Distorted, non-spherical transiting planets:
impact on the transit depth and on the radius determination.
In this paper, we quantify the systematic impact of the non-spherical shape of transiting planets, due to tidal forces and rotation, on the observed transit depth. Such a departure from sphericity leads to a bias in the derivation of the transit radius from the light curve and affects the comparison with planet structure and evolution models which assume spherical symmetry. As the tidally deformed planet projects its smallest cross section area during the transit, the measured effective radius is smaller than the one of the unperturbed spherical planet (which is the radius predicted by 1D evolution models). This effect can be corrected by calculating the theoretical shape of the observed planet.
Using a variational method and a simple polytropic assumption for the gaseous planet structure, we derive simple analytical expressions for the ellipsoidal shape of a fluid object (star or planet) accounting for both tidal and rotational deformations. We determine the characteristic polytropic indexes describing the structures of irradiated close-in planets within the mass range , at different ages, by comparing polytropic models with the inner density profiles calculated with the full evolution code. Our calculations yield a 20% effect on the transit depth, i.e. a 10% decrease of the measured radius, for the extreme case of a 1 planet orbiting a Sun-like star at 0.01AU, and the effect can be larger for smaller mass objects. For the closest planets detected so far ( AU), the effect on the radius is of the order of 1 to 10% (three times more for the mean density), by no means a negligible effect, enhancing the puzzling problem of the anomalously large bloated planets. These corrections must thus be taken into account for a correct determination of the radius from the transit light curve and when comparing theoretical models with observations.
Our analytical expressions can be easily used to calculate these corrections, due to the non-spherical shape of the planet, on the observed transit depth and thus to derive the planet’s real equilibrium radius, the one to be used when comparing models with observations. They can also be used to model ellipsoidal variations of the stellar flux now detected in the CoRoT and Kepler light curves. We also derive directly usable analytical expressions for the moment of inertia and the Love number () of a fluid planet as a function of its mass and, in case of significant rotation, for its oblateness.
As the measurement of the radii of close-in transiting planets continues to gain in accuracy, providing stringent constraints on exoplanet theoretical models, any source of errors in the radius determination must be determined with precision. Current ground and space-based photometric observations of the host stars of transiting planets enable us to address new problems. The first direct detection with Spitzer of the light emitted by the planet (DHL07) opened a new path to probe the physical properties of the surface and the atmosphere of transiting exoplanets. Among the first results of the Kepler mission, the detection of ellipsoidal variations of the host star induced by tidal interaction with a low mass companion has been claimed WOS10. More recently, CW10a; CW10b showed that light curve analysis can put direct constraints on the actual shape of transiting planets. They also investigated the impact of the precession of an oblate object with a non zero obliquity around the orbital axis on the shape and timing of the transit signal.
These observations motivate us to investigate the deformation of the planet with respect to a spherical body, because of tidal or rotational forces. While previous studies focused on the detectability of the oblateness of a flattened body, we address in the present paper the more general problem, namely the determination of the general shape of a planet (or star) distorted by both a tidal and a centrifugal potential, and its impact on the transit depth and thus on the determination of its correct radius. In order to compute the ellipsoidal shape (flattening and triaxiality) of a gaseous body, we derive in Sect. 2 a simple analytical model of the internal structure of the object (LRS94a), based on the polytropic equation of state. The polytropic indexes are calibrated in Sect. 4 by comparing with numerical models describing the structure of strongly irradiated gaseous planets (LBC09). In Sect. 3, we present directly usable analytical expressions giving the shape (oblateness and triaxiality) of a distorted planet (or star) as a function of its mass and polytropic index and compare our estimates with the numerical method outlined in Appendix A and with the measured values for the major planets of our solar system. As a by-product, our model yields analytical expression for the first gravitational moment and the love number of a self gravitating fluid body. Finally, Sect. 5 quantifies the effect of the non-sphericity of the planet on the transit depth.
We find that as the planet transits across the stellar disc, we only see the smaller cross section of its actual ellipsoidal shape so that the depth of the transit is decreased with respect to the expected signal for a spherical object, as discussed by LML10 in the case of WASP-12 b. This implies that the radius inferred from the light curve analysis, derived under the assumption of spherical planet and star, underestimates the real equilibrium radius of the object. This bias needs to be corrected for a proper comparison with theoretical 1D numerical simulations of the structure and evolution of extrasolar planets and enhances the actual discrepancy between theory and observation for the so called "bloated" planets.
2 Variational method for compressible ellipsoids
In this section, we briefly describe the energy variational method developed by LRS93 and LRS94a (hereafter LRS1 and LRS2) to construct general Darwin-Riemann equilibrium models. In Sect. 2.1 we briefly summarize the basic assumptions and the equilibrium relations are derived in Sect. 2.2. More details about the method in general, as well as the applications to compact objects, can be found in LRS1 and LRS2 and references to equations in these papers are denoted with numbers preceded by "I" and "II", respectively, in the present paper. Solutions to first order in the deformation will be derived for tidal and rotational deformations in Sect. 3.2 and 3.1, respectively.
2.1 Model description
Consider an isolated, self-gravitating fluid system in steady state. The system is characterized by conserved global quantities such as its total mass and total angular momentum . The basic idea in our method is to model our self gravitating system by a limited number of parameters , in such a way that the total energy can be written,
An equilibrium configuration is then determined by extremizing the energy according to
An expression like Eq. (1) can be written down for the total energy of a binary system (with components of mass and ). We adopt a polytropic equation of state between the pressure and the mass density ,
This defines the polytropic index and the “entropy” – both are constant within the object and sufficient (with ) to describe the mechanical structure of a given object. Under the combined effects of centrifugal and tidal forces, the objects (stars or planets) achieve nonspherical shapes. We model these shapes as triaxial ellipsoids of principal axes and , respectively. Throughout this paper, unprimed quantities refer to the component of mass while primed quantities refer to the component of mass . The three directions along which our principal axes are measured are, respectively, the line connecting the center of mass of the two components, its normal contained in the orbital plane and the direction of the orbital angular momentum vector. In the simple case of coplanar and synchronous rotation, is simply the polar radius and and are the equatorial radii of the component measured toward its companion and in the orthogonal direction, respectively.
Specifically, we assume that the surfaces of constant density within each object can be modeled as self-similar ellipsoids. The geometry is then completely specified by the three principle axes of the outer surface. Furthermore, we assume that the density profile inside each component, where is the mass interior to an isodensity surface, is identical to that of a spherical polytrope with the same volume. The velocity field, , of the fluid is modeled as either uniform rotation (corresponding to the case of a synchronized binary system), or uniform vorticity, , (for nonsynchronized systems). The vorticity vector is assumed to be everywhere parallel to the orbital rotation axis.
For an isolated rotating gaseous sphere, these assumptions are satisfied exactly when the fluid is incompressible (polytropic index ), in which case the true equilibrium configuration is a homogeneous ellipsoid (Chandrasekhar 1969). For a binary system, our assumptions are strictly valid in the incompressible limit only if we truncate the tidal interaction at the quadrupole order. We adopt this quadrupole-order truncation of the interaction potential in this paper.
Adding up the orbital separation, , our set of unknowns is or equivalently where is the central density and (when no ambiguity exists or otherwise stated, primed quantities are defined in the same manner as unprimed ones by simply making the transformation throughout the equations). The total energy can be written
is the internal energy of component 1 (cf. Eq. (I.3.1)), and
is the self-gravitational energy of component 1 (cf. Eq. (I.4.6)) with,
and are the classical variables (dimensionless radius and density taken at the surface) used to describe polytropic gaseous spheres (see Ch39) and is the gravitational constant. and are similarly defined. The kinetic energy in the inertial frame reads
where the spin kinetic energy of body 1 () is given by (cf. Eq. (I.5.6))
with being the rotational orbital velocity, is a measure of the internal rotation rate in the co-rotating frame,
is a dimensionless coefficient measuring the inertia of the body, and
is the moment of inertia with respect to the rotation axis. Tables giving values of the polytropic constants , , , and as a function of can be found in LRS1 and in Ch39. For non synchronous rotation (), our gaseous body is not in the state of solid-body rotation. A rotation rate can thus not have the usual meaning. To have a sense of the angular velocity, one can take the half of the vorticity (, where is the fluid velocity vector in the inertial frame) as a proxy111This choice has no impact on the result to first order because i) there is no cross correlation between tidal and centrifugal distortion at this level and ii) the value of only plays a role to compute the rotational distortion for which solid body rotation is ensured because . In this case, the half vorticity reduces to the usual rotation rate and . To higher order, this simply highlights the absence of a solidly rotating state and the inadequacy of the parametrization by a rotation rate in such cases.. is related to by
The orbital kinetic energy is simply
Finally, the gravitational interaction energy reads
2.2 Equilibrium relations
We can now derive the set of equilibrium relations
yielding seven algebraic equations for our seven unknowns . The details of the transformations necessary to express the total energy as a function of the unknowns and conserved quantities alone and to be able to carry out the differentiation are explained in §2.2.1 of LRS2 and just add technical details not needed here. We will thus give only the results.
Differentiation with respect to simply yields the modified Kepler’s law for the orbital mean motion
Differentiation with respect to the central density yields the virial relation,
with the mean radius and
Using expressions for and , we get the equilibrium mean radius
where is the radius of the unperturbed spherical polytrope given by (Ch39)
Finally, the differentiation with respect to and yields after some algebra (cf. Eqs. (I.8.4), (I.8.5) and (I.8.6)):
, , and is the mean density of the ellipsoid.
3 The shape of gaseous bodies
In this section, we derive analytical expressions for the deformation - induced either by centrifugal or tidal potential - of a gaseous body. To test the validity of our assumptions, we compare our predictions with the measured values for the major planets of our solar system in Sect. 3.1. Since we do not make any assumption about the masses of the two components, these equations can be used indifferently to compute the shape of the star or of the planet by choosing and when considering the planet, and vice versa when focusing on the star.
In general, the set of equations described in the previous section must be solved numerically, but we will study here the first order development of these equations at large orbital separation. This approximation corresponds to neglecting terms of order , which is consistent with our truncation of the gravitational potential at the quadrupole order and is appropriate to address close-in transiting planetary systems. In practice, this is done by setting and
in Eqs (19), (23), (2.2), (26) and their primed equivalent and by expanding these equations to first order in (). First we derive some general formulae by expanding the integrand in the definition (10):
To first order,
The principal moment of inertia of the body can also be computed and reads
The other moments of inertia can be computed by replacing 1 and 2 by the appropriate indices. The dimensionless moment of inertia for different planetary masses, age and stellar irradiation can be found in tables 3 and 6.
3.1 Rotational deformation: Maclaurin spheroids
Our set of equations also allows us to compute the effect of the centrifugal force alone on a slowly rotationg fluid object. To do so, one just has to take the limit in Eqs (19), (23), (2.2) and (26). Therefore, is a free parameter (the rotation rate of our body) and there is a degeneracy between and that allows us to choose .
We introduce the dimensionless angular velocity
as a small parameter of order in all expansions. The volume expansion factor can be calculated using
For this configuration, the usual variables are the oblateness
and the dimensionless quadrupole moment of the gravitational field given by the theory of figures to first order (ZMT73)
For practical purposes, can be computed to first order by using Eq. (27) and reads
in our geometry, where and denote the usual equatorial and polar radii. For an incompressible body () we retrieve the usual solution of the theory of planetary figures (ZT80).
Attempts have been made to constrain the oblateness and thus the rotation period of transiting planets by using the solar system planets as test cases (CW10a; CW10b). Because of the wide variety of exoplanets, it is important to have the ability to predict the flattening of fluid planets for a wider range of parameters than encountered in the solar system. Fig. 1 shows the predicted oblateness for various planet masses as a function of the rotational period .
3.2 Tidal deformation:
determination of the Love number ()
To compute the shape induced by the tidal force alone, we consider a non-rotating configuration (). From Eq. (15), this is achieved if . Then
Thus from Eq. (23) we see that there is no change of volume to lowest order,
Thus [cf. Eq. (A25) of LRS2]
As long as the hydrostatic equilibrium holds, this equation can be used to compute the shape of the planet and its host star at each point of the orbit. We recover the usual dependence of the tidal deformation in , with a factor of order unity, , which encompasses all the structural properties of the gaseous configuration.
Since we are in the linear approximation with a gravitational potential restricted to quadrupolar order, the shape of our body can be described with the usual Love number of second order, (Lov09, which is twice the apsidal motion constant often called in the stellar binary literature). Indeed, once (and for a body in hydrostatic equilibrium) is known, the external potential and the shape that a body will assume in response to any perturbing potential can be computed as detailed in Appendix A. To derive , we compute the quadrupolar term of the gravitational potential energy of the system formed by our compressible ellipsoid and a point mass, by introducing Eq. (44) in the linearized version of Eq. (17), and identify this term to the potential energy due to tides given by (Dar08)
As expected, in the limit, we retrieve the Love number of an incompressible ideal fluid planet . We can also see that is linked to the square of the dimensionless moment of inertia . This is because level surfaces are self-similar in our model and that the love number encompass both the deformation of the body () and the gravitational potential created by the deformation (). The Love number for different planetary masses, age and stellar irradiation can be found in tables 3 and 6.
We can see that the value of the Love number tends to decrease with mass above 1. This is due to the fact that more massive objects are more compressible and thus more centrally condensed (See Sect. 4). At constant mass, enrichment in heavy elements toward the center (possibly in a core) acts to decrease the value of . In general, redistributing mass from the external to the internal layers, which are less sensitive to the disturbing potential, decreases the response of the body to an exciting potential, which translates into a lower .
Our model predicts values in the range 0.3-0.6. As discussed by RW09 such values of the Love number could be inferred by the measurement of the precession rate of very Hot Jupiters on eccentric orbits. Such measurements could be carried out by Kepler for WASP-12 b analogs with an eccentricity (most favorable case) or Tres-3 b analogs with an eccentricity (for ) and lower eccentricities for higher Love number values. Such measurements would indeed be extremely valuable as they would put direct constraints on the central enrichment in heavy elements inside close Hot Jupiters, like the measurements of the gravitational moments of the solar system planets.
3.3 Synchronized planets
For values of the tidal dissipation factors inferred for Jupiter (GS66; LCB10), the timescale of pseudo synchronization of close-in giant planets is less than about a million years. The planet is thus in a state of pseudo synchronization, with a rotation rate given by (Hut81; LCB10)
in the weak friction theory, with being the eccentricity of the orbit. For the simple case of a circular orbit, the spin is thus synchronized and, either solving Eqs (19), (23), (2.2) and (26) in the synchronized case (), or simply adding the results of Eqs. (44) and (37) (there is no cross correlation terms to first order) yields
3.4 Model Validation
There are two major assumptions in the present calculations:
The absence of a central core. The aim of such an approximation is to avoid to introduce any free parameter in the model. In any case, the core mass and the global enrichment in giant extrasolar planets are yet weakly constrained (Gui05; LBC09). We will show that this approximation introduces an uncertainty of on the derived shape.
The polytropic assumption. This allows us to derive a completely analytical model. Comparison with a more detailed numerical integration (See Appendix A) shows that the deviation between the results of the two models (analytical vs numerical) is smaller than the uncertainty due to the no-core approximation.
Since the oblateness (), , , the mean radius and rotation rate are known for the major planets of our solar system, we can test our theory on these objects. The details of the calculation of the chosen polytropic indexes are presented in Sect. 4. The results are summarized in table 1, which shows the actual values of the relevant parameters for the two major planets taken from Gui05, the values of the oblateness and of calculated with our model and, for comparison, with the assumption of an incompressible body (). We see that, whereas the values of derived from the incompessible model differ from the true values by almost a factor of 2, our polytropic model predicts the -values to within . Note that higher-order terms (of order ) are not completely negligible for rapidly rotating bodies such as Jupiter and Saturn (see ZMT73; CSH92). The polytropic model, however, yields the -values that differ from the measured values by (for Jupiter) and (for Saturn). These discrepancies are mostly due to the large metal enrichment in these planet interiors, probably with the presence of a large dense core as detailed three layers models can reproduce exactly the measured moments (CSH92). Note that this no-core approximation has less relative impact on the distortion of the shape predicted by the model than on the gravitational moments because these effects scale as (See Appendix A) and respectively, being in the situations of interest. Such discontinuities in the density profile (and its derivatives) could be addressed more precisely with two different polytropes, but this would add extra free parameters and would not serve the very purpose of the present paper.
The numbers in parentheses are the
|uncertainty in the last digits of the value.|
While we decided to use a polytropic assumption to infer a fully analytical theory, the figures of a body in hydrostatic equilibrium can be derived without this assumption. As shown in Ste39 (See also ZMT73; ZT80; CSH92 for more detailed applications to the giant planets case) and outlined in Appendix A, this theory, however, requires a numerical integration even to first order. For an ideal polytropic sphere, Eq. (3.2) agrees with the numerical results of Ste39 with less that 1% error. To compare these methods in our context, we derive the values of using our analytical model (Eq. (3.2)), and by numerical integration of Eqs. (67) and (72) for our best representative models of Jupiter and Saturn in our grid (Although without cores). For Jupiter, our Eq. (3.2) gives , the numerical integration gives to be compared with the measured value of . For Saturn, these values are 0.60, 0.66 and respectively. Both models predict values than are higher than the measured ones. A direct consequence of the presence of heavy elements inside our giant planets. Comparing the models, our Eq. (3.2) yields slightly smaller values than the numerical integration which tends to mimic a central over-density (See Sect. 3.2). As discussed above a more precise modeling requires the addition of central enrichment in heavy elements whose mass fraction would be a free parameter. Without better knowledge of the internal composition of giant exoplanets, we think that the two methods yield similar results up to the sought level of accuracy. For sake of completeness, the values of computed with both methods are presented in tables 3 and 6.
4 Polytropic index in gaseous irradiated planets
To readily use the results of Sect. 3, one only needs to have a proper value for the polytropic index to be used. In this section, we derive realistic polytropic indices from numerical models of gaseous irradiated planets. All the other polytropic functions (, , …) can be derived by integrating the Lane-Emden equation and are tabulated in Ch39 and LRS1. They are given for different planetary masses, age and stellar irradiation in tables 3 and 6. We focus on the polytropic index in the planet because, in the context of transiting exoplanets, both the stellar rotation and the stellar tides have a negligible impact on the transit depth, as will be discussed in Sect. 5. The main physics inputs (equations of state, internal composition, irradiated atmosphere models, boundary conditions) used in the present calculations have been described in detail in previous papers devoted to the evolution of extrasolar giant planets (BCB03; CBB04; LBC09), and will not be repeated here.
We computed a grid of evolution models of gaseous giant planets with solar composition for various masses and incoming stellar flux . Low irradiation model can be used to infer the oblateness of long period rotating planets (such as Jupiter), while strongly irradiated models can be used to infer the shape and its impact on the transit of close-in planets. For the non irradiated case, the grid extends to . As the effect of irradiation on the internal structure decreases with the effective temperature of the object, these models computed with non irradiated boundary conditions should give a fair description of massive brown dwarfs in the range of irradiation considered. The pressure-density profile of each model is then fitted by a polytropic equation of state (Eq. (3)) at each time step and an example of the result of such a fit is shown in Fig. 2. Note that the disagreement between the actual profile and the polytrope in the lower left area of Fig. 2 is both expected and needed: This low-density region (the first 5% in mass below the atmospheric boundary surface) has a different effective polytropic index than the planetary interior. In order to capture the bulk mechnical property of the planet, we weight each shell in the internal structure profile by its mass during the fitting procedure.
This provides us with a grid tabulating the polytropic index of the planet, , and its spherical equilibrium radius, , where is the age of the object. These functions, along with other quantities (, …), are tabulated in tables 3 and 6222Electronic versions of the model grids are available at http://perso.ens-lyon.fr/jeremy.leconte/JLSite/JLsite /Exoplanets_Simulations.html. Figures 3 and 4 show the variation of with the mass for different ages with and , respectively.
As shown in Fig. 3, in the non-irradiated case, we recover qualitatively the results of CBL09: except for the early stages of the evolution, the (dimensionless) isothermal compressibility of the hydrogen/helium mixture is a monotonically increasing function of the polytropic index, , and thus of the mass of the object. In the high mass regime, slowly increases as the relative importance of ionic Coulomb effects compared with the degenerate electron pressure decreases, and approaches the limit, the expected value for a fully degenerate electron gas, when approaches the hydrogen burning minimum mass () as can be seen in table 5. In the low mass regime, the compressibility decreases with the mass because the repulsive Coulomb potential between the ions, and thus the ionic electrostatic energy becomes dominant. Ultimately, electrostatic effects dominate, leading eventually to for solid, terrestrial planets.
A new feature highlighted by the present calculations is the non-monotonic behavior occurring between 1 - 3 at early ages. This occurs when the central regions of the planet, of pressure and temperature , previously in the atomic/molecular regime, become pressure-ionized, above 1-3 Mbar and 5000-10 000K (SCV95; CSH92; SHC92), and the electrons become degenerate. An effect more consequential for the lowest mass objects, whose interiors encompass a larger molecular region. This stems from the fact that (Ch39)
Older (with smaller ) and more massive () objects have and the ionization extends all the way up to the outermost layers of the gaseous envelope, which then contains too small a mass fraction of molecular hydrogen to significantly affect the value of the polytropic index. This contrasts with younger objects around 1 - 3 , whose external molecular hydrogen envelope contains a significant fraction of the planet’s mass, leading to a larger value of the polytropic index, as molecular hydrogen is more compressible than ionized hydrogen (see e.g. Fig. 21 of SCV95). Once again, for these latter objects, the interior structure would be better described by using two different polytropes, but such a significant complication of the calculations is not needed at the presently sought level of accuracy.
As seen on Fig. 4, a strong irradiation enhances the aforementioned feature: the evolution is delayed because the irradiated atmosphere impedes the release of the internal gravothermal energy. This yields a slower contraction, thus a lower central pressure (and lower central temperature) for a longer period so that the object enters the ionization regime at a later epoch. The bump at the high mass end of the 100 Myr isochrone is due to deuterium burning which also occurs later for a given mass, because of the cooler central temperature (see above). At 100 Myr, the 20 has already burned a significant amount of its deuterium content and starts contracting again, whereas lower mass planets are still burning some deuterium supply, leading to a less compact and thus less ionized structure. This leads to the non-monotonic behaviour on the high-mass part of the diagram at 100 Myr, which reflects a similar behaviour in the mass-radius relationship.
To evaluate the uncertainty in our determination of the polytropic index, we use an alternative method to derive . As shown by Ch39, the knowledge of , and is sufficient to infer the radius of the polytrope, with the help of Eq. (24) and the central density, using
Since our numerical simulations provide both the radius, , and the central density of the object, , we can invert Eqs. (24) and (54) to compute and . This new determination of the polytropic index is compared with the previous one, obtained by fitting the profile, in Figs 3 and 4 for the 5 Gyr case: the new value corresponds to the upper envelope of the shaded area. Fig 3 shows that the two approaches yield very similar results in the non-irradiated case. For the irradiated case, the average uncertainty on our determination of lies between about 5 % and 15 % for the low mass planets.
5 Implications for transit measurements
When limb darkening is ignored, the depth of a transit is given by the ratio of the planetary and stellar projected areas. When both bodies are spherical, this simply reduces to . For close-in planet-star systems, however, both tidal and rotational deformations yield a departure from sphericity, so that what is measured is no longer the mean radius but an effective "transit radius" defined such that the cross section of the planet is equal to and similarly for the star. Thus the transit depth reads
5.1 Impact on transit depth
In general, the projected area of an ellipsoid can be computed for any orientation and then at each point of the orbit, as explained in Appendix B. Figure 5 shows the projected area of the planet () as a function of its anomaly () and inclination ()
normalized to the spherical case ().
When the planet is seen from its "side" (), the observer sees a bigger planet because the rotation of the latter on itself tends to increase its volume, as has been mentioned by LML10 for WASP-12 b. The possibility to measure these effects from the light curve is discussed in RW09 and CW10a.
For the simple case of an edge-on orbit at mid transit (, ), since the observer, the planet and the star are aligned with the long axis of the tidally deformed ellipsoid333This is still verified to first order in and as only second order terms appear.444In the following, the variables have the same meaning as in Sect. 2 and 3 with p indices when referring to the planet and to the star, and . Therefore,
where and are the respective radii the planet and the star would have in spherical equilibrium and is by definition the variation of the transit depth induced by the ellipsoidal shape of the components relative to the transit depth in the spherical case. To first order in the deformation, this is given by
The choice of the expression to be taken for the depends on the physical context. In the general case, one can use a linear combination of Eqs. (37) and (44) and get a general expression which depends on , and . However, most of the planet hosting stars have a low rotation rate compared to the orbital mean motion. This entails that the rotational deformation is negligible compared to the tidal one and can generally be neglected. As mentioned in Sect. 3.3, hot Jupiters should be pseudo synchronized early in their evolution. Therefore, we will assume such an approximation in our calculations in order not to introduce any other free parameter. The impact of the rotation alone is described Sect. 3.1. Under such an approximation,
where the parameter now denotes the mass ratio , and and are equal to for and , respectively. The first line in the above equation represents the contribution of the planet, which is always negative (for reasonable values of ). Our line of sight follows the long axis of the tidal bulge and we see the minimal cross section of the ellipsoid.
The contribution of the star is positive and, in most cases, negligible compared the planet’s contribution because
for a typical system ( for a Jupiter-Sun like system). As a consequence, the results presented hereafter do not depend on as long as realistic values of are taken.
Figure 6 portrays the relative transit depth variation computed with Eq. (5.1) for several planet masses as a function of the orbital distance, for a Sun-like parent star. While all the curves are calculated at an age of 1 Gyr, they do not change much for older ages because both the radii and the polytropic indices remain nearly unchanged after 1 Gyr (see Fig. 4). Given the accuracy of the radius determination achieved by the latest observations (1 to 10%), the transit depth variation is significant for Saturn mass objects () closer than 0.04 AU and Jupiter mass objects closer than 0.02-0.03 AU. Because we derived the equations to first order, the value of derived from our model should be taken with caution when (and are clearly not meaningful for ). In this regime, corresponding to the upper left region of Fig. 6, one should use the theory of planetary figures to higher order, but then numerical calculations become necessary, loosing the advantage of our simple analytical expressions. Figure 6 also displays the transit depth variation computed for the most distorted known transiting exoplanets, with the observationally measured parameters. The error bars reflect the uncertainties in the model and in the measured data.
5.2 Which radius?
Before going further, it is important to summarize the differences between the various radii that we have defined above. Note that, in the literature, the term "radius" is used loosely, even for non-spherical objects. Importantly enough, this can lead to discrepant normalizations throughout different studies and published values of transit radius measurements when, for example, radii are shown in units of Jupiter radii () without precisely defining the latter.
One can define , and as the distances between the center and some isobar surface along the three principal axes of inertia. For any distorted object, we can define the mean radius () as the radius of the sphere that would enclose the same volume as the described surface. In our case of a general ellipsoid, we have . If axial symmetry holds (e.g. for a rotating fluid body), we have , defining the equatorial radius, and the polar radius. Finally, is the radius of the spherical shape that the fluid body would assume if it was isolated and at rest in an inertial frame (the limiting case for which all the mentioned radii would be equal). This latter is the radius computed in usual 1D numerical evolution calculations. Note that in general because the centrifugal force has a net outward component that increases the volume of the object, as can be seen from Eq. (35).
One must be aware that only , and (reducing to and for solar system gaseous bodies) can be measured directly and are not model dependent. This is why we define as the equatorial radius of Jupiter at the 1 bar level (m, Gui05 and reference therein).
Unfortunately, transit measurements only give access to the projected opaque cross section of the planet () defining a "transit radius" which depends on the shape of the planet, its orientation during the observation and the wavelength used. To convert this transit radius inferred from the observations () to the spherical radius () - that can be compared to 1D numerical models - one must eliminate from Eqs. (55) and (57). As shown above, the stellar impact on is negligible compared to the planet’s contribution (). Then, using the first term in Eq. (5.1) and expanding the expression giving the definition of , one gets
For the most distorted known planets, the relative variation between the transit radius and the equilibrium radius
is positive and amounts to 3.00% for WASP-12 b, 2.72% for WASP-19 b, 1.21% for WASP-4 b, 1.20% for CoRot-1 b, 0.89% and OGLE-TR-56 b.555 Of course, since Eq. (60) is an implicit equation on . To obtain to the sought accuracy, a perturbative development in powers of can be obtained using recursively Eq. (60) (61) However, terms of order are of the same order than the second order corrections to the shape that we have neglected throughout.
Note that because the mean density scales as , the increase in radius implies a decrease in the mean density inferred which is about three times larger (i.e. for WASP-12 b). This is of particular importance when one wants to constrain the internal composition or enrichment of giant planets from transit measurements.
Because of the large variety of exoplanetary systems presently discovered, with many more expected in the near future, and the increasing accuracy of the observations, it is important to take into account the corrections arising from the non-spherical deformation of the planet or the star, due to rotational and/or tidal forces, as such a deformation yields a decrease of the transit depth. In order to do so, it is extremely useful to be able to compute analytically the shape of planets and stars in any configuration from the knowledge of only their mass, orbital separation and one single parameter describing their internal structure, namely the polytropic index, . Such formulae are derived in Sect. 3, and can be easily used to determine the impact of the shape of the planet on its phase curve and on the shape of the transit light curve itself (CW10a). They can also be used to model ellipsoidal variations of the stellar flux that are now detected in the CoRoT and Kepler light curves (WOS10). These formulae also give good approximations for various parameters describing the mass redistribution in the body’s interior and the response to a perturbing gravitational field, i.e. the moment of inertia, , and the Love number of second degree .
Another major implication of the present work is to show that departure from sphericity of the transiting planets produces a bias in the determination of the radius. For the closest planets detected so far ( AU), the effect on the transit depth is of the order of 1 to 10% (see Fig. 4), by no means a negligible effect. The equilibrium radius of these strongly distorted objects can thus be larger than the measured radius, inferred from the area of the (smaller) cross section presented to the observer by the planet during the transit. The analytical formulae derived in the present paper, and the characteristic polytropic index values derived for various gaseous planet masses and ages, allow to easily take such a correction into account. Interestingly, since this equilibrium radius is the one computed with the 1D structure models available in the literature, the bias reported here still enhances the magnitude of the puzzling radius anomaly (see Fig. 6 of LCB10) exhibited by the so-called bloated planets.
Acknowledgements.The authors are very grateful to Nick Cowan and Louis Shekhtman who first noticed minor errors in Appendix B that have been corrected in the present version. The authors acknowledge the hospitality of the Kavli Institute for Theoretical Physics at UCSB (funded by the NSF through Grant PHY05-51164), where this work started. This work has been supported in part by NASA Grant NNX07AG81G and NSF grants AST 0707628. We also acknowledge funding from the European Community via the P7/2007-2013 Grant Agreement no. 247060. The authors are grateful to the anonymous referee for his/her sharp and enlightening comments.
Appendix A Theory of planetary figures:
Here, we briefly outline the method described in Ste39 to compute numerically the response of a body in hydrostatic equilibrium to a perturbing potential666Here we take the convention that the force acting on a particule of mass due to a potential is . This yields some difference of signs with Ste39. and derive additional formulae. To lowest order (which is consistent with the order of approximation used throughout the present paper) the body response is linear and the total deformation is the sum of the response to each term of the decomposition of the perturbing potential. Let us consider a term of the decomposition of the potential of the form
where the are tesseral harmonics defined by
The () corresponds to positive (negative) values of m and are the usual associated Legendre polynomials. The reference axis defining and may change from one term to the other. For example, the rotation axis is best suited to treat rotational distortion and the line connecting the center of mass of each body is better to describe the tidal distortion. It is shown by Ste39 that to first order, the shape of the distorted level surface of mean radius (see ZT80 for a detailed definition) takes the form
where is the distance between the center and the level surface as a function of and and a figure function yet to be calculated. Ste39 shown that, ignoring terms of order