Exoplanet Transit (3D)

Transit method

When a planet crosses the face of its star the light dips by about the area ratio k^2 = (Rp/R*)^2: Jupiter in front of the Sun hides about 1 %, the Earth about 0.008 %. The star is darker at the edge (limb darkening, I(mu) = 1 - u1(1-mu) - u2(1-mu)^2), so a planet near the centre hides more light than one near the edge. The simulation integrates the hidden light exactly, ring by ring across the stellar disc, for the planet's true position at each moment.

Orbit and duration

The orbit comes from Kepler's third law P^2 = 4 pi^2 a^3 / (G (M* + Mp)) and Kepler's equation for eccentric orbits. The impact parameter b is how far from the stellar centre the planet's path passes, in stellar radii. The first-to-last-contact duration is T14 = (P/pi) asin[(R*/a) sqrt((1+k)^2 - b^2) / sin i] sqrt(1-e^2)/(1+e sin w). The chance that a random orbit transits at all is about R*/a: small for wide orbits, which is why most planets stay hidden.

Radial velocity

A planet and its star orbit their common centre of mass, so the star wobbles with speed amplitude K = (2 pi G / P)^(1/3) Mp sin i / (M* + Mp)^(2/3) / sqrt(1-e^2). Jupiter gives the Sun 12.5 m/s, the Earth only 0.09 m/s. Spectrographs measure this through the Doppler shift of the starlight; the radial velocity gives only Mp sin i, so a transit is needed to know the inclination and the true mass.

Habitable zone

The zone where liquid water could exist on a rocky planet's surface is computed from the star's temperature and luminosity with the polynomial fits of Kopparapu et al. (2013): the inner edge is the runaway greenhouse limit, the outer edge the maximum greenhouse limit (the green band); a wider optimistic band uses the "recent Venus" and "early Mars" limits. The star's luminosity, radius and temperature are estimated from its mass with main-sequence scaling laws. Equilibrium temperature is Teq = T* sqrt(R*/2a) (1-A)^(1/4), ignoring the greenhouse effect.

Atmospheres

Starlight filtered through the planet's thin air at the rim is absorbed at wavelengths where molecules absorb, so the planet looks slightly larger there. The atmosphere's scale height is H = k T / (mu m_u g) and the signal is roughly 2 N H Rp / R*^2 for about N = 5 scale heights. The graph models water, carbon dioxide and methane bands as Gaussians on a Rayleigh-scattering slope, with clouds flattening them. This is a teaching sketch, not a retrieval model.

Simplifications

Circular-disc planet, no moons, rings, star spots or stellar activity; noise is Gaussian and white; the star's mass-luminosity relation is approximate; the equilibrium temperature has no greenhouse term; planet density and composition do not feed back into the radius you choose. Animation speed is stretched near transit so the short event is visible, while all times shown are real.

1. Press Hot Jupiter and watch the light curve: the dip depth equals the planet-to-star area ratio. Make the planet smaller and the dip shrinks.
2. Switch to Observer (transit view) to see the planet's dark disc cross the star, and notice the limb darkening.
3. Raise the impact parameter b toward 1: the transit gets shorter and shallower until it is only a graze, then disappears.
4. Choose Earth twin: the dip is tiny and the noisy data points barely show it. The panel tells you how many transits you need for 7 sigma.
5. Pick Super-Earth, red dwarf habitable zone: see how a faint star has a habitable zone very close to it, and a short year.
6. Drag the orbit eccentricity: the orbit stretches, the radial-velocity curve becomes lopsided and the transit duration changes.
7. Look at the spectrum graph: add clouds and the molecule bands flatten; swap an air weight of 2.3 to 29 and the signal shrinks.