Issue 
A&A
Volume 538, February 2012



Article Number  A4  
Number of page(s)  8  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201116711  
Published online  26 January 2012 
Estimating transiting exoplanet masses from precise optical photometry
^{1} Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
email: misldim@ast.cam.ac.uk
^{2} LeibnizInstitut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
^{3} Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
Received: 14 February 2011
Accepted: 9 December 2011
We present a theoretical analysis of the optical light curves (LCs) for shortperiod highmass transiting extrasolar planet systems. Our method considers the primary transit, the secondary eclipse, and the overall phase shape of the LC between the occultations. Phase variations arise from (i) reflected and thermally emitted light by the planet; (ii) the ellipsoidal shape of the star due to the gravitational pull of the planet; and (iii) the Doppler shift of the stellar light as the star orbits the center of mass of the system. Our full model of the outofeclipse variations contains information about the planetary mass, orbital eccentricity, the orientation of periastron and the planet’s albedo. For a range of hypothetical systems we demonstrate that the ellipsoidal variations (ii) can be large enough to be distinguished from the remaining components and that this effect can be used to constrain the planet’s mass. To detect the ellipsoidal variations, the LC requires a minimum precision of 10^{4}, which coincides with the precision of the Kepler mission. As a test of our approach, we consider the Kepler LC of the transiting object HATP7. We are able to estimate the mass of the companion, and confirm its planetary nature solely from the LC data. Future space missions, such as PLATO and the James Webb Space Telescope with even higher photometric precision, will be able to reduce the errors in all parameters. Detailed modeling of any outofeclipse variations seen in new systems will be a useful diagnostic tool prior to the requisite ground based radial velocity followup.
Key words: techniques: photometric / methods: data analysis / planetary systems / methods: analytical / celestial mechanics
© ESO, 2012
1. Introduction
Two observational methods have dominated the study of extrasolar planets so far: radial velocity (RV) measurements and transit light curve (LC) analyses. Both have advantages and disadvantages. While RV determinations provide estimates of the planetary mass (M_{p}), the eccentricity (e) and the semimajor axis (a), they do not constrain the inclination (i) of the orbital plane with respect to the observer, thus only lower limits to M_{p} can be determined. The transit method, on the other hand, provides information on i, the ratio of the planetary to the stellar radius (R_{p}/R_{s}), and the duration of the transit (D). So far, only a combination of both strategies yields a full set of orbital and physical parameters for extrasolar planets.
Currently, two spacemissions are targeted at the detection of transiting extrasolar planets: CoRoT launched in 2006 (Deleuil et al. 1997) and Kepler launched in 2009 (Borucki et al. 1997). Their instruments are monitoring hundreds of thousands of stars, discovering thousands of planet candidates, whose RV followup will take many years (and may never be completed for the faintest candidates). Without RV followup, the most fundamental parameter of an extrasolar planet, its mass, remains undetermined from CoRoT and Kepler observations alone. The mass is the crucial parameter classifying an object as a planet, brown dwarf or a star.
Highaccuracy photometry has already been used for a number of systems to show that the planetary thermal emission, as well as the reflection of the stellar light from the planet, are detectable. In particular, Welsh et al. (2010) report the discovery of ellipsoidal variations in the Kepler LC of HATP7. This is an effect more commonly known from close stellar systems, where phasedependent light variation arises from the gravitationally distorted stars. In HATP7, the planet is close enough and massive enough to induce the same effect.
Other authors (Loeb & Gaudi 2003; Faigler & Mazeh 2011) have already presented treatments of reflected light, thermal emission, ellipsoidal variations and Doppler beaming. Our approach to the problem includes two main differences: (i) we investigate the sensitivity of our technique on the adopted phase function (Sect. 3.2); and (ii) we attempt to solve separately for the reflected light and thermal emission from the planet. For hot Jupiters around hot stars, the temperature of the planet increases dramatically and reflected light alone fails to explain the LC.
In this paper, we investigate whether the presence of ellipsoidal variations in a planetary system can be used to place meaningful constraints on the mass of the planetary companion. In Sect. 2 we describe the basic phenomena included in our simplified model of a planetary system’s LC. In Sect. 3 we compare the magnitudes of the relative contributions and consider the likely range of planetary systems for which ellipsoidal variations ought to be detectable with both CoRoT and Kepler. We reanalyze the HATP7 LC and demonstrate that the ellipsoidal variations can be used to estimate the mass of the planet to within 10% of the radialvelocity measured mass. We summarize our conclusions in Sect. 4.
2. Parametrization of photometric transits
Standard models of LCs that have been used before the advent of space missions are based on a flat curve out of transit and a limb darkening during the transit. Seager & MallénOrnelas (2003) proved analytically that for each package of planetstar characteristics there is a unique LC. Analyses of highaccuracy data from space required a revision of this simple approach. For a high accuracy LC, the model should incorporate the reflected light from the planet, thermal emission, ellipsoidal variations and Doppler boosting, which deforms the overall shape of the LC, and the secondary eclipse (Fig. 1).
Fig. 1 The standard transit model includes the primary transit f_{1} (dashed line centred at orbital phase z = 0), the secondary eclipse f_{2} (dotted line around z = 0.5) and the reflected light from the planet f_{3} (solid line). The model is based on a HAT7b twin planet (Table 2). 
In Fig. 2 we show the geometry of an arbitrary transiting system assuming an elliptical orbit. Let i be the angle between the observer’s line of sight and the orbital plane normal, while the angle between the observer’s line of sight projected onto the orbit plane and the periastron is labeled φ. The star is in the center of the reference frame and d is the distance between the star and the planet; the distance between the star and the planet during the primary transit is denoted by , during the secondary eclipse both bodies are separated by the distance .
Fig. 2 This sketch of the transiting planetary system, as seen from above, explains the variables used in our calculations. We assume clockwise rotation. 
To decode the geometry of the system from the LC we split it into three sub LCs. LC f_{1} describes the course of the primary transit, when the planet blocks the star’s light, f_{2} describes the form of the secondary eclipse, when the star blocks and planetary light, and f_{3} the rest of the LC, when both star and planet contribute to the total amount of light.
2.1. Transit and eclipse
The duration of the transit f_{1} is given by (1)where a_{d} is the semimajor axis of the system, P is the orbital period, R_{⋆} and R_{p} are the radius of the star and the planet, respectively, e is the orbital eccentricity, is the impact parameter (Ford et al. 2008) and i is the inclination of the orbital plane with respect to the observer’s line of sight. To model the shape of f_{1} we use Eq. (1) and the limb darkening equation (2)with u_{1} and u_{2} the linear and quadratic limb darkening coefficients respectively (Claret 2004; Sozzetti et al. 2007), μ as the cosine of the angle between the surface normal and the observer, and I_{0} and I_{μ} as the intensities at the stellar disk center and at μ, respectively. Once the period is known from observations, one can fit the model to the observations to deduce R_{⋆}, R_{p}, i, and . The transit of the secondary eclipse, f_{2}, is fitted with the same model but without the effects of limb darkening.
2.2. Phasedependent light curve
The shape of the LC between the transit and the eclipse (f_{3}) contains five contributions: (i) f_{⋆}, a contribution from the stellar photospheric flux, which almost certainly varies with time, but which we assume to be constant throughout this analysis; (ii) f_{ref}, a contribution from starlight reflected from the face of the planet, which is phase dependent; (iii) f_{th} contributions from the intrinsic planetary thermal emission (split by day and night), which depends sensitively on a large number of factors, including the spectral range of the observation λ, the irradiation temperature, and the atmospheric composition and structure; (iv) f_{ell}, a contribution from ellipsoidal variations of the star itself due to tidal forces from the planet; (v) flux variations f_{dop} due to a Doppler shift (Doppler boosting) of the stellar spectral energy distribution with respect to the bandpass of the instrument (Mazeh & Faigler 2010). We normalize the LC by the stellar flux, which can be determined as the minimum flux observed at phase z = 0.5. The phase LC then becomes (3)In the following, we study each of these terms in more detail.
2.2.1. Reflected light
The phase pattern of the reflected stellar flux depends on the phase angle z, i.e., the angle between star and observer as seen from the planet. Counting the orbital phase θ from primary minimum, the angles z, i and θ are related through (4)The reflected flux f_{ref} can then be expressed as (5)where α_{g} is the geometric albedo of the planet, f_{⋆} is the stellar flux at a distance d from the star, and Φ(z) is the socalled phase function. It is not entirely clear which phase function provides the most appropriate description of extrasolar planets. A popular assumption is (6)which models the planet as a Lambert sphere (Russell 1916), assuming that the intensity of the reflected light is Φ_{L} = 1/π at at phase z = π/2 (Venus case). An alternative choice is (7)which assumes that the reflected light is Φ_{C} = 0.5 at phase z = π/2. In our model, we investigate both cases.
The sum φ + θ is related to the eccentric anomaly E through (8)and E is related to the mean anomaly M through Kepler’s equation (9)If e ≠ 0 then the geometrical albedo is not constant and it is given by the equation Eq. (10) (Kane & Gelino 2010) (10)
2.2.2. Ellipsoidal variations
When a Jupiterlike planet orbits close to its host star, say a ≲ 0.1 AU, then the star will be distorted. Hence, the skyprojected shape will vary, causing flux variations. Their magnitude is given by (11)where (12)is the gravity darkening term (radiative energy transfer – von Zeipel 1924), and M_{p} and M_{⋆} are the masses of the planet and the star, respectively. As these equations show, the detection of the ellipsoidal distortion of the star provides information about the planetary mass with respect to the stellar mass.
2.2.3. Doppler boosting
Finally, the relative flux because of stellar Doppler shifts given by (13)with c as the speed of light, (14)as the star’s velocity amplitude, and ρ given by (15)Explanations of Eqs. (13–15)) and of the underlying phenomenon are given by Loeb & Gaudi (2003). If this Doppler shift can be measured in addition to the variations induced by the stellar ellipsoidal deformation, then indeed the planetary mass can be constrained independently from the stellar mass.
In the next section, we will firstly explore constraints on accuracy raised by our method and then, as test cases, we will derive the mass of the known transiting exoplanet HATP7, based on Kepler data. Finally, we predict the value of our method for the upcoming ESA mission PLATO.
2.2.4. Thermal emission
Fig. 3 Thermal emission flux maps for three different bolometric albedo values (top: α_{bol} = 0.0, middle: α_{bol} = 0.3, bottom: α_{bol} = 0.5). Thermal emission is stronger for short term period planets around hot stars. For all the plots, we assume R_{P} = 1.0R_{J}. The white part of the diagram refers to f_{th} < 10^{7}. 
The equations for the planetary emitted light are similar to those from Cowan & Agol (2011). Thus (16)and (17)T_{eff} is the effective temperature of the star, and T_{d} and T_{n} are the temperatures on the day and night side of the planet, respectively. Using the equations from Hansen (2008) and Cowan & Agol (2011), we calculate the mean temperature of the planetary day and night sides: where T_{o} = T_{eff}(R_{⋆}/d)^{0.5} and 0 ≤ ϵ ≤ 1 is the energy circulation.
Fig. 4 Color map of f_{th} / f_{ref} as a function of T_{eff} and Period for three different albedo cases (α_{g},α_{bol} = 0.05, α_{g},α_{bol} = 0.3, α_{g}, α_{bol} = 0.5 for the top, middle and bottom figure respectively). We have fixed the energy circulation to ϵ = 0.5 and R_{P} = 1.0 R_{J}. The white part of the diagram refers to f_{th}/f_{ref} < 10^{7}. 
The spectrum of the planet will strongly depend on the absence or presence of clouds, the magnitude of the greenhouse effect and nongrey opacities. For the most of our analysis we assume that the planet emits as a black body. Anyhow, the differences between a black body and a nonblack body emission are small (a factor of 1.5–3.5, see Hansen 2008). Assuming main sequence stars and planetary energy circulation ϵ = 0.0 (for the maximum thermal emission assuming black bodies), we found that for most hot Jupiters, the thermal emission from the planet is very weak in optical wavelengths. For our analysis we assume a blackbody, observed through the Kepler’s sensitivity curve, which peaks at 575 nm and covers 420 to 900 nm (Kepler Handbook). Figure 3 shows the star’s effective temperature T_{eff} versus the period of the planet for three different bolometric albedo (α_{bol}) values. Colors refer to the expected stellar relative flux.
The thermal contribution only becomes comparable to, or even larger than, the reflected light component for short period, low albedo planets around hot stars. Distinguishing the relative contributions from a LC measured in a single passband, without any knowledge of the planet’s albedo is hard, if not impossible. Both thermal and reflected components will have similar phase shapes. In Fig. 4 we compare f_{th} and f_{ref} for three albedo cases (α_{g}, α_{bol} = 0.05, α_{g}, α_{bol} = 0.3, α_{g}, α_{bol} = 0.5).
The ellipsoidal variations occur on half the orbital period, and present a rather different phase dependent LC. Even if we cannot accurately distinguish between thermal and reflected components, we show in this paper, that they can be jointly distinguished from ellipsoidal variations (Fig. 7).
3. Results
3.1. Simulations
To assess the data accuracy required by our method to yield satisfying constraints on the planetary parameters, we simulate two LCs using the equations from above. One of our model planetary systems is a hot Jupiter, analogue to the transiting planet HATP7b (Pál et al. 2009b), whereas the other one resembles the transiting superEarth CoRoT7b (Queloz et al. 2009). We customize these models in terms of the geometric albedo, for which we optimistically apply α_{g} = 0.3 in both cases (Sudarsky et al. 2000). Though observations of CoRoT7b are reconcilable with e = 0, we chose e = 0.05 to simulate enhanced phase variations in the LC. After all, we are not heading for a reconstruction of these systems but we want to estimate how accurate comparable systems could be parametrized and, as an example, if a putative small eccentricity of CoRoT7b could be determined. This is particularly important given the significant activity of CoRoT7b.
Physical parameters of our two models.
Fig. 5 Relative errors in e, ω, α_{g} and M_{P} as functions of data accuracy. The solid line denotes the HATP2b twin while the dashed line labels the CoRoT7b analog. 
To these models, we add increasingly more noise simulating data accuracies between 10^{7} and 10^{4}. The phase effect in the LCs, i.e. in f_{3}, is significant only for accuracies ≲10^{4}, which is why this phenomenon has not been detected in the LCs of CoRoT (Costes et al. 2004). We then fit the noiseless model from Sect. 2 to each of these – more or less – noisy LCs and use 1000 Monte Carlo simulations to calculate the standard deviations for each parameter in each fit using a bootstrap method explained by Alonso et al. (2008). In Fig. 5 we show the relative errors resulting from these fits for the planetary geometrical albedo α_{g} (σ_{αg}/α_{g}), eccentricity (σ_{e}/e), orientation of periastron (σ_{ω}/ω) and the mass ratio (σ_{M}/M) as a function of the root mean square (rms) of the data. With an accuracy provided by Kepler of 10^{4} (Koch et al. 2010), the eccentricity of a CoRoT7blike superEarth could merely be determined with a unsatisfactory relative error higher than 1.0 which has no physical use. For a hot Jupiter similar to HATP2b, however, the relative error in e is about 0.001 in the best case (hot Jupiter – rms 10^{7}). The orientation of periastron for the superEarth could be constrained to approximately ±4° in the worse case (rms 10^{4}) while for the hot Jupiter the accuracy is as low as ±0.001° in the best case (rms 10^{7}). Restrictions of the geometrical albedo α_{g} are ±0.001 in the best case (hot Jupiter – rms 10^{7}) to ±1.1 in the worst case (Earth like – rms 10^{4}) with no physical meaning. The respective accuracies in planet mass M_{P} are ±0.028 M_{J} for hot Jupiters (best case) and ±0.95 M_{J} for superEarths (worst case).
3.2. Relative contributions
Different phase functions (Eqs. (6) and (7)) lead to different peaks in the thermal emission and reflected LC. The ratio between thermal emission and reflected light over ellipsoidal variations becomes smaller (for the Lambert sphere) or larger (for the geometrical sphere), which is why different phase functions lead to different mass values. The mass difference (ΔM_{P}) between the two cases is a constant (36.4%). For example, if we use a wrong sphere, the mass we measure for HATP7b exoplanet is ΔM_{P} ~ 0.67 M_{J} smaller (geometrical) or larger (Lambert) than the true mass. Seeking out the most plausible phase function, we fit two models, both differing only in the assumption of either the Lambertian or the geometrical sphere, to the data. For most systems, if the components f_{ref}, f_{th} and f_{ell} are very weak, it is impossible to distinguish between the two phase functions. In this case we select the Lambert sphere in order to calculate the upper mass limit of the planet. In Fig. 6 we show the residuals between the two different phase functions. Notice that the shape and the phase (xaxis) of residuals is identical to ellipsoidal variations curve.
For most hot Jupiter systems thermal emission in optical wavelengths is much weaker than variations due to the varying projected shape of the star. For orbital periods of 1 and 5 days of a G0V star, ellipsoidal variations are 3 and 10 times larger than thermal emission, respectively, while f_{th}/f_{⋆} ~ 10^{5} and 10^{9} (Figs. 3 and 4). As the star becomes hotter (F or A type), or the period shorter (<5 days), the thermal component become to be more significant.
Thermal emission, stellar ellipsoidal deformations, and Doppler shift changes are very weak phenomena (fainter than 10^{5}), which require very highaccuracy observations to be detected. So far, it has been impossible to detect these effects from the ground but based on data from the recent spacebased mission Kepler, our system of equations offers a new gate for exoplanet mass determinations. Neither RV measurements nor transit timing variations are necessary for our procedure.
Our parametrization of reflected and thermal planetary emission are nearly identical, so our model is by definition unable to distinguish between these two effects. Both Eqs. (5) and (16) are only function of the orbital phase (Φ_{L,C}) and both components (thermal and reflected) provide exactly the same LC. We are not able to distinguish between the two components by fitting our model in a hot Jupiter system, which the planet is tidal locked by its host star. If the planetary rotation and orbital period are different (for a case of a high eccentric orbit and the simple assumption that there are no winds), then reflected light and thermal emission LCs will show different shapes.
3.3. Application to Kepler data: the case of HATP7b
Kepler is a spacebased mission, launched on Mar. 7, 2009, with the primary goal to detect Earthsized and smaller planets in the habitable zone of solarlike stars. For a m_{V} = 12 star, the photometric precision is log _{10}(rms) ~ −4.35 for 6 h combined differential photometry (Borucki et al. 2011). We choose the wellstudied transiting planet HATP7 in the Kepler field, with mass and system parameters constrained by both photometric and RV observations, to test our photometryonly method. We use all the short exposure data (1 min exposures) from Kepler’s public archive. Our sample includes 67 transits and the phase folded LC was made by ~180 000 epochs binned by a factor of ~50.
Welsh et al. (2010) already found ellipsoidal variations in HATP7. Using the established physical parameters of the system, we calculate that the effect of (f_{ell} ~ 3.1 × 10^{5}) is ~10.3 times more pronounced than the Doppler shift variations (f_{dop} ~ 3.0 × 10^{6}). Nevertheless, we include both effects in our model.
We assume that the stellar radius and effective temperature are known (T_{eff} = 6350 K). We fix the ratio of stellar and planetary radius R_{p}/R_{⋆} = 0.0778, the inclination i = 83.1° and the relative semimajor axis a/R_{⋆} = 8.22 (Welsh et al. 2010). For our first fit, we use Eq. (3), but we exclude f_{ell} and f_{dop}. We only fit f_{ref} plus f_{th} in order to check if only these two components could successfully explain the data . Our second fit includes all four components from Eq. (3) . The χ^{2} ratio between the reflected light plus the thermal emission model and the model which includes all four components is /. Using an Ftest, we found that the complete model explains the data better than the simple model of the reflected plus thermal emission. There is a 0.01% chance that the more complex model do not produce a better fit to the data (FRatio:116.28, PValue: 0.0001). In Fig. 7 we show the observed LC of HATP7 with our best fit as well as a comparison to a model, which includes only reflected stellar light from the planet, and our enhanced model. In Table 2 we list the output parameters of the fit. Because f_{th} is very weak (10^{6}–10^{7}, see Fig. 3 for the black body case), α_{bol}, T_{d} and T_{n} values can not be calculated accurately (very large errors). Welsh et al. (2010), used the secondary eclipse depth of HATP7b, in order to measure the flux from the night side of the planet (f_{th,n} = 2.2 × 10^{5}). From Fig. 3 (bottom diagram) is clear that the planet is much brighter and much hotter than expected, and other heating mechanisms might exist. We can not disentangle f_{ref} from f_{th}, however this does not affect our main result.
We find the ratio of the planetary mass over the stellar mass to be M_{p}/M_{⋆} = 1.27 M_{J}/M_{⊙}, compared to M_{p}/M_{⋆} = 1.20 M_{J}/M_{⊙} deduced by Welsh et al. (2010). Because the f_{ell} component is uncorrelated with f_{ref} and/or f_{th}, the accuracy of the mass ratio (M_{P}/M_{⋆}) can be calculated with high accuracy. Although our values are not as accurate as those from RV measurements, we can infer the planetary nature of the companion.
Fig. 6 Residuals flux between Lambert & geometrical sphere. The residuals curve is identical to the ellipsoidal variations curve. By selecting the wrong phase function we over/under estimate the planetary mass. 
Fig. 7 LC of HATP7. The solid line shows the best fit solution of our synthetic model, i.e. including stellar ellipsoidal distortions and Doppler effect variations. The dashed red line refers to the reflected light component, the blue dotted line refers to the ellipsoidal variations and the purple, almost consant dashed line refers to the thermal emission. The Lambert sphere assumption produced the best fit. 
HATP7b system parameters.
Fig. 8 Combination of stellar ellipsoidal distortion and Doppler shift effects in simulated PLATO observations. The panels show the amplitude of flux variations ΔF in the LCs for three different planetary masses: M_{P} = 0.5 M_{J} (top), 1 M_{J} (middle), and 5 M_{J} (bottom) around a range of mainsequence stars. Diamond symbols represent known transiting planets with similar masses. The white part of the diagram refers to f_{ell} < 10^{5.5}. 
3.4. PLATO mission
PLATO is a prospective ESA mission aiming at the characterization of transiting exoplanets. The mission will use 40 small telescopes covering 42 × 42 deg^{2} in total with a supposed overall accuracy of ~2.8 × 10^{5} (Claudi 2010). This will satisfy the needs of our procedure. In the following, we demonstrate the capabilities of our photometryonly method in parameterizing hypothetical exoplanet systems.
We investigate the detectability of f_{ell} and f_{dop} in a range of different systems, assuming that our target stars belong to the main sequence. For each combination we calculate the amplitude of the flux variations ΔF in the LC. In Fig. 8 we present three plots, each belonging to a different planetary mass: M_{p} = 0.5 M_{J} (top), 1 M_{J} (middle), and 5 M_{J} (bottom). For comparison we indicate some known transiting systems with similar masses. In the upper panel ΔF is dominated by f_{ell}. Here, for the sample of known systems ΔF > 10^{5}, which would be challenging to be detected. Nevertheless, with PLATO one will be able to detect f_{ell} for planets with P < 3 d around stars with spectral type K0 or later (blue zone in the top panel of Fig. 8). In the middle diagram (M_{p} = 1 M_{J}) both f_{ell} and f_{dop} affect the LC and ΔF ~ 5 × 10^{5} for most of the known systems – detectable with PLATO. In the bottom panel, where the model planet is most massive, the LCs are mostly affected by the Doppler shift variations rather than by the ellipsoidal geometry of the star. The phenomenon is fairly detectable for the most cases of the known planets while ΔF ~ 12 × 10^{5}.
4. Conclusions
The mathematical tools presented in this article can be used for a complete parametrization of transiting exoplanet systems on the basis of highaccuracy LCs only. In our model, for a specific range of systems, RV measurements are not necessary to constrain the mass of the planet (M_{p}), orbital eccentricity (e), the orientation of periastron (ω), and the geometric albedo of the planet (α_{g}). Our model also incorporates the characterization of the ratio of planetary and stellar radius (R_{p}/R_{⋆}), orbital period (P), and the orbital inclination (i). In order for this method to be fully applicable, the planet must be massive enough and orbit its host star close enough as to distort the stellar structure significantly.
With the current Kepler and potential future missions such as PLATO, we are able to measure the mass of the hot Jupiters, where the orbital periods must be ≤ 1.5, ≤ 4, and ≤ 10 days in order to measure planetary masses of 0.5 M_{J}, 1 M_{J}, and 5 M_{J} respectively. Furthermore, for host stars with spectral types earlier than FV, thermal emission component is detectable in the LC (assuming black bodies). This will affect the accuracy and even the detection of ellipsoidal variation.
As we show, the Kepler mission provides an accuracy suitable enough for our procedure to be applied for the characterization of extrasolar planets. We confirm the planetary natures of the hot Jupiters HATP7 from Kepler data alone. Our method yields masses of for HATP7b. Our results (M_{P}, α_{g}, α_{bol}, e, T_{d}, T_{n}) are in good agreement with Welsh et al. (2010) and we also confirm the ellipsoidal variations of HATP7b system. Using ellipsoidal variations we calculate
the planetary mass from the photometric LC itself, without any RV measurments. Our technique will benefit from future space missions such as PLATO and the James Webb Space Telescope (Deming et al. 2009) with rms ~ 3 × 10^{5} and ≲ 10^{6}, respectively.
Acknowledgments
The CoRoT space mission, launched on 2006 December 27, was developed and is operated by the CNES, with participation of the Science Programs of ESA, ESA’s RSSD, Austria, Belgium, Brazil, Germany and Spain. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
References
 Alonso, R., Barbieri, M., Rabus, M., et al. 2008, A&A, 487, L5 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Borucki, W. J., Koch, D. G., Dunham, E. W., & Jenkins, J. M. 1997, in Planets Beyond the Solar System and the Next Generation of Space Missions, ed. D. Soderblom, ASP Conf. Ser., 119, 153 [Google Scholar]
 Borucki, W. J., Koch, D. G., Basri, G., et al. 2011, ApJ, 736, 19 [NASA ADS] [CrossRef] [Google Scholar]
 Claret, A. 2004, A&A, 428, 1001 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Claudi, R. 2010, Ap&SS, 328, 319 [NASA ADS] [CrossRef] [Google Scholar]
 Costes, V., Bodin, P., Levacher, P., & Auvergne, M. 2004, in 5th International Conference on Space Optics, ed. B. Warmbein, ESA Spec. Publ., 554, 281 [Google Scholar]
 Cowan, N. B., & Agol, E. 2011, ApJ, 729, 54 [NASA ADS] [CrossRef] [Google Scholar]
 Deleuil, M., Barge, P., Leger, A., & Schneider, J. 1997, in Planets Beyond the Solar System and the Next Generation of Space Missions, ed. D. Soderblom, ASP Conf. Ser., 119, 259 [Google Scholar]
 Deming, D., Seager, S., Winn, J., et al. 2009, PASP, 121, 952 [NASA ADS] [CrossRef] [Google Scholar]
 Faigler, S., & Mazeh, T. 2011, MNRAS, 415, 3921 [NASA ADS] [CrossRef] [Google Scholar]
 Ford, E. B., Quinn, S. N., & Veras, D. 2008, ApJ, 678, 1407 [NASA ADS] [CrossRef] [Google Scholar]
 Hansen, B. M. S. 2008, ApJS, 179, 484 [NASA ADS] [CrossRef] [Google Scholar]
 Kane, S. R., & Gelino, D. M. 2010, ApJ, 724, 818 [NASA ADS] [CrossRef] [Google Scholar]
 Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, ApJ, 713, L79 [NASA ADS] [CrossRef] [Google Scholar]
 Loeb, A., & Gaudi, B. S. 2003, ApJ, 588, L117 [NASA ADS] [CrossRef] [Google Scholar]
 Mazeh, T., & Faigler, S. 2010, A&A, 521, L59 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pál, A., Bakos, G. Á., Noyes, R. W., & Torres, G. 2009a, in IAU Symp., 253, 428 [Google Scholar]
 Pál, A., Bakos, G. Á., Torres, G., et al. 2009b, MNRAS, 1781 [Google Scholar]
 Queloz, D., Bouchy, F., Moutou, C., et al. 2009, A&A, 506, 303 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Russell, H. N. 1916, ApJ, 43, 173 [NASA ADS] [CrossRef] [Google Scholar]
 Seager, S., & MallénOrnelas, G. 2003, ApJ, 585, 1038 [NASA ADS] [CrossRef] [Google Scholar]
 Sozzetti, A., Torres, G., Charbonneau, D., et al. 2007, ApJ, 664, 1190 [NASA ADS] [CrossRef] [Google Scholar]
 Sudarsky, D., Burrows, A., & Pinto, P. 2000, ApJ, 538, 885 [NASA ADS] [CrossRef] [Google Scholar]
 von Zeipel, H. 1924, MNRAS, 84, 665 [NASA ADS] [CrossRef] [Google Scholar]
 Welsh, W. F., Orosz, J. A., Seager, S., et al. 2010, ApJ, 713, L145 [NASA ADS] [CrossRef] [Google Scholar]
All Tables
All Figures
Fig. 1 The standard transit model includes the primary transit f_{1} (dashed line centred at orbital phase z = 0), the secondary eclipse f_{2} (dotted line around z = 0.5) and the reflected light from the planet f_{3} (solid line). The model is based on a HAT7b twin planet (Table 2). 

In the text 
Fig. 2 This sketch of the transiting planetary system, as seen from above, explains the variables used in our calculations. We assume clockwise rotation. 

In the text 
Fig. 3 Thermal emission flux maps for three different bolometric albedo values (top: α_{bol} = 0.0, middle: α_{bol} = 0.3, bottom: α_{bol} = 0.5). Thermal emission is stronger for short term period planets around hot stars. For all the plots, we assume R_{P} = 1.0R_{J}. The white part of the diagram refers to f_{th} < 10^{7}. 

In the text 
Fig. 4 Color map of f_{th} / f_{ref} as a function of T_{eff} and Period for three different albedo cases (α_{g},α_{bol} = 0.05, α_{g},α_{bol} = 0.3, α_{g}, α_{bol} = 0.5 for the top, middle and bottom figure respectively). We have fixed the energy circulation to ϵ = 0.5 and R_{P} = 1.0 R_{J}. The white part of the diagram refers to f_{th}/f_{ref} < 10^{7}. 

In the text 
Fig. 5 Relative errors in e, ω, α_{g} and M_{P} as functions of data accuracy. The solid line denotes the HATP2b twin while the dashed line labels the CoRoT7b analog. 

In the text 
Fig. 6 Residuals flux between Lambert & geometrical sphere. The residuals curve is identical to the ellipsoidal variations curve. By selecting the wrong phase function we over/under estimate the planetary mass. 

In the text 
Fig. 7 LC of HATP7. The solid line shows the best fit solution of our synthetic model, i.e. including stellar ellipsoidal distortions and Doppler effect variations. The dashed red line refers to the reflected light component, the blue dotted line refers to the ellipsoidal variations and the purple, almost consant dashed line refers to the thermal emission. The Lambert sphere assumption produced the best fit. 

In the text 
Fig. 8 Combination of stellar ellipsoidal distortion and Doppler shift effects in simulated PLATO observations. The panels show the amplitude of flux variations ΔF in the LCs for three different planetary masses: M_{P} = 0.5 M_{J} (top), 1 M_{J} (middle), and 5 M_{J} (bottom) around a range of mainsequence stars. Diamond symbols represent known transiting planets with similar masses. The white part of the diagram refers to f_{ell} < 10^{5.5}. 

In the text 
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