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Fig. 3.5 The relationship between planetary mass and distance from the star: (top) for the first 100 objects [After Udry et al. (2003)]; the limit for detection at 10m.s_1 corresponds to the sensitivity limit for instruments in 2003; (bottom) for the objects currently listed limit), and by the fact that the sensitivity of our instruments has increased, we are forced to admit that the observations in 2002-2003 remain completely valid.

An explanation of these observations may derive from the analysis of the effectiveness of the processes of migration as a function of the mass of the objects (even if in the type II migration process - i.e., that involving massive planets (cf. Sect. 6.3) - it is solely the viscosity of the protoplanetary disk that is involved, and not the mass of the planet). For the most massive objects, with a greater inertia, the migration process is less effective, and as a result they tend to remain in their initial position, at a great distance from the star. By contrast, objects of low mass where the migration process is, on the face of it, more efficient, are statistically more abundant at small distances. There is a detailed discussion of this point in Udry et al. (2003). A complete description of the planetary migration process is given in Chap. 6.

It should also be noted that a new generation of instruments has begun operation, and which has a sensitivity better than 1 m.s-1 (for example, the HARPS instrument on the ESO 3.6-m telescope at La Silla), and that certain new objects, notably of very low mass, have been discovered by this means.

3.5 Orbital Eccentricity Among Exoplanets

Determination of the eccentricity of the orbit of a planet, detected using the radialvelocity method, is carried out by analyzing the deviations of the radial velocity from the sine wave displayed by the radial velocities (cf. the previous chapter). Figure 3.6 shows the eccentricity of exoplanets as a function of their orbital period.

Fig. 3.6 Eccentricity of exoplanets as a function of their orbital period

Figure 3.6 basically shows three points:

• The short-period objects (with periods shorter than a few days) have practically circular orbits

• Above a few days, there is no obvious correlation between the period and the orbital eccentricity

• Unlike our Solar System, where eccentricities are low5, some planetary systems exhibit eccentricities that are extremely large (and may even attain 0.92)

• The distribution of eccentricities as a function of orbital period is perfectly comparable with those observed in multiple stellar systems (in binary stars, for example).

These observations may be explained by recourse to the following arguments: at short distances, the tidal effects are greater. The gravitational-potential gradient is stronger than at greater distances, which creates a differential attraction between the inner side and the outer side of the planet, which results in the formation of a bulge on the exoplanet. This bulge leads to the appearance of a torque exerted by the star on the planet, causing the dissipation of energy through friction in the planetary mantle. The dissipation of energy tends to circularize the orbit of the planet. At the same time, there should also be progressive synchronization of the periods of the planet's revolution and rotation6 as with the Moon in its orbit around the Earth. It may be shown that the time required to circularize an object whose orbital period is 7 days, amounts to about 1000 million years (Halbwachs et al., 2005). When the average age of the objects observed is taken into account, this explains the limit at a few days seen in Fig. 3.6. Above this figure of a few days, the objects' orbits are not yet completely circularized, and the distribution of eccentricities becomes more dispersed. (It should be noted that no object with a period less than 30 days has an eccentricity greater than 0.5.)

To explain the fact that the distribution of eccentricities as a function of orbital period does not display any obvious correlation for periods about a few days - unlike the case with multiple stellar systems - the following argument may be used. At a greater distance, the tidal effects (differential attraction) become negligible relative to the interaction between the planet and the protoplanetary disk in which it is forming (a theory proposed by Goldreich and Sari in 2003), or relative to the interactions between the planet and its planetary and stellar environment (a theory by Marzari and Weidenschilling in 2002). The latter interactions, however, tend to increase the orbital eccentricity, which was originally low at the time of the planet's formation, through the transfer of angular momentum (see Chap. 6).

5 The planet with the greatest eccentricity (after the eccentricity: 0.246, of Pluto, which has just lost its status as a planet), is Mercury with an eccentricity of 0.206. The other planets all have eccentricities less than 0.1, which is very low when compared with what is observed with certain exoplanets.

6 This is probably not without consequences for the planet's climate, because it always turns the same face towards the star. This point is discussed in Chap. 8.

Fig. 3.7 Orbital eccentricity of exoplanets as a function of their mass

The absence of massive objects in the immediate vicinity of the Sun may be invoked to explain the fact that eccentricities in the Solar System have remained small.

It should also be noted that certain theoretical studies predict that the eccentricity of planetary orbits increases with the mass of the planet. Figure 3.7, which represents the eccentricity of exoplanets as a function of their mass, does not, however, reveal any obvious correlation. So there is no observational support for these theoretical conclusions.

3.6 Exoplanets and Their Parent Stars

The diversity of parent-star types is one of the highlights of the currently known distribution of planetary systems. Planetary systems have been identified around a wide range of objects:

• main sequence stars for spectral types F to M

• giant (e.g. HD 122430); sub-giant (e.g. HD 47536) dwarf (e.g. HD 209458), and sub-dwarf stars (e.g. V391 Pegasi)

• protoplanetary disks

• pulsating stars

Let's consider now the metallicity of parent star. To astronomers - to put it rather dramatically - the periodic classification of the elements has, in the end, come down to just three categories:

• hydrogen, whose relative abundance (the partial fraction) by mass is denoted by X,

• helium, whose relative abundance by mass is denoted by Y,

• all the other heavier elements, known generically as 'metals', whose overall relative abundance is denoted by Z or Fe.

By definition, we have X + Y + Z = 1, and the 'metallicity' of a star is the value of Z. For the Sun, for example, X = 0.73, Y = 0.25, and Z = 0.02. (These abundances are known as 'cosmic abundances'.) For convenience, the logarithm of the abundance of iron in a star, relative to the abundance of iron in the Sun is also known as the metallicity (with no risk of confusing it with Z). It is denoted [Fe/H] and given by the ratio:

This value is zero for stars with the same metallicity as the Sun, negative for objects of lower metallicity, and positive for stars with a greater metallicity. Stars that have a higher metallicity than the solar metallicity (or cosmic abundances) are said to exhibit a 'metallicity excess'.

The metallicity of a star is measured spectroscopically in the visible or even the near UV region. The metal abundances are obtained by comparing the depths of the spectral lines associated with certain electronic transitions of the metallic atoms (Fe, Ti, etc.).

Figure 3.8 shows the distribution of metallicity of the stars that have been identified as having planets.

In examining Fig. 3.8, we can see that stars that have exoplanets are generally stars with a metallicity excess (about two-thirds of the stars in this histogram have a metallicity that is greater than the metallicity of the Sun).

This result, first published in 2002, was initially subject to some discussion, because some people saw the excess metallicity as a sign of bias in the choice of the stellar sample that had been used for the radial-velocity method. After comparison and study of various stellar samples (defined, as far as M. Mayor's team were concerned, as being all the stars of one or several, given, spectral types out to a certain distance from the Sun), this result is now accepted by the general astronomical community.

This observation is not really surprising, because in the standard model for the formation of planetary systems, planets form within a disk of dust and gas around a young star (cf. Chap. 5). The composition of this disk is similar to the composition of the star. To form planets, it is initially necessary to create rocky kernels that will form the cores of future planets. Now, these kernels consist of the metallic elements (as we have defined them). So it is not surprising that we should find planets around metallic stars.

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