## Info

Fig. 6.9 The relationship between minimum mass and period, of exoplanets, compared with the Roche limit calculated for a planet with a radius equal to that of Jupiter (aR) and twice the Roche limit (acjrc). The rectangles are planets for which the inclination (and thus the mass) is known. The two triangles correspond to the super-Neptunes GJ 436b and 55 Cnc e, whose internal structure may be different from the other, more massive, planets. If these planets were icy, this would explain how they resisted the dismemberment mechanism described in the text (After Faber et al., 2005)

Fig. 6.9 The relationship between minimum mass and period, of exoplanets, compared with the Roche limit calculated for a planet with a radius equal to that of Jupiter (aR) and twice the Roche limit (acjrc). The rectangles are planets for which the inclination (and thus the mass) is known. The two triangles correspond to the super-Neptunes GJ 436b and 55 Cnc e, whose internal structure may be different from the other, more massive, planets. If these planets were icy, this would explain how they resisted the dismemberment mechanism described in the text (After Faber et al., 2005)

1 The Roche limit is the distance from the star within which a planet is fragmented by tidal effects.

This limit is equal to 2.456.RA pM , where Rt is the radius of the star, and pp and pt are the densities of the planet and the star, respectively

### 6.3 Stability of Planetary Systems

In a sample of 179 known planetary systems, 21 have at least 2 planets, including one system around a pulsar. Of 20 multiple systems around Main-Sequence stars, 14 have 2 planets, 4 systems possess 3 planets, and 2 systems have 4 (Table 6.1). This proportion of 12 per cent multiple systems among known planetary systems is a minimum, because possible low-mass planets or ones distant from their stars cannot yet be detected. The orbital parameters of the systems are calculated from observations made over a short period of time. They correspond to elliptical orbits, which are not the planets' real orbits, particularly because the planets mutually perturb one another. So it is not possible to know the future evolution of the planetary positions by simple extrapolation from their motions. In addition, these parameters are obtained with a certain degree of inaccuracy and using specific assumptions, especially regarding the inclination of the orbits. Dynamical studies allow us to test the stability of the systems and their past and future evolution. To do so, we need to calculate the evolution of the systems taking account of the gravitational interactions between the bodies, and by exploring all the orbital parameters that are compatible with observations.

### 6.3.1 Dynamical Categories

For two planets in any given system, it is possible to look at the ratio of the orbital periods. For all the pairs of successive planets, the figures form a series running from values close to 2 to more than 300 (Fig. 6.10).