## The Dynamics of Planetary Systems

The structure of the Sun's planetary system has been the subject of numerous studies ever since it was discovered that the orbits of the planets were governed by an extremely simple law: the law of gravitation. Celestial mechanics has allowed the positions of the planets and satellites to be predicted with great precision. Moreover, the study of the stability of orbits has revealed the fundamental role of resonant interactions that govern complex configurations that are sometimes stable, and sometimes chaotic. Exosystems offer dynamicists a new field to explore or to use to test the mechanisms worked out in the Solar System.

6.1 Characteristics of the Orbits 6.1.1 Calculation of Radial Velocities

Up to now, exosystems have not been observed directly. The characteristics of the orbits of the planets are primarily deduced from analysis of the motion of the central star. The analysis techniques are the same as those used for binary-star systems. These techniques were established at the time of the first spectroscopic detections of binary stars, which date back to 1890.

The radial-velocity technique measures the velocity of the star in a reference frame containing the observer. Reduction of the observations therefore requires that the motion of the observer should also be taken into account. This has several components. Rotation of the Earth itself translates, as far as the observer is concerned, into a motion along the direction of the geographical parallel with a velocity of 460 m/s. cos(^), where 9 is the latitude of the observing site. The motion of the Earth every 28 days around the Earth—Moon barycentre translates into a motion at a rate of 13 m/s. The Earth—Moon barycentre itself orbits around the Sun, with an average velocity of 29.8 km/s and an oscillation (the orbit is an ellipse) with an amplitude of 0.5 km/s, and the Sun itself orbits the Solar System's barycentre with an average velocity of 13 m/s.

M. Ollivier et al., Planetary Systems. Astronomy and Astrophysics Library, DOI 978-3-540-75748-1.6, © Springer-Verlag Berlin Heidelberg 2009

Measurements of radial velocities are obtained with accuracies of one metre per second. Allowance for the different components of the observer's motion should, therefore, be made with a higher degree of accuracy.

There is an online tool that enables one to calculate these motions with a nominal accuracy of 2cm/s. This is VSOP87E, accessible on the Centre de Donnees Stellaires site at Strasbourg (http://cdsweb.u-strasbg.fr/ftp/cats/VI/81/).

Another effect that must be taken into account is the fact that light takes about 1000 s to go from a point on Earth to the same point when the Earth is at the opposite point of its orbit. The time of an event will not be the same depending on the position of the Earth at the time of observation. It is therefore necessary to relate the observations to a virtual clock, located at the centre of the Sun or at the Solar System's barycentre.

When the time to be measured concerns the arrival of pulses from a pulsar, the accuracy that needs to be attained also requires relativistic effects to be taken into account.

6.1.2 Orbital Characteristics from Radial-Velocity Curves

If the star is isolated, its radial velocity relative to the Sun is constant. If it has a companion (planet or star), the star revolves around the barycentre of the two bodies. In a reference frame centred on the star, the position of the planet is:

where f is the true anomaly, i.e., the angle between periapsis and the position of the companion (planet or star). jijj) are the orthogonal unit vectors in the plane of motion, i being directed towards periapsis. If r = |R|, r = <«>

1 + ecosf df 2na2

where T is the orbital period, e the eccentricity, and a the semi-major axis. The velocity of the planet is therefore:

From this equation we can deduce the velocity of the star relative to the system's barycentre:

where m is the mass of the companion, and M the mass of the star.

Fig. 6.1 The reference system, showing the plane of the sky and the plane of the planet's orbit. Angles are measured from point T(After Ferraz-Mello et al., 2006)

Fig. 6.1 The reference system, showing the plane of the sky and the plane of the planet's orbit. Angles are measured from point T(After Ferraz-Mello et al., 2006) 