Worksheet 13.2, Problem 2: Exoplanet Transits

In this problem, a different method for detecting exoplanets is introduced. In this method, an exoplanet can be found by observing the amount of light we receive from a star over time. If a planet is crossing over the star, it blocks some of the light from reaching us, and we can detect this small change. The image below shows a graph of what this would look like in observations.

Image Credit:
https://github.com/OSCAAR/OSCAAR/wiki/Introduction-to-Differential-Photometry

Draw a star projected on the sky, with a dark planet passing in front of the star along the star’s equator.

A planet of radius \(R_p\) transits a star of radius \(R_\star\).

a) How does the depth of the transit depend on the stellar and planetary physical properties? What is the depth of a Jupiter-sized planet transiting a Sun-like star?

The transit depth describes the extent to which a planet blocks the light from a star. We can already intuitively say that bigger planets will have transits of greater depth and planets transiting smaller stars will have a greater transit depth since they cover a larger proportion of the star's area. More precisely, the transit depth is the ratio of the projected area of the planet to that of the star:

\(d=\dfrac{\pi R_p^2}{\pi R_\star^2}\)

Where \(d\) is depth, \(R_p\) is the radius of the planet, and \(R_\star\) is the radius of the star. So the depth of the transit depends on the square of the ratio of the sizes of the planet and the star:

\(d=\left(\dfrac{R_p}{R_\star}\right)^2\)

For a Jupiter-sized planet transiting a Sun-like star, we know that the radius of Jupiter is approximately a tenth of the radius of the Sun, or that ten Jupiter-radii make one Solar radius. Using this fact, we find the depth of the transit to be:

\(d=\left(\dfrac{R_J}{10 R_J}\right)^2=\dfrac{1}{100}\)

So we have a depth of about 0.01 for a Jupiter-sized planet transiting the Sun, which implies that Jupiter would block 1% of the light from the Sun as it is transiting.

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?

A planet of radius \(R_p\) transits a star of radius \(R_\star\) with a transit time of \(t_t\).

To find the transit time, \(t_t\), we can use the equation \(d=rt\) where the distance traveled is the diameter of the star and the rate is the velocity of the planet's orbit. Knowing the orbital period of the planet, \(P\), and the planet's distance from the star, which we can call \(a_p\), we can find the  circumference of the orbit, and therefore the velocity of the planet:

\(v=\dfrac{2\pi a_p}{P}\)

Now, using the relation \(t=\frac{d}{v}\), we get a transit time of:

\(t_t=\dfrac{2R_\star}{\frac{2\pi a_p}{P}}=\dfrac{R_\star P}{\pi a_p}\)

So the transit time depends on the radius of the star, the distance of the planet away from the star, and the period of the planet's orbit. 

c) What is the duration of “ingress” and “egress” in terms of the physical parameters of the planetary system?

The ingress and egress are the motion of the planet crossing over the edge of the star (where ingress describes the planet as it moves into the sun, and egress describes the planet as it moves out of the sun). So the duration of the ingress and egress would be the time it takes for the planet to travel the distance of its own diameter. Again, using the equation \(d=rt\), we find that:

\(t_{in}=t_{eg}=\dfrac{2R_p}{\frac{2\pi a_p}{P}}=\dfrac{R_p P}{\pi a_p}\)

Citations:
http://kepler.nasa.gov/Science/about/characteristicsOfTransits/

I would like to thank Charles Law, Daniel Chen, and Andy Mayo for their help in solving this problem.