4.2  Planet sizes with a pocket calculator

Once you begin comparing and contrasting different giant exoplanets, you need to be a bit more precise than ‘about 1%’. The interactive application you worked with in Activity 1 performs precise calculations using Equation 1, and you can go back to it and use it for other examples. If you are confident with maths, you can also use Equation 1 yourself.

If you want to calculate the radius of a transiting planet then you need to rearrange the equation to give:

R_{\mathrm{p}} = R_{\mathrm{star}} \times\sqrt{\text{transit depth}}

means take the square root of the quantity under the sign.

If you put in a transit depth of 1% (i.e. 0.01, expressing ‘1%’ as an ordinary number) and a star of radius 1 R_Sun then this would become:

R_{\mathrm{p}} = 1\,\mathrm{R_{Sun}}\times\sqrt{0.01} R_{\mathrm{p}} = 1\,\mathrm{R_{Sun}}\times 0.1

In the step between Equations 3 and 4 we have taken the square root of 0.01. This is correct because if you square 0.1 you get 0.1 × 0.1 = 0.01, as you can check using a calculator. Equivalently, your calculator will tell you that . Try it if you are unsure!

Equation 4 is simply telling you that the radius of Jupiter is about one-tenth that of the Sun or, equivalently, that the radius of the Sun is about ten times that of Jupiter, which is just what you found in Activity 2 in Week 4. If you use a calculator to put in precise numbers for the transit depth and the radius of the star, Equation 2 will give you a precise number for the radius of the transiting planet.