## Monday, June 25, 2012

### What About Mercury?

The transit of Mercury occurs about thirteen or fourteen times per century. They are much more common than Venus transits, and thus they may get slighted a bit by avid stargazers! So, why do they happen so often?

In general, a planet’s orbital plane can have any orientation relative to our line of sight, but only a fraction of these produce a transit. In order to calculate the probability of a transit, we need to know the range of these specific orientations as a fraction of the total possible configurations.

First, let’s assume that the planet’s orbit is perfectly circular and that its radius is much smaller than the star’s. And for the time being, we assume that the star-planet system is infinitely far away, such that all lines of sight to the system are parallel. Two angles specify all possible orientations of the planet’s orbit, with respect to our line of sight, and we integrate over the ranges of these two angles that produce transits. A two dimensional angle is called a solid angle, which maps out an area in angle-space, and in spherical coordinates this integral looks like:

$\int_{0}^{2\pi}\int_{(\frac{\pi}{2}-\arcsin\frac{R_*}{a})}^{(\frac{\pi}{2}+\arcsin\frac{R_*}{a})} \sin\theta\, d\theta d\phi$

After dividing by the sphere of total angles, which is simply equal to 4π, we are left with the probability that we will observe a transit,

$\frac{R_*}{a}$

which is just a function of stellar and orbital radii!

So we see that Mercury, which has a much smaller value of a, will have a higher probability of transit than Venus. In fact, this applies for all stars and their companions. The closer the planet, the more likely a transit is!