O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O O O O O O O O O O O O O O O O B-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Software O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O

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


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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.


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

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


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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.


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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,


O B-CelestialObject O O O O O B-Wavelength I-Wavelength O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O B-ObservationalTechniques B-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques O O B-Wavelength I-Wavelength O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O B-ObservationalTechniques O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-ObservationalTechniques O O O O O B-Archive I-Archive I-Archive O B-URL O O O O O O O O B-Software O O B-URL O O B-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-ObservationalTechniques O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O


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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


O O O O O B-Wavelength B-Wavelength I-Wavelength I-Wavelength O B-Wavelength B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O O O O O B-Database O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-TextGarbage I-TextGarbage I-TextGarbage O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength B-TextGarbage O O B-Wavelength I-Wavelength B-TextGarbage O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O O O O O O O O O O O O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O B-CelestialObject O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O B-TextGarbage O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O

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


O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Location O O B-Grant O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant B-Grant B-Grant B-Grant O B-Grant O B-Organization I-Organization I-Organization I-Organization O O B-Grant O O B-Organization I-Organization I-Organization I-Organization O B-Location I-Location O O B-Grant

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).


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

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 .


O O O O O O O O B-Person I-Person O O O O O O O O O O O O O O O O B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person O O O O O O O O O O O O O O B-Software I-Software B-Software B-Software I-Software B-Software I-Software B-Software B-Software B-Software B-Software O O O B-Software O B-Software B-Software B-Software B-Software O B-Person I-Person O B-Organization I-Organization I-Organization I-Organization B-Organization O O B-Fellowship I-Fellowship B-Person I-Person I-Person O O O O O O O B-Location B-Organization I-Organization O B-Organization O O B-Grant

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.


O O O O O B-Organization B-Grant I-Grant O O B-Telescope B-Grant I-Grant I-Grant O O O O B-Telescope B-Grant I-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization B-Grant I-Grant O O O O B-Organization O O O O B-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization B-Grant I-Grant I-Grant B-Person O O O O B-Location I-Location B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person O O O O B-Organization B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Person O O O O O O O O B-Organization O O O B-Organization I-Organization I-Organization B-Person O B-Person O O O O O B-Organization I-Organization I-Organization O B-Grant I-Grant I-Grant B-Person O O O O B-Location I-Location B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O B-Organization I-Organization I-Organization I-Organization B-Grant I-Grant I-Grant B-Person O O O O B-Organization I-Organization I-Organization I-Organization B-Person O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Grant I-Grant I-Grant I-Grant O B-Grant I-Grant I-Grant I-Grant O O B-Organization I-Organization I-Organization B-Person O O O B-Organization O O B-Grant O B-Organization

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.


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

the 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


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

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


O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Location O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O B-Wavelength I-Wavelength O O O B-Survey O O O O O O B-Wavelength O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O B-Survey I-Survey O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey I-Survey I-Survey O O O O O O O O O O O B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey I-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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


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

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


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

(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


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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.


O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization B-Location O O O O O O B-Observatory I-Observatory O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O B-Organization O O O O B-Collaboration I-Collaboration O B-Organization I-Organization I-Organization B-Location O B-Organization I-Organization I-Organization B-Location B-Organization B-Location O B-Organization I-Organization I-Organization B-Location B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Location O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Location O B-Observatory O O O B-Proposal O B-Proposal O O O O O O O B-Database O O O O O B-Organization I-Organization I-Organization B-Organization O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O O O

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.


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

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 =


O O O O O O O O O O O O O O O O B-Person I-Person O B-Person I-Person O O O O O O O O O O B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person B-Person I-Person O B-Person I-Person I-Person O O O O O O B-Wavelength O B-Person O O O O B-Organization O B-Grant O O O B-Person O B-Person O O O O B-Organization O B-Location O B-Grant O B-Grant O O O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O O B-Telescope O O B-Organization O O O O O O O O O B-Organization O O O B-Organization

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.


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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.


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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.


O B-Person I-Person O O O B-Person I-Person O O O B-Organization I-Organization I-Organization O B-Grant B-Person I-Person O B-Person I-Person B-Person I-Person O B-Person I-Person O O O O O O O O B-Survey O O O O O O O O O O O O O O B-Observatory O O O B-Instrument B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Organization I-Organization I-Organization I-Organization O O O B-Survey O O O O O O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Location O B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Location I-Location O B-Organization I-Organization I-Organization I-Organization O B-Location O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O 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B-Organization I-Organization O O O O O O B-Survey I-Survey I-Survey O O O O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization B-Organization I-Organization I-Organization O B-Organization I-Organization O B-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization B-Organization B-Location B-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization I-Organization I-Organization I-Organization O B-Organization 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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.


O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Instrument O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Instrument O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Model B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O

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


O O O O O O O O O O O O O O O O O B-Person O B-Person O O O B-Organization O O B-Grant I-Grant I-Grant B-Person O O O B-Organization B-Grant I-Grant O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope O O O O O O O O B-Organization O B-Grant I-Grant I-Grant I-Grant B-Person O B-Grant B-Person O O O B-Organization I-Organization I-Organization I-Organization O O O O B-Observatory O O O O O O B-Organization O O B-Grant O O O O O O B-Organization B-Telescope I-Telescope I-Telescope O O O O B-Organization I-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization O B-Grant I-Grant O O O O O O B-Grant O O O O B-Grant O O O B-Organization O O O O O B-Organization I-Organization I-Organization I-Organization O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization O B-Grant O O O O O O B-Archive I-Archive I-Archive I-Archive B-Archive O O B-Database I-Database I-Database B-Database O B-Telescope B-Telescope O B-Survey O B-Software B-Software B-Software B-Software B-Software B-Software

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.


O O O O O O O O O O O O O O O O O O O O B-Instrument O B-Mission I-Mission O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument I-Instrument B-Instrument O B-Instrument O O O O O O O O B-Instrument O O B-CelestialObject O O O O O O B-Instrument O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission I-Mission O B-Instrument O O O O O O O O B-Instrument O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Instrument O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O B-Mission I-Mission O B-Mission O O O O O O O O O O O O B-Mission O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Model I-Model I-Model I-Model I-Model O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission I-Mission O O O O O O O O O O O O O O O O O O O O O O B-Mission I-Mission O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Mission I-Mission O O O B-Instrument O O B-Mission I-Mission B-Instrument O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O

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 ≥


O O O O O O O B-Database I-Database O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model B-Model O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Model I-Model I-Model O O O O O O B-Person O O O B-Software I-Software B-Citation I-Citation O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Database I-Database I-Database I-Database I-Database I-Database I-Database I-Database I-Database B-Database I-Database I-Database B-Citation I-Citation O O O O O O O O O O O O O O B-Software I-Software I-Software I-Software I-Software B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Database I-Database O O O O O O B-Software O B-Software B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Database I-Database O O O O O O O O O O O B-Software O B-Citation I-Citation O O O O O O O O O O B-Software B-Citation I-Citation I-Citation O B-Citation O O O O B-Model I-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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


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

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