Blog #18 - Transiting Exoplanet

Transiting Exoplanet 



1. Draw a planet passing in front of its star, with the star on the left and much larger than the planet on the right, with the observer far to the right of the planet. The planet’s semi-major axis is a.

The picture depicts the setup of the problem with the exoplanet transiting in front of the spectator at a circular orbit with semi-major axis, a. 

(a) Show that the probability that a planet transits its star is R/a, assuming Rp<<R<<a. What types of planets are most likely to transit their stars?

The probability that the star passes in front of our view is simply a matter of proportions of the surface areas of the stars entire sphere of movement and the cylindrical area taken up by the far field view of our observer. To "transit" the star, the exoplanet must move within either side of the planet, within the cylindrical gaze of the viewer. The surface area of the entire potential orbit of the planet is simply the spherical surface area at a distance of the semi-major axis: \[A_{sphere} = 4 \pi a^2\]and \[A_{view} = (2 \pi R_{\star}) * (2a) = 4 \pi R_{\star} a\]To find the probability we simply see how likely it is for the planet (on the full spherical surface area) falls within the cylindrical area of the viewpoint. \[Probability = \frac{A_{view}}{A_{sphere}} = \frac{4 \pi R_{\star} a}{4 \pi a^2} = \frac{R_{\star}}{a}\]Which gives us the expected relation of \(\frac{R_{\star}}{a}\) for the probability. 

(b) If 1% of Sun-like stars in the Galaxy have a Jupiter-sized planet in a 3-day orbit, what fraction of Sun-like stars have a transiting planet? How many stars would you need to monitor for transits if you want to detect ten transiting planets?

We begin by solving for the semi-major axis of the mentioned planet using the relation between period and semi-major axis: \[P^2 \approx a^3 \]. Using the mention of a 3-day planetary orbit we can solve for an approximate semi-major axis as \[(\frac{3 day}{365 days}) = (\frac{a}{1 AU})\]Giving us \[a \approx .04 AU \]We also know that Jupiter's radius is roughly: \[R_{\star} = .0046 AU\]Giving us a probability relation of \[\frac{R_{\star}}{a} = \frac{.0046}{.04} = .114 = 11.4 \% \] However only 1% of all Sun-like stars have the possibility of containing these Jupiter-sized planets so the total probability of observing one of these exoplanets is \[.114*.01 = .001143 = .1143 \% \]To observe 10 transiting planets, this means you would need to observe \[\frac{10}{.001143} \approx 8,748 \space planets\]