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An introduction to exoplanets
An introduction to exoplanets

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5.3  Small rocky planets

You’ve seen that the smaller the planet, the smaller the dip in brightness when a transit occurs. It is perfectly easy to detect the transit of a Jupiter-sized giant planet from the ground. But it turns out that from the ground it is nearly impossible to detect the transit of an Earth-sized planet across a Sun-sized star – you just can’t measure precise enough light curves.

Activity _unit6.5.1 Activity 4  The transit depth

Timing: Allow about 10 minutes

For a star approximately the same size as the Sun and a planet approximately the same size as Jupiter, the transit depth is roughly 1%.

What would be the transit depth if the planet radius was:

  1. Half the size of Jupiter’s radius? (Hint: think about how the cross-sectional area of the planet would change – you may find Equation 1 useful.)
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Answer

Answer: 0.25% or ¼%

The cross-sectional area of the planet depends on the square of its radius. If the radius is halved then the area will be quartered (1/2 × 1/2 = 1/4).

This means that the transit depth will also be quartered. The transit depth for Jupiter is roughly 1%, so the transit depth for this smaller planet will be about 1/4%.

If you prefer, from Equation 1, if Rp is halved and Rstar stays the same then the ratio Rp/Rstar will be halved, and so (Rp/Rstar)2 will be quartered.

  1. One-third the size of Jupiter’s radius?

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Answer

Answer: 0.111% or 1/9%

The cross-sectional area of the planet depends on the square of its radius. If the radius is multiplied by 1/3 then the area will be multiplied by 1/3 × 1/3 = 1/9.

This means that the transit depth will also be multiplied by 1/9. The transit depth for Jupiter is roughly 1%, so the transit depth for this smaller planet will be about 1/9%.

  1. One-tenth the size of Jupiter’s radius?

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Answer

Answer: 0.01% or 1/100%

The cross-sectional area of the planet depends on the square of its radius. If the radius is multiplied by 1/10 then the area will be multiplied by 1/10 × 1/10 = 1/100.

This means that the transit depth will also be multiplied by 1/100. The transit depth for Jupiter is roughly 1%, so the transit depth for this smaller planet will be about 1/100% or 0.01%.

Remember, Earth is approximately one-tenth the radius of Jupiter, so its transit would be 100 times smaller – a transit depth of just 0.01% of the light from the star! This would be pretty tricky to measure from the ground.