KEPLER-93b: A TERRESTRIAL WORLD MEASURED TO WITHIN 120 km, AND A TEST CASE FOR A NEW SPITZER OBSERVING MODE

To prepare the Kepler photometry for asteroseismic study, we first removed planetary transits from the light curve by applying a median high-pass filter (R. Handberg et al., in preparation). We then performed the asteroseismic analysis on the Fourier power spectrum of the filtered light curve. Figure 1 shows the resulting power spectrum of Kepler-93. We find a typical spectrum of solar-like oscillations, with several overtones of acoustic (pressure, or p) modes of high radial order, n, clearly detectable in the data. Solar-type stars oscillate in both radial and non-radial modes, with frequencies ν_nl, with l the angular degree. Kepler-93 shows detectable overtones of modes with l ⩽ 2. The dominant frequency spacing is the large separation Δν_nl = ν_{n + 1 l} − ν_n l between consecutive overtones n of the same degree l.

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Huber et al. (2013) reported asteroseismic properties based on the use of only average or global asteroseismic parameters. These were the average large frequency separation <Δν_nl > and the frequency of maximum oscillation power, ν_max. Here, we have performed a more detailed analysis of the asteroseismic data, using the frequencies of 30 individual p modes spanning 11 radial overtones. Using individual frequencies in the analysis enhances our ability to infer stellar properties from the seismic data. We place much more robust (and tighter) constraints on the age than is possible from using the global parameters alone.

We estimated the frequencies of the observable p modes with a "peak-bagging" Markov Chain Monte Carlo (MCMC) analysis, as per the procedures discussed in detail by (Chaplin et al. 2013; see also Carter et al. 2012). This Bayesian machinery, in addition to allowing the use of relevant priors, allows us to estimate the marginalized probability density function for each of the power spectrum model parameters. These parameters include frequencies, mode heights, mode lifetimes, rotational splitting, and inclination. This method performs well in low signal-to-noise conditions and returns reliable confidence intervals provided for the fitted parameters. We obtained a best-fitting model to the observed oscillation spectrum through optimization with an MCMC exploration of parameter space with Metropolis–Hastings sampling (e.g., see Appourchaux 2011; Handberg & Campante 2011).

We extracted estimates of the frequencies from their marginalized posterior parameter distributions. The best-fit values themselves correspond to the medians of the distributions. Table 1 lists the estimated frequencies, along with the positive and negative 68% confidence intervals. We chose not to list the estimated frequency for the l = 2 mode at ≃ 3274 μHz. This mode is possibly compromised by the presence of a prominent noise spike in the power spectrum, and the posterior distribution for the estimated frequency is bimodal. The spike may be a leftover artifact of the transit, given that the frequency of the spike lies very close to a harmonic of the planetary period. We note that omitting the estimated frequency of this mode in the modeling does not alter the final solution for the star. The lowest radial orders that we detect for the l = 0 and l = 1 modes are not sufficiently prominent for l = 2 to yield a good constraint on the frequency. We exclude them from the table.

Figure 2 is an échelle diagram of the oscillation spectrum. We produced this figure by dividing the power spectrum into equal segments of length equal to the average large frequency separation. We arranged the segments vertically, in order of ascending frequency. The diagram shows clear ridges, comprising overtones of each degree, l. The red symbols mark the locations of the best-fitting frequencies returned by the analysis described above (with l = 0 modes shown as filled circles, l = 1 as filled triangles and l = 2 as filled squares).

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We require a complementary spectral characterization, in tandem with asteroseismology, to determine the stellar properties of Kepler-93. From a spectrum of the star, we measure its effective temperature, T_eff, and its metallicity, [Fe/H]. We use these initial values, together with the two global asteroseismic parameters, to estimate the surface gravity, log g. Then, we repeat the spectroscopic analysis with log g fixed at the asteroseismic value, to yield the revised values of T_eff and [Fe/H]. We use the spectroscopic estimates of Huber et al. (2013). These values came from analysis of high-resolution optical spectra collected as part of the Kepler Follow-up Observing Program. To recap briefly: Huber et al. (2013), using the Stellar Parameter Classification pipeline (Buchhave et al. 2012), analyzed spectra of Kepler-93 from three sources. These were the HIRES spectrograph on the 10 m Keck telescope on Mauna Kea, the FIber-fed Echelle Spectrograph on the 2.5 m Nordic Optical Telescope on La Palma, and the Tull Coudé Spectrograph on the 2.7 m Harlan J. Smith Telescope at the McDonald Observatory. In that work, they employed an iterative procedure to refine the estimates of the spectroscopic parameters (see also Bruntt et al. 2012; Torres et al. 2012). They achieved convergence of the inferred properties (to within the estimated uncertainties) after just a single iteration.

Five members of the team (S. Basu, J.C.-D., T.S.M., V.S.A., and D.S.) independently used the individual oscillation frequencies and spectroscopic parameters to perform a detailed modeling of the host star. We adopted a methodology similar to that applied previously to the exoplanet host stars Kepler-50 and Kepler-65 (see Chaplin et al. 2013 and references therein). Full details may be found in the Appendix. The final properties presented in Table 2 are the solution set provided by J.C.-D. (whose values lay close to the median solutions). We include a contribution to the uncertainties from the scatter between all five sets of results. This was done by taking the chosen modeler's uncertainty for each property and adding (in quadrature) the standard deviation of the property from the different modeling results. Our updated properties are in good agreement with—and, as expected, more tightly constrained than—the properties in Huber et al. (2013).

Table 2. Star and Planet Parameters for Kepler-93

Parameter	Value and 1σ Confidence Interval
Kepler-93 (star)
Right ascensiona	19h25m4039
Declinationa	+38d40m2045
Teff (K)	5669 ± 75
R⋆ (solar radii)	0.919 ± 0.011
M⋆ (solar masses)	0.911 ± 0.033
[Fe/H)	−0.18 ± 0.10
log(g)	4.470 ± 0.004
Age (Gyr)	6.6 ± 0.9
Light curve parameters	No asteroseismic prior	With asteroseismic prior
ρ (g cm−3)	1.72$^{+0.04}_{-0.28}$	1.652 ± 0.0060
Period (days)b	4.72673978 ± 9.7 × 10−7	...
Transit epoch (BJD)b	2454944.29227 ± 0.00013	...
Rp/R⋆	0.01474 ± 0.00017	0.014751 ± 0.000059
a/R⋆	12.69$^{+0.09}_{-0.76}$	12.496 ± 0.015
Inc (deg)	89.49$^{+0.51}_{-1.1}$	89.183 ± 0.044
u1	0.442 ± 0.068	0.449 ± 0.063
u2	0.187 ± 0.091	0.188 ± 0.089
Impact parameter	0.25 ± 0.17	0.1765 ± 0.0095
Total duration (minutes)	173.42 ± 0.36	173.39 ± 0.23
Ingress duration (minutes)	2.52$^{+0.37}_{-0.06}$	2.61 ± 0.013
Kepler-93b (planet)	No asteroseismic prior	With asteroseismic prior
Rp (Earth radii)	1.483 ± 0.025	1.478 ± 0.019
Planetary Teq (K)	1039 ± 26	1037 ± 13
Mp (Earth masses)c	3.8 ± 1.5	...

Notes. ^aICRS (J2000) coordinates from the TYCHO reference catalog (Høg et al. 1998). The proper motion derived by Høg et al. (2000) −26.7 ± 1.9 in right ascension and −4.4 ± 1.8 in declination. ^bWe fit the ephemeris and period of Kepler-93 in an iterative fashion, as described in the text. We fix the other orbital parameters to fit individual transit times, and report the best linear fit to these times, rather than simultaneously fitting the transit times while imposing the asteroseismic prior. ^cMarcy et al. (2014) describe the Kepler-93 radial velocity campaign. The stated value here is a revision, with an additional year of observations, of their published value of 2.6 ± 2.0 M_⊕.

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We note the precision in the estimated stellar properties, in particular the age, which we estimate to slightly better than 15%. We estimate the central hydrogen abundance of Kepler-93 to be approximately 31%, which is similar to the estimated current value for the Sun. The age of Kepler-93 at central hydrogen exhaustion is predicted to be around 12.4 Gyr.

We estimated the uncertainty in the planetary transit parameters using the MCMC method as follows. To fit the transit light curve shape, we removed the effects of baseline drift by individually normalizing each transit. We fitted a linear function of time to the flux immediately before and after transit (specifically, from 8.6 hr to 20 minutes before first contact, equal to three transit durations, and an equivalent time after fourth contact). We employed model light curves generated with the routines in Mandel & Agol (2002), which depend upon the period P, the epoch T_c, the planet-to-star radius ratio R_p/R_⋆, the ratio of the semi-major axis to the stellar radius a/R_⋆, the impact parameter b, the eccentricity e, and the longitude of periastron, ω. We allowed two quadratic limb-darkening coefficients, u₁ and u₂, to vary. Our data set comprises 233 independent transits of Kepler-93b, all observed in SC mode (with exposure time 58.5 s, per Gilliland et al. 2010). We employed the Borucki et al. (2011) transit parameters from the first four months of observations as a starting point to refine the ephemeris. We allowed the time of transit to float for each individual transit signal, while fixing all other transit parameters to their published values. We then fitted a linear ephemeris to this set of transit times, and found an epoch of transit T₀ = BJD 2454944.29227 ± 0.00013 and a period of 4.72673978 ± 9.7 × 10⁻⁷ days. We iterated this process, using the new period to fit the transit times, and found that it converged after one iteration.

We imposed an eccentricity of zero for the light curve fit, which we justify as follows. The circularization time (for modest initial e) was reported by Goldreich & Soter (1966), where a is the semimajor axis of the planet, R_p is the planetary radius, M_p is the planetary mass, M_⋆ is the stellar mass, Q is the tidal quality factor for the planet and G is the gravitational constant:

$$\begin{equation} t_{\rm {circ}}=\frac{4}{63}\frac{1}{\sqrt{GM_{\star }^{3}}}\frac{M_{p}a^{13/2}Q}{R_{p}^{5}}. \end{equation} \tag{ 1 }$$

Planetary Q is highly uncertain, but we estimate a Q = 100 with the assumption of a terrestrial composition. This value is typical for rocky planets in our solar systems (Yoder 1995; Henning et al. 2009). We find that the circularization timescale would be 70 Myr for a 3.8 M_⊕ planet orbiting a 0.91 M_☉ star at 0.053 AU (we justify this mass value in Section 4.3). To achieve a circularization timescale comparable to the age of the star, Q would have to be on the order of 10⁴, similar to the value of Neptune. As we discuss in Section 5.1, there is only a 3% probability that Kepler-93b has maintained an extended atmosphere. This finding renders a Neptune-like Q value correspondingly unlikely. For this reason, we assume that sufficient circularization timescales have elapsed to make the e = 0 assumption valid.

To fit the orbital parameters, we employed the Metropolis–Hastings MCMC algorithm with Gibbs sampling (described in detail for astronomical use in Tegmark et al. 2004 and in Ford 2005). We first conducted this analysis using only the Kepler light curve to inform our fit. We incorporated the presence of correlated noise with the Carter & Winn (2009) wavelet parameterization. We fitted as free parameters the white and red contributions to the noise budget, σ_w and σ_r. The allowable range of stellar density we inferred from the light curve alone is much broader than the range of stellar densities consistent with our asteroseismic study. Crucially, they are consistent. When we fit the transit parameters independently of any knowledge of the star (allowing a/R_⋆ to float), we found that values of a/R_⋆ from 11.93–12.78 gave comparable fits to the light curve. This parameter varies codependently with the impact parameter (the ingress and egress times can vary within a 30 s range, from 2.5–3 minutes). We then refitted the transit light curve and apply the independent asteroseismic density constraint as follows. Rearranging Kepler's Third Law in the manner employed by Seager & Mallén-Ornelas (2003), Sozzetti et al. (2007) and Torres et al. (2008), we converted the period P (derived from photometry) and the stellar density ρ_⋆, to a ratio of the semi-major axis to the radius of the host star, a/R_⋆:

$$\begin{equation} \left(\frac{a}{R_{\star }}\right)^{3}=\frac{GP^{2}(\rho _{\star }+p^{3}\rho _{p})}{3\pi }, \end{equation} \tag{ 2 }$$

where p is the ratio of the planetary radius to the stellar radius and ρ_p is the density of the planet. We will hereafter assume p³ ≪ 1 and consider this term negligible. We mapped the stellar density constraint, ρ_⋆ = 1.652 ± 0.006 g cm⁻³, to a/R_⋆ = 12.496 ± 0.015, which we apply as a Gaussian prior in the light curve fit.

In Figure 3, we show the phased Kepler transit light curve for Kepler-93b, with the best-fit transit light curve overplotted. In Figure 4, we show the correlations between the posterior distributions of the subset of parameters in the model fit, as well as the histograms corresponding to each parameter. We depict two cases. In one case, we allowed the stellar density to float, in the other we applied a Gaussian prior to force its consistency with the asteroseismology analysis. We report the best-fit parameters and uncertainties in Table 2. We determined the range of acceptable solutions for each of the light curve parameters as follows. In the same manner as Torres et al. (2008), we report the most likely value from the mode of the posterior distribution, marginalizing over all other parameters. We quote the extent of the posterior distribution that encloses 68% of values closest to the mode as the uncertainty. To estimate an equilibrium temperature for the planet, we assumed that the planet possesses a Bond albedo (A_B) of 0.3 and that it radiates energy equal to the energy incident upon it. To generate physical quantities for the radius of the planet R_p, we multiplied the R_p/R_⋆ posterior distribution by the posterior on R_⋆. In the case where we applied the asteroseismic prior, we found a planetary radius of 1.481 ± 0.019 R_⊕ with 1σ confidence. This value includes the uncertainty in stellar radius. We note that the uncertainty of R_p/R_⋆ makes up 30% of the error budget on the planetary radius. Uncertainty on R_⋆ itself contributes the remaining 70%. Without the asteroseismic prior on a/R_⋆ (and the correspondingly less constrained measurement of R_p/R_⋆), our measurement of the planetary radius is 1.485 ± 0.27 R_⊕. The contributions are nearly reversed in this case: 68% of the error in R_p is due to uncertainty in R_p/R_⋆ and only 32% to uncertainty in the stellar radius.

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The uncertainty in planetary radius of 0.019 R_⊕ corresponds to 120 km. Kepler-93b is currently the planet with the most precisely measured extent outside of our solar system. A reanalysis of Kepler's first rocky exoplanet Kepler-10b (Batalha et al. 2011) by Fogtmann-Schulz et al. (2014) with 29 months of Kepler photometry returned a similarly precise radius measurement: 1.460 ± 0.020 R_⊕. Kepler-16b (Doyle et al. 2011), with radius 8.449$^{+0.028}_{-0.025}$ R_⊕ and Kepler-62c (Borucki et al. 2013), with radius 0.539 ± 0.030 R_⊕, are the next most precisely measured exoplanets within the published literature. The sample of these four currently comprises the only planets outside the solar system with radii measured to within 0.03 R_⊕.

The transit light curve alone provides some constraints on the host star of the planet. False-positive scenarios in which an eclipsing binary comprising (1) two stars, or (2) a star and a planet, falls in the same aperture as the Kepler target star, will produce a transit signal with a diluted depth. A comparison of the detailed shape of the transit light curve to models of putative blend scenarios, as described by Torres et al. (2004), Fressin et al. (2011), and Fressin et al. (2012), constrains the parameter space in which such a blend can reside. The likelihood of such a scenario, given the additional observational constraints of spectra and AO imaging, is then weighed against the likelihood of an authentic planetary scenario. This practice returns a robust false-positive probability. Here, we rather comment on the consistency of the transit light curve with our hypothesis that it originates from a 1.48 R_⊕ planet in orbit around a 0.92 R_☉ star.

The period and duration of the transit will not be affected by the added light of another star. The steepness of the ingress and egress in the transit signal of Kepler-93b (enabled by the wealth of SC data on the star), which lasts between 3.0 and 2.46 minutes as we report in Table 2, reasonably precludes a non-planetary object. The transiting object passes entirely onto the disk of the star too quickly to be mimicked by an object that subtends a main sequence stellar radius. We note that a white dwarf could furnish such an ingress time, but would produce an RV variation at the 4.7 day period at the level of 100 km s⁻¹, much larger than we observe (we describe the RV observations in the following subsection). We conclude that a blend scenario (1) comprising three stars is unlikely. False positive scenario (2) involves a transiting planet system within the Kepler aperture that is not in orbit around the brightest star. Rather, the diluted transit depth would conspire to make the planet appear smaller than it is in reality. In this case, transit duration still constrains the density of the host star. Certain stellar hosts are immediately ruled out, because a planet orbiting them could not produce a transit as long as observed, given the eccentricity constraint implied by the short planetary period. We constrain the host from the transit light curve parameters as follows.

Winn et al. (2009) described the approximation of the transit depth for cases in which the eccentricity e is close to zero (which we assume is valid because of the short period), where R_p ≪ R_⋆ ≪ a (where a is the semi-major axis of the planet's orbit) and where the impact parameter b ≪ 1 − R_p/R_⋆. In these cases, the transit timescale T, defined to be the total duration of the transit minus an ingress time, is related to a characteristic timescale T₀ as follows:

$$\begin{equation} T_{0}\approx \frac{T}{\sqrt{1-b^{2}}}, \end{equation} \tag{ 3 }$$

where

$$\begin{equation} T_{0}=\frac{R_{\star }P}{\pi a}\approx 13 \rm { hr }\left(\frac{P}{\rm {1 yr}} \right)^{1/3} \left(\frac{\rho _{\star }}{\rho _{\odot }} \right)^{-1/3}. \end{equation} \tag{ 4 }$$

We computed the distribution for T₀ from our posterior distributions for the total duration, ingress duration, and impact parameter. We found that values between 170 and 184 minutes are acceptable at the 1σ confidence limit. This translates to a constraint on the exoplanet's host star density of 1.72$^{+0.04}_{-0.28}$ g cm³, as given in Table 2. The density of the Kepler target star measured from asteroseismology, 1.652 ± 0.006 g cm⁻³, is within 1σ of the value inferred from the transit. This consistency lends credence to the interpretation that the bright star indeed hosts the planet.

The transit depth we report with Spitzer of R_p/R_⋆ = 0.0144 ± 0.0032 lies within 1σ of the depth measured by Kepler of 0.014751 ± 0.000057 (as we report in Table 2). The achromatic nature of the transit depth disfavors blended "false-positive" scenarios for Kepler-93b and render it more likely to be an authentic planet. J. M. Désert et al. (in preparation) consider the Spitzer transit depth, the Spitzer magnitude at 4.5 μm, and AO imaging to find a false-positive probability of 0.18%.

Marcy et al. (2014) measured the RVs of Kepler-93b. They employed Keck-HIRES spectra spanning 1132 days, from 2009 July to 2012 September. In that work, they describe their reduction and RV fitting process in detail. While they report a mass of 2.6 ± 2.0 M_⊕ in that work, the addition of another 14 spectra gathered in the 2013 observing season refine this value to 3.8 ± 1.5 M_⊕ (Geoffrey Marcy 2013, private communication). We employ this value, a 2.5σ detection of the mass, for the remainder of our analyses.

They report a linear RV trend in the Keck-HIRES observations of Kepler-93. This trend is present at a level of 10 m s⁻¹ yr⁻¹ over a baseline of three years. They conclude that the trend is caused by another object in the Kepler-93 system, and designate it Kepler-93c. They place lower limits on both its mass and period of M > 3 M_JUP and P > 5 yr. At this lower mass limit the object would be orbiting in a near-transiting geometry (though no additional transits appear in the Kepler light curve). With an additional year of observations, the linear trend is still present. Because the RVs have not yet turned over, we assert that its period must in fact be larger than twice the observing baseline, which is now three years. We assume a lower limit to the period of the perturber of six years. There exists the possibility, which Marcy et al. (2014) note in their discussion of Kepler-93, that the planet is in fact orbiting the smaller body responsible for the RV drift. In this case, the planetary properties must be revisited based upon the revised stellar host. However, while the RV data alone permit this scenario, it is ruled out based upon the density inferred from the transit light curve, in tandem with other observables. We explain our reasoning here.

In Section 4.1, we argue that the host star to the planet must possess a density within the range of 1.72$^{+0.04}_{-0.28}$ g cm⁻³ to be consistent with the transit duration. We compared this density to the Dartmouth stellar evolutionary models given by Dotter et al. (2008) to place an approximate constraint on the host star mass. We evaluated the Dartmouth models over a grid of metallicities (sampled at 0.1 dex from −1 to 0.5 dex) and a grid of ages between 500 Myr and 14 Gyr (sampled at 0.1 Gyr). We do not consider enhancement in alpha elements. We found that the 1σ range in observed density translates to a range of acceptable stellar masses from 0.75 to 1.23 M_☉. If we assume that this putative host star is physically associated with the target star (for reasons we explore in the following paragraph), and we further require that these two stars formed within 1 Gyr of each other, then the mass is constrained to between 0.82 and 1.02 M_☉. At the 3σ level of uncertainty, the density lies between 0.86 and 1.76 g cm⁻³. The range of acceptable masses within this broader confidence interval is correspondingly 0.75–1.52 M_☉. An assumption of contemporaneous formation of the two stars further constrains this range to 0.82–1.12 M_☉. Therefore, we preclude stars below 0.75 M_☉ as the planetary host. To be the perturber to the asteroseismic target star, the two stars are physically associated and we assume their formations to be coeval. A 0.75 M_☉ star, if it possessed an age within 1 Gyr of the target star, would be between 0.7 and 1.0 mag fainter in K than the brighter target star. The exact difference in magnitude, though within this range, depends upon its exact metallicity and age. With a period greater than six years and a mass >0.75 M_☉, its semimajor axis is greater than 3.9 AU.

The lack of a set of additional lines in the high-resolution spectrum brackets the upper limit on the mass and velocity of the object perturbing the bright star. Any star brighter than 0.3% the brightness of the primary star is ruled out, provided the RV difference between the bright star and the perturbing body is greater than 10 km s⁻¹. A putative star 1 mag fainter than the target star (40% its brightness) is therefore required to possess a velocity within 10 km s⁻¹ of the target star. The overlap of absorption features precludes the detection of another star if their RVs are sufficiently similar. For the sake of argument, we proceed under the assumption that the additional star satisfied this condition.

Marcy et al. (2014) use Keck AO images of Kepler-93 to rule out any companion beyond 01 within 5 mag. We compute a physical scale corresponding to this angular size. We compare the observed Kepler magnitude of 9.931 to the magnitude predicted in the Kepler bandpass for a star with the mass, metallicity, and age of Kepler-93 (listed in Table 2) by the Dartmouth stellar evolutionary models. We find a predicted Kepler magnitude at a distance of 10 pc of 4.9 ± 0.1 mag. We therefore estimate that the star is approximately 100 pc away from Earth, so that 01 is physically equivalent to 10 AU. The minimum distance between the stars of 3.9 AU (set by the lower limit on the period of the perturber) is an angular separation of 0039. The AO imaging rules out any object within 2 mag of the host star at this separation. The putative other host star to the planet, only 1 mag fainter, is therefore further constrained to lie in such a geometry and phase that it eluded AO detection. However unlikely, this scenario is plausible.

Therefore, we consider a hypothetical 0.75 M_☉ planet-host companion to the bright star, with a semimajor axis of 3.9 AU. It possessed a velocity within 10 km s⁻¹ of the host star in the high-resolution spectrum that Marcy et al. (2014) searched for a set of additional lines. At the time of the AO imaging gathered by Marcy et al. (2014), it resided at a phase in its orbit where its projected distance from the brighter star rendered it invisible. It is then the several-year baseline of RV observations that conclusively rule out the existence of this companion. If this companion existed at 3.9 AU, the brighter star would possess an RV semi-amplitude of 15.5 km s⁻¹ over the six year baseline. Given the drift in half of this period of only 45 m s⁻¹, its inclination would have to be within 018 of perfectly face-on to be consistent. If it were indeed in such a face-on orientation, it would be readily detectable with AO imaging within the stated limits. We conclude, based upon the density constraint from the transit light curve in combination with published RV and spectral constraints, that the planet indeed orbits the brighter target star. Whether or not the RV trend is due to an additional Jovian planet or to a small star remains unresolved.

There now exist eight exoplanets with radii in the 1.0–1.5 R_⊕ range with dynamically measured masses. We define the radius here as the most likely value reported by the authors, and define a mass detection as residing more than two standard deviations from a mass of zero. From smallest to largest, these are Kepler-102d (Marcy et al. 2014), Kepler-78b (Howard et al. 2013; Pepe et al. 2013), Kepler-100b (Marcy et al. 2014), Kepler-10b (Batalha et al. 2011), Kepler-406b (Marcy et al. 2014), Kepler-99b (Marcy et al. 2014), Kepler-93b, and Kepler-36b (Carter et al. 2012). The mass of Kepler-93b is detected by Marcy et al. (2014) at 3.8 ± 1.5 M_⊕. This translates to a density of 6.3 ± 2.6 g cm⁻³. Among the worlds smaller than 1.5 R_⊕, the density of Kepler-93b is statistically indistinguishable from those of Kepler-78b, Kepler-36b, and Kepler-10b.

In Figure 6, we show the mass and radius measurements of the current sample of exoplanets smaller than 2.2 R_⊕. The figure does not include the growing population of exoplanets with upper mass limits from RV measurements or dynamical stability arguments. We consider whether there exists evidence for a trend of bulk density with equilibrium temperature for this subset of planets. Their densities range from 1.3 g cm⁻³ for KOI 314.02 (Kipping et al. 2014), to 18 g cm⁻³ for Kepler-100b (Marcy et al. 2014). The exact cutoff of the transition between rocky and gaseous worlds likely resides between 1.5 and 2 R_⊕, with both observations and theory informing our understanding (Marcy et al. 2014 and Mordasini et al. 2012, respectively, among others). Rogers (2014) determined that the value at which half of planets are rocky and half gaseous occurs no higher than 1.9 R_⊕. Whether or not equilibrium temperature also affects this relationship is what we test here. Carter et al. (2012) studied the same parameter space for the known exoplanets with M < 10 M_⊕. They noted a dearth of gaseous planets (densities less than 3.5 g cm⁻³) hotter than 1250 K. They posited that evaporation plays a role in stripping the atmospheres from the planets with higher insolation. In Figure 7, we overplot in gray the region where Carter et al. (2012) observed a lack of planets. We note first that the temperatures of exoplanets have large uncertainties. This is true even for worlds such as Kepler-78b, for which Sanchis-Ojeda et al. (2013) robustly detected the light emanating from the planet alone. For the purposes of this investigation, we assign the temperature reported by the authors in their discovery papers. We use the mean value within the range of possible temperatures, if the authors quote their uncertainty. If temperature was not among the quoted parameters, we use the equilibrium temperature value from the NASA Exoplanet Archive.¹⁸ We assign the range in density uncertainty from the quoted mass and radius uncertainties. As we indicate in Figure 7, we find a flat relationship between equilibrium temperature and density, indicating that temperature is not meaningfully predictive of the densities of planets within this radius range. However, this calculation presupposes no theoretical upper limit to density. Kepler-100b, for example, must constitute 70%–100% iron by mass to be consistent with its measured properties. Marcus et al. (2010) derived an upper limit to iron mass fraction, assuming a collisional stripping history for denser worlds. This framework cannot lead to rocky planets with a fractional iron content above 75% (this fraction varies slightly with radius). We apply the hard prior that each planet cannot possess a density greater than the value computed by Marcus et al. (2010) for this mass, and recompute the best-fit line. Now, we find that the best-fit line is one that predicts increasing density with stronger insolation, but the detection is indistinguishable from a flat line. The prior does meaningfully change the mean density of worlds between 1.0 and 1.5 R_⊕. Taking their reported mass and radius measurements, the mean density of planets with 1.0 <R_⊕ < 1.5 is 10.0 ± 1.5 g cm⁻³, nearly twice as dense as Earth. This range is barely theoretical supportable, since worlds between 1.0 and 1.5 R_⊕ have maximum theoretically plausible densities of 11 g cm⁻³. We would conclude that Kepler-93b, with density of 6.3 ± 2.6 g cm⁻³, lies within the lightest 9% of planets within 1.0 <R_⊕ < 1.5. Earth, with a density of 5.5 g cm⁻³, resides 3σ from the mean density of similarly sized worlds and ought to be rare indeed. In contrast, applying a prior of plausible density returns a mean value of 7.3 ± 0.9 g cm⁻³, and we would conclude that both Earth and Kepler-93b possess average density among similarly sized worlds.

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We compare the mass and radius of Kepler-93b against the theoretical compositions computed by Zeng & Sasselov (2013) for differing budgets of water, magnesium silicate, and iron. Figure 8 depicts the compositional ternary diagram for Kepler-93b, employing the theoretical models of Zeng & Sasselov (2013). A 100% magnesium silicate planet at the radius of Kepler-93b would have a mass of 3.3 M_⊕, which is only 0.3σ removed from the measured mass of 3.8 ± 1.5 M_⊕. All mass fractions of magnesium silicate are plausible within the 1σ error bar, though a pure water and iron planet is barely theoretically supportable. Within the 1σ range in radius and mass that we report, iron mass fractions above 70% are unphysical (Marcus et al. 2010), and the entire remaining composition would need to be water for the planet to contain no rock. Given the age of the star of 6.6 ± 0.9 Gyr, we consider it possible that all water content has been lost within Kepler-93b through outgassing. If this were the case, the planet would comprise 12.5% iron by mass, and 87.5% magnesium silicate.

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To estimate of the likelihood of Kepler-93b's possessing an extended atmosphere, we first examine the criterion for Jean's escape for diatomic hydrogen. Given the temperature of 1037 ± 13 K that we infer based upon the planet's insolation (and an assumption for its albedo of 0.3), the root-mean-squared velocity within the Maxwell–Boltzmann distribution for an H₂ molecule is 3600 m s⁻¹. In contrast, the escape speed from Kepler-93b is 18 000 m s⁻¹. We assume that species with root-mean-square speeds higher than one-sixth the escape speed will be lost from the atmosphere Seager (2010), but we find $v_{\rm {rms}/v_{\rm {esc}}}\approx$ 5 for diatomic hydrogen, and so atmospheres of all kinds are theoretically plausible. We employ the metric created by Kipping et al. (2013) to assign a probability to the likelihood of an extended atmosphere, given the measured mass and radius posterior distributions. They provide a framework to calculate the posterior on the quantity R_MAH, the physical extent of the planet's atmosphere atop a minimally dense core of water as derived by Zeng & Sasselov (2013). They assume that the bulk composition of a super-Earth-sized planet cannot be made of a material lighter than water, and so if the radius of the planet is significantly larger than inferred from this lower physical limit on the density, it must possess an atmosphere to match the observed radius. We apply the radius for Kepler-93b from Table 2 of 1.478$^{+0.019}_{-0.019}$ R_⊕ and the mass of 3.8 ± 1.5 M_⊕ to find a probably of 3% that the planet possesses an extended atmosphere. Kepler-93b is therefore 97% likely to be rocky in composition. In comparison, the published radius and mass posterior distributions for Kepler-36c and Kepler-10b return a probability for an extended atmosphere of <0.01% and 0.2%, respectively. These smaller probabilities, even though Kepler-36c and Kepler-10b are nearly the same size as Kepler-93b, are attributable to the slightly higher mass estimates for the former two worlds. We conclude that there exists a small but non-zero probability that the planet possesses a significantly larger atmosphere than the two planets most similar to it.

Assuming again the Bond albedo (A_B) of 0.3, the expected relative depth due to reflected light is given by δ_ref = A_B(R_p/a)². This value is 0.4 ppm for A = 0.3, but could be as high as 1.4 ppm for an albedo of 1.0. The expected secondary eclipse depth due to the emitted light of the planet is given by δ_em = (R_p/R_⋆)²B_λ(T_p)/B_λ(T_⋆). Assuming again an albedo of 0.3, wavelength of 700 nm (in the middle of the Kepler bandpass) to estimate B_λ(T), δ_em is of order 10⁻¹⁰ and so contributes negligibly to the expected eclipse depth. In Figure 9, we show the phased light curve, binned in increments of 12 minutes, and centered on a phase of 0.5. We have treated these data similarly to the data in-transit, by fitting a linear baseline with time to the observations immediately adjacent to each predicted eclipse and dividing this line from the eclipse observations. The scatter per 15 minute error bar is 2.4 ppm, so for a 173 minute secondary eclipse, the predicted depth error bar at 1σ is 0.7 ppm. All physical depths (that is, <1.4 ppm) should therefore furnish χ² fits within 2σ of one another. We conclude that the data cannot meaningfully distinguish between eclipse depths within the physical range of albedos. Only eclipses deeper than 6 ppm are ruled out at 3σ significance, and this value is too high to be physically meaningful. We repeated the eclipse search in intervals of 1 hr from phases of 0.4–0.6 of the orbital phase (corresponding to 11 hr preceding and following the eclipse time at 0.5 the orbital phase). We produce a new eclipse light curve to test each putative eclipse time, by fitting the baseline to out-of-eclipse observations adjacent to the new ephemeris (to avoid normalizing out any authentic eclipse signal). We report no statistically significant detection of an eclipse signal over this range, and set an upper limit on the eclipse depth of 2.5 ppm with 2σ confidence.

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