KEPLER'S FIRST ROCKY PLANET: KEPLER-10b^*

The Kepler photometer is a 0.95 m effective aperture, wide field-of-view Schmidt camera in an Earth-trailing orbit. It is designed to yield 20 parts per million (ppm) relative time series precision in 6.5 hr for a 12th magnitude G2 star. The focal plane is comprised of 42 1024 × 2200 pixel science CCDs arranged together in 21 roughly square modules covering 115 deg² of sky (398 pixel⁻¹). Each pair of CCDs forming a module shares a common sapphire field-flattener lens. The coatings deposited onto the field-flattener lenses (and, to a lesser degree, the optics and quantum efficiency of the CCDs) define the effective bandpass of the otherwise filterless photometer, yielding a mean transmission of 52.6% between 423 and 897 nm (defining the 5% transmission points). Each CCD requires its own transmission function for meaningful interpretation of color-dependent behaviors that affect planet characterization (e.g., limb darkening). A description of the instrument is given in Van Cleve & Caldwell (2009) and Argabright et al. (2008), while an overview of its in-flight performance is presented in Caldwell et al. (2010) and Jenkins et al. (2010b).

Each CCD is electronically divided into two 1024 × 1100 output units defining a total of 84 readout channels. While the photometer points at a single field continuously throughout its heliocentric orbit, it rotates about the optical axis once every ∼3 months (hereafter referred to as a quarter) in order to keep the solar panels facing the Sun. Consequently, every star spends each quarter of the year on a different channel. Kepler-10 (R.A. = 19^h02^m4305, decl. = +50°14'2868) falls on channels 36 (module 11, output 4), 80 (module 23, output 4), 52 (module 15, output 4), and 8 (module 3, output 4) in quarters 0/1 (spring), 2 (summer), 3 (fall), and 4 (winter), respectively.

CCDs are read out every ∼6.5 s (6.01982 s integration and 0.51895 s read time), and every 270 readouts are co-added onboard to form 1765.5 s (∼29.4 minute) integrations (long cadence, LC). Data for up to 170, 000 stars are recorded at LC, while data for up to 512 stars are also co-added to a 58.85 s (∼1 minute) integration (9 readouts) termed short cadence (SC) as described in Jenkins et al. (2010b) and Gilliland et al. (2010). The LC photometry of Kepler-10 used in the analyses reported here was acquired between 2009 May 2 and 2010 January 9—quarters 0/1 (spring) up through the first month of quarter 4 (winter). In early January, the module containing channel 8 experienced a hardware failure that was not recoverable and observations of Kepler-10 were cut short until the subsequent spacecraft roll. Over 11,000 LC observations are used in this analysis. SC data were also collected between 2009 July 21 and August 19 (one month of quarter 2) and between 2009 September 18 and 2010 January 9 (quarter 3 and the first weeks of quarter 4). The SC data were vital in determining the fundamental stellar parameters from an asteroseismic analysis (p-mode detection) described in Section 6.2. Approximately 200,000 SC observations were collected in this time period. Both LC and SC data are used in the light curve modeling carried out to characterize the planets. Observations of Kepler-10 continue in the three quarters each year when the target is not on the failed module.

Raw flux light curves are extracted by performing an unweighted sum of calibrated pixels that have been subjected to cosmic ray removal and background subtraction (Jenkins et al. 2010a). The pixels used in the sum are those defining the optimal aperture—the set of pixels that optimizes the total signal-to-noise ratio (S/N). The optimal aperture is dependent on the local pixel response function²⁵ (PRF), measured on-orbit during the commissioning period (Bryson et al. 2010), the distribution of stellar flux on the sky near the target (crowding), and differential velocity aberration. A complete discussion of Kepler's aperture photometry pipeline (PA) is given in Twicken et al. (2010a). Systematic errors, outliers, and intra-quarter discontinuities are removed by co-trending against ancillary data products in the pre-search data conditioning (PDC) pipeline module as described in Twicken et al. (2010b). Figure 1 shows the raw (upper panel) and corrected (lower panel) flux time series for Kepler-10. Vertical lines denote the boundaries between quarters. Intra-quarter fluxes were normalized by their median flux in order to reduce the magnitude of the flux discontinuities between quarters. The largest systematic errors are the long-term drifts due to image motion (differential velocity aberration) and the thermal transients after safe mode events (e.g., that near day 115). After filtering out transit events, the measured relative standard deviation of the PDC-corrected, LC light curve is 62 ppm per (LC) cadence. An expected instrument + photon noise is computed for each flux in the time series. The mean of the per (29.4 minute) cadence noise estimates reported by the pipeline is 36 ppm. Both raw (PA) and corrected (PDC) light curves are available at the Multi-Mission Archive at Space Telescope Science Institute, MAST.²⁶

Zoom In Zoom Out Reset image size

With regards to the time stamps associated with each photometric flux, we note that Kepler's fundamental coordinate system is UTC for all time tags. Spacecraft times are converted to barycentric-corrected Julian dates at the mid-time of each cadence.

The Kepler-10b transits were identified by the transiting planet search (TPS) pipeline module that searches through each systematic error-corrected flux time series for periodic sequences of negative pulses corresponding to transit signatures. The approach is a wavelet-based, adaptive matched filter that characterizes the power spectral density of the background process (i.e., anything that is not related to the transit signal itself, such as stellar variability, instrumental signatures, differential velocity aberration-induced photometric drift, etc.) yielding the observed light curve and uses this time-variable power spectral density estimate to realize a pre-whitening filter to apply to the light curve (Jenkins et al. 2010d). TPS then convolves a transit waveform, whitened by the same pre-whitening filter as the data, with the whitened data to obtain a time series of single-event statistics. These represent the likelihood that a transit of that duration is present at each time step. The single-event statistics are then combined into multiple-event statistics by folding them at trial orbital periods ranging from 0.5 days to as long as one quarter (∼90 days). The step sizes in period and epoch are chosen to control the minimum correlation coefficient between neighboring transit models used in the search so as to maintain a high sensitivity to transit sequences in the data.

KOI-72.01 was identified by TPS in each quarter of data with a multiple event statistic >15σ. The long-period transits of KOI-72.02 were identified by manual inspection due to the fact that the current version of TPS does not operate on more than one quarter of data at a time making detection of events with periods beyond ∼30 days necessarily incomplete. Multi-quarter functionality is slated for the next software release in early 2011. The transit depth, duration, period, and epoch are derived from physical modeling (see Section 7) using all of the available data. KOI-72.01 is characterized as a 152 ± 4 ppm dimming lasting 1.811 ± 0.024 hr with transit ephemeris of T [BJD] =2454964.57375^+0.00060_−0.00082 +  N*0.837495^+0.000004_−0.000005 days. KOI-72.02 is characterized as a 376 ± 9 ppm dimming lasting 6.86 ± 0.07 hr and an ephemeris T [BJD] =2454971.6761^+0.0020_−0.0023 +  N*45.29485^+0.00065_−0.00076 days.

For each event, the even-numbered transits and odd-numbered transits are modeled independently using the techniques described in Section 7. The depth of the phase-folded even-numbered transits is compared to that of the odd-numbered transits as described in Batalha et al. (2010b). A statistically significant difference in the transit depths is an indication of a diluted or grazing eclipsing binary system. Neither of the transit events detected in the light curve of Kepler-10 shows odd–even depth differences outside of 2σ, where σ refers to the uncertainty in the transit depths reported in Section 2.3 (6 and 9 ppm for KOI-72.01 and KOI-72.02,  respectively).

The modeling allows for the presence of a secondary eclipse (or occultation event) near phase = 0.5 and reports the significance of such a signal. While its presence does not rule out the planetary interpretation, it acts as a flag for further investigation. More specifically, the flux decrease is translated into a surface temperature assuming a thermally radiating disk, and this temperature is compared to the equilibrium temperature of a low albedo (0.1) planet at the modeled distance from the parent star. There is a marginal detection (∼2σ) of a secondary eclipse associated with KOI-72.01. At just 6 ppm (see Table 7), the flux change is not severe enough to rule out the planetary interpretation (i.e., that it is due to an occultation of a planetary companion). There is no detection of a secondary eclipse associated with KOI-72.02. The binarity tests are consistent with the planet interpretation for both transit events in the light curve of Kepler-10.

To check for false positives due to background eclipsing binaries, we study the behavior of flux centroids—the center-of-light distribution in the photometric aperture—and how it behaves as a function of time, especially comparing images taken during transits with those taken outside of transit. We rely primarily on flux-weighted centroids and modeling the expected behavior based on the local distribution of point sources from the Kepler Input Catalog (KIC) and supplemented by high spatial resolution imaging (Section 4.2). Neither speckle nor AO imaging reveal any point sources in the photometric aperture that were not already listed in the KIC.

The study of centroid behavior is complicated by the fact that the Kepler-10 image is saturated in all quarters. In quarter 2, the star is on the edge of saturation, but it is apparent that there is some spilling of charge even in this case. Saturation behavior on the Kepler focal plane is known to be conservative (e.g., all charge is captured) but has strong pixel-to-pixel variation in details such as saturation level and the fraction of flux that spills up and down the CCD columns.

The high-accuracy technique of fitting the PRF to the difference image formed by subtracting the in-transit image from the out-of-transit image is inappropriate for saturated targets, because the saturated pixels do not represent the PRF of the star, and such mismatches between the pixel data and the PRF cause significant position biases in the PRF fit. As an alternative, we use a modified PRF fit technique, described below.

We compute flux-weighted centroids by creating out-of-transit and in-transit images from detrended, folded pixel time series. Separate images are created for KOI-72.01 and KOI-72.02. For each pixel time series, the de-trending operation has three steps: (1) removal of a median filtered time series with a window size equal to the larger of 48 cadences or three times the transit duration, (2) removal of a robust low-order polynomial fit, and (3) the application of a Savitzky-Golay filtered time series of order three with a width of 10 cadences (5 hr). The Savitzky-Golay filter is not applied within two cadences of a transit event, so the transits are preserved. The resulting pixel time series are folded by the transit period. Each pixel in the out-of-transit image is the average of 30 points taken from the folded time series outside the transit, 15 points on each side of the transit event. Each pixel in the in-transit image is the average of as many points in the transit as possible: three for KOI-72.01 and eleven for KOI-72.02. In the former case, though there are fewer points per transit, there are significantly more transits to draw from (several hundred for KOI-72.01 compared to just six for KOI-72.02). Consequently, KOI-72.01 yields smaller uncertainties.

A flux-weighted centroid is computed for the out-of-transit image and the in-transit-image using all of the pixels in the mask²⁷ of Kepler-10. This produces row and column centroid offsets ΔR and ΔC, and the centroid offset distance $D = \sqrt{\Delta R^2 + \Delta C^2}$.

Uncertainties of these centroids are estimated via Monte Carlo simulation, where a noise realization is injected into 48 cadence smoothed versions of the pixel time series for each trial. A total of 2000 trials are performed each for KOI-72.01 and KOI-72.02. The in- and out-of-transit images are formed using the same de-trending, folding, and averaging as the flight data. The measured uncertainties are in the range of a few times 10⁻⁵ pixels.

Table 1 shows the resulting measurements of the KOI-72.01 centroids from quarter 1, 2, 3, and 4 pixel data, along with the Monte Carlo based 1σ uncertainties. The centroids are converted into centroid offsets and offset distance with propagated uncertainties. We see that while in quarter 1 there is a >4σ observed offset, in quarters 2 and 3 the observed offset is less than 1σ. Quarter 1 had significant thermal and pointing systematics, while quarter 2 had significant pointing drift. These systematics were significantly reduced by quarter 3.

Table 2 shows the measured centroid shifts and uncertainties for KOI-72.02 in quarters 1–4. The KOI-72.02 transit uncertainties in quarter 3 are smaller because only one transit was observed in quarters 1, 2, and 4, while two transits were observed in quarter 3. Generally, the centroid shifts for KOI-72.02 are larger than KOI-72.01. This larger centroid shift may be indicative of the KOI-72.02 transit event being due to a background eclipsing binary, or the larger shift may be due to the deeper transits of KOI-72.02 (larger flux changes induce larger centroid shifts) combined with the complex behavior of pixels at or near saturation. To study this question PRF-fit centroids that ignore saturated pixels are computed for the KOI-72.02 in- and out-of-transit images, and the resulting centroid shifts are computed. These results are shown in Table 3. This technique indicates no statistically detectable centroid motion in quarter 3. The uncertainties were computed via Monte Carlo methods similar to those used to compute the flux-weighted centroid uncertainties.

While both the flux-weighted centroid and PRF fitting methods indicate statistically significant centroid motion on KOI-72.02 in quarters 1, 2, and 4, we point out that the directions of the centroid offsets are inconsistent from quarter to quarter. This indicates that these centroid offsets are not likely to be due to a background object in the sky.

The observed centroids are compared with the modeled centroids computed using point sources cataloged in the KIC (Latham et al. 2005) out to 15 pixels beyond the mask. As described in Sections 4.2.1 and 4.2.2, no additional point sources were identified in speckle or AO imaging. To generate the modeled out-of-transit image, the measured PRF is placed at each star's location on the focal plane, scaled by that star's flux. This provides the contribution of each star to the flux in the mask. For each star s_i in the mask, the depth $d_{s_i}$ of a transit is computed that reproduces the observed depth in the individual pixels. An in-transit image for each s_i is similarly but with s_i's flux suppressed by $1- d_{s_i}$. These model images are subject to errors in the PRF (Bryson et al. 2010), so they will not exactly match the sky. Further, a very simple saturation model is applied which spills saturation symmetrically up and down the column.

The modeled centroids (using the quarter 3 mask definition) are presented in Table 4. For both KOI-72.01 and KOI-72.02  the offset expected for a transit associated with Kepler-10 (and not a nearby star) is smaller than the 1σ uncertainties in D computed from the flux-weighted centroid measurements. Modeled transits on other stars in the mask predict centroid shift in excess of 10σ, which would be readily observed in our flux-weighted centroids. We can therefore rule out all known stars in the mask as responsible for the transit signal on both KOI-72.01 and KOI-72.02.

There is a radius beyond which any star capable of producing the observed transit signal would also induce a centroid shift large enough to be detected in the Kepler data when comparing in- and out-of-transit images. We estimate this confusion radius by scaling the 3σ centroid offset uncertainty by the observed transit depth as described in Section 4.1.3 of Wu et al. (2010). For this, we use the quarter 3 uncertainty in D. This radius of confusion for KOI-72.01 is 1.17 arcsec, and the radius of confusion for KOI-72.02 is 0.60 arcsec. The volume subtended by this area on the sky can be used in a BLENDER analysis (Section 5) to assess the probability of encountering an eclipsing binary (capable of producing the transit signal) in that volume of the Galaxy. However, in the case of Kepler-10  the high spatial resolution imaging described in Sections 4.2.1 and 4.2.2 provides tighter constraints on the background star population.

Spectroscopic observations on medium-class telescopes are acquired before requesting precision Doppler measurements. These "reconnaissance" spectra are used to improve upon the photometrically derived stellar classification from the KIC (T_eff = 5491 K, log g = 4.47, R_⋆ = 0.983 R_☉), identify double-lined spectroscopic binaries, and search for indications of RV variations larger than ∼1 km s⁻¹ that might suggest a stellar companion. The objective is continued false-positive elimination.

Two reconnaissance spectra were obtained with the Hamilton echelle on the Shane 3 m telescope at the Lick Observatory. High S/N observations were acquired on two successive nights in 2009 August, at heliocentric Julian dates 2455046.771 and 2455047.758, corresponding to phases 0.146 and 0.3251 for KOI-72.01, and to phases 0.658 and 0.680 for KOI-72.02. The spectral order covering about 7.0 nm centered on the Mg b lines was correlated against a library of synthetic spectra calculated by John Laird using a line list prepared by Jon Morse. The stellar parameters for the template spectrum that yielded the highest value for the peak correlation coefficient were T_eff = 5750 ±  125 K, log g = 4.5 ±  0.25, and vsin i = 0.0⁺²₋₀ km s⁻¹ for an assumed solar metallicity. The errors quoted for T_eff and log g are half the spacing of the library grid. The two Lick exposures were unusually high S/N compared to typical reconnaissance spectra, with a peak value for the correlation coefficient of 0.98. Consequently, we were able to estimate the metallicity and to interpolate to finer values of the temperature and gravity, obtaining T_eff = 5680 ± 91 K, log g = 4.33 ± 0.16, and [Fe/H] = −0.09 ± 0.04. The correlation functions for the two Lick observations showed no evidence of a composite spectrum, and the two velocities agreed within 0.1 km s^-1. Thus, there was no suggestion of a stellar companion responsible for either system of transit events, and the reconnaissance spectroscopy confirms that Kepler-10 is a Sun-like, slowly rotating main-sequence star, and supports the planetary interpretation for the transit events. Consequently, the star was scheduled for precision Doppler measurements (see Section 4.3).

The more complete our knowledge of stellar flux sources in the photometric aperture, the better we are able to assess the likelihood of a blend scenario in the interpretation of the transit event. Much of this knowledge comes from the KIC which federates point sources from the USNO-B catalog, the 2MASS catalog, and our own pre-launch Stellar Classification Program. Identification of point sources within a 15 radius requires additional imaging.

4.2.1. Speckle Imaging

Speckle imaging of Kepler-10 was obtained on the night of 2010 June 18 UT using the two-color speckle camera at the WIYN 3.5 m telescope located on Kitt Peak. The speckle camera simultaneously obtained 2000 30 ms EMCCD images in two filters: V (5620/400 Å) and R (6920/400 Å). These data were reduced and processed to produce a final reconstructed speckle image for each filter. Figure 2 shows the reconstructed R-band image. North is up and east is to the left in the image and the "cross" pattern seen in the image is an artifact of the reconstruction process. The details of the two-color EMCCD speckle camera are presented in S. B. Howell et al. (2011, in preparation).

Zoom In Zoom Out Reset image size

For the speckle data, we determine if a companion star exists within the approximately 2.5 × 2.5 arcsec box centered on the target and robustly estimate the background limit we reach in each summed, reconstructed speckle image. The two-color system allows us to believe single fringe detection (finding and modeling identical fringes in both filters) if they exist and rule out companions between 0.05 arcsec and 1.5 arcsec from Kepler-10. The speckle image was obtained with the WIYN telescope native seeing near 0.7 arcsec, and we find no Kepler-10 companion star within the speckle image separation detection limits to a magnitude limit of 6 mag in R and 4.5 mag in V below the brightness of Kepler-10.

4.2.2. AO Imaging

Near-infrared adaptive optics imaging of KOI-72 was obtained on the night of 2009 September 8 UT with the Palomar Hale 200'' telescope and the PHARO near-infrared camera (Hayward et al. 2001) behind the Palomar adaptive optics system (Troy et al. 2000). PHARO, a 1024 × 1024 HgCdTe infrared array, was utilized in 25.1 mas pixel⁻¹ mode yielding a field of view of 25 arcsec. Observations were performed using a J filter (λ₀ = 1.25 μm). The data were collected in a standard five-point quincunx dither pattern (e.g., dice pattern for number five) of 5 arcsec steps interlaced with an off-source (60 arcsec east) sky dither pattern. Data were taken at two separate times within the same night—150 frames using 1.4 s integration times and 150 frames using 2.8 s integration times—for a total on-source integration time of 10 minutes. The individual frames were reduced with a custom set of IDL routines written for the PHARO camera and were combined into a single final image. The adaptive optics system guided on the primary target itself and produced measured Strehl ratios of 0.15 at J with a central core width of FWHM = 0.075 arcsec. The final co-added image at J is shown in Figure 3.

Zoom In Zoom Out Reset image size

No additional sources were detected at J within 6.25 arcsec of the primary target. Source detection completeness was accomplished by randomly inserting fake sources of various magnitudes in steps of 0.5 mag and at varying distances in steps of 1.0 FWHM from the primary target. Identification of sources was performed both automatically with the IDL version of DAOPhot and by eye. Magnitude detection limits were set when a source was not detected by the automated FIND routine or by eye. Within a distance of 1–2 FWHM, the automated finding routine often fails even though the eye can discern two sources. Beyond that distance the two methods agreed well. A summary of the detection efficiency as a function of distance from the primary star is given in Table 5.

Table 5. Palomar AO Source Sensitivity as a Function of Distance from the Primary Target at J

Distance	Distance	ΔJ	J
(FWHM)	(arcsec)	(mag)	(mag)
1	0.075	1.5	11.4
2	0.150	3.5	13.4
3	0.225	5.0	14.9
4	0.300	5.0	14.9
5	0.375	5.5	15.4
6	0.450	5.5	15.4
7	0.525	6.5	16.4
8	0.600	7.0	16.9
9	0.675	7.5	17.4
40	3.000	9.5	19.4

Download table as:  ASCIITypeset image

We obtained 40 high-resolution spectra of Kepler-10 between 2009 August 31 and 2010 August 6 using the HIRES spectrometer on the Keck I 10 m telescope (Vogt et al. 1994). We used the same configuration of HIRES that is normally used for precise Doppler work of nearby FGK stars (Marcy et al. 2008) which yields a Doppler precision of 1.0–1.5 m s^-1 depending on spectral type and rotational vsin i. The HIRES fiber feed was not used for these observations. The standard iodine cell was placed in the telescope beam to superimpose iodine lines directly on the stellar spectrum. As the iodine lines and stellar lines are carried by exactly the same photons hitting the same optics, both sets of lines share precisely the same instrumental profile and wavelength scale. The iodine lines represent exactly the same spectrometer optics as the stellar spectrum, with no difference. We fit the composite spectrum of iodine and stellar lines simultaneously in each 100 pixel segment of spectrum, yielding a Doppler shift that automatically includes the instantaneous wavelength scale and instrumental profile. This dual fitting limits the long-term and short-term systematic Doppler errors at ∼1.0 m s^-1.

Most observations were made with the "C2 decker" entrance aperture which projects to 087 × 140 on the sky, giving a resolving power of about 60,000 at 5500 Å and enabling sky subtraction (typical seeing is 06–12). A few observations were made with the B5 decker 087 × 30 that does not permit sky subtraction. It is possible that a few of those observations suffered from minor moonlight contamination. The average exposure was 30 minutes, with some as short as 15 minutes and others as 45 minutes, depending on seeing and clouds.

The raw CCD images were reduced by subtracting an average bias, subtracting the sky counts at each wavelength just above and below the stellar spectrum, flat-fielding the spectrum with a 48 exposure sum from a quartz lamp, and extracting the spectrum with a width that includes 99.99% of the spectrum. Cosmic rays were removed from the raw image first. The pixels typically contained approximately 20,000 photons giving a Poisson-limited S/N of 140. We performed the 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 1.5–2.0 m s^-1. The resulting velocities are given in Table 6 and shown in Figure 4 as a function of time. The error bars include the internal Doppler errors and an assumed jitter of 2 m s^-1 (see below), added in quadrature. The center-of-mass velocity relative to the solar system barycenter (Gamma Velocity) for Kepler-10 is −98.93 ± 0.02 km s⁻¹ (Table 7). This is an unusually large RV, indicative of old disk or even halo membership. The low metallicity (Section 6.1), [Fe/H] = −0.15 ± 0.04, magnetic activity (Section 6.1), and asteroseismic age, 11.9 ±  4.5 Gyr (Section 6.2) also suggest old disk or halo membership.

Zoom In Zoom Out Reset image size

Nearby stars with T_eff near 5600 K and log g near 4.35 such as KOI-72 have been previously surveyed for precise Doppler work, revealing a noise-like "jitter" of ∼2.5 m s^-1 caused by surface effects including turbulence, spots on the rotating star, acoustic oscillations, and atmospheric flows associated with magnetic flux tubes. While each effect has its own time scale, it is practical to account for jitter by simply adding it in quadrature to the internal errors to yield an estimate of the total uncertainty in the star's velocity. We have included a jitter of 2.0 m s^-1 in the model fit to all of the data, photometric, and velocities.

A periodogram (Figure 5) of the velocities exhibits a tall peak at a period of 0.837 day, in agreement with the photometric period of the KOI-72.01. The coincidence between the transit and RV periods to three significant digits suggests that the RV period is physically related to the transits, as expected if the RV periodicity stems from the reflex motion of the star in response to the gravitational influence of the planet. The periodogram also shows a peak at a period near 1.2 days, which is the alias resulting from the nightly observational cadence. Similarly, there is another peak near a period of 5 days (off the figure) that is the alias caused by the beating of the 0.827 day period with the 1.0 day cadence of observations. There is no indication of power at the period of KOI-72.02.

Zoom In Zoom Out Reset image size

The velocities phased to the photometric period of KOI-72.01 (Figure 6) show a clear, continuous, and nearly sinusoidal variation consistent with a nearly circular orbit of a planetary companion. 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 0.83 day would have orbital semi-amplitudes of hundreds of kilometers per seconds, 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 is not seen here. Thus, the chance that the 0.83 day periodicity exhibited independently in both the photometry and velocities might be caused by an eclipsing binary seems quite remote.

Zoom In Zoom Out Reset image size

Precision Doppler measurements are used to constrain the mass of KOI-72.01 (Kepler-10b) as discussed in Section 7. The absence of a Doppler signal for KOI-72.02 is used to compute an upper limit to the mass of this candidate 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). In the case of Kepler-10b, there is no evidence for a correlation between the bisector spans and the RVs which would otherwise argue against the planetary interpretation (see Figure 7), and similarly for the 45 day signal of KOI-72.02. However, we note that the uncertainties in the bisector span measurements are quite large so that the rms variation of the bisector spans (10.5 m s^-1) exceeds the semi-amplitude of the RV variation (3.3^+0.8_−1.0 m s^-1). Therefore, we do not consider the bisector span measurements to be discriminating in this case.

Zoom In Zoom Out Reset image size

Given the short period of this signal, we may assume that tidal forces have circularized the orbits of any potential eclipsing binary contaminants (Mazeh 2008). We considered first the case of a hierarchical triple system. Simulations with BLENDER clearly indicate that such systems in which the eclipsing binary is composed of two stars provide poor fits to the Kepler light curve. We thus rule out this type of blend scenario. If the eclipsing binary is composed of a planet (i.e., a smaller tertiary) transiting a star, rather than two stars eclipsing each other, then it is possible to reproduce the measured light curve, but only if the secondary has very nearly the same brightness as the target star itself. In that case, the resulting size of the tertiary is $\sqrt{2}$ larger than in a model of a single star transited by a planet. However, such a bright contaminant would have been evident in our spectroscopy as a second set of lines, and this case is therefore also excluded.

We next examined the background eclipsing binary scenario, allowing the relative distance between the binary and the main star to vary over a wide range. We accounted for absorption from dust along the line of sight as described by Torres et al. (2010), adopting a representative coefficient of differential extinction of a_v = 0.5 mag kpc⁻¹. Interestingly, we found that no combination of relative distance and stellar properties for the eclipsing binary (composed in this case of two stars) gives an acceptable fit to the light curve. The reason is that all such blend configurations lead to out-of-eclipse brightness changes (ellipsoidal variations) with an amplitude so large as to be ruled out by the data. Thus, background blends of this kind can be confidently ruled out. This result is significant, because it reduces the overall likelihood of blends for KOI-72.01 considerably, as we describe below. If we allow the tertiary in the eclipsing pair to be a planet instead of a star (i.e., an object of smaller radius), then we do find a range of blend scenarios that lead to acceptable fits to the light curve, which cannot be ruled out a priori. This is illustrated in Figure 8, in which we show contours of equal goodness of fit of the light curve compared to a standard transit model fit. BLENDER indicates that these false positives can be up to about 5 mag fainter than the primary in the Kepler band, and that in all cases the secondary star is close in spectral type (and mass, or color) to the primary, or slightly earlier. Of these blends, we can further rule out those with secondaries having ΔKp < 2, which are bright enough that they would have been detected spectroscopically. This implicitly places a lower bound also on the size of the tertiaries (∼3.8 R_⊕), as tertiaries smaller than this limit only give good matches to the light curve if the contaminating star–planet pair is not too far behind the primary, and is therefore relatively bright.²⁸ Those cases would be ruled out spectroscopically, as mentioned before. The remaining blends shown in the figure above the line with ΔKp = 2 must be addressed statistically.

Zoom In Zoom Out Reset image size

For this we followed closely the methodology applied by Torres et al. (2010) for the case of Kepler-9d. We computed the mean density of stars (i.e., background contaminants) in the appropriate mass range based on Figure 8 in half-magnitude bins, using the Besançon Galactic structure models of Robin et al. (2003), and we calculated the fraction of these stars that would remain undetected after our high-resolution imaging observations described earlier (the constraints from centroid motion analysis are less stringent; see Figure 9). Some of these stars might be orbited by (transiting) planets, constituting potential blends. To estimate how many of these cases one would expect, we adopted the same frequencies of transiting Jupiters and transiting Neptunes as in Torres et al. (2010), based on the results of Borucki et al. (2010b), adjusted in the case of the Neptune-size planets to account for the lower limit of ∼3.8 R_⊕ allowed by BLENDER for the tertiaries. The outcome of these calculations is presented in Table 8. The total frequency of false positives (blend frequency, BF) we expect to find a priori for KOI-72.01 is BF = 1.4 × 10⁻⁸. Translating this into a probability statement for the planet likelihood is difficult, as argued by Torres et al. (2010), because it requires knowledge of the rate of occurrence of super-Earth-size planets, a quantity that is not yet in hand. Nevertheless, following those authors we may express the false-alarm rate for a random candidate star in the Kepler field generally as FAR = N_FP/(N_FP + N_p), in which N_FP is the number of false positives and N_p is the unknown number of planets in the sample. The number of false positives may be taken to be N_FP = BF × 156, 097 = 0.0022, the product of the BF for KOI-72.01 and the total number of Kepler targets (Borucki et al. 2010b).

Zoom In Zoom Out Reset image size

If we were to accept a confidence level of 3σ (99.73%) as sufficient for validation of a transiting planet candidate (corresponding to FAR = 2.7 × 10⁻³), then the minimum number of super-Earth-size planets required in order to be able to claim this level of confidence happens to be N_p = 1, according to the expression above. This value is so small that it gives us high confidence that the KOI-72.01 signal is not a false positive, but instead corresponds to a bona fide super-Earth-size planet. Another way to view this is that the expected number of background stars capable of producing the observed signal is ∼8 × 10⁻⁶ (see Table 8, bottom of Column 5), so the probability that what we are seeing is a background blend is ∼8 × 10⁻⁶ × P_GP/P_SE, where P_GP and P_SE are the a priori probabilities that giant planets and super-Earths occur with short-period orbits. Thus, claiming a FAR corresponding to a 3σ detection only requires that the aforementioned probability ratio does not exceed 300.

It is worth noting that the arguments above allow us to validate KOI-72.01 independently of the detection of the reflex motion of the star (RVs). This is possible in this case because we are able to rule out, using BLENDER, all false positives in which the background eclipsing binary is composed of two stars, which are predicted to induce ellipsoidal variation at a level that is not present in the photometry. Were this not the case, the BF would have come out considerably larger, requiring in turn a significantly higher value for N_p.

A similar BLENDER analysis was performed for KOI-72.02. For circular orbits, hierarchical triple systems are ruled out for the same reasons as in KOI-72.01. When considering the case of background eclipsing binaries (stellar tertiaries) with circular orbits, we find a range of blend scenarios that provide acceptable fits to the light curve, with secondaries of similar spectral type as the primary. Unlike the situation for KOI-72.01, the longer period of KOI-72.02 (45.3 days) leads to negligible ellipsoidal variation for the contaminating binaries, and this does not allow these types of blends to be excluded based on the quality of the fit, as we were able to do before. Chance alignment scenarios in which the tertiaries are planets rather than stars also lead to viable blends that can be up to 4 mag fainter than the target. Those that are brighter than ΔKp = 2 can be ruled out because they would have produced a spectroscopic signature, but fainter ones cannot be ruled out any other way if they are angularly close enough to the target to be unresolved by our imaging observations.

However, the longer period of this signal does not justify the assumption of a circular orbit. Allowing the orbit of a contaminating binary to have arbitrary eccentricity and also arbitrary orientation (longitude of periastron) can significantly increase the range of blends that provide good matches to the Kepler photometry, both for the chance alignment case and for physically associated triples. This is because the orbital speed in an eccentric orbit can be considerably larger or smaller than in a circular orbit, allowing for blends involving smaller or larger secondaries than would otherwise be permitted while still matching the observed duration of the transits, as described by Torres et al. (2010). It also increases the complexity of the problem, as the space of parameters to be explored is much larger. Additionally, because the secondaries are now not necessarily of the same spectral type as the primary star, attention must be paid to the resulting color of these blends, which could be different from the measured colors as reported in the KIC (Latham et al. 2005), in which case the blend would be excluded. Because these complications require significantly more effort to address, we are unable to provide sufficient evidence for the planetary nature of the KOI-72.02 signal at this time based on BLENDER considerations alone, and we defer such a study to a forthcoming publication. We note, however, that the transit duration and period of the two transit signals identified in the light curve give the same stellar density (1.142 ± 0.092 g cm⁻³ for KOI-72.01 versus 1.147 ±  0.096 g cm⁻³ for KOI-72.02) as derived from the transit properties using the analytic expression given in Equation (9) of Seager & Mallen-Ornelas (2003). This is a rare coincidence for a blend configuration.

We carried out an LTE spectroscopic analysis using the spectral synthesis package SME (Valenti & Piskunov 1996; Valenti & Fischer 2005) applied to a high-resolution template spectrum from Keck–HIRES of Kepler-10 to derive an effective temperature, T_eff = 5705 ± 150 K, surface gravity, log g = 4.54 ±  0.10 (cgs), metallicity, [Fe/H] = −0.15 ±  0.03, vsin i = 0.5 ±  0.5 km s^-1, and the associated error distribution for each of them. To refine the true parameters of the star, we took a novel path to constrain its surface gravity. The above effective temperature was used to constrain the fundamental stellar parameters derived via asteroseismic analysis (see Section 6.2). The asteroseismology analysis gave 4.341 ± 0.012 which is 0.2 dex lower 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. This iteration yielded values of T_eff = 5627 ± 44 K, [Fe/H] = −0.15 ± 0.04, and rotational vsin i = 0.5 ±  0.5km 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.

We also measured the Ca ii H&K emission (Isaacson & Fischer 2010), yielding a Mount Wilson S value, S = 0.180 and log R'_HK = −4.89. Thus Kepler-10 is a magnetically inactive star, consistent with its low rotational rate, vsin i= 0.5 km s^-1. Thus, Kepler-10 appears to be an old (age greater than 5 Gyr) slowly rotating inactive star, slightly above the main sequence. This is consistent with the age derived from the asteroseismology analysis (Section 6.2).

With a magnitude in the Kepler bandpass of Kp = 10.96, Kepler-10 presented itself as a promising case for asteroseismic characterization and was, consequently, placed on the SC target list before it was even identified as a planet candidate. Figure 10(a) illustrates the power density spectrum of the SC light curve. It shows a clear enhancement of power, as expected for solar-like oscillations, around a frequency of 2500 μHz. In the spectrum one can identify sequences of approximately uniformly spaced peaks. This is in accordance with the asymptotic behavior of high-order acoustic modes, according to which the cyclic frequencies ν_nl approximately satisfy

$$\begin{equation} \nu _{nl} \simeq \Delta \nu _0 (n + l/2 + \epsilon) - l (l + 1) D_0 \end{equation} \tag{ 1 }$$

(Vandakurov 1967; Tassoul 1980), where n is the radial order and l is the spherical-harmonic degree of the mode. The large frequency separation ν_nl − ν_n−1 l ≃ Δν₀ is essentially given by the inverse sound travel time across a stellar diameter; it is closely related to the mean stellar density ρ_⋆, approximately satisfying Δν₀ ∝ ρ^1/2_⋆. For main-sequence stars D₀, giving rise to the small frequency separations ν_nl − ν_n−1 l+2, is largely determined by the variation of sound speed in the core of the star and hence provides a measure of the evolutionary state of the star. Finally, is determined by conditions near the surface of the star.²⁹ (For details on the diagnostic potential of solar-like oscillations, see, for example, Christensen-Dalsgaard 2004). Photometric observations such as those carried out by Kepler are essentially restricted to degrees l ⩽ 2.

Zoom In Zoom Out Reset image size

The analysis of the observed frequency spectrum largely followed the procedures used by Christensen-Dalsgaard et al. (2010), with a pipeline developed for analysis of the Kepler p-mode data (Christensen-Dalsgaard et al. 2008; Huber et al. 2009). The first step was to carry out a correlation analysis to determine the large frequency separation Δν₀, leading to Δν₀ = 118.2 ± 0.2 μHz. Using the scaling with the mean density and the corresponding values for the Sun, this yields a first estimate of ρ_⋆ = 1.080 ± 0.006 g cm^-3. It should be noted, however, that this estimate does not take into account detailed differences between the structure of the star and the Sun.

The next step in the analysis was to identify individual modes in the frequency spectrum. The detailed structure of the spectrum is illustrated in Figure 10(b) which shows the folded spectrum, i.e., the sum of the power as a function of the frequency modulo Δν₀. As indicated, in accordance with Equation (1) there is clearly a pair of closely spaced peaks, corresponding to l = 2 and 0, as well as a single peak for l = 1. Given this identification, we were able to determine the individual frequencies of 19 modes. These are illustrated in an échelle diagram (cf. Grec et al. 1983, see below) in Figure 11.

Zoom In Zoom Out Reset image size

To determine the stellar properties, we fitted the observed frequencies to a grid of models. These were computed using the Aarhus Stellar evolution and pulsation codes (Christensen-Dalsgaard 2008a, 2008b). See also Christensen-Dalsgaard et al. (2010). Diffusion and settling of helium and heavy elements were neglected. The grid consisted of models of mass between 0.8 and 1.1 M_☉ in steps of 0.02 M_☉ and mixing length α_ML = 1.5, 1.8, and 2.1. The composition was characterized by heavy-element abundances Z = 0.011, 0.0127, and 0.0144, corresponding to the observed [Fe/H] = −0.15 ± 0.06 (see above); this assumes a solar surface ratio between the heavy-element and hydrogen abundances (Z_s/X_s)_☉ = 0.0245 (Grevesse & Noels 1993), and with the initial hydrogen abundance X₀ related to Z, from galactic chemical evolution, through X₀ = 0.7679 − 3Z₀.

For each model in the grid we calculated an evolution track extending well beyond the end of central hydrogen burning and, for the relevant models along the track, we computed adiabatic frequencies for modes of degree l = 0–2. The match between the observed and computed frequencies, ν^(obs)_nl and ν^(mod)_nl, was characterized by

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

where N is the number of observed frequencies and σ_ν is the standard error in the observed frequencies, which we estimated as 1 μHz. In the fit we also considered the observed effective temperature T_eff (see above), using as combined measure of the goodness of fit

$$\begin{equation} \chi ^2 = \chi _\nu ^2 + \left({T_{\rm eff}^{\rm (obs)} - T_{\rm eff}^{\rm (mod)} \over \sigma (T_{\rm eff})} \right)^2 \;, \end{equation} \tag{ 3 }$$

where T^(obs)_eff and T^(mod)_eff are the observed and model values and σ(T_eff) is the standard error in T_eff.

Along each evolution track we determined that model which minimized χ², among the discrete timesteps in the evolution sequence. The final best model corresponding to the track was obtained by further minimizing χ² to determine χ²_min for that track, interpolating T_eff linearly between timesteps and scaling the frequencies according to ρ^1/2_⋆. Given the resulting model values for all evolution tracks, we determined the final estimates of the stellar properties as an average over all tracks, weighted by χ⁻²_min.

A preliminary application of this procedure, using the value T_eff = 5705 ± 150 K as described in the preceding section, resulted in an average surface gravity of log g = 4.341 ± 0.012, substantially different from the spectroscopically determined value of 4.54 ± 0.10. In view of the known problems with the spectroscopic determination of log g we repeated the spectroscopic analysis, fixing log g as 4.341 ± 0.012 in accordance with the asteroseismic inference. This resulted in T_eff = 5627 ± 44 K. We then repeated the asteroseismic fits, conservatively increasing the uncertainty in T_eff to 60 K. This resulted in the final estimates of the stellar parameters presented in Table 7: M_⋆ = 0.895 ± 0.060 M_☉, R_⋆ = 1.056 ± 0.021 R_☉, ρ_⋆ = 1.068 ± 0.008 g cm⁻³, and age =11.9 ± 4.5 Gyr. The distance of 173 ± 27 pc is computed using the g-band SDSS apparent magnitude in the KIC and bolometric corrections interpolated form the tables of Girardi et al. (2008).

Table 7. Star and Planet Parameters for the Kepler-10 System

Parameter	Value	Notes
Transit and orbital parameters: Kepler-10b
Orbital period, P (days)	0.837495+0.000004−0.000005	A
Midtransit time, E (BJD)	2454964.57375+0.00060−0.00082	A
Scaled semimajor axis, a/R⋆	3.436+0.070−0.092	A
Scaled planet radius, RP/R⋆	0.01232+0.00013−0.00016	A
Impact parameter, b	0.339+0.073−0.079	A
Orbital inclination, i (deg)	84.40+1.1−1.6	A
Orbital semi-amplitude, K (m s-1)	3.3+0.8−1.0	B
Orbital eccentricity, e	0	B
Center-of-mass velocity, γ (m s-1)	−98.93 ± 0.02	B
Transit and orbital parameters: KOI-72.02
Orbital period, P (days)	45.29485+0.00065−0.00076	A
Midtransit time, E (HJD)	2454971.6761+0.0020−0.0023	A
Scaled semimajor axis, a/R⋆	49.1+1.2−1.3	A
Scaled planet radius, RP/R⋆	0.019378+0.00020−0.00024	A
Impact parameter, b	0.299+0.089−0.073	A
Orbital inclination, i (deg)	89.70+0.09−0.12	A
Observed stellar parameters
Effective temperature, Teff(K)	5627 ± 44	C
Spectroscopic gravity, log g(cgs)	4.35 ± 0.06	C
Metallicity, [Fe/H]	−0.15 ± 0.04	C
Projected rotation, vsin i(km s-1)	0.5 ± 0.5	C
Fundamental Stellar Properties
Mass, M⋆(M☉)	0.895 ± 0.060	D
Radius, R⋆(R☉)	1.056 ± 0.021	D
Surface gravity, log g⋆ (cgs)	4.341 ± 0.012	D
Luminosity, L⋆ (L☉)	1.004 ± 0.059	D
Absolute V magnitude, MV (mag)	4.746 ± 0.063	D
Age (Gyr)	11.9 ± 4.5	D
Distance (pc)	173 ± 27	D
Planetary parameters: Kepler-10b
Mass, MP (M⊕)	4.56+1.17−1.29	A, B, C, D
Radius, RP (R⊕)	1.416+0.033−0.036	A, B, C, D
Density, ρP (g cm-3)	8.8+2.1−2.9	A, B, C, D
Surface gravity, log gP (cgs)	3.35+0.11−0.13	A, B, C, D
Orbital semimajor axis, a (AU)	0.01684+0.00032−0.00034	E
Equilibrium temperature, Teq (K)	1833	F
Parameters for candidate: KOI-72.02
Mass, MP (M⊕)	<20	G
Radius, RP (R⊕)	2.227+0.052−0.057	A, D
Orbital semimajor axis, a (AU)	0.2407+0.0044−0.0053	E
Equilibrium temperature, Teq (K)	485	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: calculated assuming a Bond albedo of 0.1 and complete redistribution of heat for reradiation, and G: upper limit corresponding to three times the 68.3% credible interval from MCMC mass distribution.

Download table as:  ASCIITypeset image

To illustrate the quality of the fit, Figure 11 shows the observed frequencies and the frequencies for the best-fitting model in an échelle diagram, reflecting the asymptotic structure of the spectrum described by Equation (1). In accordance with this equation, the spectrum has been divided into segments of length Δν₀ which have been stacked. Formally this corresponds to reducing the frequencies modulo Δν₀. The asymptotic behavior is reflected in the nearly vertical columns of points, corresponding to the different values of the degree. Also, it is clear that the model provides a reasonable, although far from perfect, fit to the observations.³⁰

Kepler-10b is a high-density rocky planet. This conclusion is based on the comparison of its radius and mass, measured within 1σ, with theoretical calculations of interior structure, as illustrated in Figures 16 and 17. The conclusion accounts for known uncertainties in the theory.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 16 is the mass–radius diagram for small planets in units of Earth mass and radius (after Zeng & Sasselov 2011). Earth lies on the left edge, between two models (solid lines) of Earth-like composition with a range of Fe/Si ratios. Uranus and Neptune are out of range, above the upper right portion of the diagram. Kepler-10b (bottom 1σ ellipse) and KOI-72.02 (1σ band with upper limit in mass at ≈25M_E) are shown with models and with two other planets known in this size range: CoRoT-7b and GJ1214b. In Figure 16, the upper 1σ ellipses are for GJ 1214b (Charbonneau et al. 2009) and CoRoT-7b (Queloz et al. 2009 for mass; Bruntt et al. 2010 for radius). The uncertainty in the mass of CoRoT-7b is dependent on the methodology used to correct for the stellar activity signal that dominates the Doppler data. For example, Queloz et al. (2009) compute a mass of 4.8 ± 0.8 M_⊕, while Ferraz-Mello et al. (2010) obtain 8.5 ± 1.5 M_⊕. Hatzes et al. (2010) compute a mass of 6.9 ± 1.5 M_⊕, while Pont et al. (2010) obtain 2.3 ± 1.8 M_⊕. The mass of CoRoT-7b, provided by Pont et al. (2010), is shown by straight solid lines in Figure 16 in order to illustrate the range of mass values present in the literature.

Theoretical calculations in Figure 16 are shown as curves, which are (from top to bottom): 10% by mass H/He envelope with typical ice giant interior similar to Uranus and Neptune (short-dashed line); theoretical pure water object (dot-dashed line); 50% water planets with 34% silicate mantle and 16% Fe core (thick long-dashed line), or with a low Fe/Si ratio of 44% mantle and 6% Fe core (thin long-dashed line; and Earth-like composition with the same Fe/Si ratios (thick and thin solid lines). These models are from the grid by Zeng & Sasselov (2011), based on Valencia et al. (2007) with many updates, e.g., new water EOS by French et al. (2009). The Fe/Si ratios chosen in this grid correspond to the range observed in the solar system given a stellar Fe/Si range in the solar neighborhood (Grasset et al. 2009), where the low Fe/Si ratio corresponds to the lowest values measured in small bodies in the outer solar system, namely Ganymede. Note that the hypothetical 100% water and H/He envelope model curves are shown for illustration only; given the extremely high equilibrium temperature of Kepler-10b an extended hot atmosphere of such volatiles has to be accounted for separately, but Kepler-10b is too dense for these scenarios.

The interior structure of GJ 1214b has been modeled as an H/He/H₂O planet with a rocky core (Nettelmann et al. 2010). Indeed, it lies between the models of ice giants similar to Neptune and Uranus and the models of a 50% water planet. And while the Queloz et al. (2009) mass and radius point to a rocky composition, the lower mass of Pont et al. (2010) marginally favors a water/ice composition. The properties of Kepler-10b together with their uncertainties clearly indicate a high-density, rocky planet.

The dotted curve at the bottom of Figure 16 is the envelope corresponding to a maximum Fe core fraction expected from simulations of mantle stripping by giant impacts (Marcus et al. 2010). These simulations and planet formation scenarios indicate that pure Fe core objects cannot form in the mass range we consider. At the mass of Kepler-10b these simulations predict that the maximum attainable Fe core is about 75% by mass. For comparison, Mercury has a ≈70% Fe core.

Theoretical calculations of the interiors of high-density planets suffer from a number of uncertainties in equations of states, high-pressure phases of materials inaccessible to lab experiments, cooling and differentiation histories, etc. There is consensus in the literature (Valencia et al. 2006, 2007; Fortney et al. 2007; Seager et al. 2007; Grasset et al. 2009) regarding the general results. However, the theoretical uncertainties are compounded by a degeneracy between only two observables (radius and mass), and three or four distinct types of bulk materials: Fe core, mantle (silicates, etc.), water, and hydrogen/helium gas (see Valencia et al. 2007; Rogers & Seager 2010 for details).

At very high densities, the radius of solid planets is constrained by the lack of bulk materials with density and compression properties above that of Fe and Fe alloys (see lower-envelope dotted curve in Figure 16). Planets with high densities close to that envelope are likely composed predominantly of silicates and Fe; the degeneracy is lifted as indicated by the curvature of the iso-radius curves when they approach asymptotically the "dry" right-hand side of the ternary diagram in Figure 17.

Ternary diagrams were introduced to studies of solid planet structure by Valencia et al. (2007). The three axes of the ternary diagram represent the core, mantle, and ice fraction of a planet of a given mass (in this case, the mass derived for Kepler-10b). The axes are read by following lines parallel to the edges so that the three mass fractions sum to unity. Although there are six radiants emerging from each point in the grid, only three will result in mass fractions that add to unity. Different combinations of core, mantle, and ice fraction yield a planet radius that can be computed with theoretical models constrained by a given mass. Radius is not unique to a specific combination of mass fractions. The solid line in Figure 17 is an iso-radius. It shows the possible combinations that all yield the derived radius of Kepler-10b. The dotted lines provide bounds on the domain captured by the 1σ errors in planet radius. A ternary diagram represents a cross-section at a given mass. However, mass-dependent uncertainties are included in the error bars represented by the dotted line. The iso-radius together with the 1σ error bars are a good representation of the possible bulk compositions for a given planet.

The degeneracy in bulk composition discussed above is best illustrated in Figure 17, as a planet defined by a mass and radius is represented by a curve (solid) that can span a range of iron:mantle:water fractions. This degeneracy is practically lifted only for very high density planets near the "dry" right-hand side, close to the 100% iron core vertex, because (1) the iso-radius curves bend as they approach the 0% water level, and (2) the H₂O and OH molecules are able to be incorporated inside high-pressure silicate phases without changing their EOS much. For Kepler-10b, the combination of very high density and constraints from mantle-stripping simulations (dash-dotted line), restrict its bulk compositions to dry rocky iron-core-dominated interiors similar to Mercury. On the other hand, if the observational derivation of the mass of Kepler-10b were significantly overestimated (by more than 1σ), the planet could contain significant amount of water in its interior.

In conclusion, within 1σ–2σ in its derived radius and mass, the planet Kepler-10b is a dry rocky planet with high Fe content. Its high density does not violate the prediction for maximum mantle stripping during planet formation (Marcus et al. 2010).

The phase curve amplitude of Kepler-10b is 7.6 ± 2.0 ppm. If due to scattered light alone, this corresponds to a Kepler bandpass effective geometric albedo of 0.68. The occultation depth is 5.8 ± 2.5 ppm which corresponds to an effective geometric albedo of 0.61 ± 0.17. This is an unusually high albedo. The only known solar system bodies that are so bright are Venus (due to photochemically induced hazes) and Saturn's icy moon Enceladus (coated with fresh ice). Kepler-10b is likely too hot for any hazes and is certainly too hot for any ice. Another possibility is that Kepler-10b has silicate clouds, but they would have to be completely covering the planet's day side and have a large particle size in order to provide the required reflectivity (see, for example, Seager et al. 2000).

We prefer an interpretation that the phase curve is dominated by a thermal radiation change from the planet's day to night side. This case would be reminiscent of hot Jupiters, which have a hotter day side than night side. In the case of the Kepler bandpass, the temperature difference need not be too extreme, as long as the planet is hot enough on the day side (as is the case for Kepler-10b with a equilibrium temperature of 1833 K). This is because the Kepler bandpass is in the optical, and so the contribution of thermal radiation drops off rapidly with decreasing temperature (i.e., day-to-night side). In other words, a passband on the Wien side of the black body curve will capture significantly different fluxes with just a small change in temperature. An example of this is the phase curve of HAT-P-7 (Borucki et al. 2009; Welsh et al. 2010).

An intriguing possibility is that Kepler-10b has no atmosphere at all, having lost it over time due to atmospheric erosion. One test is to search for a hotspot directly at the substellar point, since no atmospheric winds would be available to move the hotspot off center (Seager & Deming 2009). Further Kepler data, in particular a confirmation and robust measurement of the occultation depth, of Kepler-10b will help with any interpretation.

Asteroseismic measurements of the host star Kepler-10 indicate that the planetary system is very old, allowing for more than 11 Gyr of evaporation and mass loss from the surface of Kepler-10b. Assuming, as is common, that the planet arrived at its present orbit within about 100 Myr of formation, the relevant evaporation time scales are those of water steam or a silicate surface. Any H/He envelope would have evaporated too quickly (at ≈10¹¹ g s⁻¹) to affect the further evolution of the planet, while the evaporation of the silicate mantle is too slow to remove more than 50% of the planet's original mass (Valencia et al. 2009). Therefore, it is difficult to establish if Kepler-10b is the remnant core of a water planet or an ice giant planet, though it is probable that it must have lost a significant fraction of its mass.