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The authors thank Luca Ciotti, Pasquale Londrillo, and Carlo Nipoti for sharing their NMODY code. The authors acknowledge Ortwin Gerhard for sharing the code for generating the self-consistent N -body ICs using Lucy’s method. X.W. thanks for support through NSFC grants 11503025, 11421303, Anhui NSF grant 1708085MA20, and the Fundamental Research Funds for the Central Universities. X.W. thanks for support from the Hundred Talents Project of Anhui Province. This project was partially started when X.W. was an Alexander von Humboldt Fellow at the University of Bonn.


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

gravity brightening exponent for the radiative, primary component, determined through our models, is ∼1 (von Zeipel 1924 ). For the secondary component, the posterior distribution is a slightly skewed normal distribution. By considering the peak of the skewed distribution (not the Gaussian fit to the posterior distribution), we find the gravity brightening exponent to be β = 0.59 ± 0.08. While this is not in agreement with the value of β = 0.32, suggested by Lucy ( 1967 ), we find our value in close agreement (∼2σ) with that of Claret Bloemen ( 2011 ), who computed β = 0.48 for comparable models. This value was computed using ATLAS models (Castelli Kurucz 2004 ) and the Kepler bandpass, and taking into account local gravity and convection. It is also likely that the tidal distortion affects the gravity darkening exponent, as discussed by Espinosa Lara Rieutord ( 2012 ). In our binary models, we fixed the primary and secondary rotation rates to those obtained through spectral fitting, v sin i 1 = 29(2) km s −1 and v sin i 2 = 9(2) km s −1 . Combining these values with our model-determined orbital inclination (assuming that the orbital and stellar rotational axes are aligned) and radii, for the model with pulsations, we obtained F 1 = 7.1(2) and F 2 = 4.0(1), where F is the ratio of the stellar rotational to orbital period and the subscript denotes the primary and secondary components, respectively. For both models, the one with, and the one without pulsations, we obtained the same values with slightly different uncertainties (see Table 4 ). Interestingly, under the assumption that the stellar rotational and orbital axes are aligned, the primary component is rotating slightly faster than the predicted pseudo-synchronous rotation rate (Hut 1981 ), the stellar rotation synchronous with the orbital velocity at periastron, which was calculated to be F = 5.3(2) for both components; while the secondary component is rotating slightly slower. However, if we relax the assumption that the orbital axes are aligned, it is possible to find configurations where both stars are rotating pseudo-synchronously. 5 SUMMARY AND CONCLUSIONS We have modelled the heartbeat star binary system KIC 3749404 by combining a comprehensive assortment of tools including the binary modelling software, phoebe ; emcee , a python implementation of the affine invariant mcmc techniques; and our own software to fit pulsations and Doppler boosting. With these tools, we were able to obtain two successful fits to the heartbeat star light and radial velocity curves simultaneously: one without pulsations and one with all the tidally induced pulsations greater than 20 μmag (seven pulsations). From these fits, we obtained the fundamental parameters of the binary system (see Table 3 ). The difference between the results of these two models hinges on the different inclination values: 602(1) without pulsations and 62(1)° with pulsations. This 2σ difference subsequently produced a larger


O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope I-Telescope I-Telescope O B-Person I-Person O O O O O O O O O O O B-Location 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 B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Location B-Organization I-Organization I-Organization O O O B-Grant O O O B-Organization I-Organization I-Organization I-Organization O O O O B-Person O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Grant I-Grant I-Grant B-Person O O O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O B-Location O O O B-Organization B-Organization O B-Organization B-Person O O O B-Organization I-Organization O O B-Location O O O O O B-Organization O O B-Grant I-Grant O O O B-Grant B-Person O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Grant O O B-Organization I-Organization I-Organization O O O O O O O B-Grant I-Grant B-Grant O O B-Location O O O O B-Organization I-Organization I-Organization B-Organization O O B-Location O O O O B-Organization O B-Grant I-Grant I-Grant I-Grant O O B-Location O B-Organization I-Organization B-Grant I-Grant O B-Organization O O B-Grant B-Person O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Grant B-Organization B-Location O B-Organization B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O O O O O O O O O B-Observatory O O O B-Observatory I-Observatory I-Observatory I-Observatory B-Location O O B-Proposal O O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Location B-Organization I-Organization I-Organization O O O B-Grant

We thank the technical staff of the University of Hawaii 2.2 m telescope, and Dan Birchall for his observing assistance. We recognize the significant cultural role of Maunakea within the indigenous Hawaiian community, and we appreciate the opportunity to conduct observations from this revered site. This work was supported in part by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. We thank the Gordon Betty Moore Foundation for their continuing support. X.H. acknowledges the University of San Francisco (USF) Faculty Development Fund. Z.R. was supported in part by a USF Summer Undergraduate Research Fellowship. Support in France was provided by CNRS/IN2P3, CNRS/INSU, and PNC; L.P.N.H.E. acknowledges support from LABEX ILP, supported by French state funds managed by the ANR within the Investissements d’Avenir programme under reference ANR-11-IDEX-0004-02. N.C. is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR). Support in Germany was provided by the DFG through TRR33 “The Dark Universe;” and in China from Tsinghua University 985 grant and NSFC grant No 11173017. N.S. acknowledges the support from World Premier International Research Center Initiative (WPI), MEXT, Japan, and JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers. This work is partially based on observations made with ESO Telescopes at the La Silla Paranal Observatory, Chile, under program 289.D-5035. Some results were obtained using resources and support from the National Energy Research Scientific Computing Center, supported by the Director, Office of Science, Office of Advanced Scientific Computing Research of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.


O O O O O O 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 B-Telescope O O O O B-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 B-Telescope B-Wavelength O O O B-CelestialRegion I-CelestialRegion I-CelestialRegion O O O B-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 B-Telescope O O O O O B-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 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 B-Model O O O O O O B-Model 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 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 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 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 B-Model O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Model 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

this case, the inpainting performances with DP and GAN can be affected by noise, and the reconstruction area can be visually distinguished in the square patch. Figure 12. Thumbnail images of Planck temperature maps at (a) 353 and (b) 857 GHz. The radius of the reconstructed area (black circle) is 15′. Two locations are chosen to highlight different inpainting performances with high and low SNRs of Planck 353 map, respectively, at high Galactic latitude, i.e., and at low Galactic latitude, i.e., . For a more quantitative assessment, we thus estimate the summary statistics shown in Section 4.1 and we show the results in Figure 13 . Figure 13. Summary statistics estimated on Planck dust T maps at (a) 353 and (b) 857 GHz. A clear indication that the thermal dust emission data is contaminated by instrumental noise can be inferred by comparing the shapes of Minkowski functionals estimated for the maps at 353 GHz, with the ones from signal-only simulations (Figure 4 ) or from signal-dominated maps (Figure 13 (b)). The morphology of the Minkowski functionals of the former largely resembles the functionals estimated from a Gaussian signal. Two possible candidates for the Gaussian component are (i) Planck instrumental noise, which can be approximated as white within patches, and (ii) CMB residual emission. DP inpaintings are the most affected ones in the presence of the noise at , e.g., notice how the pixel distribution (top left panel) of Figure 13 (a) noticeably departs from the ground-truth one. However, the KS test p -value performed on the DP and ground-truth samples is , i.e., not significantly low enough to reject the null hypothesis that the two samples are different. On the other hand, inpainting with GAN and NN does not show any dependence with SNR, as the performances with these methodologies are essentially similar to the ones observed with the signal-only simulated maps (e.g., Figures 7 , 8 , and 10 ). Finally, we would like to point out that for the case of inpainting with GAN, we use the weights derived from the training set composed of signal-only dust TQU simulated maps. Looking at Figure 13 we notice that GAN is able to statistically reproduce at 353 GHz the features composed by signal and noise, indicating that the network has correctly learned the features related to the intrinsic signal in the presence of noise, and injects signal plus noise features statistically coherent with the ones outside the masked area. However, when the noise is highly dominating in the patch, we can clearly distinguish smooth artifacts in some cases inpainted with GAN (see Figure 12 (a, bottom)). This is somewhat expected because GAN is trained on signal-only simulated images. Further investigations on training GAN with noisy data are needed, and we will address it in a future work. 4.3. Inpainting Maps with Point Sources In this section, we aim at showing real world applications of the three reconstruction methodologies by inpainting areas in real maps


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We thank the anonymous referee for constructive suggestions. We are grateful to Maryam Modjaz and Yuqian Liu for sending us the photospheric velocity data before the publication of their paper. This work is supported by the National Program on Key Research and Development Project of China (Grant No. 2016YFA0400801), the National Basic Research Program of China (“973” Program, Grant No. 2014CB845800), and the National Natural Science Foundation of China (grant Nos. U1331202, 11533033, U1331101, 11673006, 11573014, 11422325, and 11373022). D.X. acknowledges the support of the One-Hundred-Talent Program from the National Astronomical Observatories, Chinese Academy of Sciences), and by the Strategic Priority Research Program “Multi-wavelength Gravitational Wave Universe” of the CAS (No. XDB23040100).


O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O B-ObservationalTechniques I-ObservationalTechniques B-ObservationalTechniques O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Survey I-Survey B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Survey O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Telescope I-Telescope I-Telescope O B-Telescope B-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Telescope I-Telescope B-Telescope B-Wavelength I-Wavelength O O O O O O O O B-Survey 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 B-CelestialObject I-CelestialObject O B-Wavelength I-Wavelength O O O B-Wavelength 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 B-Survey I-Survey I-Survey I-Survey B-Survey B-Citation I-Citation I-Citation I-Citation O B-Dataset I-Dataset I-Dataset B-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-ObservationalTechniques O O O O O O O O O O O O O O B-Wavelength O O O O O B-Instrument I-Instrument O O B-Observatory I-Observatory 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 B-Wavelength I-Wavelength O O O O O O O O O O O O O O O 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Survey I-Survey 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 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

at radial gradients in stellar populations have worked within R e (e.g., Spinrad Taylor 1971 ; Mehlert et al. 2003 ; Annibali et al. 2007 ; Spolaor et al. 2010 ; Jimmy et al. 2013 ; McDermid et al. 2015 ; Goddard et al. 2017 ). In recent years, integral-field spectroscopy (IFS) has enabled stellar population measurements at a large radius (e.g., Greene et al. 2013 , 2015 ; Scott et al. 2013 ; McDermid et al. 2015 ; Boardman et al. 2017 ; Goddard et al. 2017 ; Barone et al. 2018 ; van de Sande et al. 2018 ). Here, we focus on the most massive galaxies in the present-day universe using the MASSIVE survey (Ma et al. 2014 ). We have gathered integral-field data for 90 MASSIVE galaxies and measured their stellar kinematics, including higher-order moments (Veale et al. 2017a , 2017b ), stellar velocity dispersion profiles (Veale et al. 2018 ), and kinemetry (Ene et al. 2018 ). In addition, we have Hubble Space Telescope ( HST )/Wide Field Camera 3 (WFC3) imaging for 30 of the galaxies (Goullaud et al. 2018 ) and uniform Canada–France–Hawai’i Telescope (CFHT) K -band imaging for nearly all objects. 2. Galaxy Sample MASSIVE is a volume-limited survey of 116 galaxies (see details in Veale et al. 2017b ) in the northern hemisphere within D 108 Mpc (i.e., to the distance of the Coma cluster), with K -band magnitudes of M K −25.3 mag (roughly > 10 11.5 M ⊙ ). Details of the target selection are described in Ma et al. ( 2014 ). Briefly, our targets are drawn from the Two Micron All-sky Survey (2MASS; Skrutskie et al. 2006 ) Extended Source Catalog (XSC; Jarrett et al. 2003 ) combined with distances from the high-density contrast (HDC) group catalog from Crook et al. ( 2007 ), supplemented by surface brightness fluctuations when available (Blakeslee et al. 2009 , 2010 ; Blakeslee 2013 ) and with the flow model of Mould et al. ( 2000 ) when needed. We have completed IFS observations for 90 galaxies, including complete coverage of the 75 galaxies that have M K −25.5 mag. We use the Mitchell IFS at the McDonald Observatory (Hill et al. 2008 ), which has a 107 × 107 arcsec 2 field of view, one-third of which is filled by 246 fibers of a 4′′ diameter. Each galaxy is observed at three dither positions with 20-minute exposures, interspersed with 10-minute sky observations. The spectra span 3650–5850 Å, with a spectral resolution of 4.5 Å FWHM. The data are reduced using the custom software Vaccine (Adams et al. 2011 ; Murphy et al. 2011 ). For more details, see Greene et al. ( 2015 ). A primary goal of the MASSIVE survey is to measure spatially resolved stellar kinematics for dynamical modeling (e.g., Thomas et al. 2016 ). To achieve a mean signal-to-noise ratio (S/N) of at least 20 pixel −1 , we group individual fibers into spatial bins and co-add the spectra from fibers in a given bin into a single spectrum. The binning scheme is described in detail in Veale et al. ( 2017a ), but we summarize the procedure here briefly for completeness. Central fibers with S/Ns >


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We thank Gary Ferland for helpful suggestions that led to improvement of this paper. We are grateful to Kanti Aggarwal, Guiyun Liang, Brendan McLaughlin, Swaraj Tayal, and Oleg Zatsarinny for providing inelastic cross-section data. This work made use of the references provided in the CHIANTI atomic database (Dere et al. 1997 ; Del Zanna et al. 2015 ). This research was supported in part by NSF grant AST-1408723, and in part by NSF Research Fellowship DGE 1656466 to C.D.K.


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CDF acknowledges support from the National Science Foundation collaborative grant ‘Collaborative Research: Accurate cosmology with strong gravitational lens time delays’ (AST-1312329). LVEK is supported in part through an NWO-VICI career grant (project number 639.043.308).


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

for a different simulation in Schwadron et al. 2014 ). Other intended applications have been mentioned throughout the text. We encourage interested researchers to contact us if they would like to use our simulation data for complementary investigations. Appendix A The MAS Thermodynamic MHD Model The numerical code MAS employed in this article integrates the standard viscous and resistive one-fluid MHD equations in 3D spherical coordinates. For the coronal simulation described in Sections 3.1 – 3.3 and 4 , the so-called “thermodynamic MHD model” was used, in which the standard equations are extended to include parallel electron thermal conduction, radiative losses, and parameterized coronal heating. The MAS thermodynamic MHD model has been used extensively for simulating the global corona and solar wind (e.g., Mikić et al. 1999 , 2007 ; Lionello et al. 2001 , 2009 ; Downs et al. 2013 ; Linker et al. 2017 ; Titov et al. 2017 ) and eruptive phenomena such as soft X-ray jets (Lionello et al. 2016 ; Török et al. 2016 ) and CMEs (e.g., Linker et al. 2001 , 2003 ; Mikić et al. 2013b ). In this article, we use a version of the model in which the solar wind is accelerated with Alfvén waves using a Wentzel–Kramers–Brillouin (WKB) approximation (Jacques 1977 ). A more sophisticated wave-acceleration model is under development (Lionello et al. 2014 ; Downs et al. 2016 ). In the version used here, the governing equations take the following form: where A is the magnetic vector potential, is the magnetic field, is the current density, ρ is the plasma density, T is the temperature, is the plasma pressure, v is the plasma velocity, is the direction of the magnetic field, c is the speed of light, γ = 5/3 is the adiabatic index, m p is the proton mass, k is Boltzmann’s constant, κ 0 is the coefficient of the classical Spitzer thermal conductivity, is a profile that limits the radial extent within which collisional (Spitzer’s law) thermal conduction is active, is the equivalent profile for collisionless thermal conduction, Q ( T ) is the radiative loss function, H is the coronal heating term that typically consists of a sum of empirical heating functions (Lionello et al. 2009 ), is the Alfvén-wave speed, and is the gravitational force. Note that joule heating, , is not included in Equation ( 3 ). While it is implemented in MAS, we switch it off in our simulation, otherwise the insertion of the current-carrying flux rope into the background corona (Section 3.2 ) would lead to an instantaneous, unphysical temperature increase of the rope. The last expression in Equation ( 4 ) is a semi-implicit term that is added to the equations to stabilize the algorithm for time steps larger than the fast magnetosonic wave limit (Lionello et al. 1999 ; Caplan et al. 2017 ). Equations ( 5 ) and ( 6 ) are the WKB approximation for Alfvén-wave pressure advance (Mikić et al. 1999 ), where + and − are the forward and backward Alfvén-wave energy densities. The electron density


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

of three stellar half-mass radii at infall on the individual orbits compared to the isolated run as a function of the central dark matter density slope at infall infall . The different initial dark matter density slope also affects the stellar mass-loss, since the dark matter acts as shield for the stars. In Fig. 16 , we plot the mass-loss as defined in Fig. 15 as function of the initial dark matter profile density slope but this time for stars and dark matter. In order to avoid a too crowded plot, we only show results for orbitIV , which is the most disruptive orbit for which all five satellites still have a well-defined centre. Figure 16. Dark matter (dots) and stellar (stars) mass enclosed in a constant sphere with a radius of three stellar half-mass radii at infall on orbitIV compared to the isolated run as a function of the central dark matter density slope at infall infall . In general stars are more resilient to tidal forces than dark matter, and this is due to their smaller spatial extent and larger stellar density; for example satIII is able to retain 40percent of its stars while it is practically totally stripped of dark matter. Nevertheless there is still a correlation between the stellar mass-loss and the initial dark matter slope. An exception to this relation seems to be given by satI , which has a very strong mass-loss both in the stellar and dark matter components despite having a similar infall as satIII . This is due to the different slope for the stellar density profiles between satI and satIII , while the first satellite has a slope of 1.80 (again evaluated between 1 and 2percent of the virial radius), the stellar slope for satIII (and all the other satellites) it is close to 3.0. This difference in stellar slope is most likely related to a major merger event that occurred for satI shortly before redshift one and strongly reshuffled the stellar particle orbits. Finally, we note that some satellites (especially satI ) are left at z 0 with practically no dark matter in their central region, where they are fully stellar dominated, resembling more an extended globular cluster than a dwarf galaxy, we plan to look more into this issue in a forthcoming paper. 4 DISCUSSION AND CONCLUSIONS This work is the second of a series of papers in which we are trying to understand the formation and evolution of the smallest galaxies in the universe. In the first paper ( Paper I ), we presented a series of 27 cosmological simulations of haloes in the mass range 510 8 to 10 10 , these simulations were run till z 1 and aimed to describe the properties of satellite galaxies before accretion. In this second paper, we used a subsample of these cosmological simulations as initial conditions for a series of binary hydrodynamical simulations (satellite host) with the goal of understanding the effects of accretion and environment on satellite galaxies. More specifically, we used a total of seven haloes (five luminous and two dark) with a virial mass


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ACKNOWLEDGEMENTS We would like to thank Sarah Bridle and Nicolas Tessore for bringing this problem to our attention. We also thank them for their helpful comments which greatly contributed to this paper. We would furthermore like to express our thanks to the referee, whose comments were very constructive and helped to improve this paper.


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This work was aided greatly by discussions with L. Coogan, N. Cowan, C. Goldblatt, A. Lenardic, L. Schaefer, N. Sleep, and K. Zahnle. We thank the anonymous referee for helpful comments that greatly improved the manuscript. We thank the Kavli Summer Program in Astrophysics for the setting to perform this research and the hospitality of the program members and community at the University of California, Santa Cruz. T.D.K. acknowledges support from NASA headquarters under the NASA Earth and Space Science Fellowship Program Grant PLANET14F-0038. D.S.A. acknowledges support from the NASA Astrobiology Institute Virtual Planetary Laboratory, which is supported by NASA under cooperative agreement NNH05ZDA001C.


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

we consider, the subhalo and two-halo term contributions are quite negligible. This is also the reason why we cannot place tight constraints on the subhalo fraction and subhalo mass. The contribution of the stellar mass of the BCG is only important at very small scales. Thus, the observed ESDs in this study can mainly provide us with the constraints on the host halo properties. Figure 6. The ESD measurements (shaded area) and the best-fit model predictions (black solid line) for groups in the four different mass bins. The dashed, dotted, dotted–dashed, and blue dashed lines are the contributions of an off-centered NFW profile, subhalo, stellar mass of the galaxy under consideration, and the two-halo term, respectively. Table 3 lists the best-fit parameters for groups in different mass bins. As we have seen from the likelihood distribution of parameters in Figure 5 , we can have a fairly good constraint on the halo mass of the group samples. However, if we compare the halo masses obtained from the ESDs with those obtained from the group catalog (c.f. Table 1 ), they are roughly underestimated by 0.1 to ∼0.2 dex. We will discuss this discrepancy in the following section. The overall off-center distances for our BCGs are quite small, assuring that BCGs are indeed good tracers of halo centers. On the other hand, the concentrations of the halos seem to be somewhat larger than the theoretical predictions (e.g., Zhao et al. 2009 ). However, since the concentration c and the off-center distance R sig are quite tightly correlated, if we adopt the lower value of R sig , the concentration c will drop significantly as well. Table 3 Posteriors of the Parameters We Fitted to Our Group Samples Listed in Table 4 are the best-fit parameters for the groups that are separated into the high and low X-ray luminosity subsamples. Although due to the smaller number of lens systems the error for each data point for our X-ray subsamples is somewhat larger and thus the constraints on the five parameters are somewhat weaker, we do see the prominent feature of the halo masses of the high X-ray luminosity subsamples being higher than their low luminosity counterparts in all mass bins. The difference is to 0.2 to ∼0.3 dex, which means that the high X-ray luminosity subsamples are nearly a factor of 2 more massive than their low X-ray luminosity counterparts. Table 4 Posteriors of the Parameters We Fitted to Our High and Low X-Ray Luminosity Subsamples 4. Eddington Bias of the Halo Mass Estimation Recent studies have shown that the combination of galaxy–galaxy lensing and the clustering of galaxies can be used to constrain cosmology (e.g., Cacciato et al. 2013 ; More et al. 2013 ; van den Bosch et al. 2013 ; Leauthaud et al. 2017 ), and that the halo masses estimated for groups using galaxy–galaxy lensing signals and hence the halo mass function can also be used to constrain the cosmological parameters. However, before doing so, one needs to make a careful study of the


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We thank the anonymous referee for constructive comments and suggestions. We thank Dr. Hu Zou (NAOC) for his help on optical photometry. This work is supported by the National Key RD Program of China (No. 2017YFA0402703). We thank the staff of the Nordic ALMA Regional Center (ARC) node for their support and helpful discussions. The Nordic ARC node is based at Onsala Space Observatory and funded through Swedish Research Council grant No. 2017-00648. L.F. acknowledges the support from the National Natural Science Foundation of China (NSFC, grant Nos. 11822303, 11773020, and 11433005) and Shandong Provincial Natural Science Foundation, China (ZR2017QA001, JQ201801). K.K. acknowledges support from the Knut and Alice Wallenberg Foundation and the Swedish Research Council. Y.H. acknowledges the support from NSFC (grant No. 11773063) and Natural Science Foundation of Yunnan Province (grant No. 2017FB007). Q.-H.T. acknowledges the support from the NSFC (grant No. 11803090). This paper makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.00441.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. This paper makes use of data products from the Wide-field Infrared Survey Explorer , which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This paper used public archival data from the Dark Energy Survey (DES). 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 OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas AM University. 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. Facilities: ALMA , Herschel (PACS , SPIRE) , WISE , CTIO (DECam).


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The authors benefited greatly from participation in the International Team 276 funded by the International Space Science Institute (ISSI) in Bern, Switzerland. E.D. was supported by the Grant Agency of the Czech Republic under Grant No. P209/12/1652.


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continuum of depths and durations from the many different scales of planetesimal impacts that would occur. The clustering of dips at D1500 also points to these events being correlated which is hard to reconcile with this scenario, though the planetesimals in the belts could be shepherded by planets into confined azimuthal regions (e.g. Wyatt 2003 ; Nesvorný, Vokrouhlický Morbidelli 2013 ). 5.2 Aftermath of giant impact in planetary system A possible way around the issues in Section 5.1 is to invoke dust thrown off in a single collision, perhaps analogous to the Earth–Moon system forming event (Jackson Wyatt 2012 ). In this case there need not be an underlying asteroid belt, as the collision could be between planets whose orbits recently became unstable, or between growing planetary embryos. Such events are expected to result in strong IR excesses (e.g. Jackson Wyatt 2012 ; Genda, Kobayashi Kokubo 2015 ), and are indeed seen in systems such as HD 172555 where giant impacts are the favoured explanation (Lisse et al. 2009 ). In this scenario, the putative collision would need to have occurred between the WISE observation taken in Kepler Q5 and the first large dip at D800. The dip at D1500 is then interpreted as the same material seen one orbit later, with the ∼750 d period implying an orbit at ∼1.6 au. The difference in the dip structure from D800 to D1500 could arise because the clump(s) created in the original impact are expanding and shearing out. This scenario therefore predicts that KIC 8462852 may now have a large mid-IR excess, but the most recent IR observations taken in 2015 January with Spitzer IRAC show no significant excess for KIC 8462852 (Marengo et al. 2015 ). However, non-detection of an excess would not necessarily rule this scenario out, as the dust levels derived in Section 4 (which account for the dust seen passing in front of the star) were shown to be consistent with a non-detection. A more robust prediction is that future dimming events should occur roughly every 750 d, with one in 2015 April and another in 2017 May. Two new issues arise with this scenario however. First, if the period of the orbiting material is a few years, what is the origin of the two small 0.5 per cent dips seen in the first few hundred days (D140 and D260; Table 1 ), and why did they not repeat 750 d later? It is a concern that these could require the existence of an outer planetesimal belt, which may contradict the lack of infrared emission to this star. Perhaps more problematic is the probability that this star (of unknown age) should suffer such an event that occurs within a few-year window between the WISE observation and the end of the prime Kepler mission, and that the geometry of the system is such that material orbiting at ∼1.6 au lies almost exactly between us and the star. Taking this few year window, the main-sequence lifetime, and an optimistic estimate for the scale height of giant impact debris, and the number of Kepler stars observed,


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

[α/Fe] = +0.4 Dartmouth isochrone. The dashed outline is the final adopted selection window as discussed in Section 2.2 . The agreement between the observations and the theoretical isochrone is good: for 0.6 ≤ ( g − i ) 0 ≤ 0.8 where the isochrone is approximately constant at m i = 0.029 mag, the mean of the globular cluster stars is m i = 0.033 with a standard deviation σ = 0.040 mag. The observed scatter in the globular cluster m i values is large but there is no obvious indication of a systematic offset between the observations and the theoretical isochrone. The scatter is presumably driven by uncertainties in the v magnitudes as the corresponding i 0 , ( g − i ) 0 combined colour–magnitude diagram (generated using an appropriate relative distance modulus offset) shows only a small scatter in ( g − i ) 0 : σ( g − i ) 0 = 0.021 mag about the mean RGB locus. 2.1.2 Known EMP stars An alternative approach is to investigate the location of known EMP stars in the metallicity-sensitive diagram. To perform this task we have compiled from the literature a list of stars with [Fe/H] ≤ −2.5, as determined from high-dispersion spectroscopy. The list was then cross-matched with the SkyMapper DR1.1 data base. In carrying out the match-up we have used tighter limits on the SkyMapper photometry in order to reduce the effects of photometric uncertainty on the location of the stars in the metallicity-sensitive diagram. Specifically, we required e_g_psf AND e_i_psf ≤ 0.02, e_v_psf ≤ 0.03, and E( B − V ) SFD 0.1 mag, and this resulted in a sample of 294 stars. The majority of the stars are drawn from Yong et al. ( 2013 ), Jacobson et al. ( 2015 ), Barklem et al. ( 2005 ), Roederer et al. ( 2014 ), and Cohen et al. ( 2013 ); the effective temperatures, gravities, and metallicities from the latter two sources have been adjusted to ensure consistency with the other sources. The location of the known abundance stars in the metallicity-sensitive diagram is shown in Fig. 3 . We note that the most iron-poor star known, star SMSS J031300.36−670839.3, is the dark coloured five-point star symbol at (( g − i ) 0 , m i ) = (0.62, −0.09) in the figure, while the star HE 0107−5240, which has [Fe/H] ≈ −5.4, is the dark coloured five-point star symbol at (0.59, −0.135). The subgiant star HE 1327−2326 ([Fe/H] ≈ −5.7) is located at (0.29, 0.19), and is also plotted as a dark coloured five-point star symbol in Fig. 3 . In general, the majority of the stars plotted are consistent with expectations in that they lie at notably bluer m i values at a given ( g − i ) 0 than the bulk of the population. However, this is clearly not the case for a small number of known EMP stars that fall well below their ‘expected’ location in the metallicity-sensitive diagram. As will be discussed in more detail in Section 4.1 , generally these stars are strongly enhanced in carbon. Figure 3. SkyMapper metallicity-sensitive diagram for all objects satisfying the basic selection criteria. Overplotted


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

Additional details of our numerical set-up are described in Appendix A . Mass-loss is a key ingredient in novae, which plays an important role in setting the time-scale of the eruption by removing mass from the white dwarf envelope (Livio et al. 1990 ; Kato Hachisu 1994 ). However, the exact mechanisms giving rise to the unbound outflows, as needed to produce variable internal shocks, are uncertain and not addressed in this work. Relatively slow outflows, of up to several hundreds of km s −1 , may be launched from near the binary L 2 Lagrange point. These could be driven for instance by frictional drag (e.g. Livio et al. 1990 ; Shankar, Livio Truran 1991 ; Lloyd, O’Brien Bode 1997 ; however, see Kato Hachisu 1991 ), non-axisymmetric gravitational torques from the binary (Pejcha, Metzger Tomida 2016a ), or in the outflow of a circumbinary disc formed from matter ejected from L 2 that initially remains gravitationally bound (Pejcha, Metzger Tomida 2016b ). These slow outflows, with velocities of order the binary escape speed, would plausibly be concentrated in the binary equatorial plane. Outflows can also occur from the white dwarf directly, with little or no influence from the binary. These could be driven by radiation pressure acting on the iron opacity bump (e.g. Kato Hachisu 1994 ), or by any large (near- or super-Eddington) localized energy deposition in the white dwarf envelope, e.g. from radioactive heating or damping of convectively excited waves (e.g. Quataert et al. 2016 ). Outflows launched from deeper locations in the potential well of the white dwarf would be expected to possess higher velocities ∼2000–5000 km s −1 and a more spherical geometry than the slower outflows from L 2 . Lacking a first principles model, we are motivated on phenomenological grounds to consider two outflow modes characterized by distinct mass-loss rates, , and asymptotic velocities, v w . Somewhat arbitrarily (as the precise values will vary between novae), we assume a ‘fast’ mode outflow with and a ‘slow’ outflow mode with These are spherically equivalent mass-loss rates. To obtain the true value, must be multiplied by the fraction of the total solid angle covered by each outflow component, f Ω . For instance, f Ω ≈ 1 for the fast spherical outflow, while f Ω ≈ 0.3 for the slow equatorially focused outflow. 1 The top panel of Fig. 2 shows the chosen time evolution of and v w for our two fiducial models (‘single transition’ and ‘multiple transition’). In the single-transition model (red lines in Fig. 2 ), the outflow velocity is increased from slow ( v w = v s ) to fast ( v w = v f ) only once at t ≈ 0, while in the multiple-transition model (blue lines in Fig. 2 ) the outflow velocity fluctuates between slow and fast modes several times. Local maxima (‘flares’) are observed in classical nova light curves on time-scales ranging from hours to weeks (e.g. Strope et al. 2010 ; Walter et al. 2012 ; Henze et al. 2018 ; Aydi et al. 2019 ). To generate


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

of freedom is 1. The errors on the elements of the Hessian matrix are propagated to . Since Eq. ( 9 ) is non-linear, we compute the error map on from Gaussian realizations of the errors on the coefficients of the matrix. The uncertainty in , computed within bins of values of the map, is , independent of the intensity at 353 GHz (Fig. A.1 ). Fig. A.1 Two independent estimates of the uncertainties in the determination of angles from the Hessian analysis, computed within bins of values of the map. The red squares represent the noise obtained by propagation of the errors on the coefficients of the Hessian matrix. The noise computed with the toy model (Sect. A.2 ) is plotted with black squares. Fig. A.2 Distribution functions of orientation angles for the selected ridges derived from the Hessian analysis on . They are computed within three bins of Galactic latitudes with equal numbers of data points. The mean absolute latitudes are , , and for the DFs plotted in black, orange and red, respectively. The image shows that the method does not introduce any bias as a function of Galactic latitude in estimating . Appendix A.2: Test on a toy model of the sky We have tested the Hessian algorithm on a toy model of the full sky built from a realization of a Gaussian map, , with an angular power spectrum, equal to a power law of index , computed with the procedure SYNFAST of HEALPix . The spectral index used is within the range of values found for power spectra of dust maps ( Miville-Deschênes et al. 2007 ). The mean value of is 0. We form, where , , , and are four factors chosen to fit the latitude profile, and the longitude profile at Galactic latitude , measured on the map. The values of these factors, including their sign, depend on the realization of . The toy model matches the large-scale structure of the Galactic dust emission, but it does not have its filamentary structure because it is computed from an isotropic Gaussian map. It assumes that the amplitude of the brightness fluctuations at a given scale is proportional to the brightness. This is in agreement with what has been reported for the emission at 100 m of the diffuse ISM ( Miville-Deschênes et al. 2007 ). We use the toy model to quantify the impact of the structure of the diffuse Galactic emission and data noise on the Hessian analysis. We run the Hessian analysis on the map with and without data noise computed as a Gaussian realization of the noise map on . The DFs of orientation angles, computed for the model with added noise, are flat at all Galactic latitudes, as shown in Fig. A.2 . We find no bias on the angle introduced by the large-scale gradient of the emission with Galactic latitude. We use the difference between the two angle maps computed on the map, with and without noise, to estimate the contribution of the data noise to the uncertainty in the Hessian angle (black dots in Fig. A.1 ). This uncertainty decreases for increasing values of (i.e. increasing signal-to-noise ratio).


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-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 B-Wavelength I-Wavelength O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-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-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O 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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Telescope B-Wavelength O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O 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

the cascade emission is shown. The magnetic field in the comoving frame is then . Basically, a higher B leads to a more important synchrotron radiation, while a smaller B leads to a more important IC radiation. Since electrons usually emit higher-energy photons via IC radiation than via synchrotron radiation, the flux of the cascade emission at high energies becomes higher for a smaller B . On the other hand, more high-energy photons in turn produce more e ± pairs at high energies, making a harder electron spectrum than that in the case of a large B , as is shown in Figure 7 . This explains why the spectrum is also harder in the case of B = 0.01 than in the case of B = 1 at keV energies, where the emission is dominated by synchrotron radiation of these (relatively) low-energy electrons in both cases. One can see that the gamma-ray flux around 100 MeV is not very sensitive to the value of B . This is because the injection rate of the secondary electrons that emit 100 MeV photons is dominated by the γγ -annihilation process (i.e., in the right-hand side of Equation ( 3 )), mainly relies on the photon field, and does not change with the magnetic field. On the other hand, the jet is already an electron calorimeter even if only considering IC radiation under the benchmark parameters. Therefore, we expect the total photon production rate at a certain energy is roughly equal to the energy injection rate of the electrons that emit these photons (i.e., L γ ∼ L e ). A large B only changes the way the electrons radiate their energy away but does not increase the photon production rate. Although the energy of the gamma-ray-emitting electrons is different in the IC-dominated case with a small B from that in a synchrotron-dominated case with a large B , the resultant gamma-ray flux is at the the same level, since the energy injection rate of electrons in the relevant energy range is roughly constant with respect to electron energy (i.e., ). Figure 6. Same as Figure 3 , but for the equipartition coefficient for the magnetic field in the GRB jet B . The dashed and dotted lines correspond to the IC and synchrotron radiation, respectively. Figure 7. Quasi-steady-state spectrum of the cascaded electrons in the comoving frame of the GRB jet with fiducial parameters (except for the equipartition coefficient of the magnetic field). Purple, blue, and red curves are for B = 1, 0.1, and 0.01, respectively, while solid and dashed curves are the contribution by the BH and photomeson processes, respectively. 3. Application to LAT GRBs In this section, we apply our calculations to explain the high-energy spectrum of GRB 090926A, GRB 090902B, and GRB 080916C, which represent three different types of spectral feature. The origin of prompt keV/MeV emission is beyond the scope of this work, and we just regard it as a preexisting target photon field for the BH and photomeson processes. In our calculations below, we employ a flat spectrum for the injected proton spectrum ( s =


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a revised model for a He shell detonation Type Ia SN. In any case, we think it is likely that the Mn/Fe, Ni/Fe, Cu/Fe, and Zn/Fe enhancements in G 112-43/44 and the three α -poor stars are caused by element contribution from a special kind of SN to pockets of gas from which these stars were formed. Such inhomogeneous mixing of SN products is more likely to affect the chemical evolution of dwarf galaxies than higher-mass systems because in lower-mass galaxies the gas has a relatively long cooling time, resulting in episodic star-formation bursts ( Revaz et al. 2009 ; Venn et al. 2012 ). In view of this, it is interesting that G 112-43/44, as a member of the Helmi streams, was probably born in a dwarf galaxy and that one of the three α -poor halo stars with enhancements of Mn/Fe, Ni/Fe, and Zn/Fe also has Helmi stream kinematics 6 , as seen from Fig. 1 . G 4-36 and BPS BS 16920-17 have kinematics similar to stars belonging to the Gaia -Enceladus population, but BPS CS 22966-43 lies in the Helmi box with positive V Z . Not all stars belonging to the Helmi streams have unusual abundances. Roederer et al. (2010) made a detailed abundance analysis of 12 subgiant and giant stars with Helmi stream kinematics, and Gull et al. (2021) recently added seven stars to this work. Within the typical errors of the abundance determinations (0.10–0.15 dex), [Mn/Fe], [Ni/Fe], [Cu/Fe], and [Zn/Fe] agree with the [X/Fe]-[Fe/H] trends of Galactic halo stars. The stars investigated, however, have [Fe/H] ≲ −1.5 and high [ α /Fe] ratios (e.g., [Mg/Fe] ∼ 0.3), so it remains to be seen if the Helmi streams contain a population of low- α stars at higher metallicities and if some of these stars have over-abundances of Mn, Ni, Cu, and Zn relative to Fe. Low- α stars are well known in dwarf galaxies (see the review by Tolstoy et al. 2009 ). In the Sculptor dSph galaxy, for example, [Mg/Fe] shows a declining metallicity trend from [Mg/Fe] ≃ 0.4 at [Fe/H] = −1.8 to [Mg/Fe] ≃ −0.2 at [Fe/H] = −1.0. Interestingly, Skúladóttir et al. (2017) find indications of a cosmic scatter in [Zn/Fe] of the Sculptor stars (i.e., from about −0.6 dex to +0.4 dex) and a positive correlation between [Ni/Fe] and [Zn/Fe]. At a distance of ∼85 kpc, the giants observed in Sculptor are, however, faint (17.0 V 18.5), and the errors of [Zn/Fe] are of the order of ±0.3 dex. It will require more precise [Ni/Fe] and [Zn/Fe] values as well as an improved precision of [Mn/Fe] values ( North et al. 2012 ) to verify if stars with enhanced Mn/Fe, Ni/Fe, and Zn/Fe ratios are present in Sculptor. Another interesting case is the ultra-faint dwarf galaxy Horologium I, in which Nagasawa et al. (2018) found three very metal-poor ([Fe/H] ∼ −2.5) giant stars that are α -poor ([Mg/Fe] and [Ca/Fe] ≃ 0.0) and enhanced in [Mn/Fe] by about 0.4 dex relative to Galactic halo stars. However, their [Ni/Fe] ratios seem to be normal, and their Cu and Zn abundances were not determined. Again, we need more precise abundances,