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)

Quadratic limb-darkened disk; grid-integrated F(t); TESS-like solar preset vs uniform-disk page.

Live graphs

About this model

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

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.

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