Worksheet 13.2: Problem 2: Transiting Exoplanets

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

 

 

(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 depth of the transit depends on how much the flux changes when the exoplanet crosses in front of the star. The amount of energy that is blocked depends upon both the visible area of the planet and of the star. Since luminosity is equal to the flux times the area of the star, we know that the flux received by an observer during the transit is equal to:

\[F_{t}\:=\:\frac{L_{*}}{A_{*}-A_{p}}\]

In order to find the depth of the transit, we have to compare the flux received during the transit to the flux normally received:

\[\frac{F_{t}}{F_{*}}\:=\:\frac{\frac{L_{*}}{4 \pi (R_{*}^{2}-R_{p}^{2})}}{\frac{L_{*}}{4 \pi R_{*}^{2}}}\]

\[\frac{F_{t}}{F_{*}}\:=\:(1-\frac{R_{p}^{2}}{R_{*}^{2}})\]

The depth of the transit is represented by the term \(\frac{R_{p}^{2}}{R_{*}^{2}}\).

 Since Jupiter's radius is roughly one-tenth that of the Sun, this term is equal to \(10^{-2}\).

(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? 


We know that time is equal to \(\frac{Distance}{Velocity}\). The distance that the planet's center travels is twice the radius of the star \((2R_{*}\).  We also know that the general equation for velcoity is \(V_{p}\:=\:\frac{2\pi a}{P}\). Using Kepler's Law, we can rewrite that as:

 

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


These terms describe the amount of time required for a planet to eclipse its star. Both terms repreesnt  the same amount of time. This is equal to the amount of time required to travel two radii because the distance the planet travels from the beginning to the end of the eclipse is two of it radii.We can then substitute and arrive at an expression for this time in terms of the physical parameters.