USING HIGH-RESOLUTION OPTICAL SPECTRA TO MEASURE INTRINSIC PROPERTIES OF LOW-MASS STARS: NEW PROPERTIES FOR KOI-314 AND GJ 3470

Over the past decade, the California Planet Survey (CPS) has obtained spectroscopic measurements of more than 2500 stars at Keck Observatory, monitoring their radial velocities for the characteristic signature on the host star induced by the presence of a planet (Howard et al. 2010). We make use of their High-Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) observations of 155 M dwarfs to build a high-resolution, high signal-to-noise "template" spectrum of each star. The typical CPS program HIRES setup gives a resolving power R  =  λ/Δλ  ≈  50, 000 and uses an iodine-cell for precise wavelength calibration of radial velocity measurements (Howard et al. 2010). In our application, the iodine lines are contaminants, so we can only make use of the blue and red chips of the HIRES detector, where there are no lines from the iodine cell. The blue portion of the spectrum allows us to examine between 370 nm and 480 nm, while the red portion of the spectrum covers between 650 nm and 800 nm.

The M dwarfs in the CPS sample have well-defined Two Micron All Sky Survey (2MASS) photometry and parallaxes from Hipparcos (Perryman et al. 1997; Cutri et al. 2003). This allows us to characterize the stellar sample in terms of their absolute near-IR magnitudes, ($\mathcal {M}_{{\it JHK_s}}$), and color, (V − K_s). We culled stars from the sample that were overly active or young, selected on the basis of having published rotation periods less than five days, high X-ray luminosities with a X-ray count rate >1 count s⁻¹ in ROSAT (Voges et al. 1999) or as being members of a young open cluster or moving group. We also culled stars that were known to be unresolved binaries or turned out to be missing geometric parallax measurements. We also limited the sample to stars brighter than $\mathcal {M}_{K_s}= 8$, so that the range of properties is well sampled by the CPS stars. This left us with 119 calibration stars. The sample spans a range of $\mathcal {M}_{K_s}$ from 4.5 to 7.5 as shown in the top panel of Figure 1. Using the Johnson & Apps (2009) solar-metallicity main-sequence relation, defined as $\mathcal {M}_{K_s}= \sum a_{i} (V-K_{s})^{i}$, where a = { − 9.58933, 17.3952, −8.88365, 2.22598, −0.258854, 0.0113399}, we can calculate the quantity, Δ(V − K_s), the difference between the observed color and the main-sequence color for a star of the same $\mathcal {M}_{K_s}$ such that positive values of Δ(V − K_s) correspond to redder objects. We will henceforth refer to Δ(V − K_s) as the color offset, which can be used as a proxy for metallicity (Schlaufman & Laughlin 2010). The bottom panel of Figure 1 shows that our sample spans a broad range in color offset, and hence a broad range in metallicity.

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Since our sample of calibration stars has been monitored for radial velocity shifts indicative of planets over the past 10–15 years, each star has an average of 25 observations. By combining the individual observations, we can produce high signal-to-noise, high-resolution template spectra for each of the calibration stars in our sample. We rebinned each star's spectrum onto a wavelength scale that is evenly spaced in ln λ (cadence of 1.9 × 10⁻⁶), which allows us to properly Doppler shift the spectra with respect to one another (Tonry & Davis 1979). We also corrected for small differences in the wavelength solution from night to night (on the order of a couple of pixels) for every spectral order due to changes between the cross-disperser angle and the echelle angle as well as changes in the slit illumination for any given observation.

Having aligned the spectra and removed defects (cosmic ray hits etc.), we simply co-added the flux to produce a high signal-to-noise ratio (S/N) spectrum for each star. The red portions of the spectrum yield a typical S/N per resolution element ranging from ∼800 to ∼4000, depending on the number of observations for the particular star. For the blue-chip spectra, the S/N ranges from ∼100 up to ∼600. The S/N of the red side is higher primarily because the peak of the M dwarfs' spectral energy distribution lies closer to near-infrared wavelengths than to the blue portion of the spectrum.

In order to compare the spectra, it was necessary to normalize each order to remove the effect of the blaze function of the spectrometer and to account for a pseudo-continuum. We normalize each spectral order individually. First, we separate the order into tens of "chunks," masking out telluric regions, and take the top 1%–2% of points of each chunk as representing the "continuum." We then fit a low-order polynomial to the continuum points across the full extent of the order. Lastly, we divide the spectral order by this polynomial, ignoring problematic points at the ends of the orders, to get the normalized spectrum (see the Appendix for further details). Since the echelle spectra are not flux calibrated and the blaze function distorts the shape of the spectra, a pseudo-continuum is not well defined through our normalization procedure. However, because we use the same continuum regions for all stars, this process allows for reliable, differential comparisons among stars of different types.

In Figure 2, we plot the template spectra averaged in $\mathcal {M}_{K_s}$ bins with a width of 0.1 mag. The different colors correspond to different values of $\mathcal {M}_{K_s}$, with red corresponding to cooler stars and blue to hotter stars. In the figure, we see that the absorption features are quite distinct from the "continuum," which match across the different stars. It is clear in the high signal-to-noise spectra that features such as these are quite sensitive to $\mathcal {M}_{K_s}$ and we can use the strength of the absorption for a given spectrum as indicative of the stars intrinsic luminosity. Deviations from a strict sequence are primarily due to metallicity effects. At a given $\mathcal {M}_{K_s}$, changes in metallicity will affect the strength of certain absorption features. Accounting for this second-order effect will provide us with a valuable indicator of stellar metallicity.

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The presence of spectral regions sensitive to changes in $\mathcal {M}_{{\it JHK_s}}$ motivates the development of a quantitative relationship between the strength of each region and the physical properties of a star. The regions that were found to be sensitive and useful for calibration are listed in Table 1. We identified these regions by eye, looking across the full spectrum, ignoring telluric regions, and requiring the continuous regions of the spectrum to have monotonically increasing absorption with decreasing stellar effective temperatures (see Figure 2). The useful regions consist predominately of portions of TiO and VO bands. Note that the spectral indices listed in the table are all in the red portion of the spectrum. Although there also appeared to be sensitive regions at blue wavelengths, the lower overall flux level limits their usefulness, both for the calibration procedure and for future observations. In addition to the regions listed in Table 1, there were other regions that we identified as sensitive to the physical stellar properties. However, we selected a subset that when combined provided the optimal calibration (see Section 3.3).

The CPS sample spans a representative range of properties for low-mass stars, making it useful for our calibration procedure. Since the sample is limited in size and individual stars only represent discrete points in the mass–metallicity plane, comparing the spectra directly is less than ideal, giving poor parameter resolution. Instead, it is preferable to compare the strength of sensitive features, measured from their integrated fluxes (EWs), allowing us to fit smooth functions to observed trends in EW and providing a continuous relationship between the integrated flux and the stellar properties. The integrated flux is defined as

$$\begin{eqnarray} \mathrm{EW} &=& \Delta \lambda - \int S(\lambda) d\lambda \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \quad &\approx & \Delta \lambda - \sum _{i} S_{i} h_{\lambda }, \end{eqnarray} \tag{ 2 }$$

where Δλ is the width over which the integral is computed and S(λ) is the normalized spectrum as a function of wavelength λ. Equation (2) gives the approximation for discretely sampled spectra over pixels that span Δλ evenly sampled with width h_λ. To determine the errors on our EW measurements, we randomly simulate the spectral observation using Poisson statistics with a mean in each spectral bin given by the photon counts of the actual data. We take the error on the EW measurement to be the standard deviation of the distribution of EW values in the simulation. The typical error on the EW measurements are ∼1%–5%.

In Figure 3, we plot the behavior of the EW as a function of $\mathcal {M}_{K_s}$ and Δ(V − K_s) for a particular spectral region. The colors correspond to Δ(V − K_s), with red corresponding to stars that are more metal-rich and blue corresponding to stars that are more metal-poor. For spectral regions such as this one, the strength of the feature increases with increasing metal content as well as decreasing luminosity. The behavior, expressed in Figure 3, motivates the parameterization of each EW in terms of $\mathcal {M}_{{\it JHK_s}}$ and Δ(V − K_s). For a given region, l, we calibrate the EWs against each of the absolute magnitudes and the color offset. Our calibration EW, which allows us to interpolate between the discrete sample star properties, is given by

$$\begin{equation} \mathrm{EW}_{l, \alpha } =(b + c \mathcal {M}_{\alpha }) \Delta (V-K_{s})+ \sum ^2_{i=0} a_{i} \mathcal {M}_{\alpha }^{i} \; , \end{equation} \tag{ 3 }$$

where α ∈ { J, H, K_s }, so there is a separate calibration for each passband using the same spectral region.

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We fit each passband separately instead of going directly to mass, despite tight mass–luminosity relations, because the infrared colors (e.g., J − K) are not simply functions of mass and can change with metallicity; this is in contrast to other broadband photometric studies of low-mass stars (Johnson et al. 2011, 2012). In addition to the simple polynomial terms expressed in Equation (3), we include a cross term governed by the coefficient c that accounts for differences between brighter and fainter stars in how their absorption strength responds to the addition of metals. The need for such a term is evident in how TiO features are known to saturate in late-type M dwarfs and that VO features are not apparent in early M dwarfs but appear in late-type M dwarfs (Kirkpatrick et al. 1991).

We used a Bayesian method to fit Equation (3) to each set of EWs. In addition to the coefficients in Equation (3), we also incorporated an additional parameter, σ, to take into account intrinsic scatter in our choice of parameterization. Using a Markov Chain Monte Carlo (MCMC) technique with a Metropolis–Hastings algorithm, we explore the posterior distribution for the best parameters in the calibration conditioned on the known properties of the stars in the sample. The best calibration parameters are those that maximize their respective marginal distributions, and thus maximize the probability of each parameter reproducing the stellar properties of the stars in the sample with our simple model. We report the parameter values of our calibration in Table 2. For a given spectral index, the first row of the table corresponds to the parameters for the calibration with $\mathcal {M}_{J}$, the second row for $\mathcal {M}_{H}$, and the third for $\mathcal {M}_{K_s}$.

These curves provide a calibration, which we can apply to stars of unknown properties with measured EWs and estimate $\mathcal {M}_{{\it JHK_s}}$ and Δ(V − K_s). Taking the errors in the integrated flux as normally distributed, we minimize a χ² statistic to get the stellar photometric properties. The statistic is split up into a couple components,

$$\begin{equation} \chi ^{2} = \sum _{l} \chi ^{2}_{l} + \chi ^{2}_{\mathrm{p}}, \end{equation} \tag{ 4 }$$

where the first term comes from the spectral calibration and is summed over the different indices, l, and the last term correspond to photometric constraints. The sum over all the different indices allows us to partially break the degeneracy between metallicity and mass inherent in the individual indices.

For each spectral region, l, the fitting statistic is

$$\begin{eqnarray} \chi ^{2}_{l} &=& \sum _{\alpha } \frac{ \left[ \mathrm{EW}^{{\rm obs}}_{l} - \mathrm{EW}_{l,\alpha }(\Delta (V-K_{s}), \mathcal {M}_{\alpha }) \right]^{2} }{2 \left(\sigma ^{2}_{{\rm obs},l} + \sigma _{l,\alpha }^{2} \right)}, \end{eqnarray} \tag{ 5 }$$

where EW_{obs, l} is the measured EW, with uncertainty σ_{obs, l}, and EW_{l, α} is from Equation (3), using the best calibration parameters; here, the scatter parameter, σ_{l, α}, is added in quadrature to the measurement error. The sum is over each of the three infrared passbands, α ∈ { J, H, K_s }, which have separate calibrations for the given spectral region (see Table 2).

Since many nearby stars have 2MASS photometry, we can use the distance, d, as an additional parameter by requiring that our estimates of $\mathcal {M}_{{\it JHK_s}}$ reproduce the observed photometry:

$$\begin{equation} \chi ^{2}_{\mathrm{p}} = \sum _{\alpha } \frac{ \lbrace \alpha - [\mathcal {M}_{\alpha } + 5\log _{10} (d/10\, \mbox{pc})]\rbrace ^{2}}{2 \sigma ^{2}_{p,\alpha}}, \end{equation} \tag{ 6 }$$

where α corresponds to the observed infrared magnitudes and σ_{p, α} is the corresponding measurement uncertainty.

Summing over all of the indices and the additional constraints gives the total χ² of Equation (4), which we minimize as a function of $\mathcal {M}_{{\it JHK_s}}$, Δ(V − K_s), and d to determine the best-fit stellar properties and the distance to the star.

In Figure 4, we show the results of our assessment of how well we can recover the properties of the stars in our calibration sample. The top panel of the figure shows the percentage error in reproducing the observed distance of the stars, the middle panel shows the error on the color offset, and the bottom panel shows the error in $\mathcal {M}_{K_s}$. The root-mean-scatter (RMS) for the distance, color offset, and absolute K-band magnitude are 11%, 0.18 dex, and 0.25 dex, respectively. Using established photometric relations, this scatter would correspond to 0.10–0.15 dex in [Fe/H] (depending on the literature calibration) and ∼0.05 M_☉ in mass (Schlaufman & Laughlin 2010; Delfosse et al. 2000).

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As an additional test, we examined what S/N is necessary to get consistent parameter estimates from any given stellar spectrum using our methods. We simulated a given S/N by adding noise to our template spectra using a pseudo-random number generator. Repeating this many times for each S/N, we saw what effect the noise had on our parameter estimates. In Figure 5, we plot the dispersion in our estimates as a function of S/N. The top panel is for Δ(V − K_s) and the bottom for $\mathcal {M}_{K_s}$. As the S/N increases, the initial improvements are significant. But after a S/N of ∼70, the improvements with greater signal are marginal.

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We also examined how spectral resolution affects the utility of our calibrations. By measuring EWs consistent with our original measurements, we would be able to use our calibration to recover the same set of stellar properties using lower resolution spectra. To test this, we first took the template spectra for our calibration sample and convolved the spectra down to lower resolutions (5000–45,000) in increments of 5000 from our original resolution of 50,000. We then normalized the spectra in the same manner as we did our original template spectra (see Section 2.2) and computed the EW for each of the indices of Table 1 and all of the sample stars. In Figure 6, we plot the average fractional difference in EW measurements, across all the calibration stars, between the convolved spectra and the unconvolved spectra as a function of spectral resolution, where each panel corresponds to a different index as listed in Table 1. The error bars in the plot represent the scatter in the deviation across the sample of calibration stars. For each index, the EW measurements are consistent with the original measurements for spectral resolutions above 30,000. Therefore, our calibration should not be used below this threshold without accounting for the systematic effects demonstrated in Figure 6. Although the integrated flux of a spectrum should not change at lower resolutions, the blending of pseudo-continuum with the many absorption band-heads complicates our EW measurements.

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To consider higher resolution data, we examined a subsample within our set of calibration stars that also had publicly available data from the ESO data archive using the HARPS spectrograph at a resolution of 115,000 (Mayor et al. 2003). For this subsample of 43 stars, we combined multiple observations and put together template spectra similar to our HIRES spectra (see Section 2.2). This produced a set of HARPS spectra with high S/Ns, all greater than 70. We then convolved the spectra down to the HIRES resolution of 50,000 to compare EW measurements. Normalizing the spectra in the usual manner, we then measured the EWs for the first four indices of Table 1; because of the small overlap between the HARPS spectrograph and the HIRES red chip, only these four indices were available. In Figure 7, we compare EW measurements from the HARPS subsample to the measurements from our HIRES spectral templates. The straight line in the plot corresponds to exact agreement. The measurements agreed rather well with an RMS of ∼8%, in line with the combined errors between the two measurements.

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We applied our methods to the case of the Kepler object of interest KOI-314. This M-dwarf, with a visual magnitude ∼14 and a Kepler-band magnitude K_p = 12.93, received Keck Observatory HIRES follow-up to confirm the planetary nature of the two transit signals observed in its Kepler light curve. We were able to make use of six CPS observations to construct a high-resolution spectral template to apply our method. The co-addition procedure, as discussed in Section 2, yielded a spectrum with a typical S/N of ∼250 in the red portion, plenty for our purposes. Figure 8 shows the results of our analysis with the contours for each pairing in our five parameter fit, $\mathcal {M}_{{\it JHK_s}}$, Δ(V − K_s), and d.

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Each parameter is highly correlated, leading to the oblong shaped contours of the figure. This is predominately because the absorption strength of the spectral indices is degenerate in mass and metallicity; the absorption strength can increase with a drop in the effective temperature or an increase in the metal content. Additionally, the shape of the contours between the infrared magnitudes must be consistent with the observed colors(J − H, H − K, etc.).

We can then marginalize over the full posterior probability for the five parameters to get probability distributions for the likelihood of each of the parameters. Our estimates are shown in Table 3, where we have adopted for our uncertainties the scatter we have in reproducing the stellar parameters as demonstrated in Section 3.3. We have additionally added an empirical estimate of the stellar radius based on the single star mass–radius relation of Boyajian et al. (2012) established from interferometric radii measurements of nearby low-mass stars. The table also includes select estimates from Muirhead et al. (2012a), which uses infrared spectra in conjunction with stellar models to derive properties. Additionally, in Figure 9, we plot the marginalized distributions for the stellar properties, after converting the photometric properties, $\mathcal {M}_{K_s}$ and Δ(V − K_s), to the physical properties of mass and [Fe/H]. The top left panel shows our distribution for the distance. The top right panel shows the distribution for [Fe/H], using the calibration of Neves et al. (2012) to make the conversion from Δ(V − K_s). The accompanying dashed line is a Gaussian representing the Muirhead et al. (2012a) estimate. The bottom left panel shows the distribution for mass, the solid line being our estimate, employing the Delfosse et al. (2000) relation for $\mathcal {M}_{K_s}$ with the dashed line representing the Muirhead et al. (2012a) estimate. Finally, the bottom right panel uses our mass estimate and the mass–radius relation of Boyajian et al. (2012) to estimate the radius of the star, with the dash-dot line again representing the Muirhead et al. (2012a) estimate. Our measurements match those in Muirhead et al. (2012a) within the respective uncertainties, giving us confidence in the accuracy of our derived parameters. Combining our estimates with theirs gives a mass estimate of 0.55 ± 0.04 M_☉, a radius estimate of 0.52 ± 0.04 R_☉, and a metallicity of −0.29 ± 0.08 dex.

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Table 3. Properties: KOI-314

Attribute	Muirhead et al. (2012a)	This Study
MJ	⋅⋅⋅	6.18 ± 0.25
MH	⋅⋅⋅	5.57 ± 0.25
$\mathcal {M}_{K_s}$	⋅⋅⋅	5.39 ± 0.25
Δ(V − Ks)	⋅⋅⋅	−0.20 ± 0.18
d	⋅⋅⋅	66.5 ± 7.3 pc
Massa	0.51 ± 0.06 M☉	0.57 ± 0.05 M☉
Radiusb	0.48 ± 0.06 R☉	0.54 ± 0.05 R☉
[Fe/H]c	−0.31 ± 0.13	−0.28 ± 0.10

Notes. ^aFor this work, the estimate uses the relations of Delfosse et al. (2000) to convert $\mathcal {M}_{K_s}$ to mass. ^bFor this work, the estimate uses the relations of Boyajian et al. (2012) to convert mass to radius. ^cIn converting Δ(V − K_s) to [Fe/H], the estimate uses the relation of Neves et al. (2012): [Fe/H] =0.57Δ(V − K_s) − 0.17. The listed uncertainty does not include the scatter in their relation.

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As an additional application of our methods, we consider the exoplanet hosting star GJ 3470. In the discovery paper, Bonfils et al. (2012) used HARPS radial velocities and a photometric transit detection to constrain the mass and radius of the planet, adding the system to three other M dwarfs (GL 436, GJ 1214, and KOI-254) with planets that have well-measured mass and radius estimates. The precision of the planet properties for GJ 3470b, however, was limited by the uncertainty in the stellar properties. Previous studies of GJ 3470 suggested that it is a typical field star on the main sequence, making the methods of this paper applicable (Bonfils et al. 2012).

In Section 3.4, we showed how HARPS archival spectra could be used to measure EWs compatible with the spectral index calibrations of Table 2. We applied the procedures of Section 3.4 on the HARPS spectra of GJ 3470 to obtain a spectrum with a typical S/N of 72 near the spectral indices. We measured the EWs and were able to independently estimate the distance, mass, [Fe/H], and radius of the star using entirely empirical methods; we used the mass–radius relation of Boyajian et al. (2012) to determine the radius from the mass. However, only the first four of the indices of Table 1 are useable with the HARPS spectra, reducing the precision of our measurements. Using just the four indices, we reproduced the properties of the calibration sample to an rms of 0.38 in $\mathcal {M}_{{\it JHK_s}}$, 0.27 dex in Δ(V − K_s), and 18% in distance (see methods in Section 3.3). This rms corresponds to 0.08 M_☉ in mass and 0.15 dex in metallicity using the calibrations of Delfosse et al. (2000) and Neves et al. (2012), respectively. To improve the precision of our estimates, we look for additional constraints to apply to the stellar properties.

The broadband photometric methods of Johnson et al. (2012) showed how photometric observations could be combined with transit light curve observables to provide precise estimates of the stellar properties. Following their example, we include the reduced semi-major axis of the planet orbit as an additional constraint:

$$\begin{equation} \frac{a}{R_{\star }} (M_{\star }, P) = \left(\frac{G}{4\pi ^2}\right)^{1/3} \frac{M^{1/3}_{\star }}{R_{\star }(M_{\star })} P^{2/3}, \end{equation} \tag{ 7 }$$

where the radius is given as a function of mass using the mass–radius relation of Boyajian et al. (2012) and we neglect the mass of the planet as very small compared to the stellar mass. The transit light-curve observable, d_p/R_⋆, gives the planet-star separation at the time of transit and for the case of a circular orbit matches the reduced semi-major axis of Equation (7). We take the period to be well defined as 3.33714 days, and the observed (a/R_⋆)_o as 14.9  ±  1.2 from Bonfils et al. (2012). We incorporated this constraint by adding an additional term to the total χ² of Equation (4),

$$\begin{equation} \chi ^{2}_{a} = \frac{ [ (a / R_{\star })_{o} - a/R_{\star } (M_{\star }, P) ]^{2} }{2 \sigma _{ar}^2} \;. \end{equation} \tag{ 8 }$$

In implementing the MCMC methods of Section 3.2, we use the relations of Delfosse et al. (2000) to go from $\mathcal {M}_{K_s}$ to M_⋆ in Equation (8). Our estimates, shown in Table 4, agree well with the output values derived by the planet-discovery team which combined a transit detection with radial velocity measurements, 0.54  ±  0.07 M_☉ for the mass and 0.50  ±  0.06 R_☉ for the radius, however, our stellar properties are more precise. They are also consistent with stellar parameter estimates incorporating a new infrared transit analysis (Demory et al. 2013). In Figure 10, we plot the contours for the total probability distribution in our estimate of the stellar mass as a function of observed quantities at the fixed distance and metallicity of our estimates shown in Table 4. The filled overlays show the 3σ constraints provided by each observation, with blue corresponding to a/R_⋆, green to the joint constraint in JHK, and orange given by the combined constraints of the EWs.

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Table 4. Properties: GJ 3470 System

Attribute	Bonfils et al. (2012)	This Study
Ja	⋅⋅⋅	8.794 ± 0.019
Ha	⋅⋅⋅	8.206 ± 0.023
Ka	⋅⋅⋅	7.989 ± 0.023
a/R⋆	14.9 ± 1.2	⋅⋅⋅
Period	3.33714 days	⋅⋅⋅
Rp/R⋆	0.0755 ± 0.0031	⋅⋅⋅
d	⋅⋅⋅	$29.9\pm _{3.4}^{3.7}$ pc
Mass	0.54 ± 0.07 M☉	0.53 ± 0.05 M☉
Radius	0.50 ± 0.06 R☉	0.50 ± 0.05 R☉
[Fe/H]	⋅⋅⋅	0.12 ± 0.12

Note. ^a2MASS photometry of Cutri et al. (2003).

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We can combine our stellar radius measurement with the estimate from Bonfils et al. (2012) to get a precise stellar radius of 0.50 ± 0.04 R_☉. Using the transit observable R_p/R_⋆ = 0.0755 ± 0.0031 from Bonfils et al. (2012), we get a planet radius estimate of R_p = 4.12 ± 0.37 R_⊕, slightly smaller than their estimate of R_p = 4.2 ± 0.6 R_⊕.