in the bottom panel we show distributions as a function of R and φ . Figure 2. The – plane coloured by mean φ for the simulation described in Section 2.2.1 . The bins have size Δ = 0.05 kpc, and Δ = 3 km s −1 . The bins with fewer than 10 particles are ignored. Fig. 2 shows the – plane coloured by mean φ . Stars at large radii in the – plane, i.e. stars with high E preferrentially come from stars in population 1, which have lower φ , as seen in the bottom panel of Fig. 1 . The initial DF shown in Fig. 2 is perturbed by displacing it by in . The particles are then evolved using a standard leap-frog algorithm for and the results are shown in Fig. 3 . The number density shows the spiral pattern typical of phase wrapping. Stars orbit the – plane in a clockwise sense and since those at large E have a lower vertical epicycle frequency, the DF quickly develops a trailing spiral. Also shown is the – plane coloured by mean R and φ . Here, the pattern is essentially featureless, apart from the imprint of the number density distribution: with our separable potential the in-plane and vertical epicyclic motions completely decouple. Figure 3. Number density and mean radial and azimuthal velocities in the – plane for the toy model considered in Section 2.2.1 (toy model with additively separable potential). Top panel shows the number density. The bins in the middle and lower panels are coloured by R and φ , respectively. The bin sizes and particle number threshold are the same as in Fig. 2 . 2.2.2 Kinematic spirals in a coupled potential Next we consider the addition of a R – term to the toy model potential R s is a constant radius characteristic of the distribution [if a Taylor expansion of the effective potential is performed for each L population, R s is replaced by R c ( L )]. For illustrative purposes, we choose and . These values are consistent with the cubic terms that arise in realistic Milky Way potentials. The main effect of the new term is to introduce a linear dependence in R to the vertical epicyclic frequency. In particular the difference between the vertical frequency for a population 1 star with and a population 2 star with is . As before we sample an initial distribution (equation 3 ) with 500k particles, perturb it and integrate the perturbed distribution for . The results are shown in Fig. 4 . Not surprisingly, the spirals that appear in the – number density distribution are virtually identical to the ones that appear in our separable model. However, the R and φ spirals are qualitatively different. In particular, there are gradients across the spiral arms and along their inner and outer edges. The addition of the ( R − R s ) 2 term implies a coupling between the in-plane and vertical epicyclic motions. Figure 4. Same as Fig. 3 but for the model considered in Section 2.2.2 (toy model with R – coupling included in the potential). 2.3 Realistic Milky Way potential We conclude this section on test-particle simulations by considering a realistic

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O


Right panels present the distributions of the position angles of the magnetic field orientations as a function of distance from the protostars. If multiple protostars are present in one core, the distances between the magnetic field orientations and the protostar closest to the Stokes I intensity peak are calculated. In the middle and right panels, black dashed lines denote the mean magnetic field orientations computed from the averaged Stokes Q and U emission in the area of the dense cores, and red solid and dashed–dotted (if there are multiple outflows) lines show the position angles of the outflows. The names of the protostars are labeled in the order from western to eastern sources above the middle panel in each row. Figure 12.16. L1157 Same as Figures 3 and 4, but left panels only present the magnetic orientations (small orange segments) detected in the 40′′ × 40′′ regions (dashed open squares) around the sample protostars as well as the mean orientations of the magnetic field (long thick orange segments) in the individual dense cores/clumps. Left panels show the Stokes I maps (gray scale) of the dense cores, where the sample protostars are embedded, and green contours delineate the area of the dense cores identified by Clumpf ind . Yellow dots show the locations of the protostars, and blue and red arrows present the orientations of their blue- and redshifted outflows. Black horizontal segments denote the spatial scale of 0.02 pc (or ∼4000 au). Middle panels present the distributions of the position angles of the individual magnetic field orientations detected with the JCMT POL-2 observations. Right panels present the distributions of the position angles of the magnetic field orientations as a function of distance from the protostars. If multiple protostars are present in one core, the distances between the magnetic field orientations and the protostar closest to the Stokes I intensity peak are calculated. In the middle and right panels, black dashed lines denote the mean magnetic field orientations computed from the averaged Stokes Q and U emission in the area of the dense cores, and red solid and dashed–dotted (if there are multiple outflows) lines show the position angles of the outflows. The names of the protostars are labeled in the order from western to eastern sources above the middle panel in each row. Appendix D Dependences of Misalignment on Source Properties Figure 13 compares the measured misalignment angles with the sizes, total fluxes, mean intensities, and mean polarization percentages of the dense cores and the distances to the sources. The total flux and mean intensity are expected to be proportional to the total mass and mean column density of a dense core, respectively. These comparisons show that the measured misalignments do not depend on the mass and density of the dense cores. We note that the magnetic field structures in the nearby sources can be better resolved than in the distant sources. The comparison also

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


task of solving equation ( 69 ). The problem then assumes the form of an inhomogeneous Helmholtz equation and the solution becomes particularly simple when expanding the radial dependence in modified spherical Bessel functions that fulfil the boundary conditions. Like in the classical gravity problem, Legendre polynomials remain the representation of choice for the latitudinal dependence. These basis functions allow a very efficient semi-analytical solution to the problem. Each of the calculations presented here required only a few seconds of run time on a standard 4-core notebook. There has been a discussion whether the DSG term could be neglected when inverting high precision gravity observations of Jupiter and Saturn for zonal flow properties. Our new formulation allows us to quantify the relative impact of the DSG for each gravity harmonic, practically independent of the considered zonal flow or background state. A special case arises for degree l = 1. For the background density with polytropic index unity, the l = 1 solution comprises the case where the two buoyancy contributions in the TGWE cancel. This corresponds to the homogeneous solution of the Helmholtz equation. Zonal flow and gravity perturbation then decouple, and it becomes impossible to draw any conclusion on the zonal flows from the respective gravity contribution. Kong et al. ( 2017 ) seem to have noticed the related problems without realizing their origin. However, this is of little practical interest since the origin is generally chosen to coincide with the centre of gravity so that l = 1 contributions vanish. Table 5 compares the relative DSG impact with the precision of most recent gravity measurement for Jupiter and Saturn. The even harmonics J 2 – J 6 are not listed since they are dominated by the rotational deformation of the planet. For Jupiter’s J 3 , J 5 , and J 7 coefficients, the relative impact of DSG is comparable to the error and should thus be taken into account when inverting gravity harmonics for zonal flow properties. This agrees with the results and conclusion by Kong et al. ( 2017 ). The error in the higher order observed moments may decrease as the Juno mission progresses. For Saturn, J 3 , J 5 , and J 10 seem precise enough to warrant including DSG effects. The estimates of Kong et al. ( 2017 ) and Galanti et al. ( 2019 ) for the relative impact of the DSG are compatible with our results. Including the DSG term generally increases the amplitude of the gravity coefficients. Table 5. Relative error of gravity harmonics for Jupiter (Iess et al. 2018 ) (second column) and Saturn (Iess et al. 2019 ) (third column). The fourth column shows S l , the relative impact of the DSG for radial profile f 3 also listed in Table 4. In addition to the TWE, Cao Stevenson ( 2017 ) also solve the Euler equation for a polytrope of index unity and a geostrophic equatorial zonal jet. The respective results not only provide the perturbed density structure but also the

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


analysis of photochemical models of Titan (Dobrijevic et al. 2008 ). It was found that the chemistry of C 2 H 2 and C 2 H 4 could be bistable although the underlying chemical pathways, involving neutral–neutral processes, differ from those studied here. The physical environment of Titan is significantly different from that of dense interstellar clouds; nevertheless, as H 2 is much less abundant on Titan, this work suggests that Titan’s ion–molecule chemistry could also lead to bistable solutions. Finally, the interstellar chemistry of carbon is much more complex than studied here, where we have only considered hydrocarbons up to C 2 . Autocatalytic processes in addition to those described above are likely present. For example, ethene, when viewed as a dimer (CH 2 ) 2 takes part in autocatalysis for CH 2 in reaction sequences similar to those for acetylene. Although present in our network, we found this played no role in producing bistable solutions. Analysis of larger, more extensive, carbon chemistry networks may lead to the discovery of more deeply embedded, and so less apparent, autocatalytic cycles. If one identifies the higher hydrocarbons as dimers then diacetylene (C 4 H 2 ) and the carbon tetramer (C 4 ) could act as respective intermediates for C 2 H and C 2 autocatalysis. The possibility for autocatalysis involving larger organic molecules remains to be explored; anion chemistry (e.g., Cordiner Charnley 2012 ; McElroy et al. 2013 ) may become as important as the cationic reactions considered here. 5. Conclusions We have found new bistable solutions in reduced chemical sets comparable to simplistic gas-phase chemical models of dense molecular clouds. We have identified several autocatalytic cycles in the ion–molecule chemistry of the major volatile elements: O, N, and C. In nitrogen chemistry there are two possible autocatalytic processes. An autocatalytic cycle driven by the reaction of He + with N 2 produces bistability. We find that this model has bistable solutions at high densities ( n H > 10 5 cm −3 ), which are present at all values of the elemental N abundance considered. The bistable solutions in the reduced model of the oxygen chemistry occur at low densities but are similarly present at all values of elemental O depletion, including none. Thus, bistable solutions are a fundamental property of interstellar chemistry; introduction of other elements, carbon in particular, acts to suppress these solutions by inhibiting autocatalysis through reactions with O 2 and OH. The carbon chemistry exhibits bistable solutions that appear around 10 2 cm −3 , or less, but only when an elemental depletion equivalent to that of the diffuse interstellar medium is adopted. We have identified three possible autocatalytic processes. We find that two autocatalyses occur in the acetylene chemistry, with CH, rather than atomic C, being the autocatalyst. These involve and C 2 H 2 in processes analogous to those found in the oxygen autocatalysis; their

O O O O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


and carbon-normal metal-poor stars (red crosses and blue dots, respectively). Because our model is computed for , we restrict our observed sample to stars with iron abundance within a factor of five from the model value, . This selection leaves us with CEMP, CEMP-, and CEMP- stars, that is more than of our entire carbon-rich sample. The theoretical and observed abundances differ in many aspects. 1. Observed carbon-rich and carbon-normal stars appear as two distinct and clearly separated populations. We compute the linear regression of the observed europium and barium abundances weighted with the observational errors, and we determine the following equations for the ratio of barium to europium in carbon-normal and CEMP- stars, respectively, represented in Fig. 2 with blue-dashed and magenta-solid lines. If we compute the linear fit for CEMP- stars with and CEMP- stars separately we find almost identical results as in Eq. ( 2 ). Equations ( 1 ) and ( 2 ) show that the empirical relation between [Eu/Fe] and [Ba/Fe] has almost the same slope in the two classes of stars, and CEMP- stars have [Ba/Fe] on average dex higher than carbon-normal stars at the same [Eu/Fe]. The relations described by Eqs. ( 1 ) and ( 2 ) are likely consequences of the barium-to-europium ratios produced in the - and in the -process, respectively. In contrast, in our simulation there is not a clear separation between the two groups (colour distributions in Fig. 2 ). Because - and -enrichments are independent, the initial [Eu/Fe] and [Ba/Fe] follow approximately the relation described by Eq. ( 1 ), whereas the final abundances depend on the amount of material transferred from the AGB primary stars onto their secondary companions. Higher accreted masses cause stronger abundance enhancements of carbon and barium. Because AGB stars generally do not produce high europium abundances, [Eu/Fe] is essentially determined by its initial value, except for stars with initial [Eu/Fe] less than about . Consequently, for each [Eu/Fe], which is given by the initial -enrichment, we find stars with all barium abundances up to two dex higher than predicted by a pure -process, and the correlation between [Eu/Fe] and [Ba/Fe] described by Eq. ( 2 ) is completely washed out. Lugaro et al. (2009 , 2012) and Abate et al. (2015c) criticise this scenario of independent - and -process contributions with analogous arguments. Fig. 2 [Ba/Fe] vs. [Eu/Fe] in carbon-normal and carbon-rich metal-poor stars (blue dots and red crosses, respectively). Stars enriched in -elements occupy the region above the horizontal and diagonal solid black lines, corresponding to zones A and B of the plot. According to our adopted definitions, CEMP- stars are in zone B. In zone C there are stars with , , and . CEMP- stars and carbon-normal stars with and are in zone D. The dashed-blue and solid-magenta lines are linear fits of the carbon-normal stars and CEMP- stars, respectively. The red and blue distributions represent the

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


Zanazzi Lai 2018 ). If the initial inclination angle of the disc is small, the result of the evolution is that the disc tends to be coplanar with the binary star. If the initial inclination angle of the disc is large (i.e. >40°; Martin Lubow 2017 ) and the eccentricity of the binary star is non-negligible, then the disc will evolve into a polar orbit. The presence of high-inclination discs indicates that high-inclination CBPs might exist. However, the above conclusions are based on the evolution of the isolated CBD systems themselves. Besides, they can only evolve to high-inclination orbits in specific initial configurations, such as a high initial inclination and a large eccentricity for the binary. Other mechanisms are still needed to explain why the CBD has a large inclination initially. It is worth noting that the semimajor axes of binaries with high-inclination discs found already are relatively large (several tens of au), while the discs around close binaries are nearly coplanar (Kennedy et al. 2012 ; Czekala et al. 2019 ). For the Kepler CBPs, the semimajor axes of their host binaries are small, about 0.1–0.2 au. One possibility is that the discs around well-spaced binaries are susceptible to the surrounding environment, such as th close encounters with fly-by stars. It is generally believed that most stars, and therefore most planetary systems, were born in clusters or associations (Clarke, Bonnell Hillenbrand 2000 ; Lada Lada 2003 ; Pfalzner 2013 ; Hao, Kouwenhoven Spurzem 2013 ; Cai et al. 2017 ). Some open clusters later dissolved, forming the current field stars. A notable example is that our own Solar system may have formed in an open cluster. The chemical composition of objects in the Solar system, along with the orbital elements of Sedna, suggest that the Sun formed in a cluster with roughly 10 3 –10 4 stars (Adams 2010 ). Other work even suggests that the Sun was born in a massive cluster (10 4 –10 5 stars) because a cluster of several thousand stars is much too small to produce a nearby massive supernova (Dukes Krumholz 2012 ). Malmberg, Davies Heggie ( 2011 ) considered the effect of a close encounter between stars on the planetary systems in a single star system. They found that the fly-by of a star is one of the possible mechanisms for explaining the generally high eccentricity of exoplanes (mainly discovered by radial velocity methods). It is interesting to ask how the cluster environment affects CBPs. In fact, the first CBP discovered, PSR B1620–26, is located in a globular cluster (Sigurdsson et al. 2003 ). According to the theory of star formation, it is impossible to form a close binary directly (Moe Kratter 2018 ). One of the explanations for this is that they are formed by interaction with other stars (Martin, Mazeh Fabrycky 2015 ; Hamers, Perets Portegies Zwart 2016 ; Moe Kratter 2018 ), which means that they were in a star-dense environment previously. A cluster provides such a possibility. If a CBP was born in an

B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O B-CelestialObject O O O B-CelestialObject O O O O O O O O O O O B-Citation I-Citation O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O


luminosity an excellent proxy of the star formation rate. In Arp 220, the L ′ CS(7–6) -to- L ′ CS(4–3) ratio is equal to 0.76. Using the calibration of Zhang et al. ( 2014 ), this leads to an L IR estimate of 1.6 × 10 12 L ⊙ , consistent with the L IR determined by Rangwala et al. ( 2011 ) using Herschel /Spectral and Photometric Imaging Receiver (SPIRE) data (i.e. 1.8 × 10 12 L ⊙ ). The relation linking L CS(4–3) and L IR can now be calibrated with SEPIA Band 5. H 2 O – submillimetre water line emission is considered to be an important gas cooling mechanism in warm molecular clouds and the H 2 O line (ν rest = 183.3 GHz) only requires moderate densities ( n (H 2 ) = 10 4 –10 5 cm −3 ) and low temperatures ( T ∼ 40K) to be excited (Cernicharo et al. 1994 ). We confirm the detection of H 2 O(3 3,1 –2 2,0 ) emission in Arp 220 (Fig. 1 , middle), with a peak flux of 0.14 Jy and a width at half-power of 332±31 km s −1 . These values are consistent with the peak flux (0.17 Jy) and width at half-power (310 km s −1 ) obtained with IRAM data by C06 (see section 4.1 for further analysis). CH 3 OH – extragalactic methanol maser emission has been detected towards many objects (Henkel et al. 1987 ; Sjouwerman et al. 2010 ). In particular, millimetre methanol transitions are known to be sensitive (but sometimes degenerate) tracers of temperature and density. Millimetre CH 3 OH emission has been detected in Arp 220 by Martín et al. ( 2011 ), particularly the 5–4 group (241.8 GHz). Salter et al. ( 2008 ) also report an absorption feature from the 6.7 GHz methanol line. Our SEPIA observations give us an access to the 193.5 GHz methanol (4–3) group (Fig. 1 , bottom). The peak flux is 68 mJy. The water and methanol lines have similar widths (equal to 333 km s −1 for CH 3 OH). We note that this is at the lower end of the line width range (280–590 km s −1 ) estimated from thermal molecules in Arp 220 by G09 . HC 3 N – molecules from the cyanopolyyne family like HC 3 N are standard tracers of the dense gas in Galactic star-forming regions but are also detected in extragalactic environments (Aladro et al. 2011 ; Lindberg et al. 2011 ) in many transitions (from 9–8 to 28–27). Arp 220 is one of the very few ‘HC 3 N-luminous’ galaxies (i.e. that the HC 3 N(10–9) line intensity is at least 15 per cent of the HCN(1–0) line intensity; Aalto et al. 2002 ; Martín et al. 2011 ). The HC 3 N(20–19) line is narrow and detected at 16 per cent of the HNC(2–1) luminosity. Its profile looks symmetric but its peak is shifted. This is consistent with the asymmetry observed in the 10–9 transition by Aalto et al. ( 2002 ) and could suggest that most of the emission emerges from the eastern nucleus. C 34 S – we observe a strong and broad C 34 S(4–3) line at 192.8 GHz. The line is unfortunately at the edge of the velocity range observed. An HC 3 N vibrationally excited line resides in a similar frequency range (192.6 GHz; see Fig. 1 , bottom)). Vibrationally excited emission of this

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


of gravitational wave after supernova explosion. In this study, since we consider only one progenitor model and one EOS, it is difficult to say how this relation is independent of the models, i.e. equation ( 7 ) cannot be used to analyse data from future gravitational wave observations. Even so, this relation seems to be independent of the models at least in the early phase, as shown in Fig. A2 in Appendix A . Anyway, additional models should be considered in the future. Figure 6. In the left-hand panel, frequencies of the f - and g 1 -modes are shown as a function of the square root of the PNS average density, where the thick solid and the thick dotted lines are the fitting formula given by equations ( 7 ) and ( 8 ). In the right-hand panel, frequencies of the f - and g 1 -modes are shown as a function of the surface gravity of the PNS, where the thick-solid line denote the fitting formula given by equation ( 9 ). The relation similar to equation ( 7 ) has already been proposed, as a function of x in Sotani Sumiyoshi ( 2019 ): and as a function of in the unit of M ⊙ km −2 in Torres-Forné et al. ( 2019b ): although in Torres-Forné et al. ( 2019b ) the ramp up signal is identified as g 2 -mode in their classification. Equation ( 8 ) is derived for the f -mode frequency after the avoided crossing with the g 1 -mode with the PNS models provided by the 1D numerical simulations, which are eventually collapsed into black hole. In the left-hand panel of Fig. 6 , we also plot the thick dotted line given by equation ( 8 ). From this figure, we find that the f -mode frequency is well identified by equation ( 8 ) up to f ≃ 1.4 kHz, which corresponds to T pb ≲ 0.6 s from Fig. 5 (or Fig. 2 ), but it deviates for T pb ≳ 0.6 sec. This may come from that the PNS models for deriving equation ( 8 ) become more compact in the later phase due to the massive progenitor model. On the other hand, in order to see the correspondence between the eigenmodes calculated in this study and equation ( 9 ), we show the f - and g 1 -mode frequencies as a function of in the right-hand panel of Fig. 6 . This fitting formula seems to correspond to the f -mode rather than the g 1 -mode in the very early phase and also in the late phase, at least for comparing to our results. Finally, we make a comment with respect to the previous results about the gravitational wave signals, especially obtained via the general relativistic 3D simulation (Kuroda et al. 2016 ), which is done with a 15 M ⊙ progenitor model (Woosley Weaver 1995 ) and SFHx EOS (Steiner et al. 2013 ). Unlike most of the other simulation results, several modes of gravitational wave signals have been found in this simulation. The characteristic gravitational wave frequencies extracted by the time-frequency analysis (Kawahara et al. 2018 ) are shown in the left-hand panel of Fig. 7 , where the signal A corresponds to the ramp up signals. Nevertheless, considering the results with the PNS asteroseismology shown in Fig. 2

O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation O O B-Model I-Model B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-ObservationalTechniques O O O O


HO (under laboratory conditions), which is consistent with the astronomical observation of solid CO in water-rich environments. Neill et al. ( 2014 ) performed a sensitive line survey of Sgr B2(N), and based on the HOCO species, they estimated a CO/CO ratio in the colder external envelope of between 0.010.1, implying a CO gas-phase abundance of 10–10, consistent with the finding of Minh et al. ( 1988 , 1991 ). Their lower limit to the CO gas-phase abundance of 10 relative to H indicates that the CO abundance in the gas is clearly enhanced, while most of the water is still frozen out in ices in the Sgr B2 envelope. Table 1 Properties of the observed HOCO and DOCO lines ( K). Notes. The upper limits have been computed using the rms around the line. The errors in the fluxes include the calibration and statistical uncertainties from the Gaussian fits of the lines. As part of the IRAM-30 m Large Program ASAI 1 (Lefloch et al., in prep.), we carried out a highly sensitive, unbiased spectral survey of the molecular emission of the L1544 prestellar core with high spectral resolution. This source is a prototypical starless core in the Taurus molecular cloud complex ( pc) on the verge of the gravitational collapse (Caselli et al. 2002 , and references within). It is characterised by a central high density ( cm) and a low temperature (7 K). We reported the detection of many oxygen bearing complex organic molecules produced through the release in the gas phase of methanol and ethene through non-thermal desorption processes (Vastel et al. 2014 , hereafter Paper I). The sensitivity of these IRAM observations also led to the detection of the hyperfine structure of CHCN (Vastel et al. 2015a ). In the present study we report on the detection of the HOCO ion in the L1544 prestellar core, as well as a tentative detection of its deuterated form, DOCO. 2. Observations and results The observations were performed at the IRAM-30 m towards L1544 () using the broad-band receiver EMIR at 3 mm, connected to an FTS spectrometer in its 50 kHz resolution mode. The beam of the observations is 29 and 23 at 85 and 106 GHz, respectively. Line intensities are expressed in units of main-beam brightness temperature (see Paper I for more details). The frequencies and other spectroscopic parameters of the HOCO transitions have been retrieved from the JPL database 2 from laboratory measurements by Bogey et al. ( 1988 ). Two lines of HOCO (the 4–3 and 5–4 at 85.531 and 106.913 GHz, respectively) lie in the frequency range covered by the ASAI survey for upper level energies lower than 30 K and were detected. Their spectra are shown in Fig. 1 . Table 1 reports the spectroscopic parameters as well as the properties of the two detected lines, obtained by Gaussian fitting. For the deuterated form of HOCO, DOCO, we used the JPL database with the spectroscopic parameters from Bogey et al. ( 1986 , 1988 ). Three transitions lie in the frequency range for upper level energies lower than 30 K.

O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O B-Observatory B-Telescope B-Grant I-Grant I-Grant I-Grant B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation O O O O B-Observatory O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-Observatory B-Telescope O B-CelestialObject B-TextGarbage O O O O B-Instrument O B-Wavelength I-Wavelength O O O B-Instrument I-Instrument O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Database I-Database O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O B-Survey I-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O B-Database I-Database O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O


( 2015 ) galaxies are compatible with our simulations] then the slope of the reddening law required to explain the data is much steeper than the SMC extinction law. For example, the position of the stack of undetected galaxies would require n −20, much steeper than any attenuation, or even extinction law, ever observed. Given this discrepancy, it is worth considering the potential systematics that could be affecting the Capak et al. ( 2015 ) and Bouwens et al. ( 2016 ) results. One possibility is an underestimation of the dust temperature, which is taken to be 35 K in both cases. It has been suggested that dust temperature may increase with redshift due to an increase in the intensity of the radiation field heating the dust (e.g. Béthermin et al. 2015 ), 1 and typically an upper limit of 45 K has been assumed (Schaerer et al. 2015 ). In this case, L IR will be underestimated by a factor ≈2 and the resulting IRX underestimated by ≈0.3 dex. The magnitude of this shift is illustrated by the arrow in the bottom right-hand corner of Fig. 16 . It would bring the Capak et al. ( 2015 ) and Bouwens et al. ( 2016 ) data into better agreement with our model, however it is not enough to fully account for the discrepancy. There are other potential systematic issues affecting the determination of the UV continuum slopes, and estimated upper limits on L IR , which we will discuss in detail in a future work (McLure et al. in preparation). 6.2.5 SMC-like attenuation at lower redshifts? In Fig. 16 , we have not included data at lower redshifts ( z ∼ 2–3) which also support steeper, SMC-like, attenuation laws. In particular, Reddy et al. ( 2010 ) find that galaxies with young mass-weighted ages and typically lower masses (age ≲ 100 Myr; M/M ⊙ 10 10 ) have a smaller IRX, at a given β, relative to older galaxies at the same redshift. Finally, similar conclusions have also been reached using small numbers of individual lensed galaxies (e.g. Baker et al. 2001 ; Siana et al. 2009 ). In Reddy et al. ( 2010 ), these young galaxies represented ≈13 per cent of their full star-forming galaxy sample, and in our simulated sample ≈6 per cent of galaxies have ages 100 Myr. However, since our method involves comparing to the observed LF, it can only be applied to the full galaxy population complete down to a given UV magnitude, and cannot be applied to populations split by age. Therefore, we cannot rule out the possibility that certain populations (e.g. the youngest galaxies), or any given individual galaxy, will follow a steeper attenuation curve. We also note that, since the absolute attenuations decrease towards fainter, lower mass galaxies, the uncertainties on the form of the attenuation law will correspondingly increase, as the difference in absolute attenuation between curves becomes more difficult to distinguish. Therefore, our conclusions are most robust at high masses. Nevertheless, our current analysis suggests that, on average, a Calzetti-like attenuation curve

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


unexpected to condense from a gas cooling from high temperature (Grossman et al. 2012 ). The Fe content of these minerals is an increasing function of oxidation state. Even in an oxygen-rich condensing gas, these minerals are expected to be very Fe-poor, with fayalitic contents (=Fe/(Fe+Mg)) of no more than 4% (Grossman et al. 2012 ). Before the return of samples from comet Wild 2, it was thought that oxidation of crystalline silicates might have occurred within the parent bodies of meteorites, during aqueous alteration or thermal metamorphism. However, olivines returned from comet Wild 2 show a very broad and nearly flat distribution in fayalitic content from 0 to ∼40%, and a range of fayalitic content from 0% to 97% (Frank et al. 2014 ), so the oxidation appears to have taken place in a nebular environment, through an unknown mechanism. We note that all three collections of cometary materials have the potential to have suffered oxidation state change, or the appearance of oxidation state change through sample bias, between release from the parent body and analysis in the laboratory. Samples from comet Wild 2 were captured in the coma at 6.1 km s −1 , which may have led to reduction of Fe through smelting: the arrow in Figure 3 indicates that the original material may have been less reduced than observed in laboratory analyses. Cometary materials captured in the stratosphere may suffer from biases against metal-rich, high-density particles, because of their shorter residence time. This is consistent with the metal-poor composition of individual small CP-IDPs, in contrast to the relatively metal-rich composition of particles from giant cluster IDPs. However, none of these capture or sample bias effects go in a direction to reconcile cometary oxidation state with that of the ISM. 5.2. Summary A comparison of Fe L-edge absorption spectrum of continuum X-rays from Cygnus X-1 with extinction spectra derived from synchrotron X-ray microprobe analyses of mineral standards leads to the following conclusions: 1. Observations are consistent with the hypothesis that Fe is sequestered principally in metals, at least in this line of sight. The data are marginally consistent with Fe evenly divided between sulfide and metal, a composition that would be expected if Fe and S reside entirely in metal and sulfide, e.g., in a high-temperature condensation sequence. Less than 38.5% (2 σ ) of Fe resides in amorphous silicates. 2. The observations are inconsistent with the hypothesis of Ishii et al. ( 2018b ) that interstellar silicates consist of GEMS or their subcomponents as found in chondritic-porous IDPs. This conclusion is robust even if some Fe is oxidized between residence in the ISM and observation in the laboratory. 3. Fine-grained cometary material, as represented by FGM returned by the Stardust mission from comet 81P/Wild 2, with a WD01 size distribution predicts an extinction spectrum that is inconsistent with Chandra observations. 4. Our conclusions

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


) within 3 arcsec of the new variables and (ii) manually check OGLE-II DIA light curves of nearby sources to make sure none of them show brightness trends mirroring the new variables. 3.6 Digitized photographic plates (66 Oph) Photographic images of the sky obtained in late 19th and 20th centuries contain a wealth of information about historical positions (e.g. Laycock et al. 2010 ; Vicente et al. 2010 ; Berezhnoi 2013 ; Robert et al. 2014 ) and brightness (e.g. Kolesnikova et al. 2008 ; Tang et al. 2013 ; Sokolovsky et al. 2014c ) of celestial objects. The efficient use of this information requires it to be converted to a digital form using purpose-built digitizing machines (Simcoe et al. 2006 ; De Cuyper et al. 2012 ) or a commercially available flatbed scanners capable of working with transparent materials (Vicente, Abad Garzón 2007 ; Simcoe 2009 ; Tuvikene et al. 2014 ). We use an Epson Expression 11000XL flatbed scanner operating at 2400 dpi resolution (1.4 arcsec pixel −1 , 16 bits per pixel colour depth) to digitize a 1.26 × 1.26 deg 2 area centred at α J2000 = 17:57:44.7 δ J2000 = +04:59:54 (66 Oph field; Kolesnikova et al. 2010 ) on 227 photographic plates obtained in 1976–1995 with the 40 cm astrograph. The digitized images were processed with vast following the procedure described by Sokolovsky et al. ( 2014b ). APASS B -band photometry of 1600 UCAC4 stars in the magnitude range B = 10–16 is used to calibrate the instrumental magnitude scale using the relation between aperture photographic and photoelectric magnitudes proposed by Bacher, Kimeswenger Teutsch ( 2005 ). We identify 23 variable stars including five not previously known (Table 4 , Fig. 2 ) by means of period search and visual inspection of light curves standing out in the magnitude versus σ w plot. 3.7 Light-curve filtering Often raw photometric data have to be pre-processed before computing the variability indices discussed in Section 2 . This may include (i) removing outliers from a light curve (possibly by applying iterative σ-clipping or median filtering); (ii) removing systematic effects from a set of light curves by applying local zero-point corrections (e.g. Nascimbeni et al. 2014 ) and/or the sysrem algorithm, decorrelating each light curve with external parameters such as airmass, seeing, object position on a CCD, detector temperature (e.g. Pál 2009 ; Bakos et al. 2010 ; López-Morales et al. 2010 ; Hartman et al. 2011 ; Burton et al. 2012 ; Guterman, Mazeh Faigler 2015 ; Baade et al. 2016 ) or de-trending the light curves if one is interested only in fast variability (e.g. Kovács, Bakos Noyes 2005 ; Weingrill 2015 ). A smaller-than-expected number of detections is an indirect indication of many photometry problems including the object being close to an image edge, a cosmetic defect, a bright star, a detection or saturation limit. Objects systematically affected by any of these problems can be removed from the analysis by discarding light curves having less than a

O O O O O O O O O O O O B-Survey I-Survey O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion B-CelestialObject I-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Instrument I-Instrument I-Instrument O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Survey B-Wavelength I-Wavelength O O O B-Dataset O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


stronger than the CF– L bol correlation. What causes this discrepancy? There are some possibilities, such as the difference in (i) redshift, (ii) sample selection (i.e., our sample is limited to type 1 quasars, while the sample in Ricci et al. 2017 is not), and (iii) definition of CF (i.e., is employed in this work, while a fraction of X-ray-obscured sources is employed in Ricci et al. 2017 ). It is hard to quantify, however, the influence of the difference in the sample selection and definition of the CF on the correlation strength, and it is beyond the scope of this work. We thus argue for the possibility that the difference in redshift could cause the discrepancy. Ricci et al. ( 2017 ) targeted nearby AGNs with a median redshift of z ∼ 0.037, whereas we focus on AGNs with much higher redshift up to z = 0.7. Figure 12 shows the redshift dependence of the CF– λ Edd relation. We find that the CF of AGNs with z 0.12 (which is similar to that of the sample in Ricci et al. 2017 ) tends to strongly depend on λ Edd , with a correlation coefficient of r = − 0.49 ± 0.32, while CF– λ Edd for AGNs at z > 0.2 shows weaker correlations than that at z 0.12. This indicates that the aforementioned discrepancy can be explained by the difference in redshift between the samples in Ricci et al. ( 2017 ) and this work. We also remind that our correlation analysis would be affected by the Malmquist bias, which could make the correlation strength weaker, as discussed in Section 4.2.2 . Figure 12. CF as a function of λ Edd . The blue, green, orange, and gray shaded regions represent quasars with z 0.12, 0.12 z 0.2, and z > 0.2 and the entire sample, respectively. This result also might suggest that the CF decreases with increasing redshift for at least z 0.7, a trend that is consistent with what is reported in Toba et al. ( 2014 ), although there is large uncertainty. On the other hand, the fraction of obscured X-ray AGNs increases with redshift for the same luminosity (e.g., Ueda et al. 2014 ). This discrepancy might be caused by the difference in sample selection and definition of CF. 4.6.2. BH Mass Dependence Recently, Kawakatu et al. ( 2020 ) constructed a model of a nuclear starburst disk supported by the turbulent pressure from Type II supernovae and investigated how the CF depends on AGN properties by taking account of anisotropic radiation pressure from AGNs. They found that the CF strongly depends on λ Edd for AGNs with M BH 10 8 M ⊙ , while the dependence of the CF on λ Edd is much weaker for AGNs with M BH > 10 9 −10 10 M ⊙ . Although they did not incorporate the AGN polar dust outflow, it is worth checking the dependence of the CF– λ Edd correlation on M BH . Figure 13 shows how the difference in M BH would affect the correlation strength of CF– λ Edd . The resultant r of CF– λ Edd for quasars with , , and are −0.56 ± 0.07, −0.50 ± 0.05, and −0.52 ± 0.22, respectively. This result indicates that the CF of quasars with 10 8 M ⊙ strongly depends on λ Edd .

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


magnetic field modulus of HD 50169: 5.6 kG, derived from consideration of the differential broadening of spectral lines having Zeeman patterns of different widths. We retrieved the Julian Date of this observation from Adelman (1973) , as the same spectrum was used for detailed abundance analysis. The representative point of Preston’s field determination appears as a salmon-coloured filled square in Fig. 2 . The uncertainty of this determination is difficult to assess. The most meaningful comparison between the field values derived by Preston (1971) through application of his Eq. (3) and the ⟨ B ⟩ determinations of Mathys (2017) , based on the splitting of the Fe II λ 6149.2 doublet, is for those stars for which the mean field modulus only shows small, almost insignificant variations, such as HD 2453, HD 137949, and HD 192678. For these stars, the estimates of Preston (1971) and the measurements of Mathys (2017) are in excellent agreement, within ∼100 G of each other. The comparison is less straightforward when the field modulus shows definite variability with phase, but at least for some stars (e.g. HD 12288 or HD 188041), the agreement between the values that were derived from line broadening by Preston (1971) and those that Mathys (2017) obtained from consideration of the wavelength separation of split line components, seems somewhat poorer. The error bar on ⟨ B ⟩ that we adopted for the data point from Preston (1971) in Fig. 2 , ±200 G, is only tentative, and it may well underestimate the actual uncertainty of the field values reported in this reference. Even with this “optimistic” accuracy, however, the magnetic field strength estimated nearly 50 years ago by Preston (1971) is consistent with the value of the rotation period that we derived above. With P rot = 10 600 d, HD 50169 becomes the longest-period Ap star for which magnetic field measurements have been obtained over more than a full rotation cycle. At 29.04 yr, its period is considerably longer than the next longest one, 21.75 yr for HD 9996. Furthermore, with respect to the latter, HD 50169 presents the advantage of showing resolved magnetically split lines throughout its whole rotation cycle, so that its mean magnetic field modulus can be exactly determined even around the phase of its minimum and, as a consequence, the structure of the field can be better constrained. 4. Magnetic field characterisation The observed variations of both the mean longitudinal magnetic field and the mean magnetic field modulus of HD 50169 can be well represented by the superposition of a cosine wave and of its first harmonic. The best least-squares fit solutions for P rot = 10 600 d are where the field strengths are expressed in Gauss, φ = (HJD − HJD 0 )/ P rot (mod 1) and the adopted value of HJD 0 = 2441600.0 corresponds to a maximum of the mean magnetic field modulus; ν is the number of degrees of freedom, and χ 2 / ν , the reduced χ 2 of the fit. The fitted curves are shown in Figs. 1 and 2 ; the

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(e.g. Dalal et al. 2008 ; Desjacques Sheth 2010 ). Combination of the peaks approach plus N -body simulations led to further insights, including that haloes in dense regions that do not correspond to high peaks in the initial conditions accrete more slowly than haloes that do correspond to high peaks (Ludlow Porciani 2011 ). Faster supercomputers and better codes have led to improved N -body simulations, in which the haloes and subhaloes that host galaxies, groups and clusters are resolved. Such simulations have permitted more detailed analyses of the correlations of halo properties with each other and with the halo environments measured in various ways. One of the first papers to do this was Lemson Kauffmann ( 1999 ), which used Ω 0 = 1 τCDM and Ω 0 = 0.3 ΛCDM simulations and concluded that only the halo mass distribution varies as a function of environment, with more high-mass haloes in denser environments, in reasonable agreement with the analytic calculation by Mo White ( 1996 ) based on extended Press–Schechter theory and the spherical top-hat model. While subsequent N -body calculations confirmed this density effect on the halo mass function, they implied that environmental density also affects halo major merging rates (Gottlöber, Klypin Kravtsov 2001 ) and other halo properties. Bullock et al. ( 2001 ) found that haloes in dense environments tend to have higher concentrations than isolated haloes. Sheth Tormen ( 2004 ), using the same simulations as Lemson Kauffmann ( 1999 ), found evidence that low-mass haloes form somewhat earlier in dense environments. Avila-Reese et al. ( 1999 ) found that dark matter haloes that are isolated or in intermediate-density environments have outer density profiles r −β with β ≈ 2.8 ± 0.5 while haloes in denser regions have a wider range of β up to ∼5. Avila-Reese et al. ( 2005 ), again using ΛCDM simulations that were very small by modern standards, nevertheless found that haloes in dense regions had lower spin parameters and higher concentrations and were less prolate than haloes in lower density environments, with the differences arising mostly at low redshifts z 1 from phenomena such as tidal stripping in dense environments. In this paper, we confirm and expand on these results. Maulbetsch et al. ( 2007 ) used a 512 3 particle ΛCDM simulation in a (50 h −1 Mpc) 3 volume to study the mass accretion history of galaxy-mass haloes in different environments, and found that haloes of the same final mass accreted their mass earlier in denser environments, and also accreted a significantly higher fraction of their mass in major mergers. They suggested that this could help to explain the galaxy density–morphology relation and that early-type galaxies are more common in dense environments. They also found that ∼80 per cent of haloes in higher density environments are not accreting (i.e. have ), while this fraction is only ∼20 per cent in low-density environments. They defined the environmental density on a

O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula B-Model O B-Formula I-Formula I-Formula I-Formula B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Model I-Model I-Model O O B-Model I-Model I-Model O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


indices of the thermal plasma/PUIs. The collisional thermal plasma stress tensor and PUI stress tensor in a 1D form may be expressed as , where η g is the thermal collisional viscosity coefficient and η I is the collisionless PUI viscosity coefficient associated with magnetic fluctuations (Jokipii Williams 1992 ; Zank et al. 2014 ). A 1D steady-state model in which all physical quantities depend on the x Cartesian coordinate is used and U = ( U x , 0, 0) and B = (0, 0, B z ). The 1D form of Equations ( 10 )–( 12 ) in normalized form is where is the thermal gas Mach number of the flow, , L ≡ K g / U 1 , is distance normalized to the diffusion length scale, and is the Schmidt number of the thermal plasma/PUI. The inverse compression ratio is y = U x / U x 1 and , , and . The inverse Alfvén Mach number squared of the flow far upstream is . Based on the thermal collisional term scales derived above from the expressions in Zank ( 2014 ), the dominant collisional terms in a collisional magnetized plasma can be estimated. The small mass of electrons compared to protons means they do not contribute significantly to either the thermal heat conduction or thermal viscosity. Proton–electron collisions are negligible since the mean free path is very large compared to that of proton–proton collisions. The dominant collisional terms derive from proton–proton collisions. The diffusion coefficient associated with proton–proton collisions, K pp , upstream of the observed VLISM shock is about 10 15 m 2 s −1 , and the thermal viscosity is 2.5 × 10 −8 kg m −1 s −1 , giving a thermal Schmidt number of about 0.25. Energetic PUIs outside the heliosphere are generated by secondary charge exchange, and their number density is very small in the VLISM ( n p ≈ 5 × 10 −5 cm −3 ; Zirnstein et al. 2014 ), therefore they contribute a very small pressure compared to the thermal plasma ( P I1 / P g 1 ≈ 0.05). A characteristic value for the PUI heat conduction, K I , in the VLISM is ∼10 14 m 2 s −1 . The pitch-angle scattering timescale in the VLISM is taken to be , giving 5.17 × 10 2 s, which gives a value for the PUI viscosity, η I , as ∼3.57 × 10 −13 kg m −1 s −1 , making the PUI Schmidt number about 10 −3 . The PUI Schmidt number is much smaller than that of the thermal gas (), thus PUIs and the dissipation associated with them cannot mediate the VLISM shock, as shown by Mostafavi et al. ( 2017a , 2017b ). Consequently, we can neglect PUI dissipation terms in favor of the corresponding thermal gas terms. Unlike the HTS, PUIs will therefore behave adiabatically, is constant through a broad VLISM shock, and the thermal gas will be heated preferentially. Recall that PUIs were predicted (Zank et al. 1996 ) and observed (Richardson et al. 2008 ) to be heated preferentially at the HTS, with the thermal plasma component remaining relatively unheated. The physical processes determining the structure of the HTS and the VLISM shocks are therefore different. We use the 1D form of the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Blandford 1990 ), and M 8 is the black hole mass in units of 10 8 M ⊙ . A was defined by Daly et al. ( 2018 ), A = α /( α + b ), a and b are the coefficients of the fundamental plane of black hole activity, where A = 0.43 (Merloni et al. 2003 ). L bol is the bolometric luminosity (Netzer 1990 ), L bol = 10 L BLR , and L Edd is the Eddington luminosity, L Edd = 1.3 × 10 38 ( M BH / M ⊙ ). The spin of the black hole can be estimated using the following formula (Daly 2019 ),where L j is the beam power, and g i = 0.1 and g bol = 1 are used (Daly 2019 ). The black hole spins obtained using this method can be compared with those obtained using other methods, such as the X-ray reflection method (Reynolds 2014 ). The spin values of GX 339-4 are 0.94 ± 0.02 (Miller et al. 2009 ) and (Garcĺa et al. 2009 ) using the X-ray reflection method, and the spin obtained using the method of Daly ( 2019 ) is 0.92 ± 0.06. The spin obtained using the method of Daly is consistent with those obtained with the X-ray reflection method. Because of the limitation of the observations, it is very difficult to directly measure the spin of large samples using the X-ray reflection method. Therefore, we use the method of Daly to measure the spin for our samples. The accretion disk magnetic field strength values of ∼10 4 G are also reported using other methods (Mikhailov et al. 2015 ). Thus, the accretion disk magnetic field strengths obtained here are similar to the values obtained using other methods. We use Equations ( 2 ) and ( 3 ) to estimate the magnetic field strength of the accretion disk and the spin of black hole, respectively. The values of the magnetic field strength and the spin of black hole are listed in columns (9) and (10) of Table 1 . 3.3. The Intrinsic <italic>γ</italic>-Ray Luminosity We estimate the intrinsic γ -ray luminosity using the formula (Nemmen et al. 2012 )where is the observational γ -ray luminosity. f b is the beaming factor, and f b is estimated as 1-cos(1/Γ), where Γ is the bulk Lorentz factor. The bulk Lorentz factor comes from Ghisellini et al. ( 2014 ). is the observed γ -ray luminosity from the catalog of LAT (Abdo et al. 2010 ; Ackermann et al. 2011 ). The values of and f b are listed in columns (7) and (11) of Table 1 , respectively. The values of the radiation jet power and the jet kinetic power come from Ghisellini et al. ( 2014 ). Ghisellini et al. ( 2014 ) used a simple one-zone leptonic model to get the radiation jet power and the jet kinetic power. They are listed in columns (12) and (13) of Table 1 , respectively. 4. Results and Discussions 4.1. The Distribution of Physical Parameters The density distributions of the spin of SMBHs and the magnetic field strength of the accretion disks are shown in Figure 1 . The average spins of SMBHs for FSRQs and BL Lacs are 〈 j FSRQs 〉 = 0.50 ± 0.23 and 〈 j BL Lacs 〉 = 0.44 ± 0.27, respectively. The average values of the magnetic field strength of the accretion disk for FSRQs and BL Lacs are and ,

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found by Colombo et al. ( 2014 ). Their results, furthermore, impose a cautionary warning for the study of the mass–size relationship of our own Galaxy: if we are unable to distinguish whether an observed Galactic cloud is in one or another environment, we might be producing a mass–size relationship that arises from clouds in different physical environments. It is convenient to stress that, although Colombo et al. ( 2014 ) made an effort to avoid line blending in their cloud decomposition, the situation can be quite complex, since even at the smallest scales where coherent structures can be found (i.e. cores, fibres), substantial superposition along the line of sight can occur (Ballesteros-Paredes Mac Low 2002 ; Gammie et al. 2003 ; Shetty et al. 2010 ; Zamora-Avilés et al. 2017 ; Clarke et al. 2018 ). The situation for clouds in external galaxies might deserve a full further study, since the resolution is substantially different in external Galaxies (typically, 30–50 pc; see e.g.Sun et al. 2018 ; Utomo et al. 2018 ), with only one study (Faesi et al. 2018 ) achieving resolutions of ∼10 pc. A first puzzling challenge is to understand, for instance, the origin of mass–size relation with q 2 in the interarm region of M51, since q = 2 is the limit when clouds are defined via thresholds in column density. 4.3 Fractal dimension? Different studies have interpreted the exponent of the power-law mass–size relation as representative of the fractal dimension (e.g. Elmegreen Falgarone 1996 ; Roman-Duval et al. 2010 ; Beattie, Federrath Klessen 2019 ). This approach has two hypotheses: First, the mass can be a proxy for the area. Secondly, the linear size is related also to the area in a quadratic way, since in the observational studies the size is typically considered to be proportional to the square root of the area of the cloud. In principle, the first hypothesis seems plausible: observed molecular clouds exhibit rapidly decaying N -PDFs (Kainulainen et al. 2009 ). The same is found for MCs in numerical simulations (e.g. Vázquez-Semadeni García 2001 ; Tassis et al. 2010 ; Ballesteros-Paredes et al. 2011b ; Kritsuk, Norman Wagner 2011 ). This means that the filling factor of the dense structures is quite small, which in turn, implies that the mass should be proportional to the area. However, for a collection of Galactic clouds, the superposition effects we discussed are not compatible with invoking mass as a proxy for the area. As shown in the present work, the same fact that the exponent in the mass–size relation is larger than 2 means that the mass is not proportional to the area. In addition, invoking the square root of the area as a proxy for the linear size does not tell us what will be the linear size–area correlation, which is precisely what is intended to be measured with the fractal dimension. A more appropriate estimation of the fractal dimension will be that of comparing the actual area of an ensemble of clouds with an independent estimation

O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


of a gas parcel being essentially independent of its collapse history – explaining why the asymptotic T − Δ relation holds so tightly despite the varied histories of cosmological fluid elements. (The tightness of this relation is absolutely striking in calculations that do not allow for shocking but allow for varied collapse histories. See the Zeldovich approximation calculations in Fig. 1 .) We further show that Compton cooling off of the CMB has no effect on the asymptotic slope of T (Δ) and that recombination cooling is just weak enough not to break the relation. This paper is organized as follows. We first enumerate the relevant heating and cooling processes in Section 2 . Next, in Section 3 , we derive an analytic expression for the asymptotic T − Δ relation in the limit of only photoheating and adiabatic heating/cooling. These are the processes considered in previous attempts to understand this relation analytically. Section 4 generalizes our solutions to include all relevant cooling processes. We briefly consider three applications of our results in Section 5 : (1) how quickly temperature fluctuations are erased, (2) the importance of shock heating, and (3) the thermal evolution of collapsing gas clouds. Lastly, Section 6 contrasts our results with previous attempts to understand the intergalactic T − Δ relation. Throughout we use the standard phenomenological parametrization for the temperature of the low-density IGM, where Δ is the baryonic density in units of its cosmic mean, T 0 is the gas kinetic temperature at Δ = 1, and γ − 1 is a power-law index. (Sometimes γ is called the ‘equation of state.’) When necessary, we assume a flat ΛCDM Universe with h = 0.7, Ω m = 0.3, Ω b = 0.045, and Y He = 0.25. Following Hui Gnedin ( 1997 ), we use the Zeldovich approximation to follow the density (necessary for computing the temperature) of fluid elements for certain calculations. Our Zeldovich approximation calculations draw matter elements randomly from the probability distribution of collapse eigenvalues, assuming that the gas traces the dark matter. This probability distribution only depends on the variance of the density field, which we set to 1.2. This choice is equal to the variance of a real space top hat filter containing a Lagrangian mass of 10 10 M ⊙ in our reference cosmology at z = 3 (and approximately the Jeans mass near the cosmic mean density at this redshift). 2 In addition, we also use a 20 Mpc, 512 3 gas and 512 3 dark matter particle simulation that was run with the gadget -3 smooth particle hydrodynamic code (Springel 2005 ). Reionization occurs instantaneously in this simulation at z = 9, heating all gas to 20 000 K, and the subsequent heating and cooling takes rates applicable for primordial gas if only the H i and He i are kept ionized with an ultraviolet background that is flat in specific intensity [erg s −1 Hz −1 cm −2 sr −1 ]. 2 RELEVANT HEATING AND COOLING PROCESSES The evolution of the temperature of a Lagrangian

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


O O O O O O O O B-Telescope O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-Survey I-Survey B-URL O O O O O O O O B-Telescope O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O B-Survey O B-Telescope O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O B-Wavelength I-Wavelength O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O B-Wavelength O B-Wavelength O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O B-Wavelength O B-Wavelength O O O O B-Wavelength O B-Wavelength O O O O B-Wavelength O B-Wavelength O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. The similar line of C iv λ 1550.774 appears offset to higher velocity. Both features are formed mainly in the wind of the Be star, but shell variations are present on the high velocity side of the core. Figure 3.12. Al ii λ1670. The top panel shows the individual spectra of Al ii λ 1670.7867 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.13. Al iii λ1854. The top panel shows the individual spectra of Al iii λ 1854.7164 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.14. Al iii λ1862. The top panel shows the individual spectra of Al iii λ 1862.7895 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). Thespectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. Figure 3.15. Fe iii λ1895. The top panel shows the individual spectra of Fe iii λ 1895.46 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in upper right. The lower panel shows the same interpolated in orbital phase and portrayed as a gray scale image (black corresponding to deepest absorption and white to strongest continuum flux). The actual phases of observation are indicated by arrows on the right hand side. The lines of Fe iii λ λ 1895.46, 1896.814 also appear in the figure. Figure 3.16. Mg ii λ2796. The top panel shows the individual spectra of Mg ii λ 2796.352 plotted as a function of radial velocity and orbital phase (such that the continuum is aligned with the phase of observation). The spectral flux depth relative to the continuum is indicated by the scale bar in

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