KEPLER-68: THREE PLANETS, ONE WITH A DENSITY BETWEEN THAT OF EARTH AND ICE GIANTS

The Kepler-68b transits were identified by the Transiting Planet Search (TPS) pipeline module that identifies periodic reductions in flux with a duration of hours, each corresponding to a transit of a prospective planet. The algorithm is a wavelet-based, adaptive matched filter (Jenkins et al. 2010c). TPS then identifies a time series of single "events," each having an associated "single event statistic (SES)" that represents the probability that a transit is present. The SES from each transit are combined into multiple event statistics (MES) by folding them at trial orbital periods ranging from 0.5 days to as long as half the data coverage interval.

Kepler-68b was identified by TPS in each quarter of data with an MES >15σ.

Multi-quarter searching for transits was used. The transit depth, duration, period, and epoch are derived from physical modeling (see Section 7) using all of the available data. Kepler-68b is characterized as a 345.6 ± 1.5 ppm dimming lasting 3.459 ± 0.009 hr, with transit ephemeris of T [BJD] = 2455006.85729 ± 0.00042 + N*5.398763 ± 0.000004 days. The longer-period transits of Kepler-68c were identified by non-pipeline inspections. Kepler-68c is characterized as a 53.1 ± 2.3 ppm dimming lasting 3.09 ± 0.09 hr and an ephemeris T [BJD] = 2454969.3805 ± 0.0041 + N*9.605085 ± 0.000072 days.

Two reconnaissance spectra were obtained with the Tull Coude Spectrograph of the McDonald Observatory 2.7 m Harlan J. Smith Telescope on the nights of 2010 March 25 and 28, and a third at the TRES Echelle Spectrograph of the Tillinghast 1.5 m telescope on Mt. Hopkins, also on the night of 2010 March 25. These high spectral resolution, low signal/noise spectra showed no convincing evidence for RV variability at the 0.2 km s⁻¹ level, and no hints of any contaminating spectra. These spectra were cross-correlated against a library of synthetic model stellar spectra, as described by Batalha et al. (2011), in order to derive basic stellar parameters to compare with the KIC values. These spectra were in excellent agreement, and yielded T_eff = 5750 K, log g = 4.0, and a rotational velocity of less than 4 km s⁻¹. The height of the cross-correlation peaks ranged from 0.93 to 0.96, indicating an excellent match with the library spectra. This spectroscopy suggests that Kepler-68 is a sun-like, slowly rotating main sequence star, supporting the planetary interpretation for the transit events.

There is always a possibility, especially for bright stars of high enough proper motion, for old plates to reveal the background distribution of faint stars in the current epoch. Inspection of 1953 and 1991 Sky Survey plates shows that this does not work for Kepler-68—the proper motion is far too small.

4.2.1. AO Imaging

Near-infrared adaptive optics imaging of Kepler-68 was obtained with the 6.5 m MMT telescope on Mt. Hopkins and the Arizona Infrared imager and Echelle Spectrograph (ARIES). The Kepler-68 imaging was obtained on the night of 2010 May 5 (UT) using the f/30 mode with a field of view of 20'' × 20'' and a resolution of 002085 per pixel. The AO system guided on the primary target. The FWHM of the J band combined image was 0123, while the Ks band provided 0112 imaging. A four-point dither pattern was used with a total of 16 images in each band. Further details of the MMT-ARIES high resolution imaging for Kepler follow-up support may be found in Adams et al. (2012).

Figure 5 shows a region of the ARIES Ks band image. Within the domain plotted, no other sources are visible to the 5σ depth limits as shown in Figure 6. Within the full ARIES field of view, one additional source that was 6.1 and 5.7 mag fainter in J and Ks, respectively, was identified. Using the 2MASS J and Ks magnitudes for Kepler-68 of 8.974 and 8.587 leads to estimates of 15.07 and 14.29, respectively, for the companion. Using the transformations in Appendix A of Howell et al. (2011) leads to a Kp = 17.0 ± 0.4, or a δ-magnitude of 7.0 ± 0.4 with respect to Kepler-68 in the Kepler bandpass. This AO identified source also shows up clearly in available UKIRT J direct imaging (Lucas & Samuel 2009) and corresponds to the star KIC 11295432, which does not have a specified Kp in the KIC. From a combination of Kepler data centroid consideration (Section 3) and the ARIES AO and UKIRT data, to roughly the exclusion limits reached in Figure 6, no other sources exist to relevant larger radii as well.

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4.2.2. Speckle Imaging

We obtained 52 high resolution spectra of Kepler-68 between 2010 July 19 and 2012 August 12 using the HIRES spectrometer on the Keck I 10 m telescope (Vogt et al. 1994). We configured the spectrometer, took observations, and reduced the spectra with the same method used on thousands of nearby FGKM stars (Marcy et al. 2008). This technique yields a Doppler precision of 1.0–1.5 m s⁻¹ for stars as faint as the 14th magnitude (V band), depending on spectral type and rotational vsin i. The HIRES fiber-feed was not used for these observations. An iodine cell was used to superimpose iodine lines on the stellar spectrum, providing empirical information for each exposure and each wavelength about the instantaneous wavelength scale and instrumental profile of the spectrometer.

The observations were made with the "C2 decker" entrance aperture, which projects to 087 × 140, giving a resolving power of about 60,000 at 5500 Å and enabling sky subtraction (typical seeing is 06–12). The average exposure was 11 minutes, giving a S/N per pixel of 200.

The raw CCD images were reduced in the standard way, including the subtraction of background sky counts (mostly from moonlight) at each wavelength just above and below the stellar spectrum. We used Doppler analysis with the algorithm of Johnson et al. (2009). The internal Doppler errors (the weighted uncertainty in the mean of 400 spectral segments) are typically 1.0–1.5 m s⁻¹. Our experience with hundreds of G-type main sequence stars shows that the actual errors are larger than the internal errors by ∼1.5 m s⁻¹ for such stars. Thus, we added 1.5 m s⁻¹ in quadrature to the internal uncertainties to yield our estimated uncertainty in the RV. The resulting velocities and uncertainties are given in Table 2 and shown in Figure 7 as a function of time. The error bars include the internal Doppler errors and an assumed jitter of 1.5 m s⁻¹.

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The center of mass velocity relative to the solar system barycenter (Gamma Velocity) for Kepler-68 is −20.9 ± 0.1 km s⁻¹ (Table 4). This is a typical RV for a star in the Galactic disk, indicative of a middle-aged disk star. The near solar metallicity (Section 6.1), [Fe/H] = +0.12 ± 0.074, magnetic activity (Section 6.1), and asteroseismic age, 6.3 ± 1.7 Gyr (Section 6.2), also suggest Galactic disk membership.

The primary signal from the RVs is the K = 19.9 ± 0.75 m s⁻¹ variation with a period of 580 ± 15 days, as shown first in Figure 7 and phased in Figure 8. This clear RV signature, coupled with the lack of bisector variations discussed in Section 4.4, provides discovery and confirmation of the Jovian-scale outer planet—Kepler-68d.

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The velocities phased to the photometric period of Kepler-68b in Figure 8 show a clear, continuous, and nearly sinusoidal variation consistent with a nearly circular orbit of a planetary companion. Note that the 7 mag (see Sections 3 and 4.2.1) fainter companion KIC 11295432 blended into the Kepler photometry does not fall within the HIRES slit and cannot be the source of Kepler-68b RV variations. The lack of any discontinuities in the phased velocity plot argues against a background eclipsing binary star as the explanation. Such a binary with a period of 5.4 days would have orbital semi-amplitudes of tens of kilometers per second, so large that the spectral lines would completely separate from each other and separate from the lines of the main star. Such breaks in the spectral-line blends would cause discontinuities in the velocity variation, which are not seen here. Thus, the chance that the 5.4 day periodicity exhibited independently in both the photometry and velocities might be caused by an eclipsing binary seems small.

Precision Doppler measurements are used to constrain the mass of Kepler-68b, as discussed in Section 7. The absence of a Doppler signal for Kepler-68c is used to compute an upper limit to the mass under the planet interpretation.

From the Keck spectra, we computed a mean line profile and the corresponding mean line bisector. Time-varying line asymmetries are tracked by measuring the bisector spans—the velocity difference between the top and bottom of the mean line bisector—for each spectrum (Torres et al. 2005). When RV variations are the result of a blended spectrum between a star and an eclipsing binary, we expect the bisectors to reveal a phase-modulated line asymmetry (Queloz et al. 2001; Mandushev et al. 2005).

The bisector spans in the lower panel of Figure 7 show no correlated variation with the RVs and have a scatter of 7.8 m s⁻¹, which is significantly less than the semi-amplitude of Kepler-68d. However, the uncertainties in the bisector measurements are larger than the semi-amplitude of the two smaller planets in the system and a bisector analysis is thus inconclusive with respect to Kepler-68b and Kepler-68c.

Kepler-68b was observed during two transits with Warm-Spitzer/IRAC (Werner et al. 2004; Fazio et al. 2004) at 4.5 μm (program ID 60028). The observations occurred on UT 2010 December 27 and on UT 2011 January 7. The visits lasted 7.6 hr. The data were gathered in subarray mode (32 × 32 pixels) with an exposure time of 2 s per image that yielded 15,680 images per visit.

The images used are the Basic Calibrated Data (BCD) delivered by the Spitzer archive. These files are corrected for dark current, flat-fielding, detector non-linearity and converted into flux units. The method we used to produce photometric time series from the images is described in Désert et al. (2009). We first discard the first half-hour of observations, which is affected by a significant telescope jitter before stabilization. To facilitate the evaluation of the photometric errors, we then convert the pixel intensities to electrons using the information given in the detector gain and exposure time provided in the FITS headers. We convert to UTC-based BJD following the procedure developed by Eastman et al. (2010). We correct for transient pixels in each individual image using a 20-point sliding median filter of the pixel intensity versus time. We find the centroid position of the stellar PSF and perform aperture photometry using a circular aperture with a radius of 3.5 pixels on individual BCD images; we adopt the radius that provides the smallest errors. The final number of photometric measurements used is 13, 146 data points for the first visit and 13, 262 for the second one. The raw time series are presented in the top panels of Figure 9. We find a typical S/N 260 per image, which corresponds to about 90% of the theoretical S/N.

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We used a transit light curve model multiplied by instrumental decorrelation functions to measure the transit parameters and their uncertainties from the Spitzer data, as described in Désert et al. (2011b). We compute the transit light curves with the IDL transit routine OCCULTSMALL from Mandel & Agol (2002). In the present case, this function depends on one parameter: the planet-to-star radius ratio R_p/R_⋆. The other transit parameters are fixed to the value derived from the Kepler lightcurves. The limb-darkening coefficients are set to zero, consistent with expected small values in the IR and insufficient photometric precision in the Spitzer light curves to matter.

The Spitzer/IRAC photometry is known to be systematically affected by the so-called pixel-phase effect (see, e.g., Charbonneau et al. 2005; Knutson et al. 2008). We decorrelated our signal in each channel using a linear function of time for the baseline (two parameters) and a quadratic function of the PSF position (four parameters) to correct the data for each channel. We performed a Markov Chain Monte Carlo (MCMC) analysis with six chains of length 10⁵, each providing median depth values and errors. We allow asymmetric error bars spanning 34% of the points above and below the median of the distributions to derive the 1σ uncertainties for each parameter, as described in Désert et al. (2011a).

We measured the transit depth for Kepler-68b at 4.5 μm of 350 ± 70 ppm for the first visit and 560 ± 70 ppm for the second. The weighted mean of these two values provides a transit depth of 455 ± 50 ppm. The value for the first visit is in excellent agreement with the Kepler depth of 346 ppm, suggesting that the radius-ratio of the candidate Kepler-68b to its host star is a wavelength independent function, in agreement with a dark planetary object. However, visit two is 3σ from the Kepler value. Two possibilities for this behavior, other than the small statistical probability of two such discrepant values legitimately following from the same underlying distribution, are: (1) a physically different depth in the two epochs for the transit, and (2) an inconsistency in one of the Spitzer epochs that we have failed to uncover. To investigate the first possibility, we identified the position of the two Spitzer epochs relative to Kepler coverage. Alas, despite the excellent overall duty cycle with Kepler in excess of 90%, both epochs fell into the longest downtime experienced to date with Kepler—due to a safing event before Quarter 8. We have no data with which to challenge the hypothesis that the two epochs really do have different depths, as could happen if there is an as yet unclaimed transiting planet that overlapped with the deeper Spitzer transit of Kepler-68b. We have not uncovered direct evidence for an inconsistency in the data quality or analyses sufficient to explain the discrepancy in the depths of the two Spitzer epochs. However, we do note that the scatter evident in Figure 9 for the anomalously deep transit is much larger than we normally encounter.

Given the perfect agreement of one Spitzer visit in reproducing the transit depth seen in the optical, the RV confirmation documented in the previous section, and a strong BLENDER validation to be presented in Section 5, the case for the planet interpretation of Kepler-68b remains strong.

In the absence of a dynamical confirmation of the planetary nature of the Kepler-68c signal from either a Doppler detection or a Transit Timing Variation (TTV) signature, we proceed here with a probabilistic "validation." In essence, we seek to demonstrate that the signal is much more likely to be caused by a bona-fide transiting planet than by a false positive (or "blend"). For this, we applied the BLENDER procedure, which has been described previously (Torres et al. 2004; Fressin et al. 2011, 2012b) and used to validate a number of other Kepler planets (e.g., Ballard et al. 2011; Lissauer et al. 2012; Borucki et al. 2012; Howell et al. 2012; Gautier et al. 2012). We refer the reader to these works for full details of the method.

Briefly, we performed a systematic exploration of the different types of false positives that can mimic the signal by generating large numbers of synthetic blend light curves over a wide range of parameters and comparing each of them with the Kepler photometry in a χ² sense. The photometry we used for the validation is the de-trended SC time series (one month from Quarter 2, plus the seven quarters of Quarters 5–11—886,638 measurements), which provides stronger constraints on the shape of the transit than the long cadence data, as shown later. We rejected blends that result in light curves inconsistent with the Kepler observations. We then estimated the frequency of the remaining blends by taking into account all of the available observational constraints from the follow-up observations mentioned above. Finally, we compared this frequency with the expected frequency of true planets (planet "prior") to derive the odds ratio.

The types of false positives we considered include eclipsing systems falling within the Kepler aperture that are either in the background or foreground, or that are physically associated with the target in a hierarchical triple configuration. We allowed the object producing the eclipses to be either a star or a planet. To compute the blend frequencies, we used informed estimates of the number density of stars in the background from the Besancon Galactic structure model of Robin et al. (2003), rates of occurrence of eclipsing binaries in the Kepler field from the work of Slawson et al. (2011), and frequencies of planets involved in blends based on the catalog of planet candidates (KOIs) of Batalha et al. (2013), which was constructed using observations from Quarter 1 to Quarter 6. The same catalog was also used to estimate the planet prior. This list of KOIs is bound to contain some fraction of false positives (see, e.g., Morton & Johnson 2011; Morton 2012) and it is also likely incomplete, mainly due to the difficulty in detecting shallow signals especially with long periods. Consequently, we have applied corrections for these effects following a Monte Carlo procedure described by Fressin et al. (2012a), both in computing the blend frequencies and also for the planet prior.

The observational constraints used to further reduce the number of blends included the following: (1) the color of the star as reported in the KIC (Brown et al. 2011), which allows us to rule out any simulated blends resulting in a combined color that is significantly redder or bluer than the target; (2) limits from the centroid motion analysis on the angular separation of companions that could produce the signal (Section 3); (3) brightness and angular separation limits from high-resolution adaptive optics and speckle imaging (Section 4.2); (4) limits on the brightness of unresolved companions from high-resolution spectroscopy (Section 4.1). For eclipsing systems physically associated with the target, we also considered dynamical stability constraints in hierarchical triple configurations (Holman & Wiegert 1999).

The BLENDER simulations for Kepler-68c rule out all false positive scenarios involving eclipsing binaries physically associated with the target, as the predicted light curves invariably have the wrong shape to match a planetary transit. For the scenarios involving eclipsing binaries that are in the background or foreground, we find a blend frequency of 2.8 × 10⁻⁶, and for those in which larger planets transit background stars, we estimate a much smaller frequency of 7.0 × 10⁻⁸. Hierarchical triples (a larger planet transiting a stellar companion to the target) contribute a frequency of 6.7 × 10⁻⁷. The total blend frequency is thus 3.5 × 10⁻⁶. An illustration of the constraints on false positives resulting from BLENDER as well as those from the follow-up observations is given in Figure 10.

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An estimate of the planet prior may be obtained by dividing the number of known planets of similar size as Kepler-68c from Batalha et al. (2013) by the total number of main-sequence Kepler targets observed during Quarters 1–6, which is 138,253. We find 71 KOIs that are in the same (3σ) radius range as the putative planet in Kepler-68c, of which we expect 9.4 to be false positives, following the procedures of Fressin et al. (2012a). We also expect such shallow (∼50 ppm) transit signals to be detectable around only about 9.7% of all Kepler targets, which we use to correct for incompleteness. We compute the planet prior as (71 − 9.4)/0.097/138, 253 = 4.6 × 10⁻³. This a priori planet frequency is 4.6 × 10⁻³/3.5 × 10⁻⁶ ≈ 1300 times larger than the estimated blend frequency, from which we conclude that Kepler-68c is validated as a true planet with a very high degree of confidence.

In the above calculations, we have not explicitly taken into consideration the period of the signal, which may be an important factor for small candidates such as Kepler-68c because such signals are relatively rare in the KOI list of Batalha et al. (2013). This may in principle influence both the planet prior and the blend frequencies we have just described, given that the latter also draw on the KOI list to estimate the rate of occurrence of larger planets involved in blends. Therefore, instead of allowing eclipsing binaries and transiting planets with any orbital period for the blend frequency calculations, we repeated the analysis with the more realistic approach of accepting only blends with periods near the measured period of Kepler-68c (within a factor of two). We did the same when computing the planet prior, for consistency. The total blend frequency is reduced in this way by about a factor of five, but the new planet prior is only 1.5 times smaller, resulting in a larger net odds ratio of ∼4300 that provides for an even stronger validation than before.

The above calculations neglect the fact that KOI-246.02 was found around a target that has a statistically validated transiting planet, KOI-246.01 = Kepler-68b. The planet priors used were averaged between single and multi-planet systems. Actual planet priors are about 30% smaller for single planets and more than an order of magnitude larger for multiple planets (Lissauer et al. 2012; J. J. Lissauer et al. 2013, in preparation). The presence of a known planet also increases the prior for physically associated blends, but by a smaller factor. The numbers quoted above for the likelihood of false positives are four times higher for background blends than for physically associated blends, so when the multiplicity of the system is accounted for, the odds ratios against the blends quoted above are increased by roughly an order of magnitude.

As an interesting test of the value of SC versus long cadence data for false positive discrimination, we repeated the calculations with the long-cadence time series (Quarters 1–2, Quarters 4–11, 34,556 measurements) including the period cut described above. We find an odds ratio of approximately 1500, about three times lower than when using SC, but still large enough to comfortably validate the signal. Therefore, at least for Kepler-68c, SC provides a clear advantage in ruling out blends. This is likely due to the better definition of the transit shape at ingress/egress, which is often where the main differences are between model light curves for blends and the model for a true planet. In this case, we find that the main improvement (decrease) in the blend frequencies when going from long to SC is in the scenarios involving hierarchical triples, with background eclipsing binary frequencies changing the least.

While in principle the above calculations provide a clear statistical validation of the Kepler-68c signal as a true planet, independent of the detection of the reflex motion of the star (RVs), we note that some of the blend scenarios involving background eclipsing binaries yield a fit to the Kepler photometry that is significantly better than that of a planet fit. This is a situation we have not encountered in previous validations of Kepler candidates. The light curves of these false positive scenarios all feature a very shallow (∼10 ppm) secondary eclipse that happens to match a similar dip present in the phase-folded photometry for Kepler-68c at phase 0.5. This shallow dip at phase 0.5 has a formal significance of 3σ–4σ and is the primary source for these favorable false positive fits.

The reality of the signal at phase 0.5 for Kepler-68c is in doubt; were this not so, then claiming validation in this case would be impossible. Figure 11 contrasts phase folded SC data for Kepler-68c at both the transit and offset by half phases. In assessing the reality of this putative secondary eclipse feature, we have examined the data in several ways. When averaged over widths comparable to the transit, the data show a few other excursions (both positive and negative) to deviations as large as the phase 0.5 feature; thus, the feature is obviously not highly significant. However, other aspects of data investigation do not support a straightforward dismissal of evidence consistent with the weak ∼10 ppm feature. Examination of medians over the phase bins shown in Figure 11 showed the same offsets, and thus the deviation does not result simply from a small subset of the data. Likewise, dividing the data into first and second halves before doing independent phase binnings shows evidence for a ∼10 ppm depression at phase 0.5 in both cases. In these two halves, there are again other instances of deviations of comparable width and depth; however, across the two halves, the only cases lining up are those at phase 0.5. While nothing here is convincing in terms of regarding the phase 0.5 offset as real, it is equally the case that the reality of the offset cannot be excluded. We therefore took the conservative approach of allowing BLENDER to be influenced by the apparent negative offset at phase 0.5, which still provides the favorable odds ratio required for formal validation of Kepler-68c as a planet.

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Further complicating this interpretation, these data show a small degree of correlated noise, even after folding over 64 orbits of Kepler-68c. After binning to 0.001 phase bins, the first autocorrelation value is 0.25, falling to zero within about 0.005 in phase. We formed simulated time series consisting only of Gaussian noise, stellar oscillations, and granulation (Gilliland et al. 2011a) appropriate to Kepler-68 and found that after folding this had a similar (although smaller) degree of autocorrelation, suggesting that inherent stellar variations may suffice to explain the modest correlated noise.

The robust detection of a Doppler signal at the period of Kepler-68b provides strong support for the planetary nature of that signal. There is, however, a small chance that a background eclipsing binary could mimic this spectroscopic signature of a planet (as well as the photometric signal), although this possibility was convincingly argued against in Section 4.3. The precision of our bisector span measurements in Section 4.4 is not quite high enough for a definite conclusion regarding this possibility, so we proceed here with a validation analysis along the lines of what was done for Kepler-68c.

The much deeper transit (∼350 ppm) and higher S/N of Kepler-68b result in significantly stronger constraints on the shape of the signal, and consequently in a much reduced frequency of blends that give acceptable fits to the Kepler light curve. Indeed, background eclipsing binaries are completely ruled out by BLENDER, as no such scenarios yield light curves with a transit shape that matches the observations sufficiently well. In addition, While some hierarchical triple scenarios are allowed by BLENDER, the companion stars would all be bright enough that they would have been seen in our high-resolution spectra. The only remaining blend scenarios that are viable are those involving larger planets transiting a background or foreground star. We estimate the frequency of such blends to be 1.4 × 10⁻⁸.

To estimate the a priori likelihood of a true planet, we use the fact that there are 96 candidates in the list of Batalha et al. (2013) with a planetary radius similar to that implied by Kepler-68b (within 3σ), of which an estimated six should be false positives. Signals of this kind are expected to be detectable in 58.7% of all Kepler targets. The planet prior is then (96 − 6)/0.587/138, 253 = 1.1 × 10⁻³. With this, we obtain an odds ratio of nearly 79,000 in favor of the planet interpretation, i.e., a very clear validation of Kepler-68b. The above results used the long-cadence time series for simplicity, and included the period cut described above. Use of SC would likely result in an even higher odds ratio. For completeness, we note that Morton & Johnson (2011) reported a false positive probability of 0.01 for Kepler-68b, whereas our confidence level is orders of magnitude better. Their result was based on a less sophisticated analysis, with much less Kepler data, and did not make use of any of the follow-up observations that we have utilized.

We did a spectral synthesis analysis using SME (Valenti & Piskunov 1996; Valenti & Fischer 2005) of one of our high resolution template spectra from Keck-HIRES of Kepler-68 to derive an effective temperature, T_eff = 5754 ± 44 K, surface gravity, log g = 4.2 ± 0.1 (cgs), metallicity, [Fe/H] = +0.10 ± 0.04, and vsin i = 0.5 ± 0.5 km s⁻¹.

The above effective temperature was used to constrain the fundamental stellar parameters derived via asteroseismic analysis (see Section 6.2). The asteroseismology analysis gave log g = 4.281 ± 0.008, which is 0.1 dex higher than the SME value. The asteroseismology value is likely superior because of the high sensitivity of the acoustic periods to stellar radius. Still, the asteroseismology result depended on adopting the value of T_eff from SME. We recomputed the SME analysis by freezing (adopting) the seismology value for log g. See Torres et al. (2012) for a recent discussion of such an iteration in the analogous context of high S/N transit light-curve analysis providing the log g "truth." This iteration yielded values of T_eff = 5793 ± 44 K, [Fe/H] = +0.12 ± 0.04, and rotational vsin i = 0.5 ± 0.5 km s⁻¹. The revised effective temperature was then put back into the asteroseismology calculation to further constrain the stellar radius and gravity. This iterative process converged quickly, as the log g from seismology yielded an SME value for T_eff that was only slightly different from the original unconstrained determination, and the asteroseismic log g using the iterated T_eff remained at 4.281. Likely systematic errors on T_eff and [Fe/H] of 59 K and 0.062 dex have been derived by Torres et al. (2012) in comparing results across multiple spectroscopic packages for a large number of stars. Adding these additional errors in quadrature with the errors quoted above results in more reasonable total errors of 74 K and 0.074 dex.

We measured the Ca ii H&K emission (Isaacson & Fischer 2010), yielding a Mt. Wilson S value S = 0.139 and $\log R^{\prime }_{\rm HK}$ = −5.15. These values suggest lower activity for Kepler-68 in comparison to mean solar reported as S = 0.178, $\log R^{\prime }_{\rm HK}$ = −4.90 by Lockwood et al. (2007) and S = 0.171, $\log R^{\prime }_{\rm HK}$ = −4.96 by Hall et al. (2009). Our own measure of solar activity using Ganymede as a proxy yielded S = 0.164 and $\log R^{\prime }_{\rm HK}$ = −4.97 (Isaacson & Fischer 2010) for 2009 August 31 when the Sun was still at a low state of its cycle. Thus, Kepler-68 is a magnetically inactive star, consistent with its low rotational rate, vsin i= 0.5 km s⁻¹. Kepler-68 appears to be a middle-aged (age 2–10 Gyr), slowly rotating inactive star on the main sequence. This is consistent with the age derived from the asteroseismology analysis (Section 6.2).

The activity indices, S and $\log R^{\prime }_{\rm HK}$, have modestly lower than solar values, while Figure 1 suggests significantly lower than typical levels of photometric variability compared to the Sun. Hall et al. (2009) summarize 14 years of contemporaneous photometric and spectroscopic (for activity) measurements of 28 solar analog stars with precisions in the photometry capable of detecting changes at roughly half the level seen for the Sun. Although a strong correlation between photometric variability exists with activity, they find that stars with near-solar activity indices show a range of half (not well bounded) to twice solar in photometric variability. Kepler-68 seems consistent with this lower range of photometric change at a given activity level, although not knowing what phase of a Kepler-68 activity cycle was sampled limits fidelity. Multi-year Kepler data will eventually enable robust activity—photometric variability understanding.

The utility of asteroseismology for exoplanet interpretations fundamentally rests on recognizing that both asteroseismic and transit light curve modeling (at high S/N levels and with known or assumed eccentricity) provide constraints on the same stellar parameter—ρ_⋆. Thus, high precision knowledge of ρ_⋆, which commonly results when asteroseismology is feasible, provides for a natural means of tightening the exoplanet transit light curve solution by adopting this as a prior—see Gilliland et al. (2011b) and Nutzman et al. (2011). Kepler-68 at Kp = 10.00, near solar temperature, and a KIC radius of 1.06 R_☉ was recognized early as an excellent candidate for asteroseismology, with a predicted 99% chance of success with only two months of SC data (Chaplin et al. 2011). Kepler-68 was observed at SC (Gilliland et al. 2010) for one month in Quarter 2 as a KASC survey target, then added to the science team SC targets from Quarter 5 onward to support asteroseismology and fine analyses of the short transit ingress and egress.

The power of asteroseismology in setting estimates of the stellar radius, which determines the exoplanet radius via ratio with the transit depth depending on the square of this ratio, can be seen by recounting knowledge of R_* for Kepler-68 at the time of initial analyses. Transit light curve solutions for Kepler-68b with stellar radius as a free parameter returned values of 1.63 R_☉, compared to a KIC radius of 1.06 R_☉ (with spectroscopic solutions favoring something like the KIC value). These two stellar radius values would lead to differences of a factor of 3.6 in the planetary density, emphasizing the need for better knowledge of the stellar radius.

Figure 12 shows the Kepler-68 power spectral density with input of 10 months of SC data, the near-evenly spaced peaks characteristic of solar-like oscillations are obvious. Even when first inspecting power spectra from single months of SC data, the spacing between modes was trivial to estimate at about 100 μHz compared to a solar value of 135 μHz. Since the mean stellar density is known to scale as the square of frequency spacing (Ulrich 1986), this allowed an early constraint that Kepler-68 was at 0.55 of the solar mean density, and assuming a solar mass (given the solar T_eff) provided an estimate of 1.22 R_☉ at the back-of-the-envelope level of analysis.

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The high S/N of the power spectrum shown in Figure 12 made peak-bagging (derivation of frequencies for individual modes) straightforward, and eight team members fit the modes using Maximum Likelihood Estimation, and MCMC approaches (e.g., see Fletcher et al. 2009; Handberg & Campante 2011; Appourchaux et al. 2012). The adopted frequencies listed in Table 3 came from the single set that most consistently was at the median over all eight.

The fitting technique has been reported in various versions for the analysis of HST observations of HD 17156 (Gilliland et al. 2011b) and Kepler observations of HAT-P-7 (Christensen-Dalsgaard et al. 2010) and Kepler-10 (Batalha et al. 2011). The underlying stellar evolution modeling is provided using the ASTEC code (Christensen-Dalsgaard 2008a), with eigenfrequency analyses coming from ADIPLS (Christensen-Dalsgaard 2008b).

In the present case, the stellar parameters grid includes a few values of the mixing-length parameter α_ML in addition to the mass M and the initial composition, characterized by the abundances X and Z by mass of hydrogen and heavy elements. Thus, the evolution sequences are characterized by a set of parameters $\lbrace {\cal P}_k\rbrace = \lbrace M, Z, X, \alpha _{\rm ML}\rbrace$. Details of the Kepler-68 grid are presented below.

The fit of a given model to the data is defined in terms of

$$\begin{equation} \chi ^2_\nu = {1 \over N-1} \sum _{i=1}^N \left({\nu ^{\rm (obs)}_i - \nu ^{\rm (mod)}_i \over \sigma _i } \right)^2, \end{equation} \tag{ 1 }$$

where $\nu ^{\rm (obs)}_i$ and $\nu ^{\rm (mod)}_i$ are the observed and model frequencies and σ_i is the error in the observed frequencies. It is assumed that the degree and order of the observed frequencies have already been determined. In addition, an augmented fit

$$\begin{equation} \chi ^2 = \chi ^2_\nu + \left({T_{\rm eff}^{\rm (obs)}- T_{\rm eff}^{\rm (mod)} \over \sigma (T_{\rm eff}) } \right)^2, \end{equation} \tag{ 2 }$$

including the observed and model effective temperature T_eff, is formed.

For each evolution sequence, frequencies are calculated for selected models along the sequence (typically every fifth), but such that frequencies are also available at all models in the sequence in the vicinity of the model ${\cal M}_{\rm min}^\prime$ with the smallest $\chi _\nu ^2$. Based on homology scaling, we then assume that the frequencies in the vicinity of ${\cal M}_{\rm min}^\prime$ can be obtained as $r \nu _i({\cal M}_{\rm min}^\prime)$, where $r = [R{/}R({\cal M}_{\rm min}^\prime)]^{-3/2}$, R being the surface radius of a model intermediate between the actual timesteps in the evolution sequence. The best-fitting of such model is determined by minimizing

$$\begin{equation} \chi ^2_\nu (r) = {1 \over N-1} \sum _{i=1}^N \left({\nu ^{\rm (obs)}_i - r \nu _i({\cal M}_{\rm min}^\prime) \over \sigma _i } \right)^2 \end{equation} \tag{ 3 }$$

as a function of r. The resulting value r_min of r defines an estimate R_min of the radius of the best-fitting model along the given sequence. In this way, we ensure that the scaled model is intermediate between two successive timesteps in the evolution sequence for which frequencies have been calculated. Linear interpolation to R_min then defines the final best-fitting model ${\cal M}_{\rm min}({\cal P}_k)$ (which obviously in general does not coincide with a timestep in the evolution sequence) for the given set of model parameters $\lbrace {\cal P}_k\rbrace$, and with corresponding $\chi _{\nu,\rm min}^2({\cal P}_k)$ and $\chi _{\rm min}^2({\cal P}_k)$. To obtain the final best-fitting model, we find the parameter set corresponding to the smallest $\chi _{\nu,\rm min}^2({\cal P}_k)$ (or $\chi _{\rm min}^2({\cal P}_k)$) amongst all of the evolution sequences. The best-fitting frequencies, e.g., for comparison with the observations, are obtained by applying the appropriate scaling r_min to the frequencies of the model ${\cal M}_{\rm min}^\prime$ in the minimizing sequence.

An important goal of the fit is obviously to obtain statistically well characterized estimates of the stellar properties, in particular density, radius, mass, and age. In the present analysis, these were determined as averages and standard deviations of the properties of the models, ${\cal M}_{\min }({\cal P}_k)$, over the parameters, $\lbrace {\cal P}_k\rbrace$, with the weights $\chi _{\rm min}^{-2}({\cal P}_k)$.

The calculations used the latest OPAL equation of state tables (see Rogers et al. 1996) and OPAL opacities at temperatures above 10⁴ K (Iglesias & Rogers 1996); at lower temperatures, the Ferguson et al. (2005) opacities were used. Nuclear reactions were calculated using the NACRE parameters (Angulo et al. 1999). Diffusion and settling of helium was treated using the Michaud & Proffitt (1993) approximations; diffusion and settling of heavy elements was not taken into account. Convection was treated using the Böhm-Vitense (1958) mixing-length formulation. Although some of the relevant models have a small convective core, core overshoot was not considered.

The spectroscopically determined [Fe/H] = 0.12 ± 0.04 (see Section 6.1) is related to the model quantities X and Z by

$$\begin{equation} {\rm [Fe/H]} = \log \left({Z/X \over Z_\odot /X_\odot } \right), \end{equation} \tag{ 4 }$$

where "☉" denotes solar values. To obtain the composition from this, one clearly needs to assume a value of Z_☉/X_☉ and a value of X or a relation between Z and X. For the former, the Grevesse & Noels (1993) value of Z_☉/X_☉ = 0.0245 is used. We recognize that substantially lower values have been obtained in more recent solar spectroscopic analyses (see Asplund et al. 2009 and references therein). However, these determinations lead to solar models showing a substantial increase in the discrepancy with the helioseismically determined solar structure (e.g., Bahcall et al. 2005; Christensen-Dalsgaard et al. 2009), compared with models based on the Grevesse & Noels (1993) composition. Given that the reasons for this discrepancy are so far unknown, we prefer to use as reference a solar composition that provides reasonable agreement with the helioseismic inferences. As a model of Galactic chemical evolution, ΔY = 2ΔZ has often been used, and hence (fixing the relation roughly to the Sun) X = 0.7679 − 3Z. The grid in composition allows for a spread in [Fe/H] using models with [Fe/H] = 0.02, 0.1, and 0.18, consistent with the [Fe/H] error of 0.074 after the inclusion of systematics. With the transformation discussed above, this corresponds to (Z, X) = (0.0183, 0.7130), (0.0217, 0.7029), and (0.0256, 0.6910). To avoid being restricted to a specific relation between X and Z, we have computed models for all nine resulting combinations of X and Z.

The values of the mixing length have been chosen as α_ML = 1.5, 1.8, and 2.1. Models were computed initially between 0.9 and 1.2 M_☉, with a step of 0.02 M_☉. The grid in mass was later refined, with a step of 0.01 M_☉, in the vicinity of the best-fitting models. A total of 282 evolution sequences, with typically 200–300 models in each, are considered in the fit.

The present use of adiabatic frequency calculations, and an inadequate modeling of the near-surface layers, cause errors in the resulting frequencies that must be taken into account in the fit. Here, we use a correction to the frequencies for these near-surface errors, of the form

$$\begin{equation} \delta \nu = a (\nu /\nu _0)^b \end{equation} \tag{ 5 }$$

(Kjeldsen et al. 2008), where b = 4.90 is obtained from a corresponding solar fit, ν₀ = 2071.33 μHz was fixed in the middle of the observed frequency range, and a = −1.527  μHz was obtained from a fit of a suitable reference model to the observed frequencies.

With these procedures, the best-fitting model had χ² = 5.0 (see Equation (2)). The quality of the fit is illustrated in the echelle diagram (Grec et al. 1983) shown in Figure 13. The stellar parameters and standard deviations obtained from the weighted averages over the evolution sequences are shown in Table 4.

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Table 4. Star and Planet Parameters for the Kepler-68 System

Parameter	Value	Notes
Transit and orbital parameters: Kepler-68b
Orbital period P (days)	5.398763 ± 0.000004	A
Midtransit time E (BJD)	2455006.85729 ± 0.00042	A
Scaled semimajor axis a/R⋆	10.68 ± 0.14	A
Scaled planet radius RP/R⋆	0.01700 ± 0.00046	A
Impact parameter b	0.45 ± 0.17	A
Orbital inclination i (deg)	87.60 ± 0.90	A
Orbital semi-amplitude K (m s−1)	2.63 ± 0.71	B
Orbital eccentricity e	0	B
Center-of-mass velocity γ (km s−1)	−20.9 ± 0.1	B
Transit and orbital parameters: Kepler-68c
Orbital period P (days)	9.605085 ± 0.000072	A
Midtransit time E (HJD)	2454969.3805 ± 0.0041	A
Scaled semimajor axis a/R⋆	15.68 ± 0.20	A
Scaled planet radius RP/R⋆	0.00703 ± 0.00025	A
Impact parameter b	0.84 ± 0.11	A
Orbital inclination i (deg)	86.93 ± 0.41	A
Orbital semi-amplitude K (m s−1)	$1.25^{+0.65}_{-0.95}$	B
Observed stellar parameters
Effective temperature Teff (K)	5793 ± 74	C
Spectroscopic gravity log g (cgs)	4.281 ± 0.06	C
Metallicity [Fe/H]	+0.12 ± 0.074	C
Projected rotation vsin i (km s−1)	0.5 ± 0.5	C
Fundamental Stellar Properties
Density ρ⋆ (g cm−3)	0.7903 ± 0.0054	D
Mass M⋆(M☉)	1.079 ± 0.051	D
Radius R⋆(R☉)	1.243 ± 0.019	D
Surface gravity log g⋆ (cgs)	4.281 ± 0.008	D
Luminosity L⋆ (L☉)	1.564 ± 0.141	D
Absolute V magnitude MV (mag)	4.34 ± 0.09	D
Age (Gyr)	6.3 ± 1.7	D
Distance (pc)	135 ± 10	D
Planetary parameters: Kepler-68b
Mass MP (M⊕)	$8.3^{+2.2}_{-2.4}$	A,B,C,D,F
Radius RP (R⊕)	$2.31^{+0.06}_{-0.09}$	A,B,C,D
Density ρP (g cm−3)	$3.32^{+0.86}_{-0.98}$	A,B,C,D,F
Surface gravity log gP (cgs)	3.14 ± 0.11	A,B,C,D,F
Orbital semimajor axis a (AU)	0.06170 ± 0.00056	E
Equilibrium temperature Teq (K)	1280 ± 90	G
Planetary parameters: Kepler-68c
Mass MP (M⊕)	$4.8^{+2.5}_{-3.6}$	A,B,C,D,F
Radius RP (R⊕)	$0.953^{+0.037}_{-0.042}$	A,B,C,D
Density ρP (g cm−3)	$28.^{+13.}_{-23.}$	A,B,C,D,F
Orbital semimajor axis a (AU)	0.09059 ± 0.00082	E
Planetary parameters: Kepler-68d
Orbital period P (days)	580 ± 15	F
Minimum mass MP sini (MJ)	0.947 ± 0.035	D,F
Orbital semi-amplitude K (m s−1)	19.9 ± 0.75	D,F
Orbital semimajor axis a (AU)	1.40 ± 0.03	D,F
Orbital eccentricity e	0.18 ± 0.05	F

Notes. A: Based primarily on an analysis of the photometry, B: Based on a joint analysis of the photometry and radial velocities, C: Based on an analysis by D. Fischer of the Keck/HIRES template spectrum using SME (Valenti & Piskunov 1996), D: Based on asteroseismology analysis, E: Based on Newton's revised version of Kepler's Third Law and the results from D, F: Based on radial velocities, G: Calculated assuming a random distribution of Bond albedo over 0.0–0.5, and a random set ranging from zero to full redistribution of heat from day to night sides.

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As a consistency check, the global oscillation parameter values Δν = 101.51 ± 0.09 μHz and ν_max = 2154 ± 13 μHz were derived from the best-fitting (peak bagging) frequencies and amplitudes of the most prominent peaks, providing values quite consistent with estimations from standard, automated detection codes (Hekker et al. 2010; Christensen-Dalsgaard et al. 2010; Verner et al. 2011). The resulting density, mass, and radius provided by grid-based solutions (Basu et al. 2010; Karoff et al. 2010) were consistent with our more detailed fit presented here. Likewise, fits using the Asteroseismic Modeling Portal (see Metcalfe et al. 2009 for details) gives values very close to those presented here.

Our primary asteroseismic solutions used the formal spectroscopic errors of 44 K on T_eff and allowed a spread of ±0.08 dex on [Fe/H]. We have used grid-based solutions starting with these errors, and the more appropriate errors of 74 K on T_eff and 0.074 dex on [Fe/H] to show that changes in the directly constrained stellar density are negligible, and that inferred values of stellar mass and radius change by 0.2σ and 0.3σ, respectively, in comparison to the errors already quoted in Table 4. The associated errors on stellar properties also changed little in comparison to values already in use.

The asteroseismic solutions given here are based on only about half of the SC data now available. However, the quality of the frequency extractions for Kepler-68 with the data through Quarter 7 only is already sufficiently good that residual errors from the asteroseismic solution are negligible for the inference of exoplanet parameters. Indeed, the asteroseismology for Kepler-68 will be among the very best possible with Kepler, and further results concentrating on fine details of this star will appear in the future. As an example, some of the best fitting stellar evolution models indicated a small convective core, hence it would be good to explore the inclusion of convective core overshoot. Initial exploration of core overshoot indicated that this had negligible impact on the mean stellar density required for exoplanet inferences and was not further pursued. The quality of Kepler-68 data for asteroseismology will likely support inferences on the outer convection zone depth (Mazumdar et al. 2012) and obliquity of the rotation axis (Chaplin et al. 2013) in future analyses.

Kepler-68 is a near twin of α Cen A, with inferred values of T_eff, R_⋆, M_⋆, L_⋆, and age all within 1σ of each other, despite the small error bars for both—see Bazot et al. (2012) and references therein. Of the fundamental parameters, only the [Fe/H] differs significantly, with α Cen A being more metal rich at 0.24 ± 0.02 (Neuforge-Verheecke & Magain 1997) compared to Kepler-68 at 0.12 ± 0.074. Both stars have now been interpreted with the benefit of asteroseismology. The quality of asteroseismic constraints are superior for Kepler-68, with general astronomical knowledge being better for the nearby binary α Cen A. Differential analyses of these two interesting stars may prove fruitful.

The physical and orbital properties of both transit signatures are derived by simultaneously fitting Kepler photometry and Keck RVs and by adopting the mean stellar density, R_⋆, and M_⋆ of the host star as determined by asteroseismology.

The Kepler photometry and Keck RVs are fit with non-interacting Keplerian orbits.  The model parameters are the mean stellar density (ρ_⋆) and a flux and RV zero point, and the time of transit (T0), orbital period (P), impact parameter (b), scaled planetary radius (R_P/R_⋆), RV amplitude (K) and eccentricity and argument of pericenter parameterized as ecos (w) and esin (w) for each planet. The transit was modeled using the analytic formalization of Mandel & Agol (2002) to fit photometric observations of the transit. We use the quadratic parameterization of limb darkening also described by Mandel & Agol (2002) with coefficients (0.4096, 0.2602) calculated by Claret & Bloemen (2011) for the Kepler bandpass. Model fits to the Kepler-68b lightcurve yield an eccentricity that is consistent with zero (ecos (w) = 0.02 ± 0.10; esin (w) = 20.13 ± 0.20), which is  consistent with our expectations for tidal circularization (Mazeh 2008). Given the large orbital separation of the outer planet candidate, we can not assume its orbit to be circular based on tidal circularization. The duration of the transit for the outer planet candidate is consistent with a circular orbit, but the resulting upper limit is still significant (e ≃ 0.2). For the remainder of our discussion, the models are constrained to zero eccentricity for both Kepler-68b and Kepler-68c. The RV variations are modeled by assuming non-interacting (Keplerian) orbits. With Kepler-68c validated, the relative inclination between the two orbits is likely less than 20 degrees, as larger relative inclinations would require a fortuitous alignment of the orbital nodes for both planets to transit (Ragozzine & Holman 2010).

We initially fit our observations by fixing ρ_⋆ to its asteroseismic values (see Section 6.2). Model parameters are found by chi-squared minimization using a Levenberg-Marquardt prescription. We then use the best-fit values to seed an MCMC parameter search (Ford 2005) to fit all of the model parameters with ρ_⋆ from the asteroseismic solution. We adopt the asteroseismic determined mean-stellar density as a prior of the overall solution.

For a nominal two-planet model (i.e., circular orbits with masses from Table 4), the predicted root mean square (rms) TTVs for Kepler-68b & Kepler-68c are both less than half a minute. The median timing uncertainties (based on long-cadence observations) are 3.3 and 20 minutes. Thus, it is not surprising that Kepler has not provided a TTV signal due to the interaction of Kepler-68b and Kepler-68c from initial searches. Even with a possible factor of two precision gain from SC data and an available long time-series providing the square root of the number of transits gain, predictions are that detection of a TTV signal on Kepler-68b remains marginal, and unlikely for Kepler-68c. Even increasing the masses of both planets by three times the upper "1σ" uncertainty above the estimates in Table 4, the predicted rms TTVs are less than a minute for both planets. Similarly, even models with unrealistically large eccentricities (e = 0.3 and the nominal masses) can result in rms TTVs of less than a minute. Therefore, we have not performed a detailed TTV analysis of this system.

The synthesis of RV monitoring, transit photometry, and precise asteroseismic stellar characterization reveals that Kepler-68b's mass, radius, and density are all intermediate between the properties of Earth and the solar system ice giants. Kepler-68b's bulk density ($3.32^{+0.86}_{-0.98} \rm {g\,cm^{-3}}$) is low enough to imply that volatiles (in the form of H/He or astrophysical ices) make a significant contribution to the total planet mass and volume. Kepler-68b cannot be composed of iron and silicates alone; an iron-poor silicate composition is too dense by more than 3σ. Even a carbon-rich mineralogy—which may lead to solid planets with larger radii than the Earth-like mineralogy often assumed (Madhusudhan et al. 2012)—does not account for the planet density within 1σ. Following Rogers & Seager (2010) and Rogers et al. (2011), we constrain the range of bulk compositions that are consistent with Kepler-68b's measured mass and radius. Assuming an Earth-like rocky interior composition (consisting of 32% Fe and 68% silicate by mass), Kepler-68b would need between 0.07% and 0.6% of its mass in a H/He gas layer, or, alternatively, between 21% and 76% of its mass in a water vapor envelope. Intermediate compositions with a mixture of H/He and higher mean molecular weight material from ices are also possible.

Compared to Kepler-68b, the compositions of the other planets orbiting Kepler-68 are less well constrained because the bulk planet densities are unknown. Given its Jupiter-like minimum mass, the outermost planet (Kepler-68d) is likely to be dominated by H/He. Without a transit measurement of Kepler-68d's radius, however, neither the dominant composition nor the more subtle proportion of heavy elements in the planet interior can be directly inferred. At the other extreme of the planetary mass scale, Earth-sized Kepler-68c has only a marginal RV detection. Planet interior structure models place more stringent constraints on the planet mass than the 2σ RV upper limit of 10.6 M_⊕. If Kepler-68c is a rocky body composed of iron and silicate, its mass would fall within 0.65 M_⊕ < M_p < 2.5 M_⊕—even a pure iron configuration is allowed. A residual RV precision better than 70 cm s⁻¹ is needed to constrain Kepler-68c's make-up.

A striking feature of the Kepler-68 planetary system is that it harbors one of the most strongly irradiated volatile-rich mini-Neptunes detected to date. The stellar energy flux received by Kepler-68b is more than 412 ± 34 times larger than that received by the Earth. Among low-mass (M_P < 20 M_⊕) planets with measured radii, only Kepler-10b, 55 Cnc e, CoRoT-7b, and Kepler-18b are more strongly irradiated by their host stars. These planets have higher densities than Kepler-68b, however, and are consistent with volatile-less compositions within the 1σ uncertainties on their measured masses and radii. Kepler-10b, CoRoT-7b, and Kepler-18b may be comprised solely of iron and silicates (with no H/He or astrophysical ices), and 55 Cnc e (although not dense enough to have a silicate composition) could have a carbon-rich solid composition without a volatile envelope (Madhusudhan et al. 2012). Kepler-68b's status as the most strongly irradiated mini-Neptune that unambiguously (when the 1σ uncertainties are taken into account) has a significant amount of volatiles makes it a valuable benchmark for planet mass-loss models.

Like many of the highly irradiated Kepler planets, mass loss has likely had an important influence sculpting Kepler-68b's composition over its 6.3 Gyr lifetime. Indeed, Kepler-68b lies near the edge of the empirical mass loss destruction threshold noted by several authors (Lecavelier 2007; Ehrenreich & Désert 2011; Jackson et al. 2012; Lopez et al. 2012). At 8.3 M_⊕, Kepler-68b is close to the minimum mass of 6.5 M_⊕ predicted by mass loss in Lopez et al. (2012). In order to examine the vulnerability of Kepler-68b in greater detail, we employed the coupled thermal and mass loss evolution models of Lopez et al. (2012), assuming an Earth-like core and 10% mass loss efficiency.

Although, Kepler-68b is stable against mass loss today, it is possible that it underwent substantial mass loss early in its history when radii were larger and stellar XUV fluxes were over 100× higher (Ribas et al. 2005). In fact, if Kepler-68b had a H/He envelope then, it was vulnerable to a type of runaway mass loss that occurs when the mass loss timescale becomes short compared to the cooling timescale (Baraffe et al. 2004; Lopez et al. 2012). Although less than 1% H/He today, Kepler-68b would need to have been ∼80% H/He when it was 10 Myr old in order to have a residual primordial H/He envelope today. Moreover, models that undergo this type of extreme mass loss almost always lead to a planet completely stripped of its H/He envelope (Lopez et al. 2012). The initial conditions must be carefully fine tuned in order to arrive at a planet that has such a small but non-zero H/He envelope today. This suggests that it is unlikely that Kepler-68b retains a primordial H/He envelope.

On the other hand, a steam envelope on Kepler-68b should be very stable against mass loss. If Kepler-68b is ∼50% water today, then it has only lost ∼1% of its initial water envelope since it was 10 Myr old. This suggests that it is possible that like Kepler-68d, 68b formed beyond the snow-line and migrated to its current orbit. However, another distinct possibility is that Kepler-68b does in fact have an H/He envelope, just not a primordial one. Elkins-Tanton & Seager (2008) showed that rocky planets could outgas up to 0.9% of their mass in H/He, more than sufficient to explain the envelope needed today.

On the whole, the Kepler-68 planetary system shares characteristics both with the compact Kepler multi-planet systems (e.g., Kepler-11 and Kepler-20) and with the solar system. Like Kepler-11 and Kepler-20, Kepler-68 has multiple transiting planets within 0.1 AU. In common with the solar system, Kepler-68 has a Jovian-mass planet residing outside (at greater orbital separations than) the smaller bodies in the inner system. Between 0.1 and 1.4 AU, there are no confirmed planets in the Kepler-68 system, although our census may be incomplete. The presence of a volatile-rich super Earth within 0.06 AU, combined with the presence of a luke-warm Jupiter inside the snow line, makes the Kepler-68 system an interesting case study for planet formation and migration theories.