2.1.Reflection Effect

This photometric effect, in essence, is an atmospheric phenomenon rather than a gravitational one. It mainly depends on the planet reflective capability. When a planet has a large albedo and is orbiting around a luminous star, its light variations are easier to detect in the visible range. Although the effect is small, it is significant for short-period planets in close orbits around their host stars. This is directly related to the fact that the stellar flux received by the planet decreases with distance as 1r2. Planets in close orbits are also heated by their stars, making their thermal radiation detectable. Thus, giant planets like Jupiter, with an orbital period of few days, are easier to detect by space telescopes like Kepler, since these planets collect more light from their host stars. As mentioned above, the reflection of incident stellar radiation off the planetary surface and/or atmosphere and the planet thermal emission are not easy to distinguish between them. For this reason, they are considered here as just one effect; the reflection effect. As the reflection picks up at superior conjunction (occultation) and reaches a minimum at inferior conjunction (transit), it is reasonable to modulate it by a cosinusoidal function of the phase angle φ, which describes well the planet position. The amplitude of the reflected light alone is given by: ARefl=Ageo Rpa2), where R_p is the planetary radius, and A_geo is the geometrical albedo.

Following the model described in ^{Mazeh et al. (2012)}, ^{Sudarsky et al. (2005)} and ^{Burrows & Orton (2009)}, the normalized photometric flux variation due to reflection is given by:

where the amplitude for the reflection effect, including the planetary thermal emission, is expressed as :

or in ppm units:

Here, αRefl is the reflection coefficient which depends on the albedo ( αRefl is of order unity), θ defined by θ=|ϕ-π|, is the complementary angle of φ, being φ the orbital phase angle.

The amount of reflected light does not change during their orbit for planets with circular face-on orbits from Earth’s point of view; therefore, their reflected radiation is not detected.

In summary, planets in close orbits around their host star, larger planets, and planets with higher albedo, are easier to detect as they reflect more light.

2.2.Ellipsoidal Effect

The ellipsoidal effect has its origin in the gravitational deformation of the host star by an orbiting planet (tidal distortion). Planetary gravitational tidal forces produce stellar distortions that cause photometric variations of the light curve of exoplanetary systems. This effect was presented by ^{Loeb & Gaudi (2003)} and ^{Drake (2003)}. ^{Pfahl et al. (2008)} provide a detailed theoretical investigation of the ellipsoidal deformation of the host star.

Following ^{Morris (1985)}, we describe the flux variation of the light curve due to the ellipsoidal effect. He proposed a model for the normalized flux variation that includes the first three cosinusoidal harmonics of the phase angle φ(t), as follows: ΔFFo=f2cos(2ϕ)+f1cos(ϕ)+f3cos(3ϕ); here, f ₁, f ₂ and f ₃ are the cosinusoidal amplitudes. From the results obtained by ^{Esteves et al. (2013)}, we note that the last two terms contribute less than 10% to the total ellipsoidal variations; so they are not considered in our calculations (^{Mazeh & Faigler 2010}).

The ellipsoidal variations are a gravitational effect which produces a double peak of equal height at the quarter phases of the orbit, contributing to an overall bi-modal feature in the light curve. Then, the tidal ellipsoidal normalized flux variation can be modulated in a cosinusoidal manner as:

where the ellipsoidal amplitude is given by ^{Shporer et al. (2011)}:

or, in ppm units:

M_p, M_* are the planetary and stellar masses, a, P_orb and 𝑖 are the semi-major axis, period and inclination of the planetary orbit, respectively. αEllip==0.15(15+u)(1+g)(3-u) is a coefficient which depends on the linear gravitational g and on u, the limb darkening coefficients of the host star (see ^{Mazeh & Faigler (2010)} (see for further details).

The effect of the stellar ellipsoidal distortions on the light curve can be larger than the relativistic beaming effect, which is often small, but the variation of the phase curve component is twice as fast. Furthermore, as seen in the relationships for calculating the ellipsoidal amplitude, the gravitational distortion of the star by the planet is larger if it has a low semi-major axis to stellar radius ratio and the density of the star is low. So, this ellipsoidal method can be used efficiently to find planets in evolved stars outside the main sequence. In contrast, the ellipsoidal effect is negligible for low mass planets far from the host stars.

2.3.Doppler Beaming Effect and Radial Velocity Estimation

The first theoretical contribution was presented by and the first observational contribution by ^{Hills & Dale (1974)} and the first observational contribution by ^{Maxted et al. (2000)}. ^{Loeb & Gaudi (2003)} were first to present the photometric effect in the context of exoplanet characterization. As the star and planets are orbiting around the system’s barycenter, the host star will periodically advance toward, and recede from, an observer. Thus, the brightness of the host star will vary sinusoidally at the orbital frequency of the planet (a stellar wobble is induced by the planet). Then, as the star moves toward an observer, there is an increase in the observed flux, and as it recedes the observed flux decreases. It varies with the period of the orbit, but is off by a phase from the reflected light, since the maximum boosting occurs when the planet is in its first quarter phase and the star is moving toward the observer. The relativistic Doppler beaming (boosting), now is a detectable photometric variation effect, thanks to the high-precision photometry of space telescopes like Kepler (down to ≈10 parts per million). It is not an ideal method for discovering new planets, since the effect is small, even smaller than the emitted and reflected starlight from the planet. However, with the light variations due to relativistic beaming, it is easier to detect massive planets near their host stars, since these factors increase with the movement of the star (due to the motion around the center of mass of an exoplanet system). Like the radial velocity method, this can be used to determine the orbital eccentricity and the minimum mass of the planet (which is impossible to do from reflected light alone); but the Doppler beaming effect does not require a spectrum of the star, so it can be used to study more distant stars.

The Doppler beaming effect (DBE) itself has two contributions: the first one is actually the same Doppler beaming effect which increases the luminosity toward the direction of the radial velocity of the star. The second one is the Doppler shift of the star spectrum in the Kepler observation band. These two effects could be described by the following equations (^{Rybicki & Lightman 1979}). First, considering a spherical star that radiates isotropically (or nearly isotropically) in the particle rest frame, the relativistic transformation of the received bolometric flux is given by:

where γ=1(1-β2) and  β=vc is the velocity of star in units of c, being c the speed of light. The angle θ(t) is the angle between the stellar velocity and the line of sight. From here, as the planet moves slowly compared with the speed of light, in the non-relativistic limit, the stellar normalized boosted flux is given by (Ben Placek and Kevin H. Knuth, 2015):

where αBeam is the photon-weighted integrated bandpass beaming factor, v_r the radial velocity of the host star which varies in a sinusoidal manner as vr=Ksin(ϕ), with K the semi-amplitude of the radial velocity given by (^{Cumming et al. 1999}):

Or, considering that MP<<M* and the eccentricity e = 0 for a circular orbit, in a first approximation, we can rewrite the previous expression as:

This relation, in this work, is evaluated using the well-known physical parameters, to make an analytical estimation of the radial velocity.

From here, the normalized flux variation for the Doppler beaming effect can be written as (^{Loeb & Gaudi 2003}; ^{Shporer et al. 2011})

where the beaming amplitude is found to be (^{Shporer et al. 2011}):

Therefore, the amplitude of the effect can be used to estimate the mass of the planet if the host star mass is known. The amplitude of the observed Doppler beaming photometric variation depends on the bandpass through which the planetary system is observed. The actual value of αBeam (the average spectral index) depends on the telescope band pass as well as on the type of the observed star. It can be written as αBeam∝d ln(Fν)d ln(ν). Considering a black-body effective temperature Teff, we can compute αBeam by αBeam=ex*(3-x)-3ex-1, where (^{Loeb & Gaudi 2003}), x=hνk Teff and h and k are, respectively, the Planck and Boltzmann constants.

It is worth noting that the radial velocity can be estimated from the Doppler beaming amplitude. Therefore, the mass of the planet also can be estimated from the amplitude of the Doppler boosting, as this effect is proportional to the radial velocity. Finally, it is also important to keep in mind that the Doppler variations for short period ( P≲0.2 yr) and massive ( Msini ≳MJ) planets should be a significant contributor to the variability of the exoplanetary phase curve signal (^{Loeb & Gaudi 2003}).

3.1.Removal of Systematics

First, in order to improve the signal to noise ratio, we treat the photometric data eliminating the jumps between the quarters and correcting for systematics. Since photometric variations are normally small for the primary and secondary eclipses, and even less between eclipses (due to the above mentioned photometric effects), we do not expect to have abrupt signal changes in the observed data. Then, in each data file a central moving median and a central moving average have been applied to eliminate bad points and to smooth the curve. We tested different combinations of these two steps, using 3 to 101 points, looking for the most suitable combination to smooth the curve without affecting the particularities of the signal (primary eclipse). A central moving median of 40 orders with a central moving average of 10 orders produced the best results. This procedure provided a light curve that had between 3 to 20 eclipses, depending on the planetary period and the available Kepler files. Finally, the phase curve for each planetary system was obtained by adding the light curve over parts of a period, applying the folding phase method. The phases φ = 0 in the primary eclipse and φ= π in the secondary, are taken as the starting points. The last step consists of removing the primary and secondary eclipses to obtain only the light variation curve between eclipses, which is the phase curve to be modeled.

Finally, we apply small-scale photometric variation models to find the best fitting curves to our observational data.

3.2.The Data Analysis: Phase Curve Fitting

The fitting model for a given light curve, essentially involves tuning the amplitude values for each of the three effects mentioned earlier, i.e., A_refl, A_Ellip, and A_Beam, in order to find the best fit to the corresponding photometric data of our selected planetary systems.

As was shown in the previous section, the small-scale photometric variations of these planets are proportional to trigonometric functions of the phase φ which ranges from 0 to; 2π, 0 corresponds to the transit and 𝜋 to the occultation.

For the normalized phase curve, adding the contribution of all phase effects, the following expression describes the pattern for the relative flux variation:

A₀ is expected to be of the order of unity, A_Beam positive, A_Refl and A_Ellip negative.

Only the planets with the correct amplitude effect (7 planets out of the original 60), have been presented in this work. The theoretical values for the amplitude A_Beam have been obtained from the evaluation of the relations exhibited in the Phase Curve Modeling § 2, using the parameters (mass, radius, and inclination) given in Table 1.

In Figure 2, we present the composite phase curve for KOI-13b. For this planetary system the reflection contribution is dominant, but it is modulated by the bimodal ellipsoidal effect.

Fig.2. KOI-13b phase curve fitting. The color figure can be viewed online. 

The Doppler beaming effect is also appreciable by the asymmetries in the total phase curve. However, the residual was not small enough to be considered as a good fit.

A result with a smaller residual (see Table 2) is obtained by incorporating, a priori, additional harmonics in 2φ and 3φ. Harmonics of higher order also have been considered, but no appreciable contributions was found.

Therefore, the models described below have been applied to KOI-13 and the other planets of our selected sample (Figure 3):

Model 1: Reflection effect + DBE + Ellipsoidal effect

Model 2: Reflection effect + DBE + Ellipsoidal effect + sin(2φ)

Model 3: Reflection effect + DBE + Ellipsoidal effect + sin(2φ) + cos(3φ) + sin(3φ)

Fig. 3 KOI-13b planetary light curve, phase curve, fitting model, and residuals for each of the three models indicated in the text. The color figure can be viewed online. 

In Figure 4 the phase curves with their best fitting models are presented for each of the six planetary systems that were analyzed in this contribution.

Fig. 4 Light curve, phase curve, fit model, and residual for the different planets we found with coherent effect sign. The color figure can be viewed online. 

In Table 3 we show, together with the published values, the theoretical and estimated Doppler amplitudes A_Beam, along with the values deduced for the radial velocity K.