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 B-Citation O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O B-Citation O O O O B-Citation O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O B-CelestialObject O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O B-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Database I-Database I-Database B-Citation I-Citation I-Citation I-Citation O B-URL O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Citation O O O O O O O O O O O O O O O O B-Citation O O O O O O B-Citation O O O O O O O O O O O B-Citation O O O O O O O O O O B-Database I-Database I-Database 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 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 O O O O O O B-Citation O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O

of the H 2 ortho to para spin ratio (OPR), the correlations and anticorrelations between D and 15 N found in primitive materials (van Kooten et al. 2017 ). In this paper, we report calculations that re-evaluate those of W12 in light of the revised R15 fractionation chemistry. This study is necessary because, apart from the rate coefficients adopted, there exist significant differences in the physics and chemistry of the models of W12 and R15 . The calculations of W12 considered densities similar to pre-stellar core conditions, about 10–100 times greater than those of R15 , which were based on a standard dark cloud similar to TMC-1 ( n H = 0.2–2× 10 5 cm −3 ). In contrast to W12 , R15 neither included the depletion of gas-phase species by freeze-out on to dust grains, nor the time-dependence in the H 2 OPR – both effects which have been shown to have substantial impact on the relative fractionation in nitriles and amines (here nitrogen hydrides and chemical derivatives thereof) over time ( W12 ). Furthermore, even when reactions ( 1 )–( 3 ) are not operating, there are alternative fractionation processes that could still become important in the physical conditions of the W12 model. For example, W12 identified as playing a role in the fractionation of N 2 and NH 3 , whereas Rodgers Charnley ( 2008a ) speculated that the neutral–neutral pathway could be viable and efficient if there is a barrier to the reaction However, the reaction appears to have no formation barrier, and the recommended reaction rate from the KIDA data base (Wakelam et al. 2013 , http://kida.obs.u-bordeaux1.fr ) of k (T) = 8.8 × 10 −11 (T/300) 0.42 cm 3 molecule −1 s −1 agrees with laboratory measurements down to 56 K (Daranlot et al. 2012 ). W12 assumed a 200 K barrier for reaction ( 6 ) (see Le Teuff, Millar Markwick 2000 ) and R15 found that isotopic exchange through ( 5 ) was indeed plausible, a process not considered in W12 . We have therefore updated the W12 model with respect to the revisions to nitrogen fractionation advocated by R15 , and specific reactions in the nitrogen chemistry from the KIDA data base (Wakelam et al. 2013 ). The paper is structured as follows. Revisions to the 14 N fractionation chemistry and the updates to specific reactions in the nitrogen chemistry are described in Section 2 . The chemical model is summarized in Section 3 . Results from the model calculations are given in Section 4 where they are compared to previous work. Conclusions from this study are in Section 5 . 2 REVISED CHEMICAL NETWORK 2.1 15N fractionation Table 1 lists the major reactions potentially leading to 15 N fractionation in cold, dark clouds, both ion–molecule and neutral–neutral reactions, with ZPE differences, Δ E , as determined by R15 . The resulting equilibrium coefficient, K ( T ) = k f / k r , the ratio between the forward and reverse reaction rate (e.g. Terzieva Herbst 2000 ), is listed for each reaction at 10 K, both for the new fractionation


O O O O O O O O O O O O O B-Wavelength O O O B-CelestialObject I-CelestialObject O O O O O B-Telescope B-Instrument O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-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-Telescope B-Citation I-Citation I-Citation O O O B-Wavelength O O O O B-Mission I-Mission I-Mission O O O O O O O B-Wavelength I-Wavelength O O B-Telescope 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-Wavelength O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Telescope O O O O O O O O B-Software O O O O O B-Software O O B-Identifier O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope 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 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 B-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

photons using 30′′ and 18 radius circular apertures, respectively. Table 1 Log of X-Ray Timing Observations of 1E 1207.4−5209 Notes. a Exposure times for XMM-Newton EPIC-pn does not reflect the 29% deadtime in the SmallWindow (sw) mode. b Newly reported observations. The NICER data sets are denoted by the ObsID and date of the first of the concatenated set of observations (see Section 2 for details). Table 2 Single Glitch Ephemerides for 1E 1207.4−5209 Notes. Derived parameters ( B s , , τ c ) are based on the pre-glitch timing solution. a Uncertainties in the last digits are given in parentheses. b Epoch of the glitch estimated by matching the zero phase of the two timing solutions; this assumes a constant post-glitch . The NICER (Gendreau Arzoumanian 2017 ) is an X-ray telescope attached to the International Space Station that provides submicrosecond time resolution in the 0.2–12 keV band. The NICER telescope consists of a set of 52 operational non-imaging silicon drift detectors (Prigozhin et al. 2016 ), each at the focus of an X-ray concentrator (Okajima et al. 2016 ) that subtends a 4′ radius field of view. The nominal effective area of the telescope is 1900 cm 2 at 1.5 keV. The NICER data sets were reduced and analyzed using the NICERDAD software suite distributed in the FTOOLS package, version 24Jun2019_V6.26.1, and the most up-to-date calibration files. We generated cleaned event files using the nicerl2 script that applied the standard filtering criteria. The data were further reduced by excluding detectors with anomalous count rates >5 σ above the mean rate, computed using all available detectors, in the energy range of interest. Similarly, we iteratively excluded time intervals with high background rates by comparing the event rate in 10 s steps to the mean rate. Since its launch in 2017, NICER has observed 1E 1207.4−5209 a total of 148 times to-date. Each observation is defined by a unique ObsID number and typically comprises short exposures (50% are less than 1.2 ks) spread over multiple satellite orbits, and often containing multi-day gaps within, and between. These exposures are generally too short to generate a precise pulse-phase measurement needed for our timing analysis. However, by concatenating adjacent observations we obtained eight sufficiently compact NICER data sets that contained the minimum number of events required to measure an independent pulse phase, as described below. 3. Timing Analysis For each reprocessed, cleaned event file, we folded the extracted photon arrival times on the pulsar frequency to compute the time-of-arrival (ToA) of phase zero of the pulse. To attempt a phase-connected timing solution, we fit the set of ToAs using the TEMPO software (Hobbs et al. 2006 ) to a model for the rotation phase of the pulsar including one or two of its frequency derivatives,Initially, we obtained the fold frequency from a periodogram search for the maximum power around the expected frequency, determined from the


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

CO is determined from the CO (2 − 1) emission by assuming a luminosity ratio of and α CO . Using the and 3σ for non-detections from table 6 in Boogaard et al. ( 2020 ), we estimate the α [C i ](1 − 0) with both molecular gas masses, and present in Table 6 and Fig. 8 with the coloured diamonds. Figure 8. α [C i ](1 − 0) as a function of metallicity. The black and red symbols represent each detections and 0.1 bin values for our sample. The blue boxes represent samples of high-redshift ( z = 1.9−3.4) gamma-ray burst and quasar molecular gas absorbers from Heintz Watson ( 2020 ) with , and the black line shows their best-fitting linear relation of logα [C i ](1 − 0) = −1.13 × log(Z/Z ⊙ ) + 1.33. The orange and green diamonds are ASPECS galaxies from Boogaard et al. ( 2020 ) with molecular gas masses estimated based on CO luminosity and 1.2 mm dust-continuum emission on the Rayleigh–Jeans tail, respectively. Table 6. The adopted redshift, molecular gas mass, [C i ] (1 − 0) luminosity, metallicity, and derived α [C i ](1 − 0) for the ASPECS galaxy. Note . Columns (1)–(5) are adopted from Boogaard et al. ( 2020 ). (1) and (2) are ASPECS ID and redshift, respectively. Column (3) is molecular gas mass that estimates via the 1.2 mm dust-continuum emission on the Rayleigh–Jeans tail, and (4) is the molecular gas mass determined from the CO (2 − 1) emission assuming a luminosity ratio of and α CO (see table 5 in Boogaard et al. 2020 ). Both of the molecular gas masses from Boogaard et al. ( 2020 ) include a factor of 1.36 to account for heavy elements. Column (5) lists the with 3σ for the non-detections (see table 6 in Boogaard et al. 2020 ). The α [C i ](1 − 0) in columns (6) and (7) are estimated based on molecular gas masses of M mol, RJ and M mol, CO , respectively. The metallicity in column (8) is adopted from Boogaard et al. ( 2019 ). Though our derived α [C i ](1 − 0) shows almost flat with metallicity, they distribute next to the relation of Heintz Watson ( 2020 ) as seen in Fig. 8 , and similar with the α [C i ](1 − 0) result of ASPECS galaxies as well [the α [C i ](1 − 0) values in both Heintz Watson 2020 and Boogaard et al. 2020 include a factor of 1.36 to correct the helium and heavier elements]. The metallicity in our sample mainly focuses in the range of , whereas the metallicity of the sample in Heintz Watson ( 2020 ) covers two orders of magnitude in the ranges of , which is mostly smaller than our sample. Various studies suggest that α CO increases with decreasing metallicity, turning up sharply below metallicity of (Wolfire, Hollenbach McKee 2010 ; Bolatto, Wolfire Leroy 2013 ) where CO is easily photodissociated, whereas H 2 is self-shields or is shielded by dust from UV photodissociation, and becoming shallower near subsolar metallicity (Glover Mac 2011 ; Tacconi et al. 2018 ). The flat profile between α [C i ](1 − 0) with metallicity in our sample presents that α [C i ](1 − 0) might be similar to α CO and has a fairly shallow


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Acknowledgments This research was supported by the DFG cluster of excellence Origin and Structure of the Universe (JED).


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as [see equation ( 1 )], but at the same time the additional power would increase AGN feedback, which is proportional to the luminosity. If we instead increase the BH feedback efficiency, i.e. the constant of proportionality with luminosity, the gas surrounding the BH is less cold and dense, causing BH accretion to decrease. To quantify these effects on the dual-activity observability time-scales, we compare Runs 02, C3 and C4, which differ only by the initial BH mass or feedback efficiency. Increasing the BH mass (by a factor of 2.5) increases the dual-activity time, but only by a factor ≲2. Increasing the BH feedback efficiency (by a factor of 5) tends to decrease the normalized dual-activity time, although the factor widely depends on the filter one chooses (see Fig. B8 for more details). We note that different prescriptions for BH accretion (e.g. taking into account the angular momentum of the accreting gas; Debuhr et al. 2010 ; Hopkins Quataert 2011 ; Anglés-Alcázar, Özel Davé 2013 ; Anglés-Alcázar et al. 2015 , 2017 ) or AGN feedback (e.g. Newton Kay 2013 ; Wurster Thacker 2013 ) may also affect the results. 4.4 Comparison with other work In this section, we compare our dual-activity time-scales to a few select studies, both observational and theoretical, to validate our results. 4.4.1 Comparison with observations Most surveys provide the total number of AGN pairs out of the total number of AGN (e.g. Shen et al. 2011 ). These results are difficult to compare to ours since, by construction, we do not simulate isolated galaxies that could be harbouring a single AGN. Koss et al. ( 2012 ) study the fraction of dual AGN in the all-sky Swift Burst Alert Telescope (BAT) survey. For each of the 167 BAT AGN from the survey, they search for apparent companions, finding that 77 of them have at least a companion 8 within 1–100 kpc. Of these 77 companions, 15 have an AGN, making the fraction of dual AGN systems out of interacting systems (in which at least one system has an AGN) equal to 19.5 per cent (15/77). If we divide their sample in major and minor pairs (defined as pairs with mass ratios > and ≤0.25, respectively), the fraction changes to 37.1 per cent (13/35) for the major pairs and to 4.8 per cent (2/42) for the minor pairs. Since in this case only interacting systems are considered, we can compare the numbers from Koss et al. ( 2012 ) to our results. To do that, we simply consider, for each of our mergers, the dual-activity time when the BHs are at r > 1 kpc (our galaxies, by construction, are never above r ∼ 90 kpc) and divide it by the activity time at the same projected separations. 9 In Fig. 7 , we show the normalized dual-activity time above 1 kpc, 10 kpc and 150 km s −1 , for all our simulated mergers. We also show the results of Koss et al. ( 2012 ) for their full sample of BAT AGN and their subsamples of major and minor pairs. If we neglect the 1:2 merger with high BH feedback efficiency (Run C3), which obviously has a much lower


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This work was supported by the Research Projects of the Polish Ministry of Sciences and Higher Education: 04/45/DSPB/0148 (JR and JD) and 06/65/DSPB/0516 (ME).


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We thank Marc Metchnik and Philip Pinto for their contributions to the A bacus simulation code. D.J.E., L.H.G., and D.W.F. have been supported by grant AST-1313285 from the National Science Foundation, and D.J.E. is additionally supported as a Simons Foundation Investigator.


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

70–600 keV). Therefore, the observation covers a wide energy band simultaneously and with high sensitivity making it possible to do detailed spectroscopic studies of the source. The data were reduced and processed with the standard procedures and using the latest caldb version ’20140825’. The XIS was operated in the standard data mode with the window mode that provided a timing resolution of 2 s. The nominal pointing was set to XIS. The XIS data was examined for the presence of pile-up. A central region corresponding to a pile-up fraction >4 per cent was discarded from all further analysis. 3 TIMING ANALYSIS: PULSATION SEARCH AND LIGHT CURVES We applied barycentric corrections to the event files for the timing analysis, using the FTOOLS task ‘aebarycen’. Orbital correction was not required as the observation duration is much less than the decoherence time-scale due to the orbital motion of the system (Chakrabarty et al. 1997 , decoherence time-scale is ∼54 d, equation A9). Light curves were extracted with a time resolution of 2 s and 1 s, respectively, from the XISs (0.2–12 keV) and HXD/PIN (10–70 keV). To search for pulsations in the data, pulse folding and χ 2 maximization technique was applied. Pulsations were found in the light curve at 835.29 ± 0.29 s, which was subsequently used to bin the light curves and also to create corresponding pulse profiles. Fig. 1 shows the light curves in the low (XIS: 0.2–12 keV) and high (PIN: 10–70 keV) energy bands along with the hardness ratio (HR) curve, all binned at the spin period of the source. The ‘HR’ is defined as the ratio of the count rate in the XIS band (with the three XISs added together) to the PIN band. It is clear from the figure that the count rates in both the energy bands vary by a factor up to 5 during the observation. Moreover the HR shows a variation by a similar factor with the PIN count rate (see right of Fig. 1 ) that indicates spectral variability in the source. In order to investigate the true energy dependence of the pulse profiles in this source and to construct an average model of the representative X-ray spectrum of the source, it is important to choose a time interval free from intensity or spectral variations. Therefore, in order to construct the pulse profiles and the energy spectrum of X Persei, we choose a stretch of the observation where 0.02 HR 0.03 and 0.8 PINc/s 1.4, as not much spectral variation is expected in this narrow stretch. Figure 1. Left figure shows the light curves in low- and high-energy bands with the lowest panel showing the HR. The light curves are binned at the spin period of 835.29 s. Right figure shows the HR plotted against the PIN count rate to highlight the variation in HR with the count rate in the high-energy band. 3.1 Energy dependence of the pulse profiles To investigate any possible energy dependence of the pulse profiles, we created the energy resolved pulse profiles by folding the data in different energy bands at the obtained pulse


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

centred at , . Small-diameter (7′) WISE H II regions from Anderson et al. (2014) are marked by white squares and two extended WISE H II regions (diameter > 20′) are indicated by blue dashed circles. Three very extended radio sources in both λ 21 cm and λ 6 cm images are SNR G126.2+1.6, SNR G127.1+0.5 and H II region G127.9+1.7 (DU 65, Dubout-Crillon 1976 , red rectangle), which is missing in the WISE H II region catalogue. Lower left panel : its H α emission is shown overlaid by the λ 6 cm total-intensity emission. Lower middle and lower right panels : TT-plot results for DU 65 by using the Urumqi λ 6 cm against the Effelsberg λ 21 cm data and the Effelsberg λ 11 cm data. 4.1.1. New H II regions In addition to the missing known extended H II regions in the WISE H II region catalogue, a few new Galactic H II regions and H II region candidates are also discovered in the Urumqi λ 6 cm survey. They are G68.4+0.2, G76.7−1.2, G94.4+2.7, G139.2−3.0, G139.9−2.0, G172.3+1.9 (H II region candidate), G186.7−4.0 (H II region candidate), G210.8−2.6 and G212.9−3.7. We show them in Figs. 4 , 5 and 6 . For G68.4+0.2, G76.7−1.2 and G94.4+2.7 it is difficult to compare the λ 6 cm radio emission with the WISE 12 μ m infrared emission due to the large difference in angular resolution. We extracted the 1′ resolution CGPS λ 21 cm images and found that the associated infrared emission are not for the entire radio source but for partial shell- or ridge-like structures. The TT-plots confirmed their thermal origin. For the six remaining newly identified H II regions and H II region candidates, we found supporting evidence both in the TT-plot results and the associated H α emission, except for G172.3+1.9 and G186.7−4.0. For G172.3+1.9, we did not obtain a reliable TT-plot result and for G186.7−4.0, which is not fully covered by the Effelsberg λ 21 cm survey, the sensitivity of the Effelsberg λ 11 cm data is not enough for a clear determination. The H α image quality is not high in the south-east part of G172.3+1.9, but the data show a good correlation in the north-west part. Fig. 4. New H II regions G68.4+0.2, G76.7−1.2 and G94.4+2.7 identified in the Sino-German λ 6 cm survey. We show λ 6 cm images and WISE 12 μ m emission overlaid by CGPS λ 21 cm total intensity in the upper and middle panels . We present their TT-plot results in the lower panels . Fig. 5. Newly identified H II regions G139.2−3.0, G139.9−2.0 and G172.3+1.9. H α emission overlaid by Urumqi λ 6 cm total intensity contours are shown in the upper panel . The image quality of the south-east part of G172.3+1.9 is not good; however, correlation can be clearly recognised in the north-west part. TT-plot results are displayed in the lower panels . Fig. 6. Upper panels from left to right : λ 6 cm image, H α emission overlaid by Effelsberg λ 11 cm total intensity and the TT-plot result for the new H II region G212.9−3.7. Lower panels from left to middle : H α emission overlaid by Urumqi λ 6 cm total intensity


O O O O O O O O O O O O O O O O O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Grant O B-Grant O O O B-Organization I-Organization B-Grant B-Person O O O O B-Organization I-Organization O O O B-Grant O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Grant O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O B-Location I-Location I-Location I-Location O O B-Organization I-Organization I-Organization O O B-Location B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O O B-Location B-Telescope I-Telescope I-Telescope B-Telescope O O O B-Location I-Location I-Location I-Location O O B-Organization I-Organization I-Organization O O B-Organization B-Organization I-Organization I-Organization I-Organization O O B-Location B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory I-Observatory O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O B-Location I-Location I-Location I-Location O O B-Location B-Location B-Location B-Location O B-Location O O B-Location B-Observatory I-Observatory I-Observatory I-Observatory I-Observatory I-Observatory O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O B-Instrument O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O B-Telescope I-Telescope I-Telescope O O O B-Organization O O B-Organization O O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O O O O O O O B-Observatory I-Observatory I-Observatory I-Observatory B-Organization B-Location O O O O O O O B-Database O O O O B-Organization B-Location B-Location O O O B-Database O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O B-Survey I-Survey I-Survey I-Survey I-Survey O O O O O O O B-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Organization I-Organization I-Organization

We thank the referee for the careful reading of the manuscript. This research is partially supported by the Spanish Ministerio de Ciencia e Innovación (MICINN) under grants AYA2012-39364-C02-02 and AYA2015-68012-C2-2-P, and by the Generalitat Valenciana (ACOMP/2014/129). AM acknowledges support from the Generalitat Valenciana through the grant BEST/2015/242 and from the Ministerio de Educación, Cultura y Deporte through the grant PRX15/00030. This work is based in part on the observations obtained with the Jacobus Kapteyn Telescope operated on the island of La Palma by the Isaac Newton Group, in the Spanish Observatorio Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This work is based in part on the observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This research is also based on the observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Some of the data presented here have been taken using ALFOSC, which is owned by the Instituto de Astrofisica de Andalucia (IAA) and operated at the Nordic Optical Telescope under agreement between IAA and the NBIfAFG of the Astronomical Observatory of Copenhagen. This work is based in part on the observations made at Observatoire de Haute Provence (CNRS), France. This research has made use of the SIMBAD data base, operated at CDS, Strasbourg (France) and of the WEBDA data base, operated at the Institute for Astronomy of the University of Vienna. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.


O O B-Person I-Person I-Person B-Person I-Person O B-Person I-Person O O O O O O O O O O O B-Observatory O B-Identifier B-Observatory O O O O B-Observatory O O O O B-Organization B-Location O B-Organization B-Location O O B-Organization B-Location B-Organization O B-Organization B-Location O B-Organization B-Location I-Location I-Location O O O O B-Location I-Location I-Location O B-Observatory I-Observatory I-Observatory O O O B-Observatory B-Organization O B-Observatory O B-Observatory I-Observatory I-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization O O O O B-Organization O B-Location O O O B-Grant O O B-Organization I-Organization I-Organization B-Organization O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O B-Organization I-Organization I-Organization I-Organization O O O B-Location I-Location B-Grant I-Grant I-Grant I-Grant O O O O B-Grant

We thank T. J. Haworth, J. Lesur and C. McNally for the useful discussions. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00798.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We acknowledge funding from ANR of France under contract number ANR-16-CE31-0013, from the Australian Research Council (ARC) under the Future Fellowship number FT170100040, from the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation, and from the European Union Seventh Framework Programme FP7-2011 under grant agreement No. 284405.


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

so its disruption radius is closer to the BH (see Equation ( 5 )). The longer t acc in XT2 is likely due to a lower disk thickness ratio H / R because the latter carries the most sensitive parameter dependence in Equation ( 6 ). A lower H / R in turn might be caused by a lower Eddington-normalized accretion rate. This could suggest that XT2 was a tidal stripping event because the total bound mass Δ m can be much smaller than that of a full disruption event. The isotropic-equivalent peak luminosity of CDF-S XT2, ≈3 × 10 45 erg s −1 , is extremely high, which raises the possibility of a relativistically beamed emission. Three relativistic jetted TDE candidates have been discovered so far: Sw J1644+57 (Bloom et al. 2011 ; Burrows et al. 2011 ), Sw J2058+0516 (Cenko et al. 2012 ; Pasham et al. 2015 ), and Sw J1112−8238 (Brown et al. 2015 ). The 2D model of Dai et al. ( 2018 ) suggests that the X-ray emission of TDEs may be caused by jets through the Blandfold–Znajek process. Similarly, the WD–IMBH tidal disruption/stripping discussed here may produce a jet as well. Xue et al. ( 2019 ) estimated the CDF-S XT2–like event rate density to be Gpc −3 yr −1 . For CDF-S XT1, Bauer et al. ( 2017 ) estimated a large range of the event rate, ∼10 3 Gpc −3 yr −1 for z = 0.5 to ∼1 Gpc −3 yr −1 for z = 3. Theoretically, the rates of WD disruptions are very uncertain. For IMBHs in globular clusters (GCs), Baumgardt et al. ( 2004 ) estimated a total stellar TDE rate of ∼10 −7 yr −1 via N -body simulations of GCs with an initial central BH mass of 10 3 M ⊙ . Among the disrupted stars, ∼15% are WDs, giving a rate of ∼1.5 × 10 −8 yr −1 . Adopting a number density of 34 Mpc −3 of GCs, this gives a volumetric rate of R IMBH−WD ∼ 500 yr −1 Gpc −3 (Haas et al. 2012 ; Shcherbakov et al. 2013 ). Recently, Fragione et al. ( 2018 ) semianalytically calculated the evolution of a population of GCs in a galaxy and found a rate of WD TDE ∼10 −5 yr −1 per galaxy; combining this with GC population’s dependence on redshift and galaxy types, their results show a present-day volumetric rate of R IMBH−WD ∼ 10 yr −1 Gpc −3 . For IMBHs in dwarf galaxies, MacLeod et al. ( 2014 ) calculated a rate of ∼10 −6 yr −1 per IMBH for WD disruptions via the loss-cone dynamics for the BH mass range of 10 4 –10 5 M ⊙ . This rate is ∼30 times lower than that of main-sequence stellar TDEs by supermassive black holes (e.g., Stone Metzger 2016 ). Assuming a number density of dwarf galaxies ∼10 7 Gpc −3 (Shcherbakov et al. 2013 ; MacLeod et al. 2014 ) and an occupation fraction f IMBH of IMBHs in dwarf galaxies, then the volumetric rate is yr −1 Gpc −3 . Compared with WD–IMBH TDEs, a similar case that happens more commonly is the tidal disruption/stripping of MS stars by IMBHs. Indeed, as was shown in Fragione et al. ( 2018 ), the event rate of MS–IMBH TDEs is about 30 times higher than that of WD–IMBH TDEs in most galaxies. However, the timescale (∼years) of MS–IMBH TDEs is much longer than that of


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This research was supported by NASA grants NNX13AF99G, NNX10AE46G, and NNX12AH44G, and by NSF grant AGS-0955700. X.G. was also supported by the 973 program 2012CB825602, NNSFC grant 41231067, and in part by the Specialized Research Fund for State Key Laboratories of China. We are grateful for the use of the Neutron Monitor Database at http://www.nmdb.eu , the Wilcox Solar Observatory, and the OMNI data obtained through the GSFC/SPDF OMNIWeb interface at http://omniweb.gsfc.nasa.gov .


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

keV. As these features are not commonly observed in Galactic XRBs, they imply a new spectral regime, which is usually referred to as the ‘ultraluminous state' (Roberts 2007 ; Gladstone et al. 2009 ), and may be associated with super-Eddington accretion. A few luminous ULXs have very recently been observed with NuSTAR , resulting in spectra which demonstrate that the high-energy curvature persists above 10 keV (Bachetti et al. 2013 ; Walton et al. 2013a ; Rana et al. 2015 ), confirming this result. The spectral curvature may arise either from Comptonization in a cold, optically thick corona lying above an accretion disc (e.g. Stobbart et al. 2006 ; Gladstone et al. 2009 ); from a modified accretion disc, i.e. a slim disc (Mizuno et al. 2007 ); or the hot inner regions of the disc, which are highly distorted by advection, turbulence, self-heating and spin (Beloborodov 1998 ; Suleimanov et al. 2002 ; Kawaguchi 2003 ). The soft component is thought to originate from the photosphere of a radiatively driven wind that is ejected from the spherization radius of the accretion disc down to its innermost regions, which is expected to occur at super-Eddington accretion rates (e.g. Shakura Sunyaev 1973 ; Ohsuga Mineshige 2007 ; Poutanen et al. 2007 ; Ohsuga et al. 2009 ). Partially inhomogeneous winds in ULXs may explain the puzzling short-term variability reported in some ULXs (Heil, Vaughan Roberts 2009 ), through a combination of inclination angle and accretion rate (e.g. Middleton, Sutton Roberts 2011 ; Takeuchi, Ohsuga Mineshige 2013 ; Middleton et al. 2015 ). These same factors also determine the observed energy spectrum (e.g. Sutton, Roberts Middleton 2013b ; Pintore et al. 2014 ). On the other hand, HLX-1 in ESO 243-49 (Farrell et al. 2009 ), which has a peak luminosity of ∼10 42 erg s −1 , is observed with a strikingly different set of properties than standard ULXs. It displays a spectral evolutionary pattern similar to that of the Galactic BH X-ray transients (e.g. McClintock Remillard 2006 ; Fender Belloni 2012 ), undergoing periodic outbursts (Godet et al. 2014 ; Webb et al. 2014 ) in which the source switches from a high-luminosity soft state to a fainter hard state (Servillat et al. 2011 ). Multiwavelength emission has been detected from HLX-1 as it undergoes this outburst cycle. In particular, radio emission has been detected, which is associated with discrete jet ejection events (Webb et al. 2012 ), similar to those that have been observed during hard -to- soft state transitions state in Galactic BHs. By assuming that HLX-1 was in a similar state, the authors of that study were able to refine the BH mass estimates to ∼10 3 –10 4 M ⊙ . Other good IMBH candidates include: M82 X-1 (Feng, Rao Kaaret 2010 ), for which a BH mass of ∼400 M ⊙ has been suggested based on the tentative identification of twin-peak quasi periodic oscillations (Pasham, Strohmayer Mushotzky 2014 ); and an X-ray source in the galaxy NGC 2276 (Mezcua et al. 2015 ).


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We thank the anonymous referee for a detailed report that was very helpful in improving the manuscript. A.B. is grateful to the Austrian Science Fund (FWF) for funding (projects P19300-N16 and P23946-N16). The authors thank B. Bösch (Innsbruck) for providing a Python code of the Kelson ( 2003 ) method for improved sky subtraction in the FORS2/MXU spectra. This publication is supported by the Austrian Science Fund (FWF).


B-Wavelength I-Wavelength I-Wavelength O O B-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 O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

24 μ m source in MIPSGAL data (Carey et al. 2009 ). This emission feature probably represents a large, red AGB star. Masers that are associated with neither SFRs nor AGB stars have an unknown association. In Table 2 we present data for all 325 H 2 O maser sites detected during the RAMPS pilot survey. Table 2 gives the maser positions, the errors on the positions, the velocity and intensity of the brightest maser line, and the associated environment of each maser. We found that out of 325 detected masers, 185 (57% ± 4%) are associated with an SFR, 92 (28% ± 5%) are associated with an AGB star, and 48 (15% ± 5%) have an unknown association. Figures 21 and 22 show histograms of the maser intensities and velocities for the full sample of masers and histograms of the masers separated by association, respectively. Figure 21 shows that the slopes of the maser intensity distributions look roughly like a power law past the peak of each distribution. The sharp cutoff at ∼1 K in each distribution represents our completeness limit. This number is expected, since the typical 1 σ rms in our H 2 O spectra is ∼0.4 K and we require that each detected maser has an intensity of at least three times the noise. While the intensity distributions of the various maser groups look similar, the distribution associated with SFRs shows several more masers with intensities in excess of 100 K. Figure 22 shows that we have detected masers predominantly at positive velocities, particularly the masers associated with SFRs. On the other hand, both masers associated with AGB stars and those with an unknown association have a much broader spread in their velocity distributions. Figure 21. From top to bottom, histograms of maser intensities for the full sample of masers, the masers associated with SFRs, the masers associated with AGB stars, and the masers with an unknown association, respectively. The histograms qualitatively look similar, with a roughly power-law slope at large intensities and a sharp cutoff at ∼1 K. Figure 22. From top to bottom, histograms of maser velocities for the full sample of masers, the masers associated with SFRs, the masers associated with AGB stars, and the masers with an unknown association, respectively. The distribution of masers associated with SFRs is more concentrated at positive velocities than all other distributions. This difference may indicate that some of the masers associated with AGB stars are in the Galactic halo. Table 2 RAMPS H 2 O Masers Note. Starting from the leftmost column, we present the Galactic longitude, the Galactic latitude, the error in the Galactic longitude, the error in the Galactic latitude, the velocity of the brightest channel in the spectrum, the intensity of the brightest channel in the spectrum, and the associated environment. We classify H 2 O masers as being associated with a star-forming region (SFR), an asymptotic giant branch (AGB) star, or an unknown environment. (This table is available in its


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

parameters as reported in Table C1 . The N comp = 2 results are fully consistent with those reported by Patruno et al. ( 2017 ) within their confidence intervals. Table C1. GMM results for the f spin data set for NSs in LMXBs from Patruno et al. ( 2017 ), assuming negligible measurement errors. Looking at the various statistical criteria as a function of number of components N comp = 1, 2, 3 (also illustrated in Fig. C1 ), these are also fully consistent with the findings of Patruno et al. ( 2017 ): again the likelihood ratio alone is already in favour of two components (), but only additional criteria can yield a robust decision, and on its own, the additional gain of could tempt us to use an even more complex model. Fortunately, in this case AICc and BIC agree in clearly preferring 2 components over 1 (with Δ of 7–11), and also prefer 2 over 3 by Δ of almost 5. A leave-one-out CV test (see Fig. C2 ) also shows the two-component fit to be almost as stable as a single component in this case. A MixtureInf EM-test reports a nominal p -value of 0.008 which also seems to reject a single-component hypothesis much more clearly than in the DNS case. As in Section 2.6 the exact value needs to be interpreted with caution due to the low N data , but this being only a comparison example, no further simulation tests have been conducted in this case. Figure C1. Model selection criteria for GMMs with N comp = 1, 2, 3 applied to the Patruno + data set of f spin for NSs in LMXBs. The first three panels are for sklearnGMM fits. The last panel gives the average Bayesian evidence from 10 cpnest runs at each N comp . AICc, BIC, and are plotted as differences to the N comp = 1 value. Figure C2. Illustration of leave-one-out CV on the Patruno + f spin data set. Each dotted line corresponds to a fit after leaving out one data point. In this case, the fitted distributions are relatively stable not just for N comp = 1, but also for the preferred fit with N comp = 2, and only for even more components they start becoming unstable. Figure C3. Posteriors from cpnest runs on the LMXB data set for a single Gaussian (left-hand panel) and for a two-component GMM (remaining panels). Note that σ 1 appears twice on the x -axis of the last two panels. Figure C4. Median GMM distribution functions p ( f spin ) inferred with cpnest for one (blue solid line) or two components (orange dashed line), superimposed on a data histogram, and with 90 per cent quantiles background ‘haze’. cpnest results for this data set (with uniform priors in C k , log-uniform in σ k within Hz and Gaussian μ k priors of width 70 Hz) are very consistent with the other GMM fits, except for some differences in the more degenerate N comp = 3 case. The evidence ratio is in favour by ≥16 of 2 components over 1, with 3 components preferred by a marginal factor of only 3 and much less robust results. Overall, the LMXB data set seems to be a good example for information criteria, EM-test and Bayesian evidence


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

by Weppner et al. (2015) , which presents a new cold EOS derived from the pressure derivative of the bulk modulus and the polytrope method. Where a polytrope is a structural assumption between pressure and density and it is also a solution to the Lane-Emden equation. The polytrope is defined as: where p E is the polytrope exponent. The authors show that the polytrope exponent is equivalent to the pressure derivative of the bulk modulus, 1 . This allows for an implementation of real material variables within the polytrope method. This is done by augmenting the classical Lane-Emden equation to handle zero pressure densities and a variable polytrope exponent. To correctly model the polytrope exponent, Weppner et al. (2015) also present an EOS (the Varpoly EOS) based on a variable polytrope exponent: The parameters A 0 , A 1 , and A 2 are connected to the extremes of the polytrope exponent by ambient pressure data and Thomas-Fermi Dirac (TFD) theory ( Salpeter 1967 ). In spite of its simplicity, the EOS provides an improvement in the modeling of P, B, B ′ over the other popular finite strain EOS. At extreme pressures, solids are predicted to follow TFD high pressure theory. Within TFD theory the infinite pressure asymptote of the polytrope exponent goes toward . This is the value that a finite strain EOS should go to, however, the B-M EOS goes toward a higher asymptote of and the Vinet EOS, in contrast, goes toward a lower asymptote of . The sensitivity to the compression curve with respect to the polytrope exponent is clearly shown in Weppner et al. (2015) and this highlights its importance when dealing with planetary interiors. The Varpoly EOS acts as the cold part of our EOS and in the following sections, we add a temperature dependent term to it. In addition, we develop a new model for the Gruneisen parameter, which is shown to fit well with experimental data and adheres to infinite pressure values. To correctly model the vapor transition, we introduce an expanded state model which can be seen to give good fits to prior estimates of the vapor curve. The EOS is thenfit to experimental data and compared to both the Tillotson EOS and the ANEOS model. We also test its caliber to model planets by fitting it to the Preliminary Reference Earth Model ( Dziewonski Anderson 1981 ). In the second paper, we perform numerical simulations on the lunar-forming impact, to test its merit in planetary impacts. In Sects. 2 and 3 we go through the development of the EOS. In Sect. 4 we test the EOS by fitting it to experimental data of ɛ -Fe and comparing it to the Tillotson EOS, and we also take a look at how well the EOS compares to the multiphase ANEOS model, in which we focus on the behavior of silicate material (SiO 2 , Mg 2 SiO 4 ). In Sect. 5 we discuss the EOS and our different results and finally end with some conclusions. 2 Adding a temperature dependence We begin by separating the Helmholtz free energy into three separate parts ( Zel’dovich Raizer 1967


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

effect, we find that the inclination must be i 58°. Hence, both light curve modeling and analytic estimates require i ≲ 50°otherwise we would see a reflection effect. 3.1. Joint Analysis We now perform a joint analysis that considers our spectroscopic, astrometric, and photometric constraints simultaneously. Our approach is to construct composite binary models by adding two synthetic WD spectra, properly weighted by their respective radii using the astrometric parallax constraint. We then simultaneously fit the spectroscopic Balmer line profiles and the broadband photometric measurements using the Levenberg–Marquardt algorithm (Bédard et al. 2017 ; Kilic et al. 2020 ). We use de-reddened Galaxy Evolution Explorer far-ultraviolet (FUV) and near-ultraviolet (NUV; Martin et al. 2005 ), SDSS ugriz (Abolfathi et al. 2018 ), Pan-STARRS grizy (Tonry et al. 2012 ), and UKIDSS YJ photometry (Lawrence et al. 2007 ). All four atmospheric parameters (, , , and ) are allowed to vary for a given distance. Figure 3 plots the best-fit solution: a 19,000 K DA WD plus a 8000 K DC WD, both with log g = 7. We note that a DA + DA solution yields comparable parameters, but predicts a strong double-lined H α feature that is not observed. The errors in the joint analysis are dominated by the parallax uncertainty. We estimate 1 σ errors of 1000 K in and 0.15 dex in for the DA WD, and 1500 K in and 0.25 dex in for the DC WD. Figure 3. Joint fit to the spectrum (left panel) and photometry (right panel) for the best DA + DC model. The left panel shows the synthetic model (red) overplotted on the observed MMT spectrum (black). The right panel shows the synthetic fluxes (filled circles) and observed fluxes (error bars). The red and blue lines show the contribution of each WD to the total monochromatic model flux, displayed as the black dotted line. We interpolate the atmospheric parameters through the Althaus et al. ( 2013 ) He-core WD tracks to estimate the masses. Conservatively ignoring the co-variance between and , we infer M ⊙ and M ⊙ . Our mass estimates imply that the cool WD may be less massive than the hot WD. The same result is seen in ZTF J1539 + 5027 (Burdge et al. 2019a ). This would be surprising because one expects the cool WD to have evolved first, and thus be more massive than the hot WD. However the models contain a major uncertainty linked to common envelope ejection (Li et al. 2019 , 2020 ). The thickness of the WD hydrogen envelope, assumptions about elemental diffusion and rotational mixing, the presence or absence of hydrogen shell flashes, and other issues have significant effects on the temperature, radius, and cooling age of a low-mass WD (Althaus et al. 2013 ; Istrate et al. 2016 ). To draw a significant conclusion about the WD masses and their past evolution, we need to improve our constraints, i.e., with future GW measurements. The present constraints are also consistent with J2322 + 0509 being an equal-mass binary. Given the mass


O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Model B-Model I-Model I-Model B-Model B-Model I-Model B-Model B-Model I-Model B-Model B-Model I-Model I-Model B-Model B-Model I-Model B-Model O B-Model I-Model I-Model 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 B-ObservationalTechniques I-ObservationalTechniques O O O O O O B-ObservationalTechniques I-ObservationalTechniques B-ObservationalTechniques B-Model I-Model B-Model B-Model I-Model I-Model B-Model B-Model I-Model B-Model B-Model I-Model I-Model B-Model B-Model I-Model B-Model B-Model I-Model I-Model B-Model O B-Model I-Model 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 B-Model I-Model B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Model O B-Model O O O O O O O O O O B-Model I-Model B-Model 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 B-Model I-Model I-Model B-Model B-Model I-Model I-Model B-Model O B-Model I-Model I-Model B-Model O O O O O O O O O O O O B-Model O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

the lowest error but with the longest training time, whereas the ERT can be seen to reach a very similar error much faster. 4 RESULTS The results given in Table 2 show how the seven ML algorithms performed at producing photometric redshift estimations. Furthermore, Fig. 6 displays the true spectroscopic redshifts plotted against the photometric redshift estimates for each ML algorithm. We also plotted the distributions of the redshift estimations for each of the algorithms as well as the true spectroscopic redshift in a violin plot in Fig. 7 to quickly see which algorithms were able to capture the correct distribution. Figure 6. Graphs of photometric redshift estimates against the true spectroscopic redshift where the lighter shaded contours display the more densely populated regions. From top left to bottom right – Linear Regression (LR), k -Nearest Neighbours (kNN), Multilayer Perceptron (MLP), Decision Tree (DT), Boosted Decision Tree (BDT), Random Forest (RF), and Extremely Randomized Trees (ERT). Figure 7. Violin plots showing the kernel density estimation of the underlying distributions of photometric redshift estimates of each algorithm along with the true spectroscopic redshift. From left to right – True spectroscopic redshift (zspec), Linear Regression (LR), k -Nearest Neighbours (kNN), Decision Tree (DT), Boosted Decision Tree (BDT), Random Forest (RF), Extremely Randomized Trees (ERT), and Multilayer Perceptron (MLP). From these results, we saw that all algorithms were able to successfully provide photometric redshift estimations. Using the violin plots from Fig. 7 , we could see that the rough distribution was recovered by each algorithm, with the Multilayer Perceptron (MLP) producing a slightly more similar shape to the true redshifts. However, from simply looking at the outputs shown in Figs 6 and 7 , it would be very difficult to determine which algorithm would be best to use. While the Decision Tree (DT) might be excluded due to the estimates being put into bands at set redshifts, its errors were still found to be quite low and it outperformed both the Linear Regression (LR) and MLP algorithms. Looking at the metrics in Table 2 alone, the Random Forest (RF) performed best having the lowest errors with an MAE = 0.0385 and MSE = 0.0042; however, the other algorithms k -Nearest Neighbours (kNN), Boosted Decision Tree (BDT), and Extremely Randomized Trees (ERT) all performed incredibly similarly with MAE 0.042 and MSE 0.0046. Indeed, the BDT was almost identically performing to the RF with a slightly improved precision and outlier fraction, and with such close performances of all the other algorithms, it was impossible to sufficiently determine which algorithm would be the most useful. To be able to further differentiate between them and determine which would be the best algorithm to use, it was therefore necessary to use the benchmarking results. The results of the benchmarking performed for each algorithm are plotted in Fig. 1


O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O B-Instrument O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion 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 B-Wavelength O O O O O B-Wavelength O O B-Wavelength O O O O O B-Telescope I-Telescope I-Telescope I-Telescope O B-Telescope O O O O O O B-Survey I-Survey I-Survey I-Survey I-Survey B-Survey B-Citation I-Citation I-Citation I-Citation O O O B-Wavelength B-Wavelength O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O B-Observatory O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Telescope O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O B-Wavelength I-Wavelength B-Wavelength 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

loud (GL) sample]. The RoboPol polarimeter allows for the simultaneous measurement of the Stokes q and u vectors without any moving parts. Each source on the sky is projected four times (four spots in a cross-like pattern) on the CCD camera each having its polarization orientation rotated by 67.5°. This allows for measurement of the polarization degree and EVPA of a source in a single exposure. For enhanced sensitivity and to block unwanted light, a mask is placed in the centre of the CCD, the location where the source of interest is typically positioned. A consequence of the RoboPol design is that sources outside the mask could have one (or more) of their spots overlap with a neighbouring source preventing us from confidently measuring either source’s polarization parameters (see fig. 6 in Panopoulou et al. 2015 ), especially in crowded fields. For every blazar observed with the RoboPol polarimeter, we use the automated reduction pipeline to extract the polarization degree for the sources in the 13 arcmin ×13 arcmin field of view of the RoboPol instrument excluding sources affected by such overlap (for details on the RoboPol pipeline and reduction process, see King et al. 2014 ; Panopoulou et al. 2015 ). This process yielded polarization measurements for 5837 sources. Additionally, we include 6379 sources from the Heiles ( 2000 ) catalogue of polarized stars residing above and below the galactic plane (| b | > 10° and | b | −10°) resulting in a total of 12 216 sources [hereafter γ-ray quiet (GQ) sample]. Fig. 1 shows the median polarization degree of the GL and GQ samples. Clearly, the GL sources show systematically higher polarization due to the synchrotron nature of their emission as opposed to the, most likely, dust-induced polarization of the emission in GQ sources. Figure 1. Optical polarization degree distribution for the γ-ray loud and γ-ray quiet samples. We use the Wide-field Infrared Survey Explorer ( WISE ) all-sky survey 4 and the Two Micron All Sky Survey (2MASS, Skrutskie et al. 2006 to obtain the 3.4, 4.6, and 12 μm, and H -band magnitudes for all the sources in the GL and GQ samples. The WISE colours have been successfully used in the past to find blazars among the unidentified Fermi sources (e.g. Massaro et al. 2011 ). Fig. 2 shows the WISE colour–colour diagram along with the so-called blazar strip found in Massaro et al. ( 2011 , 2012 ). While there is some minor overlap, the GL sample occupies a very distinct area in the infrared colour–colour diagram. We also estimate the H -band–4.6 μm colour. In Majewski, Zasowski Nidever ( 2011 ) stars were shown to have a narrow distribution. The GQ sample (which is mostly comprised of stars) shows a similar narrow distribution, whereas GL sources show a wider distribution shifted to higher values (Fig. 3 ). In addition, stars and nearby galactic sources could have measurable proper motion while we do not expect any significant proper motion in distant extragalactic sources (i.e.


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O 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 O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 overall null results are just due to cancellation globally. Figure 4. Additive bias c 1 for different boundary sizes ( l x , l y ). All c 1 values are units of 10 −3 . The variances are about 4 × 10 −5 . The parameter 2 cs defined in Equation ( 2 ) has a best-fit value of 408 ± 12 in the current simulation. To further check the validity of the relation in the equation, one can quantitatively check the change of the bias c 1 with the ratio of the axis sizes l x / l y for a fixed image area l x l y , or the other way around. The results are shown in Figure 5 , which are quite consistent with Equation ( 2 ). Figure 5. Change of the additive bias c 1 as a function of l x . The upper panel is under the condition of fixed axis ratio ( l x / l y ), and the lower one is for fixed total area (). The solid curves are fittings from Equation ( 2 ) using the data shown in Figure 4 . The variances of the data points are about 4 × 10 −5 . Another prediction of the model is that the boundary effect should be more significant for galaxies of larger average sizes. To observe this phenomenon, we generate another set of simulations with RW galaxies that are larger by 50% than those in the previous set of simulations. The results are shown in Figure 6 , from which we can clearly see that c 1 is larger than those reported in Figure 4 under the same condition of ( l x , l y ). Figure 6. Similar to Figure 4 . The radii of the RW galaxies used here are larger than those used in Figure 4 by 50%. All c 1 values are in units of 10 −3 . The variances are about 4 × 10 −5 . We also expect the boundary effect to be more significant for more elliptical galaxies. To observe this, we repeat the simulations twice with the Galsim galaxies. Instead of using the ellipticity probability distribution function, this time, we set the galaxy ellipticities at two specific values: e = 0.2 and e = 0.8. The results are shown in Figures 7 and 8 accordingly. One can see that the c 1 of the more elliptical cases are 10 times larger than their counterparts with smaller ellipticities. Figure 7. Similar to Figure 4 , but with Galsim galaxies. The ellipticity of these galaxies are set at 0.2. All c 1 values are in units of 10 −3 . The variances are about 1 × 10 −5 . Figure 8. Same as Figure 7 , but with more elliptical Galsim galaxies. The ellipticity of the galaxies is set to 0.8. All c 1 values are in units of 10 −3 . The variances are about 1 × 10 −5 . Overall, we can draw the conclusion that the additive bias due to the boundary is also influenced by the galaxy morphologies, such as their radii, ellipticities, etc. The results in this section only represent the bias levels for several special cases. 2.3. The Solution and Its Application to Real Data As we have shown, source selection is a statistically anisotropic process near the image boundaries. The resulting shear bias cannot be treated as a universal additive bias, as it is significant only in the neighborhood of the boundaries. To


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J.M. acknowledges support by Polish National Science Centre grants No. 2011/03/B/ST9/03281 and 2013/10/M/ST9/00729.


Acknowledgements We thank Yusuke Zemba for his great contribution to the laboratory experiments.


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Acknowledgments We thank the anonymous referee for the useful feedback. The authors would like to acknowledge Matthew J. Payne for help in obtaining this data set. JAW and ACB acknowledge support from an NSERC Discovery Grant, the Canadian Foundation for Innovation, The University of British Columbia and the European Research Council (agreement number 320620). EBF acknowledges support from the Center for Exoplanets and Habitable Worlds. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science and the Pennsylvania Space Grant Consortium. This paper makes use of the following ALMA data: ADS/JAO.ALMA[2013.1.00486.S]. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.


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We thank Fan Liu for discussions about future tests of these trends. All authors acknowledge the grant ‘The New Milky Way’ from the Knut and Alice Wallenberg Foundation. PJM acknowledges support from the Swedish National Space Board .


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X. Dumusque is extremely grateful to S. Aigrain, G. Anglada-Escudé, G. Boué, A. Correia, R. Díaz, D. A. Fischer, E. B. Ford, P. C. Gregory, A. Hatzes, J. S. Jenkins, A. Santerne, S. H. Saar, G. Scandariato, and M. Tuomi for interesting discussions about the design of the data set of the RV fitting challenge and the best way to organize it. X. Dumusque also acknowledges the Society in Science The Branco Weiss Fellowship for its financial support.


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

velocities are conveniently expressed in terms of spherical Bessel functions, j l , and spherical Harmonics, Y lm . To do that, we imagine a very large spherical volume or radius R max which entirely encompasses the data and is also sufficiently large that the effects boundary conditions are small. Expressing the density contrast as where the wavenumbers k n are fixed by the boundary conditions at R max (Fisher et al. 1995 ). We advocate the boundary conditions which yield gravity potential decaying as r − l for r > R max . Since j l Y lm are eigenfunctions of the Laplacian, these boundary conditions are equivalent to finding k n which satisfy . The numbers C ln are fixed entirely by the boundary conditions and are given in table A1 of Fisher et al. ( 1995 ). The density coefficients are Using the linear velocity–density relation one obtains (Regos Szalay 1989 ; Scharf Lahav 1993 ; Fisher et al. 1995 ), where . For velocities measured with respect to the Local Group frame the l = 1 term should be modified by subtracting 1/3 from . Using this representation for u ( r ) yields where P l is Legendre polynomial of order l and we have used . From , the cross-correlation can easily be computed . Using the expression ( 7 ) which is appropriate for definition ( 2 ) we get 2 7 DISCUSSION An aim of this contribution is to show that different approaches to the determination of are tightly related. All methods yield something which mimics the behaviour of the theoretical bulk flow. The non-trivial challenge is to contrast any estimate with predictions of cosmological model. Systematic errors related to observables of galaxy properties, contaminate various methods in different ways. More distant galaxies are more prone to these systematics, therefore, consistency checks must be performed by applying the methods subsamples of the data. Furthermore, spatial coverage of the data depends on galaxy-type and unavoidable cuts on the observables. Thus, methods should be tested on mock galaxy catalogues that mimic the observations in as much as possible, including the observed large-scale structure. The constrained simulations (Sorce et al. 2013 ) designed to match the low-redshift large-scale structure, can potentially be highly beneficial if combined with galaxy formation models. We have also emphasized the differences between the methods. Hoffman et al. ( 2015 ) provide an estimate for the bulk flow as defined in equation ( 1 ), while Feldman et al. ( 2010 ) aim at given by equation ( 2 ). These two definitions of do not coincide (Nusser 2014 ) and should not produce identical results. None the less, both methods can easily be adapted to any of the two definitions. In Section 3 , we derive a new estimator which naturally arises from the likelihood . This is a generalization of the standard ML of Kaiser ( 1988 ), but incorporates velocity modes beyond the (constant) bulk flow of the entire data catalogue. Essentially, this estimator is equivalent to


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

of the total stress-energy tensor , namely which yields Here, γ αβ = g αβ + U α U β is the projection tensor, and α is the lapse function. Latin indices take values 1, 2, 3 and Greek ones 0, 1, 2, 3. Vectors and tensors with tilde are purely spatial. For an ideal fluid with density ρ, 3-velocity , and pressure p we have We will also assume an equation of state p = p (ρ) from which we deduce the ‘speed of sound’, Finally, we will also assume ideal MHD conditions, namely In order to investigate the Rayleigh–Taylor instability in an astrophysical context, we will now consider the special case of a Kerr space–time. 2.1 Kerr space–time In Boyer–Lindquist (BL) coordinates the Kerr metric reads where M is the mass of the black hole, a is the angular momentum per unit mass (0 ≤ a ≤ M ), and (Cowling 1941 ). Notice that we work in geometrical units in which c = G = 1. For our further study we need the components of the 4-velocity of fiducial observers, now identified as zero angular momentum observers (ZAMOs), namely where In the Kerr space–time ( 11 ) with 4-velocity U μ given by equation ( 13 ), the expansion θ vanishes, the shear has two non-zero components e.g. the σ 13 and σ 23 but σ αβ γ αβ = 0 (see Thorne Macdonald 1982 , paper I, equation 2.5). The acceleration a μ is given by γ ij is the spatial metric on the space-like hypersurface x 0 ≡ t = const., with normal vector n α : where β i = γ ij g 0 j . 2.2 The perturbed MHD equations We consider only small perturbations of physical quantities as and keep only linear terms of the perturbations. In this case and In the Cowling approximation of a fixed Kerr space–time: The first-order perturbed MHD equations now become 3 The static equatorial disc We will now restrict our analysis to the investigation of a static equatorial distribution of matter of thickness h ≲ r . By ‘static’ we mean that the disc fluid is initially at rest with respect to ZAMOs, i.e. that . Our disc configuration only vaguely mimics astrophysical accretion discs. We acknowledge that neglecting the Keplerian disc rotation is an important simplification that we apply only to make some progress with the complex general relativistic formalism. However, our results may be physically relevant in MADs in which rotation plays a secondary role (see Discussion section). The problem we have in mind is a distribution of matter consisting of two regions inside and outside some radius r 0 . We will thus only consider perturbations in the immediate neighbourhood of r 0 of the form in the equatorial plane θ = π/2, where m takes integer values 1, 2, 3...For simplicity, we will henceforth ignore the index ‘0’ from the zeroth-order terms. In this case, We will also assume dipolar symmetry in the magnetic field, namely B r (π − θ) = − B r (θ), B θ (π − θ) = B θ (θ), and B φ (π − θ) = − B φ (θ). On the equatorial plane in particular, and B r , θ ∼ h 1 B θ / r , B φ, θ ∼ h 2 B θ / r , with h 1 , h 2 ≈ const. In what follows, we will set for


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

lines) are shown. The colours represent different values of the orbital periastron argument (ω) in Fig. 1 . Lower panel: Relationship between the bright time ( t bright ) and the eccentricity ( e ) given by equation ( 5 ). In the lower panel of Fig. 5 , we analyse how ω affects the relationship between t bright and e (equation 5 ). As expected, t bright is minimum for the toy model (ω = 0°) and reaches a maximum value when the summer solstice occurs at quadrature (ω = 90°, −90°). Since an optimal orbital configuration was assumed in the toy model, equation ( 8 ) and the observed t bright will underestimate the true eccentricity of the orbit (see the dashed curve in the lower panel of Fig. 5 ). However, even in a more realistic situation, equation ( 8 ) is still valid as a formula for constraining e : Also, an unknown orbital inclination contributes to either increasing or reducing F dark and F max with respect to their corresponding values in the toy model (equations 9 and 11 , respectively). In the case of F dark , ω tends to increase its true value, while i may either reduce it or increase it, thereby compensating the effect of ω. On the other hand, the constraint given by equation ( 10 ) will not be substantially modified (at least within a first-order estimation of A planet ). By comparison with the toy model, both ω and i lower the measured value of F max . This implies that equation ( 13 ) would remain valid, although the minimum A ring will probably be underestimated. To find fiducial parameters of real systems, a more general and complex model for fitting stellar light curves needs to be developed. To that end, we should include second-order effects such as the shadow/shine of the planet’s surface and atmosphere, forward scattering, the opacity of the rings, the polarization and reflection of light by dusty/icy ring particles, etc. For this optical complexity , we propose (and intend) to create a scattering model centred on the ring (most models are centred on the star), where the properties of the scattered light are computed in the sky above the planetary rings after describing the astronomical position of the star, the planet, and the observer. However, such a model is the subject of an ongoing research that will be introduced in a forthcoming paper. 5 SUMMARY AND CONCLUSIONS The first-order semi-analytical model presented here demonstrates that rings may produce detectable and distinguishable photometric signatures on the light curve of non-transiting exoplanets. We have quantified the phase-dependent amount of light that a tilted ringed planet can reflect towards a distant observer, and the model is described through four main morphological features of the so-called bright curve that help us derive some analytical constraints to orbital and physical parameters of the system. This model can be useful to detect (ringed) planets with large ring surfaces lying in face-on orbits, whose survival time-scale is ∼1 Myr (Schlichting Chang