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Fig. 2.8 Geometry of a transit

Fig. 2.8 Geometry of a transit

The duration of a transit corresponds to the time required for the planet to traverse the chord AB (Fig. 2.9). The impact parameter (denoted b and not p, to avoid confusion with notation used previously for the orbital period and the transit probability), is determined by the inclination of the orbit, i, relative to the line of sight and the relationship:

The length of the chord AB is:

The time taken to traverse it, for a circular orbit of period P, at velocity v = 2.n.ap/P. From this we may derive the duration of the transit: tab = l/v, which may be written, replacing P by its expression as a function of ap, using Kepler's Third Law, as:

2.r*v/T—b2 i/2 = (2n)2/3.2r*yT—b2 p1/3 . ^ = (G. m*)/ .P

These equations may be written more simply, by taking ap in astronomical units (the mean Earth-Sun distance), P in days, and m* and R* in solar units, to give:

500 1000 1500 2000 2500 3000 3500 4000 Orbital period (days)

Fig. 2.9 Duration of a planetary transit (in hours) as a function of the orbit's semi-major axis (left) and of the orbital period (right)

500 1000 1500 2000 2500 3000 3500 4000 Orbital period (days)

Fig. 2.9 Duration of a planetary transit (in hours) as a function of the orbit's semi-major axis (left) and of the orbital period (right)

tab = 13W1 - b2.-^. a/ = 1.8^1 - b2-^. P1/3 (2.25) mj mj

The relative amplitude of the extinction during a transit is, to a first approximation, equal to the ratio of the apparent surfaces of the planet and the star, and may be written:

F r2

So the transit of a giant planet like Jupiter causes a photometric extinction of 1 per cent, and that of a terrestrial planet like Earth an extinction of 0.01 per cent (10-4). To detect a planet by the transit method, it is therefore necessary to be able to carry out photometry of the star to better than the relative extinction. So, to detect a giant planet, a photometric accuracy of roughly a fraction of one per cent is required, while for a terrestrial planet, the photometric accuracy needs to be around some 10-5.

Although the detection of giant planets is possible from the ground, where one may achieve accuracies of around 10-3, limited by atmospheric turbulence, the detection of terrestrial-type planets may only be contemplated from space. The difference in the accuracy of an observation made from the ground and one made from space is illustrated in Fig. 2.10, which shows the same transit, that of a giant planet passing in front of HD 209458, observed from the ground and from space.

The method's potential is not limited to just the identification of planetary candidates. In fact, when combined with the radial-velocity method (where the indeterminacy of the angle of inclination relative to the plane of the sky is removed by the observation of a transit), detection of a transit enables us to derive:

• the size of the object (the amplitude of the extinction)

• its mass (by measurement of the radial velocities)

• and, as a result, the density of the object, thus allowing us to distinguish between a gaseous planet and a terrestrial-type planet.

In some cases, when the observation may be made with a good signal-to-noise ratio, by using differential spectroscopy before and during the transit, it is possible to determine the composition (or at least the presence of certain elements) in the planet's atmosphere. This point will be discussed in Chap. 7. We should also mention that the transit method has been used in the infrared, exploiting the fact that the planet is eclipsed when it passes behind the star (known as a secondary transit). Accurate knowledge of the ephemerides of the system allows us, by subtraction of the flux during the secondary transit (star alone) from that before the secondary transit (planet + star), to derive some spectral information about the object. To date, this method has been applied to two objects: TrEs1 and HD 209459, using the Spitzer space telescope. It set limits on the flux emitted by the objects in four spectral bands (see Chap. 7). It should become more generally available when the James Webb Space Telescope (JWST) enters service.

At present there are more than twenty, ground-based projects for the systematic observation of transits, generally with small, automated, wide-field telescopes (see

Fig. 2.10 A transit of HD 209458b observed (left) from the ground (Charbonneau et al., 2000) and (right) from space with the HST (Brown et al., 2001)

Chap. 8). In this case, the primary difficulties are the handling of a large quantity of data and the extraction of relative photometry of the objects, when the observations have been obtained under what are sometimes extremely varied conditions (in particular, the influence of local meteorology and the succession of day and night, which do not guarantee continuity of observation). The A-STEP project, with an observing programme from Dome-C in Antarctica is an extreme example of these ground-based observational programmes. The excellent atmospheric conditions associated with the long Antarctic night appear to be serious advantages for this programme. Nevertheless, the ultimate photometric accuracy of all these ground-based programmes appears to restrict the method to the detection of giant planets.

Two space missions designed to search for planets by the transit method are envisaged for the near future, and are discussed in detail in Chap. 8:

• the CoRoT mission, led by CNES, launched in late 2006, began operation in January 2007 and should be able to identify objects of a few terrestrial radii, orbiting close (with periods of less than 75 days) to their star;

• NASA's KEPLER mission should, from 2008, be able to detect terrestrial planets with periods up to about one Earth year.

2.2.2.2 Gravitational Microlensing

One of the most surprising aspects of Einstein's theory of relativity is the phenomenon of the gravitational lens (Einstein, 1936). In his theory, Einstein introduced energy/matter equivalence,3 with the consequence that the photon, the quantum of electromagnetic energy, is subject to gravitation, just like 'classical' baryonic matter, whose weight we all experience.

So a photon that passes at distance r from an object of mass M, which is assumed to be a point, undergoes a deflection a relative to its direction of propagation, and this is given by the following equation:

c2 r r where Rs is known as the 'Schwarzschild radius' of the gravitational lens caused by the point mass.

When a massive object lies between the observer and the object being observed, the image of the latter is thus deformed by the mass of the deflecting object. The amplitude of the phenomenon depends on the mass and position of the deflector. When the deflecting object is sufficiently massive - in which case a is greater than the resolution of the instrument being used for the observation - the effect is known as a 'macrolens': the image of the object is multiplied into an odd number of secondary images. Historically, this was the first type of lens to be observed. The image of the double quasar Q0957+561 obtained by Huchra in 1985 revealed a symmetrical multiple structure related to the presence of a galaxy on the line of sight (Huchra, 1985). When the mass of the deflecting object is low and a is less than the angular size of the observing instrument's diffraction disk, the situation is described as a 'microlens' and multiplication of the image is not observed (although it still exists), but the mean lensing effect effect instead, which takes the form of an overall amplification of the intensity of the source being observed. The first observation of a gravitational microlensing event was that of the quasar Q2237+0305 (Racine, 1992).

3 This is the famous equation E = mc2, where E is the rest energy of the particle, m its mass, and c the velocity of light. In the case of the photon, the rest energy is hv, where h is Planck's constant and v the frequency of the wave associated with the radiation.

In a completely similar manner, if a massive object passes across the line of sight to a star, the latter's brightness is increased by the effect just described, simultaneously revealing the presence of the object that serves as a lens (Fig. 2.11). The gravitational amplification effect may, therefore, allow the transient observation of an object that cannot be detected prior to the microlensing event.

Gravitational amplification is given by the following relationship (Sackett, 1999):

where u = dS/dE, with 9S the angular distance between the source and the deflector, and 0E the Einstein radius, which is itself defined as:

where G is the gravitational constant, M the mass of the lens, c the velocity of light, dL the distance between the observer and the lens, dS the distance between the observer and the source, and dSL the distance between the source and the lens.

The variation in the brightness of the object during the passage of the lens is shown in Fig. 2.12.

In the case of a simple lens, the variation in luminosity is symmetrical, and centred on the position of the lensing star. The overall set of the apparent positions of

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