Exoplanet Transit Lab (2D)

What you are looking at

A star with up to three planets on a single orbital plane, seen by a distant telescope. When a planet crosses the stellar disk it blocks a little light, and the light curve dips. Everything on the page is computed live from Newton's law of gravity: the planets pull on the star and on each other, so their transits drift away from a perfect clock. That drift is the transit timing variation (TTV).

Transit depth, radius ratio and limb darkening

For a flat, evenly lit star the fraction of light blocked is the area ratio delta = k^2, where k = Rp / Rs is the planet-to-star radius ratio. Real stars are dimmer towards the edge (limb) because we see shallower, cooler layers there. The page uses the standard quadratic law

I(mu) / I(1) = 1 - u1 (1 - mu) - u2 (1 - mu)^2, mu = sqrt(1 - r^2)

where r is the distance from the disc centre in stellar radii. A planet that crosses the bright centre removes more than k^2 of the light; one that crosses near the limb removes less. The exact blocked flux is a radial integral over the stellar disc:

F_blocked = (1 / Phi) * integral over r of I(r) * 2 alpha(r) * r dr, alpha = arccos((r^2 + z^2 - k^2) / (2 r z))

with z the separation of the planet and star centres and Phi = pi (1 - u1/3 - u2/6) the total brightness of the disc. For a small planet at separation z this reduces to delta ~ k^2 I(z) / mean(I).

Geometry of one transit

Kepler's third law gives the orbit size, a^3 = G (Ms + mp) P^2 / (4 pi^2). The impact parameter is b = a cos(i) / Rs (0 = through the middle, 1 = skimming the limb), and the time from first to last contact is

T14 = (P / pi) * arcsin( (Rs / a) * sqrt((1 + k)^2 - b^2) / sin i )

Transits occur only when b < 1 + k, which is why only about Rs / a of randomly oriented systems show one. For b > 1 - k the planet is never fully inside the disc (a grazing transit) and the bottom of the dip is round instead of flat.

N-body motion and timing variations

The star and planets are integrated with the symplectic leapfrog scheme (kick-drift-kick, step = min(P/160, 0.012 d)). Mid-transit times are found when the planet's projected position crosses the star's centre, interpolated between steps. The page fits a straight line t_n = T0 + n P and plots the residual O - C (observed minus calculated). Two planets near a first-order resonance j : j-1 exchange energy slowly, and the timing signal repeats with the super-period

P_super = 1 / | j / P_outer - (j - 1) / P_inner |

The amplitude grows with the perturber's mass and shrinks as the pair moves away from resonance (Agol et al. 2005; Lithwick, Xie and Wu 2012). An eccentric orbit (the e sliders) adds a second, faster term. This is how Kepler found non-transiting planets from the timing of transiting ones.

Honest simplifications

• All orbits are coplanar, so no planet can drift out of its transit and transit durations do not change. There is no general relativity, tidal or planet-star spin interaction.
• Limb darkening is the two-parameter quadratic law. The temperature button uses an interpolated table of typical optical values, not a fit to a stellar-atmosphere model. The star colour is a blackbody approximation. Starspots on the picture are decoration and do not change the light curve.
• When two planets overlap on the disc their blocked light is simply added (a mutual transit is not treated exactly). Planet-star and planet-planet occultations, secondary eclipses, reflected light and atmospheres are not modelled.
• Planet radii and masses are independent sliders; the page does not enforce a physical mass-radius relation, so some combinations are not real planets.
• Flux is sampled once per integration step (1 to 17 minutes of simulated time), so at high time speed the light curve is drawn with fewer points. In the 3D view planets are drawn larger than life (default 5x) except in the Observer view, which is true scale.
• Planet b starts 0.12 of its period before a transit; changing a physical parameter restarts the clock. The forecast of transit times runs 70 orbits of b (350 to 1500 days) in the background.

• Depth versus radius ratio. Pick "Hot Jupiter, central transit", then drag the radius of b from 3 to 20 Earth radii: the dip in the light curve grows roughly as k squared; the white dot in the depth graph rides on the b = 0 curve.
• Limb darkening. Press "No limb darkening (flat disk)": the bottom of the dip turns flat and the red dashed model in the transit graph matches. Press "Limb darkening from temperature" to bring the curved bottom back.
• Impact parameter. Raise b from 0 to 0.9 and watch the transit get shorter (graph T14 versus b), shallower and rounder; at b above 1 + k the planet misses the star and no dip appears.
• Find the hidden planet. Select "TTV pair near 2:1", look at the O - C graph, then drag the period ratio of c from 2.03 to 2.2: the timing wobble of b fades and the super-period changes. Raise the mass of c to make it stronger.
• Eccentricity. With planet c on, increase its eccentricity from 0 to 0.2 and see the faster ripple added to the slow TTV wave.
• Three planets. "Three-planet compact system" shows three colours of dips and three TTV curves at once.
• Look from the telescope. In 3D choose "Observer view (true scale)" for the real sky picture, or "Ride with planet b" to fly along with the planet. Use "Skip to next transit" when you do not want to wait.