THE NEPTUNE-SIZED CIRCUMBINARY PLANET KEPLER-38b^*

The details of the Kepler  mission have been presented in Borucki et al. (2010), Koch et al. (2010), Batalha et al. (2010), Caldwell et al. (2010), and Gilliland et al. (2010), and references therein. The Kepler reduction pipeline (Jenkins et al. 2010a, 2010b) provides two types of photometry, the basic simple aperture photometry (SAP), and the "pre-search data conditioned" (PDC) data in which many of the instrumental trends are removed (Smith et al. 2012; Stumpe et al. 2012). Using the PDC light curves through Q11, we conducted a visual search for small transit events in the long-period eclipsing binaries from the catalog of Slawson et al. (2011). Several candidate systems were found, including Kepler-38 (KIC 6762829, KOI-1740, and 2MASS J19071928+4216451). The nominal stellar parameters listed in the Kepler Input Catalog (KIC; Brown et al. 2011) are a temperature of T_eff = 5640 K, a surface gravity of log g = 4.47, and apparent magnitudes of r = 13.88 and Kp = 13.94. The binary period is 18.8 days, and the eclipses are relatively shallow. The depth of the primary eclipse is ≈3%, and the depth of the secondary eclipse (which is total) is ≈0.1%.

While the PDC detrended light curves are convenient for visual searches, the detrending is not always complete. In addition, the PDC process attempts to correct for light from contaminating sources in the aperture, and correct for light lost from the target that falls outside the aperture. We have found that too much correction is sometimes applied, so we therefore use the SAP (long cadence only) light curves for the analysis that is described below. The SAP light curve was detrended in a manner similar to what Bass et al. (2012) used. Briefly, each quarter was treated as an independent data set. The light curve was further divided up into separate segments, using discontinuities and data gaps as end points. Splines were fit to each segment, where the eclipses and transits were masked out using an iterative sigma-clipping routine. Once satisfactory fits were found, the data were then normalized by the splines and the segments were combined. Figure 1 shows the SAP and normalized light curves. The data span 966.8 days (2.65 years) from 2009 May to 2012 January. The Kepler spacecraft was collecting data 92.3% of the time, and in the case of Kepler-38, 93.8% of the observations collected were flagged as normal (FITS keyword SAP_QUALITY = 0).

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Initially, six transit events were noticed in the light curve. These are all due to a small body transiting the primary star. The times of mid-transit, the widths, and the depths of the transits were measured by fitting a simple "U-function" (a symmetric low-order polynomial) and the results are given in Table 1. The mean period is 103.8 days. Note, however, that the transit times are poorly described by a simple linear ephemeris (see Figure 2). Furthermore, the widths of the transits depend on the binary phase, where transits that are closer to a primary eclipse (phase ϕ = 0) are narrower than the average and transits that are closer to secondary eclipses (ϕ ≈ 0.5) are wider than the average. The large variations in the transit times and widths are a clear signature of a circumbinary object, rather than a background eclipsing binary. After a more detailed analysis was performed, two additional transits that were partially blended with primary eclipses were found, bringing the total number of transits to eight. Another transit was missed during the data download between Q7 and Q8. Figure 3 shows the transits and the best-fitting model, which is described in Section 3.1. No convincing signatures of the transits of the planet across the secondary were found in the light curve, and this point is discussed further in Section 3.1.

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We also applied a novel transit search algorithm, the "Quasi-periodic Automated Transit Search" (Carter & Agol 2012), which allows for slight variations in the inter-transit spacing. We searched through a range of periods from 50 to 300 days, and for each trial period, we computed the transit duration expected at each time in the light curve for an edge-on circular-orbiting planet. We detrended the light curve and convolved it with a boxcar with the given transit duration, and then shifted each time by the duration since (or until) the orbit would cross the barycenter of the system; this procedure makes the transit spacing and duration nearly uniform if the planet orbits with a nearly circular orbit. This algorithm detects Kepler-38b strongly with S/N ≈17, well above the 3σ false-alarm probability of ≈10. We removed the signal of Kepler-38b, re-applied the algorithm, and no other transiting bodies were detected with periods between 50 and 300 days with sizes larger than ≈0.7 R_Nep (P/105)^1/2 at >3σ significance, where R_Nep is the radius of Neptune and P is the orbital period in days.

In Kepler-16, Kepler-34, and Kepler-35, the circumbinary planets gravitationally perturb the stellar orbits, leading to phase changes of the secondary eclipse relative to the primary eclipse (Doyle et al. 2011; Welsh et al. 2012). Such phase changes can be characterized by measuring accurate eclipse times for all primary and secondary eclipses, fitting a linear ephemeris using a common period and independent zero points, and plotting the residuals on a common period observed minus computed (CPOC) diagram. In a case such as Kepler-34 where the gravitational perturbation from the planet is relatively strong, the residuals of the primary eclipse times have a different slope than the residuals of the secondary eclipses in the CPOC diagram (Welsh et al. 2012). Over long timescales, the phase changes become cyclic, and the signals in the CPOC diagram are poorly described by linear fits. However, if the timescale of the observations is short compared to the precessional timescale, then the CPOC signals of the primary and secondary will be approximately linear and fitting a linear ephemeris to the primary eclipse times and to the secondary eclipse times separately will yield different periods.

For Kepler-38, the times for the primary and secondary eclipses were measured by fitting the eclipse with a low-order cubic Hermite polynomial, as described in Steffen et al. (2011) and Welsh et al. (2012). The typical uncertainties are about 30 s for the primary eclipses and about 9 minutes for the secondary eclipses (see Table 2). Linear fits to the primary and secondary eclipse times were performed separately. The uncertainties on the individual times were scaled to give χ² ≈ N, where N is the number of primary or secondary eclipse times, and the following ephemerides were arrived at

$$\begin{eqnarray*} P &=& 18.7952679\,{\pm}\,0.0000029\,{{\rm d{\rm ays}}},\nonumber\\ T_0 &=& 2,454,952.872560\,{\pm}\,0.000090\;\;{\rm primary} \\ P &=& 18.795224\,{\pm}\,0.000051\, {{\rm d{\rm ays}}},\nonumber\\ T_0 &=& 2,454,962.2430\,{\pm}\,0.0016\;\;{\rm secondary}, \\ \end{eqnarray*}$$

where the times are barycentric Julian dates (BJD_TDB). The difference between these primary and secondary periods is 3.79  ±  4.40 s. The CPOC signals for the primary eclipses and the secondary eclipses are parallel and show no periodicities, which is an indication there are no phase changes of the secondary eclipse relative to primary eclipse. Thus, the object in the circumbinary orbit in Kepler-38 has essentially no observable gravitational effect on the binary, at least on a timescale of a few years.

Kepler-38 was observed from the McDonald Observatory with the Harlan J. Smith 2.7 m Telescope (HJST) and the Hobby-Eberly Telescope (HET) between 2012 March 28 and May 1. The HJST was equipped with the Tull Coudé Spectrograph (Tull et al. 1995), which covers the entire optical spectrum at a resolving power of R = 60, 000. At each visit we took three 1200 s exposures that were then co-added, assuming no Doppler shifts between them. The time of the observation was adjusted to the mid-time of the exposure, properly accounting for the CCD readout time. The radial velocity standard star HD 182488 was usually observed in conjunction with Kepler-38. A Th–Ar hollow cathode lamp was frequently observed to provide the wavelength calibration. In all, a total of nine HJST observations of Kepler-38 were obtained. A customized IRAF script was used to reduce the data, including the correction for the electronic bias, the scattered light subtraction, the flat-field correction, the optimal aperture extraction, and the wavelength calibration. The signal-to-noise ratio (S/N) at the peak of the order containing the Mg b feature near 5169 Å ranged from 10.3 to 15.9 pixel⁻¹. The HET was equipped with the High Resolution Spectrograph (HRS; Tull 1998), and the configuration we used had a resolving power of R = 30, 000 and a wavelength coverage of about 4800–6800 Å. During each visit to Kepler-38 we also obtained a spectrum of HD 182488 and spectra of a Th–Ar lamp. A total of six observations of Kepler-38 were obtained. The images were reduced and the spectra were extracted in a manner similar to the HJST spectra, but with software customized for the HET+HRS configuration. The S/N per pixel at the peak of the order with the Mg b feature ranged from 19.4 to 45.6

We used the "broadening function" technique (Rucinski 1992) to measure the radial velocities. The broadening functions (BFs) are essentially rotational broadening kernels, where the centroid of the peak yields the Doppler shift and where the width of the peak is a measure of the rotational broadening. In order to make full use of this technique, one must have a high S/N spectrum of a slowly rotating template star observed with the same instrumentation as the target spectra. We used observations of HD182488 (spectral type G8V) for this purpose for each respective data set (HJST and HET). The radial velocity of this star was taken to be −21.508 km s⁻¹ (Nidever et al. 2002).

The spectra were prepared for the BF analysis by merging the echelle orders using a two-step process. First, each order was normalized to its local continuum using cubic splines. The low S/N ends of each order were then trimmed so that there was only a modest wavelength overlap (≈5%–10%) between adjacent orders. The normalized orders were then co-added and interpolated to a log-linear wavelength scale. The wavelength range used in the analysis was 5137.51–5509.01 Å for the HJST spectra and 4830.00–5769.95 Å for the HET spectra.

From the BF analysis we found that the spectra were all single-lined (i.e., only one peak was evident in the BFs). Using some simple numerical simulations, we estimate that the lack of a second peak places a lower limit on the flux ratio of the two stars of F_A/F_B ≳ 25 in the HET spectral bandpass, where the subscript A refers to the brighter primary star and B refers to the fainter secondary star. A Gaussian was fit to the BF peaks to determine the peak centroids. Barycentric corrections and a correction for the template radial velocity were applied to the BF peak velocities to arrive at the final radial velocity measurements (see Table 3 and Figure 4).

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The six HET spectra were Doppler corrected to zero velocity and co-added to create a "rest-frame" spectrum covering the range 4830–6800 Å. We cross-correlated this spectrum against a library of F-, G-, and K-type dwarfs obtained with the HET/HRS (but with a different cross-disperser configuration). The spectrum of the G4V star HD 179958 provides a good match as shown in Figure 5.

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The S/N of the spectra proved to be too low to attempt a determination of the stellar temperature, gravity, and metallicity via the measurements of individual lines. Instead, we used the Stellar Parameter Classification (SPC) code (Buchhave et al. 2012) to measure the spectroscopic parameters. The SPC analysis is well suited for spectra with relatively low S/N. The observed stellar spectrum is cross-correlated against a dense grid of synthetic spectra that consists of 51,359 models covering a wide range in effective temperatures, gravities, metallicities, and rotational velocities. Since the SPC analysis uses all of the absorption lines in the wavelength region 5050–5360 Å, the metallicity will be denoted as [m/H]. However, in practice we do not expect [m/H] to be significantly different from [Fe/H] for a star that is close to solar metallicity, and in the following we will consider them to be equivalent. The HJST spectra were combined to yield a spectrum with an S/N of ≈53 at the peak of the echelle order that contains the Mg b features near 5169 Å. The HET spectra were likewise combined to yield a spectrum with an S/N of ≈196 near the Mg b feature. Separate fits were done with the gravity as a free parameter and with the gravity fixed at the dynamical value of log g = 3.926, since the dynamically determined gravity is fairly robust and has a small uncertainty. The SPC-derived parameters are given in Table 4. The gravity found by SPC agrees with the dynamical gravity at the 1σ level. The effective temperature of T_eff = 5623  ±  50 K is roughly what one expects for a spectral type of G4V. For the final adopted parameter values, we use the dynamically determined gravity, and simply average the SPC-derived values from the HJST and HET spectra with the gravity fixed.

The light and velocity curves of Kepler-38 were modeled using the photometric dynamical model described in Carter et al. (2011; see also Doyle et al. 2011; Welsh et al. 2012 for previous applications to transiting circumbinary planet systems). The code integrates the equations of motion for three bodies and, when given a reference time and viewing angle, synthesizes the light curve by accounting for eclipses and transits as necessary, assuming spherical bodies. The radial velocities of the components are also computed as a function of time. Normally, because Kepler-38 is a single-lined binary, one would have to assume a mass for the primary or assume a mass ratio to fully solve for the component masses and radii. Fortunately, the presence of transits constrain the dynamical solution since their exact timing depends in part on the binary mass ratio. On the other hand, so far the planet has had no measurable effect on the eclipse times of the primary and secondary. Given this, the stellar masses and the mass of the planet are not as tightly constrained as they were in the cases of Kepler-16, Kepler-34, and Kepler-35.

The model as applied to Kepler-38 has 34 parameters, including parameters related to the masses (the mass of star A, the binary mass ratio, and the planetary to binary mass ratio), the stellar and planetary orbits (the period, the reference time of primary eclipse, the inclination, and eccentricity/orientation parameters), radius and light parameters (the fractional stellar radii, the planetary to primary radius ratio, the stellar flux ratios, and the limb-darkening parameters), relative contaminations for each of the 11 Quarters of data, a light curve noise scaling parameter, and radial velocity zero points for the HET and HJST measurements. The model was refined using a Monte Carlo Markov Chain routine to estimate the credible intervals for the model parameters. The resulting best-fitting parameters and their uncertainties are summarized in Table 5, and derived astrophysical parameters of interest are summarized in Table 6. Figure 3 shows the transits and the best-fitting model, and Figure 6 shows schematic diagrams of the stellar and planetary orbits.

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Table 5. Fitting Parameters for Photometric Dynamical Model

Parameter	Best Fit	50%	15.8%	84.2%
Mass parameters
Mass of star A, MA (M☉)	0.949	0.941	−0.059	+0.055
Mass ratio, star B, MB/MA	0.2626	0.2631	−0.0056	+0.0067
Planetary mass ratio, Mb/MA (× 1000)	0.22	0.18	−0.11	+0.13
Planetary orbit
Orbital period, Pb (days)	105.595	105.599	−0.038	+0.053
Eccentricity parameter, $\sqrt{e_b} \cos (\omega _b)$	0.062	0.046	−0.064	+0.049
Eccentricity parameter, $\sqrt{e_b} \sin (\omega _b)$	0.040	0.004	−0.100	+0.106
Time of barycentric transit, tb (days since t0)	−37.888	−37.896	−0.022	+0.044
Orbital inclination, ib (deg)	89.446	89.442	−0.026	+0.030
Relative nodal longitude, ΔΩ (deg)	−0.012	−0.005	−0.052	+0.050
Stellar orbit
Orbital period, P1 (days)	18.79537	18.79535	−0.000051	+0.000062
Eccentricity parameter, $\sqrt{e_1} \cos (\omega _1)$	−0.0074	−0.0074	−0.0002	+0.0002
Eccentricity parameter, $\sqrt{e_1} \sin (\omega _1)$	−0.32113	−0.32266	−0.00188	+0.00185
Time of primary eclipse, t1 (days since t0)	−17.127434	−17.127462	−0.000078	+0.000072
Orbital inclination, i1 (deg)	89.265	89.256	−0.025	+0.026
Radius/light parameters
Linear limb-darkening parameter, uA	0.453	0.457	−0.007	+0.007
Quadratic limb-darkening parameter, vA	0.143	0.135	−0.018	+0.018
Stellar flux ratio, FB/FA (× 100)	0.09081	0.09059	−0.00072	+0.00071
Radius of star A, RA (R☉)	1.757	1.752	−0.034	+0.031
Radius ratio, star B, RB/RA	0.15503	0.15513	−0.00021	+0.00021
Planetary radius ratio, Rb/RA	0.02272	0.02254	−0.00030	+0.00030
Relative contamination, Fcont/FA (× 100)
Quarter 1			0 (fixed)
Quarter 2	1.40	1.37	−0.14	+0.14
Quarter 3	1.55	1.60	−0.14	+0.14
Quarter 4	1.88	1.91	−0.14	+0.14
Quarter 5	0.58	0.62	−0.14	+0.14
Quarter 6	1.21	1.17	−0.14	+0.14
Quarter 7	1.48	1.52	−0.15	+0.15
Quarter 8	1.73	1.77	−0.14	+0.14
Quarter 9	0.78	0.83	−0.14	+0.14
Quarter 10	1.26	1.25	−0.14	+0.14
Quarter 11	1.80	1.81	−0.14	+0.14
Noise parameter
Long cadence relative width, σLC (× 105)	17.25	17.19	−0.16	+0.16
Radial velocity parameters
RV offset, γ (km s−1)	18.008	17.996	−0.030	+0.031
Zero-level diff., McDonald/HET, Δγ (km s−1)	−0.083	−0.058	−0.035	+0.034

Note. The reference epoch is t₀ = 2, 454, 970 (BJD).

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The circumbinary planets in Kepler-16, Kepler-34, and Kepler-35 transit both the primary star and the secondary star. When transit events across both stars are observed, the constraints on the orbital parameters are much tighter than they are when only transits across the primary are seen. In the case of Kepler-38, the planet does not transit the secondary in the best-fitting model (Figure 6), although given the uncertainty in the nodal angle Ω, transits of the secondary might occur for some of the acceptable models derived from the Monte Carlo Markov Chain. However, individual transits of the planet across the secondary will not be observable, owing to the extreme flux ratio of star B to star A, where F_B/F_A = 9 × 10⁻⁴ in the Kepler bandpass (Table 5). The expected depth of the transit of the planet across the secondary is on the order of 21 ppm, which is a factor of 10 smaller than the noise level of ≈210 ppm. In a similar vein, occultations of the planet by star A do occur, but are undetectable given the noise level.

As an independent check on the parameters of the binary, we modeled the light and velocity curves using the Eclipsing Light Curve (ELC) code (Orosz & Hauschildt 2000) with its genetic algorithm and Monte Carlo Markov Chain optimizers. As noted above, we cannot use ELC to solve for the stellar masses or the absolute stellar radii since Kepler-38 is a single-lined binary. However, ELC can be used to find the orbital parameters (K, e, ω, i, P, and T_conj), the fractional radii R_A/a and R_B/a, the temperature ratio T_B/T_A, and the stellar limb-darkening parameters x_A and y_A for the quadratic limb-darkening law [I(μ) = I₀(1 − x(1 − μ) − y(1 − μ)²)]. In the limiting case where the stars are sufficiently separated that their shapes are spherical, ELC has a fast "analytic" mode where the equations given in Giménez (2006) are used. Table 7 gives the resulting parameters of the fit. The agreement between ELC and the photometric dynamical model is good.

The photometric dynamical model assumes the stars are spherical. We computed a model light curve using ELC, assuming "Roche" geometry (to the extent that is possible in an eccentric orbit; see Avni 1976; Wilson 1979). At periastron, the "point" radius of the primary along the line of centers differs from the polar radius by 0.021%. Thus, the assumption of spherical stars is a good one. The expected amplitude of the modulation of the out-of-eclipse part of the light curve due to reflection and ellipsoidal modulation is ≈180 ppm. If Doppler boosting (Loeb & Gaudi 2003; Zucker et al. 2007) is included, the amplitude of the combined signal from all effects is ≈270 ppm, with the maximum observed at phase ϕ ≈ 0.3. Finally, the amplitude of the signal in the radial velocity curve due to the Rossiter–McLaughlin effect during the primary eclipse (when the secondary star transits the primary) is on the order of 55 m s⁻¹ for a projected rotational velocity of 2.4 km s⁻¹.

The radius of Kepler-38b is 4.35 R_⊕ (=1.12 R_Nep or 0.39 R_Jup, using the equatorial radii), with an uncertainty of ±0.11 R_⊕ (or 2.5%). For comparison, its radius is about half of the radius of Kepler-16b (R = 0.7538  ±  0.0025 R_Jup; Doyle et al. 2011), Kepler-34b (R = 0.764  ±  0.014 R_Jup; Welsh et al. 2012), and Kepler 35b (R = 0.728  ±  0.014 R_Jup; Welsh et al. 2012). Thus, all four of the transiting circumbinary planets discovered so far have radii substantially smaller than Jupiter's. Since a Jupiter-sized planet would have a deeper transit and would therefore be easier to find (all other conditions being equal), the tendency for the circumbinary planets to be sub-Jupiter size is noteworthy. Pierens & Nelson (2008) argued that Jupiter-mass circumbinary planets in orbits relatively close to the binary should be rare owing to the various instabilities that occur during the migration phase and also to subsequent resonant interactions with the binary (Jupiter-mass circumbinary planets that orbit further out from the binary could be stable, but these would be less likely to transit owing to a larger separation). The predictions of Pierens & Nelson (2008) seem to be consistent with what is known from the first four Kepler transiting circumbinary planets.

Since Kepler-38b has not yet noticeably perturbed the stellar orbits, we have only an upper limit on its mass of M_b < 122 M_⊕ (<7.11 M_Nep or <0.384 M_Jup) at 95% confidence. While this is clearly a substellar mass, this upper limit is not particularly constraining in terms of the density, as we find ρ_b < 8.18 g cm⁻³. A reasonable mass is M_b ≈ 21 M_⊕, assuming the planet follows the empirical mass–radius relation of M_b = (R_b/R_⊕)^2.06 M_⊕ (Lissauer et al. 2011).

The gravitational interaction between the planet and the two stars causes small perturbations that will grow over time and will eventually lead to a measurable change in the phase difference between the primary and secondary eclipses. As discussed above, this change in the phase difference manifests itself as a difference between the period measured from the primary eclipses and the period measured from the secondary eclipses. In Kepler-16, Kepler-34, and Kepler-35, the timescale for a measurable period difference to occur is relatively short, as divergent periods were measured using 6 Quarters of data for Kepler-16, and 9 Quarters of data for Kepler-34 and Kepler-35. For Kepler-38, the 1σ limit on the period difference is <4.4 s using 11 Quarters. Figure 7 shows the set of acceptable planetary masses and the changes in orbital period the planet would induce, based on the Monte Carlo Markov Chain from the photometric dynamical model. The larger the planetary mass, the larger the difference in periods it causes between the primary and secondary star. The lack of any measurable period difference places an upper limit on the mass of the planet. To the left of the vertical dashed line at 122 M_⊕ is where 95% of the acceptable solutions reside. The vertical dot-dashed line marks a planetary mass of 21 M_⊕. The horizontal dashed line at 4.4 s marks the observed 1σ uncertainty in the measured value of P₂ − P₁ accumulated over the span of the current Kepler observations (52 binary star eclipses); valid planetary masses lie below this line, roughly. The dotted horizontal line at 0.9 s marks a 1σ period difference uncertainty that can be placed when ≈150 eclipses are eventually observed by Kepler, corresponding approximately to the end of the Extended Mission in the year 2017. If no period difference is measured at that time, the mass of the planet would be less than ≈20 M_⊕ with 1σ uncertainty. (The intersection of this period difference uncertainty and the 21 M_⊕ line within the set of Monte Carlo points is a coincidence.) If the planet has a normal density and a mass of 21 M_⊕, then the period difference would be ≈0.9 s, which would require ≈312 binary orbits or about 16 years to obtain a 3σ detection.

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As noted earlier, the observed timing of the planet transits and the amplitude of the radial velocity curve sets the scale of the binary, and we are able to measure masses and radii for each star. Considering the fact that Kepler-38 is a single-lined spectroscopic binary (note that the ratio of the secondary-to-primary flux in the Kepler bandpass is 9 × 10⁻⁴; see Table 5), the uncertainties in the masses and radii are fairly small (6.2% and 3.6% for the primary and secondary masses, respectively, and 1.8% for the radii). On the other hand, these uncertainties are still somewhat larger than what one would like when doing precise comparisons with stellar evolutionary models (for comparison, the stellar masses and radii are known to much better than 1% for the first three Kepler circumbinary planets; see Doyle et al. 2011; Welsh et al. 2012). Clearly, since the radius of the primary star (1.757  ±  0.034 R_☉) is much larger than the expected zero-age main-sequence radius for its mass (0.949  ±  0.059 M_☉), the primary must be significantly evolved, but is still a core hydrogen-burning star. The situation is shown in Figure 8, which gives the position of Kepler-38 in a T_eff–log g diagram. The heavy solid line is a Yonsei-Yale evolutionary track (Yi et al. 2001) for [Fe/H] = −0.11, which is our adopted spectroscopic metallicity determination (Table 4). As indicated earlier, we assume here that the iron abundance is well approximated by the [m/H] metallicity index measured with SPC. The dark shaded area is the uncertainty in the location of the track that results from the uncertainty in the mass, and the lighter shaded area also includes the uncertainty in the metallicity. The observed T_eff and log g is just outside the 1σ region of the evolutionary track. Figure 9 shows the positions of the stars in Kepler-38 on the mass–radius and mass–temperature diagrams (the temperature of the secondary was inferred from the measured temperature of the primary (Table 4) and the temperature ratio from the ELC models (Table 7)). They are compared against model isochrones from the Dartmouth series (Dotter et al. 2008), in which the physical ingredients such as the equation of state and the boundary conditions are designed to better approximate low-mass stars (see, e.g., Feiden et al. 2011). If we use the mass, radius, and metallicity of the primary, the inferred age is 13 Gyr. If, on the other hand, we use the measured temperature, the age of the system would be between 7 and 8 Gyr. Accounting for the small discrepancy between the measured temperature and the other parameters, we adopt an age of 10  ±  3 Gyr.

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The secondary star has a relatively low mass (0.249  ±  0.010 M_☉) and is one of just a handful of low-mass stars with a well-measured mass and radius. Its mass is slightly larger than those of CM Dra A (0.2310 M_☉; Morales et al. 2009) and KOI-126 B (0.20133 M_☉; Carter et al. 2011), and Kepler-16b (0.20255  ±  0.00066 M_☉; Doyle et al. 2011; see also Bender et al. 2012; Winn et al. 2011). Stars whose mass is <0.8 M_☉ typically have radii that are ∼10%–15% larger than what is predicted by stellar evolutionary models (Torres & Ribas 2002; Ribas 2006; Ribas et al. 2008; López-Morales 2007; Torres et al. 2010; Feiden et al. 2011). Relatively high levels of stellar activity induced by tidal interactions in short-period systems is one possible cause of this discrepancy (López-Morales 2007). There is a hint that the secondary star is inflated, although we note there is a small discrepancy with the models for the primary.

With Kepler data, it is frequently possible to determine the rotation period of the star from the modulations in the light curve due to starspots. In the case of Kepler-38 however, the modulations are small (rms <600 ppm for the long cadence time series, omitting the eclipses), and considerably smaller than the instrumental systematic trends in the light curve. Separating intrinsic stellar variability from instrumental artifacts is difficult, so we used the PDC light curve that has many of the instrumental trends removed. The PDC data for Kepler-38 are generally flat outside of eclipses, with the exception of Quarter 1, which we therefore omitted. The data were normalized and the primary and secondary eclipses removed from the time series. A power spectrum/periodogram was computed, and the dominant frequency present corresponds to the orbital period (18.79 days). We also patched the gaps in the light curve with a random walk and computed the auto-correlation function (ACF). The ACF revealed a broad peak at ∼18 days, consistent with the orbital period.

The fact that the orbital period is manifest in the out-of-eclipse light curve initially suggested that starspots were present and the star's spin was tidally locked with the orbital period. But the non-zero eccentricity of the orbit means that exact synchronicity is impossible (see below). Phase-folding the data on the orbital period, and then binning to reduce the noise, revealed the origin of the orbital modulation: Doppler boosting combined with reflection and ellipsoidal modulations. The amplitude is ∼300 ppm with a maximum near phase ϕ = 0.25, consistent with what is expected from the ELC models. We verified that the Doppler boost signal is also present in the SAP light curves. For more discussion of Doppler boosting in Kepler light curves, see Faigler & Mazeh (2011) and Shporer et al. (2011).

For an eccentric orbit, there is no one-spin period that can be synchronous over the entire orbit. However, there is a spin period such that, integrated over an orbit, there is no net torque on the star's spin caused by the companion star. At this "pseudosynchronous" period, the spin is in equilibrium and will not evolve (Hut 1981, 1982). Given the old age of the binary, it should have reached this pseudosynchronous equilibrium state. The ratio of orbital period to pseudosynchronous spin period is a function of the eccentricity only (Hut 1981), and for Kepler-38 this period is P_pseudo  =  17.7 days. This is close to, but slightly shorter than, the ∼18.79 day orbital period. Using this pseudosynchronous spin period and the measured stellar radius, a projected rotational velocity of V_rotsin i ≈ 4.7 km s⁻¹ is expected, if the star's spin axis is aligned with the binary orbital axis. Our spectral modeling yields V_rotsin i = 2.4  ±  0.5 km s⁻¹, which is close to the rotational velocity expected due to pseudosynchronous rotation. Given that the rotational velocity is near the spectral resolution limit, we cannot rule out systematic errors of a few km s⁻¹ caused by changes in the instrumental point-spread function, macroturbulence, etc., which could bring the measured value of V_rotsin i up to the pseudosynchronous value.

The 105 day planetary orbital period is the shortest among the first four Kepler circumbinary planets. The planet orbits the binary quite closely: the ratio of planetary to stellar orbital periods is 5.6 (ratio of semimajor axes is 3.2). Such a tight orbit is subject to dynamical perturbations, and following the analytic approximation given by Holman & Wiegert (1999), the critical orbital period in this binary below which the planet's orbit could experience an instability is 81 days. This compares favorably with the results of direct N-body integrations, which yield a critical orbit period of 75 days. Thus, while stable, the planet is only 42% above the critical period (or 26% beyond the critical semimajor axis). Kepler-38 thus joins Kepler-16 (14%), Kepler-34 (21%), and Kepler-35 (24%) as systems where the planet's orbital period is only modestly larger than the threshold for stability. The fact that the first four circumbinary planets detected by Kepler are close to the inner stability limit is an interesting orbital feature that may be explained by processes such as planetary migration and planet–planet scattering during and/or post formation of these objects. For example, it is also possible that strong planet migration will bring planets close in: migration may cease near the instability separation leading to a pile-up just outside the critical radius; or planets that continue to migrate in are dynamically ejected, leaving only those outside the critical radius (Pierens & Nelson 2008). There is also an observational bias since objects that orbit closer to their host star(s) will be more likely to transit, making them more likely to be discovered.

Regardless of the cause, the close-to-critical orbits have an interesting consequence. Many Kepler eclipsing binaries have orbital periods in the range 15–50 days and have G- and K-type stars (Prša et al. 2011; Slawson et al. 2011), and for such binaries the critical separation (roughly two to four times the binary separation, depending on the eccentricity of the binary) is close to the habitable zone.²⁰ Thus, observed circumbinary planets may preferentially lie close to their habitable zones. Kepler-16b is just slightly exterior to its habitable zone, while Kepler-34b is slightly interior (too hot). Kepler-38b is well interior to its habitable zone, with a mean equilibrium temperature of T_eq = 475 K, assuming a Bond albedo of 0.34 (similar to that of Jupiter and Saturn). Although T_eq is somewhat insensitive to the albedo, this temperature estimation neglects the atmosphere of the planet, and therefore should be considered a lower limit. It is interesting to consider the situation at a much earlier time in the past when the primary was near the zero-age main sequence. The primary's luminosity would have a factor ≈3 smaller, and the equilibrium temperature of the planet would have been T_eq ≈ 361 K, assuming a similar orbit and Bond albedo.