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Just as in the case of the variations in the radial velocity, these variations of the pulsar's period have a period of their own, which is set by the orbital characteristics of the body orbiting the pulsar.

Table 2.4 gives the value of ST* for a pulsar of one solar mass accompanied by the bodies in our Solar System at their respective distances.

The accuracy of measurements of the variations in a pulsar's period caused by the presence of one or more companions increases with the intrinsic stability of the pulsar. This is the case with the old millisecond pulsars with a stable internal structure,2 but which are revitalized by the accretion of material from a neighbouring red giant (Davis et al., 1985). So this applies to a small number of objects. Currently, the lowest-mass companions that are detectable are found in these systems, which we are able to search thanks to the stability of these pulsars.

The chronometric accuracy currently attained with these objects is far better than a millisecond (which is more than enough to detect a planet, even a terrestrial one). The method has the potential to detect planets with the mass of the Moon. In addition, the simplicity and accuracy of the method explain why it was the first to provide results. The first extrasolar planets detected, in 1992, were those around PSR 1257+12 (Wolszczan and Frail, 1992). But the discovery did not attract a lot of attention: the electromagnetic environment of the pulsar is, on the face of it, very hostile, and so leaves little hope that these planets could be habitable.

2.2.2 The Effect a Planet has on Photometry of Its Star

We have seen in the preceding section that the presence of one or more planets, orbiting a star, may be revealed indirectly by observing the movement of the star that is caused by the planet. In certain cases, we may equally hope to detect the presence of a planet by the effects produced on the luminosity of the star itself. Two principal effects may be involved: the passage of a planet in front of the star (a transit), or gravitationally induced amplification of the brightness of a background star by a multiple target.

2 Unlike young pulsars, where deformation and changes in structure lead to irregularities in the period.

2.2.2.1 Planetary Transits

A planetary system observed edge-on (where the plane of the orbit is perpendicular to the plane of the sky), is a specific, and very interesting, case. The planets may, in fact, transit the star and cause micro-eclipses, which are detectable by continuously measuring the star's flux (Fig. 2.5). The amplitude of the extinction enables us to obtain the diameter of the planet, to a close approximation, and the duration of the phenomenon, and its periodicity, enables us to derive the planet's orbital period and hence its distance from the star.

Observation of a planetary transit is possible ONLY if the system is seen very close to edge-on. To calculate the likely probability of seeing a transit, consider Fig. 2.6

The transit will be visible only if the line of sight intercepts the cylinder, constructed on the orbit, of radius ap and height 2r,. Assuming that the orientation of planetary systems with respect to the plane of the sky is essentially uniform, the probability pT of actually observing a transit is thus expressed as the ratio between the surface of the cylinder that we have just described (the total number of favourable orientations of the line of sight) to the sphere of radius ap (the total number of possible orientations of the line of sight.) For a circular orbit, the following relationship may be derived:

4 nap ap where r* is the radius of the star, and ap is the semi-major axis of the planet's orbit. This relationship may equally be expressed as a function of the planet's orbital period P, by the application of Kepler's Third Law, and becomes:

0.06 -0.04 " -0.02 0 0.02 ' 0.04" ' 0.06 Orbital phase

Fig. 2.5 Diagram illustrating the principles of a planetary transit in front of a star, and the associated photometric light-curve

0.06 -0.04 " -0.02 0 0.02 ' 0.04" ' 0.06 Orbital phase

Fig. 2.5 Diagram illustrating the principles of a planetary transit in front of a star, and the associated photometric light-curve

Fig. 2.6 The geometry for observing a transit

Fig. 2.6 The geometry for observing a transit pT

This probability is shown in Fig. 2.7, for a star that has a radius equal to the solar radius.

It will be seen that the probability decreases rapidly with the planet's orbital distance (and thus period). Detection of planets by observing their transits of a star will, therefore, independently of any considerations of detectability, be more favourable for planets at short distances from their parent star. The probability of observing a transit of a hot Jupiter is thus about 10 per cent (with a period of 3-4 days), whereas the probability of observing the transit of a planet such as the Earth (with a period of 365 days) is only about 0.5 per cent.

To calculate the duration of a transit, we will make the simplifying assumption that the orbit is circular, and consider the geometry shown in Fig. 2.8.

Fig. 2.7 Probability of a transit by a planet orbiting a solar-type star as a function of its distance from the star (left) and its orbital period (right)

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