(a) Show that the probability that a planet transits its star is \( \frac{R_*}{a} \), assuming \(R_P << R_* << a \). What types of planets are most likely to transit their stars?
In order to witness an exoplanet transit across a star, the path of the planet's orbit around the star must cross within a direct line of sight from our location on earth. For this reason, we are only able to see a limited number of exoplanets in space. In our drawing above, the necessary path is shown for a planet to be seen by an observer to the right.
To determine the probability that a planet transits a star, let's begin to quantify the system above. From the drawing, we can determine that a transit can only be observed between the two edges of the star. This region creates a solid angle described by: \[ SA = \int_0^{2 \pi} \int_{\frac{-R_*}{a}}^{\frac{R_*}{a}} sin \theta d \theta d \phi \] Since the radius of the star is so much smaller than the distance a, we can use the small angle approximation to create the bounds on the second integral of the above relationship. Evaluating this integral, we produce a solid angle of: \[SA = \frac{4 \pi R_*}{a} \] Given that the total solid angle of a sphere is \( 4 \pi \), then the probability of the visible solid angle of a transit being in line of sight is given by: \[ Probability = \frac{4 \pi R_*}{a} \times \frac{1}{4 \pi} = \frac{R_*}{a} \]
(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 I need to monitor for transits if I want to detect 10 transiting planets?
Unknowns: a, \( R_*\)
To begin, we will use Kepler's law to determine a value for a, the distance between the sun-like star and their jupiter-like planet. Similar to our work in Worksheet 13.1, Problem 1, we can achieve this by setting up a proportion with Earth's distance to the sun, which is in units of AU's. \[ \frac{ (1 year) \times ( 4 \pi^2 ) \times (AU^3) }{ GM} = \frac{ (\frac{3}{365} \, years) \times ( 4 \pi^2 ) \times (a^3) }{ GM} \] Solving for a, we get: \[ a = ( {\frac{3}{365}})^{ \frac{2}{3}}AU \] The next term that we need to define is \(R_*\). Fortunately, this has a fairly straightforward value of ~ \( \frac{1}{200}AU \). Knowing this value, we can now calculate the percent of the above declared planets that fall within our observable probability: \[ \frac{ \frac{1}{200}AU }{ ( {\frac{3}{365}})^{ \frac{2}{3}}AU } \times 1 \, percent = 5 \times 10^{-3} \] This provides the fraction of transiting planets. To determine the number of stars that would be necessary to monitor to detect 10 transiting planets, we would divide 10 by this fraction, as shown below: \[ Number \, of \, stars \, = \frac{10}{5 \times 10^{-3}} \approx 2000 \, stars \]
Acknowledgements: Team EE
Nice job
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