The star's disk is shaded with a quadratic limb-darkening law I(μ) = 1 − u₁(1−μ) − u₂(1−μ)², μ = cos of the angle to the line of sight. Each frame, the planet's silhouette is numerically integrated against that intensity field (a small grid inside the planet's disk, summed wherever it falls on the star) to get the exact fraction of starlight blocked — the same disk-overlap approach real transit-photometry pipelines use, not a lookup table.
d(θ) = √[(a·sinθ)² + (b·cosθ)²] (sky-plane separation, R★ units)
depth ≈ (Rp/R★)² (first-order, ignores limb darkening)
P = 365.25 · (a·R☉→AU)^1.5 days (Kepler's 3rd law, 1 M☉ host)
T14 ≈ (P/π)·(1/a)·√[(1+k)² − b²] (total transit duration, Winn 2010)
The bottom panel is a phase-folded light curve: flux is written into a 360-bin buffer indexed by orbital phase and redrawn every frame, so the transit dip you see is built up in place every orbit — exactly how astronomers stack real telescope data to pull a shallow transit signal out of noise. A transit only exists at all while b < 1+k; push the impact-parameter slider far enough and the dip vanishes because the planet's disk clears the star's disk entirely.
- Top-left inset — the true orbit seen from above; flattening to a line as b→0 shows the orbit going edge-on to Earth's line of sight.
- Main strip — the sky-plane view telescopes actually see: the star's disk with the planet's silhouette crossing it.