Kepler-22b: A 2.4 EARTH-RADIUS PLANET IN THE HABITABLE ZONE OF A SUN-LIKE STAR

Over 31,000 LC observations were obtained between 2009 May 13 and 2011 March 14. SC data were also collected between 2009 August 20 and 2011 March 14. The SC data were essential for determining the fundamental stellar (and thus the planet) parameters from an asteroseismic analysis (p-mode detection) described in Section 4.9. Approximately 790,000 SC observations were collected in this time period. Both LC and SC data are used in our light curve analysis.

The largest systematic errors are due to long-term image motion (differential velocity aberration) and the thermal transients after safe-mode events. After masking transit events, the measured relative standard deviation of the PDC³⁸-corrected, LC light curve is 62 ppm per LC cadence. The propagated formal uncertainty from the instrument and photon shot noise is computed for each flux measurement in the time series. The mean of the LC noise estimates reported by the pipeline is 36 ppm. The 62 ppm total measurement noise includes instrument noise, shot noise, noise introduced by the data reduction, and stellar variability. Both simple-aperture photometry (SAP) and corrected simple-aperture photometry are available at MAST.

The two upper panels in Figure 1 show the full 22 month time series before and after analysis and detrending by the Kepler pipeline, respectively (Twicken et al. 2010; Jenkins et al. 2010b). Intraquarter fluxes were normalized by their median flux in order to reduce the magnitude of the flux discontinuities between quarters. The red curve in the bottom panel is the model fit to the data. For values of the transit and orbital parameters, see Table 1.

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The depth of each transit is checked for consistency with the global model, i.e., that there is no significant evidence for the presence of a doubled-period eclipsing binary (EB). A statistically significant difference in the transit depths would be an indication of a diluted or grazing EB system (Batalha et al. 2010b). The transit events detected in the light curve of Kepler-22b are shown in the fourth panel of Figure 1 where it can be seen that all three show differences <1σ where σ refers to the uncertainty in the fitted transit depths of 10 ppm. Although only three transits are available for this test, there is no evidence of a secondary eclipse and the result of the binarity test is consistent with the planet interpretation.

We check to see whether the transit signal is due to a source other than Kepler-22b using two methods. The first method measures the center-of-light distribution in the photometric aperture and will be referred to as flux-weighted centroids. The flux-weighted centroid method measures the flux-weighted centroid of every observational cadence and fits the computed transit model multiplied by a constant amplitude to the observed flux-weighted centroid motion. The value of the constant that provides the best χ² fit is taken to be the amplitude of the centroid motion. This constant is scaled by the transit depth to estimate the location of the transit source (Jenkins et al. 2010c). The analysis shows no significant motion with a 1σ upper limit of 03 (right-hand column of Table 2).

The second technique uses the difference-image technique (Torres et al. 2011) and is referred to as pixel response function (PRF) fitting. The PRF-fitting method fits the measured Kepler PRF (Bryson et al. 2010) to a difference image. This image is formed from the average in-transit and average out-of-transit (but near-transit) pixel images. The PRF-fitted difference image centroid provides a direct measurement of the location of the transit signal in pixel space. This difference image centroid position is compared with the position of the PRF-fit centroid of the average out-of-transit image. Figure 2 shows an example of both techniques for Kepler-22b in Quarter 1, where the left column shows the observed data and difference image and the right column shows the reconstructed pixels based on the fitted PRF. The agreement between the columns shows that the fit was successful.

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Both centroiding methods begin in pixel coordinates. To perform multi-quarter analysis, the pixel-level results are projected onto the sky in R.A. and decl. coordinates. In the case of flux-weighted centroids, this projection takes place during the χ² fit. The PRF-fitted centroids are computed quarter-by-quarter and the final results are also transformed to celestial coordinates. The quarterly PRF-fitted results are then averaged (minimizing a robust χ² fit to a constant position) to account for quarterly bias due to PRF error and possible crowding (crowding is a minor issue in the case of Kepler-22b; see Section 4.1). The final multi-quarter results for the PRF-fitted results are presented in Table 3 and shown in Figure 3.

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Both centroid methods indicate that the location of the Kepler-22 transit source is less than 1.5σ from Kepler-22, a very strong indication that the transiting object is Kepler-22b. Both methods rule out a source greater than 09 away with a 3σ confidence level.

Figure 4 shows a 12 × 12 view centered on the Kepler-22 taken with the Lick Observatory 1 m Nickel Telescope to map nearby stars. The seeing is ∼15. A companion is seen 5'' to the south and is ∼5 mag fainter than the primary. An analysis of the nearby stars shows that they contribute contaminating fluxes ranging from 0.9% to 1.4%, depending on quarter. The flux light curves have been normalized to account for this contamination prior to planet search and characterization in the SOC pipeline.

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Near-infrared adaptive optics imaging of Kepler-22b was obtained on the night of 2010 July 3 UT with the Palomar Hale 5 m telescope and the Palomar High Angular Resolution Observer (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''. Observations were performed in both the J (λ = 1.25 μm) and Ks (λ = 2.145 μm) filters. The data were collected in a standard five-point quincunx dither pattern of 5'' steps interlaced with an off-source (60'' east) sky dither pattern. The integration time per source was 7.1 s at Ks and 9.9 s at J. A total of 25 frames each were acquired at Ks and J for a total on-source integration time of 3 and 4 minutes, respectively. 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; the widths of the central cores of the resulting point spread functions were FWHM = 010 at Ks and FWHM = 011 at J. The final coadded images at J and Ks are shown in Figure 5.

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Three sources were detected within 10'' of the target. The closest line-of-sight companion is separated from Kepler-22b by 55 to the south and has magnitudes of J = 16.48 ± 0.03 mag and Ks = 16.01 ± 0.02 mag. A second source was detected 55 to the northeast; that source has infrared magnitudes of J = 17.86 ± 0.05 mag and Ks = 17.05 ± 0.04 mag. Finally, a third source was marginally detected 95 to the southeast with infrared magnitudes of J = 20.2 ± 0.5 mag and Ks = 18.9 ± 0.4 mag. Based on the Kp–Ks versus J–Ks color–color relationships from Howell et al. (2011b), we derive K_p = 18.85 ± 0.1 mag, K_p = 19.9 ± 0.1 mag, and K_p = 21.8 ± 0.6 mag for 7, 8, and 10 mag fainter than the primary target in the Kepler bandpass. Together, these stars dilute the light from Kepler-22 by less than 1%.

No other significant sources were detected in the imaging. The sensitivity limits of the imaging were determined by calculating the noise in concentric rings radiating out from the centroid position of the primary target, starting at one FWHM from the target with each ring stepped one FWHM from the previous ring. The 3σ limits of the J-band and K-band imagings were approximately 19 mag and 18 mag, respectively (see Figure 6). The J-band and K-band imaging limits are approximately 9 mag fainter than the target, which corresponds to approximately 10–11 mag fainter than the target in the Kepler bandpass.

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Speckle imaging of Kepler-22 was obtained on the night of 2010 September 21 UT using the two-color speckle camera at the WIYN 3.5 m telescope located on Kitt Peak. The speckle camera obtained 2000 30 ms EMCCD images simultaneously in two filters: R (692/40 nm) and I (880/50 nm). These data were reduced and processed to produce a final reconstructed speckle image for each filter. Figure 7 shows the reconstructed R- and I-band images. 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. Seeing during the measurements was 086. The details of the two-color EMCCD speckle camera and analysis procedure are presented in Horch et al. (2009) and Howell et al. (2011b).

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For the speckle data, we determine if a companion star exists within the approximately 25 × 25 box centered on the target and robustly estimate the background limit in each summed, reconstructed speckle image. The two-color system allows us to have confidence in single fringe detection (finding and modeling identical fringes in both filters) if they exist and rule out companions between 005 and 15 from Kepler-22. We find no companion star within the speckle image separation detection limits to a magnitude limit of 4.24 mag in R (and 3.6 in mag in I), fainter than Kepler-22.

Another type of search was conducted by taking spectra of the target star. If a background EB is the source of a blend, it must be within 5.5 mag of the target or it would be too faint to produce the observed transit amplitudes. The Keck High Resolution Echelle Spectrometer (HIRES) spectra should detect the lines of the offending background star in the spectrum of the target. Keck/HIRES spectra of Kepler-22 were used to search for lines of a background star. No such lines were observed.

In addition, a Keck spectrum was taken with the goal of detecting, or placing limits on, the contributions from any additional neighboring star located within 05. The light from a closeby star, whether background or gravitationally bound to Kepler-22, would fall in the slit of the Keck spectrometer causing its spectrum to contaminate that of the main star, Kepler-22.

To detect any closeby star, we computed the cross-correlation function (CCF) for a large wavelength region of the Keck/HIRES echelle spectrum, from 360 to 620 nm. This region has few telluric lines, leaving the cross-correlation dominated by stellar lines for FGKM stars. For the template we used a spectrum of Ganymede as a solar proxy. The CCF is very smooth, at the 1% level, from such a large wavelength range, and it nicely captures the shape of the thousands of absorption lines within that wavelength region.

We then cross-correlated a library spectrum (from a vast collection of 2000 Keck spectra of FGKM stars) of similar type, in this case HD 90156 (G5 V, T_eff = 5520 K). The goal is to compute the overall shape of the CCF from such stars, enabling a comparison of the CCF from Kepler-22 to the CCF obtained with this comparison star HD 90156. Indeed, the two CCFs came out with very similar shapes within 2% of the peak of the CCF. To detect the presence of any companion in Kepler-22, we took the difference between the CCF (Kepler-22) and CCF (HD 90156) to look for differences. Again, the differences were only typically 1%–2% of the peak of the CCF, i.e., the two CCFs have nearly the same shape, as expected if there is no nearby star.

We then injected a second spectrum of a G star into the spectrum of the Kepler-22. This "fake" secondary spectrum can be adjusted to have any relative RV and any relative intensity. We then compare the difference in CCFs, CCF-(program star)−CCF-(comparison), computed with and without the "fake" secondary spectrum. The fake secondary spectrum causes a "bump" in the CCF that departs from the CCF of the comparison star. Thus, the difference between the two CCFs increases with increasing intensity of the fake secondary star.

We increased the relative intensity of the fake secondary star until the CCF of the fake binary system departs significantly from the CCF of the library comparison star. When the difference in CCF caused by the secondary star becomes larger than the systematic noise of the CCF, the secondary star is detectable.

Figure 8 is a representative plot of CCF-(Kepler-22)–CCF-(HD 90156) both as observed (dots) and also with a fake companion injected (solid line) in the spectrum of Kepler-22. For the case shown, the companion star has an intensity 0.025 of the primary star (i.e., 4 mag fainter), and it is separated by 30 km s⁻¹ (typically of either a background star or bound companion at 1 AU). This fake companion stands out against the systematic noise of the CCF and thus would be detectable. This case represents one example of a secondary spectrum that would have been detectable. Any companion 4 mag fainter (in the optical) and separated by at least 30 km s⁻¹ would have been detected at the 2σ level. At a separation of 10 km s⁻¹, the companion star would have to be brighter than delta-mag <3 to remain detectable. No nearby stars were detected at these levels.

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We obtained 16 high-resolution spectra of Kepler-22 between 2010 August 17 and 2011 August 25 using the HIRES spectrometer on the Keck I 10 m telescope (Vogt et al. 1994) and four others; one at the Nordic Optical Telescope (NOT; fiber-fed Échelle Spectrograph (FIES)), one at Fred Lawrence Whipple Observatory, and two at McDonald Observatory.

High-precision Doppler measurements are used to constrain the mass of Kepler-22b as discussed in Sections 5 and 6 (see Figure 9). The uncertainty of the RV measurements is based on the weighted uncertainty in the mean of the 700 spectral segments (0.2 nm long) that each contribute a separate Doppler measurement. The measured uncertainty is 1.4 m s⁻¹ per RV measurement. In addition, a proper assessment of precision should include an RV "jitter" of 3 m s⁻¹ for such stars, due to surface motions (Isaacson & Fischer 2010). The combination of the internal uncertainty and the jitter, added in quadrature, yields a final precision of ∼4 m s⁻¹ for each measurement (see Table 4). Although the individual RV measurements have uncertainties of ∼4 m s⁻¹, the Markov Chain Monte Carlo (MCMC) analysis (Section 5) yields a posterior upper limit for the 1σ precision of the RV variation, constrained by the known period and ephemeris of the planet, of 4.9 + 6.7/−7.4 m s⁻¹. The absence of a Doppler signal for Kepler-22b is used to compute an upper limit to the mass of this candidate under the planet interpretation.

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Warm Spitzer observations in the near-infrared can also prove useful toward validating Kepler candidates, as shown for Kepler-10c (Fressin et al. 2011) and Kepler-19b (Ballard et al. 2011). The achromaticity of the transit depth, as compared between the optical Kepler photometry and near-infrared Spitzer photometry, provides an alternate means to confirm or reject the planetary nature of the candidate, since an EB will present varying transit depths at different wavelengths unless the constituent stars have nearly identical colors (Torres et al. 2004; Tingley 2004).

We made observations of Kepler-22 during the transit of Kepler-22b using the Infrared Array Camera (IRAC; Fazio et al. 2004) on Warm Spitzer at a wavelength of 4.5 μm on 1 October 2011 UT. These observations comprise part of a GO program (ID 80117, PI: D. Charbonneau) totaling 600 hr. The observations span 17 hr, centered on the 7.4 hr long transit. We gathered the observations using the full-array mode of IRAC, with an integration time of 6 s image⁻¹. We employed the techniques described in Agol et al. (2010) for the treatment of the images before photometry. We first converted the Basic Calibrated Data products from the Spitzer IRAC pipeline (which applies corrections for dark current, flat-field variations, and detector non-linearity) from mega-Janskys per steradian to data number per second. We identified cosmic rays by performing a pixel-by-pixel median filter, using a window of 10 frames. We replace pixels that are > 4σ outliers within this window with the running median value. We also corrected for a striping artifact in some of the Warm Spitzer images, which occurred consistently in the same set of columns, by taking the median of the pixel values in the affected columns (using only rows without an overlying star) and normalizing this value to the median value of neighboring columns. Additionally, we remove the first hour of observations, while the star wandered to the position on the pixel where it spends the remaining hours of the observations.

We estimate the position of the star on the array using a flux-weighted sum of the position within a circular aperture of 3 pixels. We then performed aperture photometry on the images, using both estimates for the position and variable aperture sizes between 2.1 and 4.0 pixels, in increments of 0.1 pixels up to 2.7 pixels, and then at 3.0 and 4.0 pixels. We decided to use the position estimates using a flux-weighted sum at an aperture of 2.7 pixels, which minimized the out-of-transit rms.

We remove the effect of the IRAC intrapixel sensitivity variations, or the "pixel phase" effect (see, e.g., Charbonneau et al. 2005; Knutson et al. 2008) by assuming a polynomial functional form for the intrapixel sensitivity (which depends on the x- and y-positions of the star on the array). We denote the transit light curve f (which depends on time), and we hold all light curve parameters constant except for the transit depth. We use the light curve software of Mandel & Agol (2002) to generate the transit models. The model for the measured brightness f '(x, y) is given by

$$\begin{eqnarray} f^{\prime} &=& f\left({t,\frac{{R_p }}{{R_* }}} \right)(b_1 + b_2 (x - \bar x) + b_3 (x - \bar x)^2\nonumber\\ && + b_4 (y - \bar y) + b_5 (y - \bar y)^2 ) \end{eqnarray} \tag{ 1 }$$

where we include all of the observations (both in- and out-of-transit) to fit the polynomial coefficients and the transit depth simultaneously.

We fit for the polynomial coefficients b₁ through b₅ using a Levenberg–Marquardt χ² minimization. The Spitzer light curve contains significant correlated noise even after the best intrapixel sensitivity model is removed. We incorporate the effect of remaining correlated noise with a residual permutation analysis of the errors as described by Winn et al. (2008), wherein we find the best-fit model f ' to the light curve as given by Equation (1), subtract this model from the light curve, shift the residuals by one step in time, add the same model back to the residuals, and refit the depth and pixel sensitivity coefficients. We wrap residuals from the end of the light curve to the beginning, and in this way we cycle through every residual permutation of the data. We determine the best value from the median of this distribution, and estimate the error from the closest 68% of values to the median. We gathered 8.4 hr of observations outside transit, which is sufficient to sample the systematics on the same timescale as the 7.4 hr transit. Using the residual permutation method on the light curve treated with a polynomial, we find a best-fit transit R_p/R_⋆ to be 0.0184 ± 0.0050, a 3.7σ detection.

We note that we also treated the light curve with the weighted sensitivity function used in Ballard et al. (2010), which proved in that work to produce a time series with lower rms residuals. For this procedure, we do not assume any a priori functional form for the intrapixel sensitivity; rather, we perform a weighted sum over neighboring points for each flux measurement, and use this sum to correct each flux measurement individually. However, during these observations we observed an added component of pointing drift in the X-direction, comparable to the drift in the Y-direction of 0.1 pixels. This drift resulted in few out-of-transit observations that overlap on the pixel with in-transit observations. We found no improvement using the weighted sensitivity method (which depends strongly on the existence of out-of-transit observations to model the portion of the pixel at which the transit occurs), compared to the polynomial method—while the best-fit transit depth was similar, the error bars were 20% larger using the former method.

This value of the transit depth measured with Warm Spitzer of 340 ± 200–160 ppm is in agreement with the depth in the Kepler bandpass of 492 ± 10 ppm (corrected for limb darkening), which favors the planetary interpretation of the light curve. In Figure 10, we show the binned Spitzer light curve (by a factor of 300), with the best-fit transit model derived from the Spitzer observations and the best-fit Kepler transit model (corrected for limb darkening) overplotted. We comment further on the types of blends we rule out by BLENDER in Section 6.

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Spectroscopic observations and analysis to determine the stellar characteristics of Kepler-22 were conducted independently at several observatories. After preliminary vetting and recognizing the importance of this candidate, further analysis was conducted. LTE spectroscopic analysis using the spectral synthesis package SME (Valenti & Piskunov 1996; Valenti & Fischer 2005) was applied to a high-resolution template spectrum from Keck/HIRES to derive an effective temperature, T_eff = 5518 ± 44 K, surface gravity, log g = 4.44 ± 0.06 (cgs), metallicity, [Fe/H] = −0.29 ± 0.06, v sin i = 0.6 ±1.0 km s⁻¹, and the associated error distribution for each of them. To refine the true parameters of the star, we used these observations of the effective temperature to constrain the fundamental stellar parameters derived via asteroseismic analysis.

As an independent check of the values of the SME parameters, we also derived values by matching the spectrum to synthetic spectra (Torres et al. 2002; Buchhave et al. 2010), and in addition we employ a new fitting scheme that allows us to extract precise stellar parameters from the spectra. We report the mean of the spectroscopic classification of one HIRES spectrum, one spectrum from the FIES on the 2.5 m NOT on La Palma, Spain, one spectrum from the fiber-fed Tillinghast Reflector Échelle Spectrograph on the 1.5 m Tillinghast Reflector at the Fred Lawrence Whipple Observatory on Mt. Hopkins, Arizona and two spectra from the Tull Coudé Spectrograph on the 2.7 m the Harlan J. Smith Telescope at the McDonald Observatory Texas acquired between 2009 August and 2011 July. The analysis yields T_eff = 5642 ± 50 K, log g = 4.49 ± 0.10, [m/H] = −0.27 ± 0.08, and v sin i = 2.08 ± 0.50 km s⁻¹, which agrees with the results from the SME analysis of the HIRES spectrum within the uncertainties.

Using the high-resolution spectra acquired with HIRES at Keck Observatory, we have monitored the chromospheric emission via the Ca ii H&K lines. These lines are used to monitor stellar activity, magnetic variability, and rotation rates for main-sequence stars (Baliunas et al. 1995; Noyes et al. 1984). Using stars observed at both Mt. Wilson and Keck observatories, Isaacson & Fischer (2010) calibrated the Ca ii H&K flux measurements from Keck to the Mt. Wilson activity scale. The ratio of the flux in the cores of the Ca ii H&K lines relative to the continuum flux yields a Mt Wilson S-value equal to 0.149 ± 0.004. The S-value is parameterized as log R'_HK, the fraction of flux in the cores of the H&K lines compared to the total bolometric emission. Using log R'_HK allows for comparison of stellar activity for different stellar types regardless of continuum flux near the Ca ii H&K lines. The measured log R'_HK of −5.087 ± 0.05 indicate that the star is inactive, which is consistent with the slow rotation rate found spectroscopically. These results imply that Kepler-22 is an old star.

The data series for Kepler-22b contains 19 months of data taken at a cadence of 1 minute during Kepler-observing quarters Q2 (month 3) through Q8 (month 3; 2010 August 24 to 2010 September 22). At K_p = 11.664, the star is relatively dim, which makes detection of signatures of solar-like (p-mode) oscillations a challenging task because oscillation amplitudes are expected to be below the solar level.

The power spectrum of the light curve does not show a clear excess of power. However, based on asteroseismic analysis of the data using the pipeline developed at the Kepler Asteroseismic Science Operations Center (as described in detail by Christensen-Dalsgaard et al. 2008, 2010; Huber et al. 2009; Gilliland et al. 2011) a p-mode signal can be detected and extracted (see Figure 11). We used a matched filter approach to search for and determine a value for the average large frequency separation of the oscillations spectrum, as well as a frequency for the maximum p-mode power. The p-mode signal is located near 3.15 mHz and has a peak amplitude of approximately 3.4 ppm for radial modes. Several search algorithms were used to estimate the large separation, details of which may be found in Hekker et al. (2010) and Verner & Roxburgh (2011). The large frequency separation for p-modes, which is the prime average asteroseismic parameter (see, e.g., Gilliland et al. 2011 for details), was determined to be 137.5 ± 1.4 μHz, which is only 1.9% ± 1.0% larger than the solar large frequency separation. This value for the large separation indicates that the mean stellar density for Kepler-22 is 3%–4% larger than the solar mean density.

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Using a series of stellar models that fit the large frequency separation, combined with observed properties for metallicity and effective temperature, the stellar radius, mass, chemical composition, and the effective temperature are inferred (Stello et al. 2009; Basu et al. 2010; Metcalfe et al. 2009; Christensen-Dalsgaard et al. 2010; Quirion et al. 2010; Gai et al. 2011). The uncertainties on the estimated stellar properties provided by these fits are based on error propagation through the series of stellar models. The accepted values for the stellar surface gravity, density, mass, radius, and luminosity were derived from this analysis and are listed in Table 1.

As part of the analysis we also searched for individual p-mode frequencies which fit the detected excess power and frequency pattern. The aim of this search was to constrain the small frequency separation which would in principle allow us to estimate the age of the star and the core Helium content. Although we find frequencies that fit the expected p-mode structure, we consider the detection to be too weak to perform a detailed modeling of individual frequencies. The risk of performing a detailed frequency modeling on a weak signal is that this could provide misleading conclusions on the system age. Therefore, no age is shown in Table 1.

The tight restrictions on blends that are able to match the detailed shape of the transit allow us to estimate the expected frequency of these scenarios. We follow a procedure analogous to that described by Fressin et al. (2011). For blends with stellar tertiaries, this frequency will depend on the density of background stars near the target, the area around the target within which such stars would go undetected, and the rate of occurrence of eclipsing binaries. We perform these calculations in half-magnitude bins, with the following inputs: (1) the Galactic structure models by Robin et al. (2003) to estimate the number of stars per square degree, subject to the mass limits allowed by BLENDER; (2) results from our adaptive optics observations to estimate the maximum angular separation (ρ_max) at which companions would be missed, as a function of magnitude difference relative to the target (K_p = 11.664); and (3) the overall frequency of eclipsing binaries capable of mimicking the transits (0.78%; Fressin et al. 2011). Table 5 presents the results. Columns 1 and 2 give the magnitude range for background stars and the magnitude difference compared to the target, Columns 3 and 4 list the mean star densities and ρ_max, and Column 5 (the number of background stars we cannot detect) is the result of multiplying Column 3 by the area implied by ρ_max. Finally, the product of Column 5 and the EB frequency of 0.78% leads to the blend frequencies in Column 6. The sum of these frequencies is given at the bottom, under "Totals".

Similar calculations are performed for scenarios in which the tertiaries are planets instead of stars, and the results are presented in Columns 7–10 of the table. The planet frequencies adopted for this calculation have been taken from the census of candidates detected by Kepler, described below. Adding up the contributions from the two types of blends (0.977 × 10⁻⁶ for stellar tertiaries and 0.184 × 10⁻⁶ for planetary tertiaries), we obtain a total blend frequency of BF = 1.2 × 10⁻⁶.

We next require an estimate of the expected frequency (P_f) of true transiting planets similar to Kepler-22b ("planet prior³⁹"), to assess whether the likelihood of a blend is sufficiently smaller than that of a planet in order to validate Kepler-22b. A reasonable estimate for PF may be obtained by examining the list of candidates from the Kepler Mission itself, which currently contains over 1235 candidates (Borucki et al. 2011) detected using observations gathered from Q1 to Q6. These candidates have been subjected to various levels of vetting, including at least the following: checking for the presence of secondary eclipses that might betray a blend, making sure that the odd- and even-numbered events are of the same depth, looking for consistency in the transit depth from quarter to quarter, verifying that the shape in each quarter is transit-like, and examining the flux centroids to rule out displacements that might be due to a blended star in the photometric aperture. While these candidates have not yet been confirmed because follow-up observations are still in progress, the false-positive rate is expected to be relatively small (typically less than 10%; see Morton & Johnson 2011), so for our purposes we have assumed that all of them represent true planets. In this sample there are 437 cases that have planetary radii within 3σ of the measured value for Kepler-22b (R_p = 2.35 ± 0.12 R_⊕), where we have used this 3σ limit for consistency with a similar criterion adopted above in BLENDER. (No constraint is placed on the orbital period in either case.) Considering the total number of 190, 186 Kepler targets examined between Q1 and Q6, we obtain a planet frequency (P_f) of 437/190, 186 = 2.3 × 10⁻³, which is significantly larger than the blend frequency (B_f), by a factor of about 1900.

It may be argued, however, that the above calculation of both B_f and P_f should be restricted to planets of similar orbital period as Kepler-22b, which is fairly long (290 days), as the rate of occurrence may be different for short-period and long-period planets, and this could alter the odds ratio. If, for example, we limit the periods to be within a factor of two of 290 days, we find that the blend and planet likelihoods are both significantly smaller, and that they indeed change by different amounts. B_f is reduced to 2.9 × 10⁻⁸ (stellar tertiaries) + 1.6 × 10⁻⁸ (planetary tertiaries) = 4.5 × 10⁻⁸, and P_f is reduced to = 5/190, 186 = 2.6 × 10⁻⁵. (The numerator does not include the candidate itself.) The odds ratio then becomes P_f/B_f = 578, which is still large enough that it allows us to validate Kepler-22b with a very high degree of confidence.

Furthermore, we consider the above odds ratio of ∼600 to be a conservative estimate in the sense that it does not include corrections for the fact that shallow transits such as those of Kepler-22b can only be detected in a fraction of the 190, 186 Kepler targets. Incompleteness may affect the blend frequencies as well, but will do so to a much smaller degree because the planets involved in blends are larger (0.32–2.0 R_Jup) and have deeper transits that are easier to detect. Thus, we consider the planet prior adopted above to be a conservative estimate, which strengthens our conclusion on the true planetary nature of Kepler-22b.

Because only an upper limit to the mass of Kepler-22b is available (36 M_⊕, 1σ), any density less than 14.7 g/cc is consistent with the observations, i.e., the composition is unconstrained. Several planets with sizes similar or less than that of Kepler-22b have been discovered that have densities too low for a rocky composition (Lissauer et al. 2011a). However, others, such as Kepler-18b have a size (R_p = 1.98 R_⊕) similar to Kepler-22b and a density (4.9 ± 2.4 g/cc) sufficiently high to imply that such planets could have a solid or liquid surface. Furthermore, model studies of planetary structure often consider rocky planets with masses of 100 M_⊕ or more (Ida & Lin 2005; Fortney et al. 2007). Because there is a possibility that Kepler-22b is a planet with a surface and an atmosphere, a surface temperature will be estimated.

The HZ is often defined to be that region around a star where a rocky planet could have a surface temperature between the freezing point and boiling point of water, or the region receiving the same insolation as the Earth from the Sun (Rampino & Caldeira 1994; Kasting et al. 1993; Heath et al. 1999; Joshi 2003; Tarter et al, 2007).

The radiative equilibrium temperature for a planet is estimated from

$$\begin{equation} T_{{\rm eq}} = \left({\frac{{(1 - \alpha)(R_*)^2 }}{{4\beta {\rm a}^2 }}} \right)^{1/4} T_{{\rm eff}} = 262\;{\rm K} \end{equation} \tag{ 2 }$$

where T_eff is the effective temperature of the star (5518 K), R_* is the radius of the star, α is the planet Bond albedo, a is the planet semimajor axis, β represents the fraction of the surface of the planet that reradiates the absorbed flux (assumed to be 1.0 for a rapidly rotating body with a strongly advecting atmosphere), and T_eq is the radiative equilibrium temperature of the planet. The calculations assume a Bond albedo equal to that of the Earth (0.29). The uncertainty in the computed equilibrium temperature is approximately 22% because of uncertainties in the stellar size, mass, and temperature as well as the planetary albedo, but almost entirely due to the latter.

T_eq is the temperature at which the insolation balances the thermal radiation from the planet. It is equal to the surface temperature Ts only for a planet lacking an atmosphere. A planet with an atmosphere will have a surface temperature above T_eq because of the warming caused by the atmosphere. For example, the greenhouse effect raises Earth's surface temperature by 33 K and that of Venus by approximately 500 K. Furthermore, the spectral characteristics of the stellar flux vary strongly with T_eff. This factor affects both the atmospheric composition and the chemistry of photosynthesis (Kasting et al. 1993; Heath et al. 1999; Segura et al. 2005). Using Equation (2), and assuming a planet with a surface and an atmosphere with thermal properties similar to that of the Earth (which is unlikely) and a Bond albedo of 0.29, the surface temperature of Kepler-22b would be approximately 295 K.