ECLIPSING BINARY SCIENCE VIA THE MERGING OF TRANSIT AND DOPPLER EXOPLANET SURVEY DATA—A CASE STUDY WITH THE MARVELS PILOT PROJECT AND SuperWASP

We first identified the binaries spectroscopically using the MPP RV data. Following Fleming et al. (2010), the RV uncertainties for each star are scaled by a "quality factor" which is a multiplicative scaling that modifies the formal RV uncertainties to better account for systematics defined as

$$\begin{equation} Q = \frac{{{\rm rms}}(X - \langle {X} \rangle)}{\rm {MEDIAN}(\sigma _{X})}, \end{equation} \tag{ 2 }$$

where X represents the RV measurements and rms is the root-mean-square residual. The quality factor is derived from the rms of the RVs for the other 58 objects in the same field of view as the target, under the assumption that a majority of the objects per field should not have significant RV variability. The field containing TYC 272 has a Q = 3.44, suggesting that the formal uncertainties are underestimated. Table 1 contains the times of observations, RVs, and both the formal and scaled RV uncertainties for TYC 272. A total of 14 epochs were obtained with an average, scaled RV uncertainty of 0.345 kms⁻¹ for TYC 272.

TYC 272 was fit using the RVSIM software program (Kane et al. 2007). Figure 1 shows the best-fit orbital solution to the MPP RV data. The χ²/dof, after scaling by Q, is 2.62 with eight degrees of freedom. Because it is a short-period binary, there is additional RV jitter expected due to chromospheric activity, however, chromospheric effects are typically on the order of tens of ms⁻¹ (Santos et al. 2000). The eccentricity is consistent with a circularized orbit, as expected for the best-fit orbital period P = 5.7282 days (Mazeh 2008). The semiamplitude K ∼ 54.6 km s⁻¹ corresponds to a minimum mass of the secondary m_min ∼ 0.64 M_☉ assuming the mass of the primary is M = 1 M_☉, making this EB an excellent system to further study in the context of the mass–radius relationship. The best-fit orbital properties are listed in Table 2.

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We utilize a spectral synthesis technique as part of the process to estimate basic stellar parameters such as T_eff, log (g), and [Fe/H] for the primary. In Valenti & Fischer (2005), they outline a general procedure for deriving stellar properties using Spectroscopy Made Easy (SME; Valenti & Piskunov 1996). We follow the exact methodology described in Valenti & Fischer (2005), but include some changes that allow for a more in-depth search of χ² space. Specifically, we use the same spectral regions, line lists, and continuum regions as in Valenti & Fischer (2005), but do not allow any of the individual abundances to be free parameters.

Our SME spectral analysis pipeline is divided into two steps, because the spectral synthesis parameters are highly correlated, and therefore a range of initial guesses is required to adequately explore the χ² space. First, a forward modeling procedure is used to calculate χ² values for model spectra over a large grid of parameter space. This allows us to focus our analysis on the region of parameter space with the most likely, global, best-fit stellar parameter solution. Second, we use the SME Levenberg–Marquardt minimization algorithm as was used in Valenti & Fischer (2005) to fit observed spectra with synthetic spectra for 100 different initial input parameters. We select these different combinations of input values from the 100 grid points with the lowest chi-square values determined from the first step of our analysis pipeline. Accessing Vanderbilt University's ACCRE parallel computer facility allows us to efficiently explore the dependence of these initial parameters on our output parameters, as well as providing a good measure of the internal precision of SME's minimization procedure.

Using our implementation of SME on the ARCES spectrum of TYC 272, we derive best-fit values of T_eff = 5905 ± 177 K, log (g) = 4.83 ± 0.19, and metallicity [Fe/H] = -0.28 ± 0.12. The value for log (g) is larger than expected for stars of this temperature, but we note that log (g) is a very challenging parameter to measure spectroscopically, even at high spectral resolution, and is particularly challenging at the ARCES spectral resolution of ∼31,500.

In addition to SME spectral synthesis, we analyzed the spectrum following a standard spectroscopic method based on the requirements of excitation and ionization equilibria. Our automated analysis merges the approaches described in Porto de Mello et al. (2008) and Ghezzi et al. (2010), and will be described in a forthcoming paper of the MARVELS series (S. W. Fleming et al. 2011, in preparation). A total of 57 Fe i and three Fe ii lines were used, after a 2σ clipping was applied to remove those lines with too high or too low iron abundances. The equivalent widths (EWs) were automatically measured using the task "bplot" in IRAF. We have derived a T_eff = 5720 ± 79 K, log (g) = 4.20 ± 0.10, and metallicity [Fe/H] = -0.37 ± 0.10. We investigated the effects of a contaminating spectrum by modifying the measured EWs by a correction factor equal to the inverse of the expected flux contribution from the primary. In the case of a solar-type dwarf with a mid-K companion, the expected flux ratio is ∼10% in the V band, and the EW correction factor is ∼1.1. The general effect is to increase the derived effective temperature, surface gravity, and metallicity. However, the increase is small and well within the internal errors of our technique. We therefore conclude that a flux ratio of 10% or lower would not significantly affect our measured values based on the EW method.

We also examine the Hα line profile, which serves as a T_eff and chromospheric activity indicator, following Lyra & Porto de Mello (2005) and derive a best-fit T_eff = 5461 ± 118 K. This temperature is lower than the value determined via the Fe line EW method, but is affected by contamination from the secondary that fills in the Hα wings, resulting in an effective temperature determination that is systematically cooler. An analysis of the Hα core filling suggests that TYC 272 is much more chromospherically active than the Sun, however, there is a degeneracy between flux contamination from a companion and core emission due to activity, both of which can fill in the core of the Hα line. We further note that it is often the case that Hα-based and color-based T_eff are systematically lower than temperatures determined spectroscopically, one possible cause being chromospheric activity (Porto de Mello et al. 2008). Given the relatively short orbital period of 5.728 days, it is likely that some amount of tidal spin-up has occurred, resulting in stronger magnetic fields and increased starspot activity.

We complement the spectroscopic analysis with a variety of additional techniques to estimate stellar parameters based on other observables. To further estimate the luminosity class, we use a reduced proper motion (RPM) diagram (Collier Cameron et al. 2007b) to determine whether the host star is a likely giant or a dwarf/subgiant. The RPM diagram is excellent at distinguishing giants from dwarfs/subgiants. We tested the RPM technique on the Kepler Input Catalog (KIC),²⁰ which has had its determinations of luminosity class verified using Kepler's high-precision photometry (Koch et al. 2010). We find that only 2% of KIC giants were classified as dwarfs/subgiants using our RPM analysis. We use the proper motion values from the Tycho-2 catalog and near-IR (NIR) photometry from Two Micron All Sky Survey (2MASS; Table 3) to place TYC 272 on the RPM-J diagram (Figure 2, bottom). The solid, curved line in the RPM diagram is the dividing line between giants and dwarfs. The labels show the approximate regions where dwarfs, subgiants, and giants are located. Given how precise the RPM diagram is at identifying giant stars, the fact that TYC 272 lies in the dwarf/subgiant region of the RPM diagram means it is highly unlikely that TYC 272 is a giant star.

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We use the IRFM (Casagrande et al. 2010) to estimate the effective temperature of the primary star via a Monte Carlo approach. We sample the observed V, J, and Ks magnitudes assuming Gaussian errors, sample uniformly in [Fe/H] from −1.0 < [Fe/H] < 0.5 and uniformly in extinction from 0 < A_V < 0.075, where the upper bound is determined from the Schlegel et al. dust maps (Schlegel et al. 1998). Figure 2 shows the results of the Monte Carlo IR flux method (top). We estimate the effective temperature based on the (V − Ks) color to be T_eff = 5459(+ 233, −208) K, where the uncertainties are 1σ equivalents centered on the median of the asymmetrical T_eff distribution. The difference in temperature between the IRFM and spectroscopic results can be caused by flux contribution from the secondary. Indeed, the peak of the (J − Ks)-based temperature distribution is slightly cooler, which can occur because the flux ratio in the NIR bands is larger than in the optical bands. This IRFM T_eff is therefore a lower bound on the true T_eff of the primary due to reddening by the companion.

To further study the effective temperature of the primary, we make use of the spectral energy distribution (SED). We use the optical band measurements from HAO, near-IR measurements via the 2MASS (Skrutskie et al. 2006) Point Source Catalog, and UV measurements from Galaxy Evolution Explorer (GALEX; Martin et al. 2005), which are compiled in Table 3. NextGen models from Hauschildt et al. (1999) are used to construct the theoretical SED. Based on the results from the spectroscopic and IRFM analysis, we construct a model with T_eff = 5785, log (g) = 4.5, and [Fe/H] = -0.3 and compare with the observed fluxes. These stellar parameters were selected as representative values between the spectral synthesis, EW measurements, Hα profile analysis, and IRFM temperature calculations. Distance and extinction are free parameters

Figure 3 (top) shows the result of the SED fit and the model spectrum. The blue points are the passband-integrated model fluxes for each filter, while the red crosses are the bandpasses of each filter (horizontal bars) and the measured error in the fluxes (vertical bars). A clear excess in the near-IR is seen when we do not consider flux from the secondary. Adopting a primary mass of 1 M_☉ and a radius of 1 R_☉, we calculate a secondary mass of 0.64 M_☉ (K5 spectral-type dwarf). We then add flux contribution from a second star with T_eff = 4400 K and a radius of 0.7 R_☉ and find that the inclusion of flux from a K-type companion can explain the near-IR excess seen in the SED (Figure 3, bottom). This SED analysis does not provide an independent measurement of the stellar parameters, but rather serves as a plausibility experiment that demonstrates flux contribution from a K-type companion can explain the apparent NIR excess if the primary has a T_eff ∼ 5785 K.

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Because the secondary is too faint to have its spectral features detected in the MPP spectra, the mass of the primary cannot be directly measured via RV. Instead, we make use of Padova isochrones (Marigo et al. 2008) to estimate the mass of the TYC 272 primary. Figure 4 plots radii and masses as a function of effective temperature for two assumed ages (100 Myr and ∼9 Gyr) and a metallicity Z  = 0.01 ∼ [M/H] = -0.28. This metallicity is chosen based on the SME and EW determinations, which are in good agreement with each other. The uncertainty in [M/H] does not significantly affect the derived range of probable masses for TYC 272, therefore we elect to use a single value for the tracks' metallicities, for clarity. The derived T_eff values from the SME, EW, Hα, and IRFM are plotted as horizontal bars representing the 1σ confidence intervals. We elect to use a T_eff range equal to the EW 3σ confidence interval (5483 K < T_eff < 5957 K) as our best estimate for the primary T_eff because it is expected to be minimally affected by flux contribution from the secondary. We note however that all four T_eff are consistent to within 2σ, and in fact the SME, EW, and IRFM temperatures all agree to within 1σ. Based on this T_eff range, we find the Padova models constrain M₁ = 0.92 ± 0.1 M_☉ for log (g) > 4.0 and 8.0 < log (age) < 9.95.

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The light curve for TYC 272 is shown in Figure 5, with the best-fit model overplotted as the solid line. The top panel contains the primary eclipse, while the shallowness of the secondary eclipse can be seen in the bottom panel. The secondary eclipse of TYC 272 appears to be grazing, leading to model degeneracies between the ratio of the radii and the surface brightness ratio. This is a well-known problem for grazing EBs, and the solution is to constrain the flux ratio when fitting the light curve (Andersen et al. 1983). The flux ratio is usually constrained spectroscopically (Clausen et al. 2003; Southworth et al. 2004; Blake et al. 2008), and we utilized the ARCES spectrum to search for any evidence of a secondary spectrum. The ARCES spectrum was obtained at an orbital phase corresponding to an expected velocity shift of −27.25 km s⁻¹, however, the signal-to-noise was not sufficient to detect a secondary spectrum for all flux ratios. Figure 6 compares the ARCES spectrum with the (lower resolution, R = 12,000) Fiber Optic Echelle (FOE) (Montes et al. 1997; Montes & Martin 1998) spectrum of 9 Cet, a G2 V standard. Two different wavelength regions are examined that contain absorption features present in stars of mid-F through mid-M spectral types. No set of secondary spectra is seen, and we estimate an upper limit of <20% for the flux ratio at these wavelengths. In the absence of sufficiently high-resolution spectra, we estimate the flux ratio using the mass range expected for M₁ in Section 5.1.2 and the measured RV semiamplitude of the primary K₁. The flux ratio is constrained to be 8% ± 4%, the estimated flux ratio in V of a G-type primary and K-type companion. As shown in the SED analysis, such a flux ratio reproduces the observed SED well, particularly in the 2MASS NIR bands.

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To model the light curves, we used the jktebop²¹ implementation of the EBOP light curve model (Nelson & Davis 1972; Popper & Etzel 1981) to produce a least-squares fit to the data. We modified jktebop to include the normalization for each night of data as additional free parameters. We consulted various tabulations of linear limb-darkening coefficients (van Hamme 1993; Claret 2000, 2004) and decided to adopt a limb-darkening coefficient of 0.6 in the WASP band for the G-type primary of TYC 272. For the K-type secondary star, we used a linear limb-darkening coefficient of 0.7. An uncertainty of ±0.1 is assumed for these parameters, and the small effect of this uncertainty is accounted for in the standard errors quoted for other parameters of the light curve solution. The values and uncertainties of ecos ω and esin ω from the RV orbital solution are included as constraints to the least-squares fit to the light curve. Without including such a constraint, the best-fit value of esin ω = 0.2 ± 0.07, however, because this is determined via the widths of the eclipses, we do not get a robust estimate from the light curve data alone.

Table 4 summarizes the geometric parameters derived from fitting the photometry. To account for correlated errors ("red noise") in the photometry, uncertainties in the parameters are derived using a "prayer-bead" bootstrap Monte Carlo method. This method uses the residuals from the best fit to create synthetic light curves with noise characteristics that are similar to the real data via circular permutation. Comparison of the parameters that are common to both the RV analysis and the photometric analysis shows excellent agreement. The orbital period agrees to within 1σ, while the ecos ω and esin ω values are consistent with a circular orbit.

Table 4. Light Curve Properties—TYC 272

Parameter	Value	1σ Uncertainty
J (surface brightness ratio at disk center)	0.14	0.01
$\frac{R_{1} + R_{2}}{a}$	0.096	0.004
$\frac{R_{2}}{R_{1}}$	0.60	0.12
i (deg)	86.6	0.3
ecos ω	0.0004	0.0009
esin ω	−0.002	0.001
Period (days)	5.72840	0.00001
T0 (HJD)	2454534.7020	0.0007
$\frac{R_{1}}{a}$	0.060	0.002
$\frac{R_{2}}{a}$	0.036	0.006
$\frac{L_{2}}{L_{1}}$ (400–700 nm band)	0.048	0.026
e	0.002	0.0008
ω (deg)	284	101

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The mass of the secondary, as well as the radii of the two binary components, can be determined in an SB1 if a primary and a secondary eclipse are both observed and the mass of the primary star is known (or assumed). This is accomplished via the mass function:

$$\begin{equation} \frac{(M_2)^3}{(M_1 + M_2)^2} = \frac{4\pi ^2(a_1)^3}{{\it GP}^2}, \end{equation} \tag{ 3 }$$

where G is the gravitational constant, P is the orbital period, and a₁ is determined observationally using the definition of the RV semiamplitude K₁ from the RV equation:

$$\begin{equation} a_{j} = \frac{{K_j P \sqrt{1-e^2}}}{{2 \pi \sin {(i)}}}, j=\lbrace 1,2\rbrace, \end{equation} \tag{ 4 }$$

where e is the orbital eccentricity, i is the line-of-sight orbital inclination, and the subscript j = {1, 2} represents the primary and secondary, respectively. The radii of the two stars can be calculated directly from the geometric light curve parameters R₁a⁻¹ and R₂a⁻¹ found in Table 4 and the semimajor axis a = a₁ + a₂.

The uncertainties for M₁, M₂, R₁, and R₂ are determined as follows. The uncertainty of M₁ is taken from the independent measurement of the primary's mass, if such a determination exists, or from the range of estimates for M₁ based on the star's color and/or spectra. Using implicit partial differentiation, the propagation of error equation, and removing the cross terms since M₁ and C are independent, the uncertainty in M₂ from Equation (3) is given by

$$\begin{equation} \sigma _{M_2} = \left(\frac{4\sigma _{M_1}^2C^2(M_1+M_2)^2 + \sigma _C^2(M_1+M_2)^4}{3(M_2)^2 - 2C(M_1 + M_2)}\right)^{1/2}, \end{equation} \tag{ 5 }$$

where C is the right-hand side of Equation (3):

$$\begin{equation} C = \frac{4\pi ^2(a_1)^3}{{\it GP}^2} \end{equation} \tag{ 6 }$$

and σ_C is found by using the propagation of error equation and assuming P, K₁, e, and i are all independent. Likewise, the uncertainties in the stellar radii are calculated via the propagation of error equation using the measured uncertainties in R₁a⁻¹, R₂a⁻¹ (Table 4), and the calculated σ_a found via propagating the uncertainties in a₁ and a₂.

We adopt a primary mass of M₁ = 0.92 ± 0.1 M_☉ based on the stellar characterization presented in Section 5.1.2 and calculate M₂ = 0.610 ± 0.036 M_☉, R₁ = 0.932 ± 0.076 R_☉, and R₂ = 0.559 ± 0.102 R_☉ (Figure 7). Both the primary and secondary are consistent (1σ) with the model predictions for mass and radii. Metallicity is a minor effect on the mass and radius of the isochrones, so only the tracks corresponding to Z = 0.01 are shown.

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Although the relative errors for the masses and radii in the specific case of TYC 272 are larger than typically used to study the mass–radius relationship (∼6% for M₂, 8% and 18% for R₁ and R₂, respectively), relative errors of a few percent are possible if better constraints on the primary mass could be determined and more precise photometry of the grazing secondary eclipse could be obtained. Specifically, large-aperture telescopes and multi-dimensional correlation software (Zucker & Mazeh 1994; Zucker et al. 2003) can be used to extract RV solutions for faint companions, such that the masses of both components can be measured without relying on models or empirical relations to constrain M₁. In addition, high-precision photometric follow up of the shallow secondary eclipse will improve the precision in the radius. TYC 272 is not an ideal benchmark system because of the shallow, grazing secondary eclipse, however, the analysis presented here demonstrates the required steps to derive precise masses and radii for EBs using a combination of transit and RV exoplanet survey data. A summary of these analysis steps for EBs with low flux ratios, such as TYC 272, is described in Section 6.

TYC 1422 had a total of 16 epochs observed during the MPP, with a Q = 3.07 for its field and an average, scaled RV uncertainty of 0.304 kms⁻¹. When fitting the MPP RV measurements for TYC 1422, the residuals were found to be exceedingly large (rms ∼6 km s⁻¹). Such a large scatter is often caused by contamination of an additional spectrum from an unresolved source, a likely scenario for short-period SBs with similar fluxes.

The MPP DFDI pipeline was not designed to treat multiple sources within a single spectrum and can produce large systematic errors in those cases. It is important to note that the inability to obtain precise RV measurements for SB2s applies to the MPP instrument only. Other survey instruments, including the MARVELS survey instrument itself, will be able to treat the case of double-lined binaries. The MARVELS survey pipeline stores information about the fringe fitting that can be used to remove the fringes from the stellar DFDI spectrum and re-create a "traditional," de-fringed stellar spectrum. This spectrum can then be used to perform cross-correlation analysis and identify and measure multiple CCF components, thereby measuring RVs of double-lined binary stars.

Because the MPP instrument passed starlight through an iodine cell, a similar reconstruction of the stellar spectrum is not possible due to the iodine lines that are present. Attempting to model the iodine lines and remove them from the spectrum prior to de-fringing is also difficult, because the iodine lines are numerous and very narrow. In fact, most of the iodine lines are unresolved at the resolution of the MPP instrument. For this reason, independent RV follow-up observations (SMARTS data) were obtained to measure the RVs of the multiple components. In general, such additional follow-up is not required and RV measurements for both components of double-lined binaries can be obtained directly from the survey data itself.

Despite the large scatter in the residuals, the derived orbital period from the MPP data is in good agreement with the period derived from the SuperWASP photometry (∼6.2 days). To derive an improved orbital solution, we use the higher resolution SMARTS spectra to resolve the components and measure the Doppler reflex velocities of the individual components. We do not use the MPP data in our final analysis because it is so heavily affected by systematic errors. However, for completeness, we present the MPP RV data in Table 5.

The TYC 1422 system consists of three sources, as demonstrated in the triple set of Na D doublets from the ARCES spectrum in Figure 8. We do not attempt an SED analysis due to the complexity of the system, however, the basic stellar properties, including the HAO absolute photometry, are compiled in Table 6. The RVs for each component are phase folded to the orbital period and ephemeris as determined by the SuperWASP photometry in Figure 9. The RV measurements for each component are connected by different line types for visualization. Components 2 and 3 are identified as the EB pair. Their RVs are in antiphase, as expected for an orbiting pair, and have relative velocities that cross the zero line near the phases of primary and secondary eclipses. Component 1 is found to have a gradual linear trend of ∼ − 75 m s⁻¹ day⁻¹. Figure 10 shows the RVs from Component 1 and the best-fit linear relation. The large uncertainty for the third epoch is caused by blending. Table 7 contains the times and measured RVs for each of the three components.

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The SMARTS RVs were fit using MPRVFIT2, an IDL procedure written by N. De Lee which combines a periodogram searching algorithm and a nonlinear least-squares fitter to find the optimal Keplerian orbit for a given set of RV points. The period search algorithm consists of a modified Lomb-Scargle periodogram based on Cumming (2004) with an analytical false alarm probability derived from Baluev (2008). The software allows for simultaneous fitting of RVs from both components of a binary system. The SMARTS spectra consist of seven epochs, resulting in only one degree of freedom for a full Keplerian fit. However, because Components 2 and 3 are a physically bound pair, we can exploit the fact that several parameters are identical for the two components. Specifically, the period (P), eccentricity (e), epoch of periastron (T_p), and systemic RV (γ) are identical, while the argument of periastron (ω) differs by π radians. This results in seven degrees of freedom between the 14 combined data points, the seven parameters being {P, e, ω, T_p, K₁, K₂, γ}, where K₁ and K₂ are the RV semiamplitudes of the two components. Due to the linear trend observed in Component 1, we also include a linear term that reduces the degrees of freedom to six. Figure 11 shows the best-fit orbital solution for Components 2 and 3, and the orbital parameters are presented in Table 8. We find a best-fit period of 6.2005 days and a slightly eccentric orbit with e = 0.16 and a mass ratio of 0.78. The orbital period is in agreement with the photometric period to within 1σ. The slope of the linear term is −48 ± 18 m s⁻¹ day⁻¹, which is of comparable magnitude and direction as the slope measured on Component 1, and therefore may be residual instrument drift.

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We performed the same analysis using the jktebop software as done for TYC 272, however, third light (flux contributed from Component 1, L₃ in Table 9) is included as a free parameter in the least-squares fit to the light curve of TYC 1422. In principle, one can compare a spectroscopic flux ratio based on the EWs of spectral lines with the best-fit flux ratio derived from the light curve solution. However, in the case of TYC 1422 and the ARCES spectrum, most of the lines are blended at the resolution of the spectrograph. The Na D lines shown in Figure 8 are quite visible, however, the wings of the features are still blended, and it is challenging to find an appropriate pseudocontinuum from which to measure the EWs. Furthermore, the EWs of the Na D doublets are known to depend on T_eff (Tripicchio et al. 1997; Díaz et al. 2007), and the T_eff of the third component is not constrained in our analysis.

Table 9. Light Curve Properties—TYC 1422

Parameter	Value	1σ Uncertainty
J (surface brightness ratio at disk center)	0.88	0.01
$\frac{R_{1} + R_{2}}{a}$	0.162	0.002
$\frac{R_{2}}{R_{1}}$	0.43	0.03
i (deg)	87.7	0.5
ecos ω	−0.1283	0.0003
esin ω	0.092	0.011
L3 (contamination from third component)	0.25	0.08
Period (days)	6.199450	0.000007
T0(HJD)	2454194.3663	0.0008
$\frac{R_{1}}{a}$	0.114	0.003
$\frac{R_{2}}{a}$	0.049	0.002
$\frac{L_{2}}{L_{1}}$ (400–700 nm band)	0.16	0.02
e	0.158	0.006
ω (deg)	144	3

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We use limb-darkening coefficients of 0.55 for the primary and secondary. The third light is accounted for within the WASP band, which ranges from 400 to 700 nm and is similar to g' + r'. The light curve with the best-fit model overplotted is shown in Figure 12. Table 9 summarizes the geometric parameters derived from photometry fitting. Correlated errors are accounted for using the "prayer-bead" bootstrap Monte Carlo approach, as described in Section 5.1.3. Timing of the primary and secondary eclipse events confirms the system is in an eccentric orbit.

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Since both components of the EB pair have Doppler measurements, we can measure the masses and radii of Components 2 and 3 directly from the RV semiamplitudes and the light curve. We find the semimajor axes a₁ and a₂ using the RV semiamplitudes from Equation (4). The masses can then be determined using the center of mass equation and the complete version of Kepler's third law:

$$\begin{equation} \left(M_1 + M_2\right) = \frac{4 \pi ^2 \left(a_1+a_2\right)^3}{{\it G P}^2}, \end{equation} \tag{ 7 }$$

where M₁ and M₂ are the masses of the primary and secondary, respectively.

The radii of the two stars can be calculated directly from the geometric parameters measured from the light curves in Table 9 and the value of the semimajor axis a = a₁ + a₂. The primary star is evolved off the main sequence and has a mass of M₁ = 1.163  ±  0.034 M_☉ and a radius of R₁ = 2.063  ±  0.058 R_☉. The secondary has a mass of M₂ = 0.905 ± 0.067 M_☉ and a radius of R₂ = 0.887 ± 0.037 R_☉ and is consistent with a main-sequence G-type dwarf. The relative uncertainties for R₁ and R₂ are 2.8% and 4.2%, respectively, very close to the desired precision goal of <3%. Additional, high-precision photometry of the eclipses will be helpful to further improve the precisions. The relative uncertainties for M₁ and M₂ are 2.9% and 7.4%, respectively. The larger uncertainty for M₂ is in part due to the larger uncertainties in the RV measurements for that component. The SMARTS spectra had relatively low signal-to-noise (∼20 per resolution element), and obtaining higher signal-to-noise spectra will reduce the uncertainty in M₂ to better than 3%. Figure 13 compares the masses and radii with Padova isochrones at an age of log (t) = 9.77 and metallicities of Z = {0.01, 0.019, 0.03}. TYC 1422 is consistent with solar metallicity (solid line), whereas the dashed lines are additional metallicity tracks for comparison purposes.

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