Derivation Of the Occurrence Of Multiple Planets In Transit

All methods for extra-solar planet detection have their advantages and drawbacks. In photometry, for a planet to be detectable, it must pass in front of its star along the line of sight to the star. This may seem at first to be an unusual event, when in fact it has been shown to be equal to d*/D, or about 1% for planetary systems like our own (d* is the stellar diameter and D is the orbital diameter). Planetary scientists are interested in the complexities of an entire system rather than just a single planet. At first it might seem extremely unlikely that multiple planets would be detectable. But imagine having detected a planet in transit and that the relative inclinations, phi, of all the orbits is smaller than d*/D. Then, if you detect one planet, you should detect nearly all of them. However, for the solar system phi greater than d*/D, but still small. The exact derivation will be presented for the occurrence of multiple plane's in transit. A good approximation to the result is found to be given by (4/pi(phi)(d*/D(sub 2)), where D(sub 2) is the orbit of the second planet. For example if phi is 2 deg, then the probability for finding a second planet if it has an inclination less than phi will be equal to 36 d*/D(sub 2). For inner planets around solar-like stars, this turns out to be on the order of 20% of the systems detected. Extending this to three planets, such as for Venus, Earth and Mars, the probability would be 4%.