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Fig. 6.10 Ratio of periods of pairs of successive planets in exosystems (Table 6.1)

The systems, or to be more accurate, the pairs of planets, may be classified into three dynamical categories:

Resonant, or Class I, systems contain two planets whose orbits are very close to a mean-motion resonance of 2:1 or 3:1. These planets are strongly coupled by gravitation. Their orbits are perturbed over very short timescales and they would not be stable but for the fact that they have become trapped in a mean-motion resonance system.

Interactive, or Class II, systems: They contain planets whose orbits are not in resonance, but which are close enough to perturb one another. The ratio of their periods is greater than 4.6:1, which means that it is difficult for them to be captured in a resonance. The interactions between the planets may be strong, but conservation of angular momentum limits any variations in eccentricity. The Solar System is a member of this class. These systems may exhibit specific configurations, such as the alignment or anti-alignment of the periapses. The dynamical boundary between this class and the next one is not easy to define. The stability of orbits of Class II is, however, extremely sensitive to the values of the system's parameters, which is not the case with systems of Class III. It sometimes happens that the parameters that are deduced directly from the observations correspond to those of an unstable system and that the real parameters of the system are to be found by investigating stable systems that are close to the initial solutions and are also compatible with the observations.

Separated, or 'hierarchical', Class III systems: If the periods of the planetary orbits have a ratio greater than 30:1, any interactions between the planets are very weak. The eccentricities of the inner planets in such systems may, however, exhibit significant variations, as is the case with HD 74156 b. The evolution of these orbits is not at all sensitive to variations in the orbital elements.

HD 82943 and GJ 876 are in resonant systems. Interactive systems include 47 UMa, u And, and the Solar System. HD 168443 and HD 83443 are separated systems.

This classification enables us to understand the behaviour of systems a bit better, but it is not enough. Dynamics experts place the two inner planets of u And in Class II, because of their interactions, whereas the ratio of their periods, 52.2:1, would put them in Class III. A Class Ib has been created to include planets without significant period ratios, and having low eccentricities. In particular, this class contains the planets around pulsar PSR 1257+12 and probably 47 UMa as well.

For several of the systems observed, a systematic study of the stability over 106 years has been made, varying the orbital parameters but remaining within the error bars obtained by an analysis of the radial velocities. Each set of parameters either led to a stable or unstable configuration. Such studies have shown that the stable zones are similar within each category:

• Resonant systems have very narrow stable zones within their phase space. The stability depends on the ratio of the periods of the two planets and, to a lesser extent, on the eccentricity of the more massive of the planets.

• Interactive systems have quite broad stable zones. The stability is linked to the eccentricities of the two planets. Even in a planetary system with low eccentricities, such as the Solar System, the neighbourhood of the planets is filled with a bcd e f g h a bcd e f g h

18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 20.2 SEMI-MAJOR AXIS (AU)

Fig. 6.11 A plot of stability near Uranus. The white areas correspond to stable orbits. The hatched area corresponds to rapid ejection of the planet. The letters indicate resonances with Jupiter, Saturn, and Neptune (After Ferraz-Mello et al., 2005)

18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 20.2 SEMI-MAJOR AXIS (AU)

Fig. 6.11 A plot of stability near Uranus. The white areas correspond to stable orbits. The hatched area corresponds to rapid ejection of the planet. The letters indicate resonances with Jupiter, Saturn, and Neptune (After Ferraz-Mello et al., 2005)

a dense set of resonances. Jupiter and Saturn, for example, are close to a 5:2 ratio; Uranus is between the 7:1 resonance with Jupiter and the 2:1 resonance with Neptune on the one hand, and the 3:1 resonance with Saturn on the other. Neptune is close to the 2:1 resonance with Uranus. Figure 6.11 illustrates how Uranus is isolated within the middle of a zone of unstable orbits. • Separated systems are always stable.

Probabilities of survival to 106 years have been calculated for all the different configurations. These probabilities are 20 per cent in resonant systems, 80 per cent in interactive systems, and 100 per cent in separated systems. These figures do not express the stability of the real system but correspond to the proportion of stable systems among the virtual systems that were obtained with parameters close to the real ones.

Here are some of the systems known at the beginning of 2007. It is only a small sample of the great diversity found among exoplanetary systems. In addition, these systems may contain planets that have not yet been discovered because they are too small or have periods that are too long.

6.3.2 The GJ 876 System

The two planets are in orbits with a 2:1 resonance. Modelling the observations needs to take account of gravitational interactions between the planets because the periods are very short, 30 and 60 days, and the masses are high relative to the mass of the star, (mi + m2)/M = 0.0074. The two orbits are almost aligned, and the three angles

À1 - 2À2 + TOi, À1 - 2À2 + TO2, and À1 - 2À2 + TO1 - TO2 oscillate around 0 with a low amplitude, À1 being the mean longitude of the planet and TO1 the longitude of periapsis. A third planet has subsequently been discovered. With a period of 1.9 days, it is too far from the two others to perturb the resonant system.

This resonant configuration cannot be explained except by migration of the planets: during the course of their movement, the planets crossed a resonance zone and have remained trapped in that configuration. This explanation is incomplete, however, because this mechanism should have resulted in orbits with greater eccentricity than those observed for the planets, 0.31 and 0.05.

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