This system consists of 3 planets, of which the innermost is on a circular orbit, while the others have eccentric orbits (around 0.3). Tidal effects have circularized the innermost orbit, and the other two planets interact very strongly. The orbits are in a specific alignment: the periapses of the two planets oscillate around the same value with a low amplitude. This configuration may be caused by the migration of the planets through interaction with a gaseous disk. However, later observations favour the theory of a close encounter between u And c and a fourth planet. The models show that the eccentricity of u And d remains at a value close to 0.03 and that the eccentricity of u And c is evolving with large-amplitude changes. These parameters confirm a scenario where the eccentricity of u And d has undergone sudden variation and not a slow evolution. A possible process is a close encounter with an outer planet, which was subsequently ejected from the system (Fig. 6.12).

Modelling these observations also provides an indication of the inclination of the orbits: if the three planets are in the same plane, the system becomes unstable if sin i is less than 0.5 (where sin i = 1 corresponds to a system seen side-on). The system also becomes unstable if the inclination of the planetary orbits is greater than 40°.

Fig. 6.12 The dynamical evolution of a virtual planetary system similar to that of U Andromedae: the planets C, D, and E correspond to the planets U And c, U And d, and an additional planet. For each planet, the three lines correspond to the semi-major axis between periapsis and apoapsis. The planets are initially on quasi-circular orbits. The planet D receives an impetus that increases its eccentricity, and then the two planets C and D enter a regular cycle of interaction. The planet E is ejected, without that ejection modifying the system of C and D. The planet U And b is not taken into account in this model (After Ford et al., 2005)

Fig. 6.12 The dynamical evolution of a virtual planetary system similar to that of U Andromedae: the planets C, D, and E correspond to the planets U And c, U And d, and an additional planet. For each planet, the three lines correspond to the semi-major axis between periapsis and apoapsis. The planets are initially on quasi-circular orbits. The planet D receives an impetus that increases its eccentricity, and then the two planets C and D enter a regular cycle of interaction. The planet E is ejected, without that ejection modifying the system of C and D. The planet U And b is not taken into account in this model (After Ford et al., 2005)

6.3.5 The HD 202206 System: A Circumbinary Planet?

This is perhaps the first example of a planet around a binary system. The first companion was detected in 2002 around HD 202206. Its m.sin i is close to the limit for brown dwarfs, 16.5 MJ. Its orbit has a semi-major axis of 0.83 AU, and an eccentricity of 0.43. A second planet was announced in 2004, far less massive than the first (2.44 MJ), but farther from the star (2.55 AU). Modelling the system with three bodies, and comparing this with observations, allows us to determine the orbital parameters of the system (Table 6.2). Figure 6.13 superimposes the correlation levels with the observations (contour lines), with tones of grey representing the stability of the system. It will be seen that the parameters that best fit the observations correspond to a white zone, i.e., to unstable orbits. An external planet in such a system is ejected in 40 000 years. Close to this solution we can see a dark zone that corresponds to stable orbits. It is therefore very likely that the real system lies within this zone of stability, which only represents a small increase in the residuals. This solution corresponds to masses of 17.43MJ and 2.44MJ, respectively, and semi-major axes of 0.83 AU and 2.54 AU.

Param. |

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