We are forming a model of a planet transit in order to make some calculations what to expect. Our model will be fairly crude, but it will suffices for understanding key characteristics of light curves.

This section has some math in it. It is used to understand how a light curve will look when a planet transits. Feel free to skip to the result if you feel inclined.

A light curve is

a graph of light intensity of a celestial object or region, as a function of time.

It graphs how bright some object appears in the sky over time. Our goal is to understand what the light curve looks like when a planet transits a star.

We are modeling our planet transit in the following crude manner. We assume the star to be a square which radiates uniformly. The total luminosity is \(I_{0}\). So the luminosity per area \(\rho = \frac{I_{0}}{A}\), where \(A\) is the total area of the star.

We model our planet as a square as well. We will also assume that the planet will move with uniform speed across the stars image during the transit. When the planet is fully in front of the star, it block some of the rays of the star diminishing the luminosity to \(I_{t} = \rho \left(A - a\right)\), where \(a\) is the area of the planet.

We are interested in the relative drop in luminosity so we will divide \(I_{t}\) by \(I_{0}\) to get

\[ \frac{I_{t}}{I_{0}} = \frac{\rho \left(A - a\right)}{\rho A} = 1 - \frac{a}{A} \]

So the entire light curve looks something like this.

Even though our model is crude it does portrait important characteristics. We expect to find a dip in the luminosity when a planet transits its star. When a bigger planets transits a star we expect the dip to be more pronounced, since more of the star light is blocked from our view. Our model predicts this as well.

Astronomers have made far better models of planets, but our model will do just fine in finding the period of exo-planets.

Let's plug in some values of a star and a planet we know to see how big a dip we would expect. Jupiter orbits our Sun, so a distant observer could try to infer Jupiter's existence by observing the sun's luminosity. We will calculate the dip they can expect.

The table above lists the radius and area of the sun and Jupiter. Plugging this into our model we determine that

\[ 1 - \frac{a}{A} = 1 - \frac{1.5\mathrm{e}{+10}}{1.5\mathrm{e}{+12}} = 1 - 1\mathrm{e}{-2} \approx 0.99 \]

That is only a one percentage drop!