Fig. 2.4 The principle behind interferometry (described fully in the text)
path difference (the optical paths followed by the light gathered by both telescopes 1 and 2 being equal) is determined by observation of the interference pattern (on the bottom of Fig. 2.4). It corresponds to the point at which the envelope over the interference fringes is a maximum. The precise determination of B, by optical metrology (with an accuracy of a few nanometres), and measurement of the difference in the external path length (B.cos 9) by measurement of the displacement of the delay line relative to zero path difference (again, with an accuracy of some few nanometres), provides an accurate determination of the angle 9 between the telescope's baseline and the star on the sky.
In practice, interferometry, like the other methods, functions in a differential mode to avoid the effects of diurnal rotation and of the variation in the baseline during the observation. Here, the astrometric movement of the star is measured relative to a reference frame that is assumed to be fixed. The interferometer should therefore examine at least two objects simultaneously. This standard of performance should be obtained with the planned PRIMA (Phase Reference Imaging and Micro-arcsecond Astrometry) instrument on the European Very Large Telescope Interferometer (VLTI), under commissioning at the time this book was written.
The best astrometric measurements are currently obtained by radio interferometry with continental baselines (VLBI), where astrometric accuracy close to 100 ¡as is being attained. The optical interferometers that will shortly enter service should enable us to obtain astrometric accuracies that are at least 10 times better.
An additional stage in the search for the ultimate astrometric accuracy consists of combining interferometry with observations from space. The American SIM mission is a space-borne optical interferometer, which, around 2015, should be able to obtain an astrometric accuracy of 1 ¡as. To obtain such accuracy, it is necessary to know the length of the baseline to an accuracy of a few tens of picometres (in other words, the thickness on one atomic layer). Research and development efforts are currently under way to reach this objective, and allow the mission to be developed.
The astrometric method that we have just discussed, should enable us to measure and reconstruct the motion of the star in the plane of the sky. It omits, however, measurement of the star's motion along the line of sight, recession or approach relative to the observer. It is, however, impossible to observe, from Earth, the radial location of the star along the line of sight, because, by definition, the motion projected on the sky is non-existent. It is, however, possible to measure the radial velocity of the star by spectroscopy (see later). The expression for the radial velocity of the star may be derived from Eq. (2.4) and from the change of reference frame introduced in the preceding paragraph, which enables a conversion from the orbital reference frame to that of the sky, namely by rotations by the angles -Q, i, and c. The equation for the position of the star on the line of sight is then written as:
By using Kepler's Second Law (the law of areas), which is expressed, for an elliptical orbit of period P, semi-major axis a, and eccentricity i, by the equation:
and deriving the expression of the position of the star on the line of sight with respect to time, we obtain the equation for the star's radial velocity:
In what follows, we will set:
In the two-body case that we are considering, from the preceding equation, it is thus possible to derive an equation that is a function of the mass of the planet and the mass of the star. This equation is obtained using Kepler's Third Law, applied to the system consisting of the star and the planet:
where a = a* + ap and m*.a* = mp.ap. As a result we have:
and we then obtain, using Eqs. (2.9) and (2.12) the equation:
The above equation shows that measurement of the radial velocity (Eq. 2.9) does not allow us to simultaneously determine m* and mp. However, if we assume that the mass of the planet is negligible relative to the mass of the star, and also that the mass of the star may be estimated by some other means (for example from the position of the star on the Hertzsprung-Russel diagram, see Fig 1.12), then we may obtain an estimate of the product mp.sin(/):
It will be noted that, whatever we do, we cannot derive equations that omit the system's angle of inclination (i). So measurement of radial velocities only allows us to derive a minimum mass for the planet. Only in the specific case where the system is seen edge on, and where the planet may also be detected by its transit of the star, is it possible to determine the individual masses of the system's planets. However, by assuming that planetary systems have a random orientation relative to the plane of the sky, it is possible to ignore the orientation statistically, and obtain a mass-distribution for the planets as a whole.
In the case of a circular orbit (e = 0), Eq. (2.14) may be expressed numerically as:
mp. sin(i) « 3.510-2KP1/3 and K = — —. sin(i).ap (2.15)
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