The Uniform-Source Transit Model - COMP 521: Introduction to Scientific Computing and Data Science

The first transit model treats the star as a uniformly bright circular disk and the planet as an opaque circular disk. It asks a geometric question: what fraction of the visible stellar disk remains uncovered?

The complete-overlap starting point¶

Let

p = \frac{R_p}{R_\star},

(1)

where $R_p$ is planet radius and $R_\star$ is stellar radius. Because this is a ratio of lengths measured in the same units, $p$ has no units. If the smaller planet disk lies entirely in front of the stellar disk, the blocked fraction is the area ratio:

\delta = \frac{\pi R_p^2}{\pi R_\star^2} = p^2.

(2)

The normalized flux is therefore

\frac{F}{F_0} = 1 - p^2.

(3)

If $p=0.10$ , then $\delta=0.01$ and the normalized flux is 0.99. If $p$ is doubled to 0.20, depth becomes 0.04: area scales with the square of a length.

What Project 1 adds¶

The complete-overlap result is only one limiting case. A complete model also needs the projected center separation

z = \frac{d}{R_\star},

(4)

where $d$ is the projected distance between disk centers. Different values of $p$ and $z$ produce no overlap, partial overlap, full overlap, or complete stellar coverage. Project 1 turns those geometric branches into a testable function.

Limits of the model¶

This is a deliberately useful approximation, not a claim that stars are flat, uniform disks. It omits, among other effects:

limb darkening: stellar edges can appear dimmer than the center;
measurement noise and instrumental systematics;
starspots and other stellar variability; and
orbital details that determine the timing of $z$ .

Those omissions tell us the scope of a conclusion: the model teaches geometry and computational validation; it does not by itself fit a real observed exoplanet system.