another mechanism proposed to explain the SFH of Carina is gas accretion from the Milky Way (e.g., Lemasle et al. 2012 ; de Boer et al. 2014 ). In this view, the first episode would have ended because of gas depletion, and subsequent star formation would have been stopped for a few gigayears until the dwarf galaxy renewed its gas content thanks to an inflow from the Milky Way. Even though this is a plausible scenario, the in-falling gas should have had the same metallicity as the stars from the end of the first episode to reproduce the age–metallicity relation in Carina. A simpler way to explain why the metallicity of the stars at the end of the first episode coincides with that of the stars at the beginning of the second episode is to assume that the gas that formed the second episode in Carina was enriched within the same galaxy by the first generation of stars. Internal evolution would also be consistent with the mean metallicities and ages at different Carina regions, which according to what we see in Figure 9 are roughly constant throughout the extent of the galaxy. This would not necessarily be the case if the gas that formed those stars had been accreted at different moments and each region had different levels of influence from each inflow. For all these reasons, we conclude that the SFH of Carina must be dominated by internal evolution. Our interpretation is that, after the first episode formed the majority of the stars in Carina, the physical conditions of the gas made the galaxy unable to form stars for a few gigayears, until it contracted to activate star formation again. We claim that the temporal gap in star formation in Carina might


of the disc will be In our model, the relative change in disc mass and accretion rate in an encounter is determined by γ and the truncation radius relative to the disc characteristic radius just before the encounter, . If the disc is truncated, its viscous time-scale decreases and the disc starts evolving faster. We ignore the orientation of the discs, so the equation for truncation radius is the average truncation radius over all disc inclinations. 2.4 Numerical implementation We use the fourth-order Hermite N -body code ph4 , which is incorporated into the AMUSE framework. The softening length is and the time-step parameter is η = 10 −2 . The simulations start in virial equilibrium and last for 2 Myr. Collisional radii for each star in the cluster are defined depending on the stellar mass and the theoretical size of the disc evolving in isolation, given by equation ( 1 ). The initial collisional radius for each star is given by the distance at which the most massive star in the cluster can truncate its disc. The collisional radii of all stars are updated every 2000 yr to account for the viscous evolution of the discs. Two stars are considered to be in an encounter if the distance between them is less than the sum of their collisional radii. The encounter distance between the stars is then computed by analytically solving the two-body problem. If the encounter is strong enough as to truncate the discs, disc parameters of both stars are updated as described in Section 2.3 . After each encounter, the collisional radii of both stars are set to 0.49 r enc in order to prevent the N -body code from detecting the same encounter multiple times during the same 2000 yr window. 3 RESULTS 3.1 Initial conditions 3.1.1 Cluster properties Simulations were run for clusters with 1500 stars. The stellar masses are randomly sampled from a Kroupa initial mass distribution (Kroupa 2001 ) with lower mass limit 0.1M ⊙ and upper mass limit 100M ⊙ . The stars are initially distributed in a Plummer sphere (Plummer 1911 ) with Plummer radius 0.5 parsec. All stars are assumed to be single and coeval. All simulations were carried out for . We define three different gas scenarios for the simulations: No gas Gas presence Gas expulsion starting at Each simulation was run for two values of the turbulence parameter: α = 10 −2 and α = 5 × 10 −3 . 3.1.2 Disc properties Based on the observations of low-mass stars carried out by Isella et al. ( 2009 ) and the estimations obtained by Hartmann et al. ( 1998 ), the initial characteristic radius R c of the circumstellar discs was chosen to be with . Circumstellar discs with masses larger than 10 per cent of the mass of their star are the most likely to be gravitationally unstable (Kratter Lodato 2016 ). According to hydrodynamical simulations of collapsing gas clouds, the disc-to-star mass ratios of embedded protostars are large enough for gravitational instabilities to occur (Vorobyov 2011 ), but these instabilities lead to


from the MPA-JHU DR7 catalogue, and we obtained MBH only for galaxies classified as ellipticals (see Section 3 ). The values are given in Table 8 . MLLINERs cover the range between log (MBH/M ⊙ ) = 7.03–8.57 with a median value of log (MBH/M ⊙ ) = 7.45, while LLINERs show MBH in the range log (MBH/M ⊙ ) = 6.24–8.54 and median value of log (MBH/M ⊙ ) = 8.04. Interestingly, our MLLINERs do not contain the most massive BHs in their centres. It could be surprising that MLLINERs, having on average higher SFRs than LLINERs, show in general lower stellar masses. Different works, both observational (Kauffmann et al. 2003b ; Mateus et al. 2006 ; Leauthaud et al. 2012 ; Pérez et al. 2013 ) and numerical (Shankar et al. 2006 ; Behroozi et al. 2012 ), revealed a stellar mass of ∼6 × 10 10 M ⊙ as critical for the growth rate of stellar populations. In particular, in Pérez et al. ( 2013 ) by studying a 3D spectroscopic sample of 105 local galaxies, the authors found that in galaxies more massive than 5 × 10 10 M ⊙ the inner regions ( 0.5 R 50 ) grew as much as 50–100 per cent faster than in the lower mass galaxies. They found that the peak of relative growth rates of inner and outer galaxy regions correspond to the stellar mass of 6–7 × 10 10 M ⊙ (see their fig. 5), while for lower and higher masses the growth rate decreases and therefore SFRs (LSF) as well. The median stellar mass of our MLLINERs (6.58 × 10 10 M ⊙ ) corresponds perfectly to this region, while for most LLINERs their stellar masses are already higher and correspond to lower values of the relative growth rate (lower LSF). This explains why MLLINERs having in average lower stellar masses in comparison to LLINERs, have higher LSF. In addition, we studied the stellar mass distributions of MLLINERs and LLINERs for the three morphological groups. Table 9 shows the median stellar masses for different morphological types. As can be seen, of three morphological types the highest difference was obtained for early-type galaxies. These galaxies represent a significant fraction of MLLINERs (40 per cent, see Fig. 8 ) and their median mass corresponds exactly to the highest relative growth rate of stellar populations (according to Pérez et al. 2013 ), which then could explain their high SFR values. This is not the case for early-type LLINERs that are characterized by higher stellar masses (lower growth rates) and lower SFRs. Table 9. Median total stellar masses of MLLINERs and LLINERs in relation to morphology (given as logarithm and in M ⊙ ). 5.1.2 Extinction We found that our MLLINERs can be hosted by galaxies with a wide range of extinctions. When using the Av measurements based on the Hα and Hβ emission lines, we find that most of them reside in galaxies with high extinctions. The median Av is 1.65 mag, covering the range 0.49–3.46 mag (see Fig. 7 , top plot). For comparison, the bottom plot in Fig. 7 shows the Av distributions of LLINERs and MLLINERs when taking into account the SDSS MPA-JHU DR7 3


The authors acknowledge research funding by Deutsche Forschungsgemeinschaft (DFG) under grant SFB 963/1, project A5.


The authors would like to thank the anonymous referee for helpful comments and suggestions. The scientific results reported in this article are based on data obtained from the Chandra Data Archive. We also made use of the ROSAT data archive of the Max Planck Institute für Extraterrestrische Physik (MPE) at Garching, Germany and NRAO VLA Sky Survey archives.


This work was supported by the grant of Universiti Kebangsaan Malaysia (GUP-2018-086) and Grant-in-Aid for Scientific Research (C) 17K05673 from the Japan Society for the Promotion of Science (JSPS). W.L.T. would like to thank Takuma Nakamura and Bo-Jhou Wang for useful discussions. Special thanks to the dedicated efforts of the entire MMS mission team for data access. MMS data can be downloaded through the NASA CDAWeb at http://cdaweb.gsfc.nasa.gov/ .


environments. Our results will aid in the preparation of multicomponent ices containing acetaldehyde by allowing for calibrations and determinations of that compound in an ice sample. We also envision applications of our data to the measurement of reaction yields in experiments that might produce acetaldehyde in polar ices. Our IR intensities also can be used to estimate band strengths in spectral regions beyond the mid-IR used in the present work. See Gerakines et al. ( 2005 ) for an example. Astrochemical results based on IR band strengths published before those presented here can now be updated with our new measurements. For example, and as already described, the reference band strength of Schutte et al. ( 1999 ) for amorphous acetaldehyde's CH 3 deformation at 1350 cm −1 is A ′ = 1.5 × 10 −18 cm molecule −1 , which now is seen to be in error by nearly 80 per cent. Molecular abundances or column densities based on that older A ′ will be in error by that much, although published relative band strengths might not change substantially. Finally, we again point out that few aldehydes have been examined with the methods used here. As an example, propionaldehyde (propanal) is an interstellar molecule, and at least two laboratory papers recently have been published involving it (Jonusas, Guillemin Krim 2017 ; Qasim et al. 2019 ), but no band strength measurements are available for the solid compound. The situation is no better for H 2 CO (formaldehyde). Many laboratory papers have been published involving H 2 CO, but only estimates and indirect measurements are available for its IR intensities. Unfortunately, their accuracy is unknown. 5 SUMMARY AND CONCLUSIONS Here we have reported the first direct measurements of band intensities of solid acetaldehyde, removing the need to rely on either calculated gas-phase results or values that cannot be traced to their original source. We also have presented mid-IR spectra of the two crystalline polymorphs previously identified by neutron-diffraction studies, one stable under our conditions and one metastable. The IR spectrum of H 13 C(O) 13 CH 3 has been recorded and used to identify ketene as a radiolytic decomposition product of solid acetaldehyde for the first time. APPENDIX A The procedure used to determine absorption coefficients ( α ′) and band strengths ( A ′) is the same as in our earlier papers, and is essentially the method of Hollenberg Dows ( 1961 ). Briefly, IR spectra were recorded of ice samples having a range of thicknesses. The absorbance at each peak of interest was measured and bands of interest were integrated. Multiplication of each absorbance or band area by ln(10) ≈ 2.303 was then done to convert from a common to a natural logarithmic scale. Representative plots of the results are shown in Fig. A1 . The slope of the upper graph is α ′ and the slope of the lower one is ( A ′ ρN A / M ), where M = molar mass = 44.05 g mol −1 , N A = 6.022 × 10 23 molecules mol −1 , and density ( ρ


(Linder 2008 ). 3.4. Growing <italic>ν</italic> Mass Growing neutrino mass models, wherein the mass of the neutrino ( m ν ) increases with time and stops the dynamical evolution of the dark-energy scalar field are invoked to solve the cosmological coincidence problem, i.e., the problem that the present-day matter density and density of Λ are similar order of magnitude despite their different dependence on the scale factor (Fardon et al. 2004 ; Wetterich 2007 ). The combined dark sector (scalar field plus neutrinos) energy density (where a = 1/(1 + z )) is given by where Ω ds = 1 − Ω M is the present-day dark-energy density. The scale factor at which the neutrinos become nonrelativistic, is given by preserving continuity between the early and late-time terms (i.e., setting Equations ( 14 ) and ( 15 ) at a = a t ). The two free parameters are the early dark-energy density Ω e and the neutrino density Ω ν . The normalized Hubble parameter for this model is given by 3.5. Dark-energy Transitions at Low Redshift Mortonson et al. ( 2009 ) proposes models with large fluctuations in the dark-energy equation of state at low redshifts, typically z ≲ 0.02, that induce step-like transitions in the H ( z ). Such changes are hidden from constraints coming from Hubble flow SNe Ia (as even the lowest redshift SNe can typically be at z > z t ), but calibrators offer an additional restriction. Step-like responses in H ( z ) also evade several model-independent constraints, as they often assume smoothness of H ( z ). Phenomenologically, we can write the first Friedmann equation in these models aswhereIn this definition, z t , and δ are the position, width, and amplitude of the transition, respectively. In particular, δ = 0 corresponds to the standard ΛCDM . Hence, this model has three free parameters at a given fixed z t : , and δ . The observed Hubble constant is . In this work, we either assume z t = 0.02 or z t = 0.1; the former case motivated by Mortonson et al. ( 2009 ) and the latter case motivated so we can use all the low- z SNe at z ≈ 0.1. The adopted priors for and are equivalent to the priors shown on Table 1 for H 0 and Ω M . The difference in notation highlights the fact that and cannot be interpreted as the observed Hubble constant and the dark-matter density at redshift zero in these models (see Mortonson et al. 2009 ). 3.6. Cosmographic Expansion Along with the different dark-energy models described above, we also look into a more model-independent method, by expanding the expression for H ( z ) as a Taylor series in a cosmographic approach (for e.g., Arendse et al. 2019a , 2019b ; Feeney et al. 2019 ; Macaulay et al. 2019 ). Expanding H ( z ), we getwhere , 2 and . 4. Results In this section we present the results of fitting the different dark-energy models described in Section 4.1 to the combined calibrator and Hubble flow SNe Ia. We also discuss the impact of the systematics covariance matrix on the inferred value of H 0 in Section 4.2 .


( 1969 ) and Yoshimura ( 1975a , 1975b ) succeeded in numerically reproducing the latitudinal evolution of the toroidal magnetic field (butterfly diagram). In order to obtain the equatorward migration of the toroidal magnetic field, it is required that the angular velocity should decrease as the radius increases in the solar convection zone ( for the positive α in the northern hemisphere; : angular velocity). However, the developments of helioseismology revealed that the real radial distribution of the angular velocity in the solar convective zone does not satisfy this requirement. In addition to this point, Parker himself discussed the limitation of the α dynamo in the solar convection zone from the viewpoint of magnetic-field buoyancy (Parker 1975 ). Once the magnitude of a magnetic field has been amplified to some level, the magnetic flux tube starts moving upward from deeper to shallower regions. Since this buoyancy convection timescale, even for a very weak magnetic flux tube, is estimated to be very short as compared with the α -dynamo amplification timescale, it reaches the solar surface without being sufficiently amplified by the dynamo. This is called the magnetic buoyancy dilemma intrinsic to the magnetic flux tube. These difficulties have been considered to be obviated by assuming that the dynamo action is operated in a convectively stable thin layer between the radiative and convection zones (Parker 1975 ). This region is associated with a large differential rotation and is called the tachocline . In the tachocline dynamo picture, the solar magnetic field is generated by a strong differential rotation there and may have enough time to be amplified by the α dynamo before it is raised by the magnetic buoyancy (Spiegel Weiss 1980 ; van Ballegooijen 1982 ). As mentioned above, depending on the location of the field generation, the dynamo models are divided into two types: (i) the flux-transport dynamo, which operates in the overshoot layer below the bottom of the convection zone; and (ii) the mean-field distributed dynamo, which works in the bulk of the convection zone, in particular at the subsurface shear layer. On the surface of the Sun, large-scale meridional circulations with a speed of ∼15 m s −1 in the poleward direction have been observed. The combination of the meridional circulation and the α –Ω mean-field model demonstrated its ability to reproduce the basic features of the solar magnetic activity cycle such as the equatorward migration of the sunspots, the cycle period, and the phase difference between the poloidal and toroidal magnetic fields. In the combination with meridional circulations or convective-diffusive motions such as the Babcock–Leighton-type convection, which produces the east–west tilt of the sunspot pairs under the action of the Coriolis force on the magnetic flux tube during its rise through the convection zone, it is expected that the tachocline dynamo can reproduce several properties of the


We thank the referee and B. Czerny, C. Done, C. Knigge, J. Matthews, H. Netzer, and D. Starkey for discussions. P.H. and G.S. acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 2017-05983. K.H. acknowledges support from STFC grant ST/M001296/1. Software: MATLAB.


motion in declination direction. The proper motion of SV Cru, (μ αcos δ, Cep , μ δ, Cep ) = (− 6.43 ± 0.04, 1.31 ± 0.05) mas yr −1 , deviates 10.9σ in declination direction from the proper motion of Ruprecht 97, (μ αcos δ, cl , μ δ, cl ) = (− 6.49 ± 0.07, 0.44 ± 0.07). The mean proper motion is estimated by 23 OC members and the deviation of proper motion may be caused by the strict selection of OC members. Considering the good matching of other constraints, we regard SV Cru as an OC Cepheid. The distance and age of Ruprecht 97 are DM = 12.95 ± 0.53 mag and log ( t yr −1 ) = 8.13 ± 0.21, which are consistent with Cantat-Gaudin et al. ( 2020 ). 3.1.10 UBC 231 and WX Pup WX Pup is located at about 25.9 arcmin from the centre of UBC 231. Its proper motion is 2.2σ from the mean proper motion of UBC 231. The parameters of UBC 231 DM = 12.31 ± 0.17 mag and log ( t yr −1 ) = 7.87 ± 0.30 are in agreement with the reference values DM = 12.31 mag and log ( t yr −1 ) = 8.07, given by Cantat-Gaudin et al. ( 2020 ). 3.1.11 UBC 375 and V438 Cyg V438 Cyg is located at approximately 7.0 arcmin from the centre of UBC 375. V438 Cyg satisfies the constraints of the spatial position and parallax. The proper motion of V438 Cyg is (μ αcos δ, Cep , μ δ, Cep ) = (− 3.28 ± 0.07, −4.27 ± 0.07) mas yr −1 , which is 6.4σ away from the mean proper motion of UBC 375 (μ αcos δ, cl , μ δ, cl ) = (− 3.42 ± 0.07, −5.22 ± 0.11) mas yr −1 . The determined OC’s parameters are DM = 11.08 ± 0.21 mag and log ( t yr −1 ) = 8.03 ± 0.30, which are consistent with Cantat-Gaudin et al. ( 2020 ). Since only 32 OC members are adopted to estimate the mean proper motion, the deviation of proper motion may be attributed to the strict selection of OC members. Considering the good matching of other constraints, we still regard V438 Cyg as an OC Cepheid. 3.1.12 FSR 0158 and GQ Vul GQ Vul is located at approximately 2.4 arcmin from the centre of FSR 0158. GQ Vul agrees well with FSR 0158 in the proper motion (0.6σ) and parallax (0.1σ). This new OC Cepheid is ∼5.3 kpc from us and almost at the detection limit of our method based on proper motions and parallaxes. The far away OC Cepheids in the Milky Way’s disk are helpful to testify the probable bias in Cepheid’s PAR. 3.2 Issue about the known OC Cepheid In this subsection, we discuss known OC Cepheids not included in our catalogue. We compared with Turner ( 2010 ) who established an OC Cepheid catalogue based on optical data and found that 11 of the 24 OC Cepheids in that work are in our list. Nine of the other 13 Cepheids are in small clusters or star associations and they are not included in Cantat-Gaudin’s membership analysis catalogue. Three OC Cepheids have proper-motion differences larger than 10σ. The remaining OC Cepheid is BB Sgr in Collinder 394. 3.2.1 Collinder 394 and BB Sgr BB Sgr was treat as an OC Cepheid for a long time from Turner Pedreros ( 1985 ). However, based on Gaia , the membership is questionable. The proper motion of BB


of [2] and [3], which show FERMI spectra because they provided better statistics. We note that the corresponding RHESSI spectra at those times showed very similar results. The X-ray spectra of background-subtracted data (black solid lines) are plotted together with the fitting results for the isothermal component (red dashed lines), and the nonthermal component (blue dashed lines). The background is represented by the gray solid line. The electron temperature, T , emission measure, EM, electron distribution index, δ , and cutoff energy, E C , as well as the chi-squared of the fitting, χ 2 , are listed. Figure 4. (a)–(d) EM-weighted temperature maps during the M1.3 flare. The dotted lines represent an artificial slit along the trajectory of the jet-like feature, along which we extracted the DEM evolution. (e) Time–distance plot for EM-weighted temperatures along the slit. (f) EM distribution for the position and times marked by the colored circles in panel (e). An animation of panels (a)–(d) is available. The video begins at 21:30:01 and ends at 22:14:49. The realtime duration of the animation is 15 s. (An animation of this figure is available.) In order to infer information on the triggering of the event and its magnetic topology, we initially studied the magnetic field of AR 11944 during the flare under study as well as a few days before. We note that due to the rapidly varying photospheric magnetic field of the AR prior to the occurrence of SOL2014-01-13T21:51M1.3, its closeness to the limb, and the small-scale nature of the event, the magnetograms did not provide us with useful information for the magnetic configuration on the date of the flare occurrence. 3.3. Causal Relationship of Precursors and Flare Features In addition to the bright loop and associated chromospheric brightenings observed for P2 and P3, the impulsive flare phase starts with the appearance of an additional loop (marked by the orange dashed line in the inset of Figure 2 (d)) rooted at a chromospheric brightening (labeled (3)). This was immediately followed by the generation of flare ribbons neighboring L1 and the enhanced emission at L2 as a consequence of nonthermal bremsstrahlung (Figure 2 (e)). No X-ray sources were found at the locations where the ribbons formed encompassing L1 (Figures 3 (d), (e)), possibly due to RHESSI ’s limited dynamic range. However, strong nonthermal emission during the impulsive flare phase was registered at L2, in the form of a compact hard x-ray (HXR) source and a flat nonthermal ( δ ∼ 4.2) component in the corresponding spectrum (Figure 3 (d)). This shows that L2 was the main compact flare region. During this phase, a jet-like feature emerged from L2 (see inset in Figure 2 (e) and Figures 4 (d), (e)), reaching its maximum altitude at . The EM distribution of a position along the jet-like feature before, during and after its occurrence is shown in Figure 4 (f) (see position and corresponding times marked in Figure 4 (e)), showing that it


This work was supported in part by JSPS KAKENHI Grant Number 24540226.


F.C. acknowledges support from Swedish National Science Council grant 637-2013-7261. A.L. acknowledges support from the ERC Advanced Grant “ FEEDBACK ”. R.H.I., M.A.P.T. and A.A. acknowledge support from the Spanish MINECO through grants AYA2012-38491-C02-02 and AYA2015-63939-C2-1-P.


We thank the anonymous referee for many comments and suggestions that improved the manuscript. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has also made use of NASA's Astrophysics Data System (ADS), and the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research made use of Montage, funded by the National Aeronautics and Space Administration's Earth Science Technology Office, Computation Technologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology. Montage is maintained by the NASA/IPAC Infrared Science Archive. This publication makes use of data products from the WISE , which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. The UKIRT is supported by NASA and operated under an agreement among the University of Hawaii, the University of Arizona, and Lockheed Martin Advanced Technology Center; operations are enabled through the cooperation of the Joint Astronomy Centre of the Science and Technology Facilities Council of the UK. When some of the data reported here were acquired, reported here were acquired, UKIRT was operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the UK. Some of the data presented in this paper were obtained from the MAST. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non- HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is http://www.sdss.org/ . RC was supported by NASA Contract 1391817 and NASA ADAP grant 13-ADAP13-0155. MSY acknowledges support from the NASA ADAP grant NNX10AD64G.


We thank Andrey Kravtsov, Surhud More, Phil Hopkins, Frank van den Bosch, Andrew Hearin, Erik Tollerud, Erwin Lau, and Benedikt Diemer for useful discussions and/or comments on an early draft. We also thank the reviewer for useful comments. A.R.W. acknowledges the hospitality and stimulating environment of the Aspen Center for Physics, supported in part by the NSF. A.R.W. also gratefully acknowledges support from the Moore Center for Theoretical Cosmology and Physics at Caltech. This work was supported in part by NSF grants AST-1412768 1009811, NASA ATP grant NNX11AE07G, NASA Chandra grants GO213004B and TM4-15007X, the Research Corporation, and by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center.


GASP2D photometric decomposition of the mock images of the simulated galaxies is shown in Fig. 3 for the lenticular remnant resulting from the merger experiment gS0dE0o100 seen at an inclination . We listed the more relevant best-fitting structural parameters (i.e., the effective radius , Sérsic index , and axial ratio of the bulge, the scale-length and axial ratio of the disk, and the difference between the position angles of bulge and disk ) of the mock images of the simulated lenticular remnants seen at different inclinations in Table A.1 for the giant-giant galaxy mergers, Table A.2 for the giant-dwarf galaxy mergers, and Table A.3 for the giant S0-dwarf E0 galaxy mergers with different orbital parameters. Following Méndez-Abreu et al. (2017) , we adopted and as uncertainties on the axial ratio and position angle of both bulge and disk, respectively. 4.3. Bulge intrinsic shape of the simulated lenticular remnants seen at different inclinations We made use of our statistical method to retrieve from the mock images the probability distribution of the intrinsic axial ratios and of the bulges of the simulated lenticular remnants seen at different inclinations (, and ). We excluded from the analysis the edge-on configurations () because they do not allow us to constrain and due to the unknown orientation of the triaxial bulge in the disk plane. The probability distribution of and for the bulge of the simulated lenticular remnants resulting from the merger experiment gS0dE0o100 and seen at an inclination is shown as an example in Fig. 4 . The face-on configurations of the simulated lenticular remnants ( and ) provided the same result in terms of the probability distribution of and for all the simulated bulges, as expected if no observational and theoretical bias affected the adopted statistical method. Furthermore, we confirmed that the tightest constraints for are given when galaxies are seen face on, whereas remains unconstrained for these galaxies. We also found consistent probability distributions of and for the same simulated lenticular remnant seen at intermediate inclinations (, , and ). Because of this, we were confident of having correctly recovered the bulge intrinsic shape and suggested us to set a limit on the inclination of real galaxies () to robustly apply our statistical method to their bulges (see Sect. 2 ). In general, the probability distribution of and is tighter at with respect to or . Therefore, we considered this inclination as the ideal viewing angle for future analyses of the bulge intrinsic shape in real galaxies. The probability distributions of and of the bulge of the simulated lenticular remnant resulting from the merger experiment gS0dE0o100 seen at different inclinations are shown in Fig. 5 , while the remaining galaxies are in Fig. B.1 . The values of and derived for all the simulated lenticular remnants seen at different inclinations are listed in Table 1 . We realized that the accuracy of the photometric


This paper is dedicated to the memory of Janusz Kaluzny, founder and long-time leader of CASE, who prematurely passed away in March 2015. We thank an anonymous referee for very insightful remarks that significantly improved the final shape of the paper. A.A.P. acknowledges partial support from the Polish National Science Center (NCN) grant 2015/17/B/ST9/02082. M.R. was supported by the NCN grant DEC-2012/05/B/ST9/03931.


caused by the erupting filament plasma and the bright ribbon fronts on the outer border of the growing dimming region that come into the pixel’s FOV. Figure 7. Time evolution of the relative density changes, i.e., actual density divided by its pre-event value, derived from AIA DEM maps for pixels corresponding to the EIS spectra in Figure 6 . The segments in red indicate the phase of the impulsive density drop: its start is defined by the data steeply decreasing below 1, and its end is calculated from the time derivative of the curve (defined by the first positive increase). The green vertical lines indicate the time at which the corresponding pixel was covered by the EIS slit. To relate the evolution of the densities reconstructed from the AIA DEM maps (Figure 7 ) in the 4 pixels indicated in Figure 5 with the corresponding spectra and flows determined from the EIS scan (Figure 6 ), we need to relate the dynamics observed in AIA with the time at which the EIS slit covered the particular pixels. We first discuss the behavior of the top three pixels plotted in Figure 7 , and thereafter the bottom-most one, which is related to different structures as we will show. In the density evolution retrieved from the DEM maps (Figure 7 ), we can see that the sharp density drop due to the dimming occurs first in the leftmost (easternmost) pixel, and thereafter in the two neighboring pixels to the right. This is consistent with the expansion of the dimming region toward the west, i.e., from left to right (see Figure 8 and the associated movie). On the other hand, EIS was in scanning mode, and we may expect to see the largest upflow components in those pixels that have been activated as dimming pixels (i.e., the associated fields were “opened”) closest in time to when the EIS slit was mapping them. Figure 8. Evolution of the western dimming region in SDO /AIA 211 Å filtergrams. We show the same zoomed FOV as for the EIS maps in Figure 5 . The yellow boxes mark pixels for which we show the EIS spectral profiles in Figure 6 . The still figure shows the event at two time steps (annotated in each panel). The animated figure shows the event evolution from 2012 September 27, 23:50 UT to September 30, 01:16 UT. (An animation of this figure is available.) The EIS scan was from west to east, and the strongest upflow component is observed at the pixel ( x = 563′′, y = 44′′), which was covered by the slit at 00:04:16 UT. This is the pixel where we find that the total amount of upflowing plasma is larger than the stationary plasma component. The AIA density evolution shows that in this pixel, the strong drop in the density, indicative of the impulsive dimming onset, starts exactly at that time when it was mapped by the EIS slit. In the neighboring EIS pixel to the left, centered at x = 560′′, the impulsive density drop occurred during 23:58 to 00:06 UT, and the EIS slit covered it during this time range at 00:04:58 UT. This finding is consistent with EIS still showing a


in the ionization balance in photoionization equilibrium (Ferland et al. 1998b ; Ferland 2003 ). In order to provide reliable DR rate coefficients for use in plasma modeling, the storage ring DR experiments combined with the new calculations with the AUTOSTRUCTURE code and the FAC code have made huge progress in the past few decades. Storage ring DR experiments of highly charged iron have been performed with almost all of the charge states for Fe 7+ to Fe 22+ (Savin 2007a ; Schippers 2012 ). Following the first paper to describe the calculation goals and methodology (Badnell et al. 2003 ), a series of the papers have been published based on the calculations by the AUTOSTRUCTURE code, and the calculated data are collected into the recent database CHIANTI (Dere et al. 2009 , 2019 ). By comparison of these calculations and the storage ring experiments, DR rate coefficients for carbon-like and oxygen-like ions have been identified as the most urgent needs for astrophysical applications, and therefore precision measurements of the DR rate coefficients of these ions that are abundant in astrophysical plasmas to benchmark the theories are required. Calcium is an abundant element in the solar corona and also in high-temperature astrophysical plasmas. Emission lines from highly charged calcium ions play an important role for diagnostics and modeling in astrophysical plasmas as well as fusion plasmas (Del Zanna Mason 2018 ; Träbert et al. 2018 ). For a C-like Ca 14+ ion, the previous observations were mostly from solar flare and solar wind based on the lines that are largely in the EUV range. These EUV lines are usually from the transitions within the 2 s 2 2 p 2 –2s 2 2 p 2 and 2 s 2 2 p 2 –2 s 2 p 3 configurations. The spectral lines of 555 Å and 1098 Å from 2 s 2 2 p 2 –2s 2 2 p 2 transitions of Ca xv were measured and identified by the Solar Ultraviolet Measurements of Emitted Radiation spectrometer on the Solar and Heliospheric Observatory (SOHO). With the Skylab observation, the ratios of the 181.90 or 215.37 Å lines versus the resonance EUV line at 200.98 Å are excellent density diagnostics at high densities of solar flares (Dere et al. 1979 ). This Skylab observation of Ca xv emission lines to derive the electron density was later on confirmed by a new data analysis and a measurement from the TEXT tokamak plasma (Keenan et al. 1988 , 2003 ). The Ca xv 181.90/200.98 Å ratio was also investigated by the observations of solar coronal emission lines with the Extreme-Ultraviolet Imaging Spectrometer (Culhane et al. 2007 ) on Hinode spacecraft (Kosugi et al. 2007 ), as discussed by Warren et al. ( 2008 ) and Del Zanna ( 2008 ), and several Ca xv emission lines at EUV range are formed at about 4MK and the electron density measurements at these temperatures are potentially important for understanding the coronal heating mechanism. In addition to the EUV emission lines, transitions of C-like Ca xv within the 2 s 2 2 p 2 –2 s 2 2 p 3 d configurations


M.B.S. acknowledges a PhD student fellowship from IISER, Pune. We acknowledge insightful comments from the anonymous referee that has helped us improve the paper.


other hand (lower panel), the I b / I n ratio is only slightly dependent on C . 6 APPLICATION TO A REAL CASE: SN 1006 6.1 Why SN 1006 In the following, we shall apply the methods derived so far to a real case, refining them whenever required. The Balmer emission along the northwestern limb of SN 1006 represents for this an almost perfect case: observations with high spatial and/or spectral resolution, on which to test and use our models, are available; a shock velocity () high enough to limit the role of CE processes (this simplifies the models and makes them more reliable); structures near the limb that appear rather well ordered; no evidence of efficient CR acceleration, which justifies neglecting the effect of CRs in the analysis. With reference to the shock speed, Ghavamian et al. ( 2001 ) measured a spectral width of , from which for a low T e / T p ratio they derived a shock speed of . A lower shock velocity, , follows instead from the model of Morlino et al. ( 2013b ): for a more detailed and updated discussion, see Raymond et al. ( 2017 ). In the following analysis, we shall fix V sh to . With reference to CR acceleration, a low efficiency in the northwestern limb can be deduced from several results of observations: (1) the absence of non-thermal X-ray emission (Bamba et al. 2003 ; Winkler et al. 2014 ); (2) a non-detection of TeV (Acero et al. 2010 ) and GeV gamma-rays (Xing et al. 2016 ); moreover, from the analysis of Balmer emission, two additional pieces of information point towards a low CR acceleration efficiency: (3) the FWHM (full width at half-maximum) of the narrow line is (Sollerman et al. 2003 ), compatible with being produced by unperturbed ISM with T = 10 4 K, pointing toward the absence of a CR precursor able to heat the upstream plasma; and (4) an absence of evidence of any Hα precursor in front of the shock ( R + 07 ). 6.2 Some recent high-quality observations In this section we summarize some results of two relevant papers, Raymond et al. ( 2007 , hereafter R + 07 ) and Nikolić et al. ( 2013 , hereafter N + 13 ), in which Balmer emission from some areas along the northwestern limb of SN 1006 has been observed in great detail, and that contain a wealth of information, which will be used in the present work. The former paper is based on a very deep HST /ACS image covering most of the Hα emission from that limb, and the superb spatial resolution of that image (an FWHM of about 0.13 arcsec, corresponding to about , where d 2 is the distance of SN 1006 in units of ) allows one to probe scales shorter than the collisional mean-free paths, and therefore to resolve the physical structure of the filaments. The latter paper, instead, presents a map of a selected portion of the northwestern Hα limb of SN1006, roughly corresponding to ‘Regions 26–29’ (as dubbed by R + 07 ). The instrument used, VIMOS in the IFU mode on the Very Large Telescope (VLT), allows both good spatial resolution (with a pixel size 0.67 arcsec,


2000 ). The long-term X-ray pulse profile and pulsed fraction evolution, such as that presented in Section 4.1 , could be evidence for the slow, plastic deformation of the crust needed to produce slower regions of superfluid (Woods et al. 2004 ; Thompson et al. 2002 ). Alternatively, regions of the outer core could be involved as a result of the strong magnetic fields. Magnetar strength fields provide a strong pinning of vortices to flux tubes in the outer core, which leads to a rotation lag between the normal and superfluid components, which, when relaxed, would lead to a spin-down event (Kantor Gusakov 2014 ). In typical rotation-powered pulsars, the expected internal nature of these spin-up glitches is argued to be indicated by the lack of observed radiative changes. This “radiatively quiet” condition does not hold for many glitches in magnetars. Many magnetar glitches are accompanied by drastic changes in their X-ray flux and pulse profiles (e.g., Kaspi et al. 2003 ; Dib et al. 2009 ). This could indicate that external forces are at play in radiatively loud glitches, and these external forces may lead to the net spin-down events reported in several magnetars, including the new event reported here. We have compiled a list of all published confirmed and candidate net spin-down glitches in Table 3 . In this list, we present the magnitude of the change in spin frequency, . For net spin-down glitches that are due to overrecovering spin-up glitches, the of the glitch is presented. For events where there is no sign of a recovery, we show an upper limit for the time during which the spin-down had to occur. We also note that 6 of the 7 reported net spin-down glitches are accompanied by radiative changes. 8 8 This increases to 6 of 6 if we ignore the disputed event; see Table 3 . This is a much higher fraction of radiative activity than seen in spin-up events, where only 5 of 22 timing anomalies were accompanied by radiative changes (Dib Kaspi 2014 ). Table 3 Reported Net Spin-down Glitches Notes. a Accompanied by decaying spin-up glitch with Δ ν = 10.8(4) × 10 −5 Hz. b Possibly accompanied by second glitch event. c Accompanied by decaying spin-up glitch with Δ ν = 2.0(4) × 10 −7 Hz. d Accompanied by decaying spin-up glitch with Δ ν = 5.1(5) × 10 −8 Hz. e Disputed events, see the notes below. f Not seen in analysis of same data by Dib Kaspi ( 2014 ). For the net spin-down glitches in which the net spin down is measured to be due to an overrecovery, we also report the recovery fraction, Q . Recovering glitches are often classified by their recovery fraction, defined as . In younger pulsars, when recoveries are observed, Q tends to be larger than in older pulsars (Wang et al. 2000 ). For the new glitch we report here, we measure . Glitches with have only been seen twice before: once following the magnetar-like outburst of the pulsar PSR J1846−0258 that is powered by a high magnetic field rotation, which had (Livingstone et al. 2010 ), and once during


From the ∼56% remaining, 70% of them have a counterpart in either 1RXS or 2RXS catalogs. In this paper, we focus on and perform an extensive visual inspection and literature search of the other 30% of the systems. Of these objects, we describe in detail the analysis and characterization of a pilot sample of 13 candidates. We determined a spectroscopic redshift for each cluster candidate finding that most of them are local systems ( z ≲ 0.05), and are very extended (significant X-ray emission traced out to more than 10 arcmin radius). Moreover, all these objects have fluxes above the limit of the 1RXS and 2RXS catalogs. We argue that they might have been missed because of the source finder used in previous works, that is, the sliding-cell algorithm, which is not optimal for finding such low-flux, very extended objects. Our pilot sample also reveals that the surface-brightness distribution of the systems is flatter than the usual profile, that is β 2/3, and that the missed systems are located in the group regime given the determined masses from galaxy velocity dispersions or a L X − M scaling relation ( M 500 ≲ 10 14 M ⊙ ). In a subsequent paper, we will present and discuss in detail the remaining cluster candidates that we have found. Appendix A: Additional figure Fig. A.1. From left to right : RASS photon images ( left ), optical images, integrated count rates profiles, and spectroscopic redshift histograms ( right ) for the galaxy groups in our pilot sample. The optical images are from ATLAS (composed of g , r , i bands, group 2, 6, 10), SDSS (composed of g , r , i bands, group 1, 3, 7, 8, 9) or DSS ( r band, other groups). The images have a 1 × 1 deg 2 size, except for the ATLAS images, which are 0.30 × 0.30 deg 2 . The red cross shows the position of the group candidate. The magenta plus signs display the location of 1RXS or 2RXS sources. The cyan diamonds show the positions of galaxies used for redshift determination. The white asterisks show the positions of known optical galaxy groups/clusters obtained from NED. The green contours are obtained from the corresponding wavelet-filtered images, and the blue circle shows the extent value (i.e., the core radius) of the group. The integrated count rates profile is background subtracted (solid line). The dashed curves give the 1 σ statistical error of the count rates. The vertical and horizontal dot-dashed lines indicate the aperture radius and the count rates used for the primary photometric results. The dotted curve shows the β model of the cluster modelling used to extrapolate to r 500 (see Sect. 4.3.2 for further details). In the spectroscopic redshift histograms the redshift values are taken for all galaxies within a box of 1 × 1 deg 2 centered on the group position with z 0.10 from NED. The total number of galaxies used in each redshift determination is given by the second number displayed in the upper right corner of each panel (see also Table 3 ). The redshift distribution is fitted


the point sources matched between the two shortest wavelength IRAC bands. To illustrate the locations of detected sources in our catalog, we present in Figs. 5 through 9 the 3.6 m image of each galaxy with the positions of detected sources overplotted, followed by an image of the galaxy without sources shown. For each galaxy, red points denote the location of point sources not likely to be associated with the galaxy, while green points demarcate the regions inside isophotes for the Kron radius for all galaxies but NGC 2366, where the isophote described in de Vaucouleurs et al. (1991) was used. Consequently, the green points are more probably actual sources associated with each galaxy. Figures 10 through 14 show color-magnitude diagrams (CMDs) of [3.6] versus [3.6][4.5] for each galaxy. The points follow the color scheme from the images showing each galaxy, with green points representing sources within the isophotes for with each galaxy, and red points sources in the field of each galaxy. For each magnitude bin, the average uncertainty was computed for both the [3.6] magnitude and [3.6][4.5] color, and is plotted as an ellipse on the left hand side of the plot. For reference, lines are drawn to show the completeness limit of mag and the color [3.6][4.5] = 0, where most stars lie in the plot. In terms of absolute magnitudes, the [3.6] limit corresponds to detections of all objects brighter than in every galaxy. One other interesting feature in the plot was explained by Khan et al. (2015c) . For both NGC 55 and NGC 253, the two galaxies with the most sources of those studied in this work, one can see a “break” at about the same magnitude as the completeness limit. This feature is representative of the two different populations in the photometry, one physically associated with the galaxy, and another simply sharing the same region of sky. Khan et al. (2015c) calls the second population “contaminants”. This population contains foreground stars near the [3.6][4.5] color of 0, some Milky Way Halo late-type giants and unresolved background galaxies in other regions of the CMD. Red supergiants (RSGs) have been empirically shown to exist in the region of negative color space ( Bonanos et al. 2009 , 2010 ). This is due to a suppression of the flux in the 4.5 m photometry owing to molecular bands, most prominently CO ( Verhoelst et al. 2009 ). However, spectroscopic investigations into the individual stellar components of Local Group dwarf irregular galaxies have discovered 15% of new RSGs at positive colors in CMDs for those galaxies ( Britavskiy et al. 2014 , 2015 ). Fig. 10 Colormagnitude diagram for sources in the field of NGC 55. All photometrically measured sources are marked with a red dot, while those residing within the isophotal limits of the galaxy are represented by green dots. Overplotted are lines denoting the limiting magnitude of 18.0 and a [3.6][4.5] color of 0. On the left are ellipses corresponding to the average uncertainty for


by the FWHM. Given the nearly symmetric double-peaked profile of A370p_z1 and the small peak separation, we restrict our analysis to relatively small N H i and v exp values because it is well known that high column densities and shell velocities would significantly broaden the line and erase the blue peak, respectively (Verhamme, Schaerer Maselli 2006 ). We perform a quantitative comparison between the observed line profile and the models using the χ 2 statistic. We find that models minimizing the reduced χ 2 preferentially select low N H i (), static geometries ( v exp = 0), low dust content, small b values ( b = 20) and relatively broad input lines (). We show the best-fitting model in Fig. 4 which corresponds to the following parameter set: . We can derive the LyC escape fraction from the best-fitting column density = 99 per cent, where σ 912 = 6.35 × 10 −18 is the H i photoionization cross-section at the Lyman limit. While several models could match the positions of peak emission within the errors, the model is the only one to reproduce the shallow central depression of the profile which is the important signature of a low H i opacity (Fig. 4 ). Searching a finer parameter grid is beyond the scope of this paper, but we note that even when adopting a higher column density cm −2 , the escape fraction remains very high (94 per cent). Figure 4. Comparison of RASCAS models, smoothed by the resolution of X-Shooter, of Lyman α transfer through non-expanding shells of homogeneous gas with the observed Lyman α profile. The best-fitting model (blue) has which correspond to a high LyC escape fractions . We also show examples of models with (dashed orange, dash–dotted green), which would imply . 3.2 Did A370p_z1 self-ionize its local H <sc>ii</sc> bubble? The detection of the blue peak in the Lyman α indicates A370p_z1 sits in a large ionized bubble, otherwise the damping wing of even a partially neutral IGM would have absorbed it. Given its high escape fraction, we now consider whether A370p_z1 could have self-ionized its local H ii bubble. The blue wing extends to λ = 9479.2 Å, from the line centre, corresponding to a physical distance r H ii > 0.26 ± 0.04 pMpc. This estimate neglects any velocity offset between the Lyman α absorption dip from which we have derived the redshift and the systemic redshift as defined by more reliable tracers such as nebular absorption lines. These velocities are however found to be (Gazagnes et al. 2020 ), which, in the worst case, would therefore require a larger ionized bubble for the blue wing to be transmitted (≲ 0.5 pMpc). We also neglect peculiar velocities of the galaxy with respect to the ionized bubble gas which would redshift(blueshift) the Lyman α photons and decrease(increase) the bubble size needed for the blue wing to escape. We now estimate the volume that could have been reionized by A370p_z1 by redshift z = 6.803 and whether its radiation is sufficient to reduce the opacity of the surrounding gas to


is a mid-stage merger consisting of two face-on galaxies which appear to overlap and have a projected separation of ∼84 (6.3 kpc). Prominent rings/arms in each galaxy contain the bulk of the visible star clusters (Figure 19 ). The maximum A V adopted for this galaxy is 1.0 mag of visual extinction (see Figure 20 ; Miralles-Caballero et al. 2012 ). The observed and derived properties for the star clusters identified in this galaxy are given in Tables 15 and 16 , respectively. 2MASX J08370182-4954302 2MASX J08370182-4954302 is a mid-stage merger containing two nuclei separated by ∼066 (0.36 kpc). Surrounding the nuclei are multiple bright star clusters in a spiral ridge just northwest and west of the nuclei (Figure 21 ). The maximum A V adopted for this galaxy is 3.7 mag of visual extinction (see Figure 22 ; Rich et al. 2012 ). The observed and derived properties for the star clusters identified in this galaxy are given in Tables 17 and 18 , respectively. NGC 2623 Evans et al. ( 2008 ) discuss the detailed morphology of this galaxy at length. NGC 2623 is a late-stage merger with dust lanes running along its tidal tails into the nucleus. Several bright clusters are distributed throughout the bulge and in a “pie-wedge” concentration south of the nucleus (Figure 23 ). The maximum A V adopted for this galaxy is 1.9 mag of visual extinction (see Figure 24 ; Privon et al. 2013 ). The observed and derived properties for the star clusters identified in this galaxy are given in Tables 19 and 20 , respectively. UGC 04881 UGC 04881 is an early-stage merger containing two nuclei separated by ∼11′′ (9 kpc). Spiral dust lanes and strings of star clusters surround the NE nucleus. In the SW nucleus, a linear distribution of star clusters and a prominent dust lane are seen (Figure 25 ). The maximum A V adopted for this galaxy is 1.9 mag of visual extinction (see Figure 26 ; González-Martín et al. 2009 ). The observed and derived properties for the star clusters identified in this galaxy are given in Tables 21 and 22 , respectively. IC 2545 IC 2545 is a late-stage merger being viewed face on. Dust lanes and strings of star clusters extend from two unresolved nuclei separated by ∼08 (0.54 kpc) in the center of the galaxy. Multiple star clusters are also visible throughout the tidal tails (Figure 27 ). The maximum A V adopted for this galaxy is 4.0 mag of visual extinction (see Figure 28 ; van den Broek et al. 1991 ). The observed and derived properties for the star clusters identified in this galaxy are given in Tables 23 and 24 , respectively. NGC 3256 NGC 3256 is a late-stage merger containing a large number of star clusters along the inner (∼20′′, or 4 kpc) spiral structure of the nuclear region. The spiral dust lanes extending from the nucleus give this galaxy pockets of high and low extinction (Figure 29 ). The maximum A V adopted for this galaxy is 3.3 mag of visual extinction (see Figure 30 ; Rich et al. 2012 ). The observed and derived


field is almost horizontal and the field lines are concave-up. These are also locations where plasma can accumulate and eventually form a prominence (Aulanier Demoulin 1998 ; Lionello et al. 2002 ; van Ballegooijen 2004 ; Dudík et al. 2008 , 2012 ; Mackay van Ballegooijen 2009 ; Gibson 2015 ). The orange magnetic field lines extend up to z = 0.01 above the magnetic dips. If we scale our active region to the size of a typical active region hosting quiescent prominences, i.e., about 300 Mm, then orange magnetic field lines extend up to km above the magnetic dips. This is comparable with the (chromospheric) pressure scale height, therefore resulting in a reasonable proxy of quiescent prominences as the ones discussed in Filippov Den ( 2001 ). A comparison between each simulation suggests that different photospheric motions may result in prominences with similar shapes, but different morphologies. The side views of “Run C” show that the prominence is fragmented in its central part, i.e., it has a region devoid of dips at low altitudes, and it is also asymmetric: one of its ends is higher than the other one. This is not the case for “Run D1” where the distribution of the dips seems to follow a more homogeneous circular shape. One difference between the two cases is the asymmetry of the flows that formed the flux rope: the flows in “Run C” have a higher degree of asymmetry compared with “Run D1.” The details of the formation process may therefore influence the exact morphology of the flux rope and that of the dips, with these differences becoming more difficult to detect in the proximity of the flux rope’s axis ( ZAG15 ). The right-hand side panels show the evolution of the system at three different times. The configuration in the top panels is stable, the one in the middle is unstable in the low diffusion regime, but stable in the high diffusion regime, while the configuration in the bottom panel is unstable also in the high diffusion regime. The top-two horizontal yellow lines indicate the critical decay index and critical height for the onset of the instability under “high” and “low” diffusion conditions (see Table 1 ), while the bottom lines in all three models indicate the height where the decay index is n = 1.1. For “Run C” (respectively, “D1,”“D2”) at the time of the onset of the eruption in “low” diffusion regime, the apex of the axis of the MFR, i.e., the location where the stability criteria should be evaluated, has reached a height where the decay index is n = 1.4 (respectively, ) while the highest magnetic dips reach a height where n = 1.1. Interestingly, this value n = 1.1 is independent of the particular formation process in the three cases presented here. This suggests that the apparent instability criterion as evaluated at the location of the prominence is less sensitive to the flux rope formation process compared to when it is evaluated at the axis of the flux rope. Similarly to the flux ropes discussed in ZAG15 , these MFRs are


We thank Chris Hales, Anita Richards, Neal Jackson, Lee Whittaker and Joe Zuntz for useful discussions. We further thank Chris Hales, Eva Schinnerer and Vernesa Smolcic for providing us with the partially processed VLA data. MLB is an STFC Advanced/Halliday fellow. BT, IH and MLB are supported by an ERC Starting Grant (grant no. 280127 ).


components, respectively. The solid red curve is the sum of these two models. Fig. 8 Schematic representation of the spherically symmetric model for the molecular components of Arp 299A showing approximate spatial scales. Most of the molecular absorption is formed in the outer layers of the dense, most compact C component while the HO emission in the submillimeter seems to originate in the more optically thin C component. A tentative inflowing C component is included to account for the redshifted ground state OH doublet at m. Fig. 9 Reduced for the observed high-lying HO lines in our grid of models with . Fig. 10 for the OH at m with different values of applied to the best fit model of the inner component found using the HO lines. 3.3. Outer component Even in the best fit model for the inner component there is flux missing in the low-lying OH and and HO lines detected with PACS and, especially, the low-lying HO lines detected with SPIRE. To account for this missing flux in the models, a less excited and more extended component is required. To constrain the parameters of this outer component we have followed the same general procedure as for the inner component, but now for the low-lying lines normalized to the flux of the HO m line. In this process we have also taken the best fit model for the inner component into account. The observed line ratios are best reproduced with a dust opacity in the range . In this component collisional excitation becomes non-negligible for cm, but in these models the undetected HO m line goes into emission, favoring models with lower where collisional excitation is negligible. The reduced for our grid of models at the preferred dust opacity is presented in Fig. 11 . The line ratios are best reproduced with dust temperatures between and K, and HO columns between and cm. To also account for the absolute fluxes, the component should have a radius of pc. The best fit is achieved with , K, cm and pc. Fig. 11 Reduced for the observed low-lying HO lines in our grid of models with for the outer component, taking the contribution from the inner component into account. In Fig. 12 we present the reduced for the OH doublets in the outer component for different values of applied to the best fit model found for HO, now taking the OH contribution from the inner component into account. The OH m doublet which is redshifted by km s is not included here. The best fit is achieved with a column density of OH in the range to cm, corresponding to an OH/HO ratio in the range to . Applying a similar process to the upper limits of the OH doublets we get an upper limit to the OH column density of cm. Fig. 12 Reduced for the OH doublets, except for OH m , with different values of applied to the best fit model of the outer component found using the HO lines. 3.4. Extended component Most of the flux in both the HO and the OH lines is reproduced with the two components described in Sects. 3.2 and 3.3 . The ground state OH doublet at m is,


to the GCs in M20, we derive a dynamical mass enclosed within the outermost GC (i.e. 2.2 kpc) of 3.48 (± 0.84) × 10 8 M ⊙ , with the uncertainty given by the range in potential and orbital anisotropy described above. Our TME dynamical mass is larger than the GC-based half mass, but agrees within the uncertainties. Allowing for a reasonable range in γ would increase the uncertainty to around ±50 per cent; where assuming a lower (higher) value of γ would lead to a reduced (increased) TME mass. The resulting mass-to-light ratio from our TME analysis is M 1/2 /L 1/2 = 12.7 ± 3.7. Forbes et al. ( 2019 ) detected 20 GC candidates associated with NGC 5846_UDG1. If we assume the same mean GC luminosity function turnover magnitude (i.e. M V = –7.3) and width as found by Miller Lotz ( 2007 ) for dE galaxies in the Virgo cluster, then the inferred total GC population is around 45. For this count, the Burkert Forbes ( 2020 ) relation between the number of GCs and the halo mass implies a total halo mass of ∼2 × 10 11 M ⊙ . Given that there are several unknowns in translating the detection of GC candidates into an estimate of the total GC system, it is useful to note that M20 confirmed at least a dozen GCs associated with NGC 5846_UDG1. This gives a lower limit on the total halo mass with this relation, of 0.6 × 10 11 M ⊙ . Both of these total halo mass estimates far exceed the total stellar mass of 1.1 × 10 8 M ⊙ . For comparison, M33 has the same inferred total halo mass of 2 × 10 11 M ⊙ (Seigar 2011 ) but its stellar mass is some 50× larger. Thus, NGC 5846_UDG1 can be described as having a dwarf-like stellar mass with an overly massive dark matter halo. Assuming a total halo mass of 2 × 10 11 M ⊙ as described above, we can compare our half mass and TME mass with different halo mass profiles. We chose three profiles, namely the universal NFW profile (Navarro, Frenk White 1996 ), the mass profile resulting from the dwarf galaxy simulations of Di Cintio et al. ( 2014 ), which resembles a core-NFW profile, and the observationally motivated mass profile for dwarf galaxies from Burkert ( 1995 ). For details of the NFW and Di Cintio et al. profiles, see the appendix of Di Cintio et al. ( 2014 ). We show these three profiles, with the same total halo mass, compared to our half and TME masses measured at small radii in Fig. 5 . The plot shows that our mass measurements lie between the Di Cintio and Burkert mass profiles for a total halo mass as given by the total number of GCs. This suggests that NGC 5846_UDG1 may be better represented by a core mass profile as predicted in some simulations (e.g. Carleton et al. 2019 ) than a cuspy one. However, we cannot, at this stage, rule out other mass profiles and/or different total halo masses. Similar conclusions, favouring a cored mass profile, were reached by Gannon et al. ( 2020 ) for VCC 1287, and by Wasserman et al. ( 2019 ) for DF44. Figure 5. Halo mass profiles. Three halo mass profiles (Navarro et al. 1996 ,


ACKNOWLEDGEMENTS We thank Martin Shepherd, David Hawkins, Russ Keeney, and Erik Leitch for advice and assistance with the design and construction of the backend. The work at the California Institute of Technology and Owens Valley Radio Observatory was supported by National Science Foundation (NSF) awards AST-0607857, AST-1010024, AST-1212217, and AST-1616227, and by NASA award NNX15AF06G. The work at Oxford was supported by funding from STFC, the Royal Society and the University of Oxford. CD acknowledges support from an ERC Grant no. 307209. OGK acknowledges the support of a Dorothy Hodgkin Award in funding his studies while a student at Oxford, and the support of a W.M. Keck Institute for Space Studies Postdoctoral Fellowship at Caltech. This work formed part of MAS’s PhD thesis (Stevenson 2013 ).


The AGILE Mission is funded by the Italian Space Agency (ASI) with scientific and programmatic participation by the Italian Institute of Astrophysics (INAF) and the Italian Institute of Nuclear Physics (INFN). Investigation supported by the ASI grant I/089/06/2. We thank the ASI management for unfailing support during AGILE operations. We acknowledge the effort of ASI and industry personnel in operating the ASI ground station in Malindi (Kenya), and the data processing done at the ASI/SSDC in Rome: the success of AGILE scientific operations depend on the effectiveness of the data flow from Kenya to SSDC and on the data analysis and software management. We also acknowledge the use of innovative tools that are under development within the United Nations “Open Universe” initiative.


The authors wish to thank the anonymous referee for a constructive report, which has improved the paper. We also wish to thank the Aspen Center for Physics for hosting two summer workshops, “The Obscured Universe: Dust and Gas in Distant Starburst Galaxies” in summer 2013 and “New Frontiers in Far-infrared and Sub-millimeter Astronomy” in summer 2016, whose stimulating conversations led to this work. The Aspen Center for Physics is supported by National Science Foundation grant PHY-1066293. C.M.C. thanks the National Science Foundation for support through grant AST-1714528. Additionally, C.M.C. and J.A.Z. thank the University of Texas at Austin College of Natural Sciences for support. J.S. thanks the McDonald Observatory at the University of Texas at Austin for support through a Smith Fellowship. E.dC. gratefully acknowledges the Australian Research Council for funding support as the recipient of a Future Fellowship (FT150100079). J.A.H. acknowledges support of the VIDI research programme with project number 639.042.611, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). S.L.F. acknowledges support from an NSF AAG award AST-1518183. This work relies in part on the COSMOS survey data set. COSMOS is based on observations from the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, which is operated by AURA Inc., under NASA contract NAS 5-26555. It is also based on data collected at: the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan; the XMM-Newton and Herschel Space Observatory , ESA science missions with instruments and contributions directly funded by ESA Member States and NASA; the European Southern Observatory, Chile; Kitt Peak National Observatory, Cerro Tololo Inter-American Observatory, and the National Optical Astronomy Observatory, which are operated by AURA, Inc., under cooperative agreement with the NSF; the National Radio Astronomy Observatory, which is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc.; the Canada–France–Hawaii Telescope operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique de France and the University of Hawaii; and the James Clerk Maxwell Telescope, which is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan, Academia Sinica Institute of Astronomy and Astrophysics, the Korea Astronomy and Space Science Institute, the Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and Facility Instruments, budgeted from the Ministry of Finance (MOF) of China and administrated by the Chinese Academy of Sciences (CAS). Many of the data sets this paper and analysis rely on were obtained on the summit of Maunakea on the island of Hawai’i. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.


ACKNOWLEDGEMENTS MB and FH acknowledge partial financial support from the INFN TEONGRAV specific initiative. MB acknowledges the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support. This work was supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904. AS is supported by a University Research Fellowship of the Royal Society. This work has made use of the Horizon Cluster, hosted by the Institut d’Astrophysique de Paris. We thank Stephane Rouberol for running smoothly this cluster for us.


measure the target power spectrum , and ψ’s PDF, from the SB simulation. In using both from a single realization, the results may depend on peculiarities in that realization. The limitation is due to the lack of an available software to produce accurate fitting functions for the power spectrum and PDF of ψ. This introduced dependence is not a problem, as long as one has the aim of isolating the contribution from the PDF and the correlation of ψ to the specific realization examined, to assess how much information they convey with respect to the exact result. To emulate the final ψ field, we set the phases of the pre-translation Gaussian field to the phases of the initial density field of the realization. We do not smooth the measured ψ field, because both the power spectrum and the PDF are measured at the grid level, and one can check that the ψ field variance is already consistent with its power spectrum variance integrated up to the Nyquist frequency. In Fig. 12 , we can see that the PDF of the ψ realization from ITAM matches the one of the N -body simulation (much better than muscle , and particularly 2LPT), as it should by the virtue of the translation transform equation ( 2 ). Likewise, the Lagrangian-space ψ power spectrum of ITAM matches that of the N -body simulation better than the other prescriptions, as shown in Fig. 13 . The two standard simulation schemes have a mismatch, with 2LPT diverging the most at small scales and muscle departing the most on large scales. This confirms the detailed analysis of the power spectra of Lagrangian schemes conducted by Chan ( 2014 ), where it is also found that these techniques come short of matching the power spectrum measured in N -body simulations. Another benchmark of the goodness of a fast simulation scheme is the similarity between the resulting matter density and the N -body matter density. This is generally quantified by means of the cross-correlation spectrum, which for two fields A and B is Figure 12. PDFs of ψ for various schemes. The peak at ψ ∼ −3 is associated with collapsed halo particles. In muscle , halo particles give a Dirac-delta function at ψ = −3 (see equation 43 ). Figure 13. Transfer function of ψ, defined as the ratio of the power spectrum of ψ generated from the fast simulation scheme over the one measured from the N -body realization. The cross-correlations are plotted in the top panel of Fig. 14 . As we can see, the muscle implementation still performs better than ITAM, which in turn is better than 2LPT, at least on smaller scales. Examining this further, 2LPT is almost a local approach, in the sense that it considers the information of each voxel and its nearest neighbours (equation 37 ). ITAM by construction is non-local, as it involves the two-point correlation function. muscle is also non-local, but the non-locality is coming from a smoothing procedure on various scales centred around each voxel (equation 43 ). Considering that the one-point mapping of ITAM is very


host halo mass, and the green valley transition time-scale, τ Green , for both disc and elliptical RS satellite galaxies. We select all = 0 RS satellite galaxies that, at the time the galaxy became green ( t Green ), were also satellites (corresponding to of the satellite sample), and calculate the ratio between the satellite galaxy’s stellar mass and the host halo mass, log 10 M * / M 200 , at t Green . Throughout this section host halo mass refers to the halo mass 1 of the central galaxy within which the satellite galaxies are embedded. Fig. 7 shows the green valley transition time-scale, τ Green , as a function of log 10 M * / M 200 at t Green for RS satellite galaxies that are discs (blue dots) and ellipticals (red dots) at = 0. It can be seen that for both disc and elliptical satellite galaxies, τ Green tends to increase with M * / M 200 . This is highlighted by the best-fitting linear regression of the τ Green −log 10 M * / M 200 relation for elliptical (red dashed line) and disc (blue solid line) satellite galaxies. We find that τ Green correlates more strongly with M 200 than with M * , so that satellite galaxies embedded in cluster-sized haloes transition the green valley faster than their same-stellar mass satellite counterparts residing in lower mass haloes. Figure 7. Green valley transition time-scale, τ Green , of satellite galaxies as a function of the ratio between stellar mass and host halo mass, log 10 M * / M 200 , at the time when the galaxy left the blue cloud, t Green . The data points with error bars show the median τ Green values in bins of log 10 M * / M 200 for galaxies that have been satellites since t Green and at = 0 have κ co ≥ 0.4 (shown as blue dots) or κ co 0.4 (shown as red dots). The error bars correspond to the 25th−75th percentiles of the τ Green distribution in each mass bin. The lines correspond to the best-fitting linear regressions. The red satellite galaxies tend to have quenched more rapidly if they are elliptical and if their stellar mass is smaller relative to the total mass of their host halo. This relation is mostly driven by a dependence on halo mass, rather than on stellar mass. Fig. 7 shows that satellite galaxies embedded in clusters (log 10 M * / M 200 − 4 with haloes more massive than ) tend to cross the green valley over shorter time-scales than satellite galaxies residing in lower mass haloes, regardless of the galaxy’s morphological type. However, the spread in the τ Green distribution is large. In the case of central galaxies, although most of them reside in cluster-sized haloes, we do not find any correlation between τ Green and halo mass or M * / M 200 . This is in agreement with previous studies that suggest that environmental quenching is mostly associated with satellite galaxies rather than centrals (e.g. Peng et al. 2010 ; Darvish et al. 2016 ). Central galaxies show properties that are more associated with the galaxies’ stellar mass (i.e. on average more massive galaxies are


ACKNOWLEDGEMENTS The authors thank V. Bosch-Ramon and an anonymous referee for their useful comments. DK is supported by the Japan Society for the Promotion of Science KAKENHI Grant Numbers JP18H03722, JP24105007 and JP16H02170. MA is supported by the RIKEN Junior Research Associate Programme.


methane possesses strong and distinctive absorption signatures. An annotated list of these attempts can be found in Zahnle et al. (2011) , Knutsen et al. (2021) , and in tabulated form in Pla-Garcia et al. (2019) . Hitchcock Lovelock (1967) pointed out that the composition of the Martian atmosphere could provide clues as to the possible presence of life on the planet. Therefore, methane has long been seen as the most accessible testimony of potential signs of life ( Atreya et al. 2003 , 2007 ) or of a remnant geophysical activity on Mars. Indeed, the presence of methane has such a strong connotation of Mars being potentially an active or even a life-hosting planet, contrary to all other chemical or geophysical evidence suggesting the opposite, that it has attracted the attention of a wide community extending far beyond those interested in Martian atmospheric composition. However, the succession of failed and successful attempts to detect methane has also created a blurred picture, and a comprehensive understanding of the existence and origin of methane on Mars has remained out of reach. Several attempts that were initially presented as positive detections were later either retracted or questioned. This is the case of the Mariner 7 detection (as explained above) and the Canadian-France-Hawaii Telescope (CFHT) observations of Krasnopolsky et al. (2004) – whose spectrum quality raised the suspicion of Zahnle et al. (2011) , who also questioned the seasonal variations of methane found by Geminale et al. (2008) based on a survey carried out with the Planetary Fourier Spectrometer (PFS). Finally, the methane mapping of Fonti Marzo (2010) using the Thermal Emission Spectrometer (TES) was later retracted by the same authors in Fonti et al. (2015) . Of all the detections made between 2003 and 2010, only a few survived the scrutiny of the community. The PFS 2003 detection reported in Formisano et al. (2004) , while bearing evident shortcomings regarding its resolving power and sensitivity, has resisted criticism and remains to this date the first non-rebutted detection of methane in the Martian atmosphere. In addition, Mumma et al. (2009) published the first ground-based evidence of methane collected the same year as the PFS, the scientific implications of which were bolstered by the revelation of puzzling spatial and seasonal variations attributed to a sudden release from the Nili Fossae region. Several studies have explored the chemical ramifications ( Lefèvre Forget 2009 ) as well as the spectroscopic consistency of these detections ( Zahnle et al. 2011 ). These works converged toward the idea that our current understanding of the physics and chemistry governing the Martian atmosphere is incompatible with the observed variable behaviour of methane, which should rely on an extraordinary, yet undefined mechanism. Together with the fact that detections had long lain on the fringe of the detection capability of all instruments used at that time, the


we take the average over contributions from all pairs of cells (and the third associated cell that is uniquely determined by the triangular condition) that fall into that bin. We do not look at the bispectrum bins unless at least some combination of the wave numbers in that bin satisfies a triangular condition. To avoid counting equivalent triplets multiple times, we impose a condition k 1 > k 2 > k 3 . Fig. A1 shows the average nonlinear and linear power spectra of matter particles from Quijote simulations. We rescaled the linear power spectrum to have a matching amplitude at small wave numbers. The two are reasonably close for large-scale modes (up to k ∼ 0.1 h Mpc −1 after which the additional power in the nonlinear power spectrum becomes visible. The partial erasure of the BAO signal is also visible by eye. Fig. A2 shows the average bispectrum of matter particles from Quijote simulations at z = 0 as a function of the longest wave number. Features in the shape of this three-dimensional function provide additional opportunities for the standard ruler test. Figure A1. Non-linear power spectrum at z = 0 (blue) and a linear power spectrum (black) of Quijote matter particles. The linear power spectrum is measured from particles at z = 99 and rescaled to have a matching amplitude on large scales. Figure A2. Bispectrum of Quijote matter particles at z = 0 as a function of the largest wave number k 1 . Multiple points at the same abscissa show all bispectrum bins that share that value of k 1 but have different values of k 2 and k 3 . We multiply the bispectrum by a factor of k 1 k 2 k 3 . This factor is proportional to the number of fundamental triangles in the bin and separates different points for better visibility. To estimate the covariance of these measurements, , by computing the sample covariance over 8,000 Quijote simulations, X i l are the power spectrum and bispectrum measurements, where the first index goes over the bins and the second index goes over simulations. N sim is the number of simulations. Fig. A3 shows the correlation matrix of these measurements defined as We ordered the measurements so that the first 50 elements are binned P ( k ) up to k max = 0.5 h Mpc −1 . They are followed by binned B ( k 1 , k 2 , k 3 ) arranged in such a way that the bispectrum bin with smaller k 1 goes first (smaller k 2 when k 1 are equal, and smaller k 3 when both k 1 and k 2 are equal). The first 150 bins arranged in this way accommodate all bispectra up to k 1, max = 0.1 h Mpc −1 . The correlation between power spectra at high wave numbers is clearly visible in this plot. The bispectrum measurements are also correlated between themselves and the power spectra but to a lesser extent. Figure A3. Correlation coefficient of the power spectrum (first 50 bins, goes up to k max = 0.5 h Mpc −1 ) and the bispectrum (last 150 bins, goes up to k 1, max = 0.1 h Mpc −1 ) measurements. Colourmap is adjusted in such a way that red, blue, and white cells


We thank Andrei Mesinger for useful comments on the manuscript. AP gratefully acknowledges the use of facilities at ETH, Zürich during the early phases of this work. The research of HP is supported by the Shyama Prasad Mukherjee Fellowship of CSIR, India.