Transits. If the orbit of a planet is nearly edge-on to us, the planet passes in front of its star once an orbit and blocks a small part of the starlight. The brightness drops while it does so: the star’s light curve has a dip. The depth of the dip is the fraction of the star’s disc that is covered. With the radius of the planet R_p and the radius of the star R_s:

depth = (R_p / R_s)²

Jupiter in front of the Sun would take away about 1% of the light. The Earth would take away only 0.008%, which is why you need a space telescope. The time the transit lasts depends on the size of the orbit and how fast the planet moves:

T ≈ (P / π) × (R_s / a) × √[(1 + R_p/R_s)² − b²]

where a is the radius of the orbit (from Kepler’s third law) and b is how far from the centre of the disc the planet crosses (in units of the radius of the star). A transit that crosses nearer the edge is shorter and has a slightly different shape. The edge of a star is darker than the middle (limb darkening), so the dip is deepest in the middle and the bottom of the transit is curved. A planet that only just crosses the edge is a “grazing” transit.

Eclipsing binaries. Two stars orbiting each other, seen nearly edge-on, pass in front of each other. When the hotter star is covered (the primary eclipse) we lose more light than when the cooler star is covered (the secondary eclipse). The surface of a star gives out light in proportion to T⁴ (Stefan-Boltzmann law), so the depths tell us the ratio of the surface temperatures, and the duration of the eclipses tells us the sizes of the stars compared with their orbit. Algol, “the demon star”, has been known for centuries to fade every 2.87 days.

The wobble and the density. The transit gives the planet’s radius (from the depth) but not its mass. The Doppler wobble of the star (the radial velocity method) gives the mass (see the Doppler effect interactive). Put the two together, and you get the density:

ρ = M / (4/3 πR³)

Compare it with Jupiter (1.33 g/cm³), Saturn (0.69), Neptune (1.64) and the Earth (5.51). A planet with a density near 1 is made of hydrogen and helium; one near 5 is made of rock and iron. This is how we know that many planets are not like anything in our own Solar System, such as hot Jupiters and “puffy” planets with a lower density than water.

What the model leaves out. The orbits are circles, and the stars are shown with the same darkening of the edges that is used for the Sun. The eclipsing binary stars are treated as discs of uniform brightness that give out light in proportion to T⁴ (in a real instrument the colours matter). The planet does not shine, and nothing is added to the light by the planet. The tilt controls are limited to values that give a transit.

Things to try. (1) Choose the Earth and then Jupiter. How much deeper is the dip for Jupiter? (2) Find the tilt at which the transit just becomes shallower. (3) Make the orbit longer. What happens to the time between transits, and to the length of the transit? (4) In the second tab, make star 2 hotter than star 1: which eclipse is deeper now? (5) In the third tab, choose a planet that is 0.7 Jupiter masses and 1.38 Jupiter radii. What is its density? Is it rock or gas?