be the H ii regions misidentified as galaxies. Thus, follow-up may be needed to confirm this parallel sequence. Figure 13. Zoom-in regions of the M r / and M * / diagrams. The dashed–dotted lines indicate the separation between the fainter paralleled sequence and the main population of galaxies. 127 common galaxies from both panels are color coded by redshift, while the distinct ones are shown by the gray color. Four outliers on the left side of each panel are actually three galaxies, for which two outliers with are two H ii regions in the same galaxy. Alternatively, the distribution of the individual field for each panel in Figure 12 also has been shown in Figure 14 . Figure 14. Same as Figure 12 but for individual fields. Each galaxy has been color coded by redshift. The histograms of each unique color index are shown on the right side of the panel. 6. Summary We present here a spectroscopic redshift catalog from the LAMOST Key Project, LaCoSSPAr, which is designed to observe all sources (Galactic and extragalactic) by using repeating observations with a limiting magnitude of in two FOVs in the SGC. The project is mainly focusing on the completeness of LEGAS in the SGC, the deficiencies of source selection methods and the basic performance parameters of the LAMOST telescope. Two chosen FOVs represent the relatively lower (Field A) and higher (Field B) density of galaxies. Targets in the FOV are mainly consisted of stars, galaxies, quasars, and u -band variables, as well as complementary H ii regions. We conduct the observation from 2012 September to 2014 January. Field A contains 2519 galaxies and 13,930 stars in which 2447 () and 13,278 () of them have been observed (97%) with 5 B () and 12 F plates (), respectively. Field B contains 3104 galaxies and 11,381 stars in which 2995 () and 10,224 () of them have been observed with 6 B and 5 F plates, respectively. The raw data have been reduced with LAMOST 2D and 1D pipelines. An additional post-processing has been applied to the LAMOST 1D spectrum to remove the majority of remaining sky background residuals. More than 10,000 spectra have been visually inspected to measure redshift by using combinations of different emission/absorption features with a uncertainty of . All galaxies have been divided into two categories as emission and absorption line galaxies, depending on whether there are obvious emission lines appeared or not. In total, 1528 redshifts ( of observed galaxies; including 623 absorption and 905 emission line galaxies) have been measured in Field A with a median value of z = 0.125 and a range from 0.001 to 1.598 (99% at ). For Field B, 1570 redshifts ( of observed galaxies; including 569 absorption and 1001 emission line galaxies) have been measured with a median value of z = 0.0952 and a range from 0.006 to 3.148 (99% at ). The results show that it is possible to derive redshift from low S/N galaxies with our post-processing and visual inspection, since the S/N r is less than

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Grant I-Grant I-Grant B-Survey 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 B-Survey O O O O O O O O O O O O O O O O B-Telescope I-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 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 B-TextGarbage O O B-TextGarbage O O O O O O O O O B-TextGarbage O O O O B-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 O O O O B-Software I-Software I-Software I-Software I-Software O O O O O 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 O O O O O O O O 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 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


of the total CGM column density within ±600 km s −1 of the galaxy systemic velocity was found at ∣ v ∣ 100 km s −1 (Tumlinson et al. 2013 ; Werk et al. 2014 ). Kinematic analysis of warm gas signatures in the quasar absorption-line survey COS-GAL also shows that a substantial portion of the CGM is likely hidden at ∣ v ∣ 100 km s −1 (Zheng et al. 2019 ). In the recent CGM 2 survey, the majority of the H i column density detected in absorption around z 0.48 galaxies lies within ±250 km s −1 of the galaxy systemic velocity, despite a velocity window of 1000 km s −1 (Wilde et al. 2020 ). Additionally, Martin et al. ( 2019 ) find that the velocity of low-ionization gas around nearby galaxies is tied to the rotation of the disk out to radii of ∼70 kpc. If the Milky Way is similar to other star-forming galaxies at low redshift, these observations further reinforce the evidence suggesting that a significant fraction of Galactic baryons are yet to be detected in the low-velocity Galactic CGM. Simulations of Milky Way analogs also broadly agree that a significant fraction of circumgalactic baryons are moving at low velocities. The FOGGIE simulations, which focus on resolving the CGM around galaxies with extreme spatial and mass resolution, show with mock observations that the strongest absorption is confined to within 200 km s −1 of the systemic velocity (Peeples et al. 2019 ; Zheng et al. 2020 ). Other synthetic observations of a Milky Way−mass galaxy generated from the high-resolution cosmological simulation Enzo also predict that ∼ 65% of the Milky Way's CGM mass at Galactic latitudes ∣ b ∣ > 20° is hidden at low to intermediate velocities (∣ v ∣ 100 km s −1 ) for all gas phases and that the warm−hot component (10 5 K T 10 6 K) is primarily moving at ∣ v ∣ 150 km s −1 (Zheng et al. 2015 ; Joung et al. 2012 ). The C iv doublet is a useful probe of warm ionized gas in galaxy halos because of its strong oscillator strength (Savage Wakker 2009 ) and high covering fraction in the CGM of external galaxies (Keeney et al. 2013 ; Liang Chen 2014 ). In low-redshift galaxies that span a wide range of stellar mass (9.5 log ( M * / M ⊙ ) 11), estimates of log N C IV (cm −2 ) range from 13.5 to 14.5, with a covering fraction of ∼50% out to half the virial radius for log N C IV > 13.5 cm −2 (Bordoloi et al. 2014 ; Burchett et al. 2016 ). Using conservative ionization fraction arguments that take the maximum fraction of carbon that presents as C iv , ∼30%, for both photoionized and collisionally ionized gas, such measurements imply that > 10 6 M ⊙ of carbon sits in the CGM of external galaxies. For sub- L * galaxies with stellar masses 10 8.5 ≲ M * ( M ⊙ ) ≲ 10 10 , this estimated CGM carbon mass amounts to 50%–80% of the total ISM carbon mass (Bordoloi et al. 2014 ). Measurements of the Milky Way's CGM are needed at all velocities in order to directly probe the baryonic mass of the Milky Way, test the expectations from cosmological simulations, and compare the

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 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 B-Survey 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 B-Citation I-Citation I-Citation I-Citation O O O O B-Survey I-Survey I-Survey 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 O 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 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-CelestialObject O O O O O O O O O B-CelestialObject O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O B-Model O O O O O O O B-CelestialObject I-CelestialObject O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion 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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula 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 B-Citation I-Citation I-Citation 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 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 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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula 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 B-Formula I-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 O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObject I-CelestialObject 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


with multiple components, ν is given for the component with the highest frequency. (3) Spectrometer used to record the data. FTS and VESPA were used at resolutions of 200 kHz and 20 kHz, respectively. (4) Velocity resolution. All spectra were resampled with channels of 0.6 km s −1 (FTS) or 0.06 km s −1 (VESPA). (5) Maximum main-beam temperature of the transition in the map. (6) Approximate rms range computed from a 10 km s −1 range, at least 20 km s −1 off the line, where no astronomical signal is found. We observed 14 tiles of typically 200′′ × 180′′, and built a mosaic that covers some 130 arcmin 2 around the position . Each tile was observed multiple times and in orthogonal directions in on-the-fly (OTF) mode and position-switching mode, with a scan velocity of at most 9′′ s −1 , a dump time of 1 s, and a maximum row spacing of 12′′. The beam full width at half-maximum (FWHM) ranges from 21′′ (0.08 pc) at 115 GHz to 26′′ (0.1 pc) at 93 GHz. The off position was observed every 1 to 1.5 minutes. It was chosen at after searching the SPIRE 250 μ m map for a minimum in surface brightness. Pointing corrections and focus corrections were performed every 1.5 h and 3 h, respectively, leading to a pointing accuracy measured to be ≲5′′. We converted the antenna temperature into the main beam temperature assuming a standard telescope main-beam efficiency 3 of 0.78 for CO observations and 0.80 for N 2 H + . We also used archival data from the WISE satellite ( Wright et al. 2010 ) at 3.4, 4.6, 12, and 22 μ m with a spatial resolution ranging from 6.1′′ (0.02 pc) at the shortest wavelength to 12′′ (0.04 pc) at 22 μ m. We used these data to complement the spectral energy distributions (SEDs) of cores in the MIR range. The data were converted into surface brightness units with the conversion factors given in the explanatory supplement ( Cutri et al. 2011 ). The calibration uncertainty is approximately 6% for the 22 μ m band and less for the shorter wavelengths. 3. Method 3.1. Analysis of the molecular line data For the following analysis, we used multi-component Gaussian fits to characterise the emission lines. For compact sources, spectra in all CO isotopologues were first averaged within the ellipse of the source as defined in the GCC catalogue by Montillaud et al. (2015) . A combined single-component Gaussian fit of the 12 CO, 13 CO, and C 18 O lines with the same centroid for the three lines was attempted. If the residuals showed features with a signal-to-noise ratio (S/N) greater than five, a new fit with one more component was made, again using the same centroid for all three lines. The residuals were examined again, and new fits with additional components were attempted until the residuals were below 5 σ . We found three components at most. Examples for a few sources are shown in Appendix A.1 . The constraint that each Gaussian component must have the same centroid in all CO isotopologues was motivated by the following virial analysis, which combines

O O O O O O O O O O O O O O O O O O O O B-Instrument O 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 B-Instrument O O O O O 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 B-Wavelength I-Wavelength 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 B-Instrument 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 O O O O O O O O O O O O O O O B-Telescope O O B-Citation I-Citation I-Citation I-Citation O O B-Wavelength B-Wavelength B-Wavelength O B-Wavelength I-Wavelength I-Wavelength 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 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-Citation I-Citation I-Citation I-Citation 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 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 B-Citation I-Citation I-Citation I-Citation 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


objects typically have red optical colours ( Gulbis et al. 2006 ) with ≈30% of the population found in binaries ( Grundy et al. 2011 ). The cold component comprises ≈50% of the CKB total population ( Petit et al. 2011 ) and is believed to have formed in situ because of the presence of extremely wide binaries that otherwise would have been destroyed by close encounters with Neptune ( Parker Kavelaars 2010 ). (2) The hot classicals (HCs), with a wider range of inclinations and eccentricities compared to CCs. These represent the other 50% of the total CKB population ( Petit et al. 2011 ) 1 . Contrary to the CCs, HCs were probably implanted in their current location, evidenced by the fact that the two populations are very different from each other. Indeed, in addition to having more excited orbits, the HCs show a wider range of photometric colours and a lower binary fraction than CCs ( Noll et al. 2008 ). Resonant KBOs These are objects in mean motion resonance with Neptune, such as Plutinos (3:2), Twotinos (2:1), and 5:3, 7:4, and 5:2 populations, and so on. These resonant KBOs are estimated to make up about 25% of the total Kuiper belt population ( Gladman et al. 2012 ). Gulbis et al. (2006) found that objects in the 3:2 resonance span the full range of KBO colours. The objects in the 4:3, 5:3, 7:4, and 2:1 resonances are predominantly red and the 5:2 resonant objects are grey or neutral in colour. Moreover, the 2:1 resonance inclination distribution is much colder than the 5:2 or 3:2 resonances ( Volk et al. 2016 ). Fig. 1 Eccentricities ( upper figure ) and inclinations ( bottom figure ) of TNOs. The data are from the IAU Minor Planet Center database 2 . The dashed line and the dash-dotted line represent respectively q = 30 and q = 40 au perihelion distances. Vertical solid coloured lines indicate the main mean-motion resonances with Neptune. Scattered disc objects (SDOs) These are the objects found in the scattered disc, which extends from 30 au to roughly 1000 au and interacts with Neptune by close encounters. The SDOs have perihelion distances in between 30 au and roughly 40 au and are mainly grey (or neutral) in colour ( Sheppard 2010 ). Detached objects These have slightly larger perihelion distances than the SDOs and are no longer affected by close encounters with Neptune. The first hypothesis for the very peculiar structure of the Kuiper belt was proposed by Malhotra (1993 , 1995 ) who predicted the existence of objects in external mean-motion resonances with Neptune. In the model, these objects are trapped via adiabatic resonance sweeping during a phase of outward planetesimal-driven migration of the ice giant ( Fernandez Ip 1984 ). Indeed, resonance capture can occur when the orbital frequency of a planetesimal and/or a planet changes gradually and the resonance ‘sweeps’ across the orbit of the planetesimal. Resonance sweeping may result in the capture of the planetesimal in the resonance (if the migration direction is convergent

O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation 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 B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest 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-CelestialObject O B-Citation I-Citation I-Citation O O O B-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O B-Citation I-Citation I-Citation I-Citation O O O O O O B-EntityOfFutureInterest B-EntityOfFutureInterest O O O O O O 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-EntityOfFutureInterest O O O O O O O O O O O O O B-Citation O B-Citation I-Citation I-Citation I-Citation O O B-EntityOfFutureInterest O O O O O O O O B-CelestialObject O O B-EntityOfFutureInterest O B-EntityOfFutureInterest O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O 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 B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O 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 B-EntityOfFutureInterest O O O O O B-Database I-Database I-Database I-Database I-Database O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-CelestialObject B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest 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 B-Citation I-Citation O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-Citation I-Citation O B-Citation 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-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


We include only neighboring galaxies with stellar masses greater than M * = 10 9 M ⊙ h −1 . We define the observable Δ r as the ratio of density within 0.6 Mpc h −1 , as compared to the density within 5 Mpc h −1 . The smaller scale is chosen to represent a galaxy’s local environment, sensitive to whether or not it exists as a part of a group or a cluster of galaxies, while the larger radius is representative of its large-scale environment or neighborhood beyond its own parent halo. In the following sections, we study the properties of the host galaxies in relation to BNS mergers, as a function of the time-delay model. We then estimate the number of events required in order to make inferences about the evolutionary history of merging binaries. 3. Results 3.1. Weighted Distributions The underlying population of BNS systems is drawn from the distribution of stars in the universe. Among the first questions to be posed is whether BNS host galaxies are biased tracers of the underlying population of galaxies in the universe. For example, we may ask whether BNS mergers trace stellar mass, in which case they would occur preferentially in high M * galaxies (Artale et al. 2020 ; Ducoin et al. 2020 ; Toffano et al. 2019 ). Alternatively, BNS mergers may trace star formation rate (Phinney 1991 ) or halo mass, or they may be randomly distributed in galaxies with equal probability. We ask how large a sample of host galaxies would be required in order to distinguish these simple models, where mergers trace either the stellar mass of a galaxy, the star formation rate, or the virial mass of the parent halo in which they reside. The arguments in this section are not specific to BNS systems, and can be extrapolated to black hole binaries, or black hole–neutron star mergers, assuming that their host galaxies can be identified. Physically, we can associate a model in which BNS mergers trace the star formation rate with a zero delay time model, where binaries merge as soon as they are born. In this scenario, the merger rate would simply trace the rate at which new stars are forming in a galaxy. At the other extreme, for very long time delays on the order of 10 Gyr, we expect low-redshift BNS mergers to trace the stellar mass, as high stellar mass galaxies reached the peak of their star formation at early redshifts. Alternatively, it has been proposed that, rather than forming from isolated stellar binaries in the galactic field (Tauris et al. 2017 ; Vigna-Gómez et al. 2018 ), a significant fraction of BNSs may form via dynamical interactions in dense globular clusters (Grindlay et al. 2006 ; Andrews Mandel 2019 ). Recent studies indicate that globular cluster abundance is a good tracer of a galaxy’s host halo mass (Hudson et al. 2014 ; Harris et al. 2013 ). In this way, a BNS merger rate that traces dark matter halo mass, rather than galaxy stellar mass, could be an indication that BNS systems are preferentially formed in globular clusters. We also note that some

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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


mass. The upper and lower limits of the SNR luminosities in Fig. 6 are found in Section 5.2 and are the same as those plotted on Fig. 4 . The solid lines of Fig. 6 show the range of luminosities that can be detected by Chandra -ACIS-S with 20 ks exposures (based on flux limits of 3 × 10 −15 and 9 × 10 −15 erg cm −2 s −1 for a 10 ks exposure). The lower horizontal axes are given in terms of the maximum resolvable distance of the SOI. This is the distance at which the (diameter of the) SOI of the SMBH is just within the core of the point-spread function of Chandra . For example, the SOI of a 10 8 M ⊙ SMBH is resolvable at any distance below ∼10 Mpc and, at 10 Mpc, the lower limit of detectable soft X-ray emission is given by the red line. At any distance less than 10 Mpc, this detection threshold drops and so fainter emission from SNRs would be detectable. Figure 6. Observable limits of the combined X-ray luminosities from young SNRs as a function of SMBH mass (upper axes) or maximum resolved distance (lower axes). The maximum resolved distance is that at which the SOI of the SMBH is within the core of the PSF of Chandra . The left-hand panel shows luminosities in the soft X-ray (0.5–2 keV), while the right-hand panel shows hard luminosities (2–8 keV). SNR luminosity limits are the dashed black lines (shown also in Fig. 5 ). Flux-limited detection thresholds from Chandra exposure times of 20 ks for galaxies at the distance limit are given as solid lines. The shaded region in the left-hand panel highlights the observable range of luminosities in the soft band. From Fig. 6 , it is evident that, even at the maximum resolved distance, the upper limit of SNR emission is detectable as it is above the 20 ks threshold in the soft band through most of the M • range (although the low end of possible X-ray luminosities is possibly not within the detection threshold of Chandra ). In the hard band, the Chandra 20 ks threshold follows very closely the high limit of predicted SNR emission. This suggests that, for galaxies at the maximum resolved distance, the emission would not be detectable in the hard band. Again, however, if a galaxy is closer than this maximum distance, the threshold marked by the blue solid line will drop; therefore, hard X-ray luminosities are still potentially detectable for more nearby galaxies. To compare this with the number of SMBHs known at these distances, in Fig. 7 , we plot the distances and masses of well-established SMBHs constructed from Tables 2 and 3 in the review of Kormendy Ho ( 2013 ) and the references therein. We show in the grey region the cases where the SOI of the SMBH is not resolved by Chandra . As for the upper axes in Fig. 6 , the limit between the grey and white area is determined by the distance at which the SOI of the SMBH equals in size the core of the point-spread function of Chandra . Figure 7. Distances and masses (with errors) of known SMBHs over the same mass range considered in this paper. Data were

O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 B-Model O B-Telescope O O O O O O O O B-Wavelength I-Wavelength B-Wavelength I-Wavelength 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 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 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 B-Wavelength O O O O O O O O O B-Telescope O O O B-Wavelength I-Wavelength 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 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 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 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 B-Telescope O O O O O O O O O O O O O O O O O O O O O O


star has an initial mass within a given mass range is given by the Kroupa ( 2002 ) IMF, with a slope x = 1.7 for high masses, normalized to unity in the 0.1–100 M ⊙ range. For the starbursts, we also investigate the effects of an IMF skewed towards high masses ( x = 0.95 in the high-mass regime), similar to the one proposed by Ballero et al. ( 2007 ) for the Galactic bulge. All our computations avoid the instantaneous recycling approximation by detailed accounting of the finite stellar lifetimes. This is a necessary prerequisite for a proper treatment of elements that are produced on different time-scales by stars of different initial masses and chemical compositions. The nucleosynthetic outcomes of binary stars exploding as Type Ia supernovae (SNe Ia) and novae are included in our computations, adopting the single-degenerate scenario for their progenitors (Matteucci Recchi 2001 , and references therein) for SNe Ia and our previous work (Romano et al. 1999 , 2001 ; based on D'Antona Matteucci 1991 ) for novae. Since CNO elements are produced in negligible amounts in SN Ia explosions (Iwamoto et al. 1999 ), the exact choice of the route leading to such events does not affect the results presented in this paper. Classical novae, instead, are thought to significantly overproduce 13 C, 15 N and 17 O with respect to their solar abundances (e.g. José Hernanz 2007 ) making assumptions about their precursors a much more thorny problem (see the next section). The adopted nucleosynthesis prescriptions are summarized in Table 2 . The stellar yields for low- and intermediate-mass stars are from Karakas ( 2010 ). For massive stars, we adopt the grid of yields suggested by Nomoto, Kobayashi Tominaga ( 2013 , Models 1, 2 and 5). However, since the important effects of stellar rotation are not accounted for in this case, we also consider the grid of CNO yields from fast-rotating massive stars provided by the Geneva group (Meynet Maeder 2002b ; Hirschi, Meynet Maeder 2005 ; Hirschi 2007 ; Ekström et al. 2008 , Models 3 and 4). Detailed yields for super-AGB stars (Doherty et al. 2014a , b ) – often neglected in chemical evolution studies – are implemented in Models 2, 4 and 5, while nova nucleosynthesis – also disregarded in most studies – is included in one case (Model 5). All the adopted yields for single stars are dependent on mass and metallicity. For novae, we assume average yields, independent of mass and metallicity (see discussion in Section 4.1 ). The different sets of yields were all tested against the Milky Way data, while only the set labelled ‘1’ was used in the models for the other galaxies. Table 2. Prescriptions for nucleosynthesis. 4 RESULTS 4.1 Milky Way galaxy In this section, we present the results of our chemical evolution models for the Milky Way. These models differ in their adopted nucleosynthesis prescriptions (see Table 2 ). First, we discuss the predictions of Models 1, 2, 3 and 4, where the CNO elements come only from single stars.

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


the peak of the distribution to larger distances, from ≈5 kpc in the smallest galaxy considered here to ≈12 kpc in the largest galaxy. We note that GC distributions tend to be broader around the peak of their number distributions in elliptical galaxies relative to spiral galaxies, in which they appear more concentrated around the peak. The Sérsic index slightly affects the cluster distribution (the lower the n , the smaller the peak distance), but it has a negligible effect on the final number of GCs. Figure 2. Radial distribution of surviving GCs (with IMBH in their centers) in early-type galaxies (left) and late-type galaxies (right) with stellar mass (top), (center) and (bottom), for different Sérsic indexes. Figure 3. The final cumulative radial distribution of IMBHs in star clusters in different early-type galaxies (top) and late-type galaxies (bottom), and different Sérsic indexes. 5. TDEs We calculate the TDE rates by IMBHs in GCs in two limiting cases: one in which a density cusp cannot form due to a rapid depletion by TDEs and one in which a density cusp of stars with total mass M IMBH forms around the IMBH. In both cases, we take into account the stars accreted by the IMBH and remove stars from the host GC every time they undergo a TDE. The tidal disruption of a star by an IMBH occurs when a star on an eccentric orbit has an angular momentum smaller than the so-called loss-cone angular momentum (Peebles 1972 ; Bahcall Wolf 1976 ; Lightman Shapiro 1977 ; Baumgardt et al. 2006 )whereis the tidal disruption radius of a star of mass m * and radius R * . Within the IMBH radius of influencewhere is the velocity dispersion in the cluster core, the dynamics is dominated by the IMBH gravitational potential. Nevertheless, stars exchange energy and angular momentum through two-body interactions on the typical relaxation timescalewhere Λ is a dimensionless constant of order unity (which contains the Coulomb logarithm), and and are the velocity dispersion and density at semimajor axis a , respectively. As a consequence of the numerous small-angle two-body deflections, a star with energy E and angular momentum J will diffuse in E – J space (for a comprehensive review see Alexander 2017 , and references therein). Being a random-walk process, the change in angular momentum in a time is ( J c is the angular momentum of a circular orbit with the same a ), which implies a typical timescalewhere e is the eccentricity of the star, to change the angular momentum to be of order J . Thus, more eccentric orbits relax more rapidly than circular orbits at the same a . If (empty loss-cone regime), any star deflected into the loss-cone is disrupted within a dynamical time, while if (full loss-cone regime), stars may be scattered in and out of the loss-cone on their way from apoapse to periapse (Baumgardt et al. 2004a , 2004b , 2005 ). After a rapid initial phase characterized by the wandering of the IMBH through a sea of stars (Chatterjee et al. 2002a , 2002b

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O 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 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 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 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 O O O O O O O O O O O O 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 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


late He ii reionization). Therefore, they should not yet be interpreted as evidence in favour of the abundant high- z AGN models considered here. Combined with direct searches to better characterize the high- z faint AGN population, future measurements of the Ly forest opacity and of the IGM temperature will more tightly constrain the contribution of AGNs to the ionizing background, as well as their contribution to reionization. APPENDIX A: MODEL UNCERTAINTIES In this section, we discuss caveats for our models of the Ly forest opacity fluctuations considered in Section 3 . Below we argue that our models do not account for a number of effects that would, in combination, likely lead to a net decrease in the amplitude of eff fluctuations. We do not include the effects of inhomogeneous heating from patchy H i reionization. These temperature inhomogeneities would suppress fluctuations in eff because the overdense regions that were reionized earlier would tend to be cooler than average by z 5.5 (i.e. after reionization), with enhanced equilibrium H i densities from the T 0.7 dependence of the recombination coefficient. However, the ionizing background in those regions would tend to be stronger owing to the bias of the sources. The competition between these two effects would smooth out fluctuations in eff (recall that ). If H i reionization ends at z 6, this competition could have a significant effect on the width of the eff distribution (DAloisio etal. 2015 ). On the other hand, we also do not account for inhomogeneous heating during He ii reionization, which would likely work in the direction of enhancing the eff fluctuations, since the regions of strong ionizing background surrounding AGNs would also be the most recently photoheated. We do not take into account the finite lifetimes of AGNs. With a duty cycle of unity, our models effectively assume an infinite AGN lifetime. We argue that modelling the finite lifetimes could act to suppress eff fluctuations in at least two ways. First, for a fixed AGN abundance (which is set by the assumed luminosity function and magnitude limit), a smaller duty cycle would require that AGNs are hosted in haloes that are less massive and therefore less biased than in our models. This would lead to the AGN being less clustered overall, which would yield somewhat smaller eff fluctuations. Secondly, consider a region with an overdensity of bright AGNs in our fiducial model (see e.g. the lower left-hand corners of the panels in Fig. 3 ). These regions produce large segments of the Ly forest with low optical depth. Suppose now that only a fraction of these AGNs are active at a given redshift. In this case, the Ly opacity in this large-scale region would be less coherent an effect that would make the eff distribution narrower. On the other hand, we note that the proximity zone remnants left by AGNs that recently became dormant might mitigate these effects. We do not account for the anisotropy of emissions from AGNs. Our

O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-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


m e / m i so that the ion distribution is approximately a delta-function around p ∥ i = γ CC m i c carrying current at speed c . Details of two-stream instability are discussed in Appendix B.2 . 3.3. Ion Acceleration in the Nose We now consider the particle acceleration at the end of jets (i.e., the nose region where the jet magnetic field and current turn to the radial direction). Internal kink instability due to j z occurring in the Central Column, as analyzed in Colgate et al. ( 2015 ) and used in Section 3.2 , should persist for some distance into the nose driven by j r as the current turns radially in the nose, now less affected by curvature radiation as B φ falls like 1/ r giving curvature parameter α C ∝ 1/ r . But a new acceleration mechanism arises due to conditions in the nose region different from those in the cylindrical Central Column. One new condition concerns the sensitivity of the curvature radiation, Equation 11 (a), to R C that allows ions to be accelerated to the full voltage in the nose region, where R C = r becomes large. Another new condition concerns the escape of ions from the nose, escape itself turning out to be a necessary ingredient for the powerful acceleration mechanism not present in the Central Column. There are two ways that ions can escape from the nose. First, while ion orbits are magnetically confined in the Central Column, in the absence of an electric field ion orbits in the nose are not confined due to drifts of guiding centers in z caused by the curvature of field lines and gradients in the magnetic field intensity. Opposite drifts by electrons establish an electric charge giving E ∝ T e , the electron temperature, which can cause ions to escape by E × B motion. In addition, ions can also escape by hyper-resistive diffusion, which is what we will mainly discuss. 3.3.1. Hyper-resistive Diffusion in the Nose As in Section 3.2.1 , we represent hyper-resistivity in the nose as diffusion, noting that an electric field larger than that produced by charge separation is created if turbulence in the nose causes ions to escape by diffusion. To see this, consider the radial component of the relativistic Ohm’s law in Equations 5 (a). In the nose, with unit vector , and electron flow v e is either that due to drifts in the z -direction or that parallel to B giving so that we can drop the relativistic correction to E r . Applying Equation 5 (a) in the r -direction (perpendicular to B ) gives Equation 13 (b) giving the cosmic-ray accelerator is obtained as follows. Early in jet propagation when in the jet and nose, all of the voltage drop in the jet (giving net zero voltage around the loop) can be approximated as , which appears as an inductive electric field in our disk-centered reference frame. It is this inductive field that serves to buildup B φ behind the nose advancing at velocity . Even as the jet slows down, it continues to be true that it is that builds up B φ , giving Equation 13 (b) with in our reference

O O O O O O O O O O O O O B-Model 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 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 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 B-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 B-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-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


may marginally agree with observations within errors. Figure 10. The SFR/ M s – z relation for S-I (left-hand panel), and for S-II to S-V (right-hand panel). The solid, dashed, and dotted lines show the median trends for BGGs in observations and in the SAMs of DLB07 and G11 , respectively. The grey point shows the BGG sSFR versus redshift in observations. In Fig. 11 , we show the median sSFR dependence of BGGs to the total mass of their host groups in observations (black points) and in the SAMs of G11 (red diamonds) and DLB07 (green squares), respectively. For S-I (left-hand panel), the median sSFR rapidly decreases as a function of increasing halo mass (the best-fitting linear relations corresponds to: SFR/ M s = (−4.4 ± 1.0) × M 200 + (3.8 ± 0.8). The observed trend is steeper than the model predictions. In agreement with the SAM predictions, we find no relationship between sSFR and M 200 for S-II to S-V over 0.1 z 1.3, as shown in the right-hand panel in Fig. 11 . These results suggest that, in the group scale haloes, sSFR of BGGs is a function of redshift with no halo mass dependence. Figure 11. The SFR/ M s – M 200 relation for S-I (left-hand panel), and for S-II to S-V (right-hand panel). The solid, dashed, and dotted lines show the median trends for BGGs in observations and in the SAM of DLB07 and G11 , respectively. The grey point shows the BGG sSFR versus redshift in observations. Only, for BGGs within S-I we find that the sSFR decreases as a function of halo mass. 5 SUMMARY AND CONCLUSIONS We select the BGGs in a well-defined X-ray-selected sample of galaxy groups from the COSMOS, XMM –LSS, and AEGIS fields. This sample covers a redshift range of 0.04 z 1.3 and contains about 407 galaxy groups with a mass ranging from 10 12.8 to 10 14 M ⊙ , the largest sample for such studies in the X-ray galaxy group regime. The sample allows us to study the distribution and evolution of the stellar mass and SFR of BGGs and the relation among the stellar mass, SFR, and halo mass. We compare our results with sample drawn from the Millennium simulations with three SAMs ( G11 , DLB07 , and B06 ). We summarize our results as follows. The shape of the stellar mass distribution of the BGGs evolves towards a normal distribution as the groups evolve. This evolution is quantified by the skewness and Kurtosis of the stellar mass distribution and their redshift dependence. At all redshifts, we find that the stellar mass distribution tends to skew towards low masses. The observed fraction of the BGGs shows a strong deviation from the Gaussian fit at ≈10 10.2 M ⊙ , which are generally SF/young population. In contrast, the SAM predictions show no redshift evolution of the stellar mass distribution. Within the probed SAMs, the shape of the stellar mass distribution predicted by B06 model is more consistent with the observations at z 0.5. We show that the average stellar mass of the BGGs evolves with redshift by a factor of ∼2 from z = 1.3 to the present day. At z

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


This is a great advantage of such a Nasmyth system. Any optical designer of telescopes can observe that in this Nasmyth system the exit pupil is near the plane fold mirror, and that if some small adjustments are made to this system, an AO deformable mirror can be added. After this AO deformable mirror, a plane fold mirror can still be added to reflect light to the Nasmyth focus. In such an arrangement the length of path from the exit pupil to the pupil formed by the secondary mirror and that from the exit pupil to the Nasmyth focus should be equal. These two plane mirrors can be used, respectively, as AO deformable mirror and tip-tilt mirror for all instruments at Nasmyth platform and coudé focus. This is the second great advantage of this Nasmyth system. Such a system has been applied in the Nasmyth system of 39 m European Extremely Large Telescope (Delabre 2008 ; ESO 2011 ). However, if an AO deformable mirror is added in such a Nasmyth system, there will be some shortcomings as well. The size of the AO deformable mirror is bigger than those AO deformable mirrors set up separately in Nasmyth instruments and coudé instruments. It is difficult for a big AO deformation mirror to perform the high temporal frequency movement. Besides, an AO mirror is unnecessary for some Nasmyth instruments and coudé instruments. As it is located in the tube of the telescope, this AO mirror will produce some harmful thermal effects, especially since this mirror needs more power due to its large size. Besides, servicing also possess some difficulty. Given these circumstances, at present we only intend to use one plane fold mirror, which will serve as a tip-tilt mirror in our Nasmyth system. Nevertheless we insist that the addition of an AO deformable mirror in this Nasmyth system is an important advantage and we will continue to consider whether or not to make such an addition during future research and design of Nasmyth and coudé instruments. Excellent image quality has been obtained in our Nasmyth system. In the next section one can witness the excellent image quality also in our coudé system design. We are convinced that when we make some small readjustments for adding an AO deformable mirror, the excellent image quality can still be obtained in Nasmyth and coudé systems. In Nasmyth platforms many instruments will be installed. In order to meet this challenge, we propose (Su, Wang Cui 2004b ) the use of four Nasmyth platforms in this telescope, i.e. two Nasmyth platforms I on the upper level and two Nasmyth platforms II on the lower level. In such an option, the two light bundles to Nasmyth platform II come perpendicularly up (or down) towards this platform; for some instruments that is an advantage. Since the lengths of paths from secondary mirror to Nasmyth I foci and to Nasmyth II foci may be different, M2 and M3 will be moved to make the image arrive at the required foci. In these situations fairly large off-axis aberration may occur and FOV will be

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 I-Telescope I-Telescope I-Telescope I-Telescope I-Telescope B-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


acted at different temporal and spatial scales throughout the lifetime of the galaxies. The size of our sample does not allow for a separate analysis of whether AGNs or bars can play a role in the observed stellar metallicity. However, we observe that the mere presence of an AGN at z = 0 does not necessarily imply that it has played a role in shaping the observed galaxy stellar metallicity. An AGN can be a transient phenomenon in a galaxy. With AGN duty cycles spanning timescales in the range 10 6 − 10 8 yr ( Haehnelt Rees 1993 ; Davis et al. 2014 ; Storchi-Bergmann Schnorr-Müller 2019 ), we expect that the AGN effects on stellar metallicity should be due to past and not current AGN activity. Regarding bars, there is no consensus in the literature as to the longevity of bars, nor how many bar events the average galaxy experiences during its lifetime ( Sellwood 1999 ; Athanassoula 2002 ; Bournaud Combes 2002 ; Elmegreen et al. 2004 ; Combes 2004 ; Regan Teuben 2004 ; Pérez et al. 2008 ; James Percival 2016 ). While several authors have observed a correlation between gas abundance gradients and the presence of bars, in particular, a flattening of the gradient ( Vila-Costas Edmunds 1992 ; Martin Roy 1994 ; Zaritsky et al. 1994 ), more recent works have found no evidence of such a correlation by analysing both gas-phase metallicity and stellar metallicity ( Sánchez et al. 2012b , 2014 ; Sánchez-Blázquez et al. 2014 ; Cheung et al. 2015 ; Sánchez-Menguiano et al. 2016 ). The upper panels of Fig. 2 show the light-weighted metallicity as a function of stellar mass surface density log(Σ ⋆ ) for galaxies in the field and in groups. Field galaxies shown in Fig. 2 are a random realisation of 107 galaxies from the field sample, which were selected to have the same overall mass distribution as the group sample. Each galaxy, in either of the two samples, contributes several points to this figure. The number of points varies from galaxy to galaxy, as does the number of radial bins that each profile has. Points are colour-coded in tones of grey according to their radial distance in terms of the effective radius, r / r e . The solid curves show the number-density iso-contour levels. The stellar mass surface density and metallicity are strongly correlated although we observe significant scatter. To compare galaxies in the field and groups, the upper right panel of Fig. 2 shows the iso-contour for each environment considered. We clearly observe that galaxies in groups are more metallic for a fixed value of stellar mass surface density. Fig. 2. Top panels : light-weighted metallicity as a function of stellar mass surface density Σ ⋆ . Dots are colour-coded in tones of grey according to their distance r / r e . The left panel corresponds to galaxies in the field, the central panel shows galaxies in groups, and the right panel shows the iso-contour levels for galaxies in groups (red lines) and galaxies in the field (blue lines). The bottom panels show the median

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moon would be able to create the gap. These viscosities are notably lower than those measured by Tiscareno et al. ( 2007 ) and Colwell et al. ( 2009a ) in this region. For , the ratio has a value of 1.2 so the moon might just be able to create the Kuiper gap but for a slightly larger viscosity of, for example, , this ratio has a value of only 0.12 and the moon should definitely create a propeller and not a gap around the entire circumference of the ring. Let us consider the wide Bessel gap as another example. The diameters a spherical ice moon (or hypothetically even a silicate moon) would have to have to open this gap are and , respectively. Even for the lower limit of , the moon would have to have a nonsensical density of to fall below a diameter of . We therefore believe that single solid icy or silicate bodies can be ruled out as an explanation for the observed gaps in the C ring and the Cassini division. Lissauer et al. ( 1981 ) described the rift and gap structure of the Cassini division with a set of five embedded moons. They assumed a dispersion velocity with a spatially constant value of , which for , yields , which is much larger than the viscosities measured by Tiscareno et al. ( 2007 ) and Colwell et al. ( 2009a ), thus our chosen lower limit of . Consequently, the moons they predicted () are significantly larger than the ones we predict (see Table 1 ). Those moons might have been overlooked by Voyager , but would definitely have been detected by the cameras of the Cassini spacecraft. Ruling out the existence of single icy bodies as an explanation for the observed gaps in the C ring and the Cassini division raises the question what process could explain their formation. We believe that a swarm of icy collision shards much smaller than the dimensions presented in Table 1 , but still much larger than the largest ring particles could scatter the ring particles away, creating a gap that despite the existence of this swarm would appear completely void to the cameras of the Cassini spacecraft. However, this hypothesis cannot be tested easily with the model presented in this article and requires additional research. 4. Summary We developed a nonlinear diffusion model for gaps in planetary rings that accounts for gravitational scattering by the embedded moon and for the viscous diffusion of the ring material. This model can be used to estimate the ring’s viscosity from the radial density profile of the gap the moon creates knowing its radial position and mass and the power-law exponent β (). The mass of the moon can be derived by analyzing the wake region (Showalter et al. 1986 ; Seiß et al. 2010 ). We applied the model to the Encke and Keeler gap created by Pan and Daphnis to estimate the rings viscosity. Then, we applied the model to the gaps in the C ring and Cassini division to estimate the masses of moons that would be able to create those gaps. For , we estimate for the Encke gap, which is not an unexpected result, as Tiscareno et

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NRAO Astronomical Image Processing System 17 17 http://www.aips.nrao.edu/ package. In order to avoid false associations of radio matches, we downloaded the images from DSS1 and DSS2, which are digitized versions of several photographic astronomical surveys, in addition to radio maps, which can be found at the ESO archive. 18 18 http://archive.eso.org/dss/dss Comparison of radio and optical sources on maps from both domains, together with the NASA/IPAC Extragalactic Database, 19 19 https://ned.ipac.caltech.edu/ enabled us to distinguish incorrect matches. As some sources turned out to have extremely extended, i.e., beyond 3′, radio structures we were gradually increasing the radio search by 1′ as long as the association for the whole structure was found. Once the whole radio-emitting region was identified, the radio flux of each of the components was summed up and assigned to the given source. For sources lacking radio detections, we assigned them upper limits corresponding to the value of the sensitivity of the survey that contains the source in its footprint. While the sensitivity of NVSS and SUMSS is the same, for objects located in the area covered by both NVSS and FIRST, we decided on FIRST upper limits, as its sensitivity is lower than that of NVSS. Because the radio catalogs we used were conducted at two different frequencies, the radio fluxes at 843 MHz were recalibrated to 1.4 GHz using a radio spectral index of α r = 0.8 (with the convention of F ν ∝ ν − α ) so that the rest of our calculations are consistent. A.2. Mid-infrared Data The mid-infrared measurements were taken from the AllWISE Data Release (Cutri et al. 2013 ) which, by combining data from the cryogenic WISE (Wright et al. 2010 ) and post-cryogenic NEOWISE (“near-Earth object + WISE,” Mainzer et al. 2011 ) survey phases, resulted in a comprehensive view of the mid-infrared sky. Out of four available bands (at 3.4, 4.6, 12, and 22 μ m, corresponding to the W1, W2, W3, and W4 bands, respectively), we decided to make use of the W3 band, which was driven by the fact that at this specific wavelength the dusty torus is transparent enough to observe radiation coming directly from the central region of an AGN. At the shorter wavelengths of the W1 and W2 bands, the dusty, circumnuclear tori are optically thick and radiate anisotropically (Hönig et al. 2011 ; Netzer 2015 ). At the longer wavelengths of the W4 band, the dusty torus becomes even more transparent, but its measurements are affected by much larger errors than in W3 because (1) the W4 band traces the warm dust continuum at 22 μ m and can be contributed to by starbursts (Ichikawa et al. 2019 ); (2) the sensitivity in the W4 band is much lower than in the W3 band and its signal-to-noise ratio is the lowest of all the WISE channels, resulting in a lower detected fraction for the objects in our sample; and (3) the resolution of W4 images is worse than in other bands (e.g., 12′′ in W4 versus 65 in W3), so for some of our

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capabilities on large-aperture telescopes has revolutionized the study of high-redshift ( z ≳ 5.7) quasars (e.g., Fan et al. 2001 , 2003 , 2004 , 2006a ; Cool et al. 2006 ; Goto 2006 ; McGreer et al. 2006 ; Venemans et al. 2007 , 2015 ; Willott et al. 2007 , 2009 , 2010b ; Jiang et al. 2008 , 2009 , 2015 , 2016 ; Mortlock et al. 2009 , 2011 ; Morganson et al. 2012 ; Bañados et al. 2014 , 2016 ; Matsuoka et al. 2016 , 2018a , 2018b ; Reed et al. 2017 ; Tang et al. 2017 ). These earliest quasars, powered by supermassive black holes (SMBHs), not only constrain the physics of black hole (BH) accretion and host galaxy assembly at cosmic dawn but also provide critical information on the physical conditions of the intergalactic medium in the early universe (e.g., Fan et al. 2006a ). Since the first successful identification of z ≳ 6 quasars (Fan et al. 2001 ), the searches for these high- z quasars have matured and dramatically increased the inventory of these rare objects. There are now more than 250 quasars known at z > 5.6, with the most distant quasar reaching z ∼ 7.5 (Bañados et al. 2018b ). These impressive statistics now enable a transition from individual case studies to ensemble studies of high- z quasars. The vast majority of z ≳ 6 quasars are selected from wide-field optical+IR imaging with the dropout technique and are confirmed with optical spectroscopy (e.g., Fan et al. 2001 ). The confirmation optical spectroscopy provides limited information about these high- z quasars themselves, and to probe the physical properties of these systems (e.g., BH mass, spectral properties, broad and narrow absorption lines, etc.), near-IR spectroscopy is necessary to cover the rest-frame UV from C iv 1549 Å to Mg ii 2798 Å. Some earlier near-IR spectra of z ≳ 6 quasars already provided a glimpse of their physical properties, such as BH masses and Eddington ratios (e.g., Jiang et al. 2007 ; Kurk et al. 2007 ; Willott et al. 2010a ; De Rosa et al. 2014 ; Mazzucchelli et al. 2017 ; Kim et al. 2018 ) and chemical abundances in the broad-line region (e.g., Barth et al. 2003 ; Kurk et al. 2007 ; De Rosa et al. 2011 ). These measurements of BH properties also facilitate the study of the relations between BH growth and host galaxy assembly in the early universe, where the host galaxy properties are mostly inferred from the molecular gas emission in the millimeter regime (e.g., Walter et al. 2003 , 2009 ; Riechers et al. 2009 ; Wang et al. 2013 ; Venemans et al. 2016 ; Decarli et al. 2017 , 2018 ; Willott et al. 2017 ). However, near-IR spectroscopy of z ≳ 6 quasars is expensive, and thus only a small fraction of them have existing near-IR spectra, some of which are also of low quality (e.g., insufficient spectral coverage, signal-to-noise ratio, and spectral resolution). Given the importance of understanding the physical properties and growth of these high- z SMBHs, we have assembled a large sample of high-quality near-IR spectra for these objects, as

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


ID34) or (ID02, ID29), where one star is a Sct pulsator, but the other one is not pulsating (ID29) or pulsating only in the low-frequency domain (ID34). All the Sct stars in our sample are Sct/Dor hybrids, confirming the already known result that over 98percent of Sct stars are hybrids (Balona 2018c ). For all stars, most peaks in the low-frequency region are independent modes. In a number of stars, some of the peaks in the low-frequency region are among those of highest amplitude. Hence, the definition of a Dor star is simply a Sct star where no significant frequencies are present above 5d 1 . There appears to be no real difference between the two classes of stars. Dor, Sct, and constant stars are to be found within the same small region at the cool end of the instability strip (Balona 2018c ). Many of the Sct stars discussed above seem to exhibit rotational modulation (Table 1 ), as shown by the presence of a low-frequency peak and its harmonic, both of which tend to be variable in amplitude. This is an indication on the existence of stellar activity, either through star-spots or any other corotating surface inhomogeneity. These rotational peaks are, in many cases, among the peaks with the largest amplitudes. Moreover, rotational modulation dominates the photometric variability of some Dor stars in our sample. We detect rotation in nine out of 16 Sct stars, i.e. 56percent of the sample. In six of them the rotational peaks are among the 20 largest amplitude peaks. Moreover, we detect rotation peaks in 26 out of 34 main-sequence stars (76percent) or 16 out of 24 main-sequence stars falling within the general location of the Kepler Sct instability region (Fig. 1 ). In some cases, it seems that more than one spot could be present on the stellar surface (e.g. ID11). In other cases, like ID14, rotation seems to be almost synchronized with the orbital period in a binary system, similar to the cool rotating variable ID06. One star, ID28, seems to be a ROTD-type variable similar to those detected by Balona ( 2013 ): most likely, the sharp peak is the rotational frequency, and the hump is due to Rossby modes (Saio etal. 2018 ). There is a clear indication of activity and star-spots on the surface of the A-type stars, as previously found by Balona ( 2011 , 2013 ). This suggests the presence of magnetic fields on the stellar surfaces that are not expected in such hot stars with very thin convective envelopes. The dynamo mechanism, which is thought to be responsible for magnetic activity in cool stars, cannot operate in stars with predominantly radiative envelopes. Star-spots seem to be also common among the much hotter B stars (Balona 2016 ; Balona etal. 2019 ). This is not surprising since there is no difference in the envelope structure between the A and B stars. Amplitude variations are found among the frequencies of highest amplitude in all four stars belonging to GroupIII (see last column of Table 1 ). They show a diversity of behaviours, including

B-CelestialObject O B-CelestialObject B-CelestialObject O O O O O O O O O O O O O O B-CelestialObject 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-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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-CelestialObject 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-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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


different resolution elements (gas, DM, stars and BHs); the solution of the (magneto)hydrodynamical equations for the gaseous component; a treatment of the most important radiative cooling and heating processes; a mechanism for the conversion of gas into stars; stellar evolution and subsequent chemical enrichment of the interstellar, circumgalactic and intergalactic medium; outflows of gas from the star-forming regions of galaxies; and the formation, growth and energetic feedback of supermassive BHs in distinct low- and high-accretion rate states. In developing the TNG model, we have pursued three simultaneous goals: to bring together under a consistent framework the improvements implemented in arepo to the accuracy and robustness of the underlying numerical methods; to introduce new, rich physics; and to address the key deficiencies of the previous model. The main numerical updates are summarized in Section 2.2 and include an improved estimator for spatial gradients across gas cells, a more efficient gravity solver and time integration scheme, and a more consistent advection of the passive scalars. Together, these changes give a more accurate and better convergent hydrodynamical scheme, and will allow us to pursue even more ambitious simulations in the future. The principal new physics in the TNG model are the inclusion of MHD (see Section 2.2.1 and references therein) and a dual-mode, thermal and kinetic BH feedback model (see the companion paper by Weinberger et al. 2017 ). The new kinetic BH feedback has been shown to alleviate some of the discrepancies identified in Illustris in comparison to observations at the high end of the halo mass function ( ≳ 10 12 –10 14 M ⊙ ), in particular regulating the stellar content of massive galaxies while preserving realistic halo gas fractions. In fact, in confronting the problems identified in Illustris (see Introduction section), we aim to open new physical regimes that were impossible to robustly study in previous simulations. These include not only massive haloes emitting at X-ray wavelengths, but also the galaxies occupying the transitional green valley, and the properties of low-mass dwarfs, including cluster satellite populations. In this paper, we have focused on the intermediate- and low-mass end of the galaxy population (halo masses of a few 10 10 –10 13 M ⊙ ) and have presented an updated model for the galactic winds feedback by modifying their directionality, thermal content, velocity and energy scalings (Section 2.3.2 ). We have also updated our reference yield tables (Table 2 and Fig. 1 ) and adjusted some stellar evolution choices (Section 2.3.3 ). The TNG galactic winds (whose implementation and scalings largely follow those of Springel Hernquist 2003 ; Oppenheimer Davé 2006 ; Oppenheimer Davé 2008 ; Vogelsberger et al. 2013 ) are now launched isotropically from the star-forming regions of galaxies, bringing a simplification to the modelling that by itself does not significantly affect

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 O B-Model I-Model O O O O O O O O O O O O O O O O O B-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 O O O O O O O O O O O O O O O O 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 B-Model O O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model I-Model 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-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 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 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 B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O 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 B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation 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


1,20−19 (252,789.82 MHz), U253020, U267402, U268464, U268485, and U268833. 5. Comparison to single-dish results. Eight light curves were studied in detail in our single-dish work (He et al. 2017 ), but only six of them can be compared to the ACA results, because the other two lines (of C 4 H) suffer from the interferometric baseline problems in the ACA data, showing negative artifacts, as mentioned earlier. Two of the six line features (SiS 14−13 at 254,103 MHz and the blended lines of C 3 N+HCN+C 4 H at 267,120 MHz) follow the NIR light curve well in both the single-dish and ACA data. However, for the four line features that were anticorrelated with the NIR light in the single-dish results, none of them shows the anticorrelation in the ACA data: two of them (HCN, v 2 = 1, 3−2, l = 1 f at 267,199 MHz and the blended lines of 29 SiS+HCN, v 2 =2 0 at 267,243 MHz) show strong variations in phase with the NIR light and continuum; 30 SiO,6−5 at 254,216 MHz varies similarly but with a smaller amplitude; and the blended line feature of Na 37 Cl+CH 2 NH+HC 3 N at 254,700 MHz shows modest variation. Why are the anticorrelation cases in the single-dish data missing in the ACA results, while the correlated cases remain unchanged? The ALMA monitoring started about 7 yr (∼four stellar pulsation cycles) later than our single-dish work, and thus secular changes of stellar pulsations could have occurred. However, the different phase behaviors of the different lines cannot be explained solely by this; some unknown mechanisms operating in the circumstellar envelope (CSE) are necessary to create differences among them. The mechanisms are possibly related to the missing flux in the ACA data, as discussed earlier, and the different telescope beam sizes involved in the two works, 29′′ for the single-dish versus 4′′ × 8′′ (or 6′′) for ACA. However, a radial phase delay due to radiation transportation is unlikely, because the difference in beam radii only corresponds to ∼1552 au at a source distance of 135 pc, which can produce a phase lag of only ∼9 days with the speed of light. This is insufficient to explain the observed change from anticorrelations to correlations with the NIR or millimeter continuum light. The only known candidate mechanism is that proposed by Cernicharo et al. ( 2014 ) from their simulations in which the evacuation of molecular populations in low- J energy levels by IR excitation is different for different molecular lines at different radii of the CSE. We will analyze the time series of ALMA ACA maps and do simulations to further explore this idea in coming papers. 6. Remarks on special species. A special case is Na 37 Cl. Its J = 20−19 (254,663.465 MHz) and 21−20 (267,365.833 MHz) transitions have similarly strong broad line profiles, with peak fluxes of about 1.5 Jy beam –1 . However, while the former shows little difference between its maximum- and minimum-phase line profiles in Figure 4 , the latter has its line profile definitely varying

O B-Wavelength O B-Wavelength B-Wavelength B-Wavelength B-Wavelength O B-Wavelength 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-Telescope 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 O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength O O O O O O O O B-Telescope 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 B-Telescope O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O B-Wavelength O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength 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 B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O 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-Wavelength 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 B-Wavelength 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 O O O 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 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


while the other segments have been fading throughout the observations. This supports and quantifies the apparent asymmetries in the images in Figure 4 . Figure 6. The 3.6 μ m light curves of the four ER segments and the 3 surrounding stars, derived from the decomposition of the 04 mosaics at three epochs. Formal uncertainties on the flux densities are ≲2%. The mapping and decomposition procedures were also applied to the individual epochs of targeted observation without averaging into broader intervals. In this case, ∼25% of the 04 pixels in the images contain no data, and the decomposition is applied giving no weight to these pixels. The results are, as expected, much noisier than those illustrated above, but the same trends do emerge when similar averaging is applied after, rather than before, the decomposition. 5. Discussion Across all IR wavelengths, the light curves are reasonably well fitted by the model of Equation ( 1 ) with similar parameters (Figure 2 ). This empirical model is similar in form to those developed by Lundqvist Fransson ( 1991 ) and Dwek Felten ( 1992 ) to explain the prompt line emission from the ER. In those models, the light travel time across the ER (326 d for an ER with radius R = 239 lt-days and inclination i = 43°; Dwek Felten 1992 ) is explicitly modeled. Here, the light travel delays can be assumed to be a contributing factor to the width of the Gaussian component of our model. Irregularities in the radius and density distribution of the ER and asymmetries in the radius (or velocity) of the blast wave would be other contributing factors. The X-ray light curve is also well fitted by this model, although the date, d 0 , of the peak of the Gaussian function is somewhat later, and the exponential timescale, τ d , is significantly longer than at the IR wavelengths. Both these differences act to shift the peak of the X-ray light curve to a later date. In a very simple interpretation, and assuming that the IR emission is from dust that is collisionally heated by the X-ray emitting gas (Dwek 1986 ; Bocchio et al. 2013 ), the shorter exponential timescale for the IR emission compared to that of the X-ray emission, may suggest that dust is being destroyed faster than the gas is cooling. However, additional factors such as the development of a reverse shock, and variations in the density (and consequent post-shock temperature) of the CSM may also contribute to the differences in d 0 and τ d between the IR and X-ray light curves. Systematic differences in the IR and X-ray trends that may be indicative of dust destruction had been much more difficult to identify prior to about Day 8000 while both light curves were still rising (e.g., Dwek et al. 2008 , 2010 ). As shown in Figure 2 , we also find that the model of Equation ( 1 ) provides a good fit to the evolution of the B, R, and F502N band optical emission reported by Larsson et al. ( 2019 ) if restricted to dates after Day 5500 when the interaction with the ER is

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


magnetogram data in weak-field regions. Figure 1. Input distribution of the test configuration, displayed in longitude–latitude order. This case has m = 200, n = 400, a = π /2 − 0.1, b = π /2 + 0.1, c = 0, and d = 0.4. The input distribution is a field of random numbers distributed between −0.5 and 0.5. Figure 2. Recovered field of plotted vs. the input field of for the centered grid case shown in Figure 1 . By changing the definition of how finite-difference approximations to spatial derivatives are defined, and where different variables are located within the grid, we can improve this behavior dramatically. In the centered grid case, all variables are colocated at the same grid points. By defining the radial magnetic field and its time derivative to lie at the centers of cells, with the electric field components lying on the edges (or “rails”) surrounding the cells, the finite-difference approximations to the derivatives can be made to obey Faraday’s law to floating-point roundoff error. Figure 3 shows the analogous scatterplot shown in Figure 2 , but using the staggered grid definition described above. The relationship is a straight line. Figure 4 shows the difference between the recovered and original values of . Note that the amplitude of the error is multiplied by 1 × 10 12 , so that the error term is visible in the scatterplot. These two plots clearly show that solutions for the electric fields in a staggered grid formulation can do a far better job of representing the observed data than can the centered grid formulation. Figure 3. Recovered field of plotted vs. the input field of for the staggered grid test case. Figure 4. Difference between recovered and input as a function of the input for the staggered grid case. These figures motivate the development of our more detailed staggered grid formalism, which is described in Section 3.4 . 3.4. The Staggered Grid Formulation for PDFI_SS In three dimensions, Yee ( 1966 ) worked out a second-order-accurate finite-difference formulation for Maxwell’s equations, pointing out that if one places different variables into different locations within the grid, the governing continuum equations (the curls in Maxwell’s equations) become conservative when written down in a finite-difference form. In recent years, “mimetic methods” have been developed, which are higher-order analogs to the Yee grid, in that different variables are defined at different locations within a voxel (such as at interiors, faces, or edges), and with some internal structure in these voxel subdomains allowed. The locations of the variables depend on which integral conservation law is being applied (see, e.g., Candelaresi et al. 2014 and references therein). For our work, the second-order-accurate Yee grid is sufficient. The Yee grid is the basis for the numerical implementation of the MF code described by Cheung DeRosa ( 2012 ). In PDFI_SS, we have a slightly different situation, where most of the calculations are defined on a

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