A statistical model for the relation between exoplanets and their host stars

An extrasolar planet (or exoplanet) is a planet which orbits a star other than the Sun, and therefore belongs to a planetary system other than our Solar System. The first extrasolar planet around a main sequence star was discovered in 1995 (Mayor and Queloz, 1995). Actually more than 300 exoplanets have been documented and most of them with masses greater than Jupiter's mass (Schneider, 2009). Detecting an exoplanet is a very difficult task because they do not emit any electromagnetic radiation of their own and are completely obscured by their extremely bright host stars, that is, normal telescope observation techniques cannot be used. Thus, in order to find exoplanets, a variety of techniques like the radial velocity, pulsar timing, astrometry, gravitational lensing, spectrometry and photometry (De Pater and Lissauer, 2001) are used. The main purpose of any method is to detect the effect produced by the exoplanet on its stellar system. Besides the discoveries it is important to search for models that can explain the origin, formation and possible migration of these bodies. For example, Rice and Armitage (2005) have investigated how the statistical distribution of extrasolar planets may be combined with knowledge of the host stars' metallicity to yield constraints on the migration histories of gas giant planets. Moreover in a series of papers (Udry et al., 2003;Santos et al., 2003;Eggenberger et al., 2004;Halbwachs et al., 2005) the emerging properties of planet-host stars and characteristics of the different orbitalelement distributions of exoplanetary systems have been studied. In this work we analyze the cross-sectional data for the exoplanets detected until June 2008 through linear regression techniques. The purpose of this kind of analysis is to verify the relation between the host star and its orbiting planet. For example, if the planet's mass is strongly determined by the type of star and hence affects the planetary formation stage.

The catalog was created in February 1995 to facilitate the progress of the new field named Exoplanetology through the publication of recent detections and their associated data. The catalog is interactive and it is available in the webpage: http://exoplanet.eu . Until June 2008 the catalog contains: 303 exoplanets and 259 planetary systems (31 multiple systems). Two important considerations are: 1) the mass of the exoplanet is -at least-13 M J (Jupiter’s mass) and 2) the data source must be reliable, that is, previously published in referred journals, presented in conferences, among others.

• Stars: The stellar data are taken from well-known databases like Simbad or directly from published papers. The basic physical characteristics of a star are: radial velocity, mass, metallicity, age and distance.

• Planets: These data are taken from published papers and from the sites: Anglo-Australian Planet Search; California and Carnegie Planet Search; Geneva Extrasolar Planet Search Programmes; Transatlantic Exoplanet Survey and the Department of Astronomy at University of Texas.

We start with the following model (Model A) described by the equation:

where M P is the exoplanet’s mass and α i are the coefficients for each term. Eq. ( 1) expresses the exoplanet’s mass M P in terms of the values of the 1 where we also include the values of the t-statistics and their associated probabilities for the coefficient significance tests. From the estimated values we conclude that the only significant variable for the Model A is RS.

1. Linearity: The model passed all the Ramsey tests for linearity. We conclude that the proposed functional form is adequate.

2. Omitted Variables: According to the star formation theory, the variables M S and M ET AL must be included to explain the relation between the mass of the exoplanet and its host star.

3. Multicollineality: There is a possible weak correlation between M AG and DS.

4. Heteroskedasticity: From the White test on the residuals we conclude that they are not heteroskedastic, that means the residuals are homoskedastic.

5. Normality: From the value of the Jarque-Bera statistic we conclude that the residuals are not normally distributed.

6. Homogeneity: Defining the “dummy” variable as SIST (0 means that the exoplanet belongs to a single planetary system and 1 refers to a multiple planetary system) we conclude that the model is homogeneous.

Statistical model must satisfy all the assumptions mentioned above to be correctly specified. In our case, the Model A needs some modifications, for example, another functional form and/or the consideration of an adequate “dummy” variable. In such a case we derive the Model B:

where SM S = SIST * M S and SM ET = SIST * M ET AL are two new variables that take into account the fact that the exoplanet can belong to a single or a multiple planetary system. The parameters are estimated through OLS and the results are summarized in Table 1.

1. Linearity: The model passed all the Ramsey tests for linearity.

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