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


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


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


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


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


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.


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


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.


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.


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 .


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.


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


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


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


sum except for T = 500 K, which is obtained with the classical approximation. For more details, see the text. From Table 2 , it can be observed that the rotational contributions to the internal partition functions of the four isomers have similar values at the lowest temperatures, e.g. down to T = 10 K. For higher temperatures, above T = 225 K, the difference between the rotational partition function of fulminic acid and the values of the other three isomers (isocyanic acid, cyanic acid, and isofulminic acid), with more similar values, becomes larger. This could be explained because of the structure of fulminic acid is linear and of the other have a near-prolate asymmetric top structure. Nevertheless, the internal partition functions of isocyanic acid, cyanic acid, and isofulminic acid become completely different at higher temperatures despite the fact that these three isomers have a bent structure. This stems from the differences among the vibrational partition functions of the bent isomers. In Table 2 , the weight of the vibrational contribution to the internal partition function can also be assessed for the set of the four CHNO isomers at different temperatures. The differences between the internal partition function and the rotational partition function become noteworthy (more than 1 per cent) at temperatures higher than 225 K for HNCO, 150 K for HOCN, and T = 75 K for HCNO and HONC. Even when the vibrational partition function has been calculated with the harmonic approximation and, in some cases, using fundamentals, the values of this approximation, although underestimated, are in good agreement with the direct sum values (Carvajal et al. 2019 ). As supplementary material, the rotational, vibrational, and rovibrational partition function calculated in the present work from 1 to 500 K (in intervals of 1 K) is also provided for the following isomers: cyanic acid (HOCN); fulminic acid (HCNO); and isofulminic acid (HONC). For isocyanic acid (HNCO), same material can be found elsewhere (Carvajal et al. 2019 ). In Appendix A , we will give some details about the calculation of the partition function for the four CHNO isomers. 4.4 Chemical modelling with <sc>Nautilus</sc> In order to better understand the chemical pathways involved in the production and destruction of HNCO, we used the Nautilus code (Ruaud et al. 2016 ) to carry out a time-dependent chemical model. Nautilus allows to simulate and study grain- and gas-phase chemical processes for instance in hot and cold cores (Semenov et al. 2010 ; Reboussin et al. 2014 ; Ruaud et al. 2015 , 2016 ). The predictions of the model include the time evolution of the chemical abundances for a given set of physical and chemical parameters. For the solid-state chemistry, it considers mantle and surface as chemically active, following the formalism of Hasegawa Herbst ( 1993 ) and the experimental results of Ghesquière et al. ( 2015 ). In this sense, the code can perform a three-phase (gas plus grain


Acknowledgements D.H. acknowledges the Spanish Ministry of Economy and Competitiveness (MINECO) for the financial support under the FPI programme BES-2015-075200, R.A. under the Ramón y Cajal program RYC-2010-06519, and the three authors under the programme RETOS ESP2014-57495-C2-R. We are grateful for the hospitality of J. Cabrera and the group of H. Rauer at the DLR Institute of Planetology, where part of this work was performed.


reported 420.8 d period obtained from analysing all the MACHO data, i.e. including the orbital outburst points. Figure 1. MACHO V (a) and R (b) light curves; (c) MACHO colour variation, (d) OGLE IV light curves together with the mean MACHO V light curve (in grey) projected forward using the ephemeris of Alcock et al. ( 2001 ). The red and blue vertical lines refer to the times of the SALT and SAAO 1.9 m telescope observations, respectively. See Fig. 10 for more detail regarding the timing of the SALT spectra within the overall OGLE IV light curve. Figure 2. Lomb–Scagle periodogram of the MACHO V data showing peaks at 445.1 d, 352.7 d and 391.4 d for the first half (a), second half (b) and complete light curve (c), respectively. (d) Power spectrum of the OGLE IV data showing a peak at 423.74 d. To further explore the variation in the time-scale of the superorbital modulation, the complete MACHO data were divided into two parts. The power spectrum of the first and second half shows a peak at 446.0 ± 0.2 d and 355.4 ± 0.29 d, respectively (Fig. 2 ). The OGLE IV also shows a long-term variation but now of a much smaller amplitude, although still with a period consistent with that in the MACHO data. When all bright points related to the 16.64 d outburst were removed (as in the MACHO data) the power spectrum shows its highest peak at P sup = 423.74 ± 0.42 d and some additional peaks around 200 d (see Fig. 2 ). We have compared this behaviour to the original superorbital period by projecting the 420.8 d light curves and ephemeris by Alcock et al. ( 2001 ) into the OGLE IV data (Fig. 1 ). The optical maxima in the MACHO data are flat (i.e. no optical outbursts were seen) and last ∼200 d, meaning that we are seeing the completely disc-less B star. However in the OGLE IV phase, the optical maxima are not flat and the orbital outbursts now occur throughout the OGLE superorbital cycle, implying that during the OGLE IV optical maxima the equatorial disc was not completely dissipated. The MACHO and OGLE IV light curves folded on the superorbital period are shown in Fig. 4 . The colour variation ( V − R ) of A0538–66 shows that the source is bluer at optical maxima and becomes redder as the source fades. This behaviour is opposite to that seen in most other BeX where the source is redder when it is brighter. However, during outbursts on the orbital period associated with periastron passage of the NS, the optical brightening is accompanied by a slight increase in reddening (i.e. the source gets redder). The amplitude of the outbursts in the MACHO data clearly varies throughout the long-term 420.8 d cycle, being very strong (∼0.7 mag) at optical minima, decreasing in amplitude as the source brightens and completely disappearing at optical maxima. Similar behaviour is seen in the OGLE IV data; however, there are clearly orbital outbursts occurring even when the source is at maximum in the long-term cycle. In order to refine the orbital period, we removed the


The authors thank an anonymous referee for constructive comments, which were useful to improve the paper. D.C. thanks the MIT-India Program for travel support.


Yonsei-Yale (Spada et al. 2013 ). With the isochrones chosen, it creates synthetic populations that are then contrasted with observations. To populate the synthetic models, Talos uses a Kroupa IMF (Kroupa 2001 ) and takes into account various properties of the input data to make the models directly comparable to the observations. These properties include distance, reddening, binary fraction, photometric errors, and completeness fractions. Then, the combination of synthetic populations that best matches the observed decontaminated CMD and MDF is selected as the SFH of the object. The comparison between synthetic and observed CMDs is done using Hess diagrams, in which each point represents the number of stars at a specific color-magnitude bin. Each time a combination of synthetic populations is created, the routine calculates the differences between the star counts in the model and the observations at each color-magnitude bin, along with the difference at each metallicity bin from the MDF. With these differences, Talos calculates a Poissonian chi-squared, whose minimization gives the resulting SFH (i.e., the model that best reproduces simultaneously the observed CMD and MDF). 3.2. General Setup To run Talos and calculate the SFH, we have to determine a set of parameters associated with our data. These parameters are used to populate the synthetic models that are directly comparable to the data. From the entire CMD we used the photometry of a region that consisted of a magnitude range of 21–24.5 and a color range of −0.5–1.15, which we denominated the photometric CMD region (blue dashed box on Figure 1 ). The colors and magnitudes of the stars in the photometric CMD region are compared to the different models to fit the SFH. This region was chosen to avoid CMD locations significantly affected by completeness corrections or contamination sources, while maximizing our signal. Additionally, magnitudes brighter than g = 21 were excluded to avoid regions of late stages of stellar evolution, where the different sets of isochrones we tried (Teramo, Dartmouth, and Yonsei-Yale) differed significantly from each other. The individual bins that defined the Hess diagrams were chosen to have a size of 0.1 in magnitude and 0.025 in color. These dimensions were selected to be small enough to be able to trace the different features of the stellar populations in the CMD, and big enough to avoid significant shot noise in each bin. To construct the MDF of the data and the different models, Talos uses a region of the CMD where significant spectroscopic data exist, which we denominated the spectroscopic CMD region (green solid box on Figure 1 ). In our case, this region of the CMD had a g − r color range of 0.5–1.25 and a g- magnitude range of 18–20.5. Then, each time Talos generates a model stellar population, it constructs the associated artificial MDF with the metallicity values of all the stars from the model that lie in the spectroscopic CMD region. Analogously,


difficult to provide because we have little prior information on it. However, a parsimonious parametric model of the PSF is suited for deconvolution processes (Drummond 1998 ; Markham Conchello 1999 ; Fétick et al. 2019a ). It is thus more efficient to parametrize the PSF using the available information on the observing system. The PSF model developed in Fétick et al. ( 2019b ) is adapted to AO-corrected images as it is able to describe both the coherent core and the turbulent halo of the AO-corrected PSF. Even with the strong prior of a parsimonious PSF parametrization, the joint maximum a posteriori estimator is not ensured to converge towards a satisfactory result. Indeed, the degeneracy of this estimator was demonstrated by Blanco Mugnier ( 2011 ) with a linear combination of PSFs for application to biomedical imaging. In this paper, we show that the joint estimator is also degenerated with a non-linear PSF model, although adapted to astronomical imaging. Because of the drawbacks of the joint estimator, we consider another method, which is the marginal estimator (Berger, Liseo Wolpert 1999 ; Blanc, Mugnier Idier 2003 ; Levin et al. 2009 ). This method consists of integrating – marginalizing – the object out of the problem to estimate only the PSF parameters. This estimator is asymptotically consistent and non-biased (Lehmann 1983 ) when the number of data or the signal-to-noise ratio (S/N) increase. We apply this method to our astronomical case in order to estimate the PSF parameters. The PSF is then used in a conventional (non-blind and non-myopic) deconvolution to estimate the object. In Section 2 , we explain the reasons for our choice of a parametric PSF model adapted to AO-corrected images. The first approach would be to use this parametric model to perform joint estimation. In Section 3 , we show that the deconvolution based on a joint estimation of the PSF and the object does not converge towards the correct solution in spite of the parametric PSF model introduced. Thus, we consider the marginalized approach in Section 4 . We study the effect of the S/N and the supervision level on the PSF estimation and image deconvolution. We show that this method can be adapted for blind deconvolution, with an example using astronomical images from the Spectro-Polarimetric High-contrast Exoplanet Research (SPHERE)/Zurich Imaging Polarimeter (Zimpol) on the Very Large Telescope (Beuzit et al. 2019 ). Finally, we conclude in Section 5 . 2 PSF AND IMAGING MODELS 2.1 PSF model Because ofhighly variable atmospheric conditions and the complexity of the AO correction, the PSF must be estimated for each observation. The simple method of estimating a PSF on a bright star is not possible for many observational cases, especially for observation of rsesolved objects in a small field of view. The PSF must be estimated with another method in order to be able to perform post-processing techniques on the image. This estimation might be based on telemetry or on


and also separately on the blue ([3600, 5500] Å) and red ([5500, 7200] Å) regions. The cyan contours overlaid correspond to 1 to 5. The three maps are similar, and the location of the solution is consistent within 1. The elongated shape of the contours, in particular, on the vs. [Fe/H] plane reflects the degeneracy between the parameters. For good data and normal stars, the maps as well as the parameters do not depend on the fitted wavelength range. Figure 6 shows the maps for G 156-031, a star at the edge of the parameter space populated by the MILES stars. The values increase sharply on the low side, resulting in a dissymmetry that is potentially at the origin of a bias. We discuss below the stars labelled in Fig. 1 and listed in Table 6 . Fig. 5 maps for HD 61913 with different fitting wavelength ranges. First with the complete wavelength range used (36007400 Å), second 36005500 Å, and third 55007200 Å. The red crosses indicate the minimum, and the cyan crosses the compiled parameters. The inner contour in cyan correspond to the external errors, and the other contours correspond to the 2, 3, 4 ... levels. Table 6 Remarkable stars labelled in Fig. 1 . (1) V1855 Ori = MS 0515.4-0710 (MILES 175)is a K0 or K2 type star. V1855 Ori and MS0515.4-0710 currently appear as two distinct sources in SIM-BAD, but their cross-identification with a unique sourceleaves no doubt, and allows us to complete our compilation. Favata et al. (1997) and Ce-narro et al. (2007) determined[4570, 3.50, 0.16] using the Strömgren photometry of Moraleet al. (1996) and the calibration re-lation of Alonso et al. (1996) . Xing (2010) gives [en-tity!#x2009!]= 4908 K and = 4.45 dex,computed from the colour and calibrations of Casagrande et al. (2006) . Biazzo et al. (2012) give [5100, 4.2, 0.05], based on high-resolution spectroscopic observations. Martin et al. (1994) adopted = 5150 K from their own relation between the spectral type and the temperature. In our compilation we adopt the average parameters [4860, 4.05, 0.10]. The parameters obtained after fitting the blue range, [], and the red range, [], differ significantly, as is often the case for variable stars. Our adopted measurement is not discrepant with the analysis of Biazzo et al. (2012) and therefore we trust our measurements. (2) HD 113285 (MILES 459) is a M8 III type pulsating variable star. This star belongs to the LICK/IDS library ( Worthey et al. 1994 ). Gorgas et al. (1999) determined = 2924 K by extrapolating the versus spectral type relation of Ridgway et al. (1980) . Gorlova et al. (2003) obtained = K. The parameter was derived from the spectral type and was obtained from , luminosity, and mass. McDonald et al. (2012) derived = 2602 K using spectral energy distribution (SED) fitting, neglecting Galactic extinction and assuming a solar composition. The authors PVK adopted [], and in our compilation we adopt the average of the available measurements [], where = 0.0 dex is a reasonable guess for this type


This work was supported by the Ministry of Science and Technology of China (grant no. 2013CB837900), the National Science Foundation of China (grant nos. 11125313, 11203017, 11433002, 61271349, and 61371147), the Chinese Academy of Sciences (grant no. KJZD-EW-T01), and the Science and Technology Commission of Shanghai Municipality (grant no. 11DZ2260700).


We thank Brian Svoboda for helpful comments and suggestions. JCT and SK acknowledge an NRAO/SOS grant and NSF grant AST1411527. PC acknowledges the financial support of the European Research Council (ERC; project PALs 320620). This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00806.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. Facility: ALMA .


ACKNOWLEDGEMENTS The authors would like to especially thank Catherine Watkinson for her contributions to an earlier draft of this work. We would also like to thank Daniel Mortlock, Conor O’Riordan, Brad Greig, Andrei Mesinger, Emma Chapman, Roberto Trotta, Alan Heavens, Suman Majumdar, and the anonymous referee for their helpful conversations and suggestions. We acknowledge use of linux os, c, c ++, fortran 90, gnu, gsl,overleaf, github, python , and the python packages numpy, scipy, cornerplot, matplotlib . T. Binnie thanks Science Technology Facilities Council (STFC) for their studentship funding. J. R. Pritchard is pleased to acknowledge support from the European Research Council under ERC grant number 638743-FIRSTDAWN.


This work was supported by NASA grant NNX16AC07G through the XMM-Newton Guest Observing Program, and through grants for HST program number 14054 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The research at the Technion is supported by the I-CORE program of the Planning and Budgeting Committee (grant number 1937/12). E.B. received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 655324. SRON is supported financially by NWO, The Netherlands Organization for Scientific Research. N.A. is grateful for a visiting-professor fellowship at the Technion, granted by the Lady Davis Trust. S.B. and M.C. acknowledge financial support from the Italian Space Agency under grant ASI-INAF I/037/12/0. B.D.M. acknowledges support from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 665778 via the Polish National Science Center grant Polonez UMO-2016/21/P/ST9/04025. L.D.G. acknoweledges support from the Swiss National Science Foundation. G.P. acknowledges support by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OR 1408 and FKZ 50 OR 1604) and the Max Planck Society. P.O.P. acknowledges support from CNES and from PNHE of CNRS/INSU.


the literature. Gray symbols show Local Group dwarf galaxies from Kirby et al. ( 2013 ), and gray dashed and dotted lines represent the median and 16th and 84th percentiles of the metallicity distributions. Cyan dashed and dotted lines show the median and 16th and 84th percentiles of the metallicity distributions for various types of galaxies in Gallazzi et al. ( 2005 ). Green symbols show the stellar mass–metallicity relation for early-type galaxies binned in stellar mass (Conroy et al. 2014 ). Top right: relation between velocity dispersion and [Fe/H] for DF 44, compared with early-type galaxies binned in velocity dispersion (Conroy et al. 2014 ), and Local Group dwarf galaxies (Kirby et al. 2013 ). Bottom left: relation between metallicity and the logarithmic ratio of stellar mass to dynamical mass for DF 44, Local Group dwarf galaxies (Kirby et al. 2013 ) and early-type galaxies binned in velocity dispersion (Conroy et al. 2014 ). Previous work concluded that some UDGs, including DF 44, may live in more massive dark matter halos than would be expected for their luminosities (e.g., M vir > 10 11 M ⊙ , van Dokkum et al. 2016 ). In addition, it is well-known that the stellar properties of early-type galaxies are tightly related to their dynamical masses (e.g., Gallazzi et al. 2006 ; Graves et al. 2009 ). Therefore, we also look into the relation between their stellar properties and gravitational potential well depth. The top right panel of Figure 12 shows the relation between stellar velocity dispersion ( σ ) and metallicities. We compare DF 44 with dwarf galaxies (Kirby et al. 2013 ), massive early-type galaxies binned in velocity dispersion (Conroy et al. 2014 ). We also compare to 12 elliptical and S0 galaxies in the Coma cluster (Trager et al. 2008 ). Their iron abundances are calculated based on their metallicities and the enhanced element abundances ([Z/H] and [E/Fe]) in Trager et al. ( 2008 ) and Equation (3) in Trager et al. ( 2000 ). DF 44 stands out in this plot in the sense that it has lower metallicity for its σ compared to other galaxies. We also know that the stellar mass of DF 44 is unusually low for its σ and since it is the (massive) stars that produce metals, it is perhaps not surprising that DF 44 has low metallicity for its σ . We explore this point further in the bottom-left panel of Figure 12 , where we plot [Fe/H] as a function of the logarithmic ratio of stellar mass to dynamical mass. This ratio is proportional to the integrated star formation efficiency. In this diagram, one can imagine diagonal tracks where a galaxy in a given halo evolves along the track as it converts baryons into stars. Tracks associated with more massive halos will be offset vertically in this diagram, as a given integrated star formation efficiency (a given stellar-to-dynamical mass ratio) will result in a higher stellar metallicity for galaxies in deeper potential wells, as such systems will be able to retain a greater fraction of the


et al. ( 2012 ) analysis does not apply. This result is obviously related to the apparent first-infall status of many of these galaxies. Among the galaxies for which the binding energy is well behaved, the Rocha et al. ( 2012 ) relation suggests infall times of ∼7 Gyr for Tri II (also see Carlin et al. 2017 ), Segue 2, and Willman 1; ∼8 Gyr for Draco II and Tuc III; and >12 Gyr for Ret II. Ret II has a higher binding energy than any of the Via Lactea 2 subhalos included in the Rocha et al. sample. 6.5. Implications for the Mass of the Milky Way As discussed in previous sections and shown in Figure 6 , a number of the target galaxies of this study appear to be very loosely bound and may be on their first orbital passage around the Milky Way, which would be an unexpected result. I showed in Section 5 that this problem can be alleviated if the mass of the Milky Way is larger than assumed in the default galpy MWPotential2014 model. Relatively large Milky Way masses have been advocated by some authors (e.g., Boylan-Kolchin et al. 2013 ), but if the mass of the Galaxy is large, that will tend to exacerbate issues such as the missing satellite problem (Klypin et al. 1999 ; Moore et al. 1999 ) and the too big to fail problem (Boylan-Kolchin et al. 2011 , 2012 ). The alternative explanation is that many of the ultra-faint satellites are on their first orbit because they formed in a Magellanic group (D’Onghia Lake 2008 ) that is now infalling into the Milky Way for the first time (Besla et al. 2007 ; Kallivayalil et al. 2013 ). However, if many of the currently known satellites were originally located around the LMC, that also has adverse consequences for the missing satellite problem, because the number of dwarfs formed around the Milky Way would be even smaller. Improved estimates of the Milky Way’s mass based on dynamical modeling of the kinematics of the satellite population provided here and by Gaia Collaboration et al. ( 2018b ) will be valuable for attempting to resolve this dilemma. 7. Summary and Conclusions I have presented Gaia DR2-based proper motions, space velocities, and Galactic orbits for 17 ultra-faint satellites of the Milky Way. I cross-match lists of member stars that are confirmed via spectroscopy or variable star pulsations with the Gaia DR2 catalog to determine the motion of each system across the sky. Thanks to the exquisite astrometric performance of Gaia , even with as few as ∼3 member stars brighter than g = 20 per satellite, the proper motions are determined with a typical precision of ∼0.1 mas yr −1 per coordinate. At a median distance of 35 kpc, the uncertainties on the tangential velocities are therefore ∼25 km s −1 . I find good agreement with the proper-motion measurements reported by Gaia Collaboration et al. ( 2018b ) for Boötes I and Leo I, as well as with HST measurements of Leo I (Sohn et al. 2013 ) and Ω Centauri (Libralato et al. 2018 ). The tangential velocities of the sample of ultra-faint dwarfs analyzed


around 150 MHz 6 , the fluence limit at a 5 sigma level at zenith is 2.5 Jy ms for core stations or 1.5 Jy ms if all Dutch stations can be combined fully coherently. At that threshold, more than half of the known FRBs up to a DM of 500 pc cm −3 would be bright enough to be detected. This may be the most promising route to observe FRBs with LOFAR, since it would be possible to detect flat spectrum FRBs at 150 MHz. 6. Conclusions The FRATS project searches in real-time for millisecond-scale radio transients with the LOFAR radio telescope. Some known transients of this kind are FRBs and RRATs. The focus is to obtain an optimal sky coverage by using commensal observations and a large field of view. An innovative addition is the use of ring buffers that hold the last seconds of data from each individual antenna which ultimately allows for post-event beam-forming and imaging. This increases the positional accuracy, potentially to the full arc second resolution. This is important for pinpointing the location of FRBs. The potential of LOFAR is demonstrated by two different results. The first demonstration is the successful commensal observing from the Pilot C survey, where beamformed data were produced to search for millisecond-scale radio transients, during regular interferometric observations. Five pulsars were discovered in a 24-h scan. The second is the identification of a transient as a type II solar radio burst by localizing it in a side lobe. This could only be done by using the buffered data to create an all sky image. Besides these demonstrations we showed the results from the pilot surveys A and B, which searched up to a DM of ∼110 pc cm −3 and 500 pc cm −3 , respectively. While known pulsars were rediscovered, no unknown single-pulse sources have been detected. We have derived limits on the local FRB rates from these surveys. The first pilot survey (A) had an effective observing time of 68.51 beam h, observed from 119 to 151 MHz, the lowest frequency range on which FRBs have been searched. Taking into account the pointing direction and noise level for each observation, this sets a limit on the occurrence of FRBs of less than 1500 events per sky per day above a threshold of 1.6 kJy ms below a DM of ∼110 pc cm −3 . Pilot study B had an effective observing time of 91.2 beam h in the same band. This sets limits at 1400 events per sky per day or 134 events per Gpc 3 per day at a fluence threshold of 6.0 kJy ms for an 8-ms pulse. The expected rate is one per beam per 16 days, for a uniform extragalactic distribution, provided the source flux density follows a steep spectrum of ∼ ν −3 . In the case where the source population as a function of redshift follows the star formation rate, the rate drops to one per beam every 58 days. If the spectrum is less steep and/or a significant fraction of FRBs are broadened by scattering beyond our detection limits, a longer integration time is required. To be able to obtain the required observing time, we


stars ( Dutra et al. 2014 ; Lourenco et al. 2019 ). From the pool of models studied in Dutra et al. (2014) ; Lourenco et al. (2019) , we chose four: BSR8, BSR9, G2*, and IU-FSU – all of them allow for the DU process to set in at reasonable neutron star masses (between 1.0 and 2.0 solar masses). Furthermore, all of them present microscopic properties different enough to allow us to conclude that the results we find are most likely general. The same behavior is shared among all models studied, with the only difference being the star’s mass at which the DU process becomes available. This paper is divided as follows: in Sect. 2 we discuss the microscopic models used, Sect. 3 is devoted to the review of the thermal evolution of neutron stars, Sect. 4 contains our results for the relaxation time of neutron stars at the onset of the direct Urca process, and in Sect. 5 we present our conclusions. 2. Microscopic models Quantum hadrodynamics (QHD) is a powerful tool used to build models that represent the strongly interacting matter with hadrons being the main degrees of freedom. The first model constructed from such an approach was proposed by Walecka (1974) and Serot Walecka (1997) with the two free parameters fixed to reproduce the nuclear matter energy per particle as a function of the density, and a minimum of B 0 = −15.75 MeV at the saturation density ρ 0 = 0.19 fm −3 . However, the model also presents bad results for the effective mass ratio and incompressibilty, both at ρ = ρ 0 , namely, ( M n is the nucleon rest mass) and K 0 = 540 MeV, respectively. Over the years, many other improved versions of this model were proposed in which and K 0 are fixed to more compatible values with experimental/theoretical predictions. Furthermore, other bulk parameters at the saturation density are also used in order to constrain the free coupling constants of these microscopic models. Here, we investigate the cooling process of neutron stars based on parametrizations of a general model described by the following Lagrangian density ( Dutra et al. 2014 ; Li et al. 2008 ) in which F μ ν = ∂ μ ω ν − ∂ ν ω μ and B μ ν = ∂ μ ρ ν − ∂ ν ρ μ . ψ is the nucleon field and σ , ω μ , and ρ μ represent the fields of the mesons σ , ω , and ρ , respectively. The mean-field approximation is used in order to solve the equations of motion for the fields. This procedure, along with the energy-momentum tensor, T μ ν , allows the construction of all thermodynamics of the system since the energy density and pressure are given by E = ⟨ T 00 ⟩ and P = ⟨ T i i ⟩/3, respectively. These equations of state are evaluated, as a function of the density, by taking into account the auto-consistency of the field equations and the definition of the effective nucleon mass given by M * = M n − g σ σ . More details related to the calculations of these quantities can be found in Dutra et al. (2014) and Li et al. (2008) , and references therein. In order to study neutron stars and their thermal


(i) the same ratio between PSF size and pixel width (on the detector) at all wavelengths; (ii) the same incident flux at all wavelengths; (iii) the ratio between diagonal and off-diagonal terms. Some values can be null or negative. While it is highly unlikely that the three conditions will be satisfied exactly, there is always a chance of them almost being satisfied. In which case, instead of null eigenvalues we might see significantly smaller values compared to the others. This results in a poorly conditioned matrix. Having a large range of eigenvalues causes precision problems when trying to invert the matrix. A more careful study might be needed to explore the conditions when negative, null or very small eigenvalues can occur. 4.1.2 Results for the slicer Fig. 2 shows the result of the super-optimal procedure on a slicer simulated supernova. Using SpecSim, we simulated the observation of a Type Ia supernova at redshift 1.5 with a 2-m telescope of focal length 20m with an exposure time of 8h. The simulated errors include the zodiacal light, detector and shot noise. The high-resolution input spectrum, shown in blue, is the spectrum input to the slicer (theoretical Type Ia supernova spectrum scaled to magnitude, mirror area and exposure time). The red error bars display the 1 errors computed by the super-optimal extraction. The choice of the exposure time yields S/N of 10.4 for the Si ii absorption feature at 1.53m. Figure 2. SPECSim simulation of the Type Ia supernova spectrum and extraction with the super-optimal extractor. The input truth (blue) is simulated and reconstructed (red) with 1 error for a supernova at redshift 1.5 observed with the slicer spectrograph on a 2.0-m space telescope of 20.0-m focal length. The exposure time is 8h. The error bars estimated by the super-optimal extractor result from the zodiacal light, detector noise and shot noise. Fig. 3 shows the percentage error to compare the optimal (grey) and super-optimal (red) spectrum extraction methods highlighting the Si ii line region. The input spectrum and the observational parameters used are identical to those used in Fig. 2 . The super-optimal extraction performs significantly better than the optimal extraction. At the redshifted Si ii absorption line at 1.537m, the super-optimal extraction performs 1.6times better in the error percentage than the optimal extraction, and the residuals (deviation from theoretical) are also smaller, with their distribution compatible with the theory of chi-square fit. The goodness of fit can be tested by the pull distribution, which corresponds to the histogram of the ratio of the residuals and estimated error. Ideally, this should be a normal distribution centred at 0 and with rms of 1. This is the case for the super-optimal extraction and not for the optimal extraction. In our case, this is because the latter underestimates the errors. Figure 3. Comparison of the optimal (grey) and super-optimal (red) spectrum extraction


This study was carried out with the support by the FWF NFN project S11601-N16 ‘Pathways to Habitability: From Disks to Active Stars, Planets and Life and the related subprojects S11607-N16 and S11608-N16. HL thanks the support by FWF project ‘Characterizing Stellar and Exoplanetary Environments via Modeling of Lyman-Alpha Transit Observations of Hot Jupiters, P 27256-N27. The authors thank Dr Yamila Miguel and Dr Alessandro Mura for their valuable comments on the mineral atmospheres. RS thanks the support by the FWF project P23810-N16. The authors thank the anonymous referee for valuable comments.


This work was funded with grants from Consejo Nacional de Investigaciones Cientificas y Tecnicas de la Republica Argentina, and Universidad Nacional de La Plata (Argentina). Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (USA), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). The Gemini program ID are GS-2008B-Q-54 and GS-2009B-Q-65. This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. We thank the referee for her/his significant contributions which greatly improved this work.


from 0.13 to 0.53 M J , in resemblance to Saturn. The Jupiter mass planets simply occupy the occurrence rate of ∼10 per cent of single stars (Cumming et al. 2008 ). Similarly, for CBPs, Jupiter mass planets appear to be less common (Armstrong et al. 2014 ) as compared to single star systems. Therefore, our study, on the basis of tidal interactions between Jupiter-like planets and their host stars (Faber et al. 2005 ; Liu et al. 2013 ), cannot be applied to estimate the occurrence rate of S-type planet in close binaries. In this sense, the difficulty in more exploration is that the nowadays a dynamical tide model is not suitable for Saturn-like or Neptune-like planets any more. In particular, the inner structure of Neptune-like planets differs a lot from that of gas giants. This may eventually give rise to major differences in the dynamical tides for Saturn-like or Neptune-like planets compared to Jupiter. However, above-mentioned question is beyond the scope of this study, and we will explore this issue in future. The scenario we propose in this work is an exotic formation mechanism for S-type planets in tight binaries ( a B = 0.5 ∼ 3 au). An interesting question is whether there is a limit for the separation of a binary below which in situ formation of S-type planets is impossible. Five S-type sub-Earth-sized planets have been found in Kepler-444A(BC) system (Dupuy et al. 2016 ). Although it has a larger a B ∼ 37 au, the periastron distance is only ∼5 au ( e B ∼ 0.86). Exotic formation mechanism such as capture seems unlikely to form a multiplanet system. Due to the close pericenter passage of 5 au, the disc of Kepler-444A will be truncated to 1–2 au. Although there are many challenges for forming such a planetary system, in situ formation can be a possible scenario (Dupuy et al. 2016 ). Therefore, future planet-hunting in the binaries with a separation less than 5 au may provide vital clues to in situ or exotic formation to understand their origin. In summary, planet formation in close binaries is a fascinating and challenging subject. As mentioned previously, searching for S-type planets in eclipsing binaries arouses great scientific interest for the forthcoming space missions such as PLATO. Our work provides some suggestions for selecting target binaries in search for S-type planets in the near future. Their presence or absence will deepen our understanding of planet formation. APPENDIX A CODE TESTS We performed additional simulations to examine whether the modified codes can accurately identify pericentre passages and exert the tide effect at each pericentre passage. In order to shorten the integration time and make the dynamical tide operate immediately, we adopt the following initial conditions. We assume that the planet has entered the stable region of the primary star in the beginning. The planet is on a very high eccentric orbit with respect to the primary ( e p = 0.95, a p = 0.2). The parameters of the binary are M 1 = M ⊙ , M 2 =


We thank the anonymous referee for constructive comments which helped to improve the manuscript. We thank D. Dallacasa and F. Gastaldello for comments on an early version of the manuscript and S. Giacintucci, D. Hoang, R. Kale, T. Shimwell, T. Venturi and R. van Weeren for kindly providing us the displayed radio contours. GB acknowledges partial support from ASI/INAF n. 2017-14-H.O. The work of DR and SR was supported by the NRF of Korea through 2016R1A5A1013277 and 2017R1A2A1A05071429. The scientific results reported in this article are based in part on observations made by the Chandra X-ray Observatory and in part by XMM-Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.


The research was supported by the National Science Foundation of China grants 41174123, 41421063, WK2080000031, and WK2080000077, the Chinese Academy of Sciences grants KZCX2-YW-QN512 and KZZD-EW-01, and the Fundamental Research Funds for the Central Universities under grant WK2080000031.


We thank the anonymous referee for constructive suggestions that helped to improve the manuscript. This work is supported by National Basic Research Program of China (973 program, grant No. 2015CB857005) and National Science Foundation of China (grants No. 11421303 11503024). J.X.W. thanks support from Chinese Top-notch Young Talents Program, and CAS Frontier Science Key Research Program (QYZDJ-SSW-SLH006). Z.Y.C. acknowledges support from the China Postdoctoral Science Foundation (grant No. 2014M560515) and the Fundamental Research Funds for the Central Universities.


ACKNOWLEDGEMENTS C. Miller as well as Y. Zhang acknowledges support from Department of Energy grant DE-SC0013520. Y. Zhang thanks Alex Drlica-Wagner, Gary Bernstein, and Chris Davis for careful reading of the draft. We use DES Science Verification in this paper. We are grateful for the extraordinary contributions of our CTIO colleagues and the DECam Construction, Commissioning and Science Verification teams in achieving the excellent instrument and telescope conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the DES Data Management group. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas AM University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas AM University, and the OzDES Membership Consortium. Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support from the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.


the residuals from the planet fit, we obtain ( r , ρ) BIS = (−0.39, −0.33), and ( r , ρ) S-index = (−0.06, 0.38), which indicates no correlation among these quantities. These results, for the HARPS and CORALIE data, can be seen in Fig. 9 , with the activity indices listed in Tables A1 and A2 . We also performed the bisector analysis on K2-238, and found ( r , ρ) BIS = (0.40, 0.37), which would indicate no correlation between the BIS and the FEROS RVs. For the residuals we found ( r , ρ) BIS = (0.50, 0.36), also indicative of no strong correlation. The results are shown in Fig. 10 , and listed in Table A3 . We did not include the S-index due to the low S/N spectra obtained with FEROS, which prohibited us from measuring them reliably. Figure 10. Correlations between the RVs (top panel) and the residuals from the planetary fit (bottom panel), with the BIS from FEROS for K2-238. The colours represent the orbital phase of the RV and residuals. r and ρ are the Pearson and Spearman correlation coefficients, respectively. 3.5 Planet scenario validation In order to confirm the planetary nature of our photometric and spectroscopic measurements, we performed a blend analysis using the algorithms described in Hartman et al. ( 2011a , b ), which model the observations taking into account the possibility that they could be generated by either a planet, stellar companions physically associated with our target star or by various blend scenarios, including blended eclipsing binary and hierarchical triple systems. K2-237 b is confirmed to be a planet based solely on the photometry; it is practically impossible for the best-fitting blend scenarios to fit the observed photometry in any of the cases consistent with the spectroscopic information. For K2-238 b , the planetary interpretation is also favoured by the data: although there is a detected close-by companion in the Lick 3 m Adaptive Optics (AO) data, the light curve is not consistent with the transit/eclipses arising from the neighbour, as all the simulated light-curve signatures imply J − K colours much less than the observed J − K = 0.631 ± 0.043. Considering that the brighter source could still itself be a blend, we can reject all the blend scenarios at 2.5σ confidence based on the photometry. However, none of them are able to produce the observed 100 m s −1 sinusoidal RV variation. The best-fitting blend scenarios to the photometry also yield large bisector span variations in excess of 1 km s −1 , which are clearly ruled out by our measurements (see Fig. 10 ). We consider thus both planets to be statistically validated given our photometric and spectroscopic measurements. 3.6 Searching for additional signals in the photometry We search for additional signals in our K2 light curves, produced by other companions, orbital phase variations, or secondary eclipses by performing a Box-fitting Least-Squares periodogram (BLS, Kovács, Zucker Mazeh 2002 ) on the light curves, with the transits of the detected


We thank the anonymous referee for comments which were beneficial to the final version of this paper. M.B.P. and S.E.Z. acknowledge support from NASA through the ADAP grant NNX11AG12G. B.D.L acknowledges support from NASA ADAP NNX16AG06G. The scientific results reported in this article are based on observations made by the Chandra X-ray Observatory . Support for this work was provided by NASA through Chandra Award numbers GO5-16084A (MBP) and GO5-16084B (AK) issued by the Chandra X-ray Observatory Center, which is operated by SAO for and on behalf of the NASA under contract NAS8-03060. Based on observations made with the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program HST-GO-13942.001-A. Support for program number HST-GO-13942.001-A was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. This research has made use of NASA’s Astrophysics Data System (ADS) and the NASA/IPAC Extragalactic Database (NED). Facilities: CXO, HST , 2MASS. Software: SciPy, NumPy, matplotlib, pythonFSPS, TOPCAT, ds9.


qualitatively understood in terms of the geometry of the heliosphere (see Sect. 4.2 ). 3. Comparing the observation regions of HSTOF and Voyager 1 The flux J ENA of the ENA originating in the heliosheath can be approximately expressed as where J ion is the parent ion flux (protons for hydrogen ENA, He + for helium ENA) at the same energy as J ENA , σ cx, H , and σ cx, He are the cross sections for charge exchange between the parent ions and the neutral hydrogen or neutral helium background gas of density n H and n He , respectively, and L is the thickness of the source region along the line of sight (LOS). The values J ion and n H are averages over the distance along the LOS within the source region (the inner heliosheath). For the helium ENA, we omitted the additional contribution from double charge exchange between the He ++ and the neutral He background, which is on the order of 3% ( Swaczyna et al. 2017 ). The equation does not include the ENA losses on the way from the source, but in the energy range where most of the ENA observations are carried out (IBEX Hi, INCA, and HSTOF) these effects are relatively small (about 1% for HSTOF). Fig. 4. Solar wind flux (yearly averages based on OMNI2 data) at 1 AU as a function of time. The time interval starts earlier than the HSTOF observations discussed in this paper because of expected delay between the changes in the solar wind and in the ion distributions in the heliosheath. To compare the observations of Voyager 1 and HSTOF we use the ENA flux averaged over the HSTOF energy range: E 1 E E 2 where E 1 = 58 keV and E 2 = 88 keV. Eq. ( 1 ) then becomes where ⟨ A ⟩ denotes the average over the HSTOF energy range When used for the case of HSTOF, Eq. ( 2 ) must also be averaged over directions of the lines of sight, and over the observation time periods. As explained in Sect. 2 , we divide the HSTOF field of view into four ecliptic longitude sectors (Fig. 1 ). Because of proximity to the Voyager 1 and 2 trajectories, the forward (LISM apex) sector is most suitable for comparison with Voyager observations. A study of the energetic ion distribution in the heliosheath ( Czechowski et al. 2012 ) and the Bonn model of the heliosphere, on which this study was based ( Fahr et al. 2000 ), implies that the thickness L , the energetic ion fluxes, and the neutral density do not vary much over this region. We also consider the two flank sectors. Table 1 lists the hydrogen ENA flux values, which is also illustrated in the Fig. 2 . Table 1 includes also the estimated hydrogen ENA flux from the Voyager 1 direction. It is calculated by substituting for J ion in Eq. ( 2 ) the energetic proton flux J p, V1 at Voyager 1 and for L the distance L V1 = 28 AU between the termination shock and the cliff along the Voyager 1 trajectory. In the HSTOF energy range, Voyager 1 LECP measures only the Z ≥ 1 ion flux , which includes the protons along with the heavier ions. We assume that J p, V1 is approximately equal to . For J Z ≥


Dynamical Time, J2000.0 ecliptic and equinox). Source: JPL’s SBDB (solution date, 2020-Mar-03 08:13:59). The orbit determination in Table 1 is still uncertain and its associated evolution rather chaotic (see Section 3 ). Wiegert, Innanen Mikkola ( 1998 ) have shown that the statistical analysis of an extensive set of numerical simulations accounting for the uncertainties associated with an orbit determination can produce reliable results. In order to obtain sufficiently robust conclusions, we apply the Monte Carlo using the Covariance Matrix (MCCM) method detailed in section 3 of de la Fuente Marcos de la Fuente Marcos ( 2015b ) to generate initial conditions for our N -body simulations that have been carried out using Aarseth’s implementation of the Hermite integrator (Aarseth 2003 ); the direct N -body code is publicly available. 1 The covariance matrix required to generate initial positions and velocities for 10 4 control or clone orbits has been obtained from Jet Propulsion Laboratory’s Solar System Dynamics Group Small-Body Database (JPL’s SSDG SBDB, Giorgini 2015 ). 2 This is also the source of the data in Table 1 ; JPL’s horizons 3 ephemeris system (Giorgini Yeomans 1999 ) has been used to gather most input data used in our calculations (e.g. data in Table 2 ). Some data have been retrieved from JPL’s SBDB using the tools provided by the Python package Astroquery (Ginsburg et al. 2019 ; Mommert et al. 2019 ). Extensive details on our calculations and physical model can be found in de la Fuente Marcos de la Fuente Marcos ( 2012 ), de la Fuente Marcos de la Fuente Marcos ( 2018a ), and de la Fuente Marcos de la Fuente Marcos ( 2018b ). Table 2. Cartesian state vector of 2020 CD 3 : components and associated 1σ uncertainties. Epoch as in Table 1 . Source: JPL’s SBDB. In order to analyse the results, we produced histograms using the Matplotlib library (Hunter 2007 ) with sets of bins of constant size computed using Astropy (Astropy Collaboration 2013 , 2018 ) by applying the Freedman and Diaconis rule (Freedman Diaconis 1981 ). Instead of using frequency-based histograms, we considered counts to form a probability density so that the area under the histogram will sum to one. 3 ORBITAL EVOLUTION With our calculations, we aimed at answering four main questions: (i) When did this capture take place? (ii) When is 2020 CD 3 leaving its current geocentric path? (iii) How was the pre-capture orbit of 2020 CD 3 ? (iv) How diverse is the evolution in the neighbourhood of the initial conditions in Table 2 ? Answering the first two questions, we can find out if 2020 CD 3 is experiencing a temporarily captured fly-by or a temporarily captured orbiter episode. The third question is directly related to the origin of 2020 CD 3 and the fourth question can inform us on the reliability of our results. In order to provide robust answers to all these questions, we first focus on the initial conditions in the form of Cartesian state vectors and then we use


be expected to be smaller than both N and N̄ p . Indeed, a maximum likelihood fit υ ML ( z max ) of the PDF given by Eq. ( 13 ) to the set of simulated { z max } with N p as free parameter provides N̄ p = 2210. The right panel of Fig. 3 shows that something similar holds for the GDM since the theoretical value N * = 17 684 is greater than the number Ñ ∗ = 15330 corresponding to the maximum likelihood fit g ML ( z max ). Here, it is worth stressing that υ Th ( z max ) and g Th ( z max ), and hence the corresponding CDFs Υ Th ( z max ) and G Th ( z max ) shown in Fig. 4 are almost identical. The same holds for υ ML ( z max ) and g ML ( z max ) and the corresponding CDFs Υ ML ( z max ) and G ML ( z max ) again shown in Fig. 4 . As expected the two approaches are practically equivalent, given that both methods are essentially based on the same assumptions. Moreover, in Fig. 4 all these CDFs appear close to the sample CDF. This indicates that the value of the parameters N p and N * is not of critical importance. Fig. 3. Left panel : histogram H ( z max ) vs. the theoretical PDF υ Th ( z max ), mean PDF υ Mean ( z max ) and the maximum likelihood PDF υ ML ( z max ) of the value z max of the highest peak of a zero-mean unit-variance GRF with autocorrelation given by a circular Gaussian with dispersion set to three pixels. The numerical experiment is based on the simulation of 5 × 10 3 GRF’s of size 500 × 500 pixels. υ Th ( z max ) has been computed using Eqs. ( 18 ) and ( 8 ) with N p obtained as explained in Vio17, whereas υ mean ( z max ) has been computed with N p given by the mean number of peaks in the simulated GRF’s. In the case of υ ML ( z max ) the number N p comes out from a maximum likelihood method (see text). Right panel : corresponding g Th ( z max ), computed by means of Eq. ( 27 ) with N * given by the ratio of the total area of the GRF to the window area πl 2 with (see text). Also the PDF g ML ( z max ) has been computed with N * resulting from a maximum likelihood method. To notice that υ Th ( z max ) and g Th ( z max ) are almost identical. The same holds for υ ML ( z max ) and g ML ( z max ). Fig. 4. Cumulative distribution functions corresponding to the PDFs in Fig. 3 . A further comparison of the two methods concerns the computational burden. In this respect the GDM is superior in situations where the autocorrelation function has circular symmetry (i.e., it is characterized by only a scale parameter), since the more expensive step consists in the computation of the quantities σ j , j = 0, 1, 2, which requires the numerical computation of three one-dimensional integrals. On the other hand, PAM requires the numerical computation of a number of integrals equal to the number of peaks selected for the detection. In general the difference in computational time is on the order of a few seconds even in the case of large images. The computational superiority of GDM vanishes in the presence of autocorrelation functions when they are not


properties. This means that the stellar mass (or the dust luminosity when there is no stellar population included in the model) is not a free parameter even though it is technically possible to treat it as such. This would greatly expand the size of the parameter space by adding an extra dimension, while slowing the computation of the grid and the analysis by a similar amount, and degrade the accuracy of the estimation of the physical properties. Optionally, CIGALE can also handle fluxes for which only upper limits have been determined. We adopt the method presented in Appendix A2 of Sawicki (2012) . This affects the aforementioned computing steps in several ways. First, the computation of the χ 2 is divided between regular quantities (first three terms corresponding to Eq. ( 14 )) and those with only an upper limit (last three terms): with “erf” being the error function, f ul the flux upper limit, and the indices i , j , and k indicating respectively the fluxes, extensive properties, and intensive properties. See Eq. (A6)–(A10) of Sawicki (2012) for a full derivation 6 . The main difficulty is to determine α . For this we have to numerically solve ∂ χ 2 /∂ α = 0 for every model, which is equivalent to solving Eq. (A11) from Sawicki (2012) : We do so using the scipy.optimize.root root-finding method. Once the χ 2 are computed, the subsequent steps no longer depend on the presence or absence of upper limits. We have to note that objects in a catalogue do not all need to be fitted with the same set of data. CIGALE will automatically only consider the available data for a given object. The lack of certain data for some targets has a direct impact on some of the aforementioned computation steps. For steps 2 and 3, we simply do not include the data that have been marked as invalid in the input file (value lower than 0 or set to “nan”, by convention) in the computation of α and χ 2 . Another feature of this module is the possibility to assess whether or not physical properties can actually be estimated in a reliable way through the analysis of a mock catalogue. The idea is to compare the physical properties of the mock catalogue, which are known exactly, to the estimates from the analysis of the likelihood distribution. To build the mock catalogue we consider the best fit for each object. We then modify each quantity by adding a value taken from a Gaussian distribution with the same standard deviation as the uncertainty on the observation. This mock catalogue is then analysed in the exact same way as the original observations. Physical properties for which the exact and estimated values are similar can be estimated reliably. Applications of the pdf_analysis module to a sample of representative galaxies is presented in Sect. 5.3 . 5. Examples of <monospace>CIGALE</monospace> use cases As seen above, CIGALE has been designed to be flexible and versatile so that it may have various applications: estimation of the physical properties of an object


HD 269902 (bottom). In HD 269902, most of the power in the WWZ is contained in the lower frequencies where the stochastic low-frequency background dominates—unsurprisingly given that the star has the second largest value of α out of the five FYPS (0.024 ± 0.001). Only by altering the scaling of the image of the WWZ are we able to see the ridge associated with the lowest-frequency peak, and it is only detected with low S/N. In both stars, the lowest-frequency peak in the RPS is associated with a broad band of power in the WWZ, similar to the WWZ of HD 269953, with a secondary band appearing at day −1 in HD 269840, at the approximate location of the highest peak in the RPS. 4. Discussion 4.1. Asteroseismic Modeling The discovery of fast pulsations in YSGs has very interesting implications for the study of the interiors of evolved massive stars. The 25 M ⊙ solar metallicity Geneva model (Ekström et al. 2012 ) has a main-sequence lifetime of approximately 7 Myr. It then crosses the H-R diagram in under 1 Myr, and half a Myr later, has evolved bluewards once more to become a Wolf–Rayet star (Massey et al. 2017 ). Due to the incredibly short lifetime of YSGs on both crossings of the H-R diagram, theoretical uncertainty has long stymied our understanding of massive star evolution (Kippenhahn Weigert 1990 ). Pulsation frequencies extracted from long-baseline lightcurves assembled from space-based observations are perhaps the most precise measurements we can make; typical values of for frequencies listed in Tables A1 – A5 are . In better-studied stars, such precision allows for the diagnosis of incredibly complicated physics and is truly the benchmark of testing stellar evolution theory (Aerts 2019 ). No asteroseismic models for YSGs that reliably converge exist (Jeffery Saio 2016 ), so the era of precision YSG asteroseismology is not yet upon us. However, we can compare the frequencies observed in the FYPS with the characteristic Lamb and Brunt–Väisälä angular frequencies in a model YSG: where l is the harmonic degree, c s is the sound speed, r is the radius of the local shell, g is the gravity, and the remaining variables are defined as in Aerts et al. ( 2010 ). Modes with angular frequencies and are mostly restored by pressure ( p -modes), and modes with and are mostly restored by buoyancy ( g -modes). While rudimentary, such a comparison would illustrate the approximate regions of the stellar structure the observed pulsations will allow us to probe. To do this, we use version 12778 of Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011 , 2013 , 2015 , 2018 , 2019 ) to evolve a grid of nonrotating stellar models with initial masses between 15 and 30 M ⊙ , with a spacing of 1 M ⊙ , initial metallicity set to Z = 0.006, and initial helium mass fraction set to . We first evolve the models through the entire pre-main-sequence stage until the entirety of the star’s luminosity comes from nuclear burning and then introduce more


by the interaction of the supernova (SN) ejecta with the circumstellar medium (e.g. Ofek et al. 2007 ; Quimby et al. 2011c ; Chevalier Irwin 2011 ), (ii) pair-instability driven ( Gal-Yam et al. 2009 ; Gal-Yam Leonard 2009 ), or (iii) neutron-star driven ( Kasen Bildsten 2010 ; Dessart et al. 2012 ; Kotera et al. 2013 ; Metzger et al. 2014 ; Murase et al. 2015 ; Suzuki Maeda 2017 ). Given the variety of observed spectra, it is plausible that different processes are at play in different objects (e.g. Gal-Yam 2012a ; Nicholl et al. 2014 ). Interestingly, scenarios (i) and (iii) predict bright associated γ -ray emission in the GeV to TeV range ( Murase et al. 2011 , 2014 , 2015 ; Katz et al. 2012 ; Kotera et al. 2013 ). The search for such γ -ray emission with the Fermi Large Area Telescope (LAT) data is the scope of this paper. In the most conventional model (scenario i) SLSNe are powered by the interaction between the SN ejecta and a massive, optically thick circumstellar medium ( Smith McCray 2007 ; Smith et al. 2008 ; Miller et al. 2009 ; Benetti et al. 2014 ). SN2003ma and SN2006gy for example seem to be explained well by this phenomenology ( Smith McCray 2007 ; Ofek et al. 2007 ; Smith et al. 2010 ). Several authors ( Murase et al. 2011 , 2014 ; Katz et al. 2012 ) have demonstrated that a collisionless shock could then be formed and would host efficient cosmic-ray acceleration leading to non-thermal emission from radio-submillimeter to γ -rays. In the GeV range, this emission can escape from the system without severe attenuation, for specific shock velocities (about 4500–5000 km s −1 ) ( Murase et al. 2015 ), and at late times after the shock breakout. Ackermann et al. (2015) searched for this specific radiation with the LAT at the location of core-collapse SNe (Types IIn and Ib), with standard luminosity, spanning typical time windows of a few months to a year starting from the optical luminosity peak. No detection was reported and model-independent flux upper limits were derived. A fast-rotating central neutron star releasing its rotational energy into the SN ejecta could also drive SLSNe (model iii; e.g. Kasen Bildsten 2010 ). The rotation period has to be close to milliseconds to transfer significant energy to the ejecta. The strength of the initial dipole magnetic field of the star sets the timescale over which the energy is injected (a stronger field leads to faster decline). Magnetars ( B ~ 10 15 G) have thus been proposed as central engines for SLSNe ( Kasen Bildsten 2010 ; Dessart et al. 2012 ). Pulsars with millisecond periods at birth and milder dipole magnetic fields B ~ 10 13 G would also lead to bright peaks as well as a high-luminosity plateau lasting for several months to years ( Kotera et al. 2013 ; Murase et al. 2015 ). These authors further calculated that the young neutron-star wind nebula would present a bright X-ray and γ -ray peak, respectively, through synchrotron radiation and inverse Compton (IC) scattering,


Acknowledgements H.R. thanks a CAPES fellowship and the CAPES PDSE programme (88881.132145/2016-01). J.M. thanks support by FAPESP (2012/24392-2 and 2014/18100-4) and CNPq (Productivity fellowship). We also thank Dr. Akito Tajitsu for his important support and initial data reduction during the observations with the Subaru Telescope. This study is based (in part) on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.