The expression for the path on the plane of the sky is thus obtained in polar coordinates, whose origin is the centre of mass of the system:

which is still an elliptical path, and where the parameters defining it are the parameters of the initial orbit (in the reference frame of the orbit) and the observational geometry.

To detect an exoplanet by the astrometric method, it is thus necessary to be able to reconstruct the star's apparent path on the plane of the sky. By using Eq. (2.4), and by making the approximation that the star's true path is circular, and in the plane of the sky, it is possible to calculate the maximum value of the amplitude of the motion, which may be written as:

where mp and m* are the masses of the planet and the star, respectively; ap is the semi-major axis of the planet's orbit, and D is the distance of the exoplanetary system from Earth. The value of 89 may be calculated for different cases (Table 2.2).

This table and Eq. (2.6) show clearly that the astrometric method is more sensitive (i.e., the amplitude of the motion of the star is the greater), if the planet is:

• located at a great distance from its star.

However, the greater the distance of the planet from the star, the longer the planet's orbital period - and consequently that of the star as well - (in accordance with Kepler's Third Law). This means that the time required to detect the planet

Table 2.2 The amplitude of the apparent motion produced by the planets in the Solar System acting on the Sun, as if the latter were being observed from 5, 10, or 15 parsecs

Jupiter 1.07 1.00 0.50 0.33

Neptune 0.33 0.31 0.15 0.10

(a significant fraction of the planet's orbital period around the star) also increases drastically with the planet's distance from the star. Nevertheless, the astrometric method remains one of the rare methods sensitive to low-mass objects, orbiting far from their star. In the case where the system is multiple (in practice, as soon as there are two planets with comparable effects), the path of the star rapidly becomes complex (see Fig. 2.2), and the accuracy of the reconstruction becomes limited by the resolution of the measurements.

Several techniques are possible to determine accurately the position of stars in the sky, and consequently their proper motion. The common factor with all these techniques is the necessity of being able to use a fixed reference frame, or one that is considered as such, on the sky. The principal difficulty is obviously that of finding the position of a fixed object in the sky to a greater degree of accuracy than the astro-metric accuracy of the measurements (typically, to 1 ^arcsec for the most accurate instruments). The positions of stars are generally referred to distant point objects, if possible, extragalactic ones. Nowadays, the ultimate reference frames consist of those based on quasars (quasi-stellar objects), which are actually the active nuclei of extremely distant galaxies. In the past, the position of nearby stars (their distances being determined by measurements of their parallax) was measured relative to more distant stars lying within the same field of view.

The first astrometric measurements aimed at detecting the presence of low-mass companions coincided with the systematic use of photographic plates for astrophys-ical observations. Peter Van de Kamp, who, in 1963, announced the detection of two planets orbiting Barnard's Star (the closest star to the Sun after the Alpha Cen-tauri triple system, one component of which, Proxima Centauri, is the closest star to the Sun), based his measurements on the analysis of more than 2400 photographic plates taken between 1916 and 1963 by the various successive astronomers who used the 80-cm telescope at Sproul Observatory. In this case, a measurement consisted of locating on the photographic plate the coordinates of the photocentre of the reference objects (stars considered to be 'fixed'), and of the star for which one wanted to determine the proper motion. All that remained was to reconstruct the star's path, subtract the proper motion caused by parallax (the motion of the Earth in the Solar System), and the motion of the star in the Galaxy, to obtain the motion of the star that reflected the possible presence of a planet. The principal difficulty with this method is its low precision. The size of objects on a plate is, at best, 1 arcsec (that is the size of the physical spot forming the image with low atmospheric turbulence), and it is difficult to obtain a measurement on a photographic plate with a non-linear response, to better than one tenth of the size of the spot (i.e., 0.1 arcsec), to compare with the values in Table 2.2. Even with systematic measurements over several years, and using electronic cameras (CCD detectors), it is difficult, working from the ground and with classical telescopes, to obtain an astrometric accuracy better than 10 mas, which is too great for us to hope to detect planets - even giant ones.

The planets announced by Peter Van de Kamp remain, to this day, unconfirmed, despite the arrival of radial-velocity techniques that are far more accurate than as-trometry with photographic plates. It is highly likely that Van de Kamp did not know how to overcome all the bias involved with the method.

Apart from the systematic use of electronic cameras (CCD devices), which have appreciably increased the accuracy and reproducibility of astrometric methods, two techniques have completely revolutionized this discipline and these are:

• observation from space

• interferometry.

Observation from space allows us to avoid all the problems arising from the atmosphere. In particular, turbulence, with its accompanying differential refraction, produces fluctuations in the position of an object against the plane of the sky. The degree of these fluctuations depends on the meteorological conditions under which the observations are made, and thus on the site's astronomical quality. The European whole-sky astrometric satellite, Hipparcos, was therefore able to obtain, with an accuracy of a few mas, the position and the proper motion of more than 120000 stars in the Galaxy. This precision is at the upper limit for the detection of giant planets (see Table 2.3). The European GAIA project should, in 2012, allow us to attain an accuracy of a few ¡as. It will then be possible to search for giant planets using astrometry.

Interferometry is a technique which uses two or more telescopes, the light from which is recombined in pairs to obtain interference fringes. From measurements of these fringes, one can deduce information about the spatial structure of the object with an extremely high angular resolution (as would be obtained with a telescope having a diameter equivalent to the distance between the interferometric telescopes). In its astrometric mode, interferometry allows an extremely precise measurement of the position of an object against the sky. A diagram showing the principles behind the method is given in Fig. 2.4.

The light coming from the star of which one wants to measure the position, arrives at telescope 2 with a delay relative to the light arriving at telescope 1. This delay, caused by the angle 9 between the direction of the star and the baseline, B, between the telescopes, creates an external path difference B.cos(9) between the two rays. This path difference may be compensated by the delay line which enables the optical path followed by the light from telescope 1 to be increased or decreased. Zero

Incident light from stars

Incident light from stars

Difference in external path length B.cos(8)

Telescope 2

Telescope 1

I I Fringe detector

Telescope 1

Difference in external path length B.cos(8)

Telescope 2

I I Fringe detector

-50 -40 -30 -20 -10 0 10 20 30 Optical Path diffence (micrometres)

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