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We thank the anonymous referee for helpful comments and suggestions that improved the paper. We acknowledge the support of the National Natural Science Foundation of China (Grants No. 11125314 and No. U1531247) and the Strategic Priority Research Program (Grant No. XDB09000000) of the Chinese Academy of Sciences.


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

are very restrictive, as η is likely to be less than unity. These limits can also accommodate the gamma-ray luminosity of 3FGL J1417.5–4402, 3 × 10 34 erg s −1 at a distance of 4.4 kpc (see Section 4 ). 3.5. Evolutionary Track With a binary period of 5.4 days, PSR J1417–4402 is on the evolutionary track of low-to-intermediate mass companions that started mass transfer onto a neutron star after they left the main sequence (Podsiadlowski et al. 2002 ), and have increasing periods. This distinguishes them from the black widows, redbacks, and other low-mass X-ray binaries that have binary periods day, began mass transfer on the main sequence, and evolve toward shorter periods. For 1 M secondaries, these classes are separated by a bifurcation period of hr at the onset of mass transfer. With M , the companion of PSR J1417–4402 must have already lost most of its initial mass, only a fraction of which, accreting onto the neutron star, is sufficient to spin it up to 2.66 ms. In the models of Podsiadlowski et al. ( 2002 ), PSR J1417–4402 is not far from the ends of the tracks that terminate when the binary period is in the range 6–20 days, and the companion is a He white dwarf of 0.2–0.3 M . The neutron stars in these calculations, assumed to begin with 1.4 M , end in the range 1.8–2.2 M . Exactly how far PSR J1417–4402 is along this evolution is difficult to determine, since its companion may or may not be filling its Roche lobe, may or may not be finished accreting onto the neutron star, and may be losing mass mostly through ablation by the pulsar wind, a process that is difficult to model. 4. CONCLUSIONS We have discovered a 2.66 ms pulsar in orbit around the 5.4 day optical variable discovered by Strader et al. ( 2015 ) in the error ellipse of the high-energy gamma-ray source 3FGL J1417.5–4402. The orbital dynamics require that the companion mass is M . The presence of radio pulsations suggests that the neutron star is not currently accreting (see Section 3.2 ). The pulsar is only detected a fraction of the time it is observed (although few observations exist at orbital phases Figure 2 ), and is likely eclipsed for more than half the orbit by outflowing material from the companion; this has so far prevented us from obtaining a phase-connected timing solution. The DM distance of 1.6 kpc is in conflict with the original estimate of , which was based on the assumption that the companion fills its Roche lobe. This ambiguity then affects most of the remaining interpretation, including the precise evolutionary state of the system, whether it currently has an accretion disk, and the masses, which depend on the modeled inclination angle. The current estimate of M should really be regarded as an upper limit. There are several claims of massive neutron stars in redbacks and black widows. Modeling these systems, however, is difficult, especially in the presence of significant heating of the secondary by the pulsar (see, e.g., Romani et al. 2015 ). So far


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

for an unobscured AGN or for the Thomson-scattered emission of an obscured AGN, plus an APEC component representing the hot gas in the galaxy ( keV). We notice that the power-law component may also be partially attributed to emission from star formation and X-ray binaries, as discussed in Section 3.1.2 . The total absorption component is fixed to the Galactic value ( N H = 1.31 × 10 20 cm −2 , as given by the Colden Calculator with the NRAO data set 2 ). A similar value for the photon index () is obtained also when excluding the counts below 1.5 keV in order to avoid contamination by the hot gas. By fitting the 3.0–8.0 keV spectrum with a simple power law, we obtain a photon index . This low value is an indication that the source is heavily obscured and that the power law observed in the soft X-ray spectrum corresponds to the scattered component. However, due to the narrow energy interval considered, adding an obscured component to our fiducial model for the Chandra data does not improve the quality of the fit significantly, so that a combined fit with NuSTAR is needed to determine the real amount of obscuration. 3.1.2 Possible contribution from star formation IC 2497 is a luminous infrared galaxy (LIRG) with far-IR luminosity L FIR = 6.6 × 10 44 erg s −1 . Assuming the FIR–star formation rate relation of Kennicutt ( 1998 ), this corresponds to a star formation rate SFR ∼ 30 M ⊙ yr −1 . Using the relations of Ranalli et al. ( 2003 ) or Lehmer et al. ( 2016 ), the expected 2–10 keV luminosity is erg s −1 , which is comparable (within the large scatter of the correlations) to the observed 2–10 keV luminosity measured from the Chandra and NuSTAR spectra (we stress that all the used relations have a large scatter, so that the results have to be considered as an order of magnitude estimation). In addition, the low W1 - W2 colour, W1 - W2 = 0.3, suggests that the mid-IR emission is dominated by stellar emission (see Section 3.2.2 for a discussion of the mid-IR emission). Altogether, this points to a scenario where star formation provides an important contribution to the soft part of the X-ray spectrum. We will consider this component in the analysis below. 3.1.3 Phenomenological model Applying the unabsorbed power-law model described above to the 0.5–24.0 keV spectrum significantly underpredicts the hard X-ray counts observed with NuSTAR , especially in the 10.0–20.0 keV range (see Fig. 1 , left). This again is an indication that the source is highly obscured (e.g. Koss et al. 2016 ; Ricci et al. 2016a ). To infer the intrinsic column density N H and the X-ray luminosity of the source, we first fit the 0.5–24.0 keV spectrum with a phenomenological model which includes (1) an absorbed power law representing the intrinsic emission of the AGN, where we consider both photoelectric absorption and Compton scattering, (2) a second unabsorbed power law representing the scattered AGN component and the contribution from star formation, (3) the unabsorbed


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

parameter being set using equation (8) of Dutton Macciò ( 2014 ). When using this method for NFW profiles, we do not correct the concentration parameter from that of Dutton Macciò ( 2014 ) as is done for Di Cintio et al. ( 2014 ) profiles. Figure 9. A comparison of the halo mass estimate from GCs and the dynamical mass estimates for VCC 1287. We plot the Beasley et al. ( 2016 ) GC tracer mass estimate (circle) and our stellar velocity dispersion dynamical mass measurement (star). A total halo mass estimate based on the GC number–halo mass relation of Burkert Forbes ( 2020 ) is plotted using two differing assumptions of halo profile (blue region – NFW cusp profile; red region – Di Cintio et al. 2014 halo core profile). For both dynamical mass measurements to be consistent with the halo mass estimate from the number of GCs ( N GC = 22 ± 8; Beasley et al. 2016 ), either a lower halo concentration parameter or the presence of a dark matter core is required. Theoretical efforts to simulate the formation of UDGs frequently rely either on the creation of a cored dark matter profile (e.g. Di Cintio et al. 2017 ) or on its pre-existence (e.g. Carleton et al. 2019 ). Jiang et al. ( 2019 ) also found UDGs in the nihao simulation to lie in cored dark matter profiles along with having concentration parameters lower than that expected based on dwarf galaxy and L * galaxy control samples. It is yet to be established whether these low concentration parameters are the cause of, or result of, UDG formation (Jiang et al. 2019 ). In both, the case of a cored halo or one with a lower concentration parameter, the enclosed mass of the profile would be biased towards lower values, causing NFW profiles to represent an upper limit to the enclosed mass. For this reason, we also choose to include the Di Cintio et al. ( 2014 ) halo profile as these models naturally produce cores, which may be more representative of the actual profile of UDGs. There exists a slight tension between the dynamical masses and the halo mass, as plotted, calculated with the Burkert Forbes ( 2020 ) relation. This tension may be the result of incorrect assumptions made in converting observations into dynamical masses or, more likely, incorrect halo profile assumptions made in plotting for the full halo mass inferred from GC counts (e.g. VCC 1287 may have a lower concentration parameter than we calculate with Dutton Macciò 2014 ). In order for the dynamical mass measurements to be consistent with the halo mass estimate from the Burkert Forbes ( 2020 ) relation, the dark matter halo of VCC 1287 must either be cored (e.g. the Di Cintio et al. 2014 profile of Fig. 9 ) or have a low-concentration parameter. We note that the total halo mass from the Burkert Forbes ( 2020 ) relation implies a halo more massive than what is expected from the stellar mass–halo mass relation similar to other UDGs in the Coma Cluster (Forbes et al. 2020 ). Indeed, both the stellar mass and GC system richness of VCC


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SDSS DR7 and DR13 galaxy samples in two redshift bins for the different models of the LF described in the text. The quoted errors represent marginalized 68percent confidence intervals. Considering total bulk flow amplitudes, the DR13 estimates are generally smaller than the DR7 ones. Averaging the results over the different LF models yields values of and 48595 in units of km s 1 for the low- and high-redshift bin, respectively. In the high- z bin, the flow amplitudes are reduced by about 50200kms 1 compared to DR7, where the most significant changes appear for the Schechter model of the LF. The bulk flow components found from estimators using different LF models are in reasonable agreement, typically consistent within the quoted uncertainties. Assuming the results based on the hybrid model, the estimated bulk flows from DR13 are pointing towards ( l , b ) (315, 17)(52, 15) and (304, 22)(14, 11) for the first and second redshift bin, respectively. Similar directions are obtained for the other estimators. To better assess these measurements, we repeated the analysis for the 269 mock galaxy catalogues introduced in Section 3.2 . In contrast to the study presented in Feix etal. ( 2014 ), these mocks do not account for the photometric tilt of the SDSS DR7 calibration that severely contaminates bulk flows measured through the luminosity approach. The results based on our mocks suggest that the individual flow components obtained with our methods are not statistically independent, but subject to correlations at the level of 0.10.3. If the joint distribution of bulk flow components is approximately given by a multivariate Gaussian with covariance matrix B , the quadratic form should follow a 2 -distribution with k 3 degrees of freedom. To test the validity of this assumption, we computed directly from the SDSS mocks. For the Schechter LF model, the resulting distributions, plotted as histograms in Fig. 2 , are indeed well matched by the 2 -distribution (solid curves). Adopting the other models of the LF yields very similar results. Using the estimate of B from the mock catalogues, we may therefore assign probabilities to the bulk flow measurements listed in Table 1 . As customary, we express these in terms of confidence limits based on normally distributed data. Figure 2. Distribution of the quadratic form in two redshift bins derived from the SDSS mock catalogues, assuming the Schechter model of the LF: the histograms represent the observed (normalized) distribution and solid lines correspond to the 2 -distribution with k 3 degrees of freedom. The corresponding event levels (in units of the standard deviation ) for the inferred bulk flow amplitudes are presented in Table 2 . Compared to the DR7 results, the probability estimates of the DR13 measurements are on average considerably larger, and thus consistent with an improved photometric calibration. The most prominent change is found in the high- z bin for the Schechter model of the LF, where the


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RDB is grateful to S. Raimundo for the discussion which inspired this work. We thank the referee for constructive comments/suggestions to improve the manuscript. EB received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 655324, and from the I-CORE program of the Planning and Budgeting Committee (grant number 1937/12).


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Acknowledgements We thank the anonymous referee for their helpful comments. We are grateful to Phil Uttley for his suggestion to look into this and to Yuri Cavecchi, Daniela Huppenkothen, Anna Watts and Ralph Wijers for very useful discussions. AP acknowledges support from a Netherlands Organization for Scientific Research (NWO) Vidi Fellowship. PTO would like to acknowledge funding from STFC. This work makes use of data supplied by the UK Swift Science Data Centre at the University of Leicester and the Swift satellite. Swift , launched in November 2004, is a NASA mission in partnership with the Italian Space Agency and the UK Space Agency. Swift is managed by NASA Goddard. Penn State University controls science and flight operations from the Mission Operations Center in University Park, Pennsylvania. Los Alamos National Laboratory provides gamma-ray imaging analysis.


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IPGP contribution NS 3968. Data processing and statistical procedure described in Sect. 3 was performed by A. Shapoval and M. Shnirman under support of the Russian Science Foundation (project No 17-11-01052)


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

stars within 10 arcmin of 30 Dor. In this resulting spectroscopically selected pseudo-BBS sample, the early B-type stars outnumber the O-type stars roughly 3–1, although the implications of this are unclear since we don’t know the level of incompleteness for either. Fig. 2 shows an HR diagram with the spectroscopic OB stars plotted without (left) and with (right) an average reddening correction of E ( B − V ) = 0.13 mag applied (the average value for OB stars in the LMC adopted by A19). Also shown for comparison are LBV stars in the LMC and SMC, plotted using T eff and L Bol values from the literature (see Smith et al. 2019 ), and single-star evolutionary tracks (Brott et al. 2011 ). OB stars are placed on this HR diagram by taking their apparent or reddening corrected B − V colour as a proxy for temperature, and the luminosity comes from the V mag and a bolometric correction for the corresponding T eff value, with relations adopted from Torres ( 2010 ) and Flower ( 1996 ). These are by no means intended to be taken as accurate T eff and L Bol values; they are meant to illustrate the range of these values one might infer from apparent magnitudes and colours when only photometry is available. The red diagonal lines in Fig. 2 indicate where the V mag cut-off resides. Even though these are not the same stars as in the photometric BBS sample, there are several salient points that one can glean from Fig. 2 . Figure 2. HR diagrams comparing LBVs to inferred properties of OB stars derived from photometric colours. The T eff value used to plot each OB star is the temperature one would infer by converting the apparent B − V colour to a temperature, and the luminosity comes from the bolometric correction for that T eff value (relations adopted from Torres 2010 and Flower 1996 ) and its apparent V magnitude. The left-hand panel (a) shows the result when no reddening correction is adopted, whereas the right-hand panel (b) shows the values after applying a single average correction for E ( B − V ) = 0.13 mag to all OB stars. The black unfilled circles are spectroscopically confirmed O-type stars (similar to the sample in ST15), and the blue filled circles are spectroscopically confirmed early B-type stars, with spectral types between O9.5 and B2 (see text). Both samples are limited to spectroscopically confirmed O or early B stars brighter than V = 13.9 mag (to be consistent with the value adopted by A19), corresponding roughly to an absolute V magnitude of M V = −5. The diagonal red line in each panel shows the cut-off of V = 13.9 mag, and the four red hash marks show temperatures corresponding to different values of ( B − V ) = −0.32, −0.30, −0.20, and −0.10 mag. In both panels, the orange dashed lines are single-star evolutionary tracks from Brott et al. ( 2011 ), with initial masses in M ⊙ labelled in orange. Light yellow and blue shaded areas indicate expected effective temperature and luminosity ranges for O-type and early B-type stars


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

the difference to Cuzzi et al. ( 2010 )’s assumption that the grains would be sized to optimally couple to the smallest eddies and questioned whether the pebble concentrations in Cuzzi et al. ( 2010 )’s semi-analytic model exceeded what is physically plausible. If the assumptions in Cuzzi et al. ( 2010 ) are unrealistic and if Hopkins ( 2016 )’s modelling and assumptions capture all the important physics, then larger numbers of, and better known orbits for the ultra-wide binaries could constrain whether the turbulent concentration mechanism produces excessively large planetesimals that would have disrupted the TNOs. Pebble Accretion is a promising mechanism for growing planetesimals and protoplanets (Ormel Klahr 2010 ; Lambrechts Johansen 2012 ). It substantially enhances the cross-section for accreting small particles that are partially coupled to the gas and have their relative velocities damped by gas drag (see Johansen Lambrechts 2017 , for a review). Johansen et al. ( 2015 ) considered the evolution of a population of planetesimals growing by pebble accretion. They began with a size distribution of large bodies derived from simulations of the Streaming Instability (Youdin Goodman 2005 ; Johansen et al. 2007 ), which produces q ∼ 2.8 with a maximum mass of (Simon et al. 2016 , 2017 ). Johansen et al. ( 2015 ) considered the subsequent evolution of the planetesimals in this environment. They performed simulations of a narrow ring at 25 au and found that a single body underwent significant growth, reaching roughly an Earth mass, amounting to roughly one-third of the mass in that ring. If this result for a single 3 M ⊕ ring can be extended to a 30 M ⊕ disc spanning 20–30 au, the disc would produce ∼10 Earth mass bodies, inconsistent with the limits derived here. Whether this result is generally applicable to Pebble Accretion depends on the sensitivity of final mass-number distribution to factors such as the initial mass-number distribution, which is yet to be broadly explored. Similar to Pebble Accretion, the collisional pebble agglomeration model of Shannon, Wu Lithwick ( 2016 ) also produces a top heavy q ∼ 2 size number distribution, but with the largest bodies in the 10 s to 100 s of kilometers in size, dependent on the composition of the pebbles. Here, we use the code and general setup from that work, but consider a belt from 20 to 30 au that begins with 30 M ⊕ . With little mass in bodies larger than ∼100 km, it is compatible with the limits we have found. In addition to testing planetesimal formation models, our limits offer constraints on the dynamical history of the solar system. Several other works have invoked large bodies within the pre-Scattered disc to explain features of the Solar system. As described in Section 1 , Nesvorný Vokrouhlický ( 2016 ) invoked 1000–4000 bodies of mass or 1000 objects of mass in the pre-Scattered disc to produce the inferred capture rate of resonant objects. These choices would be incompatible with the


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We thank the anonymous referee for comments and suggestions. This paper was partially funded by the PRIN INAF 2014 grant. M.M. thanks M. Dadina for discussions about the X-IFU instrumental background.


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The authors would like to thank Janet Weiland and Chuck Bennett for providing the ILC data in a compatible format, Tom Dame and Chris Salter for helpful discussions, and the anonymous referee for guidance for improving an early version of this manuscript. No thanks are due Hurricane MARIA, which made our task that much more difficult.


O O O O O O O O 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-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 B-CelestialObject I-CelestialObject O O O B-CelestialObject I-CelestialObject O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObject I-CelestialObject 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 O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject 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-CelestialObject I-CelestialObject O O B-ObservationalTechniques I-ObservationalTechniques I-ObservationalTechniques O O O O O O O B-Software I-Software I-Software I-Software I-Software O O O O B-CelestialObject I-CelestialObject O O B-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O B-ObservationalTechniques O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation 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 B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O

with gravity modes or motions caused by subsurface convection, the cause of these variabilities is not properly understood. Thus, with the motivation to understand the dominant variability of 2.7 d in κ Cassiopeiae, we shall present here a linear non-adiabatic stability analysis of models for κ Cassiopeiae. The result of instabilities will then be determined by following them into the nonlinear regime for selected unstable models using a fully conservative numerical scheme. This paper falls into five sections where a description of the models used is given in Section 2 , and the linear stability analysis together with its results are discussed in Section 3 . The nonlinear simulations and their results are presented in Section 4 . A discussion and conclusions follow in Section 5 . 2 MODELS FOR κ CASSIOPEIAE Although the star κ Cassiopeiae has been the subject of several studies (see e.g. Underhill 1979 ; Percy 1981 ; Hayes 1984 ; Katushkina et al. 2018 ), its fundamental parameters (in particular its mass) are not precisely known. For the present investigation, we adopt the effective temperature ( T eff = 24 600 K) and luminosity (log L /L ⊙ = 5.69) estimated by Simón-Díaz et al. ( 2018 ) using high-resolution spectra of κ Cassiopeiae. These values are close to those determined by Kudritzki et al. ( 1999 ), Smartt et al. ( 2002 ), and Evans et al. ( 2004 ). Due to the uncertainty of the mass, we consider a range of models with masses between 27 and 44 M ⊙ . This mass range covers the mass of 33 M ⊙ suggested for κ Cassiopeiae by Searle et al. ( 2008 ) on the basis of stellar evolution calculations. Evolutionary tracks for models having solar chemical composition and masses of 30, 35, 40, and 45 M ⊙ respectively are shown in Fig. 1 where a thick dot corresponds to the observed position of κ Cassiopeiae in the Hertzsprung–Russell (HR) diagram. These tracks have been generated by the ‘mad star’ stellar evolution code. 1 The position of κ Cassiopeiae in the HR diagram suggests that models representing this star have masses below and close to 40 M ⊙ (see Fig. 1 ). Figure 1. Evolutionary tracks of stars with solar chemical composition and having masses in the range between 30 and 45 M ⊙ . The observed location of κ Cassiopeiae in the Hertzsprung–Russell diagram is marked by a thick dot. The amplitudes of radial eigenmodes decay exponentially from the stellar surface to the core. Thus, the latter may be disregarded when studying stellar stability (see e.g. Glatzel Kiriakidis 1993 ; Glatzel, Kiriakidis Fricke 1993 ; Saio 2011 ; Yadav Glatzel 2016 , 2017a ). Accordingly, our present study is restricted to envelope models of κ Cassiopeiae thus disregarding any influence of nuclear energy generation and nucleosynthesis. The envelope models with solar chemical composition ( X = 0.70, Y = 0.28, and Z = 0.02; where X , Y , and Z represent the mass fraction of hydrogen, helium, and heavy elements, respectively) are constructed by integrating the stellar


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I am grateful to Gabriel Martínez-Pinedo for providing stellar weak rates tables on a fine grid. Thanks are due to Robert Fisher, for interesting discussions concerning the floatability of hot bubbles, and to Ivo Seitenzahl and Friedrich Röpke for clarifying the origin of the differences between the present results and those of Parikh et al. The referee, Chris Fryer, has made interesting suggestions to improve the presentation of this paper. This work has been supported by the MINECO-FEDER grant AYA2015-63588-P.


O O O O O O O B-Organization I-Organization I-Organization I-Organization O O B-Grant O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-Person O O O O B-Organization B-Person O O B-Organization I-Organization B-Person O B-Person O B-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Person O O O B-Organization O O O O O O O B-Person I-Person I-Person O O O O O O O O O O O O O O O O B-Organization I-Organization I-Organization B-Organization O B-Telescope O B-URL O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization B-URL O O O O B-Organization O O O O O O O O O O O O O B-Collaboration I-Collaboration I-Collaboration O O O O O O O O B-Database O O O B-Organization B-Location I-Location


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

1 ). We extracted photometry for point sources in each individual image to construct light curves of Sgr A* and calibration stars. The brightness and position of all sources were measured using the point spread function (PSF)-fitting software StarFinder (Diolaiti et al. 2000 ). Details on running StarFinder on individual frames for Galactic center observations can be found in Do et al. ( 2009 ). Sgr A* is detected in almost every frame of observations on three nights. There are fewer detections of Sgr A* on 2019 May 19 due to lower-quality AO correction and faint flux levels for Sgr A*. We calibrate the point source photometry using reference stars defined by Gautam et al. ( 2019 ). We convert between K ′ magnitudes to absolute fluxes F , using the relationship mJy (Tokunaga Vacca 2005 ). To convert from K ′ fluxes to K s fluxes, we use the filter transformation computed for the color of Sgr A* (Chen et al. 2019 ). We will mainly use observed fluxes in this work to avoid confusion with the value of extinction to apply. To de-redden the flux measurements, one can use the relationship . To compare to fluxes from (Witzel et al. 2012 ), use (Schödel et al. 2010 ). We estimate the Sgr A* relative photometric uncertainties by using the relationship between flux level and flux uncertainty of nearby stars. The photometric uncertainties for stars on each night were determined by using the standard deviation of their individual flux measurements. We find that the reference stars are stable during the nights with 3% photometric errors. Following Do et al. ( 2009 ), we fit a power-law to the relationship between flux level and flux uncertainty for stars within 1′′ of Sgr A*. We then use this relationship to infer the flux uncertainty for the all flux measurements of Sgr A*. The Sgr A* relative photometric uncertainties are typically less than 5% at high flux levels and about 15% for faint flux values. 3. Results Our observations show Sgr A* to be highly variable in 2019. On 2019 May 13, Sgr A*'s flux level changed by a factor of 75 within 2 hr (from 6.19 ± 0.08 mJy to 0.08 ± 0.01 mJy). The maximum observed fluxes occurred during the beginning of the observations, suggesting that Sgr A* was likely even brighter earlier in the night. On the night of 2019 April 20, Sgr A* flux also shows large variations with measurements ranging from 1.74 ± 0.04 mJy to 0.07 ± 0.01 mJy. The two other nights of our study (2019 April 19 and 2019 May 23) show less variation at the level of one to three magnitudes of change during the night. Figures 1 – 3 show the observed light curves. Figure 2. K ′ light curves of Sgr A* (black) and a comparison star, S0–17 (white, located about 02 from Sgr A*), on four nights of observations in 2019. We use stars within 1′′ of Sgr A* to characterize the photometric error at different Sgr A* brightness levels. The photometric uncertainties are typically less than 5% at high flux levels. Observations on 2019 May 13 also include H -band


O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Telescope O O O O O O O O O O O B-Proposal O B-Person I-Person B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Proposal O B-Person I-Person O O O O O O O O B-Proposal O B-Person I-Person B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Proposal O B-Person I-Person 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 B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject O O O O O O B-Telescope O O B-Wavelength I-Wavelength I-Wavelength B-Citation I-Citation I-Citation I-Citation O O O B-Model O O O O O O O B-CelestialObject O O O O O O O O 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 B-Instrument B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Formula I-Formula I-Formula O O O O O O

be well fit by rigid body rotation at a rate of 0216 ± 0016 yr −1 . Taking into account of SPHERE’s true north uncertainty, the updated motion rate isThis pattern speed points to a driver located at au, or , from the 1.9 M central star (Figure 3 ). Figure 3. Dynamically constrained, assumed circular orbit of a single arm driver. For a 1.9 M central star, the best-fit ±1 σ orbit has a radius of ( au). Our best-fit measurement of the companion-driven spiral pattern speed is consistent with the R18 measurement within 3 σ , 18 18 Only 3 σ uncertainties are well-constrained in R18 . while our derived uncertainty is ∼40 times smaller, thanks to the use of the same instrument and the maps that are the least biased by postprocessing methods. In numerical simulations, an approximately five Jupiter mass arm driver located at ∼09 has been proposed by Baruteau et al. ( 2019 ), which is within 2 σ from our best-fit companion location assuming a 1.9 M central star, or 1 σ assuming a 1.56 M central star. 4. Discussion 4.1. Direct Imaging Constraints We obtain the direct imaging constraints on the mass of the putative planet orbiting MWC 758 with hot-start evolutionary models (i.e., Sonora, Bobcat; M. Marley et al. 2020, in preparation) using 6.73 h of Keck/NIRC2 -band archival observations: 2025 s on 2015 October 24 (Program ID: C220N2, PI: E. Serabyn, Reggiani et al. 2018 ), 11,160 s on 2016 February 12 (Program ID: U131N2, PI: E. Chiang), 3200 s on 2016 October 24 (Program ID: C221N2, PI: G. Ruane, Reggiani et al. 2018 ), and 7830 s on 2017 February 2 (Program ID: U072, PI: E. Chiang). Following the method described in Ruffio et al. ( 2018 ) and taking into account of the orbital period uncertainty for the driver, we obtain an apparent L -band magnitude of 18 at 99.9% confidence level (i.e., a 3 σ equivalent upper limit). Adopting an age of 10.9 Myr (Garufi et al. 2018 ), this corresponds to a mass of 5 M Jupiter . Using the contrast curve of Keck/NIRC2 in the M s -band (Mawet et al. 2019 ) and assuming Gaussian noise, we expect that a 5 M Jupiter planetary driver, whose mass has been predicted by Dong et al. ( 2015b ) and Baruteau et al. ( 2019 ) with the semimajor axis updated in this study, can be detected at 5 σ level if it is a hot-start planet using four half-nights of NIRC2 M s -band high contrast imaging observations (or at 3.5 σ with the Spiegel Burrows 2012 cold-start model using the same observation). 4.2. Eccentric Driver If the driver has a non-zero eccentricity e , its semimajor axis cannot be uniquely determined from an assessment of the instantaneous angular frequency due to the unknown orbital phase. Giant planets of several Jupiter masses interacting with a gaseous disk are expected to have their eccentricities quickly damped to below the disk aspect ratio, here about 20% (Dunhill et al. 2013 ; Duffell Chiang 2015 ). Simulations have also shown that density waves excited by planets with e ≳ 0.2 develop wiggles and bifurcations, as the


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We are grateful to the CSO staff for very efficient support, both on-site and during the later remote observations. Y.-W.T. is supported by the Ministry of Science and Technology (MoST) in Taiwan through grant MoST 108-2112-M-001-004-MY2. P.M.K. acknowledges support from MoST 107-2119-M-001-023 and from an Academia Sinica Career Development Award. A.D.C. acknowledges the support from the UK STFC consolidated grant ST/N000706/1. Facilities: CSO.


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 O B-Person I-Person O O O O O O O O B-Person I-Person O O O O O B-Telescope O O B-CelestialObject I-CelestialObject O O O O O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant O B-Location B-Grant O B-Organization I-Organization I-Organization I-Organization O B-Organization O B-Grant O B-Organization I-Organization I-Organization O B-Location O B-Grant O O B-Organization I-Organization I-Organization I-Organization O B-Location O O B-Grant B-Grant B-Grant O B-Grant O O O O O O O O O O O O B-Archive I-Archive I-Archive I-Archive I-Archive I-Archive I-Archive B-Archive O O O O O O B-Organization I-Organization I-Organization O B-Organization O O B-Organization I-Organization I-Organization I-Organization O O B-Observatory I-Observatory I-Observatory

Acknowledgements We appreciate Paul Wiita, James Guillochon, Tinggui Wang, Wenfei Yu, Bin Liu, Morgan MacLeod, and Dacheng Lin for helpful discussions, the referee for suggestions, and Dimitris Emmanoulopoulos for providing us with the RXTE data of NGC 7213. This work was supported in part by the National Key Research and Development Program of China (2016YFA0400804), the Youth Innovation Promotion Association of CAS (id. 2016243), the Natural Science Foundation of Shanghai (No. 17ZR1435800), and the National Natural Science Foundation of China (grant Nos. 11403074, 11773055, 11333005 and 11350110498). This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.


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 B-Software O O O O O O O O O O O O B-Software O O O O O O O O O B-Software 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 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 O O O O O O O O O O O O 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

which fall within of the HRD position of both stars and have equal composition. Fig. 5 Chi-squared distribution of central hydrogen mass fraction for the same models as in Fig. 4 . The models for the primary are indicated by the dark blue points, while the models for the secondary are indicated by the light blue points. So far, we have only taken into account the mean of the observed -mode period spacing, neglecting the effect of rotation and the information that is held in the spacing morphology. In the next step, we therefore calculate the adiabatic frequencies of our models with GYRE and compare their seismic properties to the observations. 3.2. Period spacing with GYRE We calculated the zonal dipole and quadrupole modes with GYRE for the coeval models with the lowest . As we clearly see the effect of rotation in the observed frequency spectrum (see Fig. 1 ), we apply Ledoux splitting to the zonal modes to obtain the theoretical prograde and retrograde modes and the rotational frequencies near the core for both components. We derive and and a negligible difference between the models with exponential and those with step overshooting. For these rotation rates and the co-rotating dipole-mode frequencies we find and . The comparison between the calculated modes of Models 2 and 3 and the observed period spacings is illustrated in Fig. 6 , while Fig. 7 shows the comparison of the -modes. Our main focus lies in comparing the mean period spacing values and the structure of the period spacing pattern, rather than individual frequencies. We find a satisfactory fit for both models and cannot distinguish between exponential or step overshooting. Furthermore, we are able to assign the series (yellow stars in Fig. 1 ) to the primary, as there is a clear overlap with the retrograde sectoral modes of both models, while there is a gap for the models of the secondary. From Eq. ( 3 ), we expect that the ratio of the asymptotic period spacing of the quadrupole modes to the asymptotic period spacing of the dipole modes is . We use the derived to shift the observed quadrupole sectoral modes to their zonal counterparts, and find s and . For the secondary and we find s and . These results support that the series originates in the primary. Fig. 6 Comparison of theoretical (empty symbols) to observed (filled symbols) period spacing for the primary ( top panels ) and the secondary ( bottom panels ). Ledoux splitting has been applied to the theoretical zonal modes to fit the retrograde and prograde modes. The uncertainties of the observed period spacings are smaller than the symbols. The radial orders are indicated near selected modes. Dipole modes of radial order are shown for the primary and for the secondary, while the radial orders of the quadrupole modes are and , respectively. Fig. 7 Comparison of the theoretical zonal -modes (empty symbols) to the observed modes (filled symbols) as a function of degree . Modes of the primary are shown in the top panels and


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

conceivably due to dynamical mass ejecta. Colour coding (blue-to-yellow) is proportional to amplitude defined by butterfly output ρ of time-symmetric chirp-like template correlations to data. Fig. 2 shows a continuation of GW170817 with a long-duration descending chirp during GRB170817A. It can be fitted by an exponential track ( Supplementary Information ) with s, t s = 1843.1 s, Hz, and f 0 = 98 Hz. Crucially, s shows the descending chirp to start in the 1.7 s gap between GW170817 and GRB170817A, satisfying the aforementioned causality condition. A Gaussian equivalent level of confidence greater than 3.3σ obtains for significance to the central engine of GRB170817A based on causality over 1952 s of LIGO data. Improved estimates obtain upon including the statistical significance of the strength of the signal ( Supplementary Information ). The estimated initial frequency Hz at the time of coalescence inferred from ( 2 ) is below the orbital frequency at which the stars approach the Inner Most Stable Circular Orbit (ISCO) of the system as a whole, i.e. Hz at km according to the Kerr metric (Kerr 1963 ; Bardeen, Press Teukolsky 1972 ; Shapiro Teukolsky 1983 ). At this point, a binary system of two equal mass neutron stars would have a rotational energy and a dimensionless specific angular momentum . While this allows prompt collapse to an Kerr black hole (Kerr 1963 ) any gravitational-radiation from remaining debris orbiting about the ISCO would be above 2 kHz. Instead, the 1kHz descending chirp in Fig. 2 points to radiation of from a long-lived rapidly rotating hypermassive neutron star or magnetar with a dimensionless quadrupole moment of , where R is the radius of the star, induced by dynamical and secular instabilities (Owen et al. 1998 ; Kokkotas 2008 ; Corsi Mészáros 2009 ) that may include magnetic fields (Cutler 2002 ; Abbott et al. 2017d ). Magnetar spin down indicated by the descending chirp may include magnetic interactions with surrounding matter from dynamical mass ejections from the preceding merger, in addition to gravitational radiation losses alone. These ejecta are commonly found in aforementioned numerical simulations. The interactions may result from magnetic coupling (Gosh, Pethick Lamb 1977 ; Gosh Lamb 1978 , 1979 ) and/or tidal stresses (Hotokezaka et al. 2013 ) that may include magnetic outflows from open field lines (Lovelace, Romanova Bisnovatyi-Kogan 1995 ; Parfrey, Spitkovksy Beloborodov 2016 ) and gravitational-wave emission from quadrupole moments in orbiting debris. While the descending chirp identifies the central energy reservoir of GRB170817A to be a rotating hypermassive neutron star or magnetar, its connection to GRB170817A remains uncertain. Perhaps the relatively soft second gamma-ray pulse of GRB170817A (Pozanenko et al. 2018 ) derives from outflows from debris orbiting the star, while the short-hard pulse derives from an initial outflow from the star at birth. More likely, the complex spectral evolution of


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We are sincerely grateful to an anonymous referee for instructive advice and productive suggestions that greatly helped us to improve the manuscript. This work is funded by the NSFC of China (Grant No. 11333006, 11521303, and 11403094) and by the foundation of Chinese Academy of Sciences (Grant No. XDB09010202 and Light of West China Program). We gratefully acknowledge the computing time granted by the Yunnan Observatories, and provided on the facilities at the Yunnan Observatories Supercomputing Platform. We are also very grateful to J.-J. Guo, G.-F. Lin, Q.-S. Zhang, Y.-H. Chen, and J. Su for their kind discussions and suggestions.


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This work has been supported by a grant from the Brazilian agency FAPESP (#2013/10559-5). Raissa Estrela acknowledges a CAPES fellowship.


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The authors express their thanks to Christina Ciardullo for her assistance with the graphics. M.S. acknowledges support from DFG project RA 714/9-1. M.G. acknowledges support from DFG project GO 2610/1–1. F.S. acknowledges support from DFG project SCHR 1125/3-1. Software: GARLIC (Schreier et al. 2014 , 2018a , 2018b ), HITRAN2012 (Rothman et al. 2013 ).


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ACKNOWLEDGEMENTS We would like to thank Lucy Heil for useful discussions and for providing comparison data to check our power-colour estimates, and Rudy Wijnands and Jeroen Homan for useful discussions. We also thank the anonymous referee for constructive comments. DWG acknowledges funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 617199, and support from the Netherlands Institute for Radio Astronomy (ASTRON) and the Netherlands Foundation for Scientific Research (NWO). This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. chromos makes use of astropy v1.0.3 (Astropy Collaboration et al. 2013 ), ftools (Blackburn 1995 ), numpy, pyx , and scipy .


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This work was partially supported by Consejo de Investigaciones Científicas y Técnicas (CONICET) and Secretaría de Ciencia y Técnica de la Universidad Nacional de Córdoba (SecyT). We want to thank the anonymous referee, whose very useful remarks helped us to improve this paper. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.


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 B-Person I-Person O O O O O O O B-Organization I-Organization B-Person O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization O O O O O O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship O O O O O B-Organization B-Grant I-Grant I-Grant O O O O O O O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O O O O B-Organization I-Organization I-Organization I-Organization O O O O O O O O O B-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility I-ComputingFacility B-ComputingFacility O O B-Organization I-Organization I-Organization B-Organization O B-Software B-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation O B-Software B-Citation I-Citation O B-Software B-Citation I-Citation O B-Software B-Citation I-Citation I-Citation I-Citation O

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.00773.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. Y.-L. W. is grateful to the support from the Heising-Simons Foundation. J.T. would like to acknowledge support from the Homer L. Dodge Endowed Chair. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under grant No. 2012115762. This work was supported by NSF AAG grant 1311910. The results reported herein benefitted from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate. The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma (OU). Software: pdspy (Sheehan 2018 ), CASA (McMullin et al. 2007 ), RADMC-3D (Dullemond 2012 ), emcee (Foreman-Mackey et al. 2013 ), matplotlib (Hunter 2007 ), corner (Foreman-Mackey 2016 ), GALARIO (Tazzari et al. 2018 ).


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

‘s’ in space–time, because for isotropic geodesic motion we have d s = 0 but dΣ ≠ 0. This method has been successfully used to study various non-linear processes occurring in magnetic dipole fields of pulsars and magnetars (Abishev et al. 2016 ), as well as in an electric dipole field (Vasili’ev et al. 2017 ). Calculations have shown that in this case the rays of electromagnetic waves are bent, and the electromagnetic wave itself splits at η 1 ≠ η 2 into two normal modes with mutually perpendicular polarization. These modes propagate along different rays at different speeds. At the present time, another polarization effect of non-linear electrodynamics of vacuum is observed (Taverna et al. 2014 ; Caniulef et al. 2016 ; Mignani et al. 2017 ): polarized thermal emission from X-ray-dim isolated neutron stars (the case of RX J1856.5−3754). Recent papers (Pétri 2013 , 2015 ) assume that neutron stars can also have a quadrupole magnetic field, the intensity of which can reach the intensity of dipole fields. To date, there has not been any study of the non-linear electrodynamic effect of the magnetic quadrupole field on the propagation of pulses of electromagnetic radiation. Therefore, the purpose of this article is to calculate the non-linear effect of the magnetic dipole and quadrupole fields on the propagation of electromagnetic waves. 2 FORMULATION OF THE PROBLEM We place the origin of the Cartesian coordinate system at the point where the pulsar or magnetar is located. Then, the vector of magnetic induction will have the form where B 1 is the vector of the dipole magnetic induction and B 2 is the vector of the quadrupole magnetic induction. In the Maxwellian approximation, we have where m is the magnetic dipole moment of the pulsar or magnetar. The components of the quadrupole magnetic induction vector B 2 in a spherical coordinate system r , θ, φ, as shown in Pétri ( 2015 ), will have the form where R s is the neutron star radius, B is a constant having a dimension of magnetic induction, and χ 1 ∈ [0, ] and χ 2 ∈ [0, 2] are two angles specifying the particular geometry of the quadrupole magnetic field. However, for further calculations it is more convenient for us to use a rectangular Cartesian coordinate system. Redesignating the constants B , χ 1 , and χ 2 in accordance with relations from the expression (8) we obtain where the notations were introduced f 1 = cos ξ, f 2 = sin ξcos χ 2 , f 3 = sin ξsin χ 2 , at that The B 0 constant can be expressed in terms of the magnetic induction of a quadrupole B N at a point θ = /2, φ = /4 on the surface of a neutron star: Suppose that an electromagnetic pulse is emitted from a certain point r = r s = { x s , y s , z s } at time t = t s . We assume that at the point r = r d = { x d , y d , z d } there is an electromagnetic radiation detector. We orient the axes of the Cartesian coordinate system so that the source and the electromagnetic radiation detector lie in the XOZ plane, and the Z - axis is


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

C of CH 2 and the C atoms can scramble, fractionation will occur. The dominant contribution of reaction ( 8 ) extends from somewhat greater than 10 3 yr to almost 10 5 yr, which is a much longer period than the time range when reaction ( 6 ) is dominant. Moreover, its range of dominance coincides more closely with the age range determined in multi-molecule fits to abundances in dark clouds (Wakelam et al. 2006 ). This reaction should, therefore, contribute to the differences in abundances between 13 CCH and C 13 CH, as mentioned before. Now consider the case of the isotopomer-exchange reaction:which is exothermic, as written from left to right so that the net effect is an increase in the abundance of the 12 C 13 CH isotopomer at the expense of 13 C 12 CH isotopomer. Furuya et al. ( 2011 ) used 1 × 10 −10 cm 3 s −1 as the rate coefficient of the forward reaction of the isotopomer-exchange reaction in their calculation. They also applied 8.1 K to the zero-point energy difference of 13 CCH and C 13 CH. The efficiency of this process depends upon a number of factors, including whether the exothermicity of 8.1 K is sufficient to cause the difference in observed abundance, whether the abundance of atomic hydrogen is large enough, and whether or not a barrier to reaction exists. The differences in abundances between 13 CCH and C 13 CH can be seen even at the early stages (Figure 3 in Furuya et al. 2011 ). This isotopomer-exchange reaction appears to be able to explain, at least partially, the 13 C isotopic fractionation in both L1521B and L134N. In summary, we found that C 13 CH is more abundant than 13 CCH in L1521B and L134N from our observations. This tendency agrees with the previous observations in TMC-1 and L1527 (Sakai et al. 2010 ). The higher abundance of C 13 CH compared with 13 CCH may indeed be common for dark clouds. Both reaction ( 8 ) and the isotopomer-exchange reaction likely contribute to the 13 C isotopic fractionation of CCH both in L1521B and L134N. 4.2. The Dilution of the 13C Species in Dark Clouds 4.2.1. Comparisons of the 12C/13C Ratios between CCH and HC3N among Dark Clouds It is interesting to compare the 12 C/ 13 C ratios of CCH and HC 3 N among the three dark clouds: L1521B, L134N, and TMC-1. Table 3 summarizes the 12 C/ 13 C ratios of CCH and HC 3 N in the three dark clouds and Figure 5 shows the comparisons. In the local ISM, the elemental 12 C/ 13 C ratio has been determined to be 60–70 (e.g., Milam et al. 2005 ), which we indicate as the yellow range in Figure 5 . Figure 5. Comparisons of the 12 C/ 13 C ratios of CCH and HC 3 N in the three starless cores; L1521B, L134N, and TMC-1. The errors indicate the standard deviation. The yellow range ( 12 C/ 13 C = 60–70) shows the values in the local interstellar medium. The values are the same as Table 3 . Table 3 The 12 C/ 13 C Ratios of CCH and HC 3 N in Dark Clouds Notes. a Taken from Sakai et al. ( 2010 ). b Taken from Taniguchi


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The noise is slightly lower (about 15%) than expected from the online APEX calculator because our noise measurements are made in moderately smoothed maps. 2.3. Baselining The data showed significant baseline structure, leading to large-scale correlated components in the resulting spectra. The baselines were removed over a velocity range km s -1 by the following procedure: 1. identify bright PPV regions in the line using the technique described in Sect. 3 ; 2. mask the regions identified as -bright for each other spectral line data cube; 3. fit a polynomial to the un-masked data; 4. subtract the fitted polynomial. We used a 7th-order polynomial for this process because lower order baseline removal left a significant and patterned residual. A more detailed examination of the baseline removal process is described in Appendix A . 3. Signal extraction and masking We use the method described partially in Ao et al. (2013) and more thoroughly in Dame (2011) to mask the data cubes at locations of significant signal in the brightest line. A noise map was created by computing the sample standard deviation over a 200 km s -1 range in which no signal was present. We use the line, which is the brightest of the lines, to create the mask by the following procedure: 1. Smooth the data with a Gaussian of width two pixels in each direction (spatial, , and spectral, km s -1 ). 2. Create an inclusive mask of all pixels with brightness in the smoothed map. 3. Grow the mask from the previous step by one pixel in each direction (this is known in image processing as binary dilation). The mask was then applied to the and cubes created with the same PPV gridding. There is some overlap between the line and the line in PPV space, so we shifted the mask to the velocity of the line in the cube and excluded those pixels expected to contain emission. However, inspection of the cube revealed that significant chunks of the signal were excised by this masking, so all further analysis excludes the line. Integrated intensity maps are shown in Fig. 2 a for the map masked with the above method and in Fig. 2 b for the map using the mask created from the data. Fig. 2 Integrated intensity (moment 0) maps of the a) and b) lines. The cubes were masked prior to integration using the method described in Sect. 3 . Selected region names are labeled. The thin black contours are at a column from the Herschel SED fit maps. 3.1. Ratio maps We have created ratio maps as a first step toward building temperature maps. In general, a higher indicates a higher temperature, and therefore the ratio maps are a good proxy for relative temperature. We display four different varieties of ratio map in Figs. 3 and E.1 . Figure 3 shows maps produced by creating ratio cubes, in which each voxel has the value , then averaging across velocity with constant weight for each voxel in Fig. 3 a and with the weight set by the brightness value in Fig. 3 b. In both cases, the mask described in Sect. 3 was applied to the


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( MK; Maggio et al. 2000 ). The abundances during the flares F1 and F2 are found to enhance ∼9 and ∼2 times more than those of the minimum values observed. During other superflares, abundances were found to increase between two to three times more than that of the quiescent level (Favata Schmitt 1999 ; Favata et al. 2000 ; Maggio et al. 2000 ). However, in the case of the superflare observed with Swift in EV Lac, the abundances were found to remain constant throughout the flare. From Figure 3 , it is evident that the abundance peaks after the temperature and luminosity peaks, which is consistent with a current idea in the literature (see Reale 2007 ). This could be due to the heating and evaporation of the chromospheric gas, which increases the metal abundances in the flaring loop. For both flares, the temperature peaked before the EM did. A similar delay was also observed in many other solar and stellar flares (e.g., Sylwester et al. 1993 ; Favata Schmitt 1999 ; Favata et al. 2000 ; Pandey Singh 2008 ). The temperature increases due to beam driven plasma heating and later subsequent evaporation of the plasma into upper parts of the coronal loop that increases its density, and therefore EM (). Later coronal plasma cools down by thermal conduction and then via radiative losses (e.g., Cargill Klimchuk 2004 ). 4.2. Coronal Loop Properties The derived loop lengths for the flares F1 and F2 are larger than previously observed flares by Crespo-Chacón et al. ( 2007 ) on CC Eri. The loop lengths are also in between the loop lengths derived for other G–K dwarfs, dMe stars, and RS CVn type binaries (e.g., Favata Micela 2003 ; Pandey Singh 2008 , 2012 ). Instead of ∼3 times larger peak luminosity in flare F1 than that of the flare F2, the derived loop length for flare F2 is two times larger than that of flare F1, which might be interpreted as a result of an ∼1.4 times more sustained heating rate in the decay phase of flare F1 than that of flare F2. The heating rate during flare F1 is also found to be ∼49% of the bolometric luminosity of the CC Eri system, whereas during flare F2 the heating rate is only ∼35% of the combined bolometric luminosity. The derived heating rate is also found to be more than the maximum X-ray luminosity for both flares. This result is compatible with X-ray radiation being one of the major energy loss terms during the flares. Present analysis allows us to make some relevant estimation of the magnetic field strength that would be required to accumulate the emitted energy and to keep the plasma confined in a stable magnetic loop configuration. Under the assumptions that the energy release is indeed of magnetic origin, the total non-potential magnetic field involved in a flare energy release within an active region of the star can be obtained from the relationAssuming that the loop geometry does not change during the flare, is estimated to be >12 and >2 kG for the flares F1 and F2, respectively. Bopp Evans ( 1973 ) also estimated a


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for σ u and bottom panels for σ l ; left panels for substituting ν + △ν and right panels for substituting ν − △ν into the MHD equations. In order to make a more reasonable estimate of the NS mass, we consider the error of QPOs. After substituting ν upper + △ν or ν upper − △ν into the MHD equations, the mass of the NS in 4U 0614+09 can be estimated as M 2.60 M ⊙ or M 2.50 M ⊙ from the minimum σ l . Similarly, after substituting ν lower + △ν or ν lower − △ν into the MHD equations, the mass of the NS in 4U 0614+09 can be estimated as M 2.50 M ⊙ or M 2.60 M ⊙ from the minimum σ u . The results are shown as panels (a), (b), (c) and (d) of Fig. 8 . Similar to the above process, the mass of the NS in 4U 1636–53 can be estimated as 2.05 M ⊙ M 2.31M ⊙ , 2.00 M ⊙ M 2.31M ⊙ with σ u from the panels of Fig. 9 . In summary, the mass of the NS in 4U 0614+09 is estimated as M 2.60 M ⊙ and that in 4U 1636–53 as 2.00 M ⊙ M 2.31M ⊙ . We do not recalculate the result of 4U 1608–52 because there are very limited results with errors reported. Figure 9. The minimum like-standard deviations σ l and σ u corresponding to the different masses of the NS in 4U 1636–53 when the errors of the observed kHz QPOs are considered. Top panels for σ u and bottom panels for σ l ; left panels for substituting ν + △ν and right panels for substituting ν − △ν into the MHD equations. Figure 10. Mass–radius relationships for various equations of state given in Özel Freire ( 2016 ). The dotted line, twin solid lines and twin dash–dot lines correspond to the masses M = 2.60 M ⊙ , , , respectively, which are obtained from the estimation by the twin kHz QPOs in 4U 0614+09, 4U 1636–53, 4U 1608–52. 5 DISCUSSION As the dividing line between NSs and black holes, EOSs affect the number of observable NSs and their properties, because different EOSs might lead to different mechanisms of supernova explosions (Özel Freire 2016 ). As seen from Fig. 10 , different EOSs have different maximum masses of NSs. Thus a high-mass NS can be used to rule out some EOSs. For 4U 0614+09, M 2.60 M ⊙ , and no constraint can be put on any EOS. For 4U 1636–53, 2.00 M ⊙ M 2.31 M ⊙ rules out EOSs of strange quark matter, which is consistent with the conclusion obtained from the lower limit on the NS radius (11.0 km) that was inferred from observed mHz QPOs modelled as marginally stable burning on the NS surface (Stiele, Yu Kong 2016 ). Güver et al. ( 2010 ) estimated the mass and radius of the NS in 4U 1608–52 when the most probable distance () was measured by the interstellar extinction towards the source. They obtained a mass of M = 1.74 ± 0.14 M ⊙ and a radius of . Steiner, Lattimer Brown ( 2010 ) obtained a larger radius of the NS (about ) when the photospheric radius is much larger than the radius of the NS. Poutanen et al. ( 2014 ) studied 21 photospheric radius expansion bursts from 4U 1608–52 and obtained a NS radius R larger than 12 km for 1.0 M ⊙ M 2.4 M ⊙ , and R restricted to be between 13 and 16 km for


O O O O O O O O O O O O O O O O B-Person O O O O O B-Organization I-Organization I-Organization B-Organization O B-Grant I-Grant B-Person O O O O O B-Organization I-Organization I-Organization O O B-Grant

We thank the anonymous referee for useful comments and suggestions that helped to improve the manuscript. P.C. acknowledges financial support of the European Research Council (ERC; project PALs 320620). M.J. acknowledges the support of the Academy of Finland Grant No. 285769.


O O O O O O O O O O O O O O O O O O O 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 B-Citation O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject O O O O O O O O O O O O O O O O O O

of sight lines that cross through the z -axis of the domain behind the cloud’s head. The angles between each of these sight lines and the sight line through the cloud’s head were determined from the assumption that the observer was on Earth, 12.4 kpc ( L08 ) from the SC (see Figures 1 , 2 , and 4 ). Figure 5 (a) shows as a function of this angle for Model A at various epochs. For angles ≲30°, reaches a reasonably steady state by t = 300 Myr (i.e., the variation of from epoch to epoch is relatively small compared to earlier epochs). However, there is still some time variation of the profiles, due to the fact that the flow in the model HVC’s tail is not steady, and for the last few tens of megayears of the simulation, is drifting toward larger values for angles ≲15°. For Models B and C (Figures 5 (b) and (c), respectively), we find that also reaches a reasonably steady state by around t = 300 Myr, although again there is some time variation of the profiles. In addition, the Model C profiles exhibit gaps at some angles, meaning that there is no material above the high-velocity cutoff along the sight line at those particular angles. These gaps in the high-velocity material distribution are due to the higher-density ambient medium having a stronger braking effect on the cloud material than in the other models (this braking effect was noted in footnote 1 for a model with an even higher ambient density, for which the effect was more severe than in Model C). Figure 5. Fraction of the high-velocity material originating in the halo, , as a function of angular distance behind the cloud head, extracted from various epochs of Models A–C (top to bottom; see keys). The vertical dashed lines indicate the location of F16 's sight lines relative to the SC head. Figure 6 shows averaged over multiple epochs. Within the epoch-to-epoch variations, these averaged profiles are similar for Models A and B at the cloud head and at angles ≳20° behind the cloud head. The difference between the models at intermediate angles is because the Model A cloud fragments more than the Model B cloud (as noted above), leading to more rapid mixing in the first few kiloparsecs behind the cloud head. However, other than in the first few kiloparsecs behind the cloud head, the mixing results are not very sensitive to the assumed cooling rate. In contrast, the averaged profiles from Models A and C are similar at all angles, though this is in part due to the large time variation of the Model C profiles. These results suggest that the extent of mixing is not very sensitive to the assumed ambient density. Figure 6. sampled every 1 Myr and averaged over t = 300–400 Myr from Models A, B, and C (red, blue, and green, respectively). The colored bands indicate the standard deviation. The vertical dashed lines indicate the location of F16 's sight lines relative to the SC head. Figure 7 compares the averaged profiles calculated from versions of our reference model with different maximum resolutions.