KEPLER-14b: A MASSIVE HOT JUPITER TRANSITING AN F STAR IN A CLOSE VISUAL BINARY

Speckle observations of Kepler-14 were obtained on six different nights between 2010 June and October. The observations were obtained with the dual channel WIYN telescope speckle camera recently described in Horch et al. (2011). The data collection, reduction, and image reconstruction process are described in the aforementioned paper as well as in Howell et al. (2011). The latter presents details of the 2010 season of observations for the KFOP.

A spatially close (03), nearly equal brightness (∼0.5 mag fainter) companion star was easily noted in the reconstructed speckle images. Table 1 gives the weighted mean values for the separation, position angle, and magnitude difference for our six speckle observations. The observations are weighted by the native seeing during the time of the speckle data collection as determined by the data reduction routines when fitting known single point-source speckle standard stars obtained near in time to the Kepler star observations. Figure 2 shows one of the reconstructed images of Kepler-14 with its obvious companion star.

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Near-infrared adaptive optics imaging of Kepler-14 was obtained on the night of 2010 July 3 UT with the Palomar Hale 200 inch 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''. Observations were performed in both the J (λ₀ = 1.25 μm) and K_s (λ₀ = 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 24.5 s at J and 12.75 s at K_s. A total of 20 frames were acquired at J and K_s for a total on-source integration time of 8.5 minutes and 4 minutes, respectively. The individual frames were reduced with a set of custom 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 (PSFs) were FWHM = 015 at J and FWHM = 01 at K_s. The final co-added images at J and K_s are shown in Figure 3.

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Other than the nearby object, two other sources were detected within 10'' of the primary target. The second closest object is separated from Kepler-14 by 55 to the northwest and has infrared magnitudes of J = 18.08 ± 0.04 and K_s = 17.28  ±  0.01. An additional source was detected to the southwest at a distance of 62 having infrared magnitudes of J = 19.00 ± 0.05 and K_s = 18.14 ± 0.04.

The close pair was easily resolved by the adaptive optics at both J and K_s. The pair is separated by ∼028 with a position angle of 142° east of north. The pair has magnitude differences of ΔJ = 0.34 ± 0.01 and ΔK_s = 0.40 ± 0.01. The brighter infrared (and optical) source (component A) is the northwestern star.

High-resolution AO images of Kepler-14 were obtained using the ARIES instrument on the 6.5 m MMT. ARIES is a near-infrared diffraction-limited imager and spectrograph. On 2009 November 8 it was operated in the f/15 mode, with a 40'' × 40'' field of view and a pixel scale of 004 pixel⁻¹. All images of Kepler-14 had exposure times of 10 s, with 16 images in J (in a four-point, 4'' dither pattern) and 19 images taken in K_s (16 in a four-point, 4'' dither pattern, and 3 images at other offsets). The images for each filter were calibrated using standard IRAF²⁸ procedures, and combined and sky-subtracted using the IRAF task xdimsum. The final co-added images at J and K_s are shown in Figure 4.

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In both J and K_s, the binary appearance of Kepler-14 is clear, with the fainter star, B, offset by 029 ± 001 to the southeast. The separation is comparable to the image FWHM (05 in J and 03 in K_s). The relative magnitudes were estimated by PSF fitting, yielding ΔJ = 0.398 ± 0.008 and ΔK = 0.490 ± 0.005.

The delta magnitudes from the Palomar and ARIES AO imaging are thus similar, but the difference of 0.06 mag in J and 0.09 mag in K_s suggests that the accuracy is worse than implied by the formal precision. We combined all the Speckle and AO imaging results for the assessment of the dilution from the nearby companion.

Spectroscopic observations of Kepler-14 were obtained using the FIbre-fed Échelle Spectrograph (FIES) at the 2.5 m NOT at La Palma, Spain (Djupvik & Andersen 2010) as well as High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) mounted on the Keck I telescope on Mauna Kea, Hawaii. We acquired 17 FIES spectra between 2009 August 4 and October 20, 3 of which were not used in the analysis because of very low S/N due to poor observing conditions. One HIRES template spectrum was also observed on 2009 September 10 and used to derive stellar parameters.

For HIRES, we set the spectrometer slit to 086, resulting in a resolving power of λ/Δλ ≈ 55, 000 with a wavelength coverage of ∼3800–8000 Å. We reduced the HIRES spectrum following a procedure based on that described by Butler et al. (1996).

For FIES, we used the medium- and the high-resolution fibers (13 projected diameter) with resolving powers of λ/Δλ ≈ 46, 000 and 67, 000, respectively, giving a wavelength coverage of ∼3600–7400 Å. We used the wavelength range from approximately ∼3900–5800 Å to determine the RVs. The exposure time was approximately 60 minutes yielding an S/N from 20 to 65 pixel⁻¹ (S/N of 38–120 per resolution element) over the wavelength range used. The rather large range in S/N is due to the variation in instrumental throughput and the stellar flux as a function of wavelength, and the lower throughput of the high-resolution fiber.

The FIES spectra were rectified and cross-correlated using a custom-built pipeline designed to provide precise RVs for Échelle spectrographs. The procedures are described in more detail in Buchhave et al. (2010). The science exposures were bracketed by two thorium–argon (ThAr) calibration images taken through the same fiber and extracted using the same pipeline as the science exposures. The ThAr images were then combined to form the basis for the fiducial wavelength calibration. Once the spectra had been extracted, a cross-correlation was performed order by order using the strongest exposure as the template. The orders were cross-correlated using a fast Fourier transform and the cross-correlation functions (CCFs) for all the orders were co-added and fitted with a Gaussian function to determine the RV. Uncertainties of the individual velocities were estimated by $\sigma ={\rm rms}(v)/\sqrt{N}$, where v is the RVs of the individual orders and N is the number of orders.

The light from the 0.5 mag fainter stellar companion (B) dilutes the light of the brighter star (A). In Section 5.1, we use the photometric centroid to determine that it is the brighter star (A) which is undergoing transit and thus is the planet hosting star. The very small angular separation of 029 makes it impossible to separate the two stars on the fiber for spectroscopic observations and it is thus necessary to account for the effect of the dilution on the measured RVs (see Sections 5.2 and 5.7).

In the observed FIES and HIRES spectra, we did not see a composite spectrum in any of the observations. We would easily have been able to identify two cross-correlation peaks from a composite spectrum, if the two stars did not have nearly equal RVs. The combination of the small angular separation of the two stars and the similar RV makes the probability of the two stars being a chance alignment highly unlikely, and we therefore conclude that the two stellar components are gravitationally bound in a wide orbit yielding an undetectable RV offset between the two spectra.

The RV measurements of the combined light of the two components in Kepler-14 are reported in Table 2. The RVs are relative, since they are measured relative to the strongest of the observed spectra adopted as the template. We made a separate estimate of the systemic velocity (the γ velocity) by correlating the observed spectra against the synthetic library spectrum best matching the stellar parameters. We took the mean of these velocities and subtracted the gravitational redshift of the Sun (0.636 km s⁻¹), which is not included in the calculation of the synthetic library spectra. We found the mean γ velocity of Kepler-14b to be γ = 6.53 ± 0.30 km s⁻¹.

We fitted a circular orbit to the RVs reported in Table 2, adopting the photometric ephemeris, leaving only the orbital semi-amplitude, K, and an arbitrary RV offset as free parameters. A plot of the orbital solution is shown in the top panel in Figure 5 with the residuals to the fit shown in the middle panel. The orbital parameters are listed in Table 3. Allowing the eccentricity to be a free parameter only reduced the velocity residuals by a small amount and yielded an eccentricity that was insignificant (e = 0.035 ± 0.020). However, we included the eccentricity in the light curve analysis in Section 5.4 mainly to allow for more realistic uncertainty estimates of the planetary parameters.

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We carried out a bisector span analysis (Queloz et al. 2001; Torres et al. 2007) of the FIES spectra to explore the possibility that the transit-like events are due to an eclipsing binary blended with light from a third star. The bisector spans are plotted in the bottom panel of Figure 5.

Since the observed spectrum is a composite of the planet hosting star and its fainter companion, which we assume is stationary in velocity with respect to the reflex motion induced by the planet, we expect to see a slight asymmetry in the cross-correlation peak as a function of phase. The bisector span is defined as the velocity of the bisector measured near the bottom of the CCF minus the velocity measured near the top, and we would thus expect a positive bisector span when the host star is moving toward us and a negative bisector span when the host star is moving away. In the bottom planet in Figure 5, we see a slight hint of this effect, with the bisector span being predominantly positive around phase 0.25 and predominantly negative around phase 0.75. The amplitude of the bisector spans is significantly less than the RV semi-amplitude and the hint of variation is in the expected direction, which supports the interpretation that the RV variations are due to a planetary companion.

We use a comparison of the photometric centroid in- and out-of-transit data from Kepler to determine which component contains the transit event. These centroids have been measured for quarters 1–6 using two methods: (1) a fit of the transit model to the whitened row and column centroid time series, which provides an average offset in row and column for each quarter, and (2) centroiding of quarterly average in- and out-of-transit images, where the in-transit average is constructed from all in-transit observations in a quarter and the out-of-transit average is constructed from placing the same number of points on each side of each transit event.

Both methods measured essentially identical centroid offsets. These offsets were used to reconstruct the position on the sky of the transiting object, using the methods described in Appendix A of Jenkins et al. (2010a). The final reconstructed transit source location is then the average of the reconstructed transit position over all quarters. The distance of this average reconstructed position from component A is 0.025 ± 0024 (1.04σ) and from component B is 0251 ± 0030 (8.33σ). We conclude that the transiting object is component A.

As noted in Section 3.4, we cannot separate the two stellar components on the fiber of the spectrograph and we thus observed the light from both stars in the spectra. As argued in Section 3.4, we assumed that the two stars are physically associated and that they formed together at the same time. Since the stars have nearly the same temperature due to their position on the HR diagram, we concluded that the small magnitude difference would result in an insignificant change in the host star parameters (see Section 5.3 for details).

We derived stellar atmospheric parameters from both the HIRES template spectrum and the high S/N FIES spectra used for the orbit determination, which can all be used because they are not contaminated by absorption from an iodine cell.

For the HIRES spectrum, we used an analysis package known as Spectroscopy Made Easy (SME; Valenti & Piskunov 1996), along with the atomic line database of Valenti & Fischer (2005). From the HIRES spectrum using SME, we found the following parameters: effective temperature T_eff⋆ = 6395 ± 60 K, metallicity [Fe/H] = +0.12 ± 0.06 dex, projected rotational velocity vsin i = 7.9 ± 1.0 km s⁻¹, and stellar surface gravity log g_⋆ = 4.11 ± 0.10 (cgs).

For the FIES spectra, we derived stellar parameters following Torres et al. (2002) and Buchhave et al. (2010), and in addition we employ a new fitting scheme which is currently still under development and being readied for publication, allowing us to extract more precise stellar parameters from the high S/N FIES spectra. We mention these values here as a check on the SME values, but adopt the SME values because our tools are still under development. From the FIES spectra, we found effective temperature T_eff⋆ = 6378 ± 80 K, metallicity [Fe/H] = 0.00  ±  0.12 dex, projected rotational velocity vsin i = 10.6 ± 1.0 km s⁻¹, and stellar surface gravity log g_⋆ = 4.02 ± 0.21 (cgs). All values are in good agreement with the values derived from the SME analysis, within the uncertainties, except for the value of vsin i.

Global properties of the star including the mass and radius were determined with the help of the stellar evolution models from the series by Girardi et al. (2000). Isochrones for a wide range of ages were compared against the effective temperature and metallicity from the Keck/HIRES spectra, and the mean stellar density, ρ_⋆, as an indicator of luminosity. If we assume a circular orbit, then the mean stellar density is closely related to the normalized semimajor axis a/R_⋆ (see, e.g., Seager & Mallén-Ornelas 2003; Sozzetti et al. 2007), which is one of the parameters solved for in the light curve solutions described below in Section 5.4, and is often more accurate than the spectroscopic log g. In practice we used a/R_⋆ rather than ρ_⋆, and the comparison with the isochrones was coupled with the light curve solutions, which were carried out using the Markov Chain Monte Carlo technique. Specifically, we derived a distribution of stellar properties by comparing the isochrones with each value in the a/R_⋆ chains paired with values for the temperature and metallicity drawn from Gaussian distributions centered on the spectroscopically determined values and their errors.

The presence of the visual companion detected in our high-resolution imaging adds a complication, as the extra flux reduces the depth of the transit and affects its overall shape in subtle ways, biasing the a/R_⋆ parameter. The impact of this extra dilution depends on the magnitude difference of the companion in the Kepler band (ΔKp), which we expect to be close to (but not necessarily the same as) the measured magnitude differences in other passbands (ΔV, ΔR, ΔI, ΔJ, and ΔK_s). We therefore proceeded by iteration, in parallel with the light-curve solutions. We initially ignored the dilution effect on a/R_⋆, and inferred the absolute magnitude of the target in the Kp band from the best-fit isochrone. Assuming the companion is physically associated and the two stars share the same isochrone, we then determined its mass along the isochrone with the condition that the magnitude difference in V be exactly equal to the measured value. We then read off the ΔKp value directly from the isochrone. We repeated this using each of the other magnitude difference measurements (taking those in J and K_s from the MMT and Palomar to be independent), and we averaged the resulting seven values of ΔKp to obtain 0.45 ± 0.10 mag. With the corresponding relative flux F_B/F_A a new light curve solution was carried out, leading to an improved a/R_⋆ distribution. This, in turn, was compared once again with the isochrones and led to a slightly revised brightness difference of ΔKp = 0.44 ± 0.10 mag. A further iteration did not change this significantly.

As described in Section 5.2, we have determined the host star parameters from the composite spectra of the primary star A and the fainter companion B, since it is not possible to separate the two stars on the fiber/slit of the spectrographs. We estimate that the adopted magnitude difference of ΔKp = 0.44 ± 0.10 mag does not significantly affect the derived spectroscopic stellar properties of the host star A. The companion star B is estimated to be only 30 K hotter and have a stronger surface gravity of 0.15 dex compared to the host star. We therefore choose to ignore the effect of the dilution on the stellar parameters of the host star.

The resulting properties of the host star are listed in Table 3, in which the values correspond to the mode of the distributions and the uncertainties reported are the 68.3% (1σ) confidence limits defined by the 15.8% and 84.2% percentiles in the cumulative distributions.

We estimated the distance to Kepler-14 based on isochrones by comparing against the measured magnitudes from the Kepler Input Catalog (Brown et al. 2011). We fitted the spectral energy distribution with magnitudes for the two stars taken from the Girardi isochrones resulting in a distance estimate of 980 pc. For an average angular separation of 029 the semi-major axis of the visual pair is approximately 280 AU, and with mass estimates of 1.51 M_☉ and 1.39 M_☉ for the two stars, the corresponding period is on the order of 2800 years.

We modeled the folded transit light curve assuming spherical star and planet having radius ratio R_p/R_⋆. The second star adds its light to the total light curve with the observed flux ratio between stars B and A being F_B/F_A. The planet was constrained to a circular Keplerian orbit parameterized by a period P, a normalized semi-major axis distance a/R_⋆, and an inclination to the sky plane i.

The normalized transit light curve, f(t), was calculated to be

$$\begin{eqnarray} f(t) & = & 1-\lambda \left[z(t)/R_\star,\frac{R_p}{R_\star }, u_1, u_2\right]\!{\bigg/}\!\left(1+\frac{F_B}{F_A}\right), \end{eqnarray} \tag{ 1 }$$

where z(t) is the sky-projected separation of the centers of the star and planet and λ is the fraction of the stellar disk blocked by the planet, given analytically by Mandel & Agol (2002). The limb-darkening coefficients u₁ and u₂ parameterize the radial brightness profile, I(r), of a star as

$$\begin{eqnarray} \frac{I(r)}{I(0)} & =& 1-u_1 \big(1-\sqrt{1-r^2}\big)-u_2 \big(1-\sqrt{1-r^2}\big)^2. \end{eqnarray} \tag{ 2 }$$

The continuously defined model, f(t), was numerically integrated before being compared with the long cadence Kepler light curve. In detail, for each measured time, t_j, we take n_j uniform samples t_{j, k} = t_j + kΔt_j − τ_int/2, separated by Δt_j = τ_int/n_j, over the long cadence integration interval of τ_int = 29.4 minutes. The flux at t_j was found by computing the Gaussian quadrature of the continuous model fluxes f(t_{j, k}). In practice, we took n_j = 20 for all times.

We determined the best-fit model to the data by minimizing the χ² goodness-of-fit statistic including Gaussian penalties to restrict the flux ratio F_B/F_A, ecos ω, and esin ω to agree with the observed constraints:

$$\begin{eqnarray} \chi ^2 &=& \sum _{s} \frac{\left({\mathcal F}_s-F_s\right)^2}{\sigma ^2}+\frac{\left(F_B/F_A-0.667\right)^2}{0.061^2} \nonumber \\ &&+\frac{\left(e \sin \omega -0.035\right)^2}{0.017^2}+\frac{\left(e \cos \omega -0.0006\right)^2}{0.0099^2}, \end{eqnarray} \tag{ 3 }$$

where F_s is the measured flux at time t_s and σ is the expected statistical error in the flux measurements. We selected σ = 82 ppm such that the reduced-χ² was unity for the best-fit solution.

We determined the posterior probability distribution for the fitted parameters by using a Differential Evolution Markov Chain Monte Carlo algorithm (Ter Braak 2006) with a Metropolis–Hastings jump condition and a jump acceptance probability conditional on the likelihood $\mathcal {L^{\prime }} \propto \exp (-\chi ^2/2)$. We computed chains for a parallel population of 90 members through ≈2 million generations requiring that approximately 25% of jumps were accepted on average among all population members. The chains were checked for adequate mixing and convergence by visual inspection and by observing that the number of links was much larger than the autocorrelation length (equal to the number of links at which the chain autocorrelation drops below one half) for any selected parameter. We report the 15.8% and 84.2% values of the cumulative distribution for each parameter, marginalizing over the remaining parameters.

We used a transit light-curve model multiplied by instrumental decorrelation functions to measure the transit parameters and their uncertainties from the Warm-Spitzer data as described in Désert et al. (2011b). We computed the transit light curves with the IDL transit routine OCCULTNL from Mandel & Agol (2002). This function depends on one parameter: the planet-to-star radius ratio R_p/R_⋆. The orbital semi-major axis to stellar radius ratio (system scale) a/R_⋆, the impact parameter b, and the time of mid-transit T_c were fixed to the values derived from the Kepler light curve and corrected for the dilution (see Table 3). We assumed that the limb darkening is well approximated by a nonlinear law at infrared wavelengths with four coefficients (Claret 2000) that we set to their values computed by Sing (2010).

The Spitzer/IRAC photometry is known to be systematically affected by the so-called pixel-phase effect (see, e.g., Charbonneau et al. 2005; Knutson et al. 2008). This effect is seen as oscillations in the measured fluxes with a period of approximately 70 minutes (period of the telescope pointing jitter) and an amplitude of approximately 2% peak-to-peak. We decorrelated our signal in each channel using a linear function of time for the baseline (two parameters) and a quadratic function of the PSF position (four parameters) to correct the data for each channel. We performed a simultaneous Levenberg–Marquardt least-squares fit (Markwardt 2009) to the data to determine the transit and instrumental model parameters (seven in total). The errors on each photometric point were assumed to be identical and were set to the rms of the residuals of the initial best fit obtained. To obtain an estimate of the correlated and systematic errors (Pont et al. 2006) in our measurements, we used the residual permutation bootstrap, or "Prayer Bead," method as described in Désert et al. (2009). In this method, the residuals of the initial fit are shifted systematically and sequentially by one frame, and then added to the transit light-curve model before fitting again. We allowed asymmetric error bars spanning 34% of the points above and below the median of the distributions to derive the 1σ uncertainties for each parameter as described in Désert et al. (2011a).

We compute the theoretical dilution factor by extrapolating the K_s-band measurements to the Spitzer bandpass at 4.5 μm. We estimate that 36% of the photons recorded during the observation come from the companion star. We conclude that the presence of the contaminating star decreases the effective transit depth of Kepler-14 by a factor 0.61. We measure the transit depth (limb-darkening removed) of Kepler-14 at 4.5 μm and find 1722⁺¹²⁷ _{− 138} ppm uncorrected for the dilution. This corresponds to R_p/R_⋆ = 0.0415^+0.0015 _{− 0.0017}. Applying the dilution correction, we find R_p/R_⋆ = 0.0531^+0.0019 _{− 0.0021}, which is consistent with the value derived from the Kepler photometry at better than the 2σ level. Our Spitzer observations provide an independent confirmation that the transit signal is achromatic, which supports the planetary nature of Kepler-14b.

The measured RVs of the host star (A) are affected by the light contributed by the companion star (B), because the spectrum of B is assumed to be stationary in velocity, while the spectrum of A is Doppler-shifted due to the gravitational pull of the planet. The amplitude of the observed RVs will thus be smaller than if the light from A had not been diluted, since the peak of the CCF from which we derive the RVs will be pulled toward the stationary CCF of B.

In order to assess the dilution effect on the RVs and thus the semi-amplitude of the orbit, we modeled the effect using the observed spectrum of Kepler-14. We shifted the observed spectrum in 50 m s⁻¹ increments and co-added the shifted spectrum, representing star A, with the same observed spectrum divided by a constant to simulate the stationary companion B at different flux ratios of f = F_B/F_A. We analyzed this composite spectrum using the same tools used to extract the RVs for the orbit.

The relation between the artificially induced RV shifts, V_in, and the resulting "measured" RV shifts of the composite spectrum, V_out, is linear at given flux ratio, as expected: V_out = a_fV_in, where a_f is the slope at a given flux ratio. V_in thus represents the true (corrected) RVs of the host star (V_cor) and V_out represents the observed RVs of the host star (V_obs).

We carried out this analysis at different flux ratios, fitting the linear relation between V_in and V_out, and thus obtaining the slope a_f at each flux ratio. We then fitted the slopes, a_f, themselves as a function of flux ratio with a third-order polynomial. This enables us to calculate the dilution effect for the system at any flux ratio:

$$\begin{eqnarray} &&V_{{\rm cor}} = \frac{V_{{\rm obs}}}{a_f} = \frac{V_{{\rm obs}}}{c_0 + c_1f+c_2f^2+c_3f^3}, \end{eqnarray} \tag{ 4 }$$

where f is the flux ratio of the two stars and c_i are the polynomial coefficients of the third-order polynomial. The dilution effect on the corrected RVs as a function of flux ratio can be seen in Figure 7. The horizontal dotted line represents the observed semi-amplitude of the system and the vertical dotted line represents the adopted flux ratio of the two stars. As a sanity check, we see that if the host star and companion have similar brightness, the corrected RV is about twice the observed RVs and at large magnitude differences, the corrected RV approaches the observed RVs, as expected.

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The observed orbital semi-amplitude of Kepler-14 is K_obs = 401.7 ± 7.1 m s⁻¹. Since the two stars are nearly the same temperature, the dilution changes only minutely as a function of wavelength. We thus used the magnitude difference of ΔKp = 0.44 ± 0.10 in all orders and found the corrected semi-amplitude of the orbit to be K_cor = 682.9^+26.7 _{− 24.6} m s⁻¹. We propagated the uncertainty of the adopted magnitude difference and added it in quadrature with the uncertainty of the uncorrected semi-amplitude. The correction assumes that the two stars have the same rotational broadening of the lines in their spectra and that the velocity shift between the two stars (due to motion in their wide orbit around each other) is small enough to be ignored. In the unlikely case that the companion star is rotating very rapidly, our model would overcorrect the velocity amplitude due to dilution. However, there is no evidence in the CCF for a broad secondary peak.

The dilution of the nearly equal magnitude stellar companion significantly affects the derived planetary parameters of Kepler-14b. The contamination affects the observed transit light-curve depth and therefore the inferred radius ratio. In addition, this dilution has a significant effect on the light-curve profile affecting the inferred geometric orbital parameters, most notably the normalized semi-major axis, a/R_⋆. If dilution effects are neglected, the mean stellar density estimate—which is acutely sensitive to a/R_⋆—used in conjunction with spectroscopic stellar constraints will yield significantly inaccurate derived stellar properties.

If we assume that the flux contribution from B is zero (i.e., F_B/F_A = 0 and K_obs = 401.7 ± 7.1 m s⁻¹), we find that R_{p, nocorr} = 1.036^+0.075 _{− 0.084} R_J. Using the derived magnitude difference ΔKp = 0.44 ± 0.10, however, we find the planetary radius to be R_p = 1.136^+0.073 _{− 0.054} R_J, which is almost 10% larger.

As described in Section 5.7, the orbital semi-amplitude is also significantly affected by the dilution. Using the observed orbital semi-amplitude of K_obs  =  401.7 ± 7.1 m s⁻¹, the uncorrected mass of Kepler-14b is M_{p, nocorr} = 5.14^+0.15 _{− 0.16} M_J. After correction for dilution, the semi-amplitude increases to K_corr  =  682.9^+26.7 _{− 24.6} m s⁻¹, which in turn leads to a planetary mass that is significantly larger than before (by ∼60%): M_p = 8.40^+0.35 _{− 0.34} M_J.

The effect of the dilution is much greater on the mass than on the radius of the transiting planet. As described above, the dilution of the observed transit light curve changes not only the depth of the transit, but also the light-curve profile which in turn affects the inferred stellar density estimate. The radius of the planet is thus not affected greatly by the dilution, because these two effects work against each other. The stellar mass, however, is not strongly affected by the dilution and the effect on the planetary mass therefore comes almost entirely from correction of the orbital semi-amplitude.