of neutron-rich nuclei are required in r -process simulations, and most of these nuclei lie far beyond the experimentally known region. Theoretical predictions of the properties of these neutron-rich nuclei have large uncertainties, which can further interplay with the astrophysical uncertainties. This makes it even more difficult to address the issues of the r -process. Nuclear masses are extremely important for r -process simulations, as they are needed to extract the reaction energies that go into the calculations of all involved nuclear reaction rates. During recent decades, great achievements in nuclear mass measurements have been made thanks to the development of radioactive ion beam facilities, and about 2500 nuclear masses have been measured to date (Wang et al. 2012 , 2017 , 2021 ). Nevertheless, the masses of a large number of neutron-rich nuclei relevant to the r -process remain unmeasured and cannot be reached in the foreseeable future, due to the difficulties in production, separation, and detection. Therefore, reliable theoretical predictions for nuclear masses are imperative at present. However, an accurate prediction of nuclear masses is still a great challenge for nuclear theory due to the difficulties in understanding nuclear interactions and quantum many-body systems. Tremendous efforts have been made in improving the prediction of nuclear masses and, currently, the rms deviations between theoretical mass models (Pearson et al. 1996 ; Koura et al. 2005 ; Zuker 2008 ; Goriely et al. 2009 ; Wang et al. 2014 ; Möller et al. 2016 ) and the available experimental data reach about 0.5 MeV. The rms deviations are further reduced to about 0.2 MeV by employing machine learning approaches (e.g., Wang Liu 2011 ; Niu et al. 2013 ; Utama et al. 2016 ; Neufcourt et al. 2018 ; Niu Liang 2018 ; Wu Zhao 2020 ). However, different mass models predict very different masses when moving to the region of neutron-rich nuclei and, thus, lead to obviously different abundances of nuclei in the r -process nucleosynthesis simulations (e.g., Wanajo et al. 2004 ; Sun et al. 2008 ; Farouqi et al. 2010 ; Arcones Martínez-Pinedo 2011 ; Martin et al. 2016 ; Li et al. 2019 ; Zhao Zhang 2019 ). Therefore, it is important to address how the nuclear mass uncertainties limit our knowledge of the r -process (e.g., Sun et al. 2008 ; Farouqi et al. 2010 ; Arcones Martínez-Pinedo 2011 ; Brett et al. 2012 ; Xu et al. 2013 ; Aprahamian et al. 2014 ; Martin et al. 2016 ; Mendoza-Temis et al. 2016 ; Mumpower et al. 2016 ; Li et al. 2019 ; Sprouse et al. 2020b ). Addressing the impact of mass uncertainties on r -process abundances is useful to deepen our understanding of the r -process in the present stage. For instance, can we identify the dominant r -process site with current uncertainty (e.g., Mumpower et al. 2015c ; Martin et al. 2016 )? Can we understand the origin of the rare-earth peak (e.g., Surman et al. 1997 ; Goriely et al. 2013 )? To what extent can we

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


= 6.50 a 0 and along R and φ for θ = 75° is given. These cuts show that the R stretch coordinate is strongly coupled with θ and φ . Finally, it is worth noting that this interaction potential is strongly anisotropic especially along the R , θ and φ coordinates. This strong anisotropy is expected to lead to significant values for the collisional rates of the HNS rotational (de)-excitation by He. Figure 2. Two-dimensional contour plots of the 3D HNS–He PES along the R and θ coordinates for fixed φ = 0°, 90° and 180°. We give also the cuts along the angular coordinates and along R and φ for θ = 75°. Energies are in cm −1 . Black (red) contour lines represent repulsive (attractive) interaction energies. Table 1. Equilibrium geometries and energies of the minimal structures of HNS–He 3D-PES for φ = 0°, φ = 90° and φ = 180°. 3 DYNAMICAL CALCULATION DETAILS We consider the collision between an asymmetric top molecule (HNS) with He atom. The rotational Hamiltonian of the HNS molecule is given by where I α (α = x , y , z ) are the principal moments of inertia with respect to the principal axes of inertia (Gx), (Gy) and (Gz) of HNS. (G,x,y,z) is the body-fixed reference related to the space-fixed reference (G,X,Y,Z) by the Euler angles. I x , I y and I z are related to the rotational constants according to , and . The total angular momentum of the molecule is such that . D J , D JK and D K are first-order centrifugal distortion corrections. The rotational wave function | j τ m 〉 of an asymmetric top can be defined by three quantum numbers j , τ , m and can be expressed as a linear combination of the rotational wave functions | j τ m 〉 of a symmetric top molecule such as (Townes Schawlow 1955 ): where, j, k and m denote the rotational quantum number, its projection along the molecular z -axis and its projection along the space-fixed Z -axis, respectively. τ is the pseudo-quantum number that varies between − j and j . The rotational levels of an asymmetric top molecule are conventionally labelled (adopted hereafter) where K a and K c are the projection of the rotational angular momentum along the axis of symmetry in case of prolate and oblate symmetric tops limits being related to τ through τ = K a − K c . The rotational energy levels, of HNS, are deduced from the ab initio rotational and distortional constants calculated by Ben Yaghlane et al. ( 2014 ) at the CCSD(T)-F12/cc-pVQZ-F12 level of theory which are A e = 0.63583, B e = 0.61585, C e = 19.59885, D J = 9.11 × 10 −7 , D K = 4.68 × 10 −5 , D JK = 5.30 × 10 −3 (all values are in cm −1 ). The precision of these rotational parameters should be better than ∼0.3% according to theoretical benchmarks and their comparison to experimental data dealing with sulphur compounds (see for instance Cazzoli, Puzzarini Gauss 2014 ). All data represent predictions that can help for the identification of this molecule. For scattering calculations, only the first 29 levels of HNS molecule are considered. This corresponds

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


and the observation ID are given in each plot. The soft and hard LCs (left panel), HR plot (middle panel), and the DCF plot between the soft and hard bands (right panel) of 1ES 0347−121 are shown in Figure 3 . No significant spectral change is seen from the HR plot. We checked for variations quantitatively using a standard χ 2 test,where x i is the HR value, σ i is its corresponding error, and is the mean HR value. We considered a variation in the HR to be significant only if χ 2 > χ 2 99, ν , where ν is the number of degrees of freedom (DoF) and the significance level is 0.99. These results are given for all these blazars in Table 4 , where we see that for each source , so no significant spectral variations were detected. Figure 3. Soft (3–10 keV, denoted by red filled circles) and hard (10–79 keV, denoted by black filled circles) LCs (left panels), hardness ratios (middle panels), and the discrete correlation functions between soft and hard LCs (right panels) of the blazars 1ES 0347−121, 1ES 0414+009, RGB J0710+591,1ES 1101−232, 1ES 1218+304, and H 2356−309. The source names and observation IDs are given in the left panels. Table 4 Test of Temporal Variation of HR The DCF plot is flat, which, in the presence of significant variations, would indicate no correlation between the two energy bands, but since no variations are detectable this type of DCF is expected. 4.2. 1ES 0414+009 The TeV blazar 1ES 0414+009 ( α 2000 = 04 h 16 m 530; δ 2000 = +01°05′20′′) is an HBL at z = 0.287 (Halpern et al. 1991 ). It was first detected in X-rays with the High Energy Astronomy Observatory ( HEAO 1 A-1) (Ulmer et al. 1980 ) and was classified as a BL Lac object by Ulmer et al. ( 1983 ). It was observed above 200 GeV by VERITAS with source flux equal to (5.2 ± 1.1 stat ± 2.6 sys ) × 10 −12 photons cm −2 s −1 (Aliu et al. 2012 ). 1ES 0414+009 was observed with NuSTAR on 2015 November 25 with a good exposure time of 34.16 ks. The 3–79 keV LC of 1ES 0414+009 is shown in Figure 1 . The data are both noisy and sparse, resulting in no significant IDV detection, which is consistent with the value of fractional variance given in Table 2 . Consequently, the ACF plot, given in Figure 2 , does not show any hint of variability timescale. The soft and hard LCs of 1ES 0414+009 are plotted in the left panel of Figure 3 . The HR plot in the middle panel of that figure reveals no significant spectral variations (also see Table 4 ) and the DCF plot shown in the right panel of Figure 3 again is flat. 4.3. RGB J0710+591 The high-frequency-peaked BL Lacertae object RGB J0710+591 ( α 2000 = 07 h 10 m 264; δ 2000 = +59°09′00′′) is located at a redshift of z = 0.125 (Giommi et al. 1991 ) and also was first detected by HEAO A-1 (Wood et al. 1984 ). It was detected with VERITAS with the integral flux (above 300 GeV) recorded to be (3.9 ± 0.8) × 10 −12 cm −2 s −1 (Acciari et al. 2010 ). NuSTAR observed RGB J0710+591 on 2015 September 1 for 26.48 ks. As seen from the LC in Figure 1 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-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength B-Wavelength I-Wavelength O O O O O O B-Wavelength B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Wavelength O B-CelestialObject I-CelestialObject O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O O O O B-Wavelength O O B-Observatory I-Observatory I-Observatory I-Observatory O B-Observatory I-Observatory B-Telescope 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 I-Citation I-Citation O O O O B-Wavelength I-Wavelength O B-Telescope O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject O O O B-Telescope O O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O B-CelestialObject I-CelestialObject O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Observatory B-Telescope B-Citation I-Citation I-Citation I-Citation O O O O O B-Telescope O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Telescope O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O


dynamo than precession. We must point out that this conclusion is valid only at this Ekman number. At a smaller Ekman number, the precessing flow is more complex, which favors the onset of the dynamo, and this conclusion may not hold any longer (we leave the large-scale simulations at smaller Ekman numbers for other researchers who will be interested in the result of this work). The comparison between Figures 1 (a) and 2 (a) suggests that not only the mean poloidal kinetic energy but also the fluctuation amplitude of the precessing flow are much higher than those of the convective flow. This implies that the precessional instabilities are more vigorous than the convective instabilities in the two successful dynamos. The dominant azimuthal mode in the convective flow is m = 3 (Table 1 ), indicating a shorter length scale than the dominant spin-over mode m = 1 in the precessing flow. Figure 2 (b) shows the time evolution of the magnetic energy. Compared to Figure 1 (b), the magnetic energy of the convection dynamo becomes noticeable at the time ≈5000, which is much later than the time ≈400 of the precession dynamo. It is not surprising that the magnetic energy of the convection dynamo is also much lower than that of the precession dynamo (Table 1 ) because of the lower Rm of the former. Figure 2. Convection dynamo at . Here, we show the time evolution of the poloidal kinetic energy (a) and the magnetic energy (b). After studying the dynamos driven by precession alone and by convection alone, we study the combined effect of precession and convection. In the above two dynamos, the dynamo driven by precession alone cannot be maintained at Po = −0.2 and the dynamo driven by convection alone cannot be maintained at . We test whether the preceission-convection dynamo can be maintained at Po = −0.2 and . This dynamo works! It indicates that the combined effect of precession and convection can indeed facilitate the onset of the dynamo. As we discussed, the conventional Ra differs by more than one million, which indicates that precession greatly helps the onset of the convection dynamo. Moreover, it is interesting that the Rm = 677 of the precession-convection dynamo is lower than the Rm = 694 of the failed precession dynamo, as shown in Table 1 . This suggests that the combination of precession and convection has some non-trivial effect and triggers the dynamo action at a lower Rm of the pure precession dynamo. This non-trivial effect probably arises from the resonance of precessional instability and convective instability. Figure 3 shows the time evolution of this precession-convection dynamo and its time-averaged values are listed in Table 1 . Although the of the precession-convection dynamo is lower than the of the convection dynamo, the Nu of the former is higher (Table 1 ). This is because the poloidal flow driven by precession contributes more to heat transfer than convection, i.e., of the precessing flow at Po = −0.2 is already much higher than

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


reproduced with a two-component photometric model that consists of a PN-poor inner halo with a de Vaucouleurs profile and a PN-rich exponential outer component. Relying solely on the information from surface photometry, we were unable to constrain the scale radius of the exponential profile. We constrained the shape of the exponential profile using the information from resolved stellar populations. Harris et al. (2007a) and Lee Jang (2016) published number-density profiles of MR and MP RGB stellar populations in the halo of M 105. We combined the information from these studies to infer that the density distribution of the most MP stars ([M/H] −1.0) follows an exponential profile with a central surface brightness of μ 0 = 27.7 ± 0.1 mag arcsec −2 and a scale radius of r h = 258 ± 2′′ (Sect. 5 ). Using the information from the RGB populations, we constructed a two-component photometric model, where the two components were weighted by their α -parameter values. In the PN-poor inner halo, the light distribution is dominated by intermediate-metallicity ( − 1.0 ≤ [M/H] −0.5) and MR ([M/H] ≥ −0.5) stars, while in the PN-rich outer halo the MP stars dominate the light distribution. The α -parameter values of the two components are as follows: for the inner PN-poor component, and for the outer PN-rich component. We find a difference of a factor of seven in the α -parameter values of the inner versus outer components, implying that the luminosity-specific PN number of the stellar population that traces the old and MP halo at large radii is seven times higher. We interpret the high α -parameter value and the steep faint-end PNLF slope as tell-tale signs of an additional old and MP stellar population whose contribution towards the overall light distribution increases at large radii. This interpretation is corroborated by the results of Harris et al. (2007a) and Lee Jang (2016) based on resolved stellar populations. While in earlier works ( Longobardi et al. 2013 ; Hartke et al. 2017 ) this association was made mostly by inference, we are now able to directly establish the link between the high α -parameter value, the steep PNLF slope, and the increase in the old and MP stellar population in the outer regions of M 105 (identified by Harris et al. 2007a ). The data of Harris et al. (2007a) clearly ruled out a single chemical evolution sequence for the entire halo stellar population and instead suggested a multi-stage evolutionary model. For the outer MP halo, two different scenarios were proposed: Harris et al. (2007a) argued that the MP halo was built up from pristine gas and that star-formation therein was truncated early, at the time of cosmological reionisation. In contrast, Lee Jang (2016) proposed a formation dominated by dissipationless mergers and accretion. The scenario by Harris et al. (2007a) implies that the faint MP halo reached the current extension out to 40−50 kpc at high redshift, which is at odds with the observed slow growth of the

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


data to analyse them together with our CO ( J = 3−2) observations. BD+30 3639 was the source with the best complementary CO ( J = 2−1) data, given the size of the source and the beam size of the CO ( J = 2−1) observations. A discussion of the use of these diagnostic plots is presented for different molecules in the Appendix of van der Tak et al. (2007) . Linear molecules, such as CO, are density probes at low densities, while at higher densities, the line ratios are more sensitive to temperature. However, with no other information than a single line ratio (such as our case), it is only possible to provide lower limits to density and temperature. Table 4 Size, distance, and morphology of PNe with CO emission. References. (1) Cahn et al. (1992) , (2) Robinson et al. (1992) , (3) Groenewegen et al. (2012) , (4) Frew (2008) , (5) Sahai et al. (2007) , (6) Acker et al. (1992) , (7) Tajitsu Tamura (1998) , and (8) Guzmán et al. (2006) . In Fig. 2 we show the diagnostic plot for the CO ( J = 3−2) to CO ( J = 2−1) intensity ratio, assuming 2.73 K as the background temperature and a line width of 50 km s −1 . The plot shows the results for a column density of 1 × 10 16 cm −2 , but we find that the CO line ratio is not very sensitive to column density within the range 3 × 10 15 cm −2 to ~4 × 10 16 cm −2 . The plotted ratios corresponded to the observed values for BD+30 3639 (1.6), NGC 3132 (1.3), IC 4406, and M2-9 (0.7, for the last two). However, for the latter three sources, the measured ratios are only lower limits because of different CO ( J = 2−1) and CO ( J = 3−2) beam sizes (the former larger than the latter) and the source size (larger than both beam sizes). The CO intensity ratio constrains a lower limit to the H 2 volume density, n H 2 > 10 3 –10 3.7 cm −3 , and kinetic temperature, T kin > 10–35 K. The highest lower limit values are given for IC 4406 and M2-9. 3.6 Correlation with H2 We would like to probe the relationship between the detection of CO and H 2 . We compared our sample with the sample analysed by Kastner et al. (1996) , whose study was made by imaging the molecular hydrogen emission ( λ = 2.122 μ m) using the near-infrared cryogenic optical bench mounted at Kitt Peak National Observatory, from the National Optical Astronomy Observatories. They observed 60 PNe and detected H 2 in 23 of them. When compared with out sample, 9 objects are in common. Of these 9 objects, 4 were detected in H 2 , and all are also detected in CO as part of our survey. The objects are NGC 3132, IC 4406, PN M2-9, and BD+30 3639. The remaining objects (five) were not detected in H 2 and are they detected here in CO either. These PNe are PN M3-2, PN MyCn18, NGC 6778, NGC 6790, and HD 193538. Although the overlap between PANDORA and the H 2 survey of Kastner et al. (1996) is small, we find a clear correlation between the detection of CO and the detection of H 2 . Fig. 2 Results from the LVG analysis of the CO lines. The curves represent the

O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-CelestialObject I-CelestialObject 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject I-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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength 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 O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O B-Tag O O 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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O B-Wavelength O O O O O B-Observatory I-Observatory I-Observatory I-Observatory O O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O B-Survey O O B-Survey I-Survey I-Survey 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


is in good agreement with the correlation coefficient measured from the 545 and 857GHz dust maps in the Planck Sky Model (dashed lines). Dust is more than 99percent correlated between these frequencies. In the Planck 2013 analysis of cosmological parameters, the following simple parameteric model for the dust power spectrum was adopted: The parameters , c and are fixed by matching to the mask-differenced dust power spectra. Given the universality of the dust power spectrum, neither the choice of frequency nor mask sizes are critical in fixing these parameters. Using the 857GHz mask 40mask 25 spectrum, we find 0.387, c 162.9 and 0.168, so that asymptotically . This dependence at high multipoles is compatible with previous high-resolution measurements of the dust power spectrum in high Galactic latitude cirrus clouds, C 2.7 0.1 (Miville-Deschnes, Lagache Puget 2002 ; Miville-Deschnes etal. 2010 ), although our mask-differenced results consistently give slopes at the shallower end of this range. For a given mask, the dust power in the spectrum is specified by the parameter . For the values of , c and given above, is the dust power at multipole 10. The solid black lines in Fig. 4 show the model of equation ( 4 ) compared to the mask 40mask 25 and mask 50mask 25 spectra for the 353, 545 and 857GHz spectra. The model provides a reasonably good fit but one can see evidence for an excess at 500. We have therefore fit the dust spectra to a slightly more complicated fitting function: where B , 2 , 2 , c 2 , 2 and are parameters that we fit to the 857857 mask 50mask 25 spectrum. The best-fitting parameters are B 0.74, 2 0.19, 0.58, c 2 543.1, 2 11.6 and 0.07. We normalize the template at 10, so that the dust amplitudes derived using the templates of equations ( 5 ) and ( 4 ) can be compared directly. The best-fitting template of equation( 5 ) is shown by the red lines in Fig. 4 . Equation ( 5 ) provides a better fit to both the mask 40mask 25 and mask 50mask 25 spectra than equation ( 4 ), capturing the small excess at 500. Unless otherwise stated, we use the template shape of equation ( 5 ) in our baseline analysis. In summary, using mask-differenced spectra to isolate the contribution from Galactic dust, we find that the dust power spectrum has a nearly universal shape independent of frequency and sky fraction up to mask 60. For larger sky fractions, we see relatively small departures from universality that depends on the point source mask and are most likely associated with the anisotropy of the point source masks at low Galactic latitudes. The shape of the dust power spectrum is modelled accurately by the fitting function of equation( 5 ), which provides a template shape used in our likelihood analysis. We have verified that the dust power spectra are almost perfectly correlated over the frequency range 353857GHz. We therefore solve for three dust amplitudes, , and in our likelihood analysis assuming that the dust power spectrum is completely

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It can be seen that it agrees well with the other methods tested. The three methods, whilst not in exact concurrence, are in good agreement. Thus, to separate spirals and ellipticals in our sample, we use the parameter, both because it gives the intermediate values for fraction of spirals and ellipticals, and because it is more reliable and less computationally expensive when computed for a very large number of galaxies. We also present numerical results from our tests using g r band luminosity as the divide between spiral and elliptical galaxies, with a cut defined by equation ( A1 ) (Figs A2 and A3 ). Figure A2. Similar to Fig. A1 , but here galaxies are split based on the cutoff in g r band luminosity from equation ( A1 ). Figure A3. Distribution of ETGs in the R plane, as in Fig. 3 , but using g r luminosity, rather than as the cutoff between spirals and ellipticals. We find a smaller number of ellipticals by the same selection criteria, roughly half those in the ATLAS 3D survey, and a similar distribution, with a slightly higher fraction of highly elongated or fast-spinning galaxies compared to that found with a cutoff. We also experimented with making a cutoff between star-forming and quiescent galaxies based on their magnitude in the g and r bands. Galaxies for which g r is large are redder with little star formation and these quiescent galaxies are often ellipticals, whilst smaller g r colours characterize bluer, star-forming galaxies, which in observational surveys correspond well with the population of spirals (Kormendy 2016 ). We define a cut between star-forming galaxies as those below the line and quiescent galaxies as those above, motivated by the distribution of the whole sample of galaxies in Illustris (Vogelsberger etal. 2014 ). The luminosities are found by summing the individual luminosity of all stellar particles in the galaxy. The fraction of quiescent to star-forming galaxies, by mass, is in line with observations (Muzzin etal. 2013 ) over the whole mass range examined (down to haloes with stellar mass of 10 10 M ). Fig. A3 is a comparison with the galaxy population found in the ATLAS 3D survey, as presented in Section 3.1 using a cutoff defined by . We find less galaxies classified as ellipticals in the same volume, roughly half the previous number, with a wider spread in the distribution. There are more, faster spinning galaxies included though still few as rapidly rotating as in the ATLAS 3D survey. The population of high-mass galaxies is essentially unchanged, still with particularly elongated high-mass SRs as mentioned in Section 3.1 . Fig. A2 shows the stellar mass function derived using a colour cut. Whilst the results are more in line with the observed stellar mass function at low masses, we believe that this is due to the dependence of both on galaxy mass, rather than any strong link between luminosity and kinematic behaviour. Bottrell etal. ( 2017 ) performed similar analysis on carefully constructed mock

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


also been recently reported in HH 212 in the inner 100 au with a detection of CH 3 CHO and deuterated water HDO (Codella et al. 2016 , 2018 ). Now at higher resolution, we find CH 3 CHO to reside in the disk atmosphere. Moreover, we also detect 8 other organic molecules (H 2 CCO, t-HCOOH, CH 3 OH, CH 2 DOH, 13 CH 3 OH, CH 2 DCN, HCOOCH 3 , and CH 3 CH 2 OH) in the disk atmosphere, with an excitation temperature of ∼150 ± 50 K. Adding 3 other organic molecules (D 2 CO, CH 3 SH and NH 2 CHO) from Lee et al. ( 2017c ), we have 12 organic molecules, with 9 of them being COMs, detected in the disk atmosphere. These results indicate that the hot corino in HH 212 is actually a warm atmosphere of the disk. 4.2. Formation of Organic Molecules We can study the formation of the organic molecules in the disk atmosphere by comparing their abundances to those estimated in the hot corinos around other Class 0 low-mass protostars. The abundance of CH 3 CHO, which is commonly observed in hot corinos, is estimated here to be 3.9 × 10 −9 in HH 212, similar to those found in other hot corinos, e.g., 5.8 × 10 −9 in IRAS 16293-2422B (Jørgensen et al. 2016 ), 2.4 × 10 −9 in B335 (Imai et al. 2016 ), and 4.2 × 10 −9 in NGC 1333 IRAS4A2 (López-Sepulcre et al. 2017 ). To facilitate comparison, we calculate and compare the abundances of the organic molecules relative to CH 3 CHO. Figure 4 shows the comparison of the relative abundances of seven organic molecules (excluding deuterated species and 13 C isotopologue) to those in IRAS 16293-2422B (Jørgensen et al. 2018 ), B335 (Imai et al. 2016 ), and NGC 1333 IRAS4A2 (López-Sepulcre et al. 2017 ). In this comparison, CH 3 SH is excluded because of no reliable measurement of this molecule in other hot corinos to compare with. Interestingly, the relative abundances of most molecules here in HH 212 are similar to those in other hot corinos to within a factor of few, suggesting that the formation of these molecules in the disk atmosphere could be similar to that in other hot corinos. Notice that the abundance of CH 3 SH here is also consistent with that predicted in the hot corino, which can be as high as 5 × 10 −8 (Majumdar et al. 2016 ). Moreover, the abundances here are also similar to those seen in the Class I corino SVS13-A (Bianchi et al. 2019 ) and to those seen in Comet Lovejoy, which shows similar abundances to IRAS 16293-2422B (Biver et al. 2015 ). Figure 4. Comparison of the abundances of organic molecules in the HH 212 disk atmosphere with those in the hot corinos around other low-mass protostars. As shown in Table 2 , our measurements have an uncertainty of about 50%. As discussed in Lee et al. ( 2017c ), the high degree of deuteration in methanol (with [CH 2 DOH]/[CH 3 OH] ∼0.12) and the detection of doubly deuterated formaldehyde suggest that the methanol and formaldehyde in the disk atmosphere are originally formed on icy grains and later desorbed (evaporated) into gas phase due to the heating possibly by

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x min in our research. Figure 1. Cumulative distributions of fluence, peak counts, rise time and duration for 4U 1728-34. The best-fitting results are shown by the red lines. The best-fitting parameters are given in each panel. Based on the criteria proposed by Aschwanden ( 2011 , 2014 ), which are the statistical independence of events, non-linear growth phase, random rise times and power-law distribution of these observed parameters, type I X-ray bursts can be explained by a SOC process. The SOC theory predicts that subsystems self-organize, owing to some driving force, to a critical state, at which a slight perturbation can cause a chain reaction of any size within the system (Bak, Tang Wiesenfeld 1987 ). The SOC theory also predicts a power-law distribution for the waiting times (Aschwanden 2011 ; Marković Gros 2014 ). The WTD has been widely discussed in astrophysics (Wheatland, Sturrock McTiernan 1998 ; Aschwanden McTiernan 2010 ; Wang Dai 2013 ; Li et al. 2014 ; Wang et al. 2015 ; Guidorzi et al. 2015 ; Wang Yu 2017 ). In the SOC theory, the WTD is usually described as a (non-)stationary Poisson process. The probability function is expressed by (Wheatland et al. 1998 ) where Δ t is the waiting time, T is the period and λ( t ) is the burst occurrence rate. Below, we model the occurrence rate λ. We propose that the burst rate can be modelled by λ( t ) ∝ t −γ . The reason is given below. Type I X-ray bursts are induced by a thin-shell instability. In LMXBs, the accreted H/He fuel accumulates on the stellar surface and forms a shell. The typical time for nuclear energy release is τ N ∝ Δ M / L N , where Δ M is the mass of the shell and L N is the nuclear energy generation rate (Hōshi 1968 ; Joss 1977 ). In the quiescent stage, L N remains almost unchanged. The energy will be diffused in a typical time τ d ∝ Δ M 2 , which also depends on the stellar temperature and the opacity of the shell (Hōshi 1968 ; Ayasli Joss 1982 ). The condition for triggering a burst is τ N /τ d = Δ M crit /Δ M ≤ 1, where the critical shell mass Δ M crit is obtained by τ N = τ d (Hōshi 1968 ; Ayasli Joss 1982 ). Alternative descriptions of this critical condition are also used (Fujimoto et al. 1981 , 1987 ; Narayan Heyl 2003 ). We assume that Δ M = Δ M crit happens at a typical time t 0 . When the shell grows thick enough at a time T , any perturbation affects the stellar hydrostatic structure significantly and leads to stable nuclear burning (Schwarzschild Harm 1965 ). Therefore, a burst can be triggered only in a time range t 0 ≤ t T . Since these time-scales are mainly determined by the shell mass, it is plausible to assume that the occurrence rate also mainly depends on the shell mass. Because the accreted mass is proportional to time, we can adopt for the burst rate The occurrence rate is taken as λ = 0 when t t 0 . We put all the time-dependent effects into the power-law index γ, which contains the dependence on shell mass and the possible time evolution of

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


(above 500 km) and 4 K at 200 km. Each observation was modeled independently and results for all five emission lines were combined using the technique provided in Nixon et al. ( 2008a ) Section 4 . A systematic error in the CO line intensity of 2% (Rothman et al. 2013 ) was also added to the measurement error. The best-fitting abundance for CO was found to be 49.6 1.8 ppm, which is consistent with and more accurate than previous measurements. Retrieved vertical temperature profiles from CO models were then used as fixed temperature profiles for modeling of the less abundant CO isotopologues ( 13 CO, C 18 O, and C 17 O.) For each isotopologue transition, the retrieved temperature profile which was closest in time to that observation was used as the fixed temperature profile with errors from that temperature retrieval propagated into the isotopologue retrieval. For a priori abundances, we assumed terrestrial isotopic ratios of 89.3 for 12 C/ 13 C, 498.8 for 16 O/ 18 O, and 2680.6 for 16 O/ 17 O (Lodders 2003 ). Best-fitting models of CO isotopologues were found using the same modeling procedures as CO (aside from the fixed temperature profile) and are presented in Figure 2 . The combined best-fitting 12 C/ 13 C ratio of 89.9 3.4 is consistent with similar studies of other molecules (Courtin et al. 2011 ; Mandt et al. 2012 ; Nixon et al. 2012 ), suggesting no significant carbon fractionation occurred during CO formation. Best-fitting ratios of 16 O/ 18 O and 16 O/ 17 O are 486 22 and 2917 359, respectively. Although the C 18 O (2–1) peak appears slightly under-fitted, the spectral residuals (model minus observation) for this transition indicate an optimal fit given the noise. The mixing ratios for C 18 O, derived independently from the J = 3–2 and J = 2–1 transitions, are consistent within the observational errors. Figure 2. Top: spatially integrated spectra of 13 CO with best-fitting models overlaid. Bottom: spatially integrated spectra of C 18 O and C 17 O (3–2) with best-fitting models overlaid. C 18 O (2–1) and C 17 O (3–2) are each combinations of two observations taken close in time to each other, while C 18 O (3–2) is a combination of three observations. Observations were combined in CASA before deconvolution. Additional systematic errors that have not been quantified in the final error estimate could exist. Sources of systematic errors include our assumption that Titan has the same temperature profile at all locations in our disc-averaged calculation, as well as our assumption that the Voigt profile is correct. Additional errors could exist in the line-broadening coefficients and temperature-dependence exponent used in the line database (Rothman et al. 2013 ), which could alter the line strength and shape. An accurate estimation of these systematic errors is difficult and beyond the scope of this paper, but these possible sources are mentioned for completeness. Our results constitute the most accurate determinations of atmospheric

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


stars increases rapidly toward fainter magnitudes. The relative fractions are a combinations of the intrinsic distribution of different types of sources, whereby the density of Galactic stars increases toward the disk of the Milky Way, while the distribution of extragalactic sources does not depend on the Milky Way structure, but all sources are affected by the Milky Way dust, which is mostly confined to a thin disk. The reddening therefore depends on the line of sight toward the sources going through more or less of the dust disk. This effect was dramatically illustrated by Figure 2 (bottom) of Bianchi et al. ( 2011a ): the “V-shape” region devoided of UV-source counts in their figure is essentially a direct image of the dust disk. It is also visible, though less evident, in Figure 6 . Figure 6. Top: number of sources per square degree detected in NUV (left) and in FUV (right), as a whole (circles) or by magnitude ranges (dots). Values are shown for every 5° latitude strip. Bottom: fraction of FUV detections over NUV detections (left), and fraction of sources with FUV–NUV 0.5 (middle), and 0.0 (right) among the FUV detections (no error cuts, but sources in the footprint of extended objects have been excluded from the counts). While the faint sources (largely extragalactic) dominate the total samples, the FUV-detected sources are mostly bright stars. We recall that, for average Galactic dust, the UV extinction is similar in FUV and NUV, and much higher in both bands than at optical wavelengths (Table 1 and Bianchi 2011 ). In Figure 6 we plot the density of NUV and of FUV sources in GUVcat_AIS (top plot) as a whole and divided by NUV magnitude ranges; the plots show that the sources fainter than NUV_mag = 21mag dominate the sample, despite AIS being the shallowest survey, and especially in the NUV, where extragalactic objects are more prominent. The bottom panels show the fraction of FUV detections over NUV detections, again as a function of Galactic latitude, and among these, the hot and very hot sources (FUV_mag-NUV_mag ≤ 0.5 and ≤0.0 respectively). Such UV color cuts correspond to different stellar for different types of stars (Bianchi 2009 ), but roughly hotter than ∼15,000 K. Some QSOs may intrude on these FUV–NUV color cuts, as shown by Bianchi et al. ( 2009 ): these affect the faint sources most. The different behavior of relative source densities in Figure 6 reflects the fact that brighter samples (and hotter samples) are dominated by Galactic stars, which are more numerous in the Milky Way disk (see, e.g., Bianchi et al. ( 2011a )). 8.2. Summary of Suggestions and Caveats for Using the Catalog To conclude, we distill here, in terms of practical advice for users, the relevant information on the catalogs presented in this paper. 1. GUVcat_AIS contains unique measurements of all sources from AIS observations with both FUV and NUV detectors exposed. Duplicate measurements are removed in the main catalog, however, a version called

O O 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-CelestialObject I-CelestialObject 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 B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Wavelength O O O O O O O B-Wavelength O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O B-CelestialObject O O B-Wavelength O O O O B-Wavelength O B-Wavelength O O O O O O O O B-Wavelength O O O O B-Citation I-Citation O O O O O O O O O B-Wavelength O O B-Wavelength O O B-Dataset O O O O O O O O B-Wavelength O O O O O O O O O O B-Formula I-Formula I-Formula O O O O B-Survey O O O O O O O O B-Wavelength O O O O O O O O O O O O O B-Wavelength O O B-Wavelength O O O O O O B-CelestialObject O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-Wavelength 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 B-Wavelength O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O B-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 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 B-Telescope O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O


dicarboxylic acid (Hollauer et al. 2001 ). The aforementioned study assigned the ν (C=O) + δ (COH) combination band to 1630 cm −1 . Assignment of the remaining newly emerging bands was accomplished using a study on the literature infrared spectra of the three benzene dicarboxylic acids (Arenas Marcos 1980 ). However, it is to be noted that two broad peaks, at 1776 cm −1 and 1793 cm −1 , in the spectrum at 8 K did not correspond to a single carrier. Rather, the bands were assigned to a combination of carbonyl stretches from the various carboxylic acid species in the ice. 3.1.2. Quantitative Analysis From the peak assignments in Section 3.1.1 , the column density changes throughout the irradiation can be determined for the two reactants, benzene and carbon dioxide, along with the irradiation-induced products. For carbon dioxide, the temporal development of the ν 1 + ν 3 and 2 ν 2 + ν 3 of CO 2 and the ν 3 of 13 CO 2 was monitored to quantify the carbon dioxide molecules destroyed. Exploiting the integrated absorption coefficients of the three previously mentioned bands, 1.4 × 10 −18 cm molecule −1 for the ν 1 + ν 3 of CO 2 , 4.5 × 10 −19 cm molecule −1 for the 2 ν 2 + ν 3 of CO 2 , and 7.8 × 10 −17 cm molecule −1 for the ν 3 of 13 CO 2 , we found that (1.9 ± 0.9) × 10 16 molecule cm −2 of carbon dioxide was destroyed over the course of the hour-long irradiation. Corresponding to the diminishing carbon dioxide column density were new bands assigned to carbon dioxide and carbon trioxide. The development of the two new carbon oxides was monitored using the fundamental band of carbon monoxide, assigned to 2141 cm −1 , and the ν 1 of carbon trioxide, assigned to 2044 cm −1 . Using a previously published work on the irradiation of carbon dioxide ices (Bennett et al. 2004 ), we found that the integrated absorption coefficients of the two previously mentioned carbon monoxide and carbon trioxide bands to be 1.1 × 10 −17 cm molecule −1 and 3.1 × 10 −17 cm molecule −1 , respectively. Utilizing this information, we determined that following irradiation with energetic electrons, the total increase in column density of carbon monoxide was (1.1 ± 0.1) × 10 16 molecule cm −2 , while the total increase of carbon trioxide was (7.9 ± 0.3) × 10 14 molecule cm −2 . The total column density of newly formed low-mass carbon species was therefore (1.2 ± 0.1) × 10 16 molecule cm −2 , i.e., 64 ± 3% of the total carbon dioxide destroyed was converted to carbon monoxide and trioxide. The difference between this value and the total quantity of carbon dioxide destroyed was therefore the amount reacted to form benzene carboxylic acids, i.e., (6.8 ± 1.0) × 10 15 molecule cm −2 . The destruction of benzene was monitored by the development of its ν 11 band at 690 cm −1 , with an integrated absorption coefficient of 1.5 × 10 −18 cm molecule −1 (Zhou et al. 2010 ). Comparing the initial column density of benzene with the column density following irradiation, we found that (4.4 ±

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


at 3 s resolution. Magnetic field data is provided by the MFI instrument (Lepping et al. 1995 ) and density and velocity data by the 3DP instrument (Lin et al. 1995 ) measured during a stationary slow stream. The start time of the interval is 12:00 2000 April 4 and stop time is 16:00 of 2000 April 6. During this interval the solar wind speed remained about 300 km s −1 . As it is seen from Fig. 1 , λ 0.03 except for very small τ. We suggest that this enhancement at the small scales is probably due to measurement uncertainties (which are known strongly affecting plasma parameters at high frequencies Kasper et al. 2006 ; Gogoberidze et al. 2012 ) rather than indicating a real enhancement of compressible perturbations. To support this suggestion, we present another important characteristic of compressible/non-Alfvénic perturbations (Tu Marsch 1995 ; Bruno Carbone 2013 ) by dashed line in Fig. 1 . As can be seen λ m is approximately constant and does not increase with decreasing time-scale. Figure 1. The ratio of dimensionless rms amplitudes (solid line) and (dashed line, see the text for details) as a function of time-scale for a stationary stream of the slow solar wind. The above consideration suggests that the thermal-speed spectra reported by Safrankova et al. ( 2016 ) are in fact the pseudo-thermal spectra dominated by the magnetic fluctuations rather than the temperature fluctuations. This conclusion is supported by the fact that the level of spectra reported by Safrankova et al. ( 2016 ) cannot be caused lonely by the temperature fluctuations. Indeed, the spectrum of the (pseudo-)thermal speed in their fig. 1(b) is only about three times lower than the spectrum of the bulk speed driven by Alfvén waves (their fig. 1a), whereas the authentic temperature fluctuations should be at least one order less energetic (Goldsein Siscoe 1972 ; Tu et al. 1991 ; Tu Marsch 1995 ). To be more specific, let us utilize the straightforward expression relating the bulk velocity fluctuations δ V Bn measured by Safrankova et al. ( 2016 ) and magnetic fluctuations due to Alfvén waves. By means of ( 34 ), we can eliminate from ( 31 ) thus arriving to the following estimation for the velocity spectra ratio: With reasonable values T 0∥ / T 0⊥ ∼ 0.5 or T 0∥ / T 0⊥ ∼ 1.8 and , ( 35 ) easily explains the modest ratio ≳ 1/3 between 〈|δ V Tn | 2 〉 B and deduced from figs 1(a) and (b) by Safrankova et al. ( 2016 ). With autentic temperature perturbations this ratio would be at least one order less (Goldsein Siscoe 1972 ; Tu et al. 1991 ; Tu Marsch 1995 ). 4 DISCUSSION AND CONCLUSIONS We have shown that the nature of 'temperature fluctuations' derived from the currents of Faraday cup(s) is strongly affected by the temperature anisotropy. If the temperature anisotropy is close to zero, then the derived 'temperature fluctuations' are dominated by the perturbations of real temperature (thermal speed). However, when the temperature anisotropy is finite, as is typical

O O O O O O O O O O O B-Instrument O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Instrument O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O 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 B-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O 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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


parameter in N -body simulations. Numerous studies have argued that minimizing the scatter in certain statistical descriptions of the matter field (i.e. power spectrum or two-point correlation function) among an ensemble of N -body realizations requires a softening length ε that is at least as large as the mean inter-particle separation l = L box / N 1/3 (e.g. Melott et al. 1997 ; Splinter et al. 1998 ; Romeo et al. 2008 ; Joyce, Marcos Baertschiger 2009 ). Here, L box is the (periodic) box size of the cosmological simulation and N is the total number of particles used. Most cosmological simulations, though, are run with much smaller softening lengths, typically of the order of ε/ l ∼ 0.01–0.02. For example, the Millennium simulation (Springel et al. 2005 ) has ε/ l ∼ 0.022, while the MDPL and SMDPL simulations of Klypin et al. ( 2016 ) adopt ε/ l ∼ 0.019 and ε/ l ∼ 0.014, respectively. Whether such simulations produce reliable results on scales below the (comoving) mean inter-particle separation is still a matter of fierce, ongoing debate (see e.g. Knebe et al. 2000 ; Melott 2007 ; Joyce et al. 2009 ; Power et al. 2016 ; Benhaiem, Joyce Sylos Labini 2017 , and references therein). Several studies have focused on the optimal softening length for simulating individual objects (i.e. individual DM haloes or galaxies). For example, Dehnen ( 2001 ), extending studies by Merritt ( 1996 ) and Athanassoula et al. ( 2000 ), suggests that one should aim to minimize force errors. This can be achieved by optimizing the softening length such that it simultaneously minimizes the force bias (which increases with increasing ε) and the force noise (which decreases with increasing ε). Based on such considerations, Dehnen ( 2001 ) advocates that a collisionless system with a Hernquist ( 1990 ) density profile (which has the same r −1 -cusp as a NFW profile) should be simulated with a Plummer softening , where r 0 is the scale radius of the Hernquist (or NFW) profile and N 5 = N /10 5 with N the number of particles used to model the system in question. Power et al. ( 2003 ) use a different criterion to optimize the softening length: by minimizing the impact of two-body effects on the (central) density profiles of virialized DM haloes in cosmological simulations, they advocate a much larger softening length of , where c is the NFW halo concentration parameter. Note that both of these criteria have the optimal softening length depend on the size of the system as well as the number of particles that is used to represent the system. If we adopt a concentration–mass relation of the form c ∝ M −0.1 (e.g. Dutton Macciò 2014 ) and use that the virial radii of DM haloes scale according to r vir ∝ M 1/3 ∝ N 1/3 , we find that the optimal softening length depends on halo mass according to ε opt ∝ M α , with α = +0.2 and −0.17 for the criteria of Dehnen and Power et al., respectively. This scaling is sufficiently weak so that the softening is close to optimal for haloes

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


of the inner binary progressively suppresses the effect. This is because a in decreases, bringing down the ratio in equation ( 38 ). The exact point at which K-L becomes effective also depends on the mass ratio of the intruder with respect to the inner binary, and on the eccentricities of the inner versus the outer binary. Note that as e in increases, the ratio decreases. Therefore, the K-L mechanism might eventually self-regulate itself: as it grows the inner binary more eccentric, it also makes relativistic precession more effective. The condition might therefore not be satisfied any longer, thus suppressing the K-L effect and eventually altering the overall dynamics of the system. Although our investigation is similar in spirit to that of Hoffman Loeb ( 2007 ), the above discussion highlights the importance of the differences in the two implementations. In particular, Hoffman Loeb ( 2007 ) did not include 1PN and 2PN relativistic precession in their equations. This might significantly alter the overall statistical properties of merging binaries (e.g. eccentricities, coalescence time-scales) because, as we just discussed, the conditions for triggering K-L resonances, and thus the general dynamics of the system, are different. Moreover, our treatment of the hardening in the stellar background is more accurate, since it also reproduces the eccentricity evolution in the hardening phase. This is important because (i), it affects the triggering point of K-L cycles (equation 38 ), and (ii), it has a strong impact on the stability of the hierarchical triplet and on the probability of close encounters leading to chaotic behaviour. 5.2 Astrophysical implications The main aim of this paper is to present and validate the code, and a thorough analysis of the dynamics of MBH triplets is deferred to forthcoming papers of the series we planned. Still, the few cases examined here already provide some interesting astrophysical insights that we briefly outline in the following. In all our simulations, two of the MBHs (not necessarily those originally forming the inner binary) coalesce in less than 1Gyr from the start of the dynamical friction phase of the intruder. Therefore, triple interactions might provide a viable channel to merge MBH binaries in massive, low-density galaxies, where hardening against the stellar and gaseous background might act on a time-scale of several Gyr (Khan et al. 2011 ; Sesana Khan 2015 ; Vasiliev et al. 2015 ). At least a fraction of massive elliptical galaxies, which host the most massive binaries targeted by PTAs, might realistically undergo multiple mergers at z ≲1 (see Gerosa Sesana 2015 , and references therein). If the typical coalescence time-scale of the formed MBHBs is several Gyr, then the occurrence of a second merger will bring in an intruder, typically leading to coalescence on a much shorter time-scale. Triple interactions might therefore be an important channel for merging very massive, gas-poor low-redshift

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


the LG (e.g. see updates to McConnachie 2012 ). While environmental effects (e.g. efficient destruction of satellites, dwarfdwarf mergers, stellar stripping) could alleviate some tension, these mechanisms do not appear efficient enough to resolve the discrepancy (e.g. Kirby etal. 2013 ; Deason etal. 2014 ; Garrison-Kimmel etal. 2017 ). Furthermore, the overabundance of LG dwarfs is not limited to the faintest UFDs, as it is known to persist in the classical dwarf regime (Boylan-Kolchin etal. 2015 ), which is thought to be observationally complete (e.g. Koposov etal. 2008 ). Even more challenges arise if (i) the faint end of the UVLF approaches 2.4 (i.e. many more faint galaxies); and/or (ii) such low-mass galaxies have a low star formation duty cycle (e.g. 10percent; Wyithe etal. 2014 ). In the latter case, 90percent of the haloes would be in the off state at any given time, and thus have reduced UV luminosities. Effectively, this results in higher luminosity galaxies hosted in even lower mass haloes, assuming a canonical SHM. 3.2 Minimizing the flux contribution of ultra-faint dwarfs Up to this point, we have considered the consequences of extrapolating a Schechter UVLF into the regime of UFDs. However, several numerical simulations of galaxy formation predict a turnover in the UVLF, decreasing the predicted number of very faint galaxies in the early Universe (e.g. Jaacks etal. 2013 ; OShea etal. 2015 ; Gnedin 2016 ; Liu etal. 2016 ; Ocvirk etal. 2016 ; Yue etal. 2016 ; Finlator etal. 2017 ). Although details vary, the common threads among these simulations are that not all low-mass haloes necessarily host a galaxy and that baryonic effects internal to the galaxies (e.g. supernova feedback, radiation pressure) can significantly affect galaxy formation in low-mass haloes. Alternately, a turnover in the high- z UVLF is consistent with models that suppress or eliminate small-scale structure (e.g. warm dark matter; Schultz etal. 2014 ; Dayal etal. 2015 ). A broken luminosity function at high z is also necessary in order to avoid dramatically overproducing UFDs and even classical dwarfs in the LG (Boylan-Kolchin etal. 2014 , 2015 ). Fig. 4 illustrates the effects of two plausible UVLF turnovers. For simplicity, we focus only on the z 7 case and adopt analytic UVLFs that turn over from Jaacks etal. ( 2013 ) and Boylan-Kolchin etal. ( 2015 ). The UVLF from the latter study has 2 for M UV 13 and 1.2 for M UV 13 and is designed to reproduce z 0 LG dwarf galaxy counts. We have tuned the UVLF from Jaacks etal. ( 2013 ) to match that of Boylan-Kolchin etal. ( 2015 ). Figure 4. Top panel: a comparison between the UVLFs at z 7 with and without turnovers. The steep UVLF (black; 2) is from Finkelstein etal. ( 2015 ), while the other curves show models from Boylan-Kolchin etal. ( 2015 , orange) and Jaacks etal. ( 2013 , magenta). Bottom panel: the fraction of UV flux at z 7 for each of the UVLFs shown in the top panel, when integrated to M min 3.1. A break or

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


counter rotation was produced in the first place, the formation of retrograde accretion discs has also been proposed to explain the long-term evolution of spin periods in X-ray binaries ( Nelson et al. 1997 ; Christodoulou et al. 2017 ). However, for the time window 2010–2015 we can only speculate for the evolution of Ṁ due to lack of observational coverage and because of the high absorption of the X-ray spectrum (see Fig. 3 of C18). Nevertheless, the P and Ṗ measurements during the 2014 Chandra observation are consistent with the constant Ṁ assumption (see Fig. 3 ), supporting our finding that the decrease of the observed flux between 2010 and 2015 is only due to high absorption. If Ṁ during that period was two orders of magnitude lower than after 2015 then it is possible that there was no NS spin reversal. In this case the NS spin period should have been larger than a few hundred seconds prior to the SN 2010da event. In general there are multiple ways to maintain the necessary Ṁ to fuel ULXPs. For systems fueled by Roche Lobe (RL) overflow, the accretion flow carries the angular momentum of the orbit. It is therefore implausible for an accretion disc to switch between retrograde and prograde rotation ( Chakrabarty et al. 1997 ), and any change between retrograde and prograde rotation of the NS disc system needs to be due to a change in the direction of the NS rotation. Therefore, for NGC 300 ULX1 it is more plausible that the accretion mechanism was drastically altered during the SN 2010da event, prior to which the NS could have been fueled via wind accretion. A possible explanation for such a phase transition to occur can be an episode of thermal-timescale mass transfer ( King et al. 2001 ). In that case the donor star has a radiative envelope and fills its RL as it expands across the Hertzsprung gap after the wind-fed X-ray phase ends. However, RL overflow is not the only way to explain such high accretion rates. Supergiant systems and luminous blue variables (LBV) can also exhibit mass loss rates in excess of 10 −6 M ⊙ yr −1 ( Smith 2014 ), and therefore for a binary system in a moderately close orbit the compact object can capture enough matter to sustain super-Eddington accretion. An LBV is also most probable according to Binder et al. (2016) while other authors have deemed this classification inconclusive ( Lau et al. 2016 ; Villar et al. 2016 ). We note that many of the arguments used in the classification of the system were based on the decay of the X-ray flux observed after the SN 2010da event. However, the decay in the X-ray flux was most likely a result of rapid increase in X-ray absorption (C18). 4. Conclusions We have studied the spin period evolution of NGC 300 ULX1, the most extreme spin-up NS powered by an accretion disc. We derived precise measurements of P and Ṗ for different epochs, and applied theoretical models of accretion theory in order to estimate the magnetic field strength of the NS. Even though many of the models

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


in Sect. 3.2 . We did not identify any significant gas contamination from other species in any of our spectra. Second, we selected the lines with intensities above a defined threshold (ten times the noise level). We summarize the results in Table 4 . We find that the number of U I lines differs considerably for different lamps (e.g. in the lamp 0127 we find a factor of 1.8 more lines than in the lamp 0120). The number of U II , Ne I, and Ne II lines is essentially constant. Table 4. Number of confirmed isolated U and Ne lines with S/N higher than ten found in the 12 U–Ne spectra for NIR wavelengths. The last column contains the average of the U line intensity normalized to the average of its line intensity in the 12 spectra ( I U ). We looked for the confirmed lines that can be found in all of the 12 spectra to systematically analyse the difference between the 12 lamps. In total, we detected 222 U I and 104 Ne I lines in the spectra of all lamps. We normalized the 12 spectra to the sum of the Ne lines intensities averaged in each spectrum. Then we divided the line intensity of each U line by the average line intensity of this line in the 12 spectra. Last, we calculated the average of the normalized line intensity of the 222 U I lines (named I U in Table 4 ) for each lamp. In Fig. 8 we plot the normalized line intensity of the common U I lines in the NIR spectral range for seven HCLs (for readability reasons). The dashed lines represent I U for each HCL. From Fig. 8 and Table 4 it can clearly be seen that I U varies significantly between the lamps. Lamps with the lowest U line intensities are HKH0120 and HKH0122. Lamps with the highest U line intensities are HKH0127 and HKH131. The maximum U line intensity difference is 88% (with respect to the minimum value), measured between lamps HKH0127 and HKH0120. Fig. 8. Coloured dots represent the normalized line intensity of 222 U I lines for each HCL in the spectral range from 1000 to 1700 nm. Dashed coloured lines indicate of each lamp (i.e. average of the 222 U I normalized line intensity). For readability reasons we plot only seven of the 12 HCLs. A possible explanation for the observed behaviour of the HCLs could be a different geometry of the lamps or in the light-fibre coupling efficiency, as seen in Huke et al. (2016) . This could affect the gas-metal intensity ratio. Another explanation could be that the metal line intensities are very sensitive to the internal pressure of the buffer gas ( Crosswhite et al. 1955 ). Therefore, even small changes in pressure from lamp to lamp would have an effect on the metal line intensity. We conclude that HCLs can show noticeable differences in number and intensity of metal lines even if they are manufactured in a single batch. 7. U line list We concluded in Sect. 4 that U cathode lamps are more suitable for wavelength calibrators than Th lamps from 500 to 1700 nm. To use HCLs as wavelength calibrators for high precision RV spectrographs, the wavelength of the

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


We also find a large extinction of A V = 1.56 mag for this target. HD 38043, sectors 1–3, 5–6, 8–13 The TESS light curve spans 357 d and includes all sectors except 4 and 7. The data are at 2-min cadence in all available sectors. The p modes appear to be dominated by two nearly equally spaced multiplets: the first contains the strongest three modes – the same ones for which frequencies and amplitudes are provided in Table 1 – with a spacing of ∼0.51 d −1 . The second is slightly lower in frequency and amplitude and has a spacing of ∼0.55 d −1 . These could be rotationally split multiplets of modes with similar but non-identical n and l. This star is highly multiperiodic at lower frequencies, consistent with g-mode pulsation, though we could not identify any clear period spacings. HD 41958, sectors 1, 4–5. 7–8, 11 The three strongest p modes have similar amplitudes (7.5–8.2 mmag). They do not have any obvious harmonic relationship, nor do they have the frequency ratio (0.77–0.78) expected of the fundamental mode and first radial overtone modes. HD 42503, sectors 5–7 The mutliperiodic nature of the light curve at frequencies near 5 d −1 is rather unusual. The frequency content does not strongly resemble that of γ Dor stars, but is also quite different from typical δ Sct stars. No radial modes could be identified from frequency ratios of the stronger peaks. The peaks at 0.99 and 2.00 d −1 are not perfect harmonics, and another peak at 3.94 d −1 also falls on this near-harmonic pattern. HD 43533, sectors 6–7 The star is a γ Dor variable and the light curve is dominated by low-frequency variability, with dozens of significant peaks below 6 d −1 , and this variability spills into the δ Sct frequency domain. Excluding this variability, which presumably does not include independent p modes, there are only a few peaks in the Fourier transform that make this a δ Sct star. All three extracted peaks are significant, with SNR>4, despite their low amplitudes. The low amplitudes perhaps suggest these peaks belong to a contaminating source, but the Gaia DR2 data show no likely contaminant. It is noteworthy that this star is located below the ZAMS compared to tracks of solar metallicity. HD 44930, sector 5 The Fourier transform is dominated by several p modes of mmag amplitude. A group of weaker peaks with frequencies between 2 and 3 d −1 might be g modes. The absence of peaks at even lower frequencies would then suggest these are l = 1, m = 1 prograde modes, which suggests the near-core region rotates rapidly. This star is in the Octans association and has a measured Δν of 6.03 d −1 (Bedding et al. 2020 ). The échelle diagram (Fig A1 ) suggests f 1 = 18.15 d −1 is the fundamental mode, even though this is not in agreement with the P–L prediction. HD 47425, sectors 6–7 The two dominant p modes are too close in frequency to be the fundamental mode and first radial overtone. Indeed, it is not possible to clearly identify radial modes by period ratios in

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