AN INDEPENDENT ANALYSIS OF KEPLER-4b THROUGH KEPLER-8b^*

In addition to global fits, we will search for variations in the light curve parameters for each transit. Searching for changes in the mid-transit time, commonly known as TTVs, can be used to search for perturbing planets (Agol et al. 2005; Holman & Murray 2005) or companion moons (Sartoretti & Schneider 1999; Kipping 2009a). Similarly, TDVs provide a complementary method of searching for exomoons (Kipping 2009a, 2009b). We also look for depth changes and baseline variations which may aid in the analysis of any putative signals.

Transit timing and duration measurements are made by splitting the time series into individual transits and fitting independently. In these individual fits, we only fit for p², b, T₁, t_c, and F_oot and float all other parameters around their global best-fit determined value with their a posteriori distribution. The limb-darkening coefficients are fixed to the theoretical values for these fits, since epoch to epoch variation in the LD coefficients is not expected. We choose to use method A for the individual fits as the parameter b is more likely to have a uniform prior than b² (as used in method B), from geometric considerations. As for the global fits, we perform 125,000 trials for each fit with the first 25,000 (20%) trials being used as a burn-in.

For the TDV, we define our transit duration as the time for the sky-projection of the planet's center to move between overlapping the stellar disc and exiting under the same condition, T. This value was first proposed by Carter et al. (2008) due to its inherently low correlations with other parameters. As a result, this definition of the duration may be determined to a higher precision than the other durations. T has the property that its uncertainty is exactly twice that of the mid-transit time for a trapezoid approximated light curve. In reality, limb darkening will cause this error to be slightly larger. Nevertheless, this property means that a useful definition of TDV is to halve the actual variations to give TDV = 0.5(T − 〈T〉), where 〈T〉 is the globally fitted value of T. The factor of 0.5 means that we expect the rms scatter of the TDV to be equal to, or slightly larger than, the TTV scatter. It is also worth noting that in reality we could use any factor we choose rather than 0.5, because TDV is really a measurement of the fractional variation in duration and is not absolute in the sense that TTV is.

When analyzing the TTV and TDV signals, we first compute the χ² of a model defined by a static system, i.e., constant duration and linear ephemeris. This χ² value is naturally computed from the MCMC uncertainties for each transit. In practice, these errors tend to be overestimates. This is because the MCMC only produces accurate errors if it is moving around the true global minimum. In reality, slight errors in the limb-darkening laws, the finite resolution of our integration-time compensation techniques, and the error in the assumed photometric weights themselves mean that our favorite solution is actually slightly offset from the true solution. Consequently, the MCMC jumps sample a larger volume of parameter space than if they were circling the true solution.

Despite the wide-spread use and therefore advocation of the MCMC method for transit analysis, it is also unclear whether the MCMC method does produce completely robust errors, particularly in light of the inevitable hidden systemic effects. Therefore searching for excess variance in the χ² values may not be a completely reliable method of searching for signals, echoed recently by Gibson et al. (2009). Concordantly, in all of our analyses, we compare the χ² of a static model versus that of a signal and compute the F-test. The F-test has the advantage that it does not care about the absolute χ² values, only the relative changes. Furthermore, it penalizes models which are overly complex (Occam's razor).

We therefore compute periodograms by moving through a range of periods in small steps of 1/1000 of the transit period and then fitting for a sinusoidal signal's amplitude and phase in each case. We compute the χ² of the new fit and then use an F-test to compute the false alarm probability (FAP) to give the F-test periodogram. Any peaks below 5% FAP are investigated further and we define a formal detection limit of 3σ confidence. Compared to the generalized Lomb–Scargle periodogram, the F-test is more conservative as it penalizes models for being overly complex and therefore offers a more prudent evaluation of the significance of any signal. Conceptually, we can see that the difference is that a Lomb–Scargle periodogram computes the probability of obtaining a periodic signal by chance whereas the F-test periodogram computes the statistical significance of preferring the hypothesis of a periodic signal over a null hypothesis.

In general, there are two types of periodic signals we are interested in searching for; those of period greater than the sampling rate (long-period signals) and those of periods shorter than the sampling rate (short-period signals). An example of a long-period signal would be an exterior perturbing planet and a short-period signal example would be a companion exomoon. Periodograms are well suited for detecting long period signals above the Nyquist frequency, which is equal to twice the sampling period. Short-period signals cannot be reliably seeked below the Nyquist frequency. First of all, a period equal to the sampling rate will always provide a strong peak. Periods of an integer fraction of this frequency will also present strong significances (e.g., 1/2, 1/3, 1/4, etc.). As the interval between these aliases becomes smaller, very short period signals become entwined. If a genuine short-period signal was present, it would exhibit at least partial frequency mixing with one of these peaks to create a plethora of short-period peaks. As a result, peaks in the periodogram below P_Nyquist cannot be considered as evidence for authentic signals. Therefore, we only plot our periodograms in the range of P_Nyquist to twice the total temporal coverage.

To resolve this issue, it is possible to derive an exomoon's period by taking the ratio of the TTV to TDV rms amplitudes (Kipping 2009a, 2009b). Since both of these signals are very short period, the data would ostensibly be seen as excess scatter. These excess scatters can be used to infer the rms amplitude of each signal and then derive the exomoon's period from the ratio. Excess scatter, or variance, may be searched for by applying a simple χ² test of the data, assuming a null-hypothesis of no signal being present. If a signal is found, or we wish to place constraints on the presence of putative signal amplitudes, we use the χ²-distribution due to both Gaussian noise plus an embedded high-frequency sinusoidal signal to model the variation.

An unusual aspect of the individual data sets is that consecutive data points are 30 minutes apart. Despite accounting for the long integration times in our modeling (see later in Section 3.5), we consider the possibility that the LC data could produce artificial signals in the parameter variation search. We propose that one method of monitoring the effects of the long integration time is what we denote as phasing.

Using the globally fitted ephemeris, we know the mid-transit time for all transits (assuming a non-variable system). We may compare the difference between this mid-transit time and the closest data point in a temporal sense. This time difference can take a maximum value of ±15 minutes (assuming no data gaps) and will vary from transit to transit. We label this time difference as the phasing. In all searches for parameter variations, a figure showing the phasing will also be presented. This allows us to check whether any putative signals are correlated to the phasing, which would indicate a strong probability the signal is spurious.

From Table A2 of the Appendix, the circular fit with no other free parameters is the preferred model to describe the radial velocities of Kepler-4b. We may perform other tests aside from the BIC model selection though. The global circular fit, using method A, yields a χ² = 28.54 and the eccentric fit obtains χ² = 20.56. These values correspond to the lowest χ² solution of the simplex global fit, but for the RV points only, which dominate the eccentricity determination. Assuming that the period and epoch are essentially completely driven by the photometry, the number of free parameters are two and four for the circular and eccentric fits, respectively. Therefore, based upon an F-test, the inclusion of two new degrees of freedom to describe the eccentricity is justified at the 1.7σ level (91.5% confidence). Another test we can implement is the Lucy & Sweeney (1971) test, where the significance of the eccentric fit is given by

$$\begin{equation} P(e>0) = 1 - {\rm {exp}}\Big [-\frac{\hat{e}^2}{2 \sigma _e^2}\Big ], \end{equation} \tag{ 13 }$$

where $\hat{e}$ is the modal value of e and σ_e is the error (in the negative direction). We find that this gives a significance of 88.2% or 1.6σ, in close agreement with the F-test result. We also computed the posterior distribution of the distance of the pericentre passage, in units of the stellar radius, and found (a/R_*)(1 − e) = 2.6^+1.5_−0.6 for method A. Therefore, if the eccentric fit is the true solution, Kepler-4b would make an extremely close pericentre passage.

We note that Borucki et al. (2010b) found a best-fit eccentricity of e = 0.22 (no quoted uncertainty) but the authors concluded that the eccentric fit was not statistically significant, but provided no quantification. Based upon the current observations, it is not possible to make a reliable conclusion as to whether Kepler-4b is genuinely eccentric or not. At best, it can be considered a ∼2σ marginal detection of eccentricity. The only assured thing we can say is that e < 0.43 to 95% confidence. Despite the ambiguity of the eccentricity, it is interesting to consider the consequences of this object possessing e>0.

For the eccentric fit, the extreme proximity of this planet to the star raises the issue of tidal circularization. Let us continue under the assumption that no third body is responsible for pumping the eccentricity of Kepler-4b. For a planet initially with e ∼ 1, the eccentricity decreases to 1/exp(1)ⁿ after n circularization timescales. This means the maximum number of circularization timescales which have transpired is given by n ⩽ −log(e). Since $n=\tau _{\rm {age}}/\tau _{\rm {circ}}$ (age of the system divided by circularization timescale) then we may write $\tau _{\rm {circ}} \ge \tau _{\rm {age}}/n$. Using the method of Adams & Laughlin (2006), the tidal circularization time may be written as

$$\begin{equation} \tau _{\rm {circ}} = Q_P \frac{4}{63} \frac{P}{2\pi } \frac{M_{p}}{M_\star } \Big (\frac{a}{R_\star } \frac{1}{p} \Big)^5 (1-e^2)^{13/2}. \end{equation} \tag{ 14 }$$

Given that $\tau _{\rm {circ}} \ge \tau _{\rm {age}}/n$, we may compute the minimum possible value of Q_P through rearrangement:

$$\begin{equation} Q_P \ge \frac{63}{4} \frac{2\pi }{P} \frac{M_\star }{M_{p}} \Big (\frac{p}{(a/R_\star)}\Big)^5 \frac{\tau _{\rm {age}}}{-\log (e)} \frac{1}{(1-e^2)^{13/2}}. \end{equation} \tag{ 15 }$$

The final term, (1 − e²)^13/2, may be neglected since we are interested in the minimum limit of Q_P and this term only acts to further increase the Q_P for non-zero e. The advantage of doing this is that e is a function of time and thus we can eliminate a term which would otherwise have to be integrated over time. Taking the posterior distribution of the Q_P equation for method A yields Q_P ⩾ (1.1^+1.5_−0.9) × 10⁷. We find that 99.9% of the MCMC trials correspond to Q_P ⩾ 10⁵ for the eccentric fit. Thus, if the eccentric fit was genuine, Kepler-4b would exhibit very poor tidal dissipation. This is somewhat reminiscent of GJ 436b (Deming et al. 2007) and HAT-P-11b (Bakos et al. 2009) for which a large Q_P value is also predicted and raises the interesting question as to whether hot Neptune generally possess larger Q_P values than their Jovian counterparts. We also note that WASP-18b has also been proposed to possess a greater-than-Jovian-Q_P (Hellier et al. 2009).

This discussion highlights the importance of obtaining more statistics regarding the eccentricity of transiting hot Neptunes, which may reveal some fundamental insights into the origin of these objects.

We here consider the circular model only since the eccentric fit is not unambiguously accepted. Global fits do not find any significant detection of the secondary eclipse. Secondary depths ⩾104 ppm are excluded to 3σ confidence, which corresponds to geometric albedos greater than unity and thus this places no constraints on the detectability of reflected light from the planet. The thermal emission is limited by this constraint such that $T_{P,\rm {brightness}} \le 3988$ K, which places no constraints on redistribution.

The Yonsei–Yale model isochrones from Yi et al. (2001) for metallicity [Fe/H] = +0.17 are plotted in Figure 4, with the final choice of effective temperature T_eff⋆ and a/R_⋆ marked, and encircled by the 1σ and 2σ confidence ellipsoids, both for the circular and the eccentric orbital solution from method B. Here, the MCMC distribution of a/R_⋆ comes from the global modeling of the data. Naturally, errors of the stellar parameters for the eccentric solution are larger, due to the larger error in a/R_⋆, which, in turn, is due to the uncertainty in the Lagrangian orbital parameters k and h (as opposed to fixing these to zero in the circular solution).

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The stellar evolution modeling provides color indices that may be compared against the measured values as a sanity check. The best available measurements are the near-infrared magnitudes from the 2MASS Catalog (Skrutskie et al. 2006). For Kepler-4, these are J_2MASS = 11.122 ± 0.023, H_2MASS = 10.836 ± 0.031, and K_2MASS = 10.805 ± 0.023; which we have converted to the photometric system of the models (ESO system) using the transformations by Carpenter (2001). The resulting measured color index is J − K = 0.340 ± 0.009. This is within 2σ of the predicted value from the isochrones of J − K = 0.38 ± 0.02 (see Table 2). The distance to the object may be computed from the absolute K magnitude from the models (M_K = 1.95 ± 0.35) and the 2MASS K_s magnitude, which has the advantage of being less affected by extinction than optical magnitudes. The result is 600⁺¹⁴⁰₋₆₅ pc, where the uncertainty excludes possible systematics in the model isochrones that are difficult to quantify.

The transit times and durations are listed in Table 3 and plotted in Figure 5. We find TTV residuals of 207.8 s and TDV residuals of 217.0 s using method A. Repeating the TTV analysis for method B finds best-fit times of consistently less than 0.25σ deviance (we note that similar results were found for all planets). The TTV indicates timings consistent with a linear ephemeris, producing a χ² of 5.7 for 11 degrees of freedom. This is somewhat low with a probability of 10.9% of occurring by coincidence. This may be an indication that the MCMC methods adopted here yield artificially large error bars, perhaps due to our choice of free-fitting every light curve rather than fixing certain parameters. However, we prefer to remain on the side prudence by overestimating such errors rather than underestimating them. The TDV too is consistent with a non-variable system exhibiting a χ² of 6.4 for 12 degrees of freedom (10.6% probability of random occurrence).

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Under the assumption that the timing uncertainties are accurate or overestimated, it is possible for us to determine what signal amplitudes are excluded for waveforms of a period less than the temporal baseline. For the TTV, we note that for 11 degrees of freedom, excess scatter producing a χ² of 28.5 would be detected to 3σ confidence. This therefore excludes a short-period signal of rms amplitude ⩾341.6 s. Similarly, for the TDV, we exclude scatter producing a χ² of 30.1, or a short-period rms amplitude of ⩾339.3 s, to 3σ confidence. This constitutes a 4.9% variation in the transit duration.

These limits place constraints on the presence of perturbing planets, moons, and Trojans. For the case of a 2:1 mean-motion resonance perturbing planet, the libration period would be ∼328 cycles (2.9 yr) and thus we do not possess sufficient baseline to look for such perturbers yet. However, this libration period is less than the mission lifetime of Kepler (3.5 yr) and so assuming the same timing precision for all measurements, the TTV will be sensitive to a 0.14 M_⊕ perturber.

The current TTV data rule out an exomoon at the maximum orbital radius (for retrograde revolution) of mass ⩾11.0 M_⊕. The TDV data rule an exomoon of ⩾13.3 M_⊕ at a minimum orbital distance.

Using the expressions of Ford & Holman (2007) and assuming $M_{p}\gg M_{\rm {Trojan}}$, the expected libration period of Kepler-4b induced by Trojans at L4/L5 is 50.5 cycles and therefore we do not yet possess sufficient temporal baseline to look for the TTVs. However, inspection of the folded photometry at ±P/6 from the transit center excludes Trojans of effective radius ⩾1.22 R_⊕ to 3σ confidence.

As discussed in section Section 2.3.2, we may use an F-test periodogram to search for periodic signals, as this negates any problems introduced by underestimating or overestimating timing uncertainties. While a search for excess variance is sensitive to any period below the temporal baseline, periodograms are limited to a range of $2 P_P < P_{\rm {signal}} < 2 P_{\rm {baseline}}$.

There are no peaks in the TTV periodogram below 5% FAP. The TDV does seem to exhibit a significant peak, which can also be seen by eye in the TDV plot itself, at a period of (15.5 ± 3.1) cycles, amplitude (204.9 ± 61.2) s, and FAP 2.1% (2.3σ). A linear trend in the durations could be confused with a long period signal. Fitting a simple a + bt model yields a = (170.9 ± 95.2) s and b = (−31.6 ± 13.8) s cycle⁻¹ with FAP 2.6% (2.2σ). Therefore, the two models yield very similar significances and it is difficult to distinguish between these two scenarios. The linear drift model seems improbable due to the very rapid change in duration the trend implies. The gradient equates to a change of around 1 hr per yr in the half-duration, i.e., 2 hr per yr in T. The period of the signal is too large to be caused by an exomoon and TDV is generally insensitive to perturbing planets, meaning that any planet which could produce such a large effect would dominate the RV. Other TDV effects, for example apsidal precession, are not expected to occur on timescales of 10–20 days which puts the reality of the signal in question.

The phasing of the measurements, as defined in Section 2.3.4, yields a clear periodic waveform of period 4 cycles. Therefore, one possible explanation for the TDV signal is a mixing of this phasing frequency with the Nyquist frequency to produce a signal of period 16 cycles, which is consistent with the value determined earlier. The analysis of further measurements is evidently required to assess the reality of the putative signal.

Kepler-5b was discovered by Koch et al. (2010). Our global fits reveal Kepler-5b to be a ∼1.35R_J planet on an orbit consistent with that of a circular orbit, transiting the host star at a near-equatorial impact parameter. The fitted models for method B of the phase-folded light curves are shown in Figure 6. Correlated noise was checked for in the residuals using the Durbin–Watson statistic, which finds d = 1.991, which is well within the 1% critical level of 1.867. The orbital fits to the RV points are shown in Figure 8 for method B, depicting both the circular and eccentric solutions. We obtain transit parameters broadly consistent with that of the discovery paper, except for b where method A favors an equatorial transit and method B favors a near-equatorial transit, whereas Koch et al. (2010) found b ≃ 0.4. The final parameters are summarized in Table 4. Light curve fits reveal that the theoretical limb-darkening values differ from the fitted values by a noticeable amount and the residuals of method B do reveal some structure as a consequence. Future short-cadence (SC) mode observations will pin down the limb darkening to a much higher precision.

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Table 4. Global Fits for Kepler-5b

Parameter	Discovery	Method A.c	Method A.e	Model B.c	Model B.e
Fitted parameters
P (days)	3.548460 ± 0.000032	3.548460+0.000074−0.000075	3.548460+0.000051−0.000051	3.548459 ± 0.000035	3.548459 ± 0.000035
E (BJD−2,454,000)	955.90122 ± 0.00021	955.90126+0.00049−0.00049	955.90124+0.00033−0.00033	955.90124 ± 0.00023	955.90124 ± 0.00023
T1,4 (s)	...	16580+184−165	16547+137−117	16459 ± 35	16468 ± 35
T (s)	...	15209+171−169	15232+122−118	15233 ± 23	15230 ± 25
(T1,2 ≃ T3,4) (s)	...	1309+257−84	1275+173−52	1227 ± 26	1227 ± 35
(Rp/R⋆)2 (%)	...	0.637+0.026−0.017	0.632+0.018−0.012	0.6225 ± 0.0032	0.6225 ± 0.0032
b2	...	0.066+0.143−0.059	0.042+0.106−0.038	0.021+0.027−0.012	0.022+0.030−0.013
(ζ/R⋆) (day−1)	...	11.42+0.14−0.13	11.403+0.098−0.093	11.34 ± 0.02	11.35 ± 0.02
$(F_{P,\rm {d}}/F_\star)$ (ppm)	...	29+35−35	29+24−24	26 ± 17	25 ± 25
esin ω	0a	0a	0.012+0.041−0.038	0a	0.008 ± 0.044
ecos ω	0a	0a	−0.008+0.017−0.016	0a	−0.009 ± 0.023
K (ms−1)	231.3 ± 2.8	220.6+14.2−14.2	221.1+9.3−9.4	228.1 ± 8.1	228.5 ± 8.1
γ (ms−1)	−46.7 ± 4.1	−43.1+10.0−10.0	−42.1+6.8−7.0	−44.8 ± 6.5	−43.9 ± 6.8
$\dot{\gamma }$ (ms−1 day−1)	0a	0.28+0.26−0.26	0.28+0.17−0.17	0a	0a
ttroj (days)	0a	0a	0a	0a	0a
B	1.020 ± 0.002	1.020 ± 0.002b	1.020 ± 0.002b	1.020 ± 0.002b	1.020 ± 0.002b
SME derived parameters
teff (K)	6297 ± 60	6297 ± 60	6297 ± 60	6297 ± 60	6297 ± 60
log(g/cgs)	3.96 ± 0.10	3.96 ± 0.10	3.96 ± 0.10	3.96 ± 0.10	3.96 ± 0.10
[Fe/H] (dex)	+0.04 ± 0.06	+0.04 ± 0.06	+0.04 ± 0.06	+0.04 ± 0.06	+0.04 ± 0.06
vsin i (km s−1)	4.8 ± 1.0	4.8 ± 1.0	4.8 ± 1.0	4.8 ± 1.0	4.8 ± 1.0
u1	...	0.25+0.13−0.12	0.244+0.083−0.085	0.3462a	0.3462a
u2	...	0.37+0.25−0.27	0.39+0.19−0.18	0.2972a	0.2972a
Model independent parameters
Rp/R⋆	0.08195+0.00030−0.00047	0.0798+0.0016−0.0011	0.07952+0.00111−0.00074	0.0789 ± 0.0002	0.0789 ± 0.0002
a/R⋆	6.06 ± 0.14	6.21+0.18−0.43	6.15+0.30−0.35	6.34+0.04−0.09	6.26 ± 0.29
b	0.393+0.051−0.043	0.00+0.35−0.00	0.01+0.29−0.01	0.143+0.067−0.057	0.149+0.072−0.060
i/°	86.3 ± 0.5	87.6+1.7−2.2	88.1+1.3−1.9	88.7 ± 0.6	88.6+0.6−0.7
e	0a	0a	0.034+0.029−0.018	0a	0.038 ± 0.028
ω/°	...	...	132+140−43	...	150 ± 83
RV jitter (m s−1)	...	8.2	7.7	15.5	15.0
ρ⋆/g cm−3	0.336	0.360+0.033−0.070	0.350+0.053−0.057	0.38+0.01−0.02	0.37+0.06−0.04
log(gP/cgs)	3.41 ± 0.03	3.433+0.046−0.078	3.433+0.048−0.062	3.48 ± 0.02	3.47 ± 0.04
S1,4/s	...	16580+184−165	20006+761−4329	16459 ± 35	16718 ± 1417
ts/(BJD-2,454,000)	...	961.22372+0.00049−0.00049	959.43+1.77−0.04	978.9660 ± 0.0002	975.397 ± 0.052
Derived stellar parameters
M⋆/M☉	1.374+0.040−0.059	1.370+0.050−0.036	1.372+0.048−0.041	1.352 ± 0.026	1.358 ± 0.039
R⋆/R☉	1.793+0.043−0.062	1.749+0.151−0.059	1.768+0.127−0.094	1.709+0.029−0.017	1.730 ± 0.091
log(g/cgs)	4.07 ± 0.02	4.087+0.023−0.057	4.080+0.037−0.048	4.10 ± 0.01	4.09 ± 0.04
L⋆/L☉	4.67+0.63−0.59	4.34+0.77−0.37	4.41+0.69−0.50	4.12 ± 0.22	4.21 ± 0.48
MV/mag	...	3.16+0.10−0.18	3.14+0.14−0.16	3.22 ± 0.07	3.20 ± 0.13
Age/Gyr	3.0 ± 0.6	2.83+0.23−0.25	2.82+0.24−0.24	2.9 ± 0.2	2.9 ± 0.2
Distance/pc	...	1101+94−51	1109+85−67	906 ± 17	916 ± 49
Derived planetary parameters
Mp/MJ	2.114+0.056−0.059	2.05+0.14−0.14	2.054+0.099−0.098	2.094 ± 0.079	2.101 ± 0.087
Rp/RJ	1.431+0.041−0.052	1.352+0.149−0.052	1.366+0.112−0.076	1.312+0.026−0.014	1.330 ± 0.071
ρP/g cm−3	0.894 ± 0.079	1.01+0.15−0.25	1.00+0.18−0.20	1.15 ± 0.06	1.11+0.19−0.14
a/AU	0.05064 ± 0.00070	0.05056+0.00061−0.00045	0.05059+0.00058−0.00051	0.0503 ± 0.0003	0.0504 ± 0.0005
teq/K	1868 ± 284	1790+64−33	1796+57−45	1770 ± 19	1779 ± 43

Notes. Quoted values are medians of MCMC trials with errors given by 1σ quantiles. Two independent methods are used to fit the data (A&B) with both circular and eccentric modes (c&e). ^aFixed parameter. ^bParameter was floated but not fitted.

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The global circular fit of method A yields a χ² = 20.77 for the RVs and the eccentric fit obtains χ² = 19.77. Based on an F-test, the inclusion of two new degrees of freedom to describe the eccentricity is justified below the 1σ level and is therefore not significant. We therefore conclude that the system is consistent with a circular orbit and constrain e < 0.086 to 95% confidence.

We here only consider the circular fits since the eccentric solution is not statistically significant. Method B provides the more accurate constraints due to the use of multiple baseline scalings. Proceeding with method B, the secondary eclipse is weakly detected in our analysis to 93.4% confidence, or 1.8σ, with a depth of 26⁺¹⁷₋₁₇ ppm, and is shown in Figure 7.

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The MCMC distribution of the F_p,d/F_⋆ values is shown in Figure 9. The discovery paper reported a weak 2.0σ detection of the eclipse but did not provide an amplitude for the signal. An eclipse of depth ⩾76 ppm is excluded at the 3σ level, which excludes a geometric albedo ⩾0.47 to the same level.

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If due to thermal emission, we find that by integrating a Planck function with the Kepler response function, the planetary brightness temperature would have to be 2480⁺¹⁵⁹₋₂₈₄ K, which is inconsistent with that expected for the equilibrium temperature of (1770 ± 19) K. Assuming negligible albedo and solving for the redistribution factor implies a factor of 3.7, which surpasses the maximal physically permitted value of 8/3 (corresponding to 2300 K). Internal heating also seems to be unlikely as the heat source would have to be able to generate a surface temperature of 2300 K alone or have an intrinsic luminosity of 4.6 × 10⁻⁴L_☉ which is approximately that of an M7V star. Tidal heating from eccentricity seems improbable given that the orbit is consistent with a circular orbit. However, using our best-fit eccentricity and the equations of Peale & Cassen (1978), we estimate that tidal heating could induce 8.3 × 10⁻⁸L_☉ for Q_P = 10⁵ and k_2p = 0.5. In order to get enough tidal heating, we require Q' = Q_P/k_2p = 35, which is much less than the typical values expected. Furthermore, such a low Q_P leads to very rapid circularization which somewhat nullifies the argument.

If the eclipse was due to reflection, we estimate a required geometric albedo of A_g = 0.15 ± 0.10. This latter option does offer a plausible physical origin for the secondary eclipse and therefore should be considered as a possible explanation for the observation. Although there has been a general impression that hot Jupiters must have low albedos after the remarkable MOST constraints of A_g < 0.17 for HD 209458b by Rowe et al. (2008), a recent study by Cowan & Agol (2011) finds evidence for hot Jupiters having much higher albedos (>0.4) in a statistical sample of 20 exoplanets. Therefore, a geometric albedo of 0.15 is not exceptional in such a sample and offers much more reasonable explanation than thermal emission. However, at this stage, the ∼2σ significance is too low to meet the requirements for formal detection.

The Yonsei–Yale model isochrones from Yi et al. (2001) for metallicity [Fe/H] = +0.04 are plotted in Figure 10, with the final choice of effective temperature T_eff⋆ and a/R_⋆ marked, and encircled by the 1σ and 2σ confidence ellipsoids, both for the circular and the eccentric orbital solution. Here, the MCMC distribution of a/R_⋆ comes from the global modeling of the data.

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The transit times and durations are listed in Table 5 and plotted in Figure 11. We find TTV residuals of 30.1 s and TDV residuals of 35.6 s. The TTV indicates timings consistent with a linear ephemeris, producing a χ² of 8.7 for 10 degrees of freedom. The TDV is also consistent with a non-variable system exhibiting a χ² of 10.5 for 11 degrees of freedom.

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For the TTV, we note that for 10 degrees of freedom, excess scatter producing a χ² of 26.9 would be detected to 3σ confidence. This therefore excludes a short-period signal of rms amplitude ⩾39.8 s. Similarly, for the TDV, we exclude scatter producing a χ² of 28.5, or a short-period rms amplitude of ⩾45.9 s, to 3σ confidence. This constitutes a 0.60% variation in the transit duration.

These limits place constraints on the presence of perturbing planets, moons, and Trojans. For the case of a 2:1 mean-motion resonance perturbing planet, the libration period would be ∼38.7 cycles and thus we do not possess sufficient temporal baseline to look for such perturbers yet. However, a 3:2 resonance would produce a libration period of 15.4 cycles and thus we would expect the effects to be visible over the 12 observed cycles so far. This therefore excludes a 3:2 resonant perturber of mass ⩾0.79 M_⊕ to 3σ confidence.

The current TTV data rule out an exomoon at the maximum orbital radius (for retrograde revolution) of ⩾10.6 M_⊕. The current TDV data rule an exomoon of ⩾17.3 M_⊕ at a minimum orbital distance.

Using the expressions of Ford & Holman (2007) and assuming $M_{p}\gg M_{\rm {Trojan}}$, the expected libration period of Kepler-5b induced by Trojans at L4/L5 is 10.0 cycles and therefore we can search for such TTVs. We find that such Trojans of cumulative mass ⩾3.14 M_⊕ at angular displacement 10° are excluded by our timings. Inspection of the folded photometry at ±P/6 from the transit center excludes Trojans of effective radius ⩾1.13 R_⊕ to 3σ confidence.

An F-test periodogram of the TTV reveals only one significant peaks which is very close to Nyquist frequency. We therefore disregard this frequency as being significant.

The TDV F-test periodogram reveals a significant peak at (4.0 ± 0.2)  cycles with amplitude (37 ± 11) s with significance 2.2σ. We again note that the period is not in the range which would be expected due to the known physical sources. The phasing effect has a period of 3 cycles and thus a mixing between the phasing and Nyquist frequency again offers a reasonable explanation for this signal, especially since peaks occur at 2 and 3 cycles in the TDV periodogram as well.

Kepler-6b was discovered by Dunham et al. (2010). In a similar manner to the other planets, we find consistent values for most parameters when compared to the Dunham et al. (2010) analysis. The fitted models on the phase-folded light curves for method B are shown in Figure 12. Correlated noise was checked for in the residuals using the Durbin–Watson statistic, which finds d = 1.943 which is well within the 1% critical level of 1.870. The orbital fits to the RV points using method B are shown in Figure 13, depicting both the circular and eccentric fits. The final planetary parameters are summarized at the bottom of Table 6.

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Table 6. Global Fits for Kepler-6b

Parameter	Discovery	Method A.c	Method A.e	Model B.c	Model B.e
Fitted parameters
P (days)	3.234723 ± 0.000017	3.234721+0.000043−0.000043	3.234721+0.000028−0.000028	3.234720 ± 0.000018	3.234721 ± 0.000018
E (BJD−2,454,000)	954.48636 ± 0.00014	954.48639+0.00034−0.00034	954.48637+0.00022−0.00022	954.48640 ± 0.00014	954.48641 ± 0.00015
T1,4 (s)	...	12982+160−140	12956+111−98	13003 ± 60	12977 ± 60
T (s)	...	11720+149−142	11728+100−98	11871 ± 16	11762 ± 18
(T1,2 ≃ T3,4) (s)	...	1211+234−84	1200+173−73	1227 ± 69	1218 ± 69
(Rp/R⋆)2 (%)	...	0.913+0.047−0.028	0.910+0.036−0.021	0.901 ± 0.011	0.901 ± 0.011
b2	...	0.069+0.137−0.062	0.061+0.107−0.054	0.088+0.047−0.043	0.085+0.049−0.044
(ζ/R⋆) (day−1)	...	14.80+0.19−0.19	14.80+0.14−0.13	14.67 ± 0.02	14.69 ± 0.02
$(F_{P,\rm {d}}/F_*)$ (ppm)	...	5+36−36	−4+24−24	15 ± 18	21 ± 35
esin ω	0a	0a	0.042+0.055−0.050	0a	0.117 ± 0.036
ecos ω	0a	0a	−0.019+0.020−0.019	0a	0.012+0.029−0.015
K (ms−1)	80.9 ± 2.6	78.4+6.1−6.0	81.8+4.7−4.6	78.2 ± 3.7	86.6 ± 1.6
γ (ms−1)	−18.3 ± 3.5	−18.2+4.5−4.5	−17.7+3.0−3.0	−15.9 ± 2.9	−13.6 ± 2.7
$\dot{\gamma }$ (ms−1 day−1)	0a	0a	0a	0a	0a
ttroj (days)	0a	−0.041+0.042−0.043	0a	0a	0a
B	1.033 ± 0.004	1.033 ± 0.004b	1.033 ± 0.004b	1.033 ± 0.004b	1.033 ± 0.004b
SME derived parameters
teff (K)	5647 ± 44	5647 ± 88	5647 ± 88	5647 ± 88	5647 ± 88
log(g/cgs)	4.59 ± 0.10	4.59 ± 0.10	4.59 ± 0.10	4.59 ± 0.10	4.59 ± 0.10
[Fe/H] (dex)	+0.34 ± 0.08	+0.34 ± 0.08	+0.34 ± 0.08	+0.34 ± 0.08	+0.34 ± 0.08
vsin i (km s−1)	3.0 ± 1.0	3.0 ± 1.0	3.0 ± 1.0	3.0 ± 1.0	3.0 ± 1.0
u1	...	0.55+0.13−0.11	0.544+0.084−0.072	0.4471a	0.4471a
u2	...	0.01+0.26−0.27	0.02+0.17−0.18	0.2397a	0.2397a
Model independent parameters
Rp/R⋆	0.09829+0.00014−0.00050	0.0955+0.0024−0.0015	0.0954+0.0018−0.0011	0.0257 ± 0.0009	0.0256 ± 0.0006
a/R⋆	7.05+0.11−0.06	7.32+0.22−0.48	7.00+0.44−0.46	7.21 ± 0.18	6.42 ± 0.26
b	0.398+0.020−0.039	0.26+0.19−0.26	0.25+0.16−0.25	0.296+0.068−0.099	0.292+0.071−0.102
i/°	86.8 ± 0.3	87.9+1.4−1.7	87.9+1.4−1.6	87.7+0.8−0.6	87.0 ± 0.9
e	0a	0a	0.056+0.044−0.028	0a	0.120 ± 0.036
ω/°	...	...	115+86−24	...	83 ± 12
RV jitter (m s−1)	...	0.6	0.0	3.2	2.7
ρ⋆/g cm−3	0.634	0.709+0.067−0.131	0.619+0.125−0.114	0.68 ± 0.05	0.48 ± 0.06
log(gP/cgs)	2.974+0.016−0.022	3.010+0.052−0.081	2.99+0.054−0.066	3.01 ± 0.03	2.95 ± 0.04
S1,4/s	...	12982+160−140	15371+165−2832	13003 ± 60	10022 ± 829
ts/(BJD-2,454,000)	...	959.33827+0.00034−0.00034	957.68+1.60−0.04	978.7470 ± 0.0001	978.772 ± 0.048
Derived stellar parameters
M⋆/M☉	1.209+0.044−0.038	1.127+0.045−0.065	1.137+0.057−0.076	1.136+0.034−0.072	1.170+0.064−0.087
R⋆/R☉	1.391+0.017−0.034	1.306+0.102−0.046	1.370+0.109−0.091	1.325 ± 0.042	1.503 ± 0.073
log(g/cgs)	4.236 ± 0.011	4.254+0.026−0.056	4.218+0.050−0.057	4.24 ± 0.02	4.15 ± 0.03
L⋆/L☉	1.99+0.24−0.21	1.57+0.26−0.17	1.71+0.32−0.25	1.61 ± 0.17	2.06 ± 0.28
MV/mag	...	4.35+0.14−0.17	4.26+0.18−0.19	4.33 ± 0.13	4.05 ± 0.16
Age/Gyr	3.8 ± 1.0	5.65+2.68−0.86	5.65+2.84−0.90	5.6+2.5−0.9	5.7+2.6−1.0
Distance/pc	...	604+50−37	630+59−51	621 ± 22	704 ± 37
Derived planetary parameters
Mp/MJ	0.669+0.025−0.030	0.617+0.052−0.051	0.645+0.047−0.044	0.616 ± 0.035	0.692 ± 0.032
Rp/RJ	1.323+0.026−0.029	1.208+0.129−0.049	1.271+0.116−0.091	1.224 ± 0.043	1.389 ± 0.072
ρP/g cm−3	0.352+0.018−0.022	0.426+0.069−0.105	0.389+0.084−0.081	0.41 ± 0.04	0.32 ± 0.04
a/AU	0.04567+0.00055−0.00046	0.04452+0.00058−0.00088	0.04466+0.00074−0.00101	0.0447+0.0004−0.0010	0.0451+0.0008−0.0012
teq/K	1500 ± 200	1480+51−33	1511+57−51	1488 ± 30	1579 ± 41

Notes. Quoted values are medians of MCMC trials with errors given by 1σ quantiles. Two independent methods are used to fit the data (A&B) with both circular and eccentric modes (c&e). ^aFixed parameter. ^bParameter was floated but not fitted.

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The global circular fit yields a χ² = 9.01 and the eccentric fit obtains χ² = 7.75 for the RVs. Based on an F-test, the inclusion of two new degrees of freedom to describe the eccentricity is justified with a significance of <1σ The system is therefore best described with a circular orbit with the current observations. We constrain e < 0.13 to 95% confidence.

We here only consider the circular fit since the eccentric solution is not statistically significant. We find no evidence for a secondary eclipse detection for Kepler-6b. However, our analysis does exclude secondary eclipses of depth 51.5 ppm or greater to 3σ confidence, which excludes a geometric albedo ⩾0.32 to the same level. This therefore excludes the possibility of a planet covered in reflective clouds. A brightness temperature of 2445 K is ruled out to the same level, which places no constraints on the redistribution.

The Yonsei–Yale model isochrones from Yi et al. (2001) for metallicity [Fe/H] = +0.34 are plotted in Figure 14, with the final choice of effective temperature T_eff⋆ and a/R_⋆ marked, and encircled by the 1σ and 2σ confidence ellipsoids, both for the circular and the eccentric orbital solution. Here, the MCMC distribution of a/R_⋆ comes from the global modeling of the data.

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The transit times and durations are listed in Table 7 and plotted in Figure 15. We find TTV residuals of 17.4 s and TDV residuals of 25.4 s. The TTV indicates timings consistent with a linear ephemeris, producing a χ² of 7.8 for 11 degrees of freedom. The TDV is also consistent with a non-variable system exhibiting a χ² of 11.4 for 12 degrees of freedom.

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For the TTV, we note that for 11 degrees of freedom, excess scatter producing a χ² of 28.5 would be detected to 3σ confidence. This therefore excludes a short-period signal of rms amplitude ⩾26.2 s. Similarly, for the TDV, we exclude scatter producing a χ² of 30.1, or a short-period rms amplitude of ⩾31.5 s, to 3σ confidence. This constitutes a 0.53% variation in the transit duration.

These limits place constraints on the presence of perturbing planets, moons, and Trojans. For the case of a 2:1 mean-motion resonance perturbing planet, the libration period would be ∼76.5 cycles and thus we do not possess sufficient baseline to look for such perturbers yet. However, a 4:3 resonance would produce a libration period of 17.7 cycles and thus we would expect the effects to be visible over the 14 observed cycles so far. This therefore excludes a 4:3 resonant perturber of mass ⩾0.38 M_⊕ to 3σ confidence.

The current TTV data rule out an exomoon at the maximum orbital radius (for retrograde revolution) of ⩾4.8 M_⊕. The current TDV data rule an exomoon of ⩾8.2 M_⊕ at a minimum orbital distance.

Using the expressions of Ford & Holman (2007) and assuming $M_{p}\gg M_{\rm {Trojan}}$, the expected libration period of Kepler-6b induced by Trojans at L4/L5 is 16.7 cycles and therefore we can search for such TTVs. We find such Trojans of cumulative mass ⩾0.67 M_⊕ at angular displacement 10° are excluded by our timings. Inspection of the folded photometry at ±P/6 from the transit center excludes Trojans of effective radius ⩾0.87 R_⊕ to 3σ confidence.

The TTV periodogram reveals a significant peak of period (5.34 ± 0.26) cycles with amplitude (19.7 ± 5.0) s and FAP 2.6σ, which is one of our highest significances found, yet still below our formal detection threshold of 3σ. Nevertheless, the signal warrants a more detailed investigation as it does not appear to land on any of the probable aliasing frequencies.

Given that Trojans should induce libration of period 16.7 cycles, this hypothesis can be instantly discounted. For two planets in a j:j + 1 mean motion-resonance, we may set the TTV signal's period to be equal to the expression for the expected libration period (from Agol et al. 2005) and solve for j. This yields j = 7.37^+0.28_−0.26 or a period ratio of 1.136 ± 0.005, which does not appear to be a resonant configuration. Barnes & Greenberg (2006) used the Hill criterion to provide an expression for estimating period ratios which are not guaranteed to be dynamically stable (Equation (2)). Using this expression, we estimate that period ratios of >0.75 and <1.34 are unlikely to be stable for masses above 1 M_⊕. This makes the hypothesis of a mean-motion resonance unlikely.

For an out-of-resonant signal, the period of the TTV is more variable, depending on the difference between the perturber's period and the nearest resonant position, quantified by the parameter in Agol et al. (2005). For a completely non-resonant system, = 1 and the period ratio is 6.34. As we move closer to resonance, this period ratio decreases. For the maximally non-resonant configuration, the perturber would require a mass of $M_{\rm {pert}} = (0.75 \pm 0.19) M_{\rm J}$ to generate the putative TTV signal. Given that this is approximately the same mass as the transiting planet and the period ratio is not large, it is clear that such a signal would be evident in the RV data. This makes the hypothesis of a non-resonant perturber unlikely but not completely excluded due to the possibility of out-of-the-plane perturbers (see Nesvorny & Beauge 2010; Veras et al. 2010).

If such a signal was due to an aliasing of an exomoon's orbital period, the maximum amplitude would occur for a moon at the maximum orbital separation given by Domingos et al. (2006), for a retrograde configuration. A moon of mass (1.8 ± 0.4) M_⊕ could generate such a signal and closer moons require larger masses. Such a moon should produce a photometric dip of between 40 ppm and 170 ppm depending on its composition. Following the method outlined in Kipping (2009c), we may use the phasing predicted by the TTV fit to fold the individual transit residuals for cases where maximum planet–moon separation is predicted: epochs 1, 7, and 11 and the time-reversed epochs 4 and 9. Inspection of this folded residual rules out exomoons of R>1.6R_⊕ to 2σ confidence.

An additional method of confirming a moon would be through a TDV analysis. The TDV periodogram does have one significant peak at (3.6 ± 0.2) days though but with a weaker significance of 2.2σ and amplitude (24.3 ± 7.7) s. Using the ratio of the TTV and TDV amplitude, Kipping (2009a) showed that a solution for the exomoon period is possible. Following this method and assuming both the TTV peak and the TDV peak are genuinely due to a moon, we estimate an exomoon period of (4.2 ± 1.7) hr which would place the moon at a semi-major axis of (0.8 ± 0.2) planetary radii, i.e., within the planet's volume. Although there is a large uncertainty on this, we conclude that the hypothesis of an exomoon inducing both the TTV and TDV putative signals is highly improbable.

The TDV peak is quite likely to be spurious due to the proximity of the periodogram peak to twice the Nyquist frequency. The exomoon hypothesis can be further interrogated by evaluating the range of dynamically stable moons around Kepler-6b. Assuming Jovian-like tidal dissipation values of k_2p = 0.5 and Q_P = 10⁵, the expression of Barnes & O'Brien (2002) indicates that a single moon could only be stable for the age of the system if it has a mass below 0.25 M_⊕. In contrast, the TTV could only be produced by moons >1.8 M_⊕ suggesting we would require Q_P ⩾ 10⁶ to make the two compatible. In conclusion, there is no confirming evidence for the moon hypothesis and at thus we cannot assign this explanation to the TTV peak.

The TTV signal's period of 17.3 days is similar to that what we might expect for the star's rotational period and thus could be due to the motion of star spots across the stellar surface, as observed by Alonso et al. (2008) for CoRoT-2b. Based upon the projected vsin i rotation of the star and the inferred stellar radius, we estimate the rotation period to be 23.5^+11.7_−5.9 days which is compatible with the period of the TTV signal. A bisector analysis is not possible due to the very low cadence but will be possible once SC data become available. We propose that this is the most likely explanation for the signal, based upon the current evidence.

The phasing signal has a period of 4 cycles, which is close to both periods found in the TTV and TDV periodograms. This therefore detracts credence to the hypothesis that these signals are genuine. Further observations, particularly in SC mode, where phasing will not be a significant issue, will clarify the reality of these signals.

Kepler-7b was discovered by Latham et al. (2010). The planet is noteworthy for its very low density and thus significant inflation. Our global fits find highly consistent results with the original discovery paper. We note that the Latham et al. (2010) values are the most consistent with our own values out of any of discovery papers discussed in this work. The fitted models on the phase-folded light curves are shown in Figure 16 for method B. Correlated noise was checked for in the residuals using the Durbin–Watson statistic, which finds d = 1.866, at the border of the 1% critical level of 1.868, suggesting correlated noise is marginal. The orbital fits to the RV points are shown in Figure 18, depicting both the circular and eccentric fits using method B. Unusually, both methods A and B found stellar jitter levels consistent with zero, which suggests that the publicly available RV measurements have perhaps overestimated errors. The final planetary parameters are summarized at the bottom of Table 8.

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Table 8. Global Fits for Kepler-7b

Parameter	Discovery	Method A.c	Method A.e	Model B.c	Model B.e
Fitted parameters
P (days)	4.885525 ± 0.000040	4.88552+0.00010−0.00010	4.885522+0.000062−0.000061	4.885535 ± 0.000041	4.885537 ± 0.000042
E (BJD−2,454,000)	967.27571 ± 0.00014	957.50468+0.00049−0.00049	957.50481+0.00034−0.00035	957.50461 ± 0.00020	957.50458 ± 0.00021
T1,4 (s)	...	18694+224−216	18704+149−150	18541 ± 95	18516 ± 95
T (s)	...	16951+242−237	16916+203−156	16935 ± 22	16917 ± 31
(T1,2 ≃ T3,4) (s)	...	1722+376−297	1812+245−299	1624 ± 104	1616 ± 104
(Rp/R⋆)2 (%)	...	0.661+0.038−0.034	0.671+0.024−0.035	0.6529 ± 0.0081	0.6529 ± 0.0081
b2	...	0.19+0.13−0.15	0.230+0.083−0.137	0.155+0.045−0.051	0.152+0.047−0.051
(ζ/R⋆) (day−1)	...	10.24+0.16−0.15	10.28+0.12−0.14	10.20 ± 0.01	10.21+0.02−0.02
$(F_{P,\rm {d}}/F_\star)$ (ppm)	...	24+30−30	−21+32−26	47 ± 14	41 ± 37
esin ω	0a	0a	0.00+0.13−0.13	0a	0.074 ± 0.063
ecos ω	0a	0a	0.047+0.021−0.043	0a	−0.003+0.005−0.025
K (ms−1)	42.9 ± 3.5	42.8+10.6−10.6	41.6+6.3−6.2	42.6 ± 4.3	43.1 ± 4.3
γ (ms−1)	0	−0.1+7.3−7.3	0.7+4.5−4.3	0.0 ± 2.9	−0.6 ± 2.9
$\dot{\gamma }$ (ms−1 day−1)	0a	0a	0a	0a	0a
ttroj (days)	0a	0a	0a	0a	0a
B	1.025 ± 0.004	1.025 ± 0.004b	1.025 ± 0.004b	1.025 ± 0.004b	1.025 ± 0.004b
SME derived parameters
teff (K)	5933 ± 44	5933 ± 88	5933 ± 88	5933 ± 88	5933 ± 88
log(g/cgs)	3.98 ± 0.10	3.98 ± 0.10	3.98 ± 0.10	3.98 ± 0.10	3.98 ± 0.10
[Fe/H] (dex)	+0.11 ± 0.06	+0.11 ± 0.06	+0.11 ± 0.06	+0.11 ± 0.06	+0.11 ± 0.06
vsin i (km s−1)	4.2 ± 0.5	4.2 ± 0.5	4.2 ± 0.5	4.2 ± 0.5	4.2 ± 0.5
u1	...	0.34+0.16−0.13	0.35+0.12−0.10	0.3948a	0.3948a
u2	...	0.33+0.26−0.34	0.27+0.24−0.23	0.2711a	0.2711a
Model independent parameters
Rp/R⋆	0.08241+0.00030−0.00043	0.0813+0.0023−0.0021	0.0819+0.0015−0.0021	0.0808 ± 0.0005	0.0808 ± 0.0005
a/R⋆	7.22+0.16−0.13	7.14+0.56−0.53	7.03+1.10−0.95	7.29 ± 0.20	6.78 ± 0.43
b	0.445+0.032−0.044	0.44+0.13−0.23	0.480+0.080−0.175	0.394+0.052−0.079	0.391+0.054−0.079
i/°	86.5 ± 0.4	86.5+2.0−1.4	86.3+1.7−1.9	86.9+0.7−0.5	86.4 ± 0.8
e	0a	0a	0.102+0.104−0.047	0a	0.076 ± 0.057
ω/°	...	...	129+182−80	...	93 ± 52
RV jitter (m s−1)	...	0.0	0.0	0.0	0.0
ρ⋆/g cm−3	0.304	0.288+0.073−0.060	0.276+0.151−0.097	0.31 ± 0.03	0.25 ± 0.05
log(gP/cgs)	2.691+0.038−0.045	2.69+0.13−0.15	2.66+0.14−0.16	2.71 ± 0.05	2.65 ± 0.07
S1,4/s	...	18694+224−216	15406+5065−1023	18541 ± 95	15535 ± 2350
ts/(BJD-2,454,000)	...	964.83265+0.00049−0.00049	964.662+0.074−0.158	975.8300 ± 0.0002	994.137 ± 0.058
Derived stellar parameters
M⋆/M☉	1.347+0.072−0.054	1.257+0.087−0.073	1.268+0.115−0.090	1.226+0.108−0.045	1.272+0.102−0.054
R⋆/R☉	1.843+0.048−0.066	1.83+0.17−0.14	1.86+0.34−0.28	1.791 ± 0.067	1.941+0.178−0.118
log(g/cgs)	4.030+0.018−0.019	4.012+0.061−0.063	4.00+0.12−0.12	4.03 ± 0.02	3.97 ± 0.05
L⋆/L☉	4.15+0.63−0.54	3.72+0.78−0.60	3.8+1.6−1.1	3.55 ± 0.41	4.18+0.94−0.56
MV/mag	...	3.38+0.20−0.21	3.34+0.37−0.39	3.43 ± 0.14	3.25 ± 0.19
Age/Gyr	3.5 ± 1.0	4.82+0.82−1.20	4.45+1.11−0.96	5.1+0.7−1.3	4.7+0.7−1.0
Distance/pc	...	772+79−67	784+153−122	829 ± 36	898+84−57
Derived planetary parameters
Mp/MJ	0.433+0.040−0.041	0.42+0.11−0.11	0.405+0.067−0.064	0.414 ± 0.045	0.424 ± 0.046
Rp/RJ	1.478+0.050−0.051	1.45+0.18−0.15	1.48+0.29−0.24	1.409 ± 0.058	1.527+0.144−0.097
ρP/g cm−3	0.166+0.019−0.020	0.167+0.074−0.056	0.155+0.099−0.064	0.18 ± 0.03	0.15 ± 0.03
a/AU	0.06224+0.00109−0.00084	0.0608+0.0014−0.0012	0.0609+0.0018−0.0015	0.0603+0.0017−0.0007	0.0611+0.0016−0.0009
teq/K	1540 ± 200	1569+65−58	1582+127−110	1554 ± 32	1612+68−50

Notes. Quoted values are medians of MCMC trials with errors given by 1σ quantiles. Two independent methods are used to fit the data (A&B) with both circular and eccentric modes (c&e). ^aFixed parameter. ^bParameter was floated but not fitted.

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The global circular fit of method A yields a χ² = 6.87 and the eccentric fit obtains χ² = 4.89 for the RVs. An F-test finds that the eccentric fit has a significance of 79.4%. Therefore, we conclude that the eccentric fit does not meet our detection criteria and we constrain e < 0.31 to 95% confidence.

One obvious question at this point is whether the range of allowed eccentricities could explain the bloated radius of Kepler-7b through tidal heating. The Q_P value of the planet is an element of large uncertainty here but we can constrain Q's minimum limit using the same method we adopted for Kepler-4b. We find that Q_P ⩾ 9.4^+14.0_−4.6 × 10⁶. One may argue that this is only valid under the assumption that the eccentric model is selected over the circular one, but we would counter that in the circular model case the tidal heating is zero anyway and thus the hypothesis of tidal heating being the origin of the bloated radius is invalid.

Using the posterior distribution of the minimum allowed Q_P value, we can translate this to a maximum allowed tidal energy dissipation using the expressions of Peale & Cassen (1978):

$$\begin{equation} \dot{E}_{\rm {tidal}} \ge \frac{21}{2} \frac{k_{2}}{Q_{P,\rm {min}}} \frac{G M_*^2 R_P^5 n e^2}{a^6}. \end{equation} \tag{ 16 }$$

We divide this value by the amount of power hitting the planetary surface, found by L_*R²_P/(4a²) to give us the posterior of the fraction of maximum allowed tidal heating relative to the incoming radiation and find $\dot{E}_{\rm {tidal}}/\dot{E}_{\rm {stellar}} = 1.5_{-1.0}^{+2.5} \times 10^{-12}$. We therefore conclude that tidal heating plays a negligible role in the heating of Kepler-7b and the planet's inflated radius.

As before, method B is the better tool to constrain secondaries due to its rescaling method. Method B detects a secondary eclipse in the circular fits of depth (47 ± 14) ppm and significance 3.5σ (99.96%), and is shown in Figure 17. We note that Latham et al. (2010) reported a 2.4σ weakly detected eclipse but did not report a depth estimate. The MCMC distribution of the F_p,d/F_⋆ values for method B is shown in Figure 19.

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Inspection of the method A and Latham et al. (2010) values relative to the method B result reveals a clear difference and we must consider why we obtain such different numbers. All of the fits are circular and thus the orbital solutions are nearly identical with the eclipse occurring at the same phase in each case. The only difference between them is the definition of the out-of-eclipse baseline. Method A assumes the same baseline level for both primary and secondary transits whereas method B uses a simplified phase curve model as described in Section 2.2. If the secondary eclipse local baseline is different from the primary transit baseline, method A would move toward an average solution and actually attenuate the eclipse depth. The same attenuation also occurs in the Latham et al. (2010) which utilizes the same technique as method A (D. Latham 2010, private communication).

To investigate this further, we may compare the three baseline levels found by method B: F_oot = (0.999976 ± 0.000007), F_oom = (0.999972 ± 0.000003), and F_oos = (0.999993 ± 0.000007). The first two values are compatible with a constant but the baseline surrounding the secondary eclipse is clearly larger. If the secondary eclipse baseline is larger, method A would favor a lower average baseline due to F_oot and F_oom and thus find a lower secondary eclipse. This seems to be the explanation with method A finding F_oot = (0.9999764 ± 0.0000024). We believe this explains the discrepancy between the two methods.

Using the method B results only, the difference in baselines indicates a day–night contrast. Using our MCMC trials, we estimate a nightside flux $(F_{\rm {n}}/F_\star) = 29_{-14}^{+15}$  ppm which is of 2.4σ significance. We can also calculate the day–night difference explicitly and we find $[(F_{P,\rm {d}}/F_\star) - (F_{P,\rm {n}}/F_\star)] = (17 \pm 9)$ ppm. Since nightside flux cannot be due to reflected light, this supports the hypothesis of thermal emission as a source for both the secondary eclipse and the phase curve.

There is a very strong caveat to bear in mind for the day–night contrast. The data we have analyzed have been processed by the Kepler pipeline which involves numerous data manipulation processes. Time trends of very low amplitude and of time periods of days may be affected by this processing, with the most probable behavior being that of a long-cut filter. As a result, a strong day–night contrast could be attenuated by the act of passing through this filter. It is therefore still possible that the day–night contrast is 100% (i.e., equal to the secondary eclipse depth) and thus the light we are seeing is reflected light, rather than thermal emission, but the phase curve is artificially distorted by the pipeline processing.

If the secondary eclipse was due to thermal emission, we may estimate the brightness temperature by integrating the Planck function of the planet over the Kepler response function, divided by that of the star. We repeat this for planetary brightness temperatures from 2000 K to 3000 K in 1 K steps and calculate $(F_{P,\rm {d}}/F_\star)$ in each case. We find that a best-fit brightness temperature of 2570⁺¹⁰⁸₋₈₅ K, which is much larger than the equilibrium temperature of (1554 ± 32) K. Even the nightside flux of $(F_{P,\rm {n}}/F_\star) = 29_{-14}^{+15}$ ppm would require a temperature for the nightside of 2441⁺¹²⁷₋₁₇₉ K. These very large temperatures do not seem consistent with the energy budget of the planet, particularly as tidal heating has already been shown to offer a negligible contribution to the energy budget of this planet. From stellar irradiation only, the redistribution factor would have to be 7.48 to recreate the dayside eclipse, which is greater than the maximum physically permitted value of 8/3. The only remaining possibility is internal heating, for which we find a heat source responsible for 2440 K could explain both the dayside and nightside fluxes. This would require an internal heat source of log(L/L_☉) = −3.19, which is equivalent to the luminosity of an M6-M7 star. For example, LHS 292 (M6.5) and VB8 (M7) have luminosities of log(L/L_☉) = −3.19 and log(L/L_☉) = −3.24, respectively.

While other sources of heating are possible apart from tidal, for example radioactivity, the magnitude required detracts credence from the hypothesis of thermal emission as the principal origin for the eclipse depth.

If due to reflection, method B finds a geometric albedo of A_g = (0.38 ± 0.12), which is somewhat larger than the value found for Kepler-5b of A_g = (0.15 ± 0.10), but not greatly different from the albedos of Jupiter and Saturn (0.52 and 0.47, respectively) and consistent with the study of Cowan & Agol (2011). We believe this latter hypothesis, as with Kepler-5b, is a more plausible scenario for the secondary eclipse measurement. As a consequence, we should not expect any nightside flux, which bears some questions on the apparent non-zero nightside flux detected in this study. There are two possible explanations which are consistent with reflection: (1) the nightside flux is not statistically significant at 2.4σ, (2) the phase curve has been attenuated due to the Kepler pipeline producing effects which mimic a long-cut filter, and (3) F_oos is increased away from zero nightside flux due to the phase curve of the planet, i.e., reflected light from the viewable crescent. In reality, a combination of these seems probable. Once more transits become available for analysis, a more sophisticated phase curve model can be fitted to the data set to resolve this question.

The Yonsei–Yale model isochrones from Yi et al. (2001) for metallicity [Fe/H] = +0.11 are plotted in Figure 20, with the final choice of effective temperature T_eff⋆ and a/R_⋆ marked, and encircled by the 1σ and 2σ confidence ellipsoids, both for the circular and the eccentric orbital solution. Here, the MCMC distribution of a/R_⋆ comes from the global modeling of the data.

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The transit times and residuals are listed in Table 9 and plotted in Figure 21. We find TTV residuals of 32.1 s and TDV residuals of 22.2 s. The TTV indicates timings consistent with a linear ephemeris, producing a χ² of 9.1 for 7 degrees of freedom (75.1% significance of excess). The TDV is also consistent with a non-variable system exhibiting a χ² of 4.1 for 8 degrees of freedom (84.9% significance of being too low).

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For the TTV, we note that for 7 degrees of freedom, excess scatter producing a χ² of 21.8 would be detected to 3σ confidence. This therefore excludes a short-period signal of rms amplitude ⩾38.0 s. Similarly, for the TDV, we exclude scatter producing a χ² of 23.6, or a short-period rms amplitude of ⩾42.6 s, to 3σ confidence. This constitutes a 0.50% variation in the transit duration.

These limits place constraints on the presence of perturbing planets, moons, and Trojans. For the case of a 2:1 mean-motion resonance perturbing planet, the libration period would be ∼111 cycles and thus we do not possess sufficient baseline to look for such perturbers yet. After a further 102 cycles have been obtained at the same photometric quality, the presence of a 2:1 resonant perturber of mass 0.06 M_⊕ may be constrained.

The current TTV data rule out an exomoon at the maximum orbital radius (for retrograde revolution) of ⩾2.5 M_⊕. The current TDV data rule an exomoon of ⩾5.9 M_⊕ at a minimum orbital distance.

Using the expressions of Ford & Holman (2007) and assuming $M_{p}\gg M_{\rm {Trojan}}$, the expected libration period of Kepler-7b induced by Trojans at L4/L5 is 21.4 cycles and therefore we do not yet possess sufficient baseline to search for these TTVs. Inspection of the folded photometry at ±P/6 from the transit center excludes Trojans of effective radius ⩾1.11 R_⊕ to 3σ confidence.

What is peculiar about this system is that the TTV produces a slight excess variance while the TDV does the opposite. The error on the TDV data points, and thus the scatter, should be exactly the same as the TTV points, or in fact slightly larger due to limb-darkening effects. Thus, the observation of the TTV scatter being much larger than the TDV scatter is curious. For the other planets studied, we found that the TDV scatter is greater than or equal to the TTV scatter. Also, for several planets we find the observed scatter is much lower than the uncertainties predict, suggesting we have overestimated our errors. If this is true for the Kepler-7b system, the TDV scatter indicates that the measurement uncertainties should be scaled by a factor of 0.71. This factor should be common to the TTV, which would change the observed χ² from 9.1 to 17.7, for 7 degrees of freedom. The scaled χ² would have a significance of 98.7%, or 2.5σ, which is still below our formal detection threshold. Nevertheless, we identify Kepler-7b as an important target for continued transit timing analysis.

The TTV F-test periodogram presents one significant peak very close to Nyquist frequency, which we may discard. The TDV periodogram presents no significant peaks.

Kepler-8b was discovered by Jenkins et al. (2010a). Kepler-8b posed several unique problems not encountered with the other Kepler planets. The data format of the public data was quite different from the other sets. Typical values of the photometric data were ∼0 whereas the other data sets were ∼1. This suggests that either the photometry was raw differential magnitudes or normalized fluxes with unity subtracted. We tried both assumptions and that the normalized fluxes minus unity assumption led to system parameters more consistent with the original discovery paper. This assumption has been since confirmed (J. Jenkins 2010, private communication).

Another peculiarity of the data is that the rms scatter is not constant but actually increases significantly with time, as discussed in Section 3.2. The source of this varying scatter is an instrumental effect, caused by oscillations in the photometry as a result of a heater on one of the two reaction wheel assemblies that modulates the focus by ∼1μm (see Jenkins et al. 2010c for more details). The resulting red noise induced on the photometry is shown in the periodogram of Figure 22. Since the discovery of this effect in the Q1 photometry, the Kepler Team has subsequently trimmed the bounding temperatures for this heater to reduce the amplitude and heater cycle (J. Jenkins 2010, private communication). Therefore, future data sets will not suffer from this effect as acutely as the photometry analyzed here. Concordantly, the results obtained in our analysis may not be as reliable as for the other Kepler planets.

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In our analysis, we disregard RV points occurring during the transit to avoid the Rossiter–McLaughlin effect. This is justified since our principal goal here is to produce refined system parameters rather measure the spin-orbit alignment.

We find the orbit is consistent with a circular orbit and derive a significantly lower planetary mass than that reported by Jenkins et al. (2010a). The fitted models on the phase-folded light curves are shown in Figure 23. The orbital fits to the RV points are shown in Figure 24, depicting both the circular and eccentric fits. The final planetary parameters are summarized at the bottom of Table 10.

The residuals of the light curve fits from method A (fitted limb darkening) were inspected using the Durbin–Watson statistic, which finds d = 1.344, indicating time correlated noise is certainly present. The origin of this is very likely the heater oscillations described earlier. An F-test periodogram finds the largest peak occurring at 3.27 hr with amplitude 50.6 ppm, significance 4.25σ. The rich forest of periodic lines make a correction unfeasible, as seen in Figure 23.

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Table 10. Fits for Kepler-8b

Parameter	Discovery	Method A.c	Method A.e	Model B.c	Model B.e
Fitted parameters
P (days)	3.52254+0.00003−0.00005	3.52226+0.00013−0.00013	3.522260+0.000066−0.000065	3.522418 ± 0.000057	3.522416 ± 0.000057
E (BJD−2,454,000)	954.1182+0.0003−0.0004	954.11849+0.00062−0.00062	954.1184+0.0021−0.0018	954.11912 ± 0.00040	954.11926 ± 0.00065
T1,4 (s)	...	11554+354−360	11780+139−143	11664 ± 173	11673 ± 173
T (s)	...	9837+358−373	9693+293−233	9994 ± 46	10000 ± 41
(T1,2 ≃ T3,4) (s)	...	1802+737−720	2417+274−323	1745 ± 207	1745 ± 207
(Rp/R⋆)2 (%)	...	0.887+0.091−0.106	0.942+0.092−0.041	0.870 ± 0.022	0.870 ± 0.022
b2	...	0.48+0.14−0.31	0.600+0.035−0.057	0.460+0.051−0.067	0.460+0.049−0.068
(ζ/R⋆) (day−1)	...	17.68+0.75−0.66	18.70+0.68−0.51	17.29 ± 0.08	17.28 ± 0.07
$(F_{P,\rm {d}}/F_\star)$ (ppm)	...	−12+103−105	...	−27 ± 51	−20 ± 62
esin ω	0a	0a	0.32+0.14−0.10	0a	−0.036 ± 0.062
ecos ω	0a	0a	−0.01+0.15−0.14	0a	0.030+0.081−0.108
K (ms−1)	68.4 ± 12.0	67.9+36.7−36.8	71.1+22.1−21.1	68.9 ± 18.4	68.1 ± 14.0
γ (ms−1)	−52.72 ± 0.10	−11.6+25.6−25.6	−10.8+13.7−13.9	−7.4 ± 12.4	−6.0 ± 12.1
$\dot{\gamma }$ (ms−1 day−1)	0a	0a	0a	0a	0a
ttroj (days)	0a	0a	0a	0a	0a
B	1a	1.000 ± 0.004b	1.000 ± 0.004b	1.000 ± 0.004b	1.000 ± 0.004b
SME derived parameters
teff (K)	6213 ± 150	6213 ± 150	6213 ± 150	6213 ± 150	6213 ± 150
log(g/cgs)	4.28 ± 0.10	4.28 ± 0.10	4.28 ± 0.10	4.28 ± 0.10	4.28 ± 0.10
[Fe/H] (dex)	−0.055 ± 0.06	−0.055 ± 0.06	−0.055 ± 0.06	−0.055 ± 0.06	−0.055 ± 0.06
vsin i (km s−1)	10.5 ± 0.7	10.5 ± 0.7	10.5 ± 0.7	10.5 ± 0.7	10.5 ± 0.7
u1	...	0.41+0.55−0.25	0.56+0.78−0.39	0.3585a	0.3585a
u2	...	0.11+0.44−0.83	−0.26+0.51−0.89	0.2917a	0.2917a
Model independent parameters
Rp/R⋆	0.09809+0.00040−0.00046	0.0942+0.0047−0.0058	0.0971+0.0046−0.0021	0.0933 ± 0.0012	0.0933 ± 0.0012
a/R⋆	6.97+0.20−0.24	7.16+1.57−0.82	4.71+0.51−0.61	7.12 ± 0.37	7.35 ± 0.49
b	0.724 ± 0.020	0.691+0.095−0.281	0.775+0.022−0.038	0.678+0.036−0.053	0.678+0.035−0.054
i/°	84.07 ± 0.33	84.5+2.8−1.6	75.8+3.2−6.8	84.5 ± 0.6	84.9 ± 0.6
e	0a	0a	0.35+0.15−0.11	0a	0.099 ± 0.056
ω/°	...	...	91+24−24	...	253 ± 111
RV jitter (m s−1)	...	19.7	20.7	31.5	27.1
ρ*/g cm−3	0.522	0.56+0.45−0.17	0.160+0.057−0.054	0.55+0.10−0.07	0.61+0.15−0.10
log(gP/cgs)	2.871 ± 0.119	2.91+0.27−0.36	2.50+0.15−0.18	2.92+0.11−0.17	2.94 ± 0.13
S1,4/s	...	11554+354−360	0+0−0	11664 ± 173	13046 ± 2592
ts/(BJD-2,454,000)	...	959.40166+0.00062−0.00062	957.624+1.64−0.32	955.88033 ± 0.00040	955.95 ± 0.21
Derived stellar parameters
M⋆/M☉	1.213+0.067−0.063	1.214+0.092−0.087	1.46+0.12−0.13	1.228+0.052−0.082	1.211+0.058−0.076
R⋆/R☉	1.486+0.053−0.062	1.46+0.21−0.26	2.33+0.40−0.26	1.460 ± 0.088	1.408 ± 0.107
log(g/cgs)	4.174 ± 0.026	4.192+0.149−0.094	3.860+0.084−0.110	4.19 ± 0.04	4.22 ± 0.05
L⋆/L☉	4.03+0.52−0.54	2.81+0.98−0.94	7.3+2.9−1.7	2.84 ± 0.50	2.64 ± 0.53
MV/mag	3.28 ± 0.15	3.66+0.45−0.34	2.61+0.31−0.37	3.64 ± 0.20	3.72 ± 0.23
Age/Gyr	3.84 ± 1.5	3.23+1.37−0.88	2.78+0.99−0.60	3.4+1.6−0.6	3.4+1.5−0.7
Distance/pc	1330 ± 180	935+158−174	1516+278−199	978 ± 62	943 ± 74
Derived planetary parameters
Mp/MJ	0.603+0.13−0.19	0.58+0.32−0.31	0.66+0.21−0.20	0.589 ± 0.160	0.575 ± 0.116
Rp/RJ	1.419+0.056−0.058	1.35+0.26−0.31	2.23+0.39−0.27	1.327 ± 0.092	1.278 ± 0.102
ρP/g cm−3	0.261 ± 0.071	0.28+0.36−0.17	0.0693+0.040−0.028	0.31+0.12−0.09	0.34+0.16−0.10
a/AU	0.0483+0.0006−0.0012	0.0483+0.0012−0.0012	0.0514+0.0014−0.0016	0.0485+0.0007−0.0011	0.0483+0.0007−0.0010
teq/K	1764 ± 200	1645+108−146	2059+190−126	1646 ± 58	1623 ± 67

Notes. Quoted values are medians of MCMC trials with errors given by 1σ quantiles. Two independent methods are used to fit the data (A&B) with both circular and eccentric modes (c&e). ^aFixed parameter. ^bParameter was floated but not fitted.

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The global circular fit yields a χ² = 20.96 and the eccentric fit obtains χ² = 20.55. Based on an F-test, the inclusion of two new degrees of freedom to describe the eccentricity is justified at a significance of 13.1% and thus is not accepted.

Using the circular fit, there is no evidence for a secondary eclipse; we can exclude secondary eclipses of depth 101.1 ppm to 3σ confidence, or equivalently geometric albedos ⩾0.63. We can also exclude a brightness temperature ⩾2932 K to the same confidence level, which places no constraints on the redistribution of heat around the planet.

The Yonsei–Yale model isochrones from Yi et al. (2001) for metallicity [Fe/H] = −0.055 are plotted in Figure 25, with the final choice of effective temperature T_eff⋆ and a/R_⋆ marked, and encircled by the 1σ and 2σ confidence ellipsoids, both for the circular and the eccentric orbital solution. Here, the MCMC distribution of a/R_⋆ comes from the global modeling of the data.

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The transit times and durations are listed in Table 11 and plotted in Figure 26. We find TTV residuals of 68.1 s and TDV residuals of 92.2 s. The TTV indicates timings consistent with a linear ephemeris, producing a χ² of 10.7 for 11 degrees of freedom. The TDV is also consistent with a non-variable system exhibiting a χ² of 18.8 for 12 degrees of freedom (84.9% significance of excess).

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Table 11. Mid-transit Times, Transit Durations, Transit Depths, and Out-of-transit (Normalized) Fluxes for Kepler-8b

Epoch	tc/(BJD−2,454,000)	T (s)	(Rp/R⋆)2(%)	Foot
0	953.61846+0.00043−0.00043	9887.5+88.5−86.6	0.878+0.041−0.053	0.999971+0.000022−0.000022
1	957.14108+0.00043−0.00043	10064.9+88.4−88.3	0.803+0.042−0.031	0.999999+0.000022−0.000022
2	960.66300+0.00045−0.00044	10052.9+84.7−84.9	0.766+0.040−0.024	0.999984+0.000023−0.000023
3	964.18472+0.00043−0.00044	9850.0+91.8−88.2	0.866+0.055−0.060	0.999974+0.000026−0.000026
4	967.70814+0.00054−0.00054	9865.0+123.6−130.1	0.864+0.056−0.072	0.999946+0.000027−0.000026
5	971.23018+0.00056−0.00056	9911.2+113.7−112.5	0.925+0.045−0.054	0.999951+0.000029−0.000029
6	974.75106+0.00066−0.00066	10292.7+139.2−136.6	0.820+0.054−0.055	0.999993+0.000032−0.000032
7	978.27476+0.00067−0.00065	10192.1+132.1−138.6	0.847+0.076−0.055	1.000071+0.000036−0.000036
8	981.79649+0.00076−0.00079	10083.6+168.9−160.2	0.871+0.061−0.062	0.999998+0.000039−0.000040
9	985.31931+0.00090−0.00092	$10151.9_{-_190.0}^{+183.6}$	0.832+0.073−0.069	1.000013+0.000044−0.000045
10	988.84000+0.00098−0.00095	10283.4+196.2−218.5	0.836+0.105−0.073	1.000024+0.000051−0.000051
11	992.36279+0.00102−0.00099	9740.2+220.9−206.6	0.861+0.097−0.071	0.999981+0.000056−0.000056
12	995.88753+0.00132−0.00128	9796.6+255.2−278.6	0.809+0.087−0.062	0.999909+0.000063−0.000063

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For the TTV, we note that for 11 degrees of freedom, excess scatter producing a χ² of 28.5 would be detected to 3σ confidence. This therefore excludes a short-period signal of rms amplitude ⩾58.8 s. Similarly, for the TDV, we exclude scatter producing a χ² of 30.1, or a short-period rms amplitude of ⩾73.0 s, to 3σ confidence. This constitutes a 1.46% variation in the transit duration.

These limits place constraints on the presence of perturbing planets, moons, and Trojans. For a 2:1 mean-motion resonance, the libration period would be ∼83 cycles and thus we do not possess sufficient baseline yet to look for such perturbers. However, a 4:3 resonance system would yield a libration period of ∼19 cycles and therefore we would expect to see such a signal with the 13 cycles observed. This means a 4:3 perturber of ⩾0.50 M_⊕ is ruled out to 3σ confidence by the current TTV data for this object. After 83 cycles have been obtained at the same photometric quality, the presence of a 2:1 resonant perturber of mass ⩾0.17 M_⊕ may be constrained.

The current TTV data rule out an exomoon at the maximum orbital radius (for retrograde revolution) of ⩾2.1 M_⊕. The current TDV data rule an exomoon of ⩾21.5 M_⊕ at a minimum orbital distance.

Using the expressions of Ford & Holman (2007) and assuming $M_{p}\gg M_{\rm {Trojan}}$, the expected libration period of Kepler-8b induced by Trojans at L4/L5 is 25.5 cycles and therefore we do not yet possess sufficient baseline to search for these TTVs. Inspection of the folded photometry at ±P/6 from the transit center excludes Trojans of effective radius ⩾1.58 R_⊕ to 3σ confidence.

The TTV periodogram reveals a peak at twice the Nyquist period, which we disregard. No other significant peaks are present. The TDV also has no significant peaks. The phasing signal appears to have a period of 2 cycles.