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Fig. 3.2 Histogram of the mass-distribution - more precisely, M.sin(i) - for the stellar and sub-stellar companions discovered by the radial-velocity method. It is obvious that the distribution is bimodal with planets on the left, and stars on the right. The poorly populated central zone between the two peaks (where 0.01 M0 < M.sin(i) < 0.08 M0) is known as the 'brown dwarf desert' (After Santos et al. (2002))

Fig. 3.2 Histogram of the mass-distribution - more precisely, M.sin(i) - for the stellar and sub-stellar companions discovered by the radial-velocity method. It is obvious that the distribution is bimodal with planets on the left, and stars on the right. The poorly populated central zone between the two peaks (where 0.01 M0 < M.sin(i) < 0.08 M0) is known as the 'brown dwarf desert' (After Santos et al. (2002))

Use of data from the European astrometry satellite Hipparcos has, however, enabled seven of the candidates detected by the ELODIE instrument (11 candidates in the histogram shown in Fig. 1.2), to be definitely excluded3 (Halbwachs et al. 2000). This particular result tends to show that the 'brown dwarf desert' is not only real, but, in addition, that the brown-dwarf candidates detected by radial-velocity measurements are, probably in the majority of cases, objects with significantly greater masses, where sin(i) is low. This significant separation between planets and stars may be explained - or at least understood - when the standard models for the formation of stellar and planetary bodies are considered. Stars form by the gravitational collapse of an even more massive nebula (of some 100 MQ), whereas planets, even giant ones, start in the form of embryonic nuclei and then grow by the subsequent accretion of matter. In the first case, the process is one of fragmentation from high-mass objects to less massive ones, whereas in the second case, the embryonic nucleus becomes a planet through the processes of accumulation and accretion. These two processes involve various physical phenomena which have different governing parameters, and which explain the different scale limits.

If we now consider just exoplanets, we can derive the same type of histogram for the 306 planets currently known (Fig. 3.3).

Examination of Fig. 3.3 clearly shows several points:

• although the sensitivity of the radial-velocity method (which has currently provided more than 90 per cent of candidate planets) increases as the mass of the object increases, almost half of the planets have a mass less than, or comparable with that of Jupiter. High-mass planets (several Jupiter masses) therefore seem to be rare, which gives us an indication of planetary systems' formation processes. It should, however, be mentioned that the distribution in Fig. 3.3 shows the product of the mass of the object and the sine of the angle of inclination to the line of sight (which may be derived from the radial-velocity measurement). The mass that is deduced is therefore a lower limit for the real mass of the objects. However, and making allowance for the size of the sample, it is possible to deconvolve the histogram (Santos et al., 2002), and this confirms the real tendency shown in the histogram. High-mass planets are, therefore, truly less numerous;

• The mass-distribution between 0.1 and 2 Jupiter masses is more-or-less uniform, which tends to show that, for planets, the whole range is covered. So it is obvious that there was segregation by mass over this range when the formation processes took place for these objects;

• There are more than a dozen objects whose mass is less than one tenth of a Jupiter mass. This observation is the result of the detection of objects with lower and lower masses (masses comparable with those of Uranus and Neptune), and tends to suggest that the range of masses extends as far as the domain of terrestrial masses. This result is confirmed by the first detections of several Earth-mass objects by microlensing and radial velocity measurements.

3 The astrometric data from Hipparcos enable the uncertainty over the value of sin(i) to be resolved.

Fig. 3.3 Mass distribution of exoplanets detected to date. top, the total population; bottom, detail between 0 and 2 Jupiter masses (the mass of Jupiter is about 10-3 solar masses)

3.3 The Distance-Distribution of Exoplanets

The semi-major axis of a planetary orbit may be deduced from the measurement of the orbital period by applying Kepler's Third Law, provided one can estimate the mass of the central star, which is derived from measurement of the spectral type (cf. Appendix). The distribution of the semi-major axes of exoplanets is given in Fig. 3.4.

This statistical analysis provides various items of information:

• Nearly half of the objects orbit at distances less than 0.4 AU (which is the distance between the Sun and Mercury in the Solar System). Among these, about one third orbit at distances that are extremely close to their stars, typically 0.05 AU, and thus with orbital periods of just a few days. The first planet detected by radial-velocity measurements (51 Pegasi) is the prototype for these planets, which were completely unknown until the first observation, and which have been called 'hot Jupiters' because of their size (being giant planets) and distance from their parent stars. Objects with a lower mass (a mass comparable with that of Neptune or Uranus) have since been detected in orbits that are as close to their parent star, and have thus been described as belonging to the 'hot Uranus' or 'temperate Uranus' class, as a result of their distance from the star and thus of the temperature that the object's thermal equilibrium implies. Apart from the fact that this type of object is completely unknown in the Solar System, the discovery of the first hot giant planets poses many questions about their formation and evolution. In fact, current models of protoplanetary nebulae make it impossible for these planets to have formed where they are currently found, primarily because the temperature would have been too high to allow the grains of the core to condense, and also because of the density of the disks. We must, therefore, envisage that these objects formed at greater distances from their stars and that the parameters of their orbits have evolved over the course of time, so that they attained their current position through 'orbital migration'. The question of orbital migration is considered in Chap. 6. The consequences for a planet of having an orbit close to its parent star are numerous, because the planet is permanently strongly irradiated, and the spectrum of these objects is completely different from that of a classical Jupiter-type planet, such as we know from the Solar System. In addition, taking account of the proximity of the star, the orbit of the planet is generally practically circular (or of low eccentricity), and the period of revolution is probably equal to the period of rotation (i.e., there is spin-orbit synchronization). This point is considered in Chap. 7.

• With the exception of the hot objects, the distribution of objects with distances between 0 and 5 AU is relatively uniform. These observations do not contradict all the theoretical approaches, which show that certain orbits are more stable than others in a given planetary system. The diverse range of parent stars, most notably with regard to mass, tends to smooth out the distance distribution.

• The detail shown in Fig. 3.4 for distances between 0 and 1 AU indicates that the distribution of distances is not uniform when seen at that scale. There does, in

Fig. 3.4 Distribution of the semi-major axis of the orbits of exoplanets. Bottom: detail of the region between 0 and 1 AU

fact, appear to be a relative deficit of objects around 0.4 AU. This limit seems to correspond to a distinction between objects that have migrated, and whose orbits are therefore not primitive ones, and other objects which may, potentially, have formed at the location where they are currently found. It should be pointed out that this observation is not the result of bias, because the orbital parameters are directly derived from the radial-velocity measurements, which are greater (and measurement easier), the larger the mass of the object, and the closer it is to the parent star.

Study of certain multiple systems has also revealed that various planets may occupy resonant orbits (cf. Appendix for the definition of resonance). This is particularly the case for the system Gliese 876, where three planets have been identified. One of these (Gl 876 b) has a period of 60.94 days, while another (Gl 876 c) has a period that is practically half that, 30.1 days, and the third component (which is more controversial) is a hot object with a period that is less than 2 days. Resonance effects are discussed in detail in Chap. 6.

3.4 The Relationship Between the Mass of Exoplanets and Their Distance from Their Star

In the Solar System, the most massive planets, the four giant planets, lie at great distances from the Sun, and the less massive ones are closer4. Is this separation found in exoplanetary systems? The mass/distance distribution for roughly the first 100 planets discovered and for the 209 currently known is shown in Fig. 3.5.

Analysis of the distribution for the first 100 objects clearly revealed several specific points [After Zucker and Mazeh (2002); Udry et al. (2003)]:

• the absence of high-mass objects at small distances (on the face of it, 'hot superJupiters' do not appear very numerous)

• the absence of any objects with masses below that of Jupiter between 0.5 and 5AU.

At the time, any suggestion of observational bias was ruled out by statistical arguments based on the sensitivity of the method of detection by means of measurements of the radial velocity of the parent star (which, it may be recalled, is the more sensitive, the greater the mass of the planet and the closer it is to the star).

The analysis may be repeated for the current state of detection (Fig. 3.5, right), and this shows that the situation is not really any different. Apart from a few exceptions, which may be explained by the fact that measurement of the mass is marred by an unknown in the inclination of the planetary system (which only gives a lower

4 According to its new definition, established at the General Assembly of the International Astronomical Union, in August 2006, the Solar System consists of just 8 planets, 4 terrestrial ones close to the Sun (Mercury, Venus, Earth and Mars), and 4 giants at greater distances (Jupiter, Saturn, Uranus and Neptune), with minor planets (asteroid) more-or-less scattered throughout the system.

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