Fig. 6.2 The radial-velocity curve of the star HD 65216 superimposed on a Keplerian orbital model. The parameters of the planet deduced from this model are a = 1.37 AU, e = 0.41, and ra = 198° (After Mayor et al., 2004)

Fig. 6.2 The radial-velocity curve of the star HD 65216 superimposed on a Keplerian orbital model. The parameters of the planet deduced from this model are a = 1.37 AU, e = 0.41, and ra = 198° (After Mayor et al., 2004)

The mass of the star is measured independently by spectroscopic observations and modelling. However, the best models do not enable us to determine this mass with an accuracy better than 8 per cent, which limits the accuracy of the planet's elements.

Knowledge of the mass of the star enables m.sin i to be calculated, which gives a lower limit for the mass of the planet (see Chap. 2).

6.1.3 Multiple Systems Case

If the system includes two (or more) planets, the radial velocity, Vz, becomes:

k where Kk is defined for each planet by applying Eq. (6.3). The mass function is:

It may be noted that if the two planets are coplanar, observations enable the ratio of the masses of the two planets to be calculated.

These equations assume that the planets are in Keplerian orbits around the star. This approximation is valid provided mutual perturbations between the planets are not too large. If the planets are close to one another, the observations cannot be

2000.0 2002.0 2004.0 2006.0 2008.0

Fig. 6.3 Radial velocities of the star HD 82943, calculated with a Keplerian model (grey), and with a 3-body model (black), coinciding with observations made in 2001. The line shows the difference between the models, assuming sin i = 1 (After Ferraz-Mello et al., 2005)

2000.0 2002.0 2004.0 2006.0 2008.0

Fig. 6.3 Radial velocities of the star HD 82943, calculated with a Keplerian model (grey), and with a 3-body model (black), coinciding with observations made in 2001. The line shows the difference between the models, assuming sin i = 1 (After Ferraz-Mello et al., 2005)

analyzed using this model. Comparison with a model where the planets interact would be required to calculate the system's parameters. This effect may be seen from analysis of the curve of HD 82943 (Fig. 6.3).

The observations made in 1999—2003 are in good agreement with the Keple-rian model, but longer-term observations require the three-body model. It should be noted that the model is calculated with the planetary masses equal to the minimum masses, that is, assuming sin i = 1. It is not possible to determine the angle of inclination, except in the case where the planet transits the star. Some information may be gained for other systems from comparison of observations with simulations (see later). So, in the case of the system of HD 82943, for example, if the observations diverge from the model, this would indicate that the orbit of the planets is inclined with respect to the line of sight (sin i = 1). Modelling the observations would allow us to arrive at the inclination of the system and thus at the masses of the planets. These models all assume that the orbits are coplanar.

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