No correlation between the Transit-Depth Metallicity of Kepler gas giant confirmed and candidates planets: A Bayesian Approach
NNo correlation between the Transit-DepthMetallicity of
Kepler gas giant confirmed andcandidates planets: A Bayesian Approach
Cyrine Nehm´e , and Paula Sarkis Department of Physics & Astronomy, Notre Dame University- Louaize, PO Box 72, ZoukMikael, Lebanon CEA/DRF/Irfu/SAp, F-91191 Gif-sur-Yvette, France Universit¨at Bern Space Research & Planetary Sciences Division Bern, SwitzerlandE-mail: [email protected]
Abstract.
Previous attempts to study the correlation between the transit depth and the stellarmetallicity of
Kepler gas giant planets has led to different results. A weakly significant negativecorrelation was reported from the
Kepler’s (Q1-Q12) gas giant candidates with estimated radiiof 5-20R (cid:12) and [Fe/H] values taken from the Kepler Input Catalog (KIC). With the release ofthe last
Kepler catalog (Q1-Q17), we now have the largest homogeneous sample of exoplanets.This enables a solid statistical analysis of this correlation. In the present work, we revisethis correlation, within a Bayesian framework, for two large homogeneous samples: confirmedand complete. We expand a Hierarchical model to account for false positives in the studiedsamples. Our statistical analysis reveals no correlation between the transit depth and thestellar metallicity. The fact that we found no evidence of such correlation will have implicationsfor planet formation theory and interior structure of giant planets.
1. Introduction
NASA’s Kepler Mission revolutionized the field of extra solar planets and now more then ever,it is possible to put statistical constraints on the observed planet properties and on theoriesof planet formation. Clues on the nature of giant planet formation might be revealed fromthe two correlations with stellar metallicity of main sequence stars hosting these planets. Thefirst one is the correlation of the frequency of giant planets with stellar metallicity revealed byradial velocity surveys([1]) and by transit surveys ([2]). The second correlation shows a positivetrend between the mass of heavy-elements in giant planets and the stellar metallicity ([3]). Thisexplosion of new information demonstrates the need to understand planet formation in generaland presents an opportunity to compare observed trends to the theories of planet formation andevolution. Earlier attempts to study the correlation between the transit depth and the stellarmetallicity of
Kepler’s gas giant candidates has led to different results. For instance, [4] reporteda negative correlation with a weak significant value. The author studied the transit depth of218 giant planets from [5], (Q1-Q12) catalog with estimated radii of 5-20R ⊕ and the values of[Fe/H] taken from the Kepler
Input Catalog (KIC). [4] interpreted the negative correlation asevidence that metal rich planets of a given mass are denser than their metal poor counterpartsleading to small radii ([6]). Here, we will use the latest available catalog Q1-Q17 ([7]) to study a r X i v : . [ a s t r o - ph . E P ] M a r orrelation between the transit depth and stellar metallicity. Noting that a sample of stars withtransiting planets may not accurately represent the true intrinsic distribution of the discoveredplanets. [8] reported the importance of including these effect since they can lead to biases in theproperties of transiting planets and their host stars. For these reasons, we study only a subsetof the target stars and the detected giant planets. We develop a flexible framework to accountfor uncertainties by expanding the Hierarchical Bayesian Model introduced by [9].
2. Selection criteria and complete sub-sample
We use the cumulative catalog of planets detected by the NASA
Kepler mission which, as ofApril 2015, consisted of the latest Q1-Q17 catalog ([7]). Following [4] and [10] we define gasgiant planets as planets having a radius between 5-20R ⊕ . The stellar parameters were takenfrom the Kepler stellar Q1-Q16 database ([11]). We ended up by having 84 planets confirmedand 305 candidates.With the goal of performing a robust statistical method, we prepared two different samples. Thefirst sample consists of all the 84 confirmed giant planets within the latest catalog. The secondsample contains a complete subsample of both confirmed and candidates. Hence, we performedcuts needed to take into account: the incompleteness of the catalog ([12]), the selection effectsfor host stars and planets candidates, the detection efficiency and the false positives. Afterperforming all the cuts (table 1), we retain 105 confirmed + candidates planets. We believethat thi complete subsample better represents the true intrinsic distribution of
Kepler’s giantplanets.
Table 1.
Summary of the cuts performed to obtain a complete subsample
Parameter Value
Stellar effective temperature , T eff
Stellar gravity , log g (cm/s − ) 4.0 - 5.0 Stellar Radius , R (cid:63) (cid:12)
Planetary Radius , R p ⊕ Orbital Period , P <
90 days
Detection Efficiency , SNR > Kepler magnitude , K p <
16 mag
3. Method
Hierarchical Bayesian Modeling (hereafter HBM) allows for intrinsic scatter and heteroscedasticmeasurement errors i.e the uncertainties for each data points are different. We followed anapproach similar to that proposed by [9]. We constructed the likelihood function in a simpleway in order to relate the parameters of interest to the observed data, taking into account themeasurements uncertainties. We extended the model to account for false positive by [13]. Thisupdated model the HBM is used to study the correlation between the transit depth ( δ ) andthe metallicity (FeH) of host stars. A graphical illustration of our HBM is given in Figure1. Markov chain Monte Carlo (hereafter MCMC) was performed using the Python packagePySTAN, a package for Bayesian inference. We ran models with 4 Markov Chains, with 5000iterations for the first sample and 10 000 iterations for the second one. The first 50 per centof each chain was discarded as ”burn-in”. This work is the first to perform a full HBM tostudy correlations in general and the correlation between the transit depth and the metallicityof Kepler’s giant planets, in particular. Most importantly, the quantification of the intrinsicdispersion which has not been characterized before, is now defined. igure 1.
A graphical presentation of our HBM is given in this illustration. The graynodes arethe observed parameters. The true missing parameters are in the white nodes. The blue (upperleft) nodes are the nuisance parameters and the red nodes(upper right) are the parameters ofinterest. FeH i = stellar parameter of the i th planet, σ F eH,i = uncertainty on the stellar metallicityof the i th planet, δ i = transit depth of the i th planet, σ δ,i = uncertainties on the transit depth ofthe i th planet, FeH ti = true stellar metallicity of the i th planet, δ ti = true transit depth of thei th planet, µ and τ = nuisance parameters, α , β and σ = parameters of the linear model
4. Results
The posterior distributions for each of the parameters of interest ( α , β and σ ) produced byrunning MCMC are shown in the left panel in the Figure 2, for the two samples. The equationof the ”best-fit” linear models are : δ = (0 . ± . . ± . F eH with an intrinsicscatter of σ = 0 . ± .
005 for the confirmed planets sample. For the Complet subsample thebest fit and the intrinsic scatter are δ = (0 . ± .
06) + ( − . ± . F eH and σ = 0 . ± .
5. Discussion
We presented for the first time in the exoplanet literature, and within a Bayesian framework, astudy of the correlation between the transit depth of
Kepler’s giant planets and the metallicity ofthe host star. Data from
Kepler (Q1-Q17) ([7]) allowed us to characterize the intrinsic scatter inthe relation with a robust statistical analysis. We did not assume that the observed parametershave negligible uncertainties. Moreover, we did not consider all the planets in the samplesas bona fide . We expand the hierarchical model presentend in [9], to account for the falsepositive rates. We also considered in the model the relevant selection effects. The use of MarkovChain Monte Carlo (MCMC) to fit models to observations is becoming a standard practice inastronomy. We performed MCMC using the package PyStan to estimate the parameters of thehierarchical linear model. We established that there is no correlation between the transit depthand the stellar metallicity of
Kepler’s gas giant planets. Our model indicates that there is arelatively large intrinsic scatter in the relation. Hence, the previous results could probably bean artifact which shows the importance of accounting for uncertainties and for possible falsepositives. This is an exciting result actually. It also proves the importance of accounting forselection effects and biases within the transit surveys, such as
Kepler , and the significance of igure 2. Left column are Posterior probability distribution for the parameters of our modeland as computed by the MCMC package, PySTAN, marginalized over the other parameters (leftup for the confirmed sample and left down for the complete one).
Right column graphs arethe transit depth ( δ ) of Kepler’s giant planets vs. the metallicity of the host star ([Fe/H]).Thedashed lines represent the best fit line and the light lines are samples from the MCMC chain.studying a complete subsample. [4] interpreted the negative correlation as evidence that metal-rich planets of a given mass are denser than their metal-poor counterparts, leading to smallerradii ([6]). On the contrary, with our robust statistical model, we have proven the independenceof transit depth and the stellar metallicity. It certainly warrants further investigations to checkwhat planetary formation model can explain the outcome.
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