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An introduction to exoplanets
An introduction to exoplanets

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3.3  From dips to planet size measurement

The size of the planet relative to the star is worked out by measuring the difference between the normal brightness of the star and the brightness of the star during the deepest part of the dip. This difference, expressed as a fraction or percentage, is called the transit depth and it is the key measurement taken for transiting exoplanets. If it is possible to detect the dip and measure how deep it is, then it is possible to work out the size of the planet causing it.

The amount of light blocked by a planet passing in front of its star depends on the area of the star blocked from view. The fraction of light missing, the transit depth, is calculated simply by working out the cross-sectional area of the planet, and dividing that by the cross-sectional area of the star.

Now, you know that the cross-sectional area of a sphere is a circle of the same radius, so the cross-sectional area of a planet can be written as:

Ap = πRp2,

and the cross-sectional area of a star as:

Astar = πRstar2

where Rp is the radius of a planet and Rstar is the radius of the star.

Dividing these expressions gives the transit depth, so the transit depth is equal to:

cap a sub normal p divided by cap a sub normal s times normal t times normal a times normal r or pi times cap r sub normal p squared divided by pi times cap r sub normal s times normal t times normal a times normal r squared

Because the constant π appears in both the top and bottom parts of this fraction, it cancels out and disappears.

So, the transit depth is simply:

cap r sub normal p squared divided by cap r sub normal s times normal t times normal a times normal r squared
Equation label: (Equation 3)

This can also be written as:

open cap r sub normal p divided by cap r sub normal s times normal t times normal a times normal r close squared
Equation label: (Equation 4)

Often this fraction is converted into a percentage by multiplying it by 100.

Now, this is what we need to work out the size of the planet. We can measure the transit depth from the light curve and we can estimate the radius of the star from its spectrum. This means that Rp can be calculated, and you will see how this works next week.