Conditions and conclusions

What is the probability of being able to observe a planetary transit across a star at distance a? If the star were indeed a point source, the transit would occur only if the star, the planet and the observer were in exact alignment. However, since the star has a definite radius R*, calculation shows that the probability of transit is expressed by R* /a. Simple logic: the nearer a planet is to its star, the greater the possibility of a transit; and the larger the star, the greater the chance that the planet will be seen to pass across it. In the case of a star the size of the Sun, the probability of a transit is 0.01% if the planet lies at a distance of 5 AU from its star. For a planet at 1 AU, the probability is 0.5%, rising to a not insignificant 10% if the planet orbits at 0.05 AU.

Probability of transit %

Duration of transit (hours)

Variation in flux %

Mercury

1.2

8

0.0012

Venus

0.64

11

0.0076

Earth

0.47

13

0.0084

Jupiter

0.09

30

1

Saturn

0.05

41

0.75

In the case of the solar system as seen from outside, the further the observer from the Sun, the less probable it becomes that a transit will be observable. The duration of the transit increases with the distance of the planet from the star. The variation in flux depends on the size of the planet, and is of course greatest in the case of Jupiter.

In the case of the solar system as seen from outside, the further the observer from the Sun, the less probable it becomes that a transit will be observable. The duration of the transit increases with the distance of the planet from the star. The variation in flux depends on the size of the planet, and is of course greatest in the case of Jupiter.

Probability of transit % Duration of transit (hours) Variation in flux %

The configuration of hot Jupiters - large planets close to their stars - favours the visibility of transits. The probability that the planet, as seen from Earth, will cross the face of the star attains 10%.

2.9 Observing a planetary transit 33

Space missions employing the transit method

The detection of exoEarths using ground-based techniques seems to be beyond our capabilities, since photometric accuracy of the order of 10_J) is required. For this reason, space missions ensuring the necessary photometric stability - the French COROT and the American Kepler missions - have been or will be launched in the near future. These two projects might also be able to detect possible satellites or rings of exoplanets by accurate measurement of light-curves during transits.

Deformation of transit light-curves due to the presence of a satellite. Examples of light-curves calculated for different configurations of exoplanets and their satellites. The first three curves correspond to a Jupiter-sized planet with a satellite of period 0.5 days (a) or 1.5 days (c), or without a satellite (b). Case (d) corresponds to a smaller exoplanet (2.5 Earth radii) with a slightly smaller satellite. (After Sartoretti and Schneider, A. & A. Suppl., 134, 553 (1999).)

Deformation of transit light-curves due to the presence of a satellite. Examples of light-curves calculated for different configurations of exoplanets and their satellites. The first three curves correspond to a Jupiter-sized planet with a satellite of period 0.5 days (a) or 1.5 days (c), or without a satellite (b). Case (d) corresponds to a smaller exoplanet (2.5 Earth radii) with a slightly smaller satellite. (After Sartoretti and Schneider, A. & A. Suppl., 134, 553 (1999).)

The diminution in brightness due to the interposition of the planet is easy to determine. It is simply the ratio of the apparent surfaces of the planet and the star: AF = (Rp/R*)z, where Rp is the radius of the planet. Consequently, if we are able to estimate the radius of the star we can deduce the radius of the planet. The duration of the transit depends firstly upon the period of revolution of the planet around the star (the further away the planet is from the star, the longer it will take to pass across its face), and secondly, upon the inclination of the orbit. For planets at different distances, the transit may last for periods ranging from hours to several days. Seen from outside the solar system, a transit of the Earth across the Sun would last for 13 hours. In the case of an exoplanet such as 51 Pegasi b, only 0.05 AU from its star, the duration of a transit would be just 3 hours, assuming that the star is the size of the Sun. Another factor worth noting is that it is impossible to observe an object 24 hours a day from the rotating Earth, because of the alternation of night and day - unless you are based in the polar regions! More problems to solve...

In the final analysis, from transit observations we can deduce the radius of the planet, its period, and the inclination of its orbit - but not its mass. Complementary, velocimetric observations are needed, enabling the minimum mass of the planet to be determined, and, since the orbital inclination is now known, the value of the mass of the planet. With mass, radius and distance from its star all known, some physical studies can be undertaken; but it must be said that divining the nature of exoplanets is something of a treasure hunt!

The game is worth the candle, but the difficulties are many. In order to detect an exoplanet by the transit method, the luminous flux from the star has to be metered very accurately over an extended period, in the hope of being able to observe regular fluctuations as the star dims slightly for a certain repeated amount of time. Accuracy to at least 1% in the measurement of the flux is necessary for the detection of giant planets. To increase the chances of success, fields containing millions of stars are studied, entailing the analysis of mountains of data. From this process, hundreds of candidate stars emerge, to be observed by other methods. The vast majority of these candidates will be revealed as variable stars or eclipsing binaries; but a mere handful will be labelled 'exoplanet'. So far, nine have been found or observed by this method.

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