Exoplanet Transit Calculator — radius from a light curve

The exoplanet transit calculator above works the standard chain of inferences that turns a dip in a star's brightness into a description of a planet. The depth of the dip gives the ratio of the planet's radius to the star's. The orbital period, with the star's mass, gives the size of the orbit. The duration of the dip, compared against how long a dead-central crossing would take, gives the impact parameter, which is how far off centre the planet's path across the stellar disc sits. The star's temperature and the orbital distance give an equilibrium temperature, subject to an albedo you have to assume.

Arb Digital builds free tools that make the assumptions visible instead of hiding them behind a single number. Every stellar property here is an input, because none of them come from the light curve itself — they come from spectroscopy, from parallax distances and from stellar models, and the planet parameters inherit every error in them. The page reports the impact parameter and the central-transit duration alongside the radius so you can see whether the geometry you entered is even self-consistent.

What This Exoplanet Transit Calculator Does

A transit is a small eclipse. When a planet passes between its star and a telescope, it blocks a fraction of the stellar disc, and NASA's explanation of what a transit is describes the resulting brightness dip and the light curve that records it. The blocked fraction is the ratio of the planet's disc area to the star's disc area, so the depth of the dip is the square of the radius ratio. Take the square root of the depth and you have the planet's radius in units of the star's radius, which is the single most robust result in the whole field.

Converting that ratio into kilometres, Earth radii or Jupiter radii requires the star's radius, which the light curve cannot supply. So does the orbital distance, which needs the star's mass. This is the structural feature of transit work that surprises people first: a light curve measures ratios exceptionally well and absolute quantities not at all. Every absolute number on this page is a ratio multiplied by a stellar property you provided.

The duration adds geometry. A planet crossing the exact centre of the stellar disc takes the longest possible time; one clipping the edge takes almost no time at all. The impact parameter b measures that offset in units of the stellar radius, running from zero for a central crossing to about one at the limb. Given the period, the orbital distance and the radius ratio, a measured duration pins b down, and b in turn tells you the orbital inclination.

How to Use It

1. Enter the depth in the units your photometry reports. Space-based photometry usually quotes parts per million; ground-based work often quotes per cent. A 1 per cent dip is 10,000 ppm.
2. Supply the host star radius and mass. These come from a stellar catalogue or a spectroscopic analysis, not from the transit, and they set the accuracy of everything absolute.
3. Enter the period and the duration. Duration is conventionally measured from first to last contact, when the planet's disc first touches the stellar limb and finally leaves it.
4. Choose an albedo deliberately. The equilibrium temperature scales as the fourth root of one minus albedo, so it is a weak dependence, but the result is still a scenario rather than a measurement.
5. Check the impact parameter. If it comes back near or above one, the geometry you entered describes a grazing event, and grazing transits give unreliable radii.

The Formula: How Transit Parameters Are Calculated

The radius ratio is k = R_p ÷ R_★ = √δ, where δ is the fractional transit depth. The semi-major axis comes from Kepler's third law in solar units, a = (M_★ P²)^1/3 with a in astronomical units, the mass in solar masses and the period in years. The central-transit duration is T₀ = (P ÷ π) × arcsin(R_★(1 + k) ÷ a), and for a transit of observed duration T the impact parameter follows from b² = (1 + k)² − (a ÷ R_★)² sin²(πT ÷ P). The equilibrium temperature is T_eq = T_eff √(R_★ ÷ 2a) (1 − A)^1/4.

Work the defaults by hand. A depth of 10,000 ppm is 0.01, so k is 0.1 and the planet is a tenth of the star's radius. With a solar radius of 695,700 kilometres that gives 69,570 kilometres, which is 10.92 Earth radii or 0.973 Jupiter radii. A 3.5-day period is 0.0095825 years, so a is the cube root of 1 × 0.0095825², which is the cube root of 9.1824 × 10⁻⁵, or 0.04511 astronomical units.

The stellar radius in those units is 0.0046505 astronomical units, so R_★ ÷ a is 0.10308, which is also the geometric transit probability of 10.3 per cent. The equilibrium temperature is 5,772 × √(0.10308 ÷ 2) × 0.7^0.25 = 5,772 × 0.22703 × 0.9146 = 1,199 kelvin. For the geometry, sin(π × 0.125 ÷ 3.5) = 0.11196, and dividing by 0.10308 gives 1.0862, so b² = 1.21 − 1.1798 = 0.0302 and b = 0.174. The calculator reproduces every one of those figures.

Why the Transit Method Finds So Few of the Planets That Exist

A transit only happens if the orbit is oriented so that the planet passes in front of the star from our particular viewing direction. For a randomly oriented orbit the probability of that alignment is approximately the stellar radius divided by the orbital distance, which for the default case above is about ten per cent. Move the planet out to one astronomical unit around the same star and the probability falls to less than half a per cent. Move it to Jupiter's distance and it falls below a tenth of a per cent.

This is why transit surveys are dominated by short-period planets, and why the population of known transiting planets is not the population of planets. The bias is geometric and completely understood, so it can be corrected for statistically, but it is severe: for every transiting hot Jupiter there are roughly ten identical planets around identical stars whose orbits happen to be tilted out of our line of sight.

Period length compounds the problem in a second way. Confirming a periodic signal usually requires several transits, and a planet with a 300-day period gives you one transit a year at best, and only if the survey happens to be looking. NASA's overview of how exoplanets are found and classified sets out how the transit method sits alongside radial velocity, direct imaging and microlensing, each with a different and largely complementary bias. Our Kepler's third law calculator handles the period-to-distance relationship on its own if that is all you need.

Depth Is Not Quite the Radius Ratio Squared

The clean relationship between depth and radius ratio assumes the stellar disc is uniformly bright. It is not. A star is a ball of gas with no solid surface, and looking at its centre you see deeper and hotter layers than you do at the limb, where the same line of sight passes obliquely through cooler upper material. The disc is therefore brighter in the middle and dimmer at the edge, an effect called limb darkening, and it is strong at visible wavelengths.

The consequence is that a planet crossing the centre blocks brighter-than-average light and produces a deeper dip than the area ratio alone predicts, while one crossing near the limb produces a shallower dip. It also rounds the bottom of the light curve, which would otherwise be flat. Real analyses fit a limb-darkening law simultaneously with the planet parameters, and the radius that comes out can differ from the naive square-root answer by a few per cent.

This calculator uses the uniform-disc relationship deliberately, because it is the relationship the depth definition rests on and because adding a limb-darkening law would require coefficients that depend on the star, the wavelength band and the model atmosphere used. Treat the radius here as the uniform-disc value, and expect a full fit to move it slightly.

What a Transit Cannot Tell You

The most important limit is mass. A transit measures how much light is blocked, which depends only on size. Two objects of identical radius and wildly different mass produce identical light curves, and at Jupiter's size the radius is nearly independent of mass over a large range, so a transit alone cannot distinguish a gas giant from a brown dwarf or even a small star. Mass comes from radial-velocity measurements of the star's reflex motion, or from transit-timing variations when several planets perturb each other.

Density follows only once mass and radius are both in hand, and density is what separates a rocky planet from a water world from a gas envelope. Without it, a 2-Earth-radius object is genuinely ambiguous. Our density calculator handles the arithmetic once you have both quantities, and the gravitational force calculator covers the surface gravity that follows from them.

Equilibrium temperature is also not a surface temperature. It is the temperature a body with no atmosphere would settle at if it radiated away exactly the starlight it absorbed, distributed evenly over its whole surface. An atmosphere changes this substantially through greenhouse warming and through how efficiently heat is carried from the day side to the night side. For a tidally locked planet with poor heat redistribution the day side runs hotter than the equilibrium value and the night side far colder.

Reading the Impact Parameter

The impact parameter is where a self-consistency check lives. If the period, the duration, the stellar properties and the depth you entered are mutually consistent, b comes out between zero and roughly one. If the calculation demands a negative b squared, the transit you have described lasts longer than a central crossing possibly could for that orbit, which means one of the inputs is wrong: the period, the stellar mass, the stellar radius or the duration itself.

Values close to one describe a grazing transit, where the planet never fully crosses onto the disc. Grazing events have V-shaped rather than flat-bottomed light curves and their depths systematically underestimate the planet, which is why a large b is a warning flag in any transit analysis. It is also the classic signature of a false positive: a grazing eclipsing binary star can produce a shallow V-shaped dip that mimics a planet.

The relationship between b and inclination is cos i = b R_★ ÷ a, so a small b means an orbit seen nearly edge-on. For the default figures here, b = 0.174 corresponds to an inclination of about 89 degrees. Our orbital velocity calculator gives the speed at which the planet is travelling along that orbit, and the escape velocity calculator covers the related quantity for the planet itself.

Common Mistakes to Avoid

• Mixing up ppm and per cent — a factor of 10,000 in the depth becomes a factor of 100 in the radius.
• Trusting the absolute radius more than the stellar radius — the planet size inherits the star's error in full, so quote the ratio when the star is poorly characterised.
• Reading equilibrium temperature as a surface temperature — it excludes any atmosphere, any greenhouse effect and any day-night contrast.
• Assuming a deep transit means a big planet — a small star produces a deep transit from a modest planet, which is exactly why small stars are the preferred targets.
• Ignoring a large impact parameter — a grazing geometry underestimates the radius and is a common false-positive signature.

Related Free Tools From Arb Digital

Orbital work usually needs more than one relationship at a time. The Kepler's third law calculator converts between period, orbital radius and central mass in either solar or SI units, and the orbital velocity calculator gives the speed that goes with a given orbit.

For the observing side, the angular resolution calculator gives the smallest detail an aperture can separate, and the astronomical distance converter moves between astronomical units, light years and parsecs when a catalogue quotes one and you need another.

Frequently Asked Questions

This tool is provided for educational and estimating use only. It applies the standard uniform-disc transit relations to inputs you supply and is not a substitute for a fitted light-curve analysis; published planet parameters come from full model fits with limb darkening, eccentricity and observational uncertainties treated properly.