for eccentric and radial orbits in the case M g = 3.2 × 10 11 M ⊙ . In the case e = 0.7, it is worth noting that at every passage at pericentre some stars are stripped away from the GC and tend to move on precessing ellipses whose semi-major axes increase with the number of passages at pericentre. On the other hand, when the GC experiences a ‘head-on’ collision with the SMBH, stars are scattered backwards at different velocities, depending on the impact parameter that characterizes each star–SMBH interaction. This peculiar interaction leads to the formation of a wake behind the former GC, with a structure similar to the propagation of a wave on a water plate. Figure 4. Snapshots at different times (labelled) of the cluster on an eccentric orbit in the case M g = 3.2 × 10 11 M ⊙ . The central black dot represents the BH with mass M BH = 5 × 10 8 M ⊙ . The x–y plane is the GC orbital plane. Figure 5. Same as in Fig. 4 , but for a cluster on a radial orbit. Therefore, although it is quite difficult to identify the ‘core’ of the GC, the interaction between an SMBH and an infalling GC produces debris that hides much informations about the initial orbital parameters of the GC. However, due to the fact that it is almost impossible to identify a centre for the cluster in these cases, we limited our analysis to the estimate of the amount of GC mass which remains confined to an inner region of the galaxy, around the SMBH. When considering the simulations with M g ≥ 10 11 M ⊙ , we found that eccentric and radial orbits allow a more efficient transport of mass, towards the Galactic Centre, than circular orbits do. On these orbits, the GC can carry about 20 per cent of its initial mass to the Galactic Centre, while, on circular orbits, such percentage is limited to few per cent. On the other hand, for lower galaxy masses, M g 10 11 M ⊙ , we found that the major contribution to the deposited mass comes from GC on circular orbits. We will deepen the discussion about these results in Section 4 . 4 DISCUSSION In this paper, we have presented several N -body simulations to study the interaction between a single GC and an SMBH in the Galactic Centre. The results shed light on the actual mechanisms of mass accumulation in the central region of a massive galaxy ( M g ≥ 10 11 M ⊙ ). Fig. 6 shows the GC cumulative mass profile as a function of the distance from the SMBH after the orbital decay process, for all the orbits considered and in the case of M g = 10 10 M ⊙ and M g = 3.2 × 10 10 M ⊙ . Considering our lightest galaxy model, we found that circular and radial orbits are both efficient in the transport of mass towards the Galactic Centre. In such cases, the mass left to the Galactic Centre exceeds 80 per cent of the initial mass of the GC. On the other hand, in heavier galaxy models, only GCs moving on initial nearly radial orbits deposit a non-negligible amount of their mass (∼20 per cent of the initial mass of the GC) in the inner 20 pc of the galaxy.

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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion 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 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 B-CelestialObjectRegion I-CelestialObjectRegion 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 O O O O O O O O O O O O O O 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 B-CelestialObjectRegion I-CelestialObjectRegion 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 B-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 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-CelestialObjectRegion I-CelestialObjectRegion 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


available to be detected by standard methods. These proto-clusters, likely lying at , are assumed to be in an active star-forming phase, but not yet fully virialized. To investigate these earlier evolutionary stages we need different approaches, such as the one presented in this paper. During the past decade, more and more proto-cluster candidates have been detected through different techniques, using X-ray signatures, stellar mass overdensities, Ly emission, and association with radio galaxies (e.g., Brodwin et al. 2005 ; Miley et al. 2006 ; Nesvadba et al. 2006 ; Doherty et al. 2010 ; Papovich et al. 2010 ; Hatch et al. 2011 ; Gobat et al. 2011 ; Stanford et al. 2012 ; Santos et al. 2011 , 2013 , 2014 ; Brodwin et al. 2010 , 2011 , 2013 ). However, only a few detections have been done in random fields (e.g., Steidel et al. 1998 , 2005 ; Toshikawa et al. 2012 ; Rettura et al. 2014 ), and most of these detections are biased towards radio galaxies or quasars ( Pentericci et al. 2000 ; Kurk et al. 2000 , 2004 ; Venemans et al. 2002 , 2004 , 2007 ; Galametz et al. 2010 , 2013 ; Rigby et al. 2013 ; Wylezalek et al. 2013 ; Trainor Steidel 2012 ; Cooke et al. 2014 ; Noirot et al. 2016 ), or obtained over very limited fractions of the sky, e.g., in the COSMOS field, which is 1.65 ( Capak et al. 2011 ; Cucciati et al. 2014 ; Chiang et al. 2014 ) and in the Hubble Space Telescope Ultra-Deep Field ( Beckwith et al. 2006 ; Mei et al. 2015 ) with its area. Since the expected surface density of such massive proto-clusters is fairly small, a few times ( Negrello et al. 2007 , 2010 ), performing an unbiased analysis of this population of sources requires us to explore much larger regions of the sky. This has been initiated, for example with the Spitzer SPT Deep Field survey covering 94 and has yielded the detection of 300 galaxy cluster candidates with redshifts ( Rettura et al. 2014 ). The submm and mm sky has proved to be an efficient window onto star-forming galaxies with redshifts between 1 and 6, since it allows us to detect the redshifted modified blackbody emission coming from the warm dust in galaxies. Taking advantage of the so-called negative k-correction in the submm ( Franceschini et al. 1991 ), which compensates for the cosmological dimming at high redshift in the submm, many samples of high- galaxies and also proto-cluster candidates have been identified or discovered in this frequency range in the last two decades (e.g., Lagache et al. 2005 ; Beelen et al. 2008 ; Smail et al. 2014 ). As predicted by Negrello et al. (2005) , Clements et al. (2014) showed that the proto-cluster population can be efficiently detected in the submm as overdensities of dusty star-forming galaxies, which has been made possible thanks to the the observations of larger fields in the submm and mm range. The first discoveries of strongly gravitationally lensed galaxies at very high redshift (e.g., Walsh et al. 1979 ; Soucail et al. 1987 ) opened another window onto 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 B-Wavelength O O O O B-Wavelength 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 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 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 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 O O O O O O O O B-Wavelength O O O 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 I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation 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 B-CelestialRegion I-CelestialRegion 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 B-Telescope I-Telescope I-Telescope B-CelestialRegion I-CelestialRegion 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 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 B-Survey I-Survey I-Survey I-Survey I-Survey 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-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 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-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 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 B-Wavelength 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 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


of them had a clear impact. RPC-LAP senses a large increase of plasma during each CME passage on September 11 and 19, with elevated plasma density after. We can differentiate the spiky CME signatures in Fig. 4 from the oscillations in September which correspond to seasonal effects. The spikes also do not manifest until approach into the 30 km orbits. By factoring in the CMEs studied by McKenna-Lawlor et al. ( 2016 ), we can remove two of the large spikes in the September data so they do not appear in Fig. 6 . Another possibility relates to excessive variations of the solar EUV radiation, since the current intensities collected by RPC-LAP sweeps depend on the illumination and ultimately on the solar EUV flux. Thus, a variation in EUV flux should be seen as a variation in the collected photoelectron current density by the probe and qualitatively follow each other. Until the end of 2014 August, the solar EUV flux variations match those of the RPC-LAP current measurements, with a slow increase of the photoelectron emission that can be linked directly to the increase in the flux, as the heliocentric distance between Rosetta and the Sun decreases. The 27-d periodicity in the EUV flux, due to the Sun's rotation (e.g. Woods Rottman 2002 ), has an impact on the photoelectron emission yield of the probes, resulting in the EUV flux and the I 1 current density being in qualitative agreement. After the end of 2014 August, this trend starts to be significantly altered, with the I 1 current density exhibiting large variations that do not correlate any more with the solar EUV flux variation. At the time of the transition into the HDR, the solar EUV flux variations stop driving the evolution of the I 1 currents. Thus, some other effects begin to take precedence on 2014 September 9, and the transition found in the RPC-LAP photocurrent measurements is probably not a temporally isolated artefact due to a sudden variation of the solar flux. On the temporal scale that we study in this paper, we see oscillations of the knee potential, as well as traces of this in the ion attraction current I 1 . These signals correspond to the envelopes of the 6.2-h longitudinal oscillations reported by Odelstad et al. ( 2015 ) and Edberg et al. ( 2015 ), and relate to repeated crossings of more active regions on the comet (with increased neutral outgassing). The envelopes correspond to latitudinal variations, where the Northern hemisphere of 67P is the more active one, and can be seen for electrons with RPC-IES as early as in August (Clark et al. 2015 ), but at a smaller energy and density compared to 2014 September. Although part of the reason for the spread seen at 27–29 km from the nucleus is the oversampling of the projections of active regions once Rosetta enters into the 30 km orbit, this does not explain the sudden change of magnitude upon approach to this orbit. Rosina's Double Focusing Mass Spectrometer (DFMS) measurements show increased density of O 2 following entry

O 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 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-CelestialObject B-Wavelength O O O O O O O B-Instrument O O O O O O O O O B-CelestialObject B-Wavelength 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 B-CelestialObject B-Wavelength O 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 B-Mission O O B-CelestialObject O O O O O O B-Wavelength 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 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-CelestialObject B-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 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 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 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 B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Instrument 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 B-Mission O O O O O O O O O O O O O O O O O O O O B-Instrument I-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O O O O O O O O


metallicity, we illustrate the corresponding comparison in Fig. 11 . It shows that the two samples are similar in metallicity, spanning from −1.4 dex to 0.5 dex. Comparing with LAMOST, KIC metallicities of both samples were underestimated in the metal-poor regime but overestimated in the metal-rich region. In panel (b), we clearly observe this trend with a mean and standard deviation of −0.061 and 0.210 dex, respectively, in the sense of KIC minus LAMOST. This is consistent with the metallicity trend found by Dong et al. ( 2014 ). Figure 11. Same with Fig. 9 but for metallicity. We also checked if the lack of seven independent colours (derived from eight filters: KIC griz , D51 and 2MASS JHK ) contributes to the incorrect KIC-predicted ν max values. However, only three stars do not have a full set of colours and therefore we excluded this effect as a dominant cause for the larger errors in log g . In Fig. 12 , we compare Kepler magnitudes, KIC reddening, asteroseismic masses and spatial distribution for our sample (red data) with the samples from Huber et al. ( 2011 , grey data) and Stello et al. ( 2013 , blue data). Note that we have restricted both comparison samples to seismic gravities in the same ranges as our sample (from 2.2 to 3.5 dex). Figure 12. Histograms for Kepler magnitude K p [panel (a)] and KIC reddening [panel (b)], spatial distribution in Kepler field of view [panel (c)] and histogram of asteroseismic mass [panel (d)]. The red data represents targets of our entire sample, the grey data indicates those from Huber et al. ( 2011 ) while the blue data are targets from Stello et al. ( 2013 ). From Fig. 12(a) , we can see, due to the preferential selection of bright stars by the Kepler Asteroseismic Science Consortium, our sample is systematically fainter by nearly 2 mag, compared to Huber et al. ( 2011 ). We note that Huber et al. ( 2011 ) used data spanning from Q0 to Q6 for long-cadence light curves and Q0 to Q4 for short-cadence light curves, whereas we used data from the entire nominal Kepler Mission (Q0–Q17). Longer time series enhance the probability of detecting oscillations in fainter stars. Both our sample and the Stello et al. ( 2013 ) sample show a sharp cut at K p = 14 mag due to the Kepler selection function (Batalha et al. 2010 ). Panel (b) shows most targets in our sample have reddening around 0.13 mag, which is consistent with the sample of Huber et al. ( 2011 ) but shifted by 0.03 mag compared to the Stello et al. ( 2013 ) sample. Inspection of panel (b) also reveals there are a population of stars with lower reddening peaking at 0.035 mag. To understand if this population of stars reside in special location, we plotted the spatial distribution of targets for our sample (red dots) and Stello et al. ( 2013 , blue dots) in the entire Kepler field of view. We see fewer targets located in the three modules at the bottom-right corner, which are closest to the Galactic plane and have less stars because of the Kepler

O O O O O 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-Dataset O O O O O O O O O O O O O O O O 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-Dataset O B-Telescope 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 B-Dataset B-Wavelength O B-Wavelength O B-Survey B-Wavelength O O O O O B-Dataset O O O O O O O O O O O O O O O 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 B-Dataset 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 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 B-Telescope O B-Wavelength I-Wavelength O O O B-Dataset 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Collaboration I-Collaboration I-Collaboration I-Collaboration 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Dataset O B-Dataset O O O O O B-Dataset O B-Dataset O O O O O O O O O O O O B-Mission I-Mission B-Dataset 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-Formula I-Formula I-Formula I-Formula I-Formula O O 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Citation I-Citation I-Citation I-Citation I-Citation 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 B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O B-Telescope


range of 1.58, 1.60, and 1.57, respectively. However, the power spectra have flattened in the dissipation range for the whole ICME and the previous AF-rich region, with the spectral indexes of 1.12 and 0.82, respectively, which may be aliased by the instrument noises. The power spectrum for the previous AF-lack region still has a steepening in the dissipation range, with the spectral index of 1.84. Similarly, the break frequencies are also comparable to the frequency corresponding to the ion inertial length. Figure 3. Power spectral density of magnetic fluctuations inside the ICME: (A) the whole ICME observed by Wind at 1.0 au; (B) the AF-rich region of the ICME at 1.0 au; (C) the AF-lack region of the ICME at 1.0 au; (D) the whole ICME observed by Ulysses at 5.4 au; (B) the AF-rich region of the ICME at 5.4 au; (C) the AF-lack region of the ICME at 5.4 au. The dashed colored lines are the power-law fitting results. The red ones are in the inertial range, while the green ones are in the dissipation range. The vertical dashed lines represent the proton gyro-frequency, f pc , the break frequency, f b , and the frequency corresponding to the ion inertial length, f di . The energy cascade rate then can be calculated from Equation ( 1 ). For the whole ICME, the energy cascade rates at 1.0 au are derived to be about 2688.6 J kg −1 s −1 , which is close to the lower limit (2970 ∼ 65,320) estimated by Vasquez et al. ( 2007 ) when the solar wind speed is between 350 and 400 km s −1 . When the ICME propagates to 5.4 au, this value reduces to 274.7 J kg −1 s −1 , suggesting that the capacity of turbulence cascade reduces as the ICME propagates outward. For the AF-rich region, the energy cascade rate drops from 3312.2 to 89.7 J kg −1 s −1 accompanied with the disappearance of AFs, when the ICME propagates from Wind to Ulysses . For comparison, the energy cascade rates for the AF-lack region can be maintained at a certain level, and the values at 1.0 and 5.4 au are 526.9 and 737.3 J kg −1 s −1 , respectively. In addition, Smith et al. ( 2006 ) have found the dependence of the dissipation range spectrum on the rate of energy cascade through the inertial range. Our results are also in agreement with their findings that the steeper spectral in the dissipation range forms from greater cascade rates. Liu et al. ( 2006 ) have taken into account the Coulomb energy transfer between protons and alphas, and derived the equations of the heating rate required for protons ( ε p ) and alphas ( ε α ) to produce the observed temperature profile. The specific expressions are as follows where k B is the Boltzmann constant, T p and T α are the temperature of protons and alphas, m α is the alpha mass, τ pα and τ αp are the Coulomb collision timescales and have a relationship of , n α is the alpha number density, and τ e is the expansion time of the plasma. For more details, please refer to Appendix A in Liu et al. ( 2006 ). The total required heating rate ( ε re ) 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O 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 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 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 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 I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O


within the assumed cosmological prior for the parameters . As specified above, in this investigation we assume very conservative prior h ∈ [0.6, 0.85] and Ω m ∈ [0.04, 1] (fixing ). Therefore, F − ( z ) = 0.6[0.04(1 + z ) 3 + 0.96] 1/2 and F + ( z ) = 0.85(1 + z ) 3/2 . Secondly, because of peculiar velocities v p , the apparent redshift z obs of the GW host might be different from the cosmological redshift z determined by the Hubble flow, thus adding a further uncertainty term to the estimated redshift. To account for this effect, for each event defined by D l ± Δ D l , we assign a possible host redshift range . We assume v p = 500 km s −1 , consistent with the standard deviation of the radial peculiar velocity distribution observed in the Millennium run (Springel et al. 2005 ). We neglect additional contributions to the apparent redshift of the GW event coming from kicks at the compact binary formation, which can be as high as 360 km s −1 (Kelley et al. 2010 ), thus comparable with our choice of the peculiar velocity. The support Θ that enters in the evaluation of the galaxy likelihood function in equation ( 8 ) is the error-box defined by . To perform the proposed statistical D l – z reconstruction, we now need to attach a realistic distribution of galaxy to each of those error-boxes. We use the simulated observed light cones of Henriques et al. ( 2012 ) constructed from the Millennium run. The simulation assumes a Λ CDM cosmology with (which justifies the cosmological parameters assumed above to construct the BHB population). Because of the resolution of the simulation, galaxy distributions extracted from it are reliable only down to M * = 10 9 M ⊙ . We checked that at the magnitude cut imposed by Henriques et al. ( 2012 ), their reconstruction of the all-sky galaxy map is complete down to this mass limit out to z ≈ 0.15. The all-sky galaxy map provides sky location, stellar mass, cosmological redshift z , and observed redshift z obs (i.e. inclusive of peculiar velocities) for each galaxy. To select galaxy counterpart candidates to each GW event, we proceed as follow: we fix the cosmological parameters to the values of the Millennium run and we select all galaxies with a cosmological redshift (i.e. excluding peculiar velocities) in the volume shell z ± Δ z consistent with the LISA measurement D l ± Δ D l ; we randomly choose one of those galaxies to be the ‘true host’ of the GW signal. Note that this amounts to say that the true host can be anywhere in the interval D l ± Δ D l , i.e. we are not centring the error-box around the true host; we consider the error-box defined by , where ΔΩ is centred at the sky location of the selected ‘true host’; and we list all the galaxies with M * > 10 9 M ⊙ that fall within the error-box. In doing this, we make sure that their observed redshift z obs lies within the selected range. Each galaxy z obs is the relevant z j that enters in equation ( 8 ). In fact, in a real cosmological measurement one can

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


collisions in this specific case. Table 1. Percentage of collision, escape survivors and the cases of secondary disruption for every mass ratio q . This percentage is out of 90 different initial obliquity θ. Survivors mean cases that the lifetime is larger than 200 yr. We simulated 1260 different initial conditions and found that secondary fission should occur in 16 per cent of the cases and collisions in 22 per cent of cases. So, a significant part of the results are candidates to become a stable binary system. Interesting results are the spin fission phenomena in cases with q > 0.20. The cases with q > 0.20 have at least a few percentage of secondary disruption. Jacobson Scheeres ( 2011 ) did not find secondary spin fission in cases of q > 0.20 and their conclusion is that the chaotic systems with q > 0.20 can only become a stable binary through tidal process or BYORP process, but with our results we show that for the non-planar case this is not true. In other words, it is possible get a stable binary for q > 0.20 through a dynamical process (secondary disruption), but with low probability. It means that non-planar systems with large mass ratio can become a stable binary following the small ratio mechanisms outlined in Jacobson Scheeres ( 2011 ), expanding this route to forming binaries. We also analysed the rotational mode of the secondary characterized by the dynamic inertia I D (Scheeres et al. 2000 ) for all escape cases, where I D = H 2 /2 E k , H is the magnitude of rotational angular momentum and E k is the rotational kinetic energy. The parameter I D tells whether the body is in a tumbling or non-tumbling state, and in the tumbling case whether it is in a long-axis mode (LAM) or short-axis mode (SAM). Uniform rotation (non-tumbling) can be about the largest moment of inertia I D = C or about the smallest moment of inertia I D = A , and for all values of I D between A and C the body is tumbling and will have precession and nutation. The condition for a body to be in a SAM rotation mode is ( B I D C ), while the condition for it to be in an LAM rotation mode is ( A I D B ). We choose nine equal regions between the values of A and C and computed the percentage of cases that the values of I D finished in each region, so we made a plot of these percentage of cases as a function of for all mass ratios studied, as shown in Fig. 14 . We can conclude that all of the bodies are in some state of tumbling after escape. Most of them are in SAM, and are closer to uniform rotation about the maximum moment of inertia. The bodies between B and A are in a LAM, and will likely take longer to settle into the principal axis rotation. We note that the most probable state seems to be close to rotation about the intermediate axis, followed by a SAM state and followed by a LAM state. In contrast we find all of the primary body spin states to be in a SAM state, with small excitation. Figure 14. A bar graph of percentage cases as a function of . The vertical

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


typically multi-periodic oscillators that can show very rich pulsation frequency spectra that challenge asteroseismic analyses and the theoretical interpretation of the observed pulsation frequencies. A- and F-type stars can also show a combination of pressure ( p -), gravity ( g -), and Rossby ( r -) modes (e.g. Uytterhoeven et al. 2011 ; Saio et al. 2018 ). Hence, we investigated all available photometric data sets of β Cas for the presence of g - and r- modes in addition to the already described p -modes. g -modes would be found in the range of ∼3.3 to 0.3 d −1 in frequency (corresponding to ∼0.3 to 3.0 days in period; Kaye et al. 1999 ); r- modes appear as a group of peaks below m times the rotational frequency. BRITE-Constellation and SMEI data are known to allow the detection of g -mode pulsations when the standard procedures for data reduction are applied (e.g. Kallinger et al. 2017 ; Zwintz et al. 2017 ). The BRITE-Constellation and SMEI data for β Cas do not show any signs for g -mode pulsations. We also checked the TESS data for the potential presence of g -modes before we applied the Gaussian filter to remove the instrumental signal, and also did not find any evidence for them (see Fig. B.1 ). In all our data sets, we do not detect a frequency that can be attributed to the rotation of β Cas or a group of peaks typical for r -modes. There are two prominent physical effects that complicate the asteroseismic analysis of δ Scuti stars. One effect is moderate to fast rotation, which can cause a splitting of the pulsation modes with the same n and l values but different m values (e.g. Lignières et al. 2006 ). The second effect is the presence of magnetic fields, which is a rather recent discovery for δ Scuti stars (e.g. Neiner Lampens 2015 ; Neiner et al. 2017 ). In this context, β Cas is a special case because observationally it shows only three independent p -mode pulsation frequencies and not several dozen to hundreds of modes. The reason might be connected to the inclination angle of only 19° ( Che et al. 2011 ) and the presence of the magnetic field. Despite the rather high rotational speed of β Cas, no sign of a rotational splitting can be detected in the present observational material. We calculated the pulsation constant, Q , based on our values for the three independent frequencies, F 1 , F 2 , and F 4 , and our fundamental parameters, T eff = 6920 K and log g = 3.53 cgs. The pulsation constant Q ( Petersen 1976 ; Stellingwerf 1979 ) can be expressed as: where P is the pulsation period in days and M bol is the bolometric magnitude, which is defined as M bol = M V + BC. The absolute magnitude of 1.14 mag is calculated from the distance of 16.8 pc, and the bolometric correction, BC, can be calculated from the relation given by Reed (1998) . The pulsation constant has a value of 0.033 for fundamental mode pulsation ( n = 0), 0.025 for the first overtone ( n = 1), 0.020 for the second overtone ( n = 2), and 0.017 for the third

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


F336W, F438W) = 0.0 and Δ(F275W−F814W) = 0.0 in the chromosome maps. The results are plotted in the two panels of Fig. 2 . We display on the horizontal axis the different chemical compositions investigated. The vertical axis displays the indices ΔC(F275W, F336W, F438W) and Δ(F275W−F814W) for all our synthetic populations. The results are quite clear. The index Δ(F275W−F814W) is affected essentially only by variations in Y; instead, the ΔC(F275W, F336W, F438W) index is in comparison weakly sensitive to Y, but is mainly affected by variations in N (e.g. Piotto et al. 2015). The range of Δ(F275W−F814W) values for P1 stars observed in several clusters like M 3 and NGC 2808 discussed here is therefore due to a range of Y abundances. This is however not found in other clusters, for example NGC 288. The extended P1 sequences in the chromosome maps also display a tilt towards increasing ΔC(F275W, F336W, F438W) for decreasing Δ(F275W−F814W) (e.g. Milone et al. 2017 ). Based on Fig. 2 , this can possibly be explained by a narrow range of N amongst P1 stars. We conclude that the spread in P1 stars indicates that a relatively wide spread of Y is present within this population, possibly accompanied by a narrow range in N. Fig. 2. Expected chromosome map indices of RGB stars with P1 chemical composition (squares), P1 with He enhanced by ΔY = 0.05 (triangles up), P1 with He enhanced by ΔY = 0.15 (triangles down), P2 extreme (crossed squares), P2 extreme with ΔY = 0.05 (stars), and P2 moderate with ΔY = 0.05 (crossed circle); see text for details. In summary, P1 stars are defined by their location on the chromosome maps, which is consistent with the position of stars with homogeneous and field-like [O/Fe] and [Na/Fe] measurements in the clusters like M 4 and NGC 2808 investigated here. These P1 stars are often called first-generation stars in the literature, according to the proposed cluster self-enrichment scenarios. We have shown here that P1 stars in clusters with extended P1 in their chromosome maps, such as NGC 2808 and M 3, show a spread of initial Y and a small range of N. In other cases, like NGC 288 with a non-extended P1 in the chromosome map, the Y and N abundances are also homogeneous, like O and Na. The former behaviour is entirely unexpected in the context of all scenarios for the origin of MPs. 3.2. Distribution of the subpopulations along the RGB An independent test for He-abundance variations among P1 stars is to look at the optical colour distributions of each of the identified subpopulations. Bolometric corrections in optical filters are negligibly affected when moving from typical P1 to P2 metal distributions (e.g. Sbordone et al. 2011 ), but an increase in Y shifts the RGB towards bluer colours because isochrones with enhanced He have hotter T eff values. For each cluster, we first identified different RGB subpop-ulations from the chromosome maps in Fig. 1 . The mean values of the ΔC(F275W, F336W, F438W) and Δ(F275W−F814W)

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


spiral and elliptic galaxies, with the type assigned by visual recognition from archival optical images. For all 31 sources we found the spectral signature of the re-emission by an obscuring torus. From the peak frequency and luminosity we can derive a mean dust temperature of T T ∼ 300 K and a mean covering factor of q T = cos θ T = 0.3; most of the sources have q T 0.4. Therefore, from Eq. ( 7 ), we obtain a mean radius of R T ∼ 5 parsec. To best describe the SED of all the 20 unknown AGN sources with data in the millimeter and submillimeter wavelength band, we had to consider a second emitting toroidal structure at a larger distance from the black hole. It is reasonable to interpret this component as a dust belt inside the galaxy rather than a proper dusty torus surrounding the black hole. The mean obtained value for the temperature distribution is T DB ∼ 50 K, the mean covering factor q DB = 0.6, and the distance distribution shows a peak at ∼150 pc, where some sources reach ∼1 kpc. In Fig. 8 we show the distribution of the emitting structures distance. In evaluating the absorption and consequent re-emission on this scale, we must take into account the contribution of stars. It is reasonable to think that a large fraction of the radiation heating the dust does not come from the AGN, but from nearby stars. Nonetheless, since the dust belt blackbody has a peak luminosity that is, on average, equal or larger than the host galaxy peak luminosity, we can comfortably assume that the stars alone cannot provide enough energy to explain the measured emission. Furthermore, a heavily obscured accretion disk, whose presence is highly suggested by the X-ray spectral emission, could easily provide the necessary energy to heat the dust emitting this spectral feature. This is a clear clue that the host galaxy environment plays a key role in obscuring the AGN. Fig. 8. Distance distribution from the black hole of the IR and submillimeter emitting structures found in our work. The blue columns represent the radius of the classical obscuring torus in the AGN inner region; the red line indicates the radius of an emitting dust belt, of which we have found traces in 20 out of 31 sources. The distances are derived from Eq. ( 7 ). Furthermore, we can make two considerations about the general slope of the SED in these 31 sources. First, all but five sources show a clear sign of absorption in the soft-X ray energy range. Those that do not are SWIFT J0317.1+1542, SWIFT J0329.4−2157, SWIFT J1643.4+0986, SWIFT J2004.6−1111, and SWIFT J0302.2+6853. Of these, only the last source shows no signs of re-emission in the submillimeter band, which in our model is interpreted as the emission from a dust belt. These sources need more in-depth studies, in particular obtaining optical spectra could be very useful. Also, regarding the low-energy emission, most of the sources with data at submillimeter wavelengths show an IR-to-optical luminosity ratio of about one, with a few

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 B-Formula I-Formula I-Formula I-Formula I-Formula 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 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 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 B-Formula I-Formula I-Formula I-Formula I-Formula 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 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-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 B-Wavelength I-Wavelength O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject 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 O O O O O O O O O O O O B-Wavelength O O O B-Wavelength O O O O O O O O


the differing resonance profiles converging on to differing threshold positions. The present calculation has shifted the thresholds to lie in their exact observed positions listed in Table 1 . In Fig. 6 we present the effective collision strengths for some higher lying transitions, the 3d 7 4 F 9/2 –3d 7 2 G 9/2 (1 → 8), 3d 7 2 P 1/2 –3d 7 2 H 9/2 (11 → 13) and 3d 7 2 G 7/2 –3d 7 2 H 9/2 (9 → 13). Excellent agreement is evident for all three transitions when a comparison is made with the data of Storey Sochi ( 2016 ) across all temperatures where a comparison is possible. The collision strengths and effective collision strengths for all 42 486 forbidden and allowed lines are available from the authors on request but we present in Table 4 the effective collision strengths for all transitions among the lowest 11 3d 7 fine-structure levels across nine temperatures of astrophysical importance. 1 Figure 6. The effective collision strengths, Υ 1 → 8 (top), Υ 11 → 13 (middle) and Υ 9 → 13 (bottom), presented as a function of temperature in K (bottom). Solid black lines with stars are the current data set and the dashed line with crosses represent the results from Storey Sochi ( 2016 ). Table 4. Effective collision strengths as defined by equation ( 5 ) are presented between an upper ( j ) and lower state ( i ), across a range of nine temperatures (K), 3800 T 25 100 K. We have denoted ( a ) ≡ ×10 a . 3.3 Line ratios Combining the electron-impact excitation rates with the decay rates ( A -values), it is possible to investigate important infrared and visible line ratios. For simplicity and purpose of this study, we neglect the recombination and ionization process from the neighbouring ion stages and focus on the populating mechanisms solely of Co iii . Assuming local thermal equilibrium and detailed balance, we can derive the collisional radiative modelling approach detailed in (Griffin et al. 1997 ). We can therefore investigate selected transitions that have been suggested as temperature or density diagnostics. We present in Table 5 a comparison of emissivities (ρ ij ) between our current results, and those from Storey Sochi ( 2016 ) for particular transitions at electron densities N e = 10 4 cm −3 and N e = 10 7 cm −3 . The emissivities are calculated relative to the H β line [see equation ( 1 ) in Storey Sochi ( 2016 )], where the effective recombination coefficients can be found in Hummer Storey ( 1987 ) and Storey Hummer ( 1995 ). The wavelengths are those from National Institute of Standards and Technology (NIST) and we use our current set of A -values. In general there is excellent agreement for all emissivities considered with the major outlier ρ 1 → 2 found to be around 30 per cent at N e = 10 4 cm −3 . This is evident from the strong Υ 1 → 2 from Fig. 5 (a) at T = 10 4 K. By replacing this one effective collision strength to Υ 1 → 2 ( T = 10 000) = 4.18, then the ρ 1 → 2 differences are within a much more agreeable 8 per cent. A similar

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


therefore agree. At 5.5 z 5.9, the GOLD distribution lies between the bounds shown in Fig. 8 . At z > 5.9, the difference is attributable to the small sample size of the sub-samples (see the text). The differences between the SILVER and GOLD samples at 5.7 z 5.9 are due to the small sample size of the GOLD sample. The difference in number of lines of sight in this range is a factor of 2.5, which reflects the fact that very high-quality spectra are harder to obtain for quasars at z > 6.0. The SILVER and GOLD sample are marginally consistent according to a KS test( p 0.2). On the other hand, the sizes of the SILVER and GOLD samples are similar at 5.9 z 6.1, and agree with each other well. However, the additional lines of sight included in the BRONZE sample are of insufficient quality to distinguish opacity beyond τ eff ≳ 3. Since the mean opacity measured in the SILVER sample is roughly τ eff ∼ 4.5, these S/N 5 spectra yield upper limits (equal to twice the error) which are very poor in comparison. The small sample size in this bin (8 measurements in the GOLD sample, and 16 in the BRONZE one) mitigates the significance of this discrepancy, with the KS level indicating only mild discrepancy ( p 0.2). In addition, two of the BRONZE lines of sight do show tentative signs of residual transmission. The z = 6.34 quasar J1148+0702 displays a transmission peak within 6 Å of the formal end of the proximity zone, while the z = 6.23 quasar J2325 displays such a peak outside of its proximity zone but has an S/N of 1.8, making it impossible to definitely rule of reduction issues. Further scrutiny of these objects is required in order to determine whether these peaks may be related to particularly long and sporadic proximity zones. In light of these results, we decide to adopt the SILVER distribution in every instance in this paper where the data are presented or analysed in the form of a CDF. The maximum-likelihood method described above has been explicitly designed to account for observational errors and sensitivity, and we accordingly use the BRONZE sample only in Section 6 . 5 COMPARISON WITH MODELS In this section, we compare the updated CDFs of Lyman-α transmission in 50 cMpc h −1 windows with three simulations. Here, we briefly describe these simulations and outline their most relevant features. When comparing predictions from these numerical simulations to observational results, it is important to keep in mind a few caveats. First, all of the following numerical models explicitly rescale the ionizing background intensities to match the observed mean effective optical depth of the observations presented in this paper, 〈 F 〉 obs , by freely choosing a parameter A suchthat: At low redshift ( z ≲ 5.7), this rescaling is small and is somewhat justified by the difficulty of self-consistently generating the ionizing UVB. In the following analysis, all simulation snapshots have been rescaled in this way to observed 〈 F 〉 obs for each redshift bin. The

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and Fig. 6 , we have shown that the radial density profile of the ejecta is well described by a power-law function of the velocity v with an exponent 6. This distribution is shallower than those of expanding envelopes in normal SNe (from n 9 to 12 as assumed in our model), which is clearly a consequence of the additional energy injection. The density distribution of freely expanding ejecta is a key to distinguishing existing models of SLSNe and other extraordinary SNe. In the following, we discuss how ejecta with such density structure form in the presence of the central energy source. For simplicity, we restrict ourselves to the Newtonian limit. We consider the following idealistic case. In normal SNe, the strong blast wave driven by a point explosion is the only way to transport the explosion energy to the outermost layer. As a result, only a small fraction of the energy is deposited in the outer envelope, while the inner ejecta possess the bulk of the energy, leading to a steep kinetic energy distribution (Matzner McKee 1999 ). In contrast to normal SNe, the central engine deposits an energy much larger than the supernvova ejecta for a time-scale much longer than the expansion time-scale of the ejecta. In our setting, the additional energy is injected at a constant rate L . Therefore, if the energy is continuously transferred throughout all the layers of the ejecta without loss or stagnation, the kinetic energy flux v 3 of the ejecta powered by the energy injection at a radius R 0 roughly proportional to the energy flux . The presence of relativistic flows leaking from the hot bubble makes the efficient energy transport possible. The flows can directly bring and deposit the additional energy throughout different layers. Thus, the following relation, is expected to hold for SN ejecta with a sufficiently long-term energy supply. In other words, the kinetic luminosity of the flow is constant. This relation gives the density distribution immediately after the ejecta are affected by the energy injection. We denote the time when the ejecta following this relation are created by t 0 . The ejecta are not freely expanding at this time. As the ejecta expand to the surrounding space, the density distribution would gradually be modified. Our goal is to derive the density distribution at the freely expanding stage. We regard the radial velocity v as a Lagrangian coordinate and derive the density distribution as a function of v . We consider a concentric shell with inner and outer velocity coordinates v and v v . The inner and outer boundaries are located at R R 0 ( v ) and R R 0 ( v v ) at t 0 . The width l s of the shell is initially given by, where the last expression is obtained for a sufficiently small v . The boundaries travel at the velocities v and v v with time and reach R R 0 ( v ) v ( t t 0 ) and R R 0 ( v v )( v v )( t t 0 ) at t . For t much longer than t 0 , the initial radius R 0 ( v ) can be neglected and the inner and outer radii are

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


iron. Finally, the microturbulent velocity (ξ t ) is established by erasing any trend between the abundance from Fe i and EWr. The same approach to constrain ξ t is used consistently in the three independent analysis. The atmospheric parameters are then set in an iterative fashion, first setting the temperature, and then revisiting T eff as the gravity and microturbulence are tweaked. The error associated with the determination of ξ t is estimated by propagating the uncertainty in the slope in the rEW versus Fe abundance plane. This uncertainty is of the order of ≃0.07 km s −1 . The derived T eff are typically based on more than ≃70–130 Fe i lines and have internal uncertainties of about 50–60 K, while the internal uncertainties in log g are of the order of ≃0.06 dex. The derived atmospheric parameters and [Fe i /H] and [Fe ii /H] abundances are listed in Table 2 , along with their associated uncertainties. The left-hand panel of Fig. 2 illustrates how they compare to Y14 results. We observe that our T eff , log g , and ξ t estimates are in excellent agreement with those listed in Y14 , being the average difference between our and Y14 estimates Δ T eff = −19 ± 8 K, Δlog g = −0.0 ± 0.0 dex, Δξ t = −0.09 ± 0.02 km s −1 . The agreement between our [Fe i /H] and [Fe ii /H] abundances and those listed in Y14 is also very good. In particular, we note that the metal-poor stars identified by Y14 , i.e. see Table 1 , are also the most metal-poor stars in our analysis (see Table 2 ). Additionally, the four stars with [Fe/H] ≃ −1.0 dex are the same metal-rich stars in Y14 . In the left-hand panel of Fig. 3 , we plotted the histograms of the [Fe i /H] and [Fe ii /H] distributions when the full spectroscopic analysis is performed. Our ability to portrait the [Fe i /H] and [Fe ii /H] distributions for this small data set critically depends on the adopted bin size (Δ). To adequately represent the shape of the underlying distribution, in Fig. 3 we selected Δ as the value which minimize C (Δ) = (2 me − va )/Δ 2 ; where me and va are the mean and variance of the data set. As one can see, we obtain an iron dispersion that mirrors the dispersion measured by Y14 . We find a large (Δ[Fe/H] ≃ 0.8 dex) iron distribution with two prominent peaks at [Fe/H] ≃ −1.7 and −1.0 dex and a smaller component at [Fe/H] ≃ −1.5 dex. The iron dispersion within the metal-poor stars is larger (Δ[Fe/H] ≃ 0.4 dex) than that expected from observational errors alone, indicating the presence of two separated subpopulations with different metallicity. Photometric analysis . We use VI WFI photometry to derive photometric T eff and log g * estimates. 2 Input T eff values have been computed by means of the ( V − I ) 0 – T eff transformation by Alonso, Arribas Martínez-Roger ( 1999 ) adopting a colour excess E ( B − V ) = 0.02 mag taken from the most update version (2010) of the McMaster catalogue (Harris 1996 ). Surface gravities have been computed assuming the photometric T eff , 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 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 B-Citation O O O O O O O O B-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula 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 B-Citation O O O O O O O O O O O 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 B-Formula I-Formula I-Formula I-Formula O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Citation O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula 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 B-Wavelength 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 B-Model I-Model I-Model I-Model I-Model I-Model I-Model I-Model I-Model I-Model O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Dataset I-Dataset B-Citation I-Citation O O O O O O O O O O O O O


pressure wherever a shock is detected (see Fig. 2 ) since shocks are the acceleration sites for CRs. The following conditions (Gupta, Nath Sharma 2018b ) have been used to detect whether a particular region within the simulation box is shocked or not: ; Δ x |∇ p |/ p > δ tolerance ; ∇ T .∇ρ > 0. Figure 2. Comparison of density and thermal pressure between TH and TH+CR runs for M12 galaxy at ≈20 Myr simulation time. The three sets of plots (from left) are for SFR = 0.1, 1, and 10 . We also show the CR pressure in a separate plot. Green dots show the shock locations. In the TH+CR runs, these are the locations where CR energy is injected. For all our simulations, we have used δ tolerance = 0.5. Wherever a shock is detected, we redistribute the total pressure between thermal and CR component following the CR pressure component fraction w = p cr /( p th + p cr ). In all our models, we have used w = 0.2. 3 RESULTS To understand the properties of outflowing gas between the models with and without CRs, we investigate (i) the morphology of the outflowing gas, (ii) outer shock position, (iii) mass-loading factor, and (iv) temperature distribution of the gas. These quantities have been used in previous studies to quantify the dynamical impact of CRs. Here we present the results from our simulations of three different galaxies with halo masses of , and using three different constant SFRs: 1 0.1, 1, and 10 . To represent runs with and without CRs, we have used two different labels: ‘TH+CR’ and ‘TH’, respectively. Note that we have used our simulation for M11 galaxy as the fiducial run. The used parameters are given in Table 1 . 3.1 Outflow morphology In Figs 2 , 3 , and 4 , we show the morphology of outflowing gas for three different halo masses for TH and TH+CR runs. For all runs, we find that the shape of the outflow is nearly spherical in nature. For M8, the shape is expected to be more spherical because, in this case, the galactic disc is much smaller than the outflow length-scale and the disc is not included in our simulation. For all cases, the forward and reverse shocks are prominently distinguishable, except for SFR = 0.1 in M12 galaxy. In this case, the outflow has an irregular shape and does not show a prominent reverse shock; the locations of shocks are rather distributed inside the bubble. This can be seen from the shock locations, as displayed by the green dots in the second rows of Figs 2 , 3 , and 4 . Figure 3. Same as Fig. 2 , for M vir = 10 11 M ⊙ . Figure 4. Same as Fig. 2 , for M vir = 10 8 M ⊙ . For SFR = 0.1 (first column of Figs 2 , 3 , and 4 ), a comparison of different snapshots among three haloes shows that for M12 galaxy, before the outflow breaks out of the central region, an acoustic wave propagates from the central region. This can be understood as follows. In our set-up, CRs are initially confined in the galactic disc and the halo has very low CR pressure. Due to the pressure gradient, the CRs start to diffuse out of the disc

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


expects stalled bubbles (and consequently, a size distribution with slope ≈−2.7) in the presence of cooling, or if this depends on the ambient pressure. In order to answer the question, we have run our simulations for two different pressures. Fig. 12 shows the scatter plot of bubbles (similar to Figs 6 and 8 for our fiducial ISM pressure) for two different values of ISM pressures, at 30 Myr. Essentially, we have decreased the non-thermal contribution in the ambient pressure, by reducing the equivalent temperature from 4 × 10 4 to 2 × 10 4 K (left-hand panel) and then to 10 4 K (right-hand panel). Comparing with Figs 6 (adiabatic case) and 8 , we find that the bubbles in these two cases are not dominated by stalled bubbles, as was the case for Fig. 8 . For the left-hand panel, we find that bubbles with L ∼ 5 × 10 36 erg s −1 stall after a time-scale of ∼7 Myr, and bubbles bigger than that at latter times. When the pressure is further lowered, in the right-hand panel, all bubbles keep growing until 30 Myr, and the circumstance is similar to Fig. 6 . We recall that, in these cases of domination by growing bubbles, the size distribution is likely to be steeper than 1 − 2β, as we confirm below. Figure 12. Scatter plot for bubbles for ambient pressure p 0 = 1.38 × 10 −12 dyne cm −2 (left), and 6.9 × 10 −13 dyne cm −2 (right), at 30 Myr. We show in Fig. 13 the size distribution of the bubbles for these values of pressure at two different times after the onset of star formation (10 and 30 Myr). In general, we find that lowering pressure steepens the size distribution, by allowing low -luminosity bubbles to grow to larger sizes. This trend is shown in Fig. 14 , for two different epochs, 10 and 30 Myr. At lower pressures, there is an evolution of the slope between these time-scales, since bubbles keep growing in low ambient pressure, compared to high-pressure environments. Figure 13. Size distribution of bubbles for ambient pressure p 0 = 2.76 × 10 −12 dyne cm −2 (blue), 1.38 × 10 −12 dyne cm −2 (red), and 6.9 × 10 −13 dynes cm −2 (green), at 10 and 30 Myr. Figure 14. The slope of size distribution as a function of ISM pressure, for two different epochs, 10 Myr (green) and 30 Myr (red). The distributions at lower (non-thermal) pressures also show a positive part at small sizes, in addition to the falling numbers at large sizes as we have seen earlier. The rising part comes from growing young bubbles whose age is smaller than the stalling time of the bubble with the lowest luminosity. This generates a peak in the distribution. The peak depends upon the radius of the stalled bubble with the lowest luminosity at the time being observed. Fig. 13 shows that at higher pressures the peak radius is independent of time but starts varying with time as soon as we start decreasing the pressure. This occurs due to the fact that at higher pressures (such as, at 2.76 × 10 −12 dyne cm −2 ), the bubbles with low luminosities stall much before 10 Myr hence the peak

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


to a predicted final binding energy equal to E bin ∼ 2.4 E bin,0 in very good agreement with what shown in Fig. 7 . 3.3 Radial variation of the binary fraction Fig. 8 shows the radial profiles of the binary fraction (defined as the ratio of the number of binaries for each population to the total number of binaries and single stars in the cluster) for all binaries, for FG binaries, and for SG binaries. The low ionization rate of the compact binaries we are considering, facilitates the central accumulation of binaries segregating from the outer regions and results into an increasing binary fraction in the cluster inner regions. It is interesting to note that the evolved binary fraction radial profiles are in general qualitative agreement with those found in the recent observational study of Milone et al. ( 2016 ). Figure 8. Time evolution of the radial profiles of the binary fraction (defined as the ratio of the number of binaries to the total number of binaries and single stars) for the model MPr5f1x3-800. Each profile is calculated by combining three snapshots around the time indicated in the legend. Top, middle, and bottom panels show, respectively, the profiles for all the binaries, for FG binaries, and for SG binaries. The horizontal dashed line in the top panel shows the initial binary fraction. The three panels of Fig. 9 show the cumulative and the differential radial profiles of SG-to-FG binary ratio and the differential SG-to-FG ratio for single stars. The interplay among ionization, ejection and segregation towards the inner regions results into a progressive decrease of the SG-to-FG binary number ratio in the cluster inner regions. The global SG-to-FG binary number ratio decreases with time due to the dominant effect of the preferential ionization or ejection of SG binaries; it is interesting to note that this is opposite to what happens to the global SG-to-FG number ratio for single stars which increases with time because for single stars the preferential evaporation of FG stars (see also Vesperini et al. 2013 ) drives the evolution of this number ratio. Figure 9. Time evolution of the radial profile of the cumulative number ratio of SG to FG binaries (top panel), the differential SG-to-FG binary number ratio (middle panel), and the differential number ratio of SG to FG single stars (bottom panel) for the model MPr5f1x3-800. Lines with different colour and style correspond to the different times indicated in the legend, and the horizontal lines show the global SG-to-FG ratio for binaries (middle panel) and single stars (bottom panel). (Each profile has been calculated using three combined snapshots around the times indicated in the legend.) As shown in Fig. 9 , the SG-to-FG binary number ratio depends on the distance from the cluster centre. It is important to identify the radius R glob , where the local SG-to-FG ratio is equal to the global value of this ratio. In paper I , we have already shown that R glob depends on the initial

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


in Sect. 3 are indicated as blue circles filled with gray, while values for the average stacks in Table A.1 are indicated as open blue circles. The correlation between EW(CIII]) and EW(Ly α ) appears to be very strong ( r P = 0.99, r S = 1) and hides the wide range of properties of CIII] vs. Ly α . As a comparison with the literature, we plot the observed values from the different Ly α samples of Shapley et al. (2003) (green stars, average for Ly α samples), Rigby et al. (2015) (purple diamonds), and Stark et al. (2014 , 2015) (orange squares). This behavior is somewhat expected as the radiative transfer of Ly α or CIII] photons is markedly different. As a striking example we point to object 520391642 (second from the top in Fig. 1 ), which shows strong CIII] (EW > 20), as well as CIV, HeII, or NV, but only weak Ly α emission. Our results compare well with the literature when the large spread of the relation between EW(CIII]) and EW(Ly α ), the selection, and the size of different samples is taken into account. The average properties from stacks show that our results agree well with those of Shapley et al. (2003) . VUDS galaxies span a broad range of properties, including galaxies with and without Ly α in emission, expanding from smaller and possibly biased samples for which a stronger correlation between Ly α and CIII] is reported ( Stark et al. 2014 , 2015 ; Rigby et al. 2015 ; Guaita et al. 2017 ). We find that the chance of emitting CIII] and CIV at an appreciable level (EW > 3 Å) for galaxies with EW(Ly α ) > 25 (the broadly used definition of a LAE) is 42% and 20%, respectively. For galaxies that emit both Ly α and CIII] at these levels, the median EW(CIII]) is 7.7 Å, compared to a median EW(CIV) of 3.4 Å for the average LAE emitting CIV. The overall weak correlation that is observed between CIII] and Ly α emission strength for individual galaxies is in line with findings from photoionization models (e.g., Jaskot Ravindranath 2016 ; Zackrisson et al. 2017 ; Nakajima et al. 2018 ) when a broad range of conditions such as ionizing fields, geometry, and dust opacity is taken into account. 5. Distribution of EW(CIII]) To answer the question of how frequently CIII] emission is observed in UV-selected galaxies, we seek to establish the fraction of CIII] emitters in the VUDS sample as a function of EW(CIII]). While it is straightforward to identify strong CIII] emitters, it is more difficult to separate weak emitters from noise. We therefore proceed with a statistical analysis of the distribution of CIII] emission for the 3899 VUDS galaxies with 2 z 3.8. If CIII] emission is absent and in the presence of noise, the line measurement resulting from the process presented in Sect. 2.2 could result in either a false-positive or negative EW. As CIII] is only expected in emission, we use the distribution of EW measurements that indicates absorption as an estimate of how many CIII] emission lines could be produced from noise spikes, under 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 B-Wavelength I-Wavelength 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 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 B-Wavelength I-Wavelength O O B-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength I-Wavelength O B-Citation I-Citation I-Citation I-Citation 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 B-Wavelength I-Wavelength 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 B-Formula I-Formula I-Formula 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 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 B-Citation I-Citation I-Citation I-Citation O B-Survey 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 B-Wavelength I-Wavelength 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 B-Citation I-Citation I-Citation I-Citation 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 B-Wavelength 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 B-Wavelength 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 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 B-Wavelength 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 O 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 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 O O O O O O O O O O O O O O O O O O O O O O O


the minimum comoving scale R = 8 h −1 Mpc. This cut is translated to an angular harmonic mode on each tomographic bin using the Limber relation l max = k max × χ(〈 z i 〉), where is the mean redshift for the i th tomographic bin of the analysis and χ( z ) the comoving distance computed on the fiducial cosmology of the analysis. Similar approaches were taken in Nicola et al. ( 2020 ) and Doux et al. ( 2020 ). We fix , which yields l max = 80, 127, 172, 215, and 246 for the five redshift bins for DES-Y1 cosmology. We checked these cuts do not change significantly when using a FLASK cosmology. We also keep l min = 43 for all bins based on the impact of binning for our modelling. In order to verify whether these scale cuts are effective to mitigate the effects of nonlinear bias we perform a simple χ 2 test comparing two data vectors: a fiducial data vector generated with a linear bias and a data vector contaminated with nonlinear bias. More specifically, we generate a contaminated data vector with an additional quadratic bias parametrized as a function of the linear bias as in Lazeyras et al. ( 2016 ). Using the FLASK covariance matrix we find that Δχ 2 = 0.2 for the physical cuts, where well below the criterion Δχ 2 1 adopted in Abbott et al. ( 2019a ). Hence the nonlinear biases as modelled above are mitigated by the physical scale cuts. Just for comparison, for the naive scale cuts, we find that Δχ 2 = 14.31 and therefore this further justifies our fiducial analysis choice for the physical scale cuts. 7 RESULTS ON MOCKS In order to validate our pipeline, we ran the implemented nested sampling algorithm using as input a random mock realization as well as the average of the set of mocks. We explored the parameter space: where b i is the constant linear galaxy bias in the redshift shell z i with a flat prior (0.8 b i 3.0), and Δ z i is the respective shift in the mean of the photometric redshift distribution for each redshift bin i : n i ( z ) → n i ( z − Δ z i ). The Gaussian priors are centred in Δ z i = 0 (for the synthetic data used in this section) and width as in Elvin-Poole et al. ( 2018 ). We use the FLASK covariance matrix as the fiducial one. We show that the CosmoLike theoretical covariance matrix yields very similar results in Fig. 4 . 11 In this section, we use the average of the FLASK mocks as a proxy for a noiseless data vector, as used in simulated likelihood analyses (Krause et al. 2017 ). The use of the FLASK covariance matrix gives the uncertainty one would expect for the real data likelihood analyses, which is the goal of the comparison between the harmonic and configuration space performed in the DES-Y1 data in Section 8 . The recovery of unbiased cosmological parameters in this case also shows that the Gaussian likelihood approximation is adequate. Figure 4. Comparison of the bias constraints to the average of the FLASK mocks using either FLASK (DES-Y1 run) or CosmoLike (quick run) covariance matrices with physical scale cuts

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


the AGN luminosity, or the distance from the BH. For simplicity we vary only one of these, the density, while keeping the others fixed. As discussed in following sections, this change, while simple, does serve to illustrate the types of SEDs that will filter through the wind. Changes in the mass-loss rate of the wind can cause changes in the hydrogen density of the equatorial obscurer. We examine the effects of such variations upon the transmitted SED in Figure 3 , which shows three typical SEDs. As the figure shows, the shape of the SED is highly sensitive to the value of the hydrogen density. Figure 3. SED transmitted through equatorial obscurer and incident upon the BLR is shown for three different values of the hydrogen density. The unextinguished SED is also shown. The SED is dramatically dependent on the hydrogen density of the obscurer. High hydrogen densities produce strong absorption in the XUV region and strong emission in the FUV/optical regions. The density and flux parameters chosen here do not matter in detail. The transmitted SED actually depends on the ionization parameter, which is the ratio of the ionizing flux to the hydrogen density (Osterbrock Ferland 2006 ). Increasing the hydrogen density lowers the ionization parameter inversely. Particular values of the density and flux do not matter as long as the ratio giving the ionization parameter is kept constant. As the ionization parameter increases the level of ionization of the gas increases. The gas opacity decreases as the number of bound electrons decreases. The ionization structure changes in ways that produce the three characteristic SEDs shown in Figure 3 . These are the three cases: 1. Case 1 has the lowest density and the highest ionization, and is shown in black. This wind is fully ionized, has no H or He ionization fronts, and nearly fully transmits the entire incident SED. 2. Case 2 has an intermediate density and is shown in blue. This has a He 2+ − He + ionization-front but no H ionization-front. The incident SED is heavily absorbed for the XUV energies, 11 11 We refer to the region 6–13.6 eV (912–2000 Å) as FUV, 13.6–54.4 eV (228–912 Å) as EUV, and 54.4 eV to few hundred eV (less than 228 Å) as XUV. although most of the hydrogen-ionizing radiation is transmitted. 3. Finally, Case 3 is shown with the red line and has the highest density. The wind has both H and He ionization-fronts, and much of the light in the EUV and XUV regions is absorbed. 4. The Response of the BLR to Changes of the Transmitted Continuum We now show how the EWs of the BLR lines in Figure 2 are affected by changes in the transmitted SED of the equatorial obscurer. Figure 4 shows how the EW of the strongest observed lines reacts as the density, n (H), of the equatorial obscurer varies. These changes are due to variations in the SED filtering through the equatorial obscurer. The three general types of SED shown in Figure 3 produce the three different BLR regimes shown in Figure 4 . We

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