The Ins And Outs Of Velocimetric Detection

A sensitive method, but with a few drawbacks

The presence of a planet causes a variation in the radial velocity of a star. This depends on several criteria: the mass of the star, the mass of the planet, the period of the planet's orbit, and the orbital eccentricity. An approximation might begin with the simplest case: that of a planet in a circular orbit. Here, variations in velocity are sinusoidal, with a period matching the orbital period P of the planet. Their amplitude K in m/s is expressed by

The Ins and outs of velocimetric detection 25

using 'natural' units: the period of revolution P in years, the mass of the planet in Jupiter masses, and the mass of the star in solar masses. If Jupiter were 1 AU from the Sun it would cause a variation in radial velocity of our star of 28.4 m/s, whereas in the case of the Earth the value is only 0.1 m/s! It should be noted that it is not the mass of the planet which is directly involved, but the quantity Mp sin i, where the angle i defines the inclination of the orbit of the planet as seen from

Definition of angle /', defining how an observer on Earth sees the orbit of the planet observed. When /' is 90° the orbit is seen edge-on, and when it is 0° it is seen face-on.

The relationship between minimum mass/period of revolution of extrasolar planets (in red) and some solar system planets (in blue). The velocimetric method is incapable of detecting giant planets at distances from their stars comparable to those of terrestrial planets (except in the case of Jupiter). It is therefore no surprise that astronomers have not yet detected a system similar to our own.

The principle of the velocimetric method of detecting extrasolar planets, (a) The planet orbits its star, or more precisely, they both orbit about C, the barycentre of the star-planet pair, (b) The Doppler effect enables Earth-bound observers to detect the star's motion as induced by a large enough planet or a collection of planets. The alternating shifts towards the red and the blue, characteristic of stars being perturbed by their planet(s), may be graphically represented by a sinusoidal curve.

The principle of the velocimetric method of detecting extrasolar planets, (a) The planet orbits its star, or more precisely, they both orbit about C, the barycentre of the star-planet pair, (b) The Doppler effect enables Earth-bound observers to detect the star's motion as induced by a large enough planet or a collection of planets. The alternating shifts towards the red and the blue, characteristic of stars being perturbed by their planet(s), may be graphically represented by a sinusoidal curve.

Earth (see figure on p. 25). If this angle is 90°, the orbit is seen edge-on. In this case, sin i is 1, and the variation in velocity is maximised. If the angle is 0°, then the orbit is seen face-on, and sin i is 0, causing no variation in radial velocity. In the usual case, where the orbit is presented at an intermediate inclination, angle i is between 0° and 90°, and Mp sin i lies between 0 and the actual value of the mass of the planet. With no other source of information than the measurement of radial velocities, neither the inclination of the orbit nor the value of sin i are known. This is the great drawback of this method: it can be used to determine not the mass of the planet, but the minimum value of that mass. The parameters obtained are therefore the minimum mass, the period (directly linked with the semimajor axis a of the orbit, according to Kepler's third law), and the eccentricity of the orbit if not circular.

The method is not without its caveats. The more massive the planet, the more marked is the perturbation of the radial velocity. The same is true in the case of short periods. The velocimetric method is therefore well suited to the study of massive, short-period planets, close to their stars; in short, 'hot Jupiters'. Another factor working against the discovery of 'outer' planets is that, for the unambiguous detection of such a planet, the duration of the study should be at least equal to its orbital period.

Remember that the theoretical limit of 1 m/s is difficult to achieve, since some of the stars in question are variable, as is the case with the Sun. These variations introduce inevitable 'noise' into the observations. Moreover, a star's intrinsic periodic variations also have to be taken into account (the Sun, for example, has a cycle of activity of ~11 years), as such variations might be misinterpreted as being caused by planets. As with the detection of pulsar planets, the method is not dependent upon the distances of stars, but in reality it is difficult to achieve accurate measurements of radial velocity for very distant (and therefore fainter) stars.

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