CONSTRAINING THE OBLATENESS OF KEPLER PLANETS

The shape of an oblate planet can be quantified by the flattening, or oblateness, parameter f,

$$\begin{equation} f = \frac{R_{\rm eq}-R_{\rm pol}}{R_{\rm eq}} , \end{equation} \tag{ 1 }$$

where R_eq and R_pol are the equatorial and polar radii, respectively (Murray & Dermott 1999). We also define the effective mean radius to be

$$\begin{equation*} R_{p} \equiv \sqrt{R_{\rm eq} R_{\rm pol}}, \end{equation*}$$

which is the radius of a spherical planet of the same cross-sectional area. The planet's spin obliquity, θ, is defined as the angle between the polar axis of the planet and the orbital angular momentum vector (Carter & Winn 2010a).

The flattening f can be related to the rotation period P_rot of a planet with mass M_p by

$$\begin{equation} P_{\rm rot} = 2\pi \sqrt{\frac{R_{\rm eq}^3}{GM_p (2f-3J_2)}} , \end{equation} \tag{ 2 }$$

where J₂ is the quadrupole moment (Carter & Winn 2010a).

However, what we can measure from transit is not the true flattening and obliquity, but their projected components on the plane of the sky, f_⊥ and θ_⊥, meaning that the constrained oblateness from transit is only a lower limit on the true oblateness. The relations between f, θ and f_⊥, θ_⊥ can be found in Carter & Winn (2010a). The definition for the sign of θ_⊥ is shown in the upper panel of Figure 2.

The transit light curve of an oblate planet is described by the intersection between an ellipse and a circle. For the ingress and egress regions of the light curve, where the deviation from a spherical planet model light curve is greatest, we compute the intersection region using the quasi-Monte Carlo integration algorithm introduced by Carter & Winn (2010a).⁶ For the in-transit regions, where the planet is fully within the stellar disk, we replace the oblate planet signal with a spherical planet with the same cross-sectional area. Since R_p  ≪  R_⋆, we assume the spatial limb darkening variation within the area covered by the planet ellipse is negligible. In fact, the in-transit signal induced by limb darkening is <10 ppm even for planets with substantially large R_p/R_⋆ (Carter & Winn 2010a), which is insignificant for the purpose of model fitting compared with the noise level (∼500 ppm) of Kepler short-cadence light curves. By replacing the oblate planet with a spherical planet for the in-transit region, we are able to speed up the original algorithm introduced by Carter & Winn (2010a) by a factor of 10. We demonstrate in Figure 1 the theoretical oblateness-induced signal for a Jupiter-size planet in a 100 day orbit around a Sun-like star with uniform surface brightness (i.e., no limb darkening effect). We assume the planet has a Saturn-like oblateness (f_⊥ = 0.1, note that f_saturn = 0.098). Signals from different projected planet spin obliquities θ_⊥ (0°, 45°, 90°) are plotted for comparison. We also add an example signal from the oblate planet transiting a limb-darkened stellar surface in Figure 1, with the in-transit signal considered, to show the influence of the limb darkening effect.

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As Figure 1 shows, the oblateness-induced signal is at least an order of magnitude greater over ingress and egress than the in-transit part. The theoretical maximum amplitude of the signal at the ingress/egress part can be estimated by

$$\begin{equation} {\rm Signal}_{\rm max} \approx \frac{f_\perp }{2 \pi } \left(\frac{R_{p}}{R_\star }\right)^2 = 160\ {\rm ppm} \left(\frac{R_{p}/R_\star }{0.1}\right)^2 \left(\frac{f_\perp }{0.1} \right), \end{equation} \tag{ 3 }$$

which can be achieved when the impact parameter satisfies the following condition:

$$\begin{equation} b = -\min [{\rm abs}(\sin {\theta _\perp }), {\rm abs}(\cos {\theta _\perp })]. \end{equation} \tag{ 4 }$$

The derivation for this expression of the maximum achieved signal is shown in detail in Appendix A. Although Equation (3) is useful in estimating the amplitude of an oblateness-induced signal, one should keep in mind that the recovered signal from light-curve modeling may have a smaller amplitude due to slight changes in those from other fitting parameters, especially when the oblateness-induced signal is mirror symmetric (Barnes & Fortney 2003). Space-based observations, such as Kepler, can achieve photometric precision comparable to this signal amplitude (Jenkins et al. 2010; Gilliland et al. 2010; Murphy 2012), making possible the detection of planetary oblateness when multiple transits are observed.

It is also useful to have a comprehensive understanding of the amplitude of the oblateness signal for arbitrary projected obliquity angles, θ_⊥, and impact parameters, b₀. To achieve this, we simulated theoretical light curves on a grid of θ_⊥ and b₀ with fixed f_⊥ and R_p/R_⋆. For most of the ranges of θ_⊥ and b₀ we consider, assuming a moderate projected oblateness f_⊥ (≲ 0.2), that Equation (3) can be written as

$$\begin{equation} {\rm Signal}_{\rm max} (R_{p}/R_\star,f_\perp,\theta _\perp,b_0) \approx \frac{f_\perp }{2 \pi } \left(\frac{R_{p}}{R_\star }\right)^2 g(\theta _\perp,b_0). \end{equation} \tag{ 5 }$$

In Figure 2 we show an example of Signal_max(R_p/R_⋆, f_⊥, θ_⊥, b₀) with R_p/R_⋆ = 0.1 and f_⊥ = 0.1. This figure tells us that the amplitude of an oblateness signal reaches maximum at θ_⊥ ∼ 0° for a system that is very close to edge-on, and at larger (absolute) projected obliquities as the system becomes more inclined. Particularly, for b₀ ≈ ±0.7, the signal will have maximum amplitude at θ_⊥ ≈ ±45°, where the shape of the signal becomes the most asymmetric (see Figure 1). This is why the detectability of oblateness is maximized at b ≈ 0.7, as determined numerically by Barnes & Fortney (2003). The values of θ_⊥ where the maximum amplitude of signal is achieved are also those where the oblateness can be most tightly constrained. We will further demonstrate this point in the modeling of simulated signal and real data.

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If limb darkening is considered, the maximum amplitude decreases by a factor of 6(1 − u₁ − u₂)/(6 − 2u₁ − u₂), where u₁ and u₂ are the quadratic limb darkening coefficients (Claret 2000).

As discussed in Section 2.1, for a Jupiter-size planet (R_p/R_⋆ = 0.1) with Saturn-like oblateness (f = 0.1), the maximum signal amplitude due to oblateness is ∼160 ppm.

The signal-to-noise ratio (S/N) for a single transit in the white noise limit can be estimated by

$$\begin{equation} {\rm S/N} = \left(\frac{f_\perp }{0.1}\right) \left(\frac{R_p/R_*}{0.1}\right)^2 \left(\frac{\rm ootv}{160\ {\rm ppm}}\right)^{-1} \sqrt{2N_{\rm p}}, \end{equation} \tag{ 6 }$$

in which, ootv is the out-of-transit variation amplitude (per point) and N_p denotes the number of observations during the ingress/egress time. The duration of the ingress/egress time τ can be estimated by

$$\begin{eqnarray} \tau &\approx & \frac{2R_{p}}{v_{\rm orb}} = 50.4\ {\rm min{\rm ute}s} \left(\frac{R_{p}/R_\star }{0.1} \right) \left(\frac{R_\star }{R_\odot }\right) \nonumber\\ && \times \left(\frac{P_{\rm orb}}{100\ \rm days}\right)^{1/3} \left(\frac{M_\star }{M_\odot }\right)^{-1/3} \ . \end{eqnarray} \tag{ 7 }$$

For a 12 magnitude star, the Kepler long-cadence data can typically achieve a photometric precision of 100 ppm. Assuming white noise, the equivalent out-of-transit scattering (ootv) of the corresponding short-cadence data is ∼550 ppm, which corresponds to S/N ∼3. We note that this S/N estimation is almost exact for short-cadence data, with the oblateness-induced signal sufficiently mapped out by the 1 minute cadence. Transits having only long-cadence data will typically have a much shallower signal than the theoretical value, given that the signal duration is roughly comparable to the long integration time (30 minutes). For example, for a Jupiter-size planet in a 15 day orbit, the ingress/egress duration is ∼30 minutes, and thus the oblateness signal is strongly suppressed in the long-cadence data. For this reason, we do not consider the long-cadence data in the present work.

With Kepler light curves from Q1–Q16 (∼1400 days), we expect a planet with a 100 day orbital period and a Saturn-like oblateness to be detected with a S/N ∼11 in the white noise limit. In the red noise limit, S/N would scale with the number of transits instead of the number of data points observed, which would reduce the expected S/N severely.

To demonstrate the detectability of the oblateness signal, we perform a signal injection and recovery exercise using out-of-transit light-curve segments from the KOI 368 system. KOI 368 is a photometrically quiet A dwarf hosting a transiting M dwarf with P = 110.32 days, R_p/R_* = 0.084, i = 89.36 (b = 0.64), and limb darkening coefficients u₁ = 0.23, u₂ = 0.29 (Zhou & Huang 2013). The KOI 368 light curves and system properties resemble the transits of an optimal case, long-period planet candidate. We inject 12 transits of an oblate planet with the same system parameters as KOI 368.01, and f_⊥ = 0.1 and θ_⊥ = 45°, onto random segments of the SAP short-cadence light curves of KOI 368. The injected transit yields an oblateness signal with amplitude 80 ppm. In contrast, the out-of-transit variation amplitude of the light curves is 300 ppm.

The simulated light curve is then processed by our standard detrending pipeline (see Section 3) and fitted with the MCMC method. For comparison, we also report the results with standard Mandel–Agol transits injected in the same segments and detrended in the same way. In each case, the simulated data is fit first by a standard transit model and then an oblate planet model. To demonstrate the least and most optimal cases, we present the injection and recovery for the case of a single transit and the case of all 12 full transits for the KOI 368 system. The light curves and the residuals to the standard and oblate planet models (for the 12 transits case) are shown in Figure 3.

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The signal is detected at a significant confidence level in our oblate planet model fitting. The original injected oblateness f_⊥ is recovered in the single transit case. The left panels in Figure 4 show the posterior distributions from the fitting of a single injected short-cadence transit that contains the oblateness signal. Although the projected obliquity suffers the degeneracy between f_⊥ and θ_⊥, and is therefore not well recovered, the asymmetry between θ_⊥ > 0 and θ_⊥ < 0 implies that the true value should be positive.

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For comparison, the posteriors of the null case are shown in the right panels of Figure 4. The null case is also an example that can be used to demonstrate the ability of constraining the oblateness with Kepler data. From the fitting of a single injected transit, an oblateness as large as that of Saturn can be ruled out if the planet has moderate obliquity. However, the constraint we can place at θ_⊥ = 0°, ±90° is very weak, because for this case b = 0.64 and the oblateness signal can still be very small even for large oblateness, as shown in Figure 2.

The correlations between oblateness parameters f_⊥, θ_⊥, and other transit parameters derived from the MCMC fitting of a single injected oblate planet transit are shown in Figure 5. f_⊥ appears to be correlated with (R_p + R_⋆)/a, R_p/R_⋆, and i₀, and shows long tails toward large oblateness in each plot, because θ_⊥ is only weakly constrained. These correlations suggest that one might get systematic biases on the planetary parameters if an oblate planet is fit with a spherical model. It is worth noting that the f_⊥ is hardly correlated with the limb darkening coefficients q₁ and q₂.

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When the number of transits is increased to 12, the injected signal is recovered with a significantly higher confidence level, as shown in Figure 6. We can see the oblateness signal visually in the residual (from all 12 transits) of the fitting using only the standard Mandel–Agol model in Figure 3. For the null case, we can successfully rule out an oblateness as small as that of Uranus for most θ_⊥. The injected parameters, and the recovered parameters from standard model and oblateness model are shown in Table 1.

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HAT-P-7b orbits around a slightly evolved F star in a 2.2 day orbit (Pál et al. 2008). A planet in such a close-in orbit is believed to be tidally locked, and therefore HAT-P-7b should not exhibit rotation-induced oblateness. We include this planet in our sample for a number of reasons. First, this planet has been observed with short-cadence mode for the entire Kepler operation time. There are in total 591 full transits of HAT-P-7b within the Q1 to Q16 short-cadence data. The 1 minute cadence out-of-transit variation is 234 ppm. This allows us to directly compare our modeling result with that of HD 189733b, the only other hot-Jupiter with its oblateness constrained by observations (Carter & Winn 2010a). Furthermore, there are various other photometric effects present in the HAT-P-7 light curves, such as the intrinsic stellar variability, planetary phase variations, stellar ellipsoidal variations (Borucki et al. 2009), and possible planet-induced gravity darkening effects (Morris et al. 2013). An analysis of HAT-P-7b can also help us understand how these effects influence our oblateness model fitting procedure. We list the physical parameters of the host star in Table 2.

Table 2. Physical Parameters about the Host Star HAT-P-7 (KOI 2.01, KIC 10666592)

Parameters	Values
Teff (K)	6350 ± 80a
log g (dex)	$4.07^{+0.04}_{-0.08}$a
[Fe/H] (dex)	+0.26 ± 0.08a
M⋆ (M☉)	$1.47^{+0.08}_{-0.05}$a
R⋆ (R☉)	$1.84^{+0.23}_{-0.11}$a
vsin i⋆ (km s−1)	3.8 ± 0.5a
J-band mag	9.555
H-band mag	9.344
K-band mag	9.334

Note. ^aAdopted from Pál et al. (2008).

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Fitting all 591 short-cadence transits is extremely computationally expensive. It may also produce questionable results given the systematic transit depth variations between different quarters (Van Eylen et al. 2013). We therefore fit the light curve of HAT-P-7b in three groups of 100 consecutive orbits each. The MCMC chains from the three groups are combined afterward to arrive at the global best-fit parameters. Examining the individual group results will also allow us to better judge the effect of stellar variability and red noise on the consistency of our results. Our best-fit parameters for both standard and oblateness fittings, together with the initial parameters from Kepler, are listed in Table 3. We show in Figure 7 the upper limits of f_⊥ for different θ_⊥. The overall 68% upper limit on the oblateness is 0.067. The oblateness is only weakly constrained around θ_⊥ = 0° and ±90°, which results in the small bump around 0.1 in the posterior distribution of f_⊥. However, if the planet is moderately tilted, an oblateness as large as that of Neptune (0.017) can be ruled out with 95% confidence. This constraint is comparable to that placed on HD189733b by Carter & Winn (2010a), who ruled out a Uranus-like oblateness (0.023; 95% confidence) for most of the projected obliquity angles and placed an overall 95% upper limit of 0.058 on the planet.

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Table 3. Fitting Parameters of HAT-P-7b

Parameters	Kepler	Standard Model	Oblate Model
q1a	0.28 ± 2b	0.28 ± 1	0.28 ± 1
q2a	0.33 ± 1b	0.33 ± 1	0.33 ± 1
Porb (days)	2.2047354 ± 1	2.2047350 ± 3	2.2047349 ± 3
t0 (BJDUTC-2454000)	954.35780 ± 2	954.35860 ± 2	$954.358608^{+16}_{-15}$
(Rp + R⋆)/a	⋅⋅⋅	0.2595 ± 4	$0.2597^{+16}_{-4}$
a/R⋆	4.7	⋅⋅⋅	⋅⋅⋅
Rp/R⋆	0.07545 ± 2	0.07748 ± 4	$0.07750^{+9}_{-7}$
i0 (deg)	88.2	83.13 ± 5	$83.11^{+9}_{-18}$
f⊥	⋅⋅⋅	⋅⋅⋅	<0.067

Notes. ^aq₁ and q₂ are calculated using Equations (8) for the "Kepler model." ^bAdopted from Van Eylen et al. (2013).

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KOI 686.01 (KIC 7906882) is a planetary candidate with an orbital period of 52.51 days and a planet-to-star radius ratio R_p/R_⋆ = 0.12, orbiting around a G-type star with a Kepler-band magnitude of 13.58. The physical parameters of the host star are listed in Table 4.

We assign epoch zero to the first transit observed by Kepler. KOI 686.01 has only one short-cadence full transit at epoch +12. The out-of-transit variation amplitude of this transit is 851 ppm. After reducing the data according to Section 3, we fit this short-cadence transit to a standard transit model, using the parameters given by Kepler as initials. With the best-fit parameters given by the standard transit model, we then set f_⊥ and θ_⊥ free to allow the oblateness fitting. Our best-fit parameters for both standard and oblateness fittings, together with the initial parameters from Kepler, are listed in Table 5.

Table 5. Fitting Parameters of KOI 686

Parameters	Kepler	Standard Model	Oblate Model
q1	0.46 ± 7	0.29 ± 15	$0.32^{+24}_{-11}$
q2	0.34 ± 2	0.39 ± 29	$0.34^{+40}_{-26}$
t0 (BJDUTC-2454000)	1004.6737 ± 1	1004.6743 ± 2	1004.6743 ± 2
(Rp + R⋆)/a	⋅⋅⋅	0.0108 ± 2	$0.0107^{+3}_{-2}$
a/R⋆	109.3 ± 4.2	⋅⋅⋅	⋅⋅⋅
Rp/R⋆	0.1166 ± 5	0.121 ± 2	0.12 ± 2
i0 (deg)	89.4	89.60 ± 2	$89.61^{+4}_{-2}$
f⊥	⋅⋅⋅	⋅⋅⋅	<0.251

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We show the posterior plots from the oblateness fitting in Figure 8. Based on the best-fit parameters, the impact parameter b₀ of KOI 686.01 transiting the host is determined to be 0.64. For this b₀, the oblateness should be weakly constrained at θ_⊥ = 0° and ±90° according to Figure 2. Therefore the result is consistent with our expectation.

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No oblateness signature is detected for KOI 686.01. The overall projected oblateness f_⊥ can only be constrained to be <0.25 (68% limit). For most θ_⊥ ≠ 0°, ±90°, we can rule out a planet more oblate than Saturn. For comparison, the posterior distributions of f_⊥ and θ_⊥ for the simulated standard transit light curve are shown in the right panels of Figure 8. The 68% confidence upper limit of the projected oblateness for the injected null case is 0.26. The comparison between the null-case result and data-fit posterior suggests that KOI 686.01 is consistent with a spherical shape. However, the upper limit we can set on the oblateness is very weak, so an oblateness as large as that of Saturn can only be ruled out in 1σ level if the planet is moderately tilted.

KOI 197.01 is a planetary candidate⁹ with an orbital period of 17.28 days and a planet-to-star radius ratio R_p/R_⋆ = 0.091, orbiting around a K-type star whose Kepler band magnitude is 14.02. Physical parameters about the host star are listed in Table 6.

There are 18 short-cadence full transits of KOI 197.01 at epochs from +13 to +30. The averaged out-of-transit variation amplitude is 1196 ppm. After reducing the data according to Section 3, we fit the standard transit model to the normalized light curves using the parameters given by Kepler as initials. With the best-fit parameters given by this standard transit model, we then set f_⊥ and θ_⊥ free to allow the oblate model fitting. Our best-fit parameters for both standard and oblateness fittings, together with the initial parameters from Kepler, are listed in Table 7.

Table 7. Fitting Parameters of KOI 197

Parameters	Kepler	Standard Model	Oblate Model
q1	0.50 ± 6	0.41 ± 6	$0.41^{+7}_{-6}$
q2	0.40 ± 1	0.48 ± 7	$0.49^{+7}_{-6}$
Porb (days)	17.276290 ± 7	17.27629 ± 2	17.27629 ± 2
t0 (BJDUTC−2454000)	966.8386 ± 2	966.8395 ± 4	$966.8396^{+3}_{-4}$
(Rp + R⋆)/a	⋅⋅⋅	0.0301 ± 4	$0.0299^{+7}_{-4}$
a/R⋆	36.6 ± 3.8	⋅⋅⋅	⋅⋅⋅
Rp/R⋆	0.091 ± 1	0.0914 ± 6	$0.0914^{+7}_{-4}$
i0 (deg)	90.0	89.88 ± 28	$89.87^{+24}_{-25}$
f⊥	⋅⋅⋅	⋅⋅⋅	<0.186

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We show the posterior plots of the oblateness fitting in Figure 9. Based on the best-fit parameters, the impact parameter b₀ of KOI 197.01 is 0.076. For this b₀, the oblateness is weakly constrained at θ_⊥ ≈ ±45° according to Figure 2. Therefore the result is consistent with our expectations.

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We detected no oblateness signal for KOI 197.01. The overall projected oblateness f_⊥ is constrained to be <0.19 (68% limit). An oblateness as large as that of Saturn can be ruled out in 1σ level if the projected obliquity θ_⊥ is close to 0°. For comparison, the posterior distribution of f_⊥ and θ_⊥ for the injected null-case standard transit light curves are shown on the right panel of Figure 9. The 68% confidence upper limit of the injected null-case projected oblateness is 0.18. We conclude that KOI 197.01 is consistent with a spherical shape, but with the oblateness only weakly constrained.

KOI 423.01 is a confirmed planet/brown dwarf (for simplicity, we only note it as a planet hereafter) with mass 18 M_J (Bouchy et al. 2011). The planet has an orbital period of 21.09 days, orbiting around an F-type star that has a Kepler-band magnitude of 14.33. The planet-to-star radius ratio of this system is R_p/R_⋆ = 0.087. Physical parameters about the host star are listed in Table 8.

Table 8. Physical Parameters about the Host Star KOI 423 (KIC 9478990)

Parameters	Values
Teff (K)	6260 ± 140a
log g (dex)	4.1 ± 0.2a
[Fe/H] (dex)	−0.29 ± 0.10a
M⋆ (M☉)	$1.10^{+0.07}_{-0.06}$a
R⋆ (R☉)	$1.39^{+0.11}_{-0.10}$a
vsin i⋆ (km s−1)	16 ± 2.5a
Age (Gyr)	5.1 ± 1.5a
J-band mag	13.236
H-band mag	12.999
K-band mag	12.918

Note. ^aAdopted from Bouchy et al. (2011).

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We use the 12 full short-cadence transits of KOI 423.01 available, from epochs +43 to +55, with +48 absent, in our analysis. The averaged out-of-transit-variation amplitude is 1256 ppm. After reducing the data according to Section 3, we fit the standard transit model to the normalized light curves using the parameters given by Kepler as initials. With the best-fit parameters given by this standard transit model, we then set f_⊥ and θ_⊥ free to fit for oblateness. The best-fit parameters for both the standard and oblateness fittings, together with the initial parameters from Kepler, are listed in Table 9.

Table 9. Fitting Parameters of KOI 423

Parameters	Kepler	Standard Model	Oblate Model
q1	0.40 ± 4	0.24 ± 8	0.23 ± 5
q2	0.27 ± 1	0.32 ± 13	$0.34^{+11}_{-9}$
Porb (days)	21.08717 ± 2	21.08722 ± 4	21.08722 ± 3
t0 (BJDUTC−2454000)	1035.8578 ± 3	1035.8574 ± 19	0.8575 ± 17
(Rp + R⋆)/a	⋅⋅⋅	0.0391 ± 11	$0.0392^{+9}_{-11}$
a/R⋆	29.6 ± 3.5	⋅⋅⋅	⋅⋅⋅
Rp/R⋆	0.086 ± 1	0.0888 ± 8	0.0889 ± 6
i0 (deg)	90.0	89.26 ± 33	$89.25^{+30}_{-13}$
f⊥	⋅⋅⋅	⋅⋅⋅	$0.22^{+11}_{-11}$

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The posterior plots from the oblateness fitting are shown in Figure 10, the light curve and residuals of standard fit and oblateness fit are shown on the upper panel of Figure 11. The residuals to each individual transit are shown in the lower panel of Figure 11.

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The marginalized distribution of f_⊥, shown in the lower left panel of Figure 10, suggests that KOI 423.01 may have an oblateness of $f_\perp =0.22^{+0.11}_{-0.11}$, which is substantially larger than any planet in the solar system. The total S/N of all 12 transits is 7.4 (Equation (6)). The S/N of each transit is also shown in the residual plots of Figure 11.

To check for the robustness of such a signal, we first perform the injection and recovery of the null signal case, with a spherical planet, into the out-of-transit light curves. The recovered posteriors of the null signal case are plotted on the right panels of Figure 10. The null detection case produces no oblateness signal, with the f_⊥ constrained to <0.2 at the 1σ level.

We then inject a simulated oblate planet transit, with the same transit parameters as KOI 423.01, and f_⊥ = 0.22 and θ_⊥ = −40°, into the out-of-transit sections of the light curve. The injected transits are shifted to the true transit epochs and fitted similarly to the observed transits. The recovered posterior probability distributions are plotted in Figure 12. We successfully recovered the injected oblateness factor f_⊥, arriving at a posterior distribution that is similar to what we see with the observed data. This suggests that the observed oblateness signal is unlikely to be due to the red noise and stellar oscillations in the light curves.

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To check for the consistency of such a signal between epochs, we choose 11 out of the 12 available observed transits, with epoch +45 excluded due to a spot-crossing event (see the lower panel of Figure 11), and divide them into three subgroups, with three transits in the first subgroup and four in each of the other two, to see if such a detection appears in each subgroup. We fit these subgroups simultaneously, such that all subgroups share the same system parameters, but each has its own independent f_⊥ and θ_⊥ values. In total, we fit 13 free parameters. The result of this fitting is shown in Figure 13. Subgroups 2 and 3 both indicate oblateness detections and are consistent within 1σ level. Subgroup 1 does not show noticeable detection according to the marginalized posterior distribution of f_⊥, but the asymmetry in the marginalized distribution of θ_⊥, as well as the two-dimensional posterior contour between f_⊥ and θ_⊥ suggests a potential oblateness detection, which is also consistent with that from the other two subgroups within 1σ level.

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However, we note that similar exercises performed on the simulated data sets can show inconsistency between epochs. As an example, shown in Figure 14, in the null case one of the three subgroups indicates that a large oblateness is detected, whereas in the signal-injected case one subgroup does not show any oblateness detection. These results suggest that the noise level per subgroup of transits is too high to allow for a robust consistency analysis. Therefore, we conclude that although we cannot validate the consistency of the observed oblateness signal based on the current data, we cannot dismiss such a potential signal either.

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