of the CCH/HC 5 N ratio between the observed values and the model calculations (Taniguchi et al. 2019a ). The red and black curves show the modeled CCH/HC 5 N ratio and temperature, respectively, in (a) the slow warm-up period model with ζ = 1.3 × 10 −17 s −1 , (b) the fast warm-up period (5 × 10 4 yr) model with ζ = 1.3 × 10 −17 s −1 , and (c) the fast warm-up period model with ζ = 4.0 × 10 −14 s −1 . The orange line indicates the observed value in G28.28–0.36 and the yellow range covers the observed values in the three MYSOs including the standard deviation. The blue line in panel (a) indicates the observed value in L1527 and the blue range covers the error in L1527. Figure 6. Production fraction of formation pathways of HC 3 N during the warm-up period (Taniguchi et al. 2019a ). JHC 3 N denotes molecules on the dust surface. As discussed in Section 4.2 , the observed CCH/HC 5 N ratios in MYSOs agree with the model at a temperature of ∼85 K. This temperature corresponds to 1.08 × 10 6 yr, 16 16 This age does not mean a physical age of an MYSO, because we derive the age from the chemical network simulations and the age will change by the assumed warm-up timescales. as shown in panel (a) of Figure 5 . Just before this time ( t 1.07 × 10 6 yr), the main formation pathway of HC 3 N is the reaction between C 2 H 2 and CN (panel (a) of Figure 6 ). This is consistent with the reaction proposed based on the 13 C isotopic fractionation in G28.28–0.36. The electron recombination reaction of HC 3 NH + is dominant in the model with the high cosmic-ray ionization rate during the whole warm-up period (panel (b) of Figure 6 ). Thus, the 13 C isotopic fractionation of HC 3 N in G12.89+0.49 and G16.86–2.16 can be explained by this high cosmic-ray ionization rate model. These results suggest that the cosmic-ray ionization rates in G12.89+0.49 and G16.86–2.16 may be higher than that in G28.28–0.36 or physical structures in G12.89+0.49 and G16.86–2.16 allow cosmic rays to penetrate into regions where carbon-chain species are formed. The observed CCH/HC 5 N ratios in MYSOs agree with the age of (4.7–4.86)×10 5 yr as indicated in panel (c) of Figure 5 . In summary, the observed CCH/HC 5 N ratios around MYSOs can be reproduced in the chemical network simulations at a given temperature and time. Taking the 13 C isotopic fractionation of HC 3 N into consideration, G12.89+0.49 and G16.86–2.16 may prefer the model with the high cosmic-ray ionization rate ( ζ = 4 × 10 −14 s −1 ), while G28.28–0.36 agrees with the model with the standard cosmic-ray ionization rate ( ζ = 1.3 × 10 −17 s −1 ). These results imply that carbon-chain chemistry in G12.89+0.49 and G16.86–2.16 may resemble that in the OMC-2 FIR 4 young intermediate-mass protoclusters (Fontani et al. 2017 ; Favre et al. 2018 ), but that in G28.28–0.36 is different from the other MYSOs and OMC-2 FIR 4. The clear 13 C isotopic fractionation of HC 3 N in G28.28–0.36 supports its bottom-up formation during the

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


for each of the late-phase loops, the energy input in late-phase loops reaches ∼7.7 × 10 29 erg. For the main flaring loops (as mentioned in Sect. 3.1 ), both of the two-stage energy release processes contribute to the heating. Hence, the heating duration of the main flaring loops is double compared with the late-phase loops. Assuming that the amplitude of heating rate is equal to H 0 , the energy input in the main flaring loops is estimated to be (1.5−3.8) × 10 29 erg. Based on the EBTEL simulation, the energy input in the post-flare loops amounts to (9.2−11.5) × 10 29 erg. 3.6. Nonthermal energy in flare-accelerated electrons Under the thick-target assumption ( Fletcher et al. 2013 ), HXR emissions at the footpoints are produced by collisional bremsstrahlung during the passage of flare-accelerated nonthermal electrons through a dense target, where the electrons are stopped by Coulomb collisions and lose energy in the chromosphere. The localized plasma can be heated up to ∼10 MK accompanied by chromospheric evaporation and condensation ( Tian et al. 2014 ; Li et al. 2015 ; Zhang et al. 2016a , b ; Zhou et al. 2020 ). The distribution of injected nonthermal electrons above a low-energy cutoff ( E c ) is assumed to be in the form of single power law with a spectral index of δ , , where A 0 is a normalized parameter of the total electron flux (in units of 10 35 electrons s −1 ). In Fig. 6 , three characteristic HXR spectra made from detector 9 of RHESSI during the first stage of energy release are displayed. In each panel, the fitted thermal component is drawn with a red line, while the nonthermal component is drawn with a blue line. It is obvious that the isothermal temperatures from RHESSI are close to those derived from GOES. Fig. 6. Three characteristic HXR spectra with an integration time of 20 s (black lines) and the fitted thermal (red lines) and nonthermal (blue lines) components. T e and EM of the thermal component are labeled. The low-energy cutoff ( E c ), spectral index ( δ ) of nonthermal electrons, and of the residuals are also labeled. The delivered energy by nonthermal electrons above E c is calculated by integrating the injected power during 12:40−12:46 UT, where A nt is the electron injection area, E H is set to be 30 MeV ( Warmuth Mann 2016a ). The nonthermal energy in flare-accelerated electrons during the first stage is estimated to be ∼1.74 × 10 30 erg. Considering that the HXR observation from RHESSI was unavailable during the second stage of energy release around 12:47:30 UT, it is impossible to calculate the nonthermal energy of electrons from RHESSI during the second stage. In Fig. 2 b, the second peak of GOES derivative accounts for ∼1/4 of the first peak, implying that energy input during the second stage accounts for ∼1/4 of the first stage. In this way, the nonthermal energy in the flare-accelerated electrons during the second stage is crudely estimated to be 0.44 × 10 30 erg. Therefore, the total nonthermal energy

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


such as the Schmidt–Kennicutt relationship between the surface density and the star formation rate (Schmidt 1959 ; Kennicutt 1998 ) or the density threshold for star formation (Lada, Lombardi Alves 2010 ). In particular, the observed star formation rates for several of these clouds are an order of magnitude below those predicted by these relationships between the density and the rate (Longmore et al. 2013a ). A case in point is the cloud GCM-0.02-0.07 ( M ∼ 10 5 M ⊙ ; Johnston et al. 2014 ) nicknamed the Brick because of its high optical depth in the infrared and lack of internal infrared emission from star formation. Only 3–4 per cent of the mass of this cloud is in dense star-forming regions often called high-mass cores. The elliptical ring also includes the massive star-forming region Sgr B2 with some of the highest rates of star formation in the galaxy and the region Sgr C rich with H ii regions indicating earlier active star formation. Fig. 5 shows the locations of the clouds on their elliptical orbit. Figure 5. Schematic of the clouds in elliptical orbit in the Galactic Centre adapted from Kruijssen et al. ( 2014 ). The view is of the plane of the Galaxy from above. [Kruijssen, Dale Longmore ( 2015 ) suggest that the orbit is not exactly elliptical in that the path from Sgr C may not close back to the location of the Brick.] The clouds on the ring orbit in the direction from the Brick toward Sgr B2. The Brick and other clouds with low star formation rates are near the pericentre, 45 pc from the supermassive black hole Sgr A* (Johnston et al. 2014 ), while Sgr B2 is at the apocentre, twice as distant at 100 pc. Longmore et al. ( 2013b ) suggest that the Brick and nearby clouds are currently undergoing tidal compression that will result in intense star formation by the time the clouds arrive at the location of Sgr B2. The mini-starburst Sgr B2 would then be the result of compression in its earlier passage through the pericentre of the orbit. Observations indicate that the clouds on the orbit between the Brick and Sgr B2, the dust ridge clouds, may be just beginning to fragment (Walker et al. 2018 ). Alternatively, fragmentation and ultimately the star formation rate may be controlled by the turbulent-entropic instability. If orbital shear slows the effective turbulent dissipation rate by supplying fresh turbulent energy, then the difference in orbital shear along the elliptical orbit from pericentre to apocentre may prevent star formation near the pericentre and allow it at the apocentre. Following Kruijssen et al. ( 2014 ), we derive a rotation curve, v circ = 23.5 r 0.38 km s −1 from the mass distribution measured around the Galactic Centre by Launhardt, Zylka Mezger ( 2002 ). At the radial distance from the Brick to the Galactic Centre, 45 pc, the orbital shear across a 10 pc diameter cloud is Δ v = 8.4 km s −1 , while the one-dimensional turbulent velocity dispersion in the Brick is 4.3 km s −1 (Johnston et al. 2014 ). At the

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


from September 22. Additionally, in the image related to the observation from September 22, there is a radial polarization pattern across the Southern half of the core region. This suggests that the magnetic field B is roughly circular and centred on the brightness peak of the core, as one might expect from a helical field when one views it down the axis. This polarization pattern remained through 2016 March. The γ-ray LC from the Fermi -LAT and X-ray LC from the Swift -BAT do not show any obvious flux enhancement during the time of the radio flaring activity, although they show some activity (both BAT and LAT) about 40 d before the radio flare. During this time, there were no optical or VHE observations because of the Sun. Figure 4. A sequence of total (contours) and polarized (colour scale) intensity images of Mrk 421 at 43 GHz obtained with the VLBA. The image is convolved with an FWHM of 0.24 × 0.15 mas 2 along PA = −10°. The global total intensity peak is 329 mJy/beam, and the contours 0.35, 0.70, 1.4, etc. up to 89.6 per cent of the global intensity peak. The colour scale is the polarized intensity, and the black line segments within each image show the direction of the polarization electric vector, with the length of the segment being proportional to the polarized intensity values. The black vertical line indicates the position of the core. See Section 3.2 for details. The low polarization fraction at 43 GHz on 2015 August 1 implies that the magnetic field was very highly disordered in the core at this epoch. A radial polarization pattern, as measured at 43 GHz on 2015 September 22, can result from turbulent plasma flowing across a conical standing shock, as found in the simulations of Cawthorne, Jorstad Marscher ( 2013 ) and Marscher ( 2016 ). However, in such a scenario the linear polarization pattern is always present, since it is created by the partial ordering of the magnetic field by the shock front. Periods of polarization across the entire core should not be observed. An alternative picture ascribes the radial polarization pattern to the circular appearance of a magnetic field with a helical or toroidal geometry that is viewed within ∼0.2/Γ radians of the axis of the jet (Marscher et al. 2002 ), where Γ is the bulk Lorentz factor of the emitting plasma. In this case, the ratio of the observed polarization measured in the image, , to the value for a uniform magnetic field direction, , 6 implies that the helical field is superposed on a highly disordered field component that is ∼10 times stronger. If the helical field becomes disrupted by a current-driven kink instability, particle acceleration could cause a flare (Nalewajko 2017 ; Zhang et al. 2017 ; Alves, Zrake Fiuza 2018 ). The polarization pattern then becomes complex, with the possibility that the polarization becomes very low at some point (Dong, Zhang Giannios 2020 ). Such a flare would be expected to start at X-ray and VHE γ-ray energies upstream of the core, then

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


open circles joined with dotted line are for shock free accretion solutions. See the text for details. Until now, we have studied the various properties of accretion shock waves around black holes. These shocks are non-dissipative in nature as the specific energy remains conserved across the shock front (Chakrabarti 1989 ). Indeed, shocks of this kind are radiatively inefficient as well. Apparently, the realistic shock waves are likely to be dissipative, where a part of accreting energy is escaped from the shock location through the disc surfaces. This essentially causes the reduction of the overall specific energy profile in the PSC region (Singh Chakrabarti 2011 ). The plausible mechanism that perhaps regulates the energy dissipation at the shock is the thermal Comptonization process (Chakrabarti Titarchuk 1995 ) that eventually reduces the thermal energy in the PSC. Based on the above consideration, we assume that the loss of energy across the shock front is proportional to the temperature difference between the immediate pre-shock and post-shock flow and therefore, the energy loss across the shock front is calculated as (Das, Chakrabarti Mondal 2010 ) where a and a denote the immediate pre-shock and post-shock sound speed and f represents the fraction of the available thermal energy lost at shock. In this analysis, we treat f as a free parameter and chose its value as f 0.998 for representation. Evidently, this is the measure of energy dissipation across the shock (Das etal. 2010 ; Singh Chakrabarti 2011 ; Sarkar Das 2013 ; Kumar Chattopadhyay 2013 ). We then compute the maximum energy dissipation () by freely varying all the flow parameters and plot it as function of in for various values of accretion rates in Fig. 12 (a). Here, solid, dotted and dashed curves represent the results corresponding to , 0.075 and 0.1, respectively. We observe that for a given accretion rate (), initially shows gradual increment with in , which eventually approaches towards a saturation value in the gas pressure dominated domain. Meanwhile, it was shown by numerical simulation (Machida etal. 2006 ) that in the magnetically dominated state, accretion disc becomes cool as compared to a gas pressure dominated disc. This clearly indicates that the gas pressure dominated flow is expected to possess higher thermal energy content and accordingly, we obtain enhanced when in is larger. Moreover, as is increased, cooling becomes more effective in the flow resulting the lowering of thermal energy content. Therefore, for a given in , remains smaller for flows having higher . When in is large, the effect of synchrotron cooling due to the increase of accretion rate becomes practically insignificant. Hence, when in 10 4 , asymptotes to a saturation value irrespective to the flow accretion rate. Figure 12. Variation of maximum energy dissipation at shock () as a function of at inner sonic point ( in ) for three different values of accretion rates, , 0.075 and 0.1 denoted by

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


( 75 ) and ( 78 )) normalized by the total number of mergers. In these two examples, we set χ 1 = χ 2 . The top panel is for a 0 = 100 au (see Figure 19 ) and we include merging systems with the number of mergers is 673 for e out = 0, 790 for e out = 0.3, 1159 for e out = 0.6, and 2828 for e out = 0.9, so that N max = 5450. The lower panel is for a 0 = 20 au, and we include systems with the number of mergers is 146 ( e out = 0), 180 ( e out = 0.3), 227 ( e out = 0.6), and 411 ( e out = 0.9), so that N max = 964. The other parameters are the same as in Figure 19 . If the distributions of and are uncorrelated, as we may expect to be the case for , when the octupole effect is significant (see Figure 9 and the discussion in the last paragraph of Section 4.2 ), the distribution of χ eff can be derived directly from and , the distribution functions of and . Define where is the maximum possible value of χ eff for given m 1 χ 1 and m 2 χ 2 (this maximum is achieved at ). Then Equation ( 75 ) becomeswhere and . Note that and . Given P 1 ( μ 1 ) and P 2 ( μ 2 ), the distribution function of is In the special case when μ 1 and μ 2 are uniformly distributed, we have P 1 = P 2 = 1/2, and Equation ( 80 ) giveswhere we have assumed without loss of generality. Thus, even for uniform distributions of and (see the case of e out = 0.9 in Figure 19 ), the effective spin parameter χ eff is preferentially distributed around χ eff = 0 (see Figure 20 ). Note that the spin–orbit misalignment distribution and χ eff distribution obtained above refer to relatively wide BH binary systems ( a 0 ≳ 10–100 au) that experience a merger due to large LK eccentricity excitation. Such systems necessarily have . For BH binaries with smaller separations ( a 0 ≲ 1 au) and not much less than unity, the spin–orbit misalignment distribution can be quite different (see Liu Lai 2017 ). Antonini et al. ( 2018 ; see also Rodriguez Antonini 2018 ) have carried population studies of BH mergers in triple systems (based on DA secular equations) and have found a similar peak around χ eff = 0 in the χ eff distribution. Rodriguez Antonini ( 2018 ) also showed an example of the final spin–orbit misalignment distribution with a peak around 90°, in qualitative agreement with our result. They did not distinguish the difference in the spin–orbit misalignment distributions between largely quadrupole systems (small ε oct ) and strong octupole systems (large ε oct ). We do not agree with the reason(s) they gave for the peak in the χ eff distribution. In particular, it is important to recognize that, during the orbital decay, the adiabaticity parameter (Equation ( 61 )) transitions from ≪1 to ≫1, and this transition determines . 5. Summary and Discussion In this paper we have studied BH binary mergers in triple systems. A sufficiently inclined tertiary companion excites large eccentricity in the BH binary orbit through gravitational perturbations (the LK mechanism), significantly shortening its

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


conclude that the errors assigned by Reid et al. (2014) to and are conservative, and there do not appear to be any bias, given the available maser samples so far. For the rotation curve, the situation is somewhat different. Although the values found for are very close to the initial values adopted, the statistical spread is larger than expected. Reid et al. (2014) reported an error of 0.4 in the rotation curve which is optimistic compared with our findings. From our simulations, we would constrain the rotation curve value as given the BeSSeL data sample selection. The larger error possibly indicates that our assumed velocity distributions are too wide. Fig. 10 Marginalized posterior probability density distributions for correlated circular velocity parameters from 100 simulated galaxies mimicking the BeSSeL sample. Left panel : circular orbital speed of the Sun. Middle panel : orbital angular solar speed. Right panel : difference between the circular solar motion and average source peculiar motions. When more sources are added to the sample selection (sets A and B), Fig. 8 shows an initial improvement in the accuracy of the fundamental Galactic parameters. For set A, the errors in , , and can improve up to , , and , respectively, when more northern hemisphere sources are added. In contrast when southern hemisphere sources are also added (set B), the errors can decrease to , , and , respectively. Further improvements would require much better astrometry for weak sources, requiring for example much more sensitive observations. 4.2.2. Solar motion and average source peculiar motions Figure 7 shows the distribution obtained for solar velocity components (, , and ). The uncertainties derived were around 2 for all solar velocity components with respect to the initial parameters. For , the standard deviation found was 1.2 , which compares well to the results published by Reid et al. (2014) (i.e., ). In addition, the spread for and are narrower with values of 2.4 and 0.5 , respectively. Compared to the BeSSeL results (i.e., and ), we can affirm that the solar motion results published by Reid et al. (2014) have conservative estimates. For the radial and tangential average peculiar motions ( and ), Table 4 shows that indeed these peculiar velocities can be fitted with high accuracy using the simulations; however, the relative spreads are high compared with other parameters (see Fig. 7 ). Even so, compared to the BeSSeL results (i.e., and ), the errors in the simulated sample selection are still lower (i.e., and ). The reason for higher dispersions for and could be related to the number of parameters that must fit independently. The parameters and have largely the same effect on the observations for nearby sources and therefore are directly correlated (see Table 5 ). For , the correlation is high with two components (, ), which affects the fitting and hence the estimated accuracy. 4.2.3. Parameter correlations Pearson product-moment correlation

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 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 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 B-Survey O O O O O O O O O B-CelestialObject O O O O O B-CelestialObject 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 B-CelestialObject O O O O O O O B-TextGarbage I-TextGarbage O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O B-CelestialObject 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O


The real-time duration of the video is 9 s. (An animation of this figure is available.) The first column of Figure 2 presents the evolution of the photospheric continuum I c . We can observe that this expanding structure begins to form a central dark spot (Δ t = 252 s), which eventually grows in size while the EG splits into two smaller granules (Δ t = 504 s). Then, the southernmost descendant granule located at undergoes the same evolution (Δ t = 756 s), until it splits into four granules (Δ t = 1008 s). The second column of Figure 2 shows the integrated signal of V measured . The emerging flux region first appears at Δ t = 0 s, being located at [5′′, 4′′]. Magnetic field patches with apparent opposite polarity appear close to each other. These magnetic patches expand and occupy the EG area (Δ t = 252 s), showing a pattern of alternate polarities (salt-and-pepper or sea-serpent configuration). Some of these flux kernels intensify: notably, this occurs in the region where the first dark spot is forming (Δ t = 504 s). Later, this structure becomes an intergranular lane (Δ t = 756 s). Doppler velocity maps, displayed in Figure 2 (third column), highlight the plasma motions in the region. They indicate that the structure is generally characterized by conspicuous blueshifts, corresponding to upflows up to 1–1.5 km s −1 . In the central dark patch, downflows of ≃0.5 km s −1 are found (Δ t = 504 s and, partly, at Δ t = 1008 s), in agreement with previous observations. The same coherent behavior is also visible in the online movie. The maps of magnetic flux density (Figure 2 , fourth column) reveal that the distribution of the magnetic areas derived from V integrated is representative of the real magnetic flux patches. Indeed, a mixture of polarities is found where V integrated maps show an apparent salt-and-pepper configuration, suggesting once again that the quantity described in Equation ( 3 ) is a good qualitative proxy for the longitudinal component of the magnetic field, . The evolution of magnetic flux density qualitatively follows that of V integrated , as mentioned above. A polarimetric analysis of the dead-calm region (see Martínez González et al. 2012b ) in FoV QS , used as a control box for comparison, is presented in Figure 3 . This region of very quiet Sun entered the IMaX FoV after the first 200 s of the EG evolution. In Figure 3 (top panels) we present the maps of the integrated signal of V measured , at the same representative times shown for FoV EG in Figure 2 (second column). No new magnetic structures are seen to emerge during the analyzed time series. Figure 3 (bottom panel) shows a plot of the mean polarization signal averaged over FoV EG (black symbols) within the region of interest occupied by the EG inside FoV EG (red symbols), and over FoV QS (empty circles), along the sequence. The signal in FoV EG is on average 2.7 times stronger than in FoV QS . Moreover, the polarization signal in FoV QS remains almost constant during the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Instrument O O O O O O O O O O O O O O O O O O O B-CelestialObject 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 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 B-Instrument O O O O O O O O O O B-Instrument O O B-Instrument O O O O O B-Instrument O O O O O O O O O B-Instrument O O O O O O O O O B-Instrument O O O O O O O B-Instrument O O O O O


formation of multiple populations, as we are going to outline in the next. Figure 17. Left-hand panel: Difference between the mass-loss of 2Ge and 1G stars, , against the F 275 W − F 814 W colour extension of the HB. The groups of M3- and M13-like clusters are represented as orange and blue dots, as labelled. Right-hand panel: Present-day and initial cluster mass against the colour extension of the HB (top and bottom panels, respectively). 5.3 Extra-mass-loss in the 2Ge as a tracer of the multiple populations formation The discovery that 2Ge stars of all GCs lose more mass than the 1G (Fig. 11 , see also Tailo et al. 2015 , 2019b , a ) and, in particular, the correlations between the extra-mass-loss Δμ e and the extra-helium content of the 2Ge population δ Y max (Fig. 12 ) and the total (present-day and initial) mass of the clusters (Fig. 13 ) represent powerful constraints for the mechanisms of formation of multiple stellar populations. Since evolutionary effects alone cannot provide such increase (see Appendix D ) the need of an additional mechanism arises, allowing us to gain information on the environment where the distinct generations were born. Tailo et al. ( 2015 ) discuss, in the specific context of the blue hook morphology of ω Cen, how the formation environment of the different multiple populations can affect the stellar evolution during the RGB phase, mainly by affecting the initial stellar rotation. Here, we apply the same scenario to the less extreme environments encountered by the 2Ge of most clusters (see the schematic illustration in Fig. 18 ). Very young low-mass stars ( M /M ⊙ ∼ 0.1−1.0) behave as the T -Tauri stars in the Galactic field, pre-main sequence stars in the convective phase of contraction towards the main sequence. T -Tauri’s typically rotate with a period of 1–12 d, and their rotation rate is maintained constant in time by the magnetic coupling of the star with the accretion disc (magnetic disc-locking, see Armitage Livio 1996 ). Bouvier et al. ( 1997a ) and Bouvier, Forestini Allain ( 1997b ) developed a model to explain why main sequence stars in young clusters (such as α Persei) show a wide distribution of rotation velocities. They showed that, the earlier would pre-main sequence stars break the magnetic disc-locking, the faster would be their rotation when they reach the main sequence. Afterwards, during the main-sequence phase, the surface rotation rate (together with the rotation of the whole convective envelope) slows down thanks to the angular momentum loss associated with the stellar winds, but the core mostly preserves its fast rotation. Figure 18. Schematic interpretation of the different rotation of 1G and 2G stars. During the pre-main-sequence evolution, the rotation of the central contracting star is magnetically locked to the rotation of its residual accretion disc, having a diameter of 50–100 au. While the 1G formation occurs in an ambient of moderate density (yellow region), where the disc–disc

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


good for travel distances of about 8°–20° with measurement errors increasing with increasing travel distance. At very low travel distances (about Δ 6°), it can be seen that the forward-modeled travel times do not fit the measured ones. The reason for this effect lies in the fact that the simulated data only incorporates modes with harmonic degree l ≤ 170. This cut-off acts as an additional filter at a harmonic degree at which the filter with the lowest phase speed (L170) is centered. The modes selected by the filter thus do not form a proper wave packet, which may lead to artifacts in the cross-correlations (see, e.g., Zharkov et al. 2006 ). Additionally, it is noteworthy that the quality of the match of the power spectrum (compare matched versus unmatched) does not have a large effect on the travel times at higher travel distances (Δ > 8°). At smaller travel distances (Δ 8°), however, one can observe a substantial effect of the quality of the match of the power spectrum on the travel times. We note here that this conclusion may be different for other flow models. We also note that the absence of a proper averaging of multiple kernels in distance and latitude has a considerable effect for smaller travel distances (Δ 10°) but seems to vanish for the largest distances in the case of the flow model considered here. For comparison, forward travel times using ray kernels are also shown in Figures 5 and 6 . It can be seen that the magnitude of the ray kernel travel times is, in general, very similar to those obtained using Born kernels. However, Born kernels seem to better reproduce some features in the measured travel times, e.g., jumps in the magnitude of the travel times from one filter to another such as from Δ = 221 to Δ = 241 (see the middle panel in Figure 6 ). Because the given flow model varies on rather large length scales, a relatively good performance of the ray kernels is expected. This may be different for meridional flow models, which vary on smaller length scales, see Section 1 . We also note here that the measured travel times obtained using the filter with the highest phase speed (L035) encounter a spurious constant offset, see Figure 7 . This offset was corrected for by substracting the mean travel time at each distance. Such an offset is not present in real solar data (Kholikov et al. 2014 ; Jackiewicz et al. 2015 ). In Jackiewicz et al. ( 2015 ), this offset was not noticed in the simulated data, as a few distances had been dropped from the analysis, which resulted in an equivalent correction. The origin of this offset is not completely understood but may be connected to the relatively coarse spatial resolution of the simulated data (256 pixels on 180°). Figure 7. Measured travel times, without offset correction. Left and middle panels: same travel times as displayed in Figure 5 (top left and top middle panels). Right panel: measured GB04 and forward-modeled travel times from the Born kernels for all travel distances, at

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


al. (2011) . The percentages in Fig. 8 are therefore a minimum estimate only. Despite this, we can see in cases (ii) and (iii), a higher percentage of the flux remains at its original connection for the duration of the simulations. The change in connectivity of the field lines results in strong outflow velocities from the end of the current layers. White arrows showing the projected horizontal velocity components in the central region () in the mid-plane are displayed in Fig. 9 for case (i) at . These show large velocities from the narrow current layer along , where the maximum velocity is , where is the maximum Alfvén velocity in the mid-plane (at ), an indication of reconnection occurring. We note, that the magnetic field is 3D and reconnection is not confined to the mid-plane, but use the current contours and velocities in the mid-plane to illustrate the differences between the cases. Fig. 9 Contour of the magnitude of the current density in the mid-plane between at for case (i) with 2 sources. The arrows show the projected velocity in the plane. Fig. 10 Contours of the magnitude of the current density in the mid-plane between at a) and b) for case (ii) with 4 weak sources. The arrows show the projected velocity in the plane. Similarly, in case (ii), strong outflow velocities are seen from the current “wings” shown in Fig. 10 a at and very similar behaviour is observed for case (iii) (not shown). Later in cases (ii) and (iii), the current contours in Fig. 10 b show the “X” current formation to fragment and then (in addition to the outwards velocity towards the edge of the domain) there are inwards velocity outflows of reconnected field lines towards the centre of the mid-plane. This leads to further current being built up in the centre of the domain. The maximum velocity (in the mid-plane) for cases (ii) and (iii) are and , respectively, and occurs at , after this fragmentation of the “X” current formation. This is much smaller than in case (i) and suggests that the reconnection is weaker. The location of the sources in the case with two flux tubes appears highly idealised to produce the concentrated single current layer in the centre of the domain. In comparison, in the four-source case, as they are evenly distributed around the centre of rotation, a shear is first created between the neighbouring flux tubes, which due to the relative angles of the magnetic field lines will always be less than that induced in the two-source case and produces reduced current values and results in a smaller percentage of flux changing connectivity. 3.1.2. Energetics The energy evolution is shown by the volume integrated energy densities in Fig. 11 for the cases with two, four weak and four compact sources, displayed in pink, blue and black, respectively and can be described as follows: when the driving velocity acts on the field, the field lines are advected and the magnetic field becomes twisted, gradually increasing the magnetic energy in the domain (see

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


which was managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The fits had an average rms residual of 0.15 km s −1 and an average maximum residual of 0.3 km s −1 . Two observations have larger uncertainties. The signal-to-noise ratio (S/N) for the 814 cm −1 setting from 2007 was too low to use telluric lines in the GV Tau N spectrum. Instead, the dispersion function was fit to lines in the telluric standard (rms 0.06 km s −1 ), and this solution was applied to GV Tau N. Three measurable telluric lines in the GV Tau N spectrum were used to assign a zero-point (wavelength shift) uncertainty of 0.5 km s −1 . The 932 cm −1 setting from 2007 only contains three telluric lines; hence, an independent dispersion solution was not possible. In this case, the pipeline wavelength scale was retained, with an assumed uncertainty of 1 km s −1 . Model telluric transmission spectra were utilized for a few of the settings. Because the S/N for the 780 cm −1 setting was degraded as a result of insufficient S/N in the telluric calibrator, corrections for atmospheric lines were made using the software package TERRASPEC (Bender 2010 ). TERRASPEC was also used to improve the correction for telluric water lines in the 767 and 807 cm −1 settings. For all of the observations, additional continuum normalization was carried out for each order of interest by fitting a low-order polynomial to the continuum outside the intervals of absorption from known molecular transitions in GV Tau N. 3. Analysis In the TEXES spectra we detect numerous absorption lines of NH 3 , HCN, and C 2 H 2 ; a few lines of H 2 O in absorption; and H 2 in emission (Table 2 ). The H 2 emission from GV Tau was previously reported in Bitner et al. ( 2008 ) and is not discussed further here. Most of the HCN and C 2 H 2 lines were measured in 2006, with additional lines measured in 2007. The unexpected detection in 2006 of numerous NH 3 absorption lines was followed up in 2007 with grating settings that probed lower-energy NH 3 transitions. Weak absorption lines of water were also detected in the 2007 spectra. The entire set of pipeline-reduced spectra for GV Tau N is shown in Figure A1 . The positions and identifications of the detected lines are indicated. Table 2 All Lines Notes. a *: detection; x: nondetection; ...: not covered; t: telluric line; e: order edge/roll-off; u: high uncertainty; d: blended with other line. b ν 4 + ν 5 – ν 4 band, i.e., absorption out of the ν 4 level. c The 813.85 cm −1 ortho- and 813.83 cm −1 para-H 2 O lines were treated as a single line with combined g l . While the H 2 O lines are pure rotational transitions within the ground vibrational state, the NH 3 , HCN, and C 2 H 2 absorption lines are predominantly rovibrational transitions out of the ground vibrational state to a higher vibrational state; the C 2 H 2 ν 4 + ν 5 – ν 4 line is an exception (see Section 3.1 ). The HCN ν 2

O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O O O B-Organization I-Organization I-Organization O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Software B-Citation I-Citation O B-Software O O O O O O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O B-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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


of spectra for 10 K, 50 K, and 300 K, while series F[001]a lacks a 10 K spectrum. Figure 1 shows the measured transmission spectra ( T meas , solid and dashed lines) plotted over the wavenumber ( in cm −1 ); the corresponding wavelengths (30 μ m–500 μ m) can be inferred from the upper axes. Spectra for a-polarization are given in panel (a), b-polarization in (b), c-polarization in (c). The two incomplete series F[001]a and F[010]c are plotted as dashed lines. The former has been scaled by a factor of 1.8 in order to demonstrate the good agreement to series F[010]a after multiplicative scaling (see Sect. 3.2 ). The strong temperature dependence of the transmission of the forsterite crystal is easily recognizable in all three polarization directions. While measurable transmission is detected only below 50 cm −1 at room temperature, the crystal becomes transparent already below 250 cm −1 at 10 K. For each temperature, the transmission increases towards smaller wavenumbers, apart from absorption bands positioned at 202 cm −1 (particularly strong in a-polarization, but weaker also in c-polarization) and 145 cm −1 (in b-polarization, very weak in a- and c-polarization). These bands are the already mentioned 49 μ m and 69 μ m single-phonon bands of forsterite, respectively. They are excited exclusively by a- and b-polarized radiation, respectively, the weaker appearances in other polarization directions are due to the small error of the polarizer alignment and to the beam divergence of the FTIR spectrometer (±7.5 deg). Apart from these bands, a relatively weak minimum at ∼90 cm −1 also appears in several of the spectra for a-polarization. As there is no such long-wavelength infrared band predicted by theory ( Iishi 1978 ) and the band is not present at 10 K, we think that it is an artifact of the measurement setup. We note that the position of the band is one of the intensity minima created by multiple reflection in the inner cryostat windows. These minima occurring with a period of 18 cm −1 are sometimes not well compensated by the reference measurements and lead either to periodic structures in the spectrum (e.g., in the 10 K and 50 K spectra in Fig. 1 b), or coincide for unknown reasons with erratic peaks (mainly maxima) in the enstatite spectra (see Sect. 3.3 ). Fig. 1. Temperature-dependent transmission spectra of the forsterite crystal for orientation of the electric field parallel to the a-axis ( panel a ), b-axis ( panel b ), and c-axis ( panel c ). Complete temperature series (F[010]a, F[001]b, F[100]c) are plotted as solid lines, the two incomplete series as dashed lines. The F[001]a curves have been scaled by a factor of 1.8; they approximately match the F[010]a curves for the same temperatures. Dotted lines: calculated maximum transmission curves taking into account the intensity reflected by the crystal surfaces. 3.2. Maximum transmission curves and measurement errors The dotted lines in Fig. 1 indicate the theoretical maximum

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


limits were calculated from the misfit function when all solutions with a fit at least of 99percent of the best fit were accepted. Figure 2. Inversion for optimum cosmological parameters of (a) the CDM model and (b) the opaque universe model. The colour shows the mean of the absolute distance modulus residua between the predicted model and the SN Ia data as a function of (a) the Hubble constant H 0 and the dark energy , and (b) the Hubble constant H 0 and the B -band intergalactic opacity B . Only the most accurate 552 SNe Ia with z 1.4 and an error less than 0.50 mag are used. The optimum solutions marked by the white plus signs are defined by (a) and ( 0.122 and 2.466) and (b) and ( 0.088 and 1.920). The residual Hubble plots are shown in Fig. 3 . The optimum solutions have roughly comparable misfits for both models: 0.16 for the CDM model (Fig. 3a ) and 0.17 for the model of the opaque universe (Fig. 3b ). However, as illustrated in Figs 3(c) and (d), the CDM model performs slightly better than the opaque universe model, because of a better fit of the binned residua for redshifts less than 0.6. Nevertheless, the model of the opaque universe also accounts for the observed dimming of the SNe Ia within the confidence level of 95percent (Fig. 3d ). Figure 3. Residual Hubble plots for (a, b) the individual SN Ia data and (c, d) the binned SN Ia data. (a, c) The flat CDM model with and . (b, d) The opaque universe model with and . The error bars in (c, d) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Importantly, the optimum solutions are sensitive to the redshift interval of the SNe Ia used in the inversion. If the optimum model of the opaque universe is searched using the SNe Ia with redshifts z 0.6 only, and the SNe Ia are grouped in the following redshift bins: z [0, 0.025, 0.05, 0.1, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.5, 0.6], we get and (Fig. 4 ). We see in Fig. 4(b) that the model fits the SN Ia measurements at z 0.6 much better than in Fig. 3(d) , but a significant discrepancy appears at higher redshifts. A similar result was obtained by Goobar etal. ( 2002 ) and Riess etal. ( 2004 ), who concluded that the discrepancy at high redshifts excludes the SN Ia dimming to be produced by intergalactic dust. Figure 4. Residual Hubble plots for (a) the individual and (b) binned SN Ia data for the opaque universe model with and . The vertical dashed line in (b) denotes the upper redshift limit of the SN Ia data considered in the inversion. The error bars in (b) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Note that the best estimate of H 0 for the CDM model obtained by Riess etal. ( 2016 ) using the SN Ia data is . The precision of the distance scale was further improved by a factor of 2.5 by Riess etal. ( 2018 ). This value is within the uncertainty of the estimate of H 0 for the CDM model obtained in this paper and shown in Fig. 2(a) . The accuracy of the

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


larger intensity variations in the range and a patchy distribution peaked at the location of recent star formation events. The patchy ionization structure yields an escape fraction of , a value consistent with observations of lower redshift galaxies with similar brightness (e.g. Bouwens et al. 2016 ; Grazian et al. 2017 ), and averaged values of simulated galaxies with similar mass (Xu et al. 2016 ). We note, however, that large fluctuations are expected between different evolutionary stages (Trebitsch et al. 2017 ). The [C ii ] line luminosity of Freesia is . The emission extends on a few kpc scales, with peaks around Freesia-A, Freesia-B, and dense non-star-forming clumps found near the galaxy. The emission mostly comes from gas with and illuminated by an ISRF intensity . Instead, [O iii ] emission is concentrated around Freesia-A, the star-bursting component of the system, and its emission is dominated by gas with a wider range of properties, i.e. and U ≃ (6 ± 2) × 10 −3 . At the Freesia-A location, the oxygen line is very bright, . However, the smaller extent of the [O iii ] emitting region implies a lower galaxy-integrated emission, i.e. . With respect to [C ii ], the [O iii ] show both a spatial offset () and a spectral shift of (), reminiscent of similar evidence found in some systems at high z (Carniani et al. 2017 ). Freesia lies below the local [C ii ]–SFR relation (De Looze et al. 2014 , within ≃ 2σ), as it is in a starburst phase; that is, it sits above the KS relation (Schmidt 1959 ; Kennicutt 1998 , within ≃ 2σ). Spatial analysis reveals that patches of the galaxy that are above the resolved local – De Looze et al. ( 2014 ) relation are located below KS, and vice versa. In particular, due to the recent starburst, the dense centre () of Freesia lies above the KS () relation. At such high star formation surface densities the increases with less rapidly than expected from the local relation De Looze et al. ( 2014 ), which however only covers the low end of the range. Thus, the observed [C ii ]–SFR deficit can be primarily ascribed to negative stellar feedback during starburst phases, disrupting MCs around star formation sites. This interpretation was originally proposed in Vallini et al. ( 2015 ); our results fully endorse it. Metallicity effects have a weaker impact on the origin of [C ii ]–SFR deviations. Gas with extremely low ( Z ≲ 10 −2 Z ⊙ ) metallicities fails below both the local – and the KS relations, because of lack of C + ions and inefficient H 2 formation, respectively. However, gas with Z ≲ 10 −2 Z ⊙ is scarce in Freesia and its combined contribution is sub-dominant when galaxy integrated relations are considered. This situation is common to most of the galaxies in our sample. A metallicity effect might be relevant for galaxies that cannot retain their metal production (Xu et al. 2016 ; Jeon Bromm 2019 ), which are expected to be smaller ( M ⋆ ≲ 5 × 10 6 M ⊙ ), and with a lower star formation (), for which however

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


bias models that distinguish between centrals and satellites in upcoming work. 2.5 Incompleteness factor <italic>f</italic>ic The final parameter in our extended HOD model is the incompleteness factor (e.g. Leauthaud et al. 2016 ; Rodríguez-Torres et al. 2016 ; Guo, Yang Lu 2018 ). The inclusion of incompleteness is partially motivated by detection of incompleteness in the BOSS CMASS and LOWZ galaxy samples compared with theoretical stellar mass functions, but it is also needed to marginalize over uncertainties in the galaxy number densities since we do not model its full redshift dependence. We define f ic as a modification to in equation ( 1 ): where 0 f ic ≤ 1. Our implementation simply uniformly downsamples the mock galaxies by f ic to produce the desired galaxy number density. To summarize, our extended HOD model contains the five baseline parameters: M cut , M 1 , σ, α, and κ; six extended parameters: s, s p , s v , α c , A , and A e ; and one incompleteness factor f ic , for a total of 12 parameters. 3 DATA AND SIMULATIONS 3.1 BOSS CMASS galaxy sample The Baryon Oscillation Spectroscopic Survey (BOSS; Bolton et al. 2012 ; Dawson et al. 2013 ) is part of the SDSS-III programme (Eisenstein et al. 2011 ). BOSS Data Release 12 (DR12) provides redshifts for 1.5 million galaxies in an effective area of 9329 deg 2 divided into two samples: LOWZ and CMASS. The LOWZ galaxies are selected to be the brightest and reddest of the low-redshift galaxy population at z 0.4, whereas the CMASS sample is designed to isolate galaxies of approximately constant mass at higher redshift ( z > 0.4), most of them being also luminous red galaxies (LRGs; Reid et al. 2016 ; Rodríguez-Torres et al. 2016 ). The survey footprint is divided into chunks that are covered in overlapping plates of radius . Each plate can house up to 1000 fibres, but due to the finite size of the fibre housing, no two fibres can be placed closer than 62 arcsec, referred to as the fibre collision scale (Guo, Zehavi Zheng 2012 ). For this paper, we limit our measurements to the galaxy sample in the redshift range 0.46 z 0.6 in DR12. We choose this moderate redshift range for completeness and to minimize the systematics due to redshift evolution. Applying this redshift range to both the North and South Galactic Caps gives a total of approximately 600 000 galaxies in our sample. We showcase the number density variation over redshift in Fig. 1 . The average galaxy number density is given by n data = (3.01 ± 0.03) × 10 −4 h 3 Mpc −3 . Figure 1. The CMASS DR12 galaxy comoving number density distribution across our redshift range. The red dashed line corresponds to the North Galactic Cap, whereas the blue dotted line corresponds to the South Galactic Cap. The green solid line shows the combined number density. The two vertical lines mark z = 0.46 and z = 0.6, respectively. Fig. 2 shows the redshift-space 2PCF of the same BOSS sample and its corresponding correlation matrix,

O O O O O O O O O O O O O O B-Tag O O O O O B-Model I-Model I-Model O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-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 O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O 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 B-Dataset I-Dataset I-Dataset I-Dataset O B-Survey I-Survey I-Survey I-Survey B-Survey B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Survey O B-Citation I-Citation I-Citation I-Citation O B-Dataset I-Dataset I-Dataset I-Dataset B-Dataset O O O O O O O O O O O O O O O O O O B-Dataset O B-Dataset O B-Dataset I-Dataset 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 B-Formula I-Formula I-Formula O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Formula I-Formula I-Formula 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 B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O B-Model O O O B-Dataset I-Dataset O O O O O


since 2003. WIRCam is mounted at the prime focus of the 3.6 m CFH telescope. It is equipped with an image stabilization unit, which consists of a tip-tilt glass plate in front of the camera activated by the signal read-out from small 14 × 14 pixel regions centred on bright stars ( Puget et al. 2004 ). The camera cryostat includes the eight-lens field corrector and an entrance window. The detector focal plane consists of a mosaic of four 2 × 2 k HAWAII2-RG detectors separated by 7 mm. The sampling on sky is 0.′′3 per 18 μm pixel, providing a total field of view of 20′ × 20′ with a 2′ wide central cross between the four detectors. The transmission curve of the Y -band filter used in this work is shown on Fig. 1 , together with the MegaCam filter transmission curves. Fig. 1 Instrument and telescope total efficiency (optics and detector) in the MegaCam u *, g ’, r ’, i ’, z ’ filters and WIRCam Y -band. The survey observations consisted of four dithered 75 s Y -band images of the same field. They were split into two visits, with dithering within a visit and between visits of 2′′. The two visits were separated by at least 20 days in order to discard slow moving objects and strongly variable objects. The time difference between visits of the same field varied strongly between the four CFHTLS fields. For W1 and W4, the observations were performed very rapidly at the beginning of the survey, while for W2 and W3 (winter and spring fields), they suffered from poorer weather conditions and were executed over much longer periods of time. The interval between visits of the same fields were in the range of 20–40 days for 50% (W1) and 90% (W4) of the observations, whereas for the W2 and W3 fields, most of the visits were separated by approximately 1 yr. The mapping of the fields was organized so as to match the CFHTLS one-square-degree tiles, with nine WIRCam pointings per CFHTLS tile. The observations were defined so as to prevent the execution of the second-visit observations in the same sequence as for the first visits, which could have generated similar persistence patterns in the data. A total of approximately 150 h of observations were conducted during four CFHT semesters, 2010B, 2011B, 2012A, and 2012B. The observations were carefully checked by our team in the days following the observations using the Elixir-IR pre-processing package (see next section), and observations that were executed under high background conditions and/or poor seeing conditions were not validated and were subsequently re-executed. 3 Data processing 3.1 Needfor a custom pipeline: Elixir-IR The CFHT offers to all its WIRCam users a data detrending and calibration service, the “I”iwi pipeline ( Thanjavur et al. 2011 ), which aims to remove the instrumental signature and delivers an absolute photometric calibration within 5%. This pipeline relies on generic recipes, such as acquiring flat fields over a whole observing run in order to create master detrending frames that are applied

O O B-Instrument O O O O O O O O O O B-Telescope I-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Instrument I-Instrument I-Instrument I-Instrument I-Instrument I-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-Wavelength I-Wavelength 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 B-Instrument B-Wavelength I-Wavelength B-Wavelength I-Wavelength B-Wavelength I-Wavelength B-Wavelength I-Wavelength B-Wavelength O O O B-Instrument B-Wavelength I-Wavelength O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion O B-CelestialRegion O B-CelestialRegion O O O O O O O O O O O O O O B-CelestialRegion O B-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion O B-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O B-Instrument 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 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 B-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O B-Telescope O O O O B-Instrument O O O O O O O O B-Software I-Software 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


(), by using equation ( 36 ) for a clear atmosphere The red curve in Fig. 4 depicts our approximation, which follows the solution of Lecavelier des Etangs etal. ( 2008 ) when s , and is zero otherwise. Fig. 5 shows the amount by which our approximation underestimates the effective atmospheric thickness. It is at most E 1 (e ) 0.492 when s 0.561. We can apply the same line of reasoning to quickly determine which CIA features rise above the surface continuum. We use instead of equation ( 43 ), where k is the modified mean collision-induced cross-section with its minimum set at the surface collision-induced cross-section ( k s ), given by We then determine the spectral modulation between any two spectral regions with equation ( 34 ). The surface decreases the spectral modulation only if s or k s k in one of the two spectral regions. Figure 5. Amount by which the effective thickness of the atmosphere (in units of scaleheight) is underestimated for q 1 (see equations 11 and 12 ) with the approximation outlined in Section 3.4 as a function of the surface slant optical depth. Can we use this approximation to quickly determine the location of a surface from observations? If the observed spectral modulation is due to a single species and is less than that expected in a clear atmosphere, we can relate the observed spectral modulation, ( h / H ) obs , with respect to a local maximum in its cross-section ( i ) by Unfortunately, the surface location cannot be unambiguously determined because the mole fraction ( f i ) of the species is unknown. Indeed, equation ( 46 ) shows that the spectral modulation of a species increases with abundance in the presence of a surface, as demonstrated numerically by Btrmieux ( 2016 ) for a refractive boundary. The only way to break this degeneracy are two-fold: either the spectral signature is that of a major species (i.e. f i 1) and we can determine the location of the surface, or we can predict the location of the surface and then determine the abundance of the minor species. Given that refractive boundaries are dependent on the bulk properties of the exoplanetary atmosphere as well as the planetstar geometry, quantities that are retrieved during the analysis of transit data, location of refractive boundaries can be predicted in principle and used to place a lower limit on the abundance of detected minor species (Btrmieux 2016 ). 3.5 Transmission spectrum versus formalism To illustrate the impact of the choice of formalism on a computed transmission spectrum, we apply the following formalisms on simulations by Btrmieux ( 2016 ): the formalism of Lecavelier des Etangs etal. ( 2008 ), the exact solution embodied by equation ( 35 ) with q 1, and the approximation discussed in Section 3.4 . The chosen atmospheric model is that of a Jovian-like planet with an H 2 He atmosphere identical to Jupiters, a 600K isothermal profile, and a constant H 2 O abundance with altitude with a mole fraction of 10 8 . We choose 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 B-Citation I-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-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 B-EntityOfFutureInterest I-EntityOfFutureInterest 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


indeed become an important ionization source in T Tauri disks. However, our model should be seen as a first attempt at a comprehensive modelling of SP ionization in protoplanetary disks. In the following we discuss further important aspects like variability, non-stellar origin of SPs, the importance of magnetic fields and future prospects for SP modelling in protoplanetary disks. 4.3.1. Flares and variability In our models we assumed continuous (i.e. time-averaged) particle fluxes and X-ray luminosities. Although this is a reasonable assumption (see Sect. 2.2 ) it is likely that the disk is also hit by singular powerful X-ray and/or SP flares. From the X-ray COUP survey of the ONC cloud, Feigelson et al. (2002) and Wolk et al. (2005) derived a median X-ray flare luminosity of , comparable to the X-ray luminosity in our high X-ray models, and peak flare luminosities up to . The duration of such flares can last from hours up to three days with a typical frequency of roughly one powerful flare per week. Ilgner Nelson (2006) argue that for such flare properties the disk ion chemistry responds to time-averaged ionization rates. However, in case the duration between two flares is longer than the recombination time scale in the disk, such singular and strong flares would produce asymmetric features in molecular ion emission (i.e. a singular flare only affects a certain fraction of the disk). The spatially resolved SMA observations of TW Hya show indeed such an asymmetric structure ( Cleeves et al. 2015 ). However, as discussed by Cleeves et al. (2015) these features could also have a different origin like spiral arms or an hidden planet locally heating the disk. If such asymmetric structures are caused by stellar flares they would provide complementary constraints on the X-ray/SP activity of the star. Multi-epoch data of spatially resolved molecular ion emission is required to prove the flare scenario (i.e. the features should disappear quickly). The current (sub)mm interferometers like ALMA, NOEMA and SMA provide the required spatial resolution and might even allow for monitoring of disks in molecular ion lines on a daily/weekly basis in the future. Modelling of such observations does not necessarily require 3D chemical models. Assuming that the disk physical structure is not affected by flares, radial cuts through the disk can be modelled with 2D (time-dependent) thermo-chemical models as presented here. 4.3.2. Magnetic fields In our model we neglect the impact of magnetic fields on the SP transport. Stellar and disk magnetic fields are in particular relevant for the question if particles actually hit the disk (see Feigelson et al. 2002 , for a discussion). Magnetic fields can either drag the particles away from the disk (e.g. like in a wind) but could also funnel the particles and concentrate the ionizing flux in particular regions of the disk. In the first scenario SPs can still have an impact on the upper layers of the disk but certainly not on

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Wavelength O O O O O B-Wavelength B-Survey I-Survey O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-Wavelength O O O O O O O B-Wavelength O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O 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 B-Telescope O O B-CelestialObject I-CelestialObject O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O 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 B-Wavelength O O B-Observatory B-Telescope O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


have a constant absolute magnitude of −7 mag. The investigation on decline times for all novae in the catalogue indicates that the novae with D- and J- type light-curve shapes may not have linear decline. If so, only considering the decline times ( t 2 or t 3 ) may not characterize the absolute magnitude at outburst maximum and the decline well. All of these arise uncertainties when using of MMRD relation if the type of nova is not clarified. Furthermore, the luminosities of novae may differ depending on their stellar populations or their evolutionary states, i.e. on the stellar age and metallicity. For the scatters found in other distance indicators, e.g. Cepheids, it has been suggested that a different metallicity of the host galaxy may significantly affect the zero-point of the relation (Freedman et al. 2001 ). In this case, there is a possibility on any environmental dependence on MMRD relations as well. Since our and the other studies in literature depend on disc novae in the Galaxy, the contribution of other populations (bulge, halo, even thick disc) is not well known. Our study does not dispel the concerns regarding the validity the MMRD relation, but has raised open questions. To understand the MMRD relation better, the novae in galaxies, which have reliable distances (and extinctions), should be studied with high quality photometric and spectroscopic data. A larger sample of them would also be very welcome as well as corresponding parameters and well-determined light curves that allow us to classify their light-curve shapes. Our catalogue may also help to investigate the nature of MMRD diagrams in the long-term. Studying the MMRD relation is important to understand the origin of the nova eruption itself, besides using it as a tool to estimate absolute magnitudes. In this study, we did not find any signs that the spectral classification represents the stellar population of the Galactic novae, as suggested by Della Valle Livio ( 1998 ). However, our decline time histogram implies that the He/N+Hybrid novae are fast systems with a peak time at t 3 = 13 d, while the slow Fe ii novae have a peak decline time at t 3 = 42 d. In contrast to the suggestion of Della Valle Livio ( 1998 ), we found that both Fe ii and He/N novae or fast and slow novae are located in the whole range of vertical distances, but mainly concentrated throughout the Galactic disc. The differences between the results in these two studies may arise from the samples used in the analyses. The sample in Della Valle Livio ( 1998 ) contains a smaller number of novae with spectral classification, which reach only to a vertical distance of ∼2 kpc, whereas our sample have much more systems reaching up to two times farther in vertical direction from the Galactic plane. A relatively large number of slow novae located close to the Galactic plane and fast novae farther from Galactic plane must have been undetected in their analyses. The approach proposed by Della Valle Livio ( 1998

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


INT-G (intermediate gravity) and FLD-G (field gravity) classes defined by Allers Liu ( 2013 ). In Section 2 , we describe the selected sample of objects presented in this paper. In Section 3 , we provide details of the X-shooter observations and data reduction. In Section 4 , we explain the analysis carried out and our results. In Section 5 , we derive the physical parameters for our sample using evolutionary models for substellar objects. Finally, in Section 6 we summarize our results. 2 SAMPLE SELECTION We selected a sample of 20 benchmark objects with confirmed membership of open clusters or associations, with known ages, metallicities and distances. We chose a sample with photometry and optical spectral types confirming membership. In the case of the Pleiades and Praesepe, members are confirmed not only by photometry and optical spectroscopy, but also by proper motion (Lodieu, Deacon Hambly 2012b ; Boudreault et al. 2012 ; Bouy et al. 2015 ). We selected objects in each of the associations or open clusters with magnitude ranging from J = 14.9 to J = 18.1 mag, which allowed us to obtain spectroscopy with a signal-to-noise ratio of between 4 and 53 in the near-infrared, using exposure times of between 40 min and 1.5 h. Our sample has spectral types between M5.5 and L3.5, with at least one target per open cluster or association of each spectral type between M5.5 and L0. To complete our sample in age range, we added three members from the UppSco association, and four from the Cha I region from proposals: 093.C-0109(A) (P.I. Van der Plas), 093.C-0769(A) (P.I. Bonnefoy) and 095.C-0378(A) (P.I. Testi). We list the objects in our sample with their main characteristics in Table 1 . Table 1. List of observed targets with their full name, coordinates, magnitude, spectral types, cluster or association membership, and flag for disc-bearing sources and binary candidates. Notes . References: [1] Boudreault et al. ( 2010 ), [2] Hodgkin et al. ( 1999 ), [3] Boudreault et al. ( 2012 ), [4] Bouvier et al. ( 1998 ), [5] Stauffer et al. ( 1998 ), [6] Bouvier et al. ( 1998 ), [7] Bouvier et al. ( 1998 ), [8] Nagashima et al. ( 2003 ), [9] Lodieu, Deacon Hambly ( 2012a ), [10] Bihain et al. ( 2006 ), [11] Ardila, Martín Basri ( 2000 ), [12] Martín, Delfosse Guieu ( 2004 ), [13] Lodieu et al. ( 2007 ), [14] Persi et al. ( 2000 ), [15] López Martí et al. ( 2004 ). References of disks detections: [16] Luhman Muench ( 2008 ), [17] van der Plas et al. ( 2016 ), [18] Herczeg et al. ( 2009 ), [19] Long et al. ( 2017 ). a NR: the existence of a disc has not been reported for this object. References for spectral type: [20] Alves de Oliveira et al. ( 2012 ), [21] Luhman ( 2007 ), [22] Luhman ( 2004 ), [23] Lodieu et al. ( 2018 ), [24] Slesnick et al. ( 2008 ), [25] Bihain et al. ( 2010 ), [26] Martín et al. ( 1996 ), [27] Stauffer et al. ( 1998 ), [28] Boudreault Lodieu ( 2013 ), [29] West et al. ( 2011 ), [30] Manara et al. ( 2017 ). References for binary candidates:

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