CHANGING PHASES OF ALIEN WORLDS: PROBING ATMOSPHERES OF KEPLER PLANETS WITH HIGH-PRECISION PHOTOMETRY

In our analysis, we used both the Kepler LC and short cadence (SC) simple aperture data (see Table 1). In the only multiplanet system, Kepler-10b, we removed the second planet's transit. Instrumental signals were removed by performing a linear least squares fit³ of the first eight cotrending basis vectors (CBVs) to the time-series of each quarter individually. Before cotrending, we removed any bad data points flagged by Kepler in the SAP or CBV files and to prevent contamination we only fit the CBVs to the out-of-transit time-series. The fitted basis vectors were then divided out of the quarter, in order to preserve the amplitude of the physical signals of interest. Since CBVs are only provided for the LC data, we interpolated onto the SC time-stamps using cubic splines.

The CBVs are unique to each quarter and channel of the detector. They are ordered such that the first CBV is the dominant systematic signal of the channel and the subsequent CBVs decrease in signal strength. However, every target within a given channel is not similarly affected by instrumental systematics and therefore it is possible for the relative strengths of each CBV to vary. Determining which CBVs to fit in order to best remove systematics and preserve real astronomical signals is out of the scope of this paper. However, we do investigate the effect of varying the number of CBVs and find that there is little variation in the strength of the first four CBVs when additional CBVs are added. For each target and quarter we opt to fit the first eight CBVs, the maximum recommended by the Kepler Data Processing Handbook, as we cannot fully assess the affect that excluding CBVs would have on our analysis.

In order to remove quarter-to-quarter discontinuities, we normalized each quarter to its out-of-transit median. After cotrending and combining all quarters, we removed outliers by calculating a running median and standard deviation of 21 measurements around each point and rejecting measurements that differed by more than 5σ. For each planet, the raw Kepler simple aperture photometry, the cotrended light curve and the cotrended out-of-transit light curve after outlier removal can be found in Figures 11–17.

Kepler's large pixel size, with a width of 398, allows for the possibility of dilution from a background or foreground star or a nearby stellar companion. In the literature we found that several of our 14 systems have one or more nearby stellar companions (see Table 2). However, only KOI-13 is significantly diluted by its companion. For KOI-13, studies find a large range of dilutions: 38–48% (Szabó et al. 2011; Adams et al. 2012; Shporer et al. 2014). In our analysis, we adopt the Shporer et al. (2014) value of 48%, corresponding to a dilution factor of 1.913 ± 0.019. We also account for the quarter-to-quarter third light fraction provided by the Kepler Input Catalog, the average of which can be found in Tables 5–8.

Our model fits for $\sqrt{e}{\rm cos} \;\omega$ and $\sqrt{e}{\rm sin} \;\omega$ (e.g., Triaud et al. 2011), where e is the orbital eccentricity and ω is the argument of periapsis.

For eccentric orbits, unlike for circular orbits, the true anomaly (ν) does not change linearly in time and the star-planet separation (d) is time dependent. To adjust our phase curve model to account for this, the true anomaly and the separation were calculated in a manner similar to Bonavita et al. (2012, 2013), derived using the ephemeris formulae of Heintz (1978), as

$$\begin{eqnarray}&&{{E}_{0}}=M+e\;{\rm sin} \;M+\frac{{{e}^{2}}}{2}\;{\rm sin} (2M)\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{E}_{n+1}}={{E}_{n}}+\frac{M-{{E}_{n}}+e\;{\rm sin} \;{{E}_{n}}}{1-e{\rm cos} {{E}_{n}}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\nu =\frac{1}{\pi }\;{\rm arctan} \left\{ \sqrt{\frac{1+e}{1-e}}{\rm tan} \left( \frac{E}{2} \right) \right\}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&d=a\frac{1-{{e}^{2}}}{1+e\;{\rm cos} \;\nu }\end{eqnarray} \tag{ 5 }$$

where M is mean anomaly, E is the eccentric anomaly and a, the semimajor axis, can be calculated from d and the planet-star separation during transit.

The projected star-planet separation (z₀) can then be calculated by

$$\begin{eqnarray}&&{{z}_{0}}=\frac{d}{{{R}_{\star }}}\sqrt{1-{{{\rm sin} }^{2}}(\nu +\omega ){{{\rm sin} }^{2}}i}\end{eqnarray} \tag{ 6 }$$

where i is the orbital inclination.

Note that the mean anomaly is defined from periastron passage as

$$\begin{eqnarray}&&M=2\pi \frac{t-{{T}_{{\rm peri}}}}{P}\end{eqnarray} \tag{ 7 }$$

where ${{T}_{{\rm peri}}}$ is the time of periastron passage. This is important because the orbital phase used in our models is defined from mid-transit as

$$\begin{eqnarray}&&\phi =2\pi \frac{t-{{T}_{{\rm mid}}}}{P}\end{eqnarray} \tag{ 8 }$$

where ${{T}_{{\rm mid}}}$ is the time of mid-transit. As a result, there is a phase offset between M and ϕ. This phase offset is taken into account by requiring that $\nu +\omega$ at mid-transit be 90°. Subsequently all parameters computed from M ($\nu +\omega$, d and z₀) must be similarly offset.

In Sections 3.2–3.4 we describe the circular phase curve models. The eccentric models can be obtained by substituting ϕ for $\nu +\omega$ and a for d.

We model the secondary eclipse using the formalism from Mandel & Agol (2002) for a uniform source and normalize the model by R$_{{\rm p}}^{2}$/R$_{\star }^{2}$. Here an eclipse model amplitude of one corresponds to a fully occulted planet while a grazing eclipse results in an amplitude less than one. The planetary brightness is modeled as a Lambert sphere described by

$$\begin{eqnarray}&&{{F}_{{\rm p}}}={{A}_{{\rm p}}}\frac{{\rm sin} \;z+(\pi -z){\rm cos} \;z}{\pi }\end{eqnarray} \tag{ 9 }$$

where ${{A}_{{\rm p}}}$ is the peak brightness amplitude, θ is a phase offset in peak brightness and z is defined by

$$\begin{eqnarray}&&{\rm cos} (z)=-{\rm sin} \;i\;{\rm cos} (2\pi [\phi +\theta ])\end{eqnarray} \tag{ 10 }$$

Assuming only reflected light from the planet, the geometric albedo in the Kepler bandpass (${{A}_{{\rm g}}}$) can then be calculated by

$$\begin{eqnarray}&&{{F}_{{\rm ecl}}}={{A}_{{\rm g}}}{{\left( \frac{{{R}_{p}}}{a} \right)}^{2}}\end{eqnarray} \tag{ 11 }$$

Planet mass (${{M}_{{\rm p}}}$) was measured through the analysis of Doppler boosting and ellipsoidal variations. These signals, in-phase with planet's period, are stellar brightness fluctuations induced by the planet's gravity.

Doppler boosting is a combination of a bolometric and a bandpass-dependent effect. The bolometric effect is the result of non-relativistic Doppler boosting of the stellar light in the direction of the star's radial velocity (RV). The observed periodic brightness change is proportional to the star's RV, which is a function of the planet's distance and mass (Barclay et al. 2012). While the bandpass-dependent effect is a periodic red/blueshift of the star's spectrum, which results in a periodic change of the measured brightness as parts of the star's spectrum move in and out of the observed bandpass (Barclay et al. 2012).

Ellipsoidal variations, on the other hand, are periodic changes in observed stellar flux caused by fluctuations of the star's visible surface area as the stellar tide, created by the planet, rotates in and out of view of the observer (Mislis et al. 2012). If there is no tidal lag, the star's visible surface area and ellipsoidal variations are at maximum when the direction of the tidal bulge is perpendicular to the observer's line of sight (i.e., ϕ of 0.25 and 0.75) and at minimum during the transit and the secondary eclipse.

The Doppler (first term in Equation (12)) and ellipsoidal (second term in Equation (12)) contributions can be described by

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{F}_{{\rm m}}} & = & {{M}_{{\rm p}}}\cdot \left\{ {{\left( \frac{2\pi G}{P} \right)}^{1/3}}\frac{{{\alpha }_{{\rm d}}}\;{\rm sin} \;i}{c\cdot M_{\star }^{2/3}} \right. \\ {} & {} & \times \left. \left( \frac{1+e\;{\rm cos} \;\omega }{\sqrt{1-{{e}^{2}}}} \right){{f}_{{\rm d}}}-\frac{{{\alpha }_{2}}\;{{{\rm sin} }^{2}}i}{{{M}_{\star }}}{{f}_{e}} \right\} \\ \end{array}\end{eqnarray} \tag{ 12 }$$

where ${{M}_{\star }}$ is the host star mass, G is the universal gravitational constant, c is the speed of light and ${{\alpha }_{{\rm d}}}$ is the photon-weighted bandpass-integrated beaming factor. Here we have assumed ${{M}_{{\rm p}}}\ll {{M}_{\star }}$ and similar to E13 and Barclay et al. (2012) we calculate ${{\alpha }_{{\rm d}}}$ in the manner described by Bloemen et al. (2011) and Loeb & Gaudi (2003).

In Equation (12), f_d and f_e are the phase dependent modulations of the Doppler boosting and ellipsoidal signals, respectively. They can be described by

$$\begin{eqnarray}&&{{f}_{d}}={\rm sin} (2\pi \phi )\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{f}_{e}} & = & {{\left( \frac{a}{{{R}_{\star }}} \right)}^{-3}}{\rm cos} (2\cdot 2\pi \phi ) \\ {} & {} & +{{\left( \frac{a}{{{R}_{\star }}} \right)}^{-4}}{{f}_{1}}{\rm cos} (2\pi \phi ) \\ {} & {} & +{{\left( \frac{a}{{{R}_{\star }}} \right)}^{-4}}{{f}_{2}}{\rm cos} (3\cdot 2\pi \phi ) \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where f₁ and f₂ are constants used to determine the amplitude of the higher-order ellipsoidal variations (Morris 1985)

$$\begin{eqnarray}&&{{f}_{1}}=3{{\alpha }_{1}}\frac{5\;{{{\rm sin} }^{2}}i-4}{{\rm sin} \;i}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{f}_{2}}=5{{\alpha }_{1}}\;{\rm sin} \;i\end{eqnarray} \tag{ 16 }$$

In Equations (12)–(16), ${{\alpha }_{1}}$ and ${{\alpha }_{2}}$ are functions of the linear limb darkening and gravity darkening parameters (Claret & Bloemen 2011), which we calculate in the same manner as E13 and Barclay et al. (2012). The values of f₁, f₂, ${{\alpha }_{{\rm d}}}$, ${{\alpha }_{1}}$ and ${{\alpha }_{2}}$ can be found in Table 3.

Table 3.  Relevant Constants for Doppler Boosting and Ellipsoidal Modeling

	${{\alpha }_{d}}$	${{\alpha }_{1}}$	${{\alpha }_{2}}$	${{f}_{1}}$	${{f}_{2}}$	$u$	$y$
Kepler-5b	3.44	0.0649	1.23	0.194	0.324	0.545	0.290
Kepler-6b	3.84	0.0718	1.38	0.215	0.359	0.628	0.398
Kepler-7b	3.68	0.0679	1.30	0.197	0.338	0.582	0.344
Kepler-8b	3.51	0.0651	1.24	0.186	0.324	0.549	0.299
Kepler-10b	3.85	0.0692	1.36	0.201	0.345	0.603	0.391
Kepler-12b	3.66	0.0677	1.30	0.203	0.338	0.580	0.341
Kepler-41b	3.85	0.0718	1.39	0.197	0.356	0.629	0.402
Kepler-43b	3.61	0.0687	1.29	0.198	0.342	0.586	0.329
Kepler-76b	3.38	0.0637	1.20	0.150	0.311	0.532	0.273
Kepler-91b	4.72	0.0802	1.60	0.102	0.376	0.734	0.540
Kepler-412b	3.77	0.0706	1.35	0.186	0.348	0.613	0.379
TrES-2b	3.71	0.0674	1.31	0.192	0.335	0.580	0.354
HAT-P-7b	3.41	0.0657	1.22	0.184	0.326	0.551	0.282
KOI-13ba	2.77	0.0550	1.29	0.163	0.274	0.477	0.406
KOI-13bb	2.27	0.0492	1.39	0.146	0.246	0.443	0.539
KOI-13bc	2.43	0.0517	1.49	0.153	0.258	0.476	0.624

Notes. Although ${{\alpha }_{1}}$, u, and y are not explicitly mentioned in this paper their description and use in calculating ${{\alpha }_{2}}$, f₁, and f₂ can be found in Esteves et al. (2013). Derived using stellar parameters from:

^aShporer et al. (2014). ^bHuber et al. (2014). ^cSzabó et al. (2011).

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The phase curve is also fit with planet mass fixed to the RV derived value calculated via

$$\begin{eqnarray}&&K={{\left( \frac{2\pi G}{P} \right)}^{1/3}}\frac{{{M}_{{\rm p}}}\;{\rm sin} \;i}{M_{\star }^{2/3}\sqrt{1-{{e}^{2}}}}\end{eqnarray} \tag{ 17 }$$

where K is the published RV semi-amplitude (see Tables 5–8).

In E13, we discovered of a phase-shifted 6.7 ± 0.3 ppm third harmonic in the residual of KOI-13b's phase curve. To investigate if this signal is present for other planets our model also includes a cosine third harmonic, which we allow to vary in amplitude (A₃) and phase (${{\theta }_{3}}$) and is described by

$$\begin{eqnarray}&&{{F}_{3}}={{A}_{3}}{\rm cos} (3\cdot 2\pi (\phi -{{\theta }_{3}}))\end{eqnarray} \tag{ 18 }$$

For each system, we compute and analyze the light curve autocorrelation as described by McQuillan et al. (2013). For each target, the autocorrelation indicates the presence of correlated variability and/or evidence for systematic effects such as linear trends and jumps. For seven planets (Kepler-7b, -8b, -41b, -43b, -76b, -412b, and TrES-2b), the peak and shape of the autocorrelation shows evidence for sinusoidal light curve variability with an evolving phase and amplitude, most likely by stellar variability (e.g., starspots rotating in and out of view once per stellar rotation period). For TrES-2b and Kepler-412b we find a stellar rotation period at 9 and 10 times the planets period, respectively. The latter of which is in agreement with the stellar rotation period derived from Deleuil et al. (2014). For the others we find that the rotation period is significantly offset from multiples of the orbital period.

For Kepler-91b, the autocorrelation exhibits stochastic variability, with a dominant timescale equal to the planet's period, and does not exhibit repeatability or resemble the signal expected from rotation. McQuillan et al. (2013) find these signal characteristics typical of red giant stars, like Kepler-91b's host star Lillo-Box et al. (2014), as they are known to show significant correlated noise on timescales of hours to weeks due to stellar granulation (Mathur et al. 2012; Barclay et al. 2015). The affect of Kepler-91b's orbital period phased stellar variability is discussed in Section 5.3. For the six other systems we did not find the stellar rotation period as the autocorrelation did not exhibit a clear dominant signal.

Since the phase and ampitude of this stellar variability evolves with time it is unclear how it will affect our fitted phase curve parameters. To assess this we performed a bootstrap, using a simple regression analysis, to measure ${{F}_{{\rm ecl}}}$, ${{A}_{{\rm p}}}$, and ${{M}_{{\rm p}}}$ for subsets of the data. To simplify the model, the peak offset and third harmonic were fixed to the best-fit parameters from our MCMC analysis. For each fit we randomly selected half of the available individual orbits, allowing for orbits to be redrawn. However, to prevent skewing from incomplete phase curves we only selected orbits where at least 80% of the data was available. For each planet we fit 5000 subsets for the amplitudes of ${{F}_{{\rm ecl}}}$, ${{A}_{{\rm p}}}$, and ${{M}_{{\rm p}}}$. We then compared the peak and standard deviation of the bootstrap parameter distribution to the best-fit and 1σ values from our MCMC analysis.

If only random noise is present we would expect bootstrap uncertainties $\sqrt{2}$ times larger than our MCMC uncertainties. However, we find that, although the peak value of the bootstrap distribution is in agreement with our best-fit MCMC values, the uncertainties derived from the bootstrap are consistently larger than thoses from the MCMC. Since the MCMC derived errors fail to reflect the time-variable noise characteristics of the data, we chose to adopt the larger uncertainty values derived from our bootstrap analysis.

With the exception of Kepler-43b, the increase in uncertainty does not greatly alter the significance level of our fitted parameters. While for Kepler-43b the significance of our measurements of M_p, A_p, and F_ecl is reduced to 1, 1, and 0σ, respectively. The larger uncertainty in the bootstrap parameters reflect variations in the phase curve amplitude and/or shape between different subsection of the data. The effect of this increased phase curve variability on is discussed in Section 5.3.

Previous studies of Kepler-91b's (KOI-2133) transit and phase curve have yielded differing orbital and planet parameters. The Kepler transit fit, performed by E13 yielded orbital parameters that are used, in conjunction with the eclipse depth, to conclude that, due to its large albedo (>1), Kepler-91b is a false positive. Sliski & Kipping (2014) also use Kepler data to find a similar set of orbital parameters, but use an asteroseismic analysis to conclude that the stellar density is not consistent with their measured orbital parameters. Lillo-Box et al. (2014) find a drastically different set of orbital parameters and confirm the planetary nature of Kepler-91b using high-resolution images and through the reanalysis of the Kepler data, including transit, phase curve, and asteroseismic analysis. Recently, Barclay et al. (2015) simultaneously model RV and Kepler data using a Gaussian process to find a set of orbital parameters similar to those found by Lillo-Box et al. (2014). Barclay et al. (2015) hypothesize that the false positive conclusions found by previous studies are due to a temporally correlated stellar noise component, caused by stellar granulation on Kepler-91b's red giant host star.

In this analysis we find orbital and planet parameters consistent with those found by Barclay et al. (2015) and with the planetary nature of Kepler-91b (see Table 7). Possible causes for the differing results between our studies is the inclusion of an additional three quarters of data, leading to an increase in accuracy as variations are averaged over longer timescales. In our current analysis we also simultaneously fit for the phase-curve and the transit, and fix the planetary mass to that derived from RV measurements by Barclay et al. (2015).

Table 7.  Stellar and Planetary Parameters

Parameter	Kepler-76b	Kepler-91b	Kepler-412b	TrES-2b
KOI	1658.01	2133.01	202.01	1.01
KIC	4570949	8219268	7877496	11446443
Period (days)c	1.5449298 ± 0.0000004	6.24658 ± 0.00008	1.7208604 ± 0.0000003	2.47061317 ± 0.00000009
${{T}_{\star }}$ (K)	6409 ± 95e	4550 ± 75f	5750 ± 90h	5850 ± 50i
${\rm log} \;g$ (cgs)	4.2 ± 0.3e	2.953 ± 0.007f	4.30 ± 0.07h	4.426$_{-0.023}^{+0.021}$ i
$[{\rm Fe}/{\rm H}]$	-0.1 ± 0.2e	0.11 ± 0.07f	0.27 ± 0.12h	-0.15 ± 0.10i
${{R}_{\star }}/{{R}_{\odot }}$	1.32 ± 0.08e	6.30 ± 0.16f	1.287 ± 0.035h	1.000$_{-0.033}^{+0.036}$ i
${{M}_{\star }}/{{M}_{\odot }}$	1.2 ± 0.2e	1.31 ± 0.10f	1.167 ± 0.091h	0.980 ± 0.062i
K (m s−1)	306 ± 20e	70 ± 11g	142 ± 11h	181.3 ± 2.6j
${{M}_{{\rm p}}}$ from RV $({{M}_{{\rm J}}})$	2.01$_{-0.35}^{+0.37}$	0.81$_{-0.17}^{+0.18}$	0.941$_{-0.019}^{+0.125}$	1.197 ± 0.068
Avg 3rd light (%)d	5.55	1.00	5.96	0.729
Transit fit
T0(BJD-2454833)	133.54841$_{-0.00002}^{+0.00001}$	136.3958 ± 0.0002	133.02122 ± 0.00002	122.763360$_{-0.000002}^{+0.000003}$
b	0.96226$_{-0.0049}^{+0.0040}$	0.8667$_{-0.0105}^{+0.0077}$	0.7942$_{-0.0032}^{+0.0028}$	0.84359$_{-0.00062}^{+0.00075}$
${{R}_{{\rm p}}}/{{R}_{\star }}$	0.1033$_{-0.0030}^{+0.0024}$	0.02181$_{-0.00042}^{+0.00054}$	0.10474$_{-0.00077}^{+0.00054}$	0.12539$_{-0.00035}^{+0.00049}$
$a/{{R}_{\star }}$	4.464$_{-0.041}^{+0.049}$	2.496$_{-0.043}^{+0.050}$	4.841$_{-0.023}^{+0.024}$	7.903$_{-0.016}^{+0.019}$
i (degrees)	77.55$_{-0.17}^{+0.20}$	69.68$_{-0.57}^{+0.67}$	80.559$_{-0.079}^{+0.084}$	83.872$_{-0.018}^{+0.020}$
u1	0.762$_{-0.139}^{+0.073}$	0.73$_{-0.19}^{+0.13}$	0.36$_{-0.15}^{+0.11}$	0.330$_{-0.060}^{+0.080}$
u2	−0.346$_{-0.047}^{+0.067}$	−0.002$_{-0.1}^{+0.2}$	0.29$_{-0.12}^{+0.18}$	0.285$_{-0.087}^{+0.062}$
Phase curve fit
${{F}_{{\rm ecl}}}$ (ppm)	131.6$_{-8.0}^{+8.7}$	35 ± 18b	53$_{-24}^{+21}$	7.7$_{-2.6}^{+2.4}$
${{A}_{{\rm p}}}$ (ppm)	106.9$_{-5.6}^{+6.2}$	25$_{-19}^{+18}$ b	31$_{-19}^{+20}$	4.1$_{-1.8}^{+1.7}$
θ (in phase)	0.0230 ± 0.0034	−0.132$_{-0.047}^{+0.050}$ b	0a	0a
${{F}_{{\rm n}}}$ (ppm)	28 ± 14	17$_{-30}^{+31}$ b	22$_{-44}^{+40}$	3.6$_{-4.3}^{+4.2}$
${{M}_{{\rm p}}}$ $({{M}_{{\rm J}}})$	2.01a	0.81a,b	0.941a	1.197a
Derived parameters
${{R}_{{\rm p}}}$ $({{R}_{{\rm J}}})$	1.36 ± 0.12	1.367$_{-0.060}^{+0.069}$ b	1.341$_{-0.046}^{+0.044}$	1.247$_{-0.045}^{+0.050}$
a (Au)	0.0274$_{-0.0019}^{+0.0020}$	0.0731$_{-0.0031}^{+0.0034}$ b	0.02897$_{-0.00092}^{+0.00093}$	0.0367$_{-0.0013}^{+0.0014}$
${{A}_{{\rm g},{\rm ecl}}}$	0.246$_{-0.029}^{+0.038}$	0.46$_{-0.26}^{+0.30}$ b	0.113$_{-0.053}^{+0.049}$	0.031$_{-0.010}^{+0.0099}$
${{T}_{{\rm eq},{\rm max} }}$ (K)	2740$_{-60}^{+50}$	2600 ± 70b	2360 ± 40	1880 ± 20
${{T}_{{\rm eq},{\rm hom} }}$ (K)	2140 ± 40	2040 ± 50b	1850 ± 30	1470 ± 10
${{T}_{{\rm B},{\rm day}}}$ (K)	2890$_{-60}^{+70}$	3000$_{-300}^{+200}$ b	2400$_{-200}^{+100}$	1910$_{-80}^{+60}$
${{T}_{{\rm B},{\rm night}}}$ (K)	2380$_{-200}^{+150}$	$_{\lt 3400(2\sigma )}^{\lt 3200(1\sigma )}$ b	$_{\lt 2600(2\sigma )}^{\lt 2500(1\sigma )}$	$_{\lt 2000(2\sigma )}^{\lt 1900(1\sigma )}$
${{\rho }_{p}}$ (g cm−3)	1.1$_{-0.4}^{+0.6}$	0.4$_{-0.1}^{+0.2}$ b	0.52$_{-0.04}^{+0.13}$	0.8 ± 0.1
${\rm log} \;{{g}_{p}}$ (cgs)	3.5 ± 0.2	3.0 ± 0.1b	3.13$_{-0.02}^{+0.08}$	3.30 ± 0.06

Notes. Depth has been scaled to account for the partial eclipse of Kepler-76b, ${{F}_{{\rm ecl},{\rm unscaled}}}=95.19$ ppm (see Section 3.2).

^aBest-fit model favored fixed values for these parameters. ^bPlanet's with evidence for non-planetary phase curve modulations (see Section 5.3). ^cFrom the NASA Exoplanet Archive (Akeson et al. 2013). ^dFrom the Kepler Input Catalog. ^eFrom Faigler et al. (2013). ^fFrom Lillo-Box et al. (2014). ^gFrom Barclay et al. (2015). ^hFrom Deleuil et al. (2014). ⁱFrom Sozzetti et al. (2007). ^jFrom O'Donovan et al. (2006).

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Similar to previous studies, our Kepler-91b phase curve exhibits dips and jumps with amplitudes similar to those of our measured eclipse depth. We find that the amplitudes of these variations are actually larger than the error estimates derived from our bootstrap, as correlated noise on timescales of the Kepler-91b's planetary period is coherent over a significant number of transits.

Kepler-43b's autocorrelation reveals light curve variations with a period of 12.8 days, which is also visible by eye in the unfolded time-series (see Figure 4). These variations, which exhibit a double-dip feature, are indicative of a bimodel brightness distribution, due to concentrations of active regions on opposite hemispheres McQuillan et al. (2013).

Of our targets, Kepler-43b was the only planet where our bootstrap analysis considerably reduced the significance of our detection (see Section 5.2), which we attribute to large star spot induced variations. Further evidence for Kepler-43b's phase curve contamination can be found in residuals, where correlated noise, with an amplitude similar to the eclipse depth, can be seen. Furthermore, we find that the peak planetary brightness is several times larger than the eclipse depth. This is only possible if the planet does not fully pass behind the star, which would require a highly inclined and eccentric orbit both of which are not found by our phase curve analysis or the RV analysis from Bonomo et al. (2012).

Since Kepler-43b and Kepler-91b show compelling evidence for a non-planetary phase curve signal, we chose to exclude them from our discussion in Section 6.

For KOI-13b, also referred to in the literature as Kepler-13b, we ran our fits using three sets of stellar parameters(Szabó et al. 2011; Huber et al. 2014; Shporer et al. 2014). We do this only for KOI-13b because the reported stellar parameters differ greatly from one another and more importantly because the choice of stellar parameters strongly influences our derived equilibrium temperature. This is of particular importance for KOI-13b as its blackbody peaks very close to the edge of the Kepler bandpass. Therefore a small increase in equilibrium temperature can lead to a significant increase in the contribution from thermal emission in the Kepler bandpass, while a small decrease in temperature will require a larger amount of reflected light to explain the observed secondary eclipse depth. Further discussion of KOI-13b's equilibrium temperature and albedo can be found in Section 5.7.

The determination of KOI-13b's host star temperature is complicated by the fact that it has a stellar companion at 112 with almost the same apparent brightness. For the rest of our discussion we adopt the results obtained using the most recent spectroscopically derived values from Shporer et al. (2014) as our best-fit values, but we note that the results for this planet strongly depend on the assumed stellar parameters.

Only three planet's in our sample (HAT-P-7b and Kepler-43b) did not favor a model with a fixed mass. However, for both planets our light curve derived mass is within 2σ of the RV derived value. In Table 9 we present all >1σ published Kepler eclipse depths, planetary brightness, ellipsoidal (A$_{{\rm e}}$) and Doppler boosting (A$_{{\rm d}}$) amplitudes for the 14 planets in our sample. In this table we have presented all planetary brightness measurements as peak-to-peak amplitudes and all ellipsoidal and Doppler measurements as semi-amplitudes. In addition we have adjusted the amplitudes for KOI-13b from E13, Angerhausen et al. (2014), and Shporer et al. (2011), to account for dilution due to a nearby companion (see Section 2.2).

For all published measurements, where errors were reported, we calculate the σ deviation from our results. A histogram of the differences can be found in Figure 6. For planets where a fixed mass was favored we report RV derived amplitudes and use their associated errors in our comparison to published values.

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In the majority of studies, the eclipse depths, planetary brightness and ellipsoidal amplitude are within 1σ of our values, while for Doppler boosting most previous values lie within 2σ and are skewed toward higher values. There are also some notable outliers. For HAT-P-7b and KOI-13b early phase curve studies, using significantly less data, differ by more than 5σ. While more recent studies of Kepler-76b and HAT-P-7b(E13; Faigler et al. 2013; Angerhausen et al. 2014), find significantly higher Doppler boosting amplitudes. We hypothesize that the inclusion of a planetary brightness offset is the source of this discrepancy between Doppler measurements and that pre-eclipse brightness shifts were previously measured as increased Doppler boosting.

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To test if a degeneracy exist, we refit HAT-P-7b and Kepler-76b with ellipsoidal variations and Doppler boosting as separate free parameters with no priors. From our posterior distributions (see top panel of Figure 8), we find that there is a correlation between the Doppler boosting and brightness offset, where a smaller positive offset (i.e., a pre-eclipse shift) results in a larger Doppler amplitude. We also find that reducing the offset results in a lower planetary brightness amplitude (see bottom panel of Figure 8), which could account for the significantly larger brightness amplitude we find for HAT-P-7b (see Table 9). For the other phase curve parameters, eclipse depth and ellipsoidal amplitude, we do not find a correlation with brightness offset.

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For two of our targets, HAT-P-7b and KOI-13b, we found that the favored model included a phase-shifted cosine third harmonic signal (see Section 3.4 for description and Table 8 for values). For KOI-13b we measure a third harmonic amplitude in agreement with our previous value from E13 and ∼3 ppm less than the amplitude found by Shporer et al. (2014). This third harmonic could be due to the movement of the stellar tidal bulge raised by the planet (i.e., the source of ellipsoidal variations) across areas of the star with different surface brightnesses. The motivation for this reasoning is the asymmetry in KOI-13b's transit caused by a spin–orbit misalignment (Barnes et al. 2011; Szabó et al. 2011) and significant gravity darkening due to rapid stellar rotation ($v\;{\rm sin} \;i$ = 65–70 km s⁻¹; Szabó et al. 2011).

Table 8.  Stellar and Planetary Parameters

Parameter	HAT-P-7b		KOI-13b
		Model 1	Model 2	Model 3
KOI	2.01	⋯	13.01	⋯
KIC	10666592	⋯	9941662	⋯
Period (days)b	2.2047354 ± 0.0000001	⋯	1.763588 ± 0.000001	⋯
${{T}_{\star }}$ (K)	6350 ± 80d	7650 ± 250e	9107$_{-425}^{+257}$ f	8511 ± 1g
${\rm log} \;g$ (cgs)	4.07$_{-0.08}^{+0.04}$ d	4.2 ± 0.5e	3.867$_{-0.148}^{+0.235}$ f	3.9 ± 0.1g
$[{\rm Fe}/{\rm H}]$	0.26 ± 0.08d	0.2 ± 0.2e	0.070$_{-0.650}^{+0.140}$ f	0.2 ± 0.1g
${{R}_{\star }}/{{R}_{\odot }}$	1.84$_{-0.11}^{+0.23}$ d	1.74 ± 0.04e	3.031$_{-0.944}^{+1.198}$ f	2.55 ± 0.1g
${{M}_{\star }}/{{M}_{\odot }}$	1.47$_{-0.05}^{+0.08}$ d	1.72 ± 0.10e	2.47$_{-0.72}^{+0.45}$ f	2.05 ± 0.1g
K (m s−1)	213.5 ± 1.9d		...
${{M}_{{\rm p}}}$ from RV $({{M}_{{\rm J}}})$	1.781$_{-0.056}^{+0.081}$	⋯	...	⋯
Avg 3rd light (%)c	0.184	⋯	0.142	⋯
Transit fit
T0 (BJD-2454833)	121.358572 ± 0.000004	⋯	120.56596$_{-0.00003}^{+0.00002}$	⋯
b	0.4960$_{-0.0013}^{+0.0011}$	0.2536$_{-0.0035}^{+0.0038}$	0.2537$_{-0.0034}^{+0.0037}$	0.2535$_{-0.0034}^{+0.0033}$
${{R}_{{\rm p}}}/{{R}_{\star }}$	0.077524$_{-0.000022}^{+0.000017}$	0.087373$_{-0.000024}^{+0.000023}$	0.087373$_{-0.000022}^{+0.000025}$	0.087371$_{-0.000023}^{+0.000022}$
$a/{{R}_{\star }}$	4.1545$_{-0.0025}^{+0.0029}$	4.5007$_{-0.0040}^{+0.0039}$	4.5008$_{-0.0042}^{+0.0039}$	4.4987$_{-0.0054}^{+0.0046}$
i (degrees)	83.143$_{-0.020}^{+0.023}$	86.770$_{-0.052}^{+0.048}$	86.769$_{-0.050}^{+0.046}$	86.770$_{-0.046}^{+0.047}$
u1	0.3497$_{-0.0035}^{+0.0026}$	0.3183 ± 0.0019	0.3183$_{-0.0021}^{+0.0020}$	0.3183 ± 0.0019
u2	0.1741$_{-0.0044}^{+0.0057}$	0.2024 ± 0.0039	0.2024 ± 0.0042	0.2024$_{-0.0037}^{+0.0039}$
Phase curve fit
${{F}_{{\rm ecl}}}$ (ppm)	71.2$_{-2.2}^{+1.9}$	172.0$_{-1.6}^{+1.7}$	170.8$_{-1.5}^{+1.6}$	170.8$_{-1.5}^{+1.6}$
${{A}_{{\rm p}}}$ (ppm)	73.3 ± 4.0	151.9$_{-3.2}^{+3.3}$	150.4$_{-3.6}^{+3.1}$	149.9$_{-3.2}^{+3.3}$
θ (in phase)	0.01935$_{-0.00082}^{+0.00080}$	0a	0.00178 ± 0.00052	0.00280 ± 0.00053
${{F}_{{\rm n}}}$ (ppm)	−1.1$_{-6.2}^{+5.8}$	20.2$_{-4.9}^{+4.8}$	20.6$_{-4.6}^{+5.2}$	21.2 ± 4.8
${{M}_{{\rm p}}}$ $({{M}_{{\rm J}}})$	1.63 ± 0.13	9.28 ± 0.16	12.83$_{-0.23}^{+0.22}$	9.963 ± 0.17
${{A}_{3}}$ (ppm)	−1.93 ± 0.23	−7.03 ± 0.27	−7.14 ± 0.26	−7.33 ± 0.27
${{\theta }_{3}}$ (in phase)	0.0163$_{-0.0052}^{+0.0054}$	−0.0840 ± 0.0024	−0.09314$_{-0.00141}^{+0.00053}$	−0.09880$_{-0.0026}^{+0.0025}$
e	0a	0.00064$_{-0.00016}^{+0.00012}$	0.00064$_{-0.000098}^{+0.00012}$	0.00074$_{-0.00016}^{+0.00110}$
ω (degrees)	0a	5$_{-10}^{+8}$	−1$_{-7}^{+13}$	9$_{-7}^{+26}$
Derived parameters
${{R}_{{\rm p}}}$ $({{R}_{{\rm J}}})$	1.419$_{-0.085}^{+0.178}$	1.512 ± 0.035	2.63$_{-0.82}^{+1.04}$	2.216 ± 0.087
a (AU)	0.0355$_{-0.0021}^{+0.0045}$	0.03641 ± 0.00087	0.063$_{-0.020}^{+0.025}$	0.0533$_{-0.0022}^{+0.0021}$
${{A}_{{\rm g},{\rm ecl}}}$	0.2044$_{-0.0067}^{+0.0058}$	0.4565$_{-0.0052}^{+0.0054}$	0.4532$_{-0.0051}^{+0.0054}$	0.4529$_{-0.0052}^{+0.0055}$
${{T}_{{\rm eq},{\rm max} }}$ (K)	2820 ± 40	3300 ± 100	3900$_{-200}^{+100}$	3626$_{-2}^{+3}$
${{T}_{{\rm eq},{\rm hom} }}$ (K)	2200 ± 30	2550 ± 80	3040$_{-140}^{+90}$	2837 ± 2
${{T}_{{\rm B},{\rm day}}}$ (K)	2860 ± 30	3490$_{-70}^{+60}$	3830$_{-90}^{+60}$	3715 ± 7
${{T}_{{\rm B},{\rm night}}}$ (K)	$_{\lt 2300(2\sigma )}^{\lt 2100(1\sigma )}$	2590$_{-120}^{+110}$	2790$_{-140}^{+120}$	2734$_{-89}^{+76}$
${{\rho }_{p}}$ (g cm−3)	0.8$_{-0.3}^{+0.2}$	3.6 ± 0.3	0.9$_{-0.6}^{+2.0}$	1.2 ± 0.2
${\rm log} \;{{g}_{p}}$ (cgs)	3.32$_{-0.14}^{+0.09}$	4.02 ± 0.03	3.7 ± 0.3	3.72 ± 0.04

Notes. For KOI-13b, values were reported for three sets of stellar parameters from the literature, where our chosen values from Shporer et al. (2014). A stellar mass uncertainty of ±0.1 M$_{\odot }$ and a stellar radius uncertainity of ±0.1 R$_{\odot }$ was assumed when not given in the literature.

^aBest-fit model favored fixed values for these parameters. ^bFrom the NASA Exoplanet Archive (Akeson et al. 2013). ^cFrom the Kepler Input Catalog. ^dFrom Pál et al. (2008). ^eFrom Shporer et al. (2014). ^fFrom Huber et al. (2014). ^gFrom Szabó et al. (2011).

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Table 9.  Literature Phase Curve Parameters

References	${{F}_{{\rm ecl}}}$	${{A}_{{\rm p}}}$	${{A}_{{\rm e}}}$	${{A}_{{\rm d}}}$
KOI-13b
This work	172.0$_{-{\rm 1}{\rm .6}}^{+{\rm 1}{\rm .7}}$	151.9$_{-{\rm 3}{\rm .2}}^{+{\rm 3}{\rm .3}}$	74.1$_{-{\rm 5}{\rm .5}}^{+{\rm 6}{\rm .1}}$	10.02$_{-{\rm 0}{\rm .54}}^{+{\rm 0}{\rm .58}}$
Ang14	162 ± 10f	151 ± 10e	71 ± 10f	0
Shp14	173.7 ± 1.8f	156.4 ± 2.1g	60.1 ± 1.0	9.45 ± 0.49
Est13	169.6$_{-2.3}^{+2.2}$ f	143.5$_{-1.8}^{+1.7}$ f	72.31 ± 0.81f	8.58 ± 0.31f
Maz12	163.8 ± 3.8	142 ± 3g	66.8 ± 1.6	8.6 ± 1.1
Mis&Hod12	...b	...b	...d	...c
Cou12	124.3$_{-7.8}^{+6.9}$	...	...	...
Shp11	...	152.2 ± 2.5f,g	57.9 ± 1.3f	10.1 ± 0.8f
Sza11	120 ± 10	...	...	...
Kepler-7b
This work	39 ± 11	48$_{-{\rm 17}}^{+{\rm 16}}$	1a	0.5a
Ang14	46.6 ± 3.9	47.8 ± 5.2	0	3.9 ± 4.0
Dem13	48 ± 3	50 ± 2	0	0
Cou12	53.2$_{-13.0}^{+14.1}$	0	...	...
Dem11	44 ± 5	42 ± 4	0	...
Kip&Bak11	47 ± 14	30 ± 17h	...	...
Kepler-5b
This work	18.6$_{-{\rm 5}{\rm .3}}^{+{\rm 5}{\rm .1}}$	19.3$_{-{\rm 7}{\rm .6}}^{+{\rm 9}{\rm .5}}$	3.1$_{-{\rm 3}{\rm .4}}^{+{\rm 3}{\rm .5}}$	1.1$_{-{\rm 1}{\rm .3}}^{+{\rm 1}{\rm .2}}$
Ang14	19.8 ± 3.6	8.3 ± 2.6	3.1 ± 1.4	0
Est13	18.8 ± 3.7	16.5 ± 2.0	4.7$_{-1.1}^{+1.0}$	0
Dés11	21 ± 6	0	...	...
Kip&Bak11	26 ± 17	0	...	...
Kepler-41b
This work	44$_{-{\rm 16}}^{+{\rm 15}}$	69 ± 25	5a	1a
Ang14	46.2 ± 8.7	46.3 ± 7.9	2.8 ± 4.3	0
Qui13	60 ± 11	37.4 ± 6.3	4.5 ± 3.3	0
San11	64$_{-12}^{+10}$	64$_{-12}^{+10}$	0	0
Cou12	77 ± 24	0	...	...
Kepler-412b
This work	53$_{-{\rm 24}}^{+{\rm 21}}$	31$_{-{\rm 19}}^{+{\rm 20}}$	9a	2a
Ang14	40.2 ± 9.0	17.3 ± 7.4	6.3 ± 3.9	2.7 ± 2.2
Del14	47.4 ± 7.4	28.8 ± 3.2	10.2 ± 2.5	1.81 ± 0.14
Cou12	69$_{-30}^{+31}$	0	...	...
Kepler-12b
This work	20.2$_{-{\rm 7}{\rm .6}}^{+{\rm 8}{\rm .3}}$	22.9$_{-{\rm 6}{\rm .0}}^{+{\rm 6}{\rm .2}}$	0.9a	0.6a
Ang14	18.7 ± 4.9	16.7 ± 2.6	0	0
For11	31 ± 8	0	...	...
Kepler-43b
This work	11$_{-{\rm 37}}^{+{\rm 40}}$	71$_{-{\rm 60}}^{+{\rm 56}}$	18$_{-{\rm 20}}^{+{\rm 26}}$	9 ± 10
Ang14	17.0 ± 5.3	51.7 ± 3.3	16.9 ± 2.2	0
HAT-P-7b
This work	71.2$_{-{\rm 2}{\rm .2}}^{+{\rm 1}{\rm .9}}$	73.3 ± 4.0	19.0$_{-{\rm 2}{\rm .4}}^{+{\rm 2}{\rm .3}}$	2.22$_{-{\rm 0}{\rm .24}}^{+{\rm 0}{\rm .23}}$
Ang14	69.3 ± 0.5	60.8 ± 0.5	16.8 ± 0.3	5.3 ± 0.1
Est13	68.31 ± 0.69	65.75 ± 0.48	19.09 ± 0.25	5.80 ± 0.19
Mor13	69.1 ± 3.8	...	...	...
Van12	71.85 ± 0.23	...b	18.9 ± 0.3c	...
Jac12	61 ± 3	61	...d	...d
Mis&Hel12	...a	...b	15.5e	3.0
Cou12	75.0$_{-8.1}^{+9.5}$	63.2$_{-12.6}^{+13.6}$	...	...
Wel10	85.8	63.7	18.65e	...
Bor09	130 ± 11	122	...	...
TrES-2b
This work	7.7$_{-{\rm 2}{\rm .6}}^{+{\rm 2}{\rm .4}}$	4.1$_{-{\rm 1}{\rm .8}}^{+{\rm 1}{\rm .7}}$	3a	2a
Ang14	10.9 ± 2.2	3.0 ± 0.8	2.9 ± 0.5	2.0 ± 0.2
Est13	7.5 ± 1.7	4.77$_{-0.63}^{+0.65}$	3.67 ± 0.33	2.40 ± 0.30
Bar12	6.5$_{-1.8}^{+1.7}$	3.41$_{-0.82}^{+0.55}$	2.79$_{0.62}^{+0.44}$	3.44$_{-0.33}^{+0.35}$
Kip&Spi11	0	6.5 ± 1.9	1.50$_{-0.93}^{+0.92}$	0.22$_{-0.87}^{+0.88}$
Kepler-10b
This work	7.5$_{-{\rm 2}{\rm .1}}^{+{\rm 2}{\rm .0}}$	9.79$_{-{\rm 3}{\rm .2}}^{+{\rm 3}{\rm .4}}$	0.5a	0.04a
Dem14	7.4$_{-1.0}^{+1.1}$	0	0	0
Fog14	9.91 ± 1.01	8.13 ± 0.68	0	0
San14	7.5 ± 1.4	...	...	...
Rou11	5.6 ± 2.0	5.6 ± 2.0	0	0
Bat11	5.8 ± 2.5	7.6 ± 2.0	0	0
Kepler-6b
This work	11.1$_{-{\rm 5}{\rm .3}}^{+{\rm 4}{\rm .8}}$	17.2$_{-{\rm 5}{\rm .1}}^{+{\rm 6}{\rm .3}}$	2a	1a
Ang14	11.3 ± 4.2	9.5 ± 2.7	0	1.0 ± 0.9
Est13	8.9 ± 3.8	12.4 ± 2.0	2.7 ± 1.0	0
Dés11	22 ± 7	0	...	...
Kepler-8b
This work	26±10	24$_{-{\rm 15}}^{+{\rm 16}}$	2a	0.8a
Ang14	16.5 ± 4.4	13.8 ± 3.7	3.7 ± 2.0	0
Est13	26.2 ± 5.6	25.3$_{-2.6}^{+2.7}$	2.5 ± 1.2	4.0 ± 1.4
Kepler-76b
This work	131.6$_{-{\rm 8}{\rm .0}}^{+{\rm 8}{\rm .7}}$	106.9$_{-{\rm 5}{\rm .6}}^{+6.2}$	22a	3a
Ang14	75.6 ± 5.6	101.3 ± 3.6	22.6 ± 1.9	11.4 ± 1.0
Fai13	98.9 ± 7.1	112 ± 5.0g	21.5 ± 1.7	15.6 ± 2.2
Kepler-91b
This work	35±18	25$_{-{\rm 19}}^{+{\rm 18}}$	51a	1a
Lil14	0	25 ± 15	60.5$_{-17}^{+16}$ e	1.5$_{-1.0}^{+0.5}$ e
Est13	38.7 ± 8.2	13.1$_{-6.0}^{+5.8}$	45.2$_{-2.1}^{+3.1}$	0
Bar14	49 ± 15	27.4$_{-16}^{+21}$	50.8$_{-5.0}^{+4.8}$	...i

Notes. Zero values indicate non-detections.

^a A$_{{\rm e}}$ and A$_{{\rm d}}$ were fixed in our best-fit model. ^bA$_{{\rm g}}$ reported instead of an amplitude. ^cAdjusted by 1/π. ^dM$_{{\rm p}}$ reported instead of an amplitude. ^eConverted to a semi-amplitude. ^fCorrected by a dilution factor of 1.913 ± 0.019 (see Section 2.2). ^gConverted to a peak-to-peak amplitude. ^hA$_{{\rm p}}$ derived from reported F$_{{\rm ecl}}$ and F$_{{\rm n}}$. ⁱA$_{{\rm d}}$ fixed to RV derived value.

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For HAT-P-7b we find that the BIC favors a model including a 8σ detection of a 1.93 ppm third harmonic. Interestingly, similar to E13, Van Eylen et al. (2013) and Morris et al. (2013), we find an asymmetry in HAT-P-7b's transit. Analysis of this asymmetry is outside the scope of this paper, but, like KOI-13b, it could be related to the system's significant spin–orbit misalignment (Narita et al. 2009; Winn et al. 2009). Although, unlike KOI-13b's host star, HAT-P-7 has an unusually low, and somewhat disputed, $v\ {\rm sin} \ i$ of 2–6 km s⁻¹, indicating a nearly pole-on view (Narita et al. 2009; Winn et al. 2009; Albrecht et al. 2012; Torres et al. 2012).

Recently, Cowan et al. (2013) have shown that phase curves can contain contributions in odd harmonics due to the brightness map of the planet. However, we favor a stellar origin as the signal is still present when KOI-13b is obscured during secondary eclipse (see Figure 7).

With our eclipse depths we calculate each planet's geometric Kepler albedo (${{A}_{{\rm g},{\rm ecl}}}$), which assumes no thermal brightness contribution, and the brightness temperature of the day and night-side (${{T}_{{\rm B},{\rm day}}}$ and ${{T}_{{\rm B},{\rm night}}}$), which assumes no contribution from reflected light and applies to the Kepler bandpass. We also calculate the equilibrium temperature, which assumes ${{A}_{{\rm B}}}=0$, for the two limiting cases, instant re-radiation (f = 2/3, ${{T}_{{\rm eq},{\rm max} }}$) and homogeneous re-distribution (f = 1/4, ${{T}_{{\rm eq},{\rm hom} }}$). These values can be found in Tables 5–8 and their calculation described in E13 and references therein.

For Kepler-10b, Kepler-91b, and KOI-13b we find ${{A}_{{\rm g},{\rm ecl}}}$ between 0.4 and 0.6, while for all the other planets we find albedos <0.25. Kepler-10b's very high albedo is a clear outlier from our sample, as is its small radius, at only 1.4R$_{\oplus }$, ultra short period and rocky composition (Batalha et al. 2011). Of our targets Kepler-10b is only planet where the presence of an atmosphere is not required by its mass and radius measurements. If this is the case then it is possible that Kepler-10b's high albedo, when compared to the hot-Jupiter albedos, is actually common of close-in rocky planets.

For Kepler-91b our noise analysis finds correlated stellar variability that varies on timescales equal to the planet period (see Sections 5.1 and 5.3). We suspect that this variability is contaminating our measurement of Kepler-91b's eclipse depth, resulting in an unusually high albedo.

KOI-13b's high albedo is most likely the result of blackbody emission leaking into the optical as KOI-13b's equilibrium temperature, at 3300–3900 K, peaks very close to the edge of the Kepler bandpass (see Section 5.4 for further discussion).

Since the eclipse depths at optical wavelengths are likely a combination of reflected light and thermal emission, we self-consistently solve for ${{A}_{{\rm g}}}$ by assuming a Bond albedo of $\frac{3}{2}{{A}_{{\rm g}}}$ and taking into account both contributions using

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{F}_{{\rm ecl}}} & = & {{\left( \frac{{{R}_{{\rm p}}}}{{{R}_{\star }}} \right)}^{2}}\frac{\int {{B}_{\lambda }}\left\{ {{T}_{\star }}{{\left( \frac{a}{{{R}_{\star }}} \right)}^{-1/2}}{{[f(1-\frac{3}{2}{{A}_{{\rm g}}})]}^{1/4}} \right\}\;{{T}_{{\rm K}}}\;d\lambda }{\int ({{T}_{{\rm K}}}{{F}_{\lambda ,\star }}d\lambda )} \\ {} & {} & +\;{{A}_{{\rm g}}}{{\left( \frac{{{R}_{{\rm p}}}}{a} \right)}^{2}} \\ \end{array}\end{eqnarray} \tag{ 19 }$$

where ${{B}_{\lambda }}$ is the Planck function of the equilibrium temperature expression in parentheses, ${{T}_{{\rm K}}}$ is the Kepler transmission function and ${{F}_{\lambda ,\star }}$ is the stellar flux computed using the NEXTGEN model spectra (Hauschildt et al. 1999).

We calculate each target's self-consistent albedo and corresponding temperature for the two limiting cases, denoted max and hom, and present the values in Table 10. For several planets we find that the self-consistent albedo, in the instant re-radiation limit, is marginally consistent with zero, meaning that thermal emission alone can explain their observed eclipse depths. This can be seen in the upper left panel of Figure 9 where we compare the maximum equilibrium temperature to the reflection only geometric albedo (${{A}_{{\rm g},{\rm max} }}$) and in the upper right panel of Figure 9 where we compare the equilibrium temperature to the self-consistent geometric albedo (i.e., considering both reflected and thermal emission).

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Table 10.  Self-consistent Albedos and Temperatures

	Self-consistent				Reflected Light Fraction
	A$_{{\rm g},{\rm max} }$	T$_{{\rm max} }$	A$_{{\rm g},{\rm hom} }$	T$_{{\rm hom} }$	f$_{{\rm ref},{\rm max} }$	f$_{{\rm ref},{\rm hom} }$
Kepler-5b	0.067$_{-0.058}^{+0.050}$	2182$_{-70}^{+77}$	0.118$_{-0.037}^{+0.035}$	1670$_{-46}^{+47}$	0.549$_{-0.058}^{+0.451}$	0.9706$_{-0.037}^{+0.029}$
Kepler-6b	0.049$_{-0.041}^{+0.037}$	1827$_{-44}^{+47}$	0.069$_{-0.035}^{+0.032}$	1419$_{-32}^{+33}$	0.706$_{-0.041}^{+0.294}$	0.9849$_{-0.035}^{+0.015}$
Kepler-7b	0.232$_{-0.083}^{+0.081}$	1870$_{-110}^{+100}$	0.247$_{-0.071}^{+0.073}$	1450$_{-80}^{+72}$	0.933$_{-0.083}^{+0.067}$	0.99643$_{-0.0714}^{+0.0036}$
Kepler-8b	0.094$_{-0.090}^{+0.072}$	2060$_{-120}^{+130}$	0.131$_{-0.055}^{+0.054}$	1589 ± 81	0.708$_{-0.090}^{+0.292}$	0.9820$_{-0.055}^{+0.018}$
Kepler-10b	0.583$_{-0.345}^{+0.083}$	1600$_{-1600}^{+960}$	0.584$_{-0.260}^{+0.082}$	1270$_{-1270}^{+640}$	0.99794$_{-0.3452}^{+0.0021}$	1.00$_{-0.26}^{+0.00}$
Kepler-12b	0.071 ± 0.046	1845$_{-69}^{+68}$	0.091$_{-0.035}^{+0.039}$	1432$_{-50}^{+47}$	0.775$_{-0.046}^{+0.225}$	0.9882$_{-0.035}^{+0.012}$
Kepler-41b	0.044$_{-0.044}^{+0.070}$	2220$_{-120}^{+99}$	0.104$_{-0.043}^{+0.042}$	1695$_{-77}^{+78}$	0.407$_{-0.044}^{+0.593}$	0.9611$_{-0.043}^{+0.039}$
Kepler-43ba	0.030$_{-0.030}^{+0.303}$	2043$_{-352}^{+79}$	0.069$_{-0.069}^{+0.270}$	1574$_{-255}^{+86}$	0.423$_{-0.030}^{+0.577}$	0.9649$_{-0.069}^{+0.035}$
Kepler-76b	0.144$_{-0.103}^{+0.082}$	2580$_{-160}^{+170}$	0.239$_{-0.033}^{+0.041}$	1920$_{-86}^{+74}$	0.58$_{-0.10}^{+0.42}$	0.9711$_{-0.033}^{+0.029}$
Kepler-91ba	0.44$_{-0.44}^{+0.22}$	1980$_{-1980}^{+690}$	0.46$_{-0.27}^{+0.21}$	1520$_{-1520}^{+390}$	0.9688$_{-0.445}^{+0.031}$	0.99825$_{-0.2655}^{+0.0018}$
Kepler-412b	0.031$_{-0.031}^{+0.082}$	2334$_{-120}^{+70}$	0.107$_{-0.057}^{+0.051}$	1769$_{-73}^{+76}$	0.271$_{-0.031}^{+0.729}$	0.9506$_{-0.057}^{+0.049}$
TrES- 2b	0.0049$_{-0.0049}^{+0.0144}$	1877$_{-28}^{+21}$	0.029$_{-0.011}^{+0.010}$	1455 ± 20	0.1591$_{-0.0049}^{+0.7918}$	0.9548$_{-0.011}^{+0.045}$
HAT-P-7b	0.047$_{-0.047}^{+0.046}$	2764$_{-88}^{+87}$	0.1937$_{-0.0093}^{+0.0078}$	2022$_{-34}^{+36}$	0.228$_{-0.047}^{+0.243}$	0.9478$_{-0.0093}^{+0.0522}$
KOI-13bb	0.404$_{-0.078}^{+0.030}$	2580$_{-160}^{+260}$	0.4529$_{-0.0077}^{+0.0069}$	1919$_{-79}^{+81}$	0.884$_{-0.078}^{+0.077}$	0.99206$_{-0.0077}^{+0.0079}$
KOI-13bc	0.00$_{-0.00}^{+0.37}$	3880$_{-850}^{+110}$	0.441$_{-0.013}^{+0.011}$	2310$_{-140}^{+99}$	0.00$_{-0.00}^{+0.82}$	0.9736$_{-0.013}^{+0.026}$
KOI- 13bd	0.316$_{-0.019}^{+0.018}$	3088 ± 43	0.4453$_{-0.0057}^{+0.0060}$	2154$_{-16}^{+15}$	0.698$_{-0.019}^{+0.049}$	0.98332$_{-0.0057}^{+0.0167}$

Notes. Derived using stellar parameters from:

^aPlanet's with evidence for non-planetary phase curve modulations (see Section 5.3). ^bShporer et al. (2014). ^cHuber et al. (2014). ^dSzabó et al. (2011).

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For KOI-13b, we calculate the albedos and temperatures using three sets of stellar parameters reported in the literature (see Table 8). These three studies report a stellar temperature between 7650 and 9100 K, corresponding to an equilibrium temperature from 3300 to 3900 K and leading to a variation in the self-consistent albedo from 0.0 to 0.4 (see Table 10).

In the bottom left panel of Figure 9 we compare the day-side brightness temperature (${{T}_{{\rm B},{\rm day}}}$) to the maximum equilibrium temperature (${{T}_{{\rm eq},{\rm max} }}$) and find that for several planets the brightness temperature is significantly higher than the equilibrium temperature. However, the majority of this excess brightness temperature can again be accounted for if we include a reflected light contribution as, at optical wavelengths, even low albedos can produce a planet brightness dominated by reflected light. For all planets we find self-consistent albedos consistent with the planetary nature of our targets (i.e., <1).

There does not appear to be a correlation between the excess brightness temperature and equilibrium temperature as an excess is seen in both the hottest and coolest planets in our sample. It is possible is that this extra flux is seen because we are probing significantly hotter layers of the planet's atmosphere, which is supported by our finding that for Kepler-76b and KOI-13b we need an increase in brightness to explain both their observed day- and night-side flux (see Section 5.8).

It is possible to constrain the planet's night-side brightness using the day-side brightness provided by the eclipse depth and the planetary brightness amplitude, which measures the day–night flux difference. Note that for the planets with a peak brightness offset we calculate the difference between the eclipse depth and brightness amplitude at ϕ = 0.5, not the peak amplitude. For Kepler-76b and KOI-13b we find a 2 and 4σ night-side flux detection, respectively, while for all other planets we find a night-side flux consistent with zero. For Kepler-76b the eclipse appears shallower than the brightness amplitude (see Figure 4). However, after compensating for Kepler-76b's partial secondary eclipse we find that the eclipse depth is a significantly larger than the planet's brightness.

By combining the day and night-side measurements it is possible to place a constraint on the planet's re-distribution factor (f') by simultaneously solving for f' and ${{A}_{{\rm g},{\rm max} }}$ in the following equations

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{F}_{{\rm ecl}}} & = & {{\left( \frac{{{R}_{{\rm p}}}}{{{R}_{\star }}} \right)}^{2}} \\ {} & {} & \times \frac{\int {{B}_{\lambda }}\left\{ {{T}_{\star }}{{\left( \frac{a}{{{R}_{\star }}} \right)}^{-1/2}}{{[(\frac{1}{4}+f^{\prime} )(1-\frac{3}{2}{{A}_{{\rm g}}})]}^{1/4}} \right\}\;{{T}_{{\rm K}}}\;d\lambda }{\int ({{T}_{{\rm K}}}{{F}_{\lambda ,\star }}d\lambda )} \\ {} & {} & +\;{{A}_{{\rm g}}}{{\left( \frac{{{R}_{{\rm p}}}}{a} \right)}^{2}} \\ \end{array}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\begin{array}{ccccccccccccccc} {{F}_{{\rm n}}}={{\left( \frac{{{R}_{{\rm p}}}}{{{R}_{\star }}} \right)}^{2}} \\ \times \frac{\int {{B}_{\lambda }}\left\{ {{T}_{\star }}{{\left( \frac{a}{{{R}_{\star }}} \right)}^{-1/2}}{{[(\frac{1}{4}-f^{\prime} )(1-\frac{3}{2}{{A}_{{\rm g}}})]}^{1/4}} \right\}\;{{T}_{{\rm K}}}\;d\lambda }{\int ({{T}_{{\rm K}}}{{F}_{\lambda ,\star }}d\lambda )} \\ \end{array}\end{eqnarray} \tag{ 21 }$$

where f' = 1/4 corresponds maximum day-side and zero night-side temperature, while f' = 0 is an equal day–night temperature. For planets with a marginal night-side detection, Kepler-8b, Kepler-12b, Kepler-412b and TrES-2b, we find albedos of 0.13 ± 0.06, 0.09 ± 0.04, 0.11 ± 0.06, and 0.03 ± 0.01, respectively, but do not find any constraint on f'.

For the planets with a significant night-side flux (Kepler-76b and KOI-13b) our simple model fails, and we do not find any solution. This is most likely due to the assumption that the atmosphere is isothermal, with two temperatures, one for the day-side and one for the night-side. However, this can be accounted for if we are probing a significantly hotter layer of the planet's atmosphere in the Kepler bandpass.

The self-consistent albedo measurements give strong evidence that, for most planets, the light from the planet in the Kepler band is a combination of both reflected light and thermal emission. Since Kepler observes in a single broad optical band, it is difficult to constrain the relative contributions from each source. However, using our self-consistent albedo equation we can estimate the reflected light fraction (f$_{{\rm ref}}$) by dividing the observed geometric albedo by our self-consistent albedo. The values for the two limiting, instant re-radiation or homogeneous re-distribution of absorbed stellar energy, cases can be found in Table 10.

In the bottom right panel of Figure 9 we compare the reflected light fraction and maximum equilibrium temperature and do not find a significant correlation. This is not surprising as most, if not all, values are not well constrained and several are completely unconstrained.

For eight of our fourteen targets, our model favored the inclusion of an offset of the peak planetary bright from mid-eclipse. Kepler-7b, -8b, -12b, -41b, -43b, and -91b exhibit brightness peaks after eclipse, referred to as a negative offset in our formalism, while the Kepler-76b and HAT-P-7b reach peak brightness before eclipse (positive offset).

Of our sample, only Kepler-7b, -12b, and -43b have published offset measurements, all with the peak after the eclipse (Demory et al. 2013; Angerhausen et al. 2014). For Kepler-7b, studies, including this work, find an offset ranging from −0.07 to −0.11 (Demory et al. 2013; Angerhausen et al. 2014). While for the two other planets, both this work and Angerhausen et al. (2014) find significant negative offsets, ranging from −0.08 to −0.10 for Kepler-12b and −0.12 to −0.19 for Kepler-43b.

The other planets in our sample for which we find an offset, Kepler-8b, Kepler-41b, Kepler-76b, and HAT-P-7b, are not found in other work (Angerhausen et al. 2014). The latter two of which are possibly due to a model degeneracy between Doppler boosting and a pre-eclipse shift in planetary brightness (see Section 5.5).

In the left panel of Figure 10, we plot peak offset as a function of equilibrium temperature (${{T}_{{\rm eq},{\rm max} }}$) for all planet's, including those where a zero fixed offset was favored. We find that when all planets are considered there appears to be no correlation between offset and temperature. However, if we only consider planets where a peak offset was found, we find that the six cooler planet's exhibit negative offsets ranging from −0.7 to −0.12, while the hotter two have smaller positive offsets of ∼0.02.

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When we exclude planet's where our analysis shows compelling evidence that correlated noise, most likely due to the star, is dominating the phase curve signal (see Section 5.3). The remaining planets (Kepler-6b, -7b, -8b, -41b, -76b, and HAT-P-7b) fall into two regions, cooler planets (T$_{{\rm eq},{\rm max} }$ <2300 K) with large post-eclipse shifts and hotter planets (T$_{{\rm eq},{\rm max} }\gt$2300 K) with small pre-eclipse shifts (see black circles in Figure 10). This bimodal distribution in peak offset and temperature could be a result of two distinct mechanisms, which we discuss in the following section.