Transit depth formula: from a light curve to an exoplanet's radius

The transit method in one formula

Seen from far away, a star is a bright disc of area π R*². A planet crossing it covers a dark disc of area π Rp². If the star were equally bright everywhere, the fraction of light lost would be exactly the ratio of the areas:

δ = ΔF ÷ F = π Rp² ÷ π R*² = (Rp ÷ R*)²

Rearranged for the planet: Rp = R* × √δ. Two things follow. First, the transit only gives the planet's size relative to its star, so you need the star's radius from other observations. Second, the dip grows with the square of the ratio, so small planets around large stars are very hard to see. This is the method that space telescopes such as Kepler and TESS used to find most of the known exoplanets.

Measuring the depth from a light curve

A light curve is the star's brightness over time, usually normalised so the usual brightness is 1. To measure the depth:

• Average the points outside the transit for the baseline: =AVERAGEIF(C2:C14,"out",B2:B14) = 1.0000.
• Average the points at the bottom of the dip: =AVERAGEIF(C2:C14,"in",B2:B14) = 0.9956. Leave out the sloped edges (ingress and egress), where the planet is only partly in front of the star.
• Depth = 1 − in ÷ baseline = 0.0044 = 0.44%, or 4,400 parts per million (ppm).

Averaging several points beats reading off the single lowest one, because every measurement has noise. Dividing by the baseline instead of assuming exactly 1 removes small calibration offsets.

Worked example: from a light curve to the planet's radius

Rows 2 to 14 are thirteen brightness measurements of a star around one transit, normalised so the star's usual brightness is about 1. Column C marks each point as out of transit, on the edge (ingress or egress) or in transit. Rows 16 to 27 turn the dip into the transit depth and the planet's size.

Worked example: from a light curve to the planet's radius

	A	B	C
1	Time from mid-transit (h)	Relative flux	Phase
2	-3.0	1.0002	out
3	-2.5	0.9998	out
4	-2.0	1.0001	out
5	-1.5	0.9999	out
6	-1.0	0.9978	edge
7	-0.5	0.9957	in
8	0.0	0.9955	in
9	0.5	0.9956	in
10	1.0	0.9979	edge
11	1.5	1.0000	out
12	2.0	1.0001	out
13	2.5	0.9999	out
14	3.0	1.0000	out
15
16	Star radius (solar radii)		0.85
17	Baseline flux (mean of "out")		1.0000
18	Flux in transit (mean of "in")		0.9956
19	Transit depth δ = 1 − in ÷ baseline		0.44%
20	Depth in ppm		4,400
21	Radius ratio Rp/R* = √δ		0.0663
22	Planet radius (Earth radii)		6.15
23	Planet radius (Jupiter radii)		0.55
24	Planet radius (km)		39,225
25	Depth of Earth crossing the Sun (ppm)		84
26	Depth of Jupiter crossing the Sun		1.06%
27	Depth of an Earth-size planet at this star (ppm)		116

Simplified, fictional light curve with no noise model, limb darkening or orbit fit. Educational example, not affiliated with any space agency or mission. Radius conversions use the IAU nominal solar radius (695,700 km), Earth's equatorial radius (6,378.1 km) and Jupiter's equatorial radius (71,492 km).

From depth to radius in Earth, Jupiter and km

The radius ratio is =SQRT(depth) = 0.0663: the planet is 6.6% as wide as its star. Multiply by the star's radius in the unit you want:

• In Earth radii: ratio × R* in solar radii × 109.1 (one solar radius is about 109.1 Earth radii) = 0.0663 × 0.85 × 109.1 = 6.15.
• In Jupiter radii: divide by 11.21 = 0.55.
• In kilometres: ratio × R* × 695,700 km = about 39,225 km.

For a sense of scale: Earth crossing the Sun would cause a dip of only 84 ppm, Jupiter about 1.06%. Around the 0.85 solar-radius star of the example, an Earth-size planet would dim it by 116 ppm, which is why the smaller the star, the easier it is to find small planets.

What makes real light curves harder

• Limb darkening. Stars are dimmer at the edge than in the center, so the bottom of the transit is rounded and the simple formula is only an approximation. Professional fits model the darkening.
• Grazing transits. If the planet only clips the edge of the star, the dip is V-shaped and shallower, and √δ underestimates the planet.
• Extra light. A second star in the same pixel dilutes the dip and makes the planet look smaller. An eclipsing binary star in the background can even mimic a planet.
• The star's radius. Rp is proportional to R*, so a 10% error in the star's radius is a 10% error in the planet's.
• Noise. A dip of 100 ppm needs very precise photometry and many stacked transits to stand out.

Beyond the depth: period, orbit and density

One dip is a candidate. Repeated dips at a regular interval give the orbital period, and with the star's mass Kepler's third law turns the period into the size of the orbit, from which you can estimate the planet's temperature and whether it lies in the habitable zone. Combine the radius from the transit with a mass from radial-velocity measurements and you get the planet's density: rocky, watery or gassy. The same geometry also explains why most planets are never seen to transit: the orbit has to be lined up almost edge-on to us. For more sky calculations in a spreadsheet, see the astronomy calculations course.

Common mistakes

• Forgetting the square root. The depth is the ratio of areas. The ratio of radii is √δ, not δ.
• Using the lowest single point. Noise makes one point too deep or too shallow. Average the flat bottom of the transit.
• Including ingress and egress. The sloped edges are partial transits and make the depth too small.
• Mixing percent and ppm. 0.44% is 0.0044 or 4,400 ppm. Take the square root of the fraction, not of the percentage.
• Ignoring the star's size. The same depth means a much larger planet around a large star. Always multiply by R*.
• Calling every dip a planet. Grazing eclipsing binaries, starspots and instrument effects also cause dips. Planets need repeated, consistent transits and follow-up.