Worksheet 13.2, Problem 2: Transiting Planets

The Problem: Now draw the star projected on the sky, with a dark planet passing in front of the star along the star’s equator.

Part A: How does the depth of the transit depend on the stellar and planetary physical properties?

For this problem we will be trying to find \( \Delta F \) which we can find using \( \frac{F_T}{F_{\star}} \) \[F_{\star}= L_{\star} A_{\star} = L_{\star} \pi R_{\star}^2 \] \[F_T = L_{\star} (A_{\star} - A_{Planet}) = L_{\star}(\pi R_{\star}^2 - \pi R_P^2) \] Combining these gives: \[ \frac{F_T}{F_{\star}}  = \frac{L_{\star}(\pi R_{\star}^2 - \pi R_P^2)}{ L_{\star} \pi R_{\star}^2 } = 1 - \frac{ R_P^2}{R_{\star}^2 } \]
What is the depth of a Jupiter-sized planet transiting a Sun-like star?

We know \( \frac{R_J}{R_{\odot} } \approx \frac{1}{10} \) so: \[ \frac{F_T}{F_{\odot}} \approx 1 - \frac{ 1}{10^2}  \approx \frac{99}{100}\] Which means that the flux of the Sun when Jupiter is transiting is about 99% of its normal flux.

Part B: In terms of the physical properties of the planetary system, what is the transit duration, defined as the time for the planet’s center to pass from one limb of the star to the other?
\[ t = \frac{d}{v} \] So we need distance and velocity. Distance would just be twice the radius of the star and velocity would be: \[ v_p = \frac{2 \pi a }{P} \] Combining those we get: \[t = \frac{2R_{\star}}{\frac{2 \pi a }{P} } = \frac{PR_{\star}}{\pi a } \]

Part C: What is the duration of “ingress” and “egress” in terms of the physical parameters of the planetary system?

Once again, we can use: \[ t = \frac{d}{v} =  \frac{2R_{P}}{\frac{2 \pi a }{P} } = \frac{PR_{P}}{\pi a } \] 

Acknowledgements: I worked with Sean, April, and Barra on this problem.