Why not match TESS data exactly?

Real light curves fold in passband-dependent LD coefficients, stellar granulation, spots, systematics, and often joint fits with RV or asteroseismology. This page isolates the geometric + quadratic-LD effect on a toy grid.

Is the grid integration “correct”?

It is a coarse numerical quadrature for pedagogy. Finer grids or analytic overlap formulas would smooth small numerical ripples but preserve the same qualitative limb rounding.

Exoplanet Transit (limb darkening u₁, u₂)

The same circular planet–star geometry as the uniform-disk transit page is used, but the stellar intensity follows quadratic limb darkening I(μ) ∝ 1 − u₁(1−μ) − u₂(1−μ)² with μ = √(1 − r²) on the projected disk (r in units of R_*). Observed flux is the ratio of disk integrals of I over visible area after subtracting the opaque planetary circle, approximated on a fixed square pixel grid inside the unit disk. Larger u₁ and u₂ dim the limb relative to disk center, rounding ingress and egress and changing grazing transit shapes compared with a uniform disk. Spots, blending, eccentricity, and full passband-integrated LD are not modeled.

Who it's for: Follows the uniform-disk transit page; introductory photometry before fitting TESS/Kepler light curves with stellar models.

Key terms

limb darkening
quadratic law
transit photometry
impact parameter
ingress

Live graphs

How it works

This page keeps the same circular geometry as the uniform-disk transit simulator, but replaces the star with a quadratic limb-darkening intensity I(μ) ∝ 1 − u₁(1−μ) − u₂(1−μ)², μ = √(1 − r²) on the projected disk (r in units of R_*). Observed flux is the disk integral of I over visible surface area, computed on a fixed pixel grid (coarse but fast). Limb darkening makes ingress/egress rounder than a uniform disk and slightly changes the bottom shape for grazing transits. Star spots, blending, eccentric orbits, and LD degeneracy with impact parameter are still ignored — use this next to the uniform-disk page for contrast.

Key equations

I(μ)/I₀ = 1 − u₁(1−μ) − u₂(1−μ)² , μ = √(1 − r²) (projected disk)

F/F₀ = (∬_{disk} I dA − ∬_{overlap} I dA) / ∬_{disk} I dA (planet disk opaque)

Frequently asked questions

Why not match TESS data exactly?: Real light curves fold in passband-dependent LD coefficients, stellar granulation, spots, systematics, and often joint fits with RV or asteroseismology. This page isolates the geometric + quadratic-LD effect on a toy grid.
Is the grid integration “correct”?: It is a coarse numerical quadrature for pedagogy. Finer grids or analytic overlap formulas would smooth small numerical ripples but preserve the same qualitative limb rounding.