Escape to Planet 55
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A stronger correlation between the planetary mass and radius potentially provided by a better estimate of the transit depth would significantly improve interior characterization and reduce drastically the uncertainty on the gas envelope properties. Transiting planets are particularly interesting because their radius can be determined from the transit depth.
On top of this, transmission spectroscopy can provide insights on their gas layers, if any. Despite an inherent degeneracy, the ability to characterize the interiors of exoplanets improves with higher precision on mass and radius. To date, objects have both a mass and a radius in the exoplanets. High-precision data are the challenge of the next decade. In many cases, the uncertainty on planetary parameters is dominated by the uncertainties in mass and radius which are generally of several percent of the host star. We will never know a planet better than its host star.
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In particular, one of the most important parameters needed to characterize exoplanets is the stellar radius see e. One of the few bright stars hosting transiting planets known today is 55 Cnc a. This star is the main component of a wide binary system, and hosts a system of five planets, detected with the radial velocity technique Fischer et al. One of them 55 Cnc e, the closest to the star is transiting and has been detected independently by Winn et al.
As one of the first transiting super-Earths, it has received a lot of attention, and many studies have already attempted to determine its composition. Previous studies employed infrared and optical observations of transits, occultations, and phase curves Demory et al. The implication for a possible gas layer is an optically thick layer with inefficient heat redistribution. The presence of a hydrogen-rich layer is unlikely, since it would not sustain stellar evaporation and in fact no extended hydrogen atmosphere has been detected Ehrenreich et al. Furthermore, the study of 55 Cnc e's thermal evolution and atmospheric evaporation by Lopez suggests either a bare rocky planet or a water-rich interior.
The composition of 55 Cnc e is a matter of debate and a consistent explanation of all observations is yet to come. The most recent interferometric study of 55 Cnc was performed by Ligi et al. Since 55 Cnc hosts a transiting exoplanet, the density of the star can be determined using the transit light curve of Maxted et al. It is therefore timely to use these new data to constrain the internal structure of the transiting planet. In this paper, we present in Sections 2 and 3.
As much as possible, we use analytical derivations of the probability density functions PDFs of the parameters of interest from those of the observed quantities. We apply these numerically to the case of 55 Cnc and its transiting planet, and show that we can reduce the uncertainty on the planetary density. In Section 3. Compared to previous applications of the model Dorn et al. The results are then compared to a scenario where the mass—radius correlation is neglected, and to a scenario where constraints on refractory element abundances are used.
Thereby, we can quantify the information content of the different data inputs on the planetary interior. Eventually, we provide the most precise interior estimates while rigorously accounting for data uncertainties. Section 4 is devoted to a summary and conclusion. In this section, we focus on the parameters of the host star, 55 Cnc.
We combine them to retrieve the parameters of interest luminosity L , effective temperature , mass M , radius R. More specifically, we provide analytically the joint PDF of these parameters from that of the observable quantities. A joint PDF shows the correlations; from the way the parameters are derived, correlations are strong and inevitable, and provide valuable information, as will be illustrated in this paper. Also, multiplying by a prior may lead to non-Gaussian final distributions.
Before determining the mass and radius of 55 Cnc, we first evaluate prior knowledge on stellar parameters that will help to improve the interpretation of observational data. More specifically, we look for possibilities of excluding sets of parameters that would correspond to the less populated regions of the Hertzsprung—Russell hereafter H-R diagram.
We take a Bayesian approach in order to estimate L and. In essence, this approach accounts for both the probability distribution of L and for 55 Cnc as deduced from observations of the star, and the prior distribution of L and for stars in general as derived from the H-R diagram. In the following, we discuss the approach in more detail and explain how it can affect the estimate of the stellar radius. The stellar radius R is the product of the angular radius , in radian with the distance d , which is proportional to the inverse of the parallax :.
It gives directly the PDF of R as a function of the observables. They are spread along a diagonal direction along , that is equal radius lines because both are increasing functions of F bol see also the Appendix of Ligi et al. In this case, the correlation would be 1. This curve corresponds to varying F bol while keeping the stellar radius and distance fixed. The uncertainty on the stellar radius and distance smears the PDF around this curve. Here, the coefficient of correlation of L and is 0. Figure 1. Contour lines: likelihood of the luminosity and effective temperature of 55 Cnc as given by Equation 7 based on observations by Ligi et al.
Background grayscale: density of stars in the Hipparcos catalog in this region; in this box, the minimum and maximum of are respectively 23 light gray and dark. Hence, one can estimate a priori regions in the H-R diagram where 55 Cnc has more chances to be found, and regions where it should not.
This is a prior PDF in the plane.
To build this prior, we have downloaded the Hipparcos catalog hip2. In Figure 1 , the background grayscale maps , the number density of stars in the Hipparcos catalog light for low density, dark for high density, linear arbitrary scale. The main sequence goes down steeply from the top left corner.
Inside the largest ellipse shown, the ratio of the maximum to minimum is 1. The star 55 Cnc appears to be in the vicinity of the main sequence.
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Eventually, the joint PDF of L and is. It should be noted that L is so well constrained by the observations that the multiplication by the prior has almost no effect on the PDF of L : we estimate from and from f HR as well. As for the temperature, while the expected value of from is with a standard deviation of , the found from f HR is:.
The data are very informative, and we are not dominated by the prior. Using Equations 2 and 3 gives m, f R being a Gaussian, as in Ligi et al. In Appendix A. No information is lost, and no uncertainty is added by moving to the H-R plane. Hence, using Equations 5 and 6 with f HR given by Equation 8 shows only the effect of the prior. Integrating this numerically, we find.
Figure 2. Top: joint probability density function of the mass and radius of the star 55 Cnc. The nine plain thick contour lines separate 10 equal-sized intervals between 0 and the maximum of Equation 9. The dashed blue contour lines show the same for the case where one mistakenly considers M and R as independent. Bottom: marginal PDFs of R and M plain lines ; the dashed blue line is the Gaussian obtained without the use of the prior in the case of R , and is a Gaussian curve of the same mean and standard deviation, for comparison, in the case of M.
Maxted et al. Then, the joint likelihood of M and R can be expressed analytically:. Using f R given by Equations 2 and 3 , the result is , with a correlation coefficient with of 0. The level curves of this distribution are shown in Figure 2 as the tilted solid ellipses. Using the prior in the H-R diagram, one gets , with a correlation coefficient with of 0.
Our results are summarized and compared to the ones of Ligi et al. We find that the prior from the Hipparcos catalog does not change significantly the joint PDF of ,. The interferometric observations are precise enough to constrain the stellar parameters. In what follows, we thus use the analytical expressions Equations 2 , 3 , and 9.
If correlation is neglected and M and R are directly taken with their uncertainties as independent variables, their joint PDF becomes a 2D Gaussian distribution represented by the dashed ellipses with horizontal and vertical axes in Figure 2. In doing so, one would have correct marginal distributions they are close to Gaussian. Obviously, taking the correlation into account reduces the area to explore in the mass—radius parameter plane, and should help constrain the structure and composition of the transiting planet, as we will see in the next section.
L and of 55 Cnc being known, one could fit them with stellar evolution models to infer the corresponding mass, age, and other parameters like the radius.
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Stellar models are a precious tool to estimate stellar parameters that are not measurable, provided observational constraints are tight enough. Nonetheless, this method should be used with care, for the following reasons. This highlights the difficulty of using stellar models to derive accurately the mass and radius of an individual star with reliable uncertainties. Of course, accuracy is difficult to assess; however, the variability of estimates yields a proxy for it.
Here, the different values from stellar models are only in rough agreement with one another, so it would be inappropriate to just pick one, neglecting the uncertainty on the parameters of the model. Note that the mass range we find using the Bayesian approach above encompasses the various stellar models mentioned here for the young solution see also Ligi et al. Although the interferometric radius disagrees with the radius found by asteroseismology for some stars which opens the question of possible bias for one of these methods , it overcomes assumptions that are otherwise introduced by the use of stellar models.
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Hence, reassured by the agreement with stellar models, in the following we adopt the estimate of the mass and radius for 55 Cnc given in Section 2. We stress that our error bar is larger than the brutal use of a single stellar model could provide, but we think it is the best possibility so far for 55 Cnc. In this section, we apply the previous results on the host star to the transiting planet 55 Cnc e. This planet has attracted a lot of attention already, being one of the first discovered transiting super-Earth, as explained in Section 1.
It is therefore an excellent case to test the power of our method. From the PDF of the mass and radius of the star, we deduce that of the planet analytically. For any M p , M , one can define the associated semi-amplitude of the radial velocity signal K , following a classical formula resulting from Kepler's law: where P is the orbital period, and we have assumed that the eccentricity is zero 9.
Similarly, for a pair R p , R , the corresponding transit depth is. Therefore, the PDF associated to any fixed planetary mass and radius is. The correlation is visible, as the cloud is elongated in a direction parallel to isodensity lines. An Earth-like composition is almost excluded, while a pure rocky interior appears possible. The blue dots in Figure 3 correspond to the case where would be replaced in the expression of by a PDF of M , R that would neglect their correlation shown as short dashed lines in Figure 2.
In this case, an Earth-like composition could be excluded with less confidence. Figure 3. Mass and radius data samples for O , OC , and the OH that mostly differ in terms of the correlation between mass and radius. In comparison, two idealized mass—radius relationships for pure MgSiO 3 and Earth-like interiors are plotted. MgSiO 3 represents the least dense end-member of purely rocky interiors. Therefore, purely rocky interiors cannot be exluded in cases of O and OC , whereas in the case of the hypothetical high correlation OH , the interior must be rich in volatiles.
See the text for details. It is particularly interesting to consider the correlation in order to estimate the density of the planet. From our joint PDF, we find. The limiting factor here is the uncertainty on , which is mainly responsible for the correlation between mass and radius to be much smaller for the planet 0. More precise observations of the transit would be very useful in this particular case and would allow us to increase significantly the gain on the density precision. In the particular case of 55 Cnc, the best way to do so would be to better constrain its density by obtaining a finer light curve.
In the next subsection, we use this joint PDF to characterize the interior of 55 Cnc e, including a test scenario where and K e would be known with negligible uncertainty, which is shown in Figure 3 as the pale dots labeled OH ; in this case, one recovers the 0. The estimates of planetary mass and radius are subsequently used to characterize the interior of 55 Cnc e.
To do so, we use the generalized Bayesian inference analysis of Dorn et al. This method allows us to rigorously quantify the degeneracy of the following interior parameters for a general planet interior:. Unlike Dorn et al. For the highly irradiated planet 55 Cnc e, any water layer would be in a vapour or super-critical state.
The prior distributions of the interior parameters are listed in Table 2. The priors are chosen conservatively. The cubic uniform priors on r core and reflect equal weighing of masses for both core and mantle. Prior bounds on and are determined by the host star's photospheric abundance proxies, whenever abundance constraints are considered. Since iron is distributed between core and mantle, only sets an upper bound on. A log-uniform prior is set for and. For example, for the data scenario O , we consider planetary mass and radius as well as other data, but we neglect mass—radius correlation and abundance constraints.
The structural model for the interior uses self-consistent thermodynamics for core, mantle, and to some extent also the gas layer. For the core density profile, we use the equation of state EoS fit of iron in the hexagonal close-packed structure provided by Bouchet et al. For the silicate mantle, we compute equilibrium mineralogy and density as a function of pressure, temperature, and bulk composition by minimizing the Gibbs free energy Connolly We assume an adiabatic temperature profile within core and mantle.
For the gas layer, we solve the equations of hydrostatic equilibrium, mass conservation, and energy transport. The metallicity is the mass fraction of C and O in the gas layer, which can range from 0 to 1. For the gas layer, we assume an irradiated layer on top of a convection-dominated layer, for which we assume a semi-gray, analytic, global temperature averaged profile Guillot ; Heng et al. The boundary between the irradiated layer and the underlying layer is defined where the optical depth in visible wavelength is Jin et al. Within the convection-dominated layer, the usual Schwarzschild criterion is used to determine where in the layer convection or radiation is more efficient.
The planet radius is defined where the chord optical depth becomes 0.
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We refer the reader to model I in Dorn et al. For each scenario, we have used the generalized MCMC method to calculate a large number of sampled models that represent the posterior distribution of possible interior models. The resulting posterior distributions are shown in Figure 4 , which displays cumulative distribution functions cdf.
The thin black line is the initial prior distribution. The colored lines correspond to the different data scenarios. They indicate how the ability to estimate interiors changes by considering different data. A summary of interior parameter estimates is stated in Table 3. Figure 4. The prior distributions are shown in black. OCA , OA and are not shown. In the first scenario O , the uncorrelated planetary mass and radius given in Table 1 are considered, as well as the orbital radius and stellar luminosity.
These data help to constrain the mass and radius fraction of the gas layer, the size of the rocky interior and the core, while intrinsic luminosity, gas metallicity, and mantle composition are poorly constrained. In the second scenario OC , we add the correlation coefficient of M p and R p. In the OA scenario, we add constraints on refractory element ratios compared to the scenario O with uncorrelated mass and radius.
Thereby the density of the rocky interior is better constrained which also affects the estimates of , , and by a few percent. The information value of abundance constraints is discussed by Dorn et al. If abundance constraints are considered, the effect of adding the mass—radius correlation is more pronounced. This can be seen by comparing scenario OA with OCA , in which the latter accounts for both the correlation and the abundance constraints.
The additional correlation mostly improves , , and. Note that neglecting the uncertainty on the planet-to-star radius and mass ratios also leads to reducing significantly the uncertainties on M p and more importantly R p : we get where the slight but negligible difference in the expected value with the previous case is due to the non-use of the Hipparcos prior here. This is true for , , and. In this scenario, we can exclude the possibility of a purely rocky interior and find gas layers with radius fractions larger than 0.
This hypothetical case illustrates the high value in both a high radius precision and mass—radius-correlation for interior characterization. The OCA scenario represents our most complete data set given the considered interferometric data. Figure 5 shows the posterior distribution of the OCA scenario in more detail. The one-dimensional posterior functions illustrate that only some interior parameters can be constrained by data, since prior and posterior distributions significantly differ: gas mass fraction , , r core , and.
The gas layer properties of metallicity and intrinsic luminosity are very degenerate and the data considered here do not allow us to constrain them. We find that the gas layer has a radius fraction of and a mass fraction about 10 times larger than for Earth, although with large uncertainty see Table 3. The gas metallicity is weakly constrained; however, low metallicities are less likely i. Figure 5.
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