Magnetic Inflation and Stellar Mass. I. Revised Parameters for the Component Stars of the Kepler Low-mass Eclipsing Binary T-Cyg1-12664

We obtained Kepler light curve data for quarters 1 through 17 from the Mikulski Archive for Space Telescopes (MAST). Long-cadence data recorded at regular intervals and with exposure times of 1766 s are available for all quarters of the primary Kepler mission except for quarters 7, 11, and 15. No short-cadence data are available, which is contradictory to what is reported in Çakırlı et al. (2013). On inspection, the Kepler light curves show roughly 100 primary and secondary eclipses with a period consistent with the orbital period reported in Çakırlı et al. (2013). The light curves show out-of-eclipse modulation that is nearly synchronous with the system orbital period, reaching a maximum peak-to-peak amplitude of $\sim 3 \%$. We attribute the modulation to starspots on the primary star combined with synchronous stellar rotation. We used the PDCSAP_FLUX data, which is corrected for effects from instrumental and spacecraft variation (Smith et al. 2012; Stumpe et al. 2012). We also removed obvious outliers by hand. Figure 1 shows the first three primary and secondary eclipse pairs in quarter 6 of Kepler data.

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2.2.1. IGRINS Observations

We observed T-Cyg1-12664 using the the Immersion GRating INfrared Spectrometer (IGRINS, Yuk et al. 2010) on the 4.3 m Discovery Channel Telescope (DCT) on the nights of UT 2016 October 16 through UT 2016 October 18. IGRINS is a cross-dispersed, high-resolution near-infrared spectrograph with wavelength coverage from 1.45 to 2.5 $\mu {\rm{m}}$. IGRINS has a spectral resolution of $R=\lambda /{\rm{\Delta }}\lambda \,=$ 45,000 and allows for simultaneous observations of both H- and K-band in a single exposure (Yuk et al. 2010; Park et al. 2014; Mace et al. 2016). The exposure times were calculated to achieve a signal-to-noise ratio of ∼10 or higher per wavelength bin. We observed A0V standard stars (HR 7098 and HD 228448) that are within 0.2 airmasses of T-Cyg1-12664, before or after target observations, for the purpose of telluric corrections. There is a publicly available reduction pipeline for the IGRINS (Sim et al. 2014), and we used this pipeline to process all of our spectra. IGRINS is a visiting instrument at the DCT whose principal site is the McDonald Observatory in Texas.

Cross-correlation templates with spectral types between G1 and M4 were also observed with IGRINS on the 2.7 m Harlan J. Smith Telescope at McDonald Observatory, and reduced in the same manner as T-Cyg1-12664. Radial velocities for the template stars were determined using the method summarized in Mace et al. (2016) and are precise to 0.5 km s⁻¹.

IGRINS' H- and K-band data contain 28 and 25 orders, respectively. The pipeline performs dark subtraction and flat-fielding first, followed by an AB subtraction to remove the OH airglow emission lines, and finally extracts the spectrum. The pipeline uses telluric airglow emission lines for wavelength calibration. However, the current pipeline version does not support careful removal of telluric absorption lines, so we further processed the pipeline extracted spectra. For this task, we used xtellcor_general, a software tool designed to remove telluric lines from near-infrared spectra (Vacca et al. 2003). The software accepts a measured spectrum of an A0V star and a target spectrum. It uses a model spectrum of Vega (an A0V star) to construct the telluric spectrum, calculates the relative shift between the observed A0V standard and the target spectrum, and applies the shift to the constructed telluric spectrum. In its final step, xtellcor_general divides the telluric spectrum from the target spectrum.

The middle panel in Figure 2 shows a sample IGRINS H-band telluric-corrected spectrum. We processed all of the target spectra and the radial velocity standard spectra to remove telluric lines. We selected the radial velocity standards in consideration of previously reported spectral types for each component star. Çakırlı et al. (2013) reported spectral types of K5 and M3 and Iglesias-Marzoa et al. (2017) reported spectral types of G6 and M3. We used a G5 template as the radial velocity standard for the primary component (HIP 102574, with radial velocity of −69.8 km s⁻¹) and an M3 template as a radial velocity standard for the secondary component (GJ 752, with radial velocity of 36.6 km s⁻¹).

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We only used IGRINS H-band data as the sky background in the K-band reduced the signal-to-noise of the reduced spectra. Of the 28 orders in the H-band spectra, we selected the 8th through the 14th, due to their signal to noise. These orders gave us a wavelength coverage of 1.59–1.70 $\mu {\rm{m}}$.

To measure the radial velocity, we cross-correlated the target spectra with the template spectra. Before cross-correlating, we transformed the wavelength scale of the spectra from regular intervals to logarithmically increasing intervals, so that radial velocity shifts in the spectra are equivalent to the same fractional shift in the wavelength interval regardless of the order. We interpolated the spectra onto the logarithmically increasing wavelength grid using a linear spline function. Next, we used the Two-dimensional CORrelation software package (TODCOR; Zucker & Mazeh 1994) to measure radial velocities of each component star. TODCOR simultaneously calculates the radial velocities of each component by cross-correlating the target spectrum against the two templates over a range of radial velocities. This produces a two-dimensional cross-correlation function, with the peak location corresponding to each component's measured radial velocity. We ran TODCOR on each order, using the mean between the orders as our measured radial velocities. We estimated the uncertainties by calculating the root mean square of the radial velocities across the orders and divided by the square root of number of orders used. Figure 3 shows an example of a two-dimensional cross-correlation function. We calculated the barycentric Julian date for each observation, converted the radial velocities into the reference frame of the solar system barycenter for both the target and the radial velocity template, and reported those as the final radial velocity measurements.

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2.2.2. NIRSPEC Observations

2.2.3. HIRES Observations

As discovered by Çakırlı et al. (2013), a faint and slightly redder object appears blended with T-Cyg1-12664 in seeing-limited images. To determine the role of this object in the Kepler light curve and corresponding EB parameters, we acquired visible-light and infrared AO imaging of T-Cyg1-12664 using the Robo-AO system on the 60 inch Telescope at Palomar Observatory (Baranec et al. 2013, 2014; Law et al. 2014). We observed T-Cyg1-12664 on UT 2014 June 17 using a clear anti-reflective coated filter. The camera response function is spectrally limited by the E2V CCD201-20 detector response, with a steep drop off short-ward of 400 nm and long-ward of 950 nm. This closely matches the response of the Kepler camera, which employs no filters and is primarily dictated by the CCD response. The individual images were combined using post-facto shift-and-add processing using T-Cyg1-12664 as the tip-tilt star. We detected the faint object at a separation of of $4\buildrel{\prime\prime}\over{.} 12\pm 0\buildrel{\prime\prime}\over{.} 03$ and a position angle of 283° ± 2° with respect to T-Cyg1-12664 (see Figure 4). We measured a contrast of 3.91 ± 0.10 mag in the Kepler band (K_P) between the EB and the third object.

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On UT 2014 September 3, we observed T-Cyg1-12664 in the H-band with Robo-AO, using an engineering grade Selex ES Infrared SAPHIRA detector (Finger et al. 2014) in a GL Scientific cryostat mounted to the Robo-AO near-infrared camera port (Atkinson et al. 2016) with active infrared tip-tilt guiding using T-Cyg1-12664 as the reference star (Baranec et al. 2015). The contrast in the H-band was measured to be 2.74 ± 0.10 mag.

Archival photometry of T-Cyg1-12664 from the Kepler Input Catalog (KIC, Batalha et al. 2010) and 2MASS (Cutri et al. 2003) list magnitudes of K_P = 13.100 ± 0.03 and H = 11.582 ± 0.015 for the blended objects. Combining these measurements with the Robo-AO contrast measurements, we calculated magnitudes of K_P = 13.129 ± 0.031 and H = 11.666 ± 0.017 for the EB and K_P = 17.04 ± 0.10 and H = 14.41 ± 0.09 for the third object. The color (${K}_{P}-H=2.63\pm 0.14$) of the third object is consistent with an early M dwarf star or a distant, intrinsically bright, and reddened evolved star. Calculating a photometric parallax, we find that if the third object were a dwarf, it would reside roughly 150 pc more distant than the EB, though photometric parallaxes are highly uncertain. We note that widely separated, physically associated stars are common near EBs and are consistent with proposed scenarios for the formation of close binaries via Kozai cycles (Fabrycky & Tremaine 2007; Tokovinin 2017).

To study the eclipses in detail, first we modeled the out-of-eclipse modulations in order to remove their effects from the eclipse events. We discuss the causes of these modulations in more detail in Section 4. We used "george", a Gaussian processes module written in Python (Ambikasaran et al. 2014). Gaussian processes are a generalization of the normal (Gaussian) probability distribution. The technique assumes that every point in the time series is associated with a normally distributed random variable and covariance between data points is constant over the data set. The george software package employs "kernels" to measure the covariance between data points in the time series. The uncertainty is calculated by taking the determinant of the n × n covariance matrix where n is the number of data points in the time series.

The out-of-eclipse modulations evident in the Kepler light curve show quasi-periodic behavior, that is within 3% of the orbital period of the system. We used the exponential-squared and the exponential-sine-squared kernels in george to describe the following behaviors observed in the light curve modulations: amplitude, decay/growth, and the period. We obtained the model light curve for the out-of-eclipse modulation by combining the two kernels through multiplication and fit it to the Kepler data using a Levenberg–Marquardt algorithm implemented in Python (mpfit Markwardt 2009). Figure 5 shows the out-of-eclipse light curve of the quarter 6 data and the best-fit detrending model obtained from george, and the resulting residuals. After detrending, we normalized the flux by dividing by the median value.

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To model the eclipse events, we used the publicly available eclipsing binary modeling code "eb" written for detached eclipsing binaries by Irwin et al. (2011). The eb software package creates model eclipse light curves based on 37 parameters, each of which are described in Irwin et al. (2011). For a given set of parameters and time stamps, eb generates a synthetic light curve and a synthetic radial velocity curve. We chose to fit for 16 parameters and fixed the remaining 21 parameters, which describe starspots, gravity darkening, and reflection effects. The 16 free parameters that were fitted are listed in Table 1. We smoothed the eb light curve model to account for the Kepler long-cadence integration time. Coughlin et al. (2011) investigated the effect of Kepler long-cadence integration time in light curves of eclipsing binaries and found that the shape of light curve model can be changed significantly if the long-cadence integration time is not accounted for, which will result in erroneous measurements in the stellar parameters. However, this effect is only shown in the systems with small relative radii sum (<0.1) and short orbital periods (<2.5 days), and we note that this is not the case for T-Cyg1-12664.

Table 1.  Modeling Parameters

Parameter	Description
J	Central Surface Brightness ratio
$({R}_{1}+{R}_{2})/a$	Fractional sum of the radii over the semimajor axis
${R}_{2}/{R}_{1}$	Radii ratio
$\cos i$	Cosine of orbital inclination
P (days)	Orbital period in days
T0 (BJD)	Primary mid-eclipse
$e\cos \omega$	Orbital eccentricity × cosine of argument of periastron
$e\sin \omega$	Orbital eccentricity × sine of argument of periastron
L3	Third light contribution from a nearby companion
γ (km s−1)	Center of mass velocity of the system
q	Mass ratio (M2/M1)
Ktot	Sum of the radial velocity semi-amplitude
${u}_{{K}_{p}}$	Linear limb-darkening coefficient in Kepler band
$u{{\prime} }_{{K}_{p}}$	Square root limb-darkening coefficient in Kepler band

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Table 2.  Measured Radial Velocities for the Primary and the Secondary Stars

BJD	V1 (km s−1)	${\sigma }_{1}$ (km s−1)	V2 (km s−1)	${\sigma }_{2}$ (km s−1)	Instrument
2457678.606272	−43.4	1.0	49.1	5.8	IGRINS
2457678.667269	−48.5	0.5	60.2	7.0	IGRINS
2457679.604463	−50.4	5.9	63.9	5.8	IGRINS
2457679.702549	−48.1	0.5	57.0	0.4	IGRINS
2457680.570521	9.0	1.6	−51.7	3.8	IGRINS
2457680.697738	17.1	1.1	−63.5	5.8	IGRINS
2457680.773412	21.7	5.9	−77.3	9.4	IGRINS
2454311.884614	34.1	5.9	−91.0	1.9	NIRSPEC
2456845.013664	−55.6	10.7	82.6	5.5	NIRSPEC
2456851.948214	29.9	2.6	−73.3	10.9	NIRSPEC
2456827.110291	34.4	0.4	−92.0	3.7	HIRES
2456829.040765	49.3	1.1	66.4	6.8	HIRES
2456829.918712	4.9	0.8	−35.1	6.8	HIRES
2456831.098019	40.4	0.8	−103.5	5.9	HIRES
2456843.064820	40.6	0.5	−104.8	6.8	HIRES
2456844.062487	1.9	1.7	−28.7	6.8	HIRES
2456845.058524	−60.6	0.7	⋯	⋯	HIRES
2456846.116576	−17.5	0.6	4.6	6.8	HIRES
2456846.946306	32.8	1.14	⋯	⋯	HIRES
2456849.053336	−58.5	0.6	⋯	⋯	HIRES
2456849.986511	−32.8	0.6	⋯	⋯	HIRES
2456851.894018	35.0	0.8	−87.7	6.1	HIRES
2456854.057728	−37.9	1.5	⋯	⋯	HIRES

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After we detrended, normalized, and phase-folded the Kepler light curves, we employed the Levenberg–Marquardt technique and performed chi-square minimization, which was done via Python's external package, "mpfit" (Markwardt 2009). We ran the mpfit algorithm three times in order to determine the best-fit values: once just varying the period and the epoch of the primary eclipse, once varying the rest of the parameters described in Table 1 except for the four limb-darkening parameters, and then finally varying all 16 parameters. Excluding for the first run, we performed each mpfit run using the best-fit parameters obtained from the previous run. For the first two mpfit runs, we did not fit the limb-darkening parameters, as limb-darkening is a higher-order effect on the shape of the light curve. Instead, we adopted the square root limb-darkening law, as demonstrated in Claret (1998) to be superior for low-mass stars, like M dwarfs. We set u1 = 0.63 and u2 = 0.6043 for the primary and u1 = 0.4580 and u2 = 0.6508 for the secondary, accounting for the effective temperatures (Claret & Bloemen 2011) based on the spectral types of Çakırlı et al. (2013). After the second mpfit run, we determined that the best-fit was close enough to treat the limb-darkening coefficients as free parameters. For the mpfit to return a good-fit, we had to be careful with choosing the step sizes. We excluded the majority of the out-of-eclipse light curves, as they were the dominant noise source in the ${\chi }^{2}$ calculation and as the flattened out-of-eclipse fluxes had no information on the component stars.

To further refine the fit and determine reliable uncertainties for the individual parameters, we employed a Markov chain Monte Carlo (MCMC) algorithm. We used Python's external MCMC package, emcee, written by Foreman-Mackey et al. (2013). We used the best-fit parameters from the last mpfit fit as the starting parameters in the MCMC chains. We employed 100 chains, each with 10,000 steps, and assumed uniform priors on all parameters. We varied T₀, P, J, $\cos i$, $e\cos \omega$, $e\sin \omega$, $(R1+R2)/a$, $R2/R1$, and four limb-darkening parameters, two for each component. For the limb-darkening parameters, we stepped in the q₁ and q₂ parameterization of limb-darkening, developed by Kipping (2013), rather than the linear and quadratic coefficients. The q₁ and q₂ parameterization of limb-darkening forces all possible combinations of the parameters to be physical, as long as both values are between 0 and 1. We set the third light, L3, to the value we directly measured from the AO imaging.

Although the eb model takes $e\cos \omega$ and $e\sin \omega$ as free parameters, we stepped in $\sqrt{e}\cos \omega$ and $\sqrt{e}\sin \omega$ as Eastman et al. (2013) argue this is more efficient. Once completed, we discarded the "burn-in" and took the maximum likelihood parameters as the best-fit values, and the standard deviations of the parameter distributions as the uncertainties. We report our best-fit values and uncertainties in Table 3.

Table 3.  Parameters for T-Cyg1-12664 (this work)

Fitted	Primary	Secondary
J	0.0675 ± 0.0069
$({R}_{1}+{R}_{2})/a$	0.1138 ± 0.0023
${R}_{2}/{R}_{1}$	0.5161 ± 0.0224
cosi	0.0666 ± 0.0034
P (days)	4.12879671 ± 0.00000003
T0 (BJD)	2454957.32116092 ± 0.0000051
$e\cos \omega$	-0.00176 ± 0.00002
$e\sin \omega$	${0.0438}_{-0.0089}^{+0.0121}$
L3	0.0265 ± 0.0025 (fixed)
γ (km s−1)	−8.6 ± 0.4
q	0.54 ± 0.01
Ktot	149.6 ± 1.6
${u}_{{K}_{P}}$	${0.053}_{-0.030}^{+0.039}$	${0.757}_{-0.171}^{+0.242}$
$u{{\prime} }_{{K}_{P}}$	${0.944}_{-0.057}^{+0.033}$	${0.025}_{-0.025}^{+0.107}$
Calculated	Primary	Secondary
e	${0.0439}_{-0.0026}^{+0.0024}$
i (°)	86.20 ± 0.20
${a}_{\mathrm{tot}}$ (${R}_{\odot }$)	12.23 ± 0.16
K (km s−1)	52.3 ± 1.2	97.3 ± 1.3
M (${M}_{\odot }$)	0.92 ± 0.05	0.50 ± 0.03
R (${R}_{\odot }$)	0.92 ± 0.03	0.47 ± 0.04
KP (mag)	13.141 ± 0.031	18.066 ± 0.031

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Table 2 shows the extracted radial velocity measurements from all of our data. To independently measure the masses of each component, we did not combine our radial velocity data with the previously published radial velocity data. Moreover, instead of fitting the light curve and the radial velocity data simultaneously, we chose to fit the photometric and the spectroscopic data individually. We found that simultaneous fitting resulted in poor fits to the radial velocity data, because the number of data points in the Kepler data far outweigh the radial velocity data.

As we briefly mentioned in Section 3.1, the eb software package outputs an RV model, which we can use when performing the radial velocity fit of the 16 free parameters in Table 1. Parameters that affect the RV model are the orbital period (P), the epoch of the primary mid-eclipse (t₀), the mass ratio (q), ${K}_{\mathrm{tot}}/c$, the systematic velocity (γ), $e\cos \omega$, and $e\sin \omega$. The orbital period (P) and the epoch of the primary mid-eclipse (t₀) were fixed to the values from the best-fit values of the light curve. Because the light curve has more data points than the radial velocity data and samples in a finer step, for $e\cos \omega$ and $e\sin \omega$, we took their posterior distribution from the light curve fit as the priors for the MCMC run in the radial velocity fitting.

Unlike in the light curve fitting, we did not employ mpfit, as we had good starting points for all of the parameters from the light curve fit. In order to confirm the validity of our code, we tried fitting the Devor (2008), Çakırlı et al. (2013), and the Iglesias-Marzoa et al. (2017) radial velocity points, which compose the same set of radial velocity data that Iglesias-Marzoa et al. (2017) had in their fit. Our calculated mass ratio, individual mass, the radial velocity semi-amplitudes, and the sum of the semi-amplitudes match well with those of Iglesias-Marzoa et al. (2017) within the uncertainties. Following this validity check, we fitted IGRINS H-band, NIRSPEC K-band, and HIRES radial velocity points alone, ignoring the measurements of Çakırlı et al. (2013) or Iglesias-Marzoa et al. (2017).

Figures 6, 7, and 9 show the best-fit models we obtained using eb. Figure 6 shows the phase-folded Kepler light curve data with the best-fit model plotted in red. Figure 7 shows the zoomed-in region around each eclipse with the best-fit (top panel) and the residuals (bottom panel). We report the parameters from the model with the highest likelihood as our best-fit values. We adopted the standard deviation of the MCMC chains as the uncertainty for all of the parameters with symmetric posterior distribution. However, for esinw, the distributions are not symmetric and we choose to use the 34.1 percentile around the highest likelihood value. Figures 8 and 10 show the triangle plots from the light curve and radial velocity MCMC run, respectively. Figure 9 shows the primary (in blue) and the secondary (in green) data points from all available radial velocity data with the best-fit model in red.

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Table 3 shows the fitted and the calculated parameters from the light curve and radial velocity fitting. We measured ${K}_{1}=52.3\,\pm 1.2$ km s⁻¹ and ${K}_{2}=97.3\pm 1.3$ km s⁻¹. For the masses and the radii, we measured ${M}_{1}=0.92\pm 0.05{M}_{\odot }$ and ${R}_{1}=0.92\,\pm 0.03{R}_{\odot }$ for the primary and ${M}_{2}=0.50\pm 0.03{M}_{\odot }$ and ${R}_{2}\,=0.47\pm 0.04{R}_{\odot }$ for the secondary. Our measured masses and radii are different from the previous two groups' results, which we discuss in the following section.

Comparing with the stellar parameters of Kepler targets in Huber et al. (2014), the resulting K_P magnitudes for the primary and secondary are consistent with the fitted masses and radii and indicate a mid G and early M dwarf EB at a distance of ∼460 pc, significantly further than the estimated distance of Çakırlı et al. (2013). The color for the third object is consistent with an early M dwarf at a distance of ∼610 pc, or a distant, evolved, and reddened star. In either case, the third object would not be associated with the EB; however, the uncertainty in distance is large, as it is based entirely on a single color that is expected to degenerate with stellar metallicity. There remains a distinct possibility that the third object is an associated early M dwarf star. The third light contribution we measured is different by ∼1.4σ in comparison with what Iglesias-Marzoa et al. (2017) reported. We attribute this difference to our inclusion of AO imaging, which provides a direct measurement of the third light, rather than fitting it as a model parameter in the light curve.

In the Kepler light curve, the causes of modulations in out-of-eclipse data can be starspots, reflection effects, ellipsoidal variations, beaming effects, and gravity darkening. The reflection, ellipsoidal, and beaming signals for our best-fit parameters are at least one order of magnitude less than the observed modulations (Lillo-Box et al. 2016). Gravity darkening, according to von Zeipel Theorem (von Zeipel 1924), is significant in stars that are hot enough to have radiative envelopes (earlier than F type), which is not the case for T-Cyg1-12664. Therefore, the dominant cause of out-of-eclipse modulation must be starspots.

The eccentricity of the orbit is non-zero. The effect of non-zero eccentricity is shown in the midtime of the secondary eclipse, which slightly departs from 0.5 in orbital phase. Eccentricity and the argument of periastron (ω) determine the time interval between the primary and the secondary eclipse (e cos ω) and the duration of eclipse (e sin ω). For stars with a convective envelope, the circularization timescale (${\tau }_{\mathrm{circ}}$) and the synchronization (${\tau }_{\mathrm{sync}}$) timescale are proportional to ${(a/{R}_{1})}^{8}$ and ${(a/{R}_{1})}^{6}$, respectively, where a is the binary semimajor axis and R₁ is the radius of the primary component (Zahn 1975). For T-Cyg1-12664, these timescales are ${\tau }_{\mathrm{sync}}\simeq 5.8$ Myr and ${\tau }_{\mathrm{circ}}\simeq 1.1$ Gyr. As evident from the out-of-eclipse modulations in the Kepler light curve, the binary orbit is nearly synchronized. However, the binary orbit is not circularized, as the eccentricity of the orbit is non-zero.

Our reported masses and radii differ from the previous two publications. We attribute the discrepancies to the difference in the radial velocity measurements. Our spectroscopic data completely cover the orbital period and allow the radial velocity measurements of both components. In fact, Iglesias-Marzoa et al. (2017) stated that future radial velocity observations, specifically near-infrared observations, would likely further refine the system parameters.

T-Cyg1-12664 is a spotted system. Starspots on a rotating photosphere can introduce radial velocity variations (e.g., Andersen & Korhonen 2015). Gagné et al. (2016) investigated the effect of starspots on radial velocity measurements and found that an active star shows a long-term radial velocity variation of 25–50 m s⁻¹ in the near-infrared. In the near-infrared, the spot-induced radial velocity signal is significantly reduced due to the lower contrast between spots on the photosphere at longer wavelengths. Reiners et al. (2010) showed that spot-induced radial velocity variations have a ${\lambda }^{-1}$ dependence, where λ is the observed wavelength. Both combined, the radial velocity signal from spots in our measurements are not significant and are well within the measurement uncertainties, even for the visible-wavelength HIRES observations.

The primary and the secondary components are different in spectral type. A mismatch in radial velocity templates can cause offsets in the radial velocity zero-point. For the IGRINS and NIRSPEC spectra, we used a combination of a G5 and M3 template, and for the HIRES spectra, we used a combination of a K1 and M3.5 template. The data sets have consistent radial velocity zero-points, despite using different templates. For this reason, we do not believe this is responsible for the discrepancy with the two previous studies.

Çakırlı et al. (2013) report the orbital inclination angle of $i=83\buildrel{\circ}\over{.} 84\pm 0\buildrel{\circ}\over{.} 04$, corresponding to grazing eclipses. For grazing eclipses, the extracted radius ratio from the photometry alone is not well constrained and is degenerate with other parameters. To better constrain the radius ratio, spectroscopic light ratios must be provided, which was not the case in Çakırlı et al. (2013). From our analysis, we measured $i\,=86\buildrel{\circ}\over{.} 20\pm 0\buildrel{\circ}\over{.} 20$, which is nearly an edge-on orbital configuration, and our result is much less affected by the degenerate radius ratio. However, we were not able to measure the spectroscopic light ratio from our data and this remains as a caveat to our measurement.

The measured masses and radii for both components indicate that neither of the stars is inflated and the values agree well with the predictions of stellar evolutionary models. Iglesias-Marzoa et al. (2017) reported the spectral type of the primary to be a G6 main-sequence star. However, their reported primary mass is 0.680M_☉, which is low for a typical main-sequence G6 type star. Our measurement for the primary component mass indicates the primary star is a solar type star. For the secondary component, our measurement corresponds to an early M dwarf star.

We note that unlike other eclipsing-binary fitting procedures, our method did not fit for the effective temperatures of the component stars, which would result in measured spectral types and a semi-empirical distance to the system. We purposefully do not fit for effective temperature, instead fitting for the central surface brightness ratio between the two stars in the Kepler band. We see this as an advantage. To determine the effective temperatures of the component stars, we would have to invoke bolometric corrections that depend on accurate atmospheric models of the stars. Atmospheric models of low-mass stars are known to disagree with spectroscopic observations due to the many molecular opacities required (Allard et al. 2012). A recent investigation by Veyette et al. (2016) showed that the carbon-to-oxygen ratio of a low-mass star can dramatically change model spectra of low-mass stars, even at a fixed effective temperature, which would also affect the reported effective temperature. Instead, by specifically reporting the ratio of component stars' central surface brightnesses in a well defined band, we remove the assumptions about metallicity, carbon-to-oxygen ratio, and particular molecular opacity tables.