m* + mp where GOp denotes the distance between G and Op and O* Op the distance between O* and Op.
Each of the bodies thus follows an elliptical orbit, with the centre of mass of the system at one of the foci. If a is the semi-major axis of the planet's orbit around the star, the semi-major axes of the stellar and planetary orbits relative to the barycentre of the system at one of the foci may be written:
In the general case, the 'central' star also describes an orbit that is more or less complex depending on the number of planets in the system, certain properties of which may be described by observing:
• stellar motion projected on the plane of the sky. Here, we observe a variation in the position of the star relative to a fixed reference frame (consisting of very distant bodies, such as quasars). This method is known as 'astrometry'.
• stellar motion along the line of sight (radial motion). Here, we measure the star's velocity of approach or recession as a result of the motion around the centre of mass. This measurement method is known as 'radial velocimetry' or 'Doppler velocimetry' (after the experimental method used to measure the radial velocity).
In certain instances, when the star emits a periodic signal (as with a pulsar, for example), the motion of the source may be deduced from the changes in the pulsar's period as it is measured here on Earth. We shall return to this technique at the end of this section.
As the planet or planets move around the centre of mass, the central star also describes an orbit, the complexity of which depends on the number of planets in the system. In the case of our Solar System, the motion of the Sun is primarily caused by the presence of Jupiter, but also reveals the presence of other, less-massive planets (Fig. 2.2).
In the case of a system with a single planet (or where one of the objects dominates the others by a wide margin to such an extent that its effect masks those of all the other planets), the motion of the star is an ellipse. We shall be particularly concerned with such orbits, the general equation for which may be expressed, using polar coordinates having their origin at the focus, and in the orbital plane, by:
where a is the semi-major axis, e the eccentricity, and v the angular position of the object relative to an origin given by the orbit's periastron.
The motion, projected onto the plane of the sky is deduced from the preceding expression by a change of reference frame that takes into account the orientation of the orbital plane relative to that of the sky, and also the conventions regarding the notation of the associated angles.
To describe the orientation of the orbit relative to the plane of the sky, it is normal to use the astronomical equivalent of Euler angles. The following are thus defined successively:
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