Magnetic Inflation and Stellar Mass. III. Revised Parameters for the Component Stars of NSVS 07394765

We accessed the NASA Exoplanet Archive (Akeson et al. 2013) to download publicly available data for NSVS 0739 from the Wide Angle Search for Planets (WASP; Butters et al. 2010). The WASP passband ranges from ∼400–700 nm, roughly encompassing the Sloan g and r bands. The observations were made between 2004 September and 2008 April for a total of 6718 photometric data points. With a V-band magnitude of 13.0, NSVS 0739 is near the limiting magnitude of the survey, introducing noticeable noise into the light curve. Nonetheless, Figure 2 shows that WASP clearly detected both eclipses for the system. See Table 1 for the details of the WASP observations.

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Table 1.  Description of WASP Observations for NSVS 0739

Coordinates (R.A. Decl.)	8h25m51894, 24°27'460
WASP magnitude	13.17819
Start time (BJD)	2453261.742889
End time (BJD)	2454575.437458
Number of points	6718

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We removed two nights of WASP data for which there were data points significantly deviating from the expected in-eclipse value despite having a phase corresponding to an eclipse. These points were likely caused by adverse weather conditions during these nights. We also established a maximum relative flux limit of 1.1 to exclude extreme increases in flux corresponding to weather or flares on the component stars. We converted time units from the archive-supplied HJD to BJD_TDB using a calculator by Eastman et al. (2010).

We also obtained a new NSVS 0739 primary eclipse observation on UT 2017 February 1 using the 32 cm Dall-Kirkham telescope at CMO in Tempe, AZ. The detector is a thermo-electrically cooled SBIG ST-6303On CCD. The night was photometric, and we acquired 180-second exposures in the Johnson V band. We used the commercial software package MPO Canopus to perform aperture photometry on NSVS 0739 and four reference stars with 15'' apertures and sky annulus subtraction. This software specializes in asteroid and variable star analyses, offering a graphical interface for image calibration, astrometry, and photometry. We used the AAVSO Photometric All-Sky Survey (APASS) DR9 to supply V magnitudes for the reference stars. Because we observed multiple targets on this night, there are gaps in coverage of the NSVS 0739 eclipse (Figure 3).

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We observed another primary eclipse (Figure 4) on UT 2019 April 14 with the 0.7 m telescope at Thacher Observatory in Ojai, CA (O'Neill et al. 2017; Swift & Vyhnal 2018). The new observation was made in the Johnson V band with integration times of 1 minute. We reduced the data using the astropy utilities in Python (Astropy Collaboration et al. 2013, 2018). We performed aperture photometry on NSVS 0739 and two nearby reference stars that we tested for stability and high S/N. Given the variability of the seeing, we optimized the aperture used in each image to maximize the S/N on NSVS 0739, and we chose sky radii from stacked images to avoid background sources in areas outside the wings of the point-spread function (PSF). The final light curve was stable and did not require us to fit the out-of-eclipse data for a trend in airmass or time.

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We used the Immersion Grating Infrared Spectrograph (IGRINS; Park et al. 2014; Prato 2017) at the Discovery Channel Telescope to obtain spectra for NSVS 0739 at five different times. IGRINS is a cross-dispersed, near-infrared, high-resolution spectrometer covering wavelengths between 1.45 and 2.45 μm (H and K bands) at R ∼ 45,000. Calculated exposure times were intended to provide an S/N of at least 10. See Figure 5 for an example H-band spectrum. Given that the orbital period of NSVS 0739 was on the order of two days, useful observations only needed to be separated by a few hours.

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We reduced the IGRINS data following the procedures described in Han et al. (2017). Briefly, the first step involved feeding the raw data through the IGRINS pipeline (Sim et al. 2014). We then used observations of a nearby A0 star to calibrate for telluric lines. We observed the A0 star on the same night as each target observation, under similar weather conditions. The telluric correction was done using the xtellcor_general data reduction software package written in IDL (Vacca et al. 2003). The software propagates uncertainties in the telluric correction to the final uncertainties in the spectra.

Using TODCOR (Zucker & Mazeh 1994), we performed a two-dimensional cross-correlation between two template BT-Settl model spectra (Allard et al. 2012; Baraffe et al. 2015) for stars with 3100 and 3300 K effective temperatures (Figure 5) and high-S/N spectral orders 8–14 of our H-band IGRINS data to find the radial velocity of both stars in the system (e.g., Figure 6). Orders near the edges of the detector experience distortions that diminish the quality of the derived radial velocities, and other unused orders contain large telluric features that are not sufficiently corrected by the A0 calibration spectrum. We obtained uncertainties on the radial velocities by performing cross-correlation with each order independently and taking a standard deviation of the mean of the results. We did not use the lower-S/N K-band data from IGRINS because the cross-correlation functions were not as definitive in this band. To account for the motion of the Earth around the Sun, we performed a barycentric correction using the tools of Wright & Eastman (2014). Table 2 lists the five new radial velocity points.

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Under the assumption that the passbands of WASP, the Command Module Observatory (CMO), and Thacher data overlapped, we fit a model to the photometry and radial velocities based on the eb software by Irwin et al. (2011). This code generates simulated photometry and radial velocity curves. The upper rows of Table 3 show the model parameters that were fitted. Under the assumption of no third light, and neglecting the effect of light-travel time due to the system's nearly equal-mass components and low eccentricity, we made an initial least-squares fit using mpfit (Markwardt 2009). Initiating uniform priors for each parameter centered on our results from mpfit, we performed a Markov chain Monte Carlo (MCMC) exploration of parameter space using emcee (Foreman-Mackey et al. 2013). For the MCMC, we established a normal likelihood function (LF) of the form

$$\begin{eqnarray}&&\mathrm{ln}(\mathrm{LF})=-0.5\ast \displaystyle \sum _{i=1}^{{n}_{\mathrm{points}}}\left[\displaystyle \frac{{\left({y}_{i}-{\hat{y}}_{i}\right)}^{2}}{{\sigma }_{i}^{2}}+\ \mathrm{ln}(2\pi {\sigma }_{i}^{2})\right],\end{eqnarray} \tag{ 1 }$$

summing over all points (n_points) in the phase-folded data (y_i) in comparison with model points (${\hat{y}}_{i}$) generated by eb and the uncertainty of each data point, σ_i. For each step out of the total 50,000 and each chain out of 100, we performed Affine-Invariant sampling of the fitted parameters with the EnsembleSampler class of emcee. After visual inspection of the chains for each parameter, we discarded a burn-in of the first 10,000 steps to prevent our results from being biased toward the prior values.

Table 3.  NSVS 0739 Fitted and Calculated Parameter Descriptions, Maximum-likelihood Values, and 1σ Uncertainties

Fitted Parameter	Description	NSVS 07394765
(1)	(2)	(3)
J	Central surface brightness ratio	${0.66}_{-0.06}^{+0.20}$
(R1 + R2)/a	Fractional radii sum over semimajor axis	${0.1555}_{-0.0006}^{+0.0022}$
R2/R1	Radius ratio	${1.043}_{-0.093}^{+0.053}$
$\cos i$	Cosine of orbital inclination	${0.0041}_{-0.0036}^{+0.0020}$
P	Orbital period	${2.26537743}_{-0.00000005}^{+0.00000021}$ days
T0	Time of primary mid-eclipse	${2454573.45195}_{-0.00029}^{+0.00011}\ {\mathrm{BJD}}_{\mathrm{TDB}}$
$e\cos \omega$	Eccentricity × cosine of argument of periastron	$-{0.00011}_{-0.00020}^{+0.00035}$
$e\sin \omega$	Eccentricity × sine of argument of periastron	$-{0.001}_{-0.010}^{+0.027}$
γ	Center of mass system velocity	$-{3.5}_{-1.3}^{+0.5}$ km s−1
q	Mass ratio (M2/M1)	${0.921}_{-0.014}^{+0.017}$
Ktot/c	Sum of radial velocity semi-amplitudes/speed of light	${0.0005860}_{-0.0000096}^{+0.0000015}$
u11	Linear limb-darkening coefficient, star 1	$-{0.22}_{-0.54}^{+0.60}$
u21	Square-root limb-darkening coefficient, star 1	${0.43}_{-0.87}^{+0.45}$
u12	Linear limb-darkening coefficient, star 2	${0.9}_{-1.4}^{+0.7}$
u22	Square-root limb-darkening coefficient, star 2	${0.1}_{-0.6}^{+1.0}$
Calculated Parameter	Description	NSVS 07394765
e	Eccentricity	${0.001}_{-0.001}^{+0.017}$
i	Orbital inclination	${89.76}_{-0.12}^{+0.21}\ \mathrm{degrees}$
a	Semimajor axis	${0.03651}_{-0.00060}^{+0.00009}$ au
K1	Radial velocity semi-amplitude, star 1	${84.3}_{-1.3}^{+0.2}$ km s−1
K2	Radial velocity semi-amplitude, star 2	${91.5}_{-2.1}^{+0.7}$ km s−1
M1	Mass, star 1	${0.661}_{-0.036}^{+0.008}\,{M}_{\odot }$
M2	Mass, star 2	${0.608}_{-0.028}^{+0.003}\,{M}_{\odot }$
R1	Radius, star 1	${0.599}_{-0.019}^{+0.032}\,{R}_{\odot }$
R2	Radius, star 2	${0.625}_{-0.027}^{+0.012}\,{R}_{\odot }$
log g1	Log of surface gravity, star 1 (cgs)	${4.705}_{-0.051}^{+0.018}$
log g2	Log of surface gravity, star 2 (cgs)	${4.632}_{-0.024}^{+0.028}$
L2/L1	Orbit-averaged photometric light ratio	${0.72}_{-0.18}^{+0.31}$

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Due to the small number of radial velocity data points compared to the plentiful photometric points, we performed the latter fit separately from the former, with the same number of steps, chains, and burn-in. This separation of fits ensured that the overall determination of goodness-of-fit was not dominated only by photometry (see Section 3.2 of Han et al. 2017, who also used this fitting method). We show the final photometric fit (including WASP, CMO, and Thacher eclipses) and residuals in Figure 7. The residual structure visible in the CMO and Thacher primary eclipse fits shows the limitation of the assumption that the WASP passband overlaps with these V-band observations. Nonetheless, these new data helped to better constrain the period and time of mid-primary eclipse by roughly quadrupling the time baseline of eclipse observations and supplementing WASP data with higher-cadence coverage. The primary and secondary radial velocity fits appear in Figure 8.

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We fit limb-darkening coefficients for each star using the square-root model demonstrated to be effective for low-mass M dwarfs in Claret (1998). During fitting, we parameterized limb-darkening in terms of q1 and q2 from Kipping (2013). Though the final fit did not provide strong constraints on these coefficients, the sampling of a wide variety of limb-darkening coefficients induced additional variation in the best-fit parameters for each step, widening the distribution of the calculated radius values compared to a fit with better-constrained limb-darkening coefficients. Therefore, limb-darkening uncertainties are incorporated into the error of the other fitted parameters.

The MCMC run yielded 100 chains of 40,000 values for every fitted parameter (after discarding the burn-in steps). To solve for the desired results and their uncertainties, we calculated each final parameter from its distribution of all 4 × 10⁶ values. We adopted each distribution's maximum-likelihood value to be the reported parameter value and computed its difference from the 16th and 84th percentiles of each distribution to establish the 1σ confidence intervals reported as our uncertainties. For the eccentricity parameter, we instead used the 0th and 68th percentiles for error bars, because the nearly circular system does not have a normal distribution about the highest-likelihood value. Table 3 lists the maximum-likelihood fitted and calculated parameters with their 1σ uncertainties. Note that to minimize the results' dependence on stellar atmospheric models, we do not compute the stellar effective temperature or luminosity ratio. See Section 4.5 of Han et al. (2017) for a further explanation. We show triangle plots (Foreman-Mackey 2016) for the photometric and radial velocity fits in Figures 9 and 10, respectively. We note that our ephemeris predicts future eclipses at significantly different times than the discovery papers.

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After our analysis, the stars now fall into statistical agreement with the empirical and theoretical mass–radius trends for M dwarfs (Figure 11). We discuss our confidence in the new masses and radii, along with a possible cause for the initial hyper-inflated results, in Section 4.

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