🪐 Exoplanet Transit Light Curve

An exoplanet crosses its host star blocking Δ(F/F_star) = (R_p/R_star)² of starlight. Adjust orbital inclination, planet radius, and limb darkening to see the Mandel-Agol transit shape.

SpaceInteractive
Top: star + planet view · Bottom: flux light curve · Limb darkening shown by color gradient

How it Works

The simulation computes the quadratic limb-darkening transit light curve. The stellar surface brightness follows I(μ) = 1 - u₁(1-μ) - u₂(1-μ)² where μ = cos(θ) is the angle from disk center. The planet, modelled as an opaque circle of radius k·R★, moves across the disk.

At each time step the overlap integral of the planet disk with the limb-darkened stellar disk is computed numerically. The normalized flux F/F₀ = 1 - ΔF where ΔF depends on the position z of the planet center relative to the star center.

Transit depth: δ = (R_p / R_★)² = k² Impact param: b = (a/R_★) cos i Duration: T_14 = (P/π) arcsin[(R_★/a)√((1+k)²-b²)/sin i] Limb dark: I(μ) = 1 - u₁(1-μ) - u₂(1-μ)²

Frequently Asked Questions

What is a transit light curve?

A transit light curve shows the flux of a star over time as an orbiting planet passes in front of it. The U-shaped dip has depth proportional to (R_planet/R_star)² and duration related to orbital speed and impact parameter.

How do we measure exoplanet radii from transits?

The fractional flux drop during transit equals (R_p/R_star)². By measuring this depth precisely with space telescopes like Kepler or TESS, astronomers determine the planet's radius once the stellar radius is known.

What is limb darkening?

Limb darkening is the phenomenon where a star appears darker near its edges than at its center. We see deeper, hotter layers at center and cooler, shallower layers toward the edge. It affects the shape of transit light curves significantly.

What is the impact parameter b?

The impact parameter b = (a/R_star)·cos(i) measures how centrally the planet crosses the star. b=0 means central transit; b=1 means the planet just grazes the stellar limb. When b > 1+k no transit occurs.

How does orbital inclination affect the transit?

For a transit to occur the inclination must be close to 90°. Small deviations increase the impact parameter b, making the transit shorter. At i such that b > 1+k (k=R_p/R_star) no transit occurs.

What is the Mandel-Agol model?

The Mandel-Agol (2002) model provides analytic expressions for the transit light curve including limb darkening effects. It uses elliptic integrals to compute the exact overlap area between the planet disk and limb-darkened stellar disk.

What missions find exoplanet transits?

Key transit missions include Kepler (2009-2018, found 2600+ planets), K2 (2014-2018), TESS (2018-present), and the upcoming PLATO mission. Ground-based surveys like WASP and HATNet also find transiting planets.

Can we detect planetary atmospheres from transits?

Yes, via transmission spectroscopy: at different wavelengths, the planet appears slightly larger where its atmosphere absorbs light. JWST is now detecting water, CO2, and methane in exoplanet atmospheres this way.

What causes the flat bottom of a transit light curve?

When the planet is fully on the stellar disk (second to third contact), the flux drop is constant and equal to (R_p/R_star)². The flat bottom ends when the planet begins to exit at third contact.

What are ingress and egress in a transit?

Ingress is from first contact (planet touches star) to second contact (planet fully on star). Egress is third to fourth contact when the planet is leaving. Their duration gives information about the planet size and orbital speed.

About this simulation

This simulation renders an orbiting planet crossing a quadratically limb-darkened star, computing the resulting flux dip frame by frame from an analytic overlap-area model. The top view shows the star's brightness gradient and the planet's silhouette moving along its orbit, while the bottom panel builds up the classic U-shaped transit light curve in real time as the planet transits.

🔬 What it shows

A star with realistic quadratic limb darkening I(μ) = 1−u₁(1−μ)−u₂(1−μ)², an opaque planet of radius k·R★ crossing it, and a live light curve showing the resulting flux dip, alongside computed Transit depth, Impact parameter b, transit duration T_14, minimum flux, and whether a transit occurs at all.

🎮 How to use

Adjust Planet radius k, Orbital inclination i, Limb darkening u₁ and u₂, and Orbital period, then click Reset to rebuild the light curve and restart the animated transit. Watch how each parameter reshapes the light curve's depth, duration, and bottom flatness.

💡 Did you know?

The transit method is so sensitive that Kepler and TESS can detect a flux dip as small as (R_p/R_star)² — for an Earth-sized planet crossing a Sun-like star that's a dip of only about 0.008%, roughly the equivalent of measuring a flea crawling across a car headlight from miles away.

Frequently asked questions

Why does increasing Planet radius k make the transit dip so much deeper?

Transit depth scales as k² = (R_p/R_star)², so doubling the planet's radius quadruples the fraction of starlight blocked — this squared relationship is exactly how astronomers infer planet size from a measured dip depth once the star's radius is known.

Why does the Transit? readout switch to "No" as I lower Orbital inclination i?

As inclination moves away from 90°, the impact parameter b = (a/R★)cos(i) grows, meaning the planet's projected path passes farther from the star's center; once b exceeds 1+k the planet's disk never overlaps the star's disk at all, so no transit is observed from our vantage point.

Why does the bottom of the light curve stay perfectly flat for large k but not for small k?

A flat bottom occurs when the planet is fully within the stellar disk (between second and third contact) with a fully overlapping shadow of constant area; whether and how long that flat section lasts depends on both k and the impact parameter b, since a grazing (high-b) transit may never reach a fully-flat phase.

Why does raising Limb darkening u₁ and u₂ change the light curve's shape near ingress and egress?

Limb darkening makes the star's edge dimmer than its center, so when the planet blocks light near the limb it removes less flux than when centered — this creates the smoothly rounded ingress/egress shape rather than sharp linear ramps, and stronger limb darkening (higher u₁, u₂) accentuates that rounding.

Why does changing Orbital period only affect T_14 and not the transit depth?

Transit depth depends purely on the geometric ratio k² of planet-to-star radius, which is unrelated to orbital period; period instead sets how fast the planet moves across the disk, so a longer period (slower orbital speed at fixed geometry) stretches the transit duration T_14 without changing how deep the dip is.