A transit light curve can reveal an exoplanet’s size with a small set of repeatable measurements. This guide shows how to calculate transit depth, estimate the planet-to-star radius ratio, convert that ratio into a planet radius, and report simple measurement uncertainty without overstating what the data can tell you.
Overview
When an exoplanet passes between its star and a telescope, the planet blocks a small fraction of the star’s light. The light curve records this change as a temporary dip in brightness. The depth of that dip is related primarily to the relative areas of the planet and star:
Transit depth, δ ≈ (Rp/R★)2
Here, Rp is the planet’s radius and R★ is the star’s radius. Rearranging gives:
Rp/R★ ≈ √δ
Once the stellar radius is known or assumed, the planet radius follows:
Rp ≈ R★√δ
This is an introductory estimate, not a complete transit-model fit. The simple area relationship works best for a clean, flat-bottomed transit observed in normalized flux. Real analyses may also account for stellar limb darkening, the planet’s path across the star, exposure time, instrumental effects, star spots, and uncertainties in the star’s physical properties.
For background on the wider detection process, see how transit, radial velocity, and imaging methods compare. If you are starting with an actual dataset, the public-data light curve activity provides a useful companion exercise.
How to estimate
1. Identify the baseline and transit minimum
Use the light curve’s out-of-transit measurements to estimate the baseline flux, Fbase. This is the typical brightness before and after the dip. Then estimate the minimum in-transit flux, Fmin. Avoid choosing a single unusually low data point; use an average of several points near the bottom or a fitted minimum if the dataset supports one.
If the light curve is already normalized, the baseline may be close to 1. In that case, the calculation becomes especially simple. If the baseline is not 1, use the measured baseline rather than assuming it.
2. Calculate the fractional transit depth
Use:
δ = (Fbase − Fmin)/Fbase
Transit depth is a fraction. Multiply by 100 to express it as a percentage:
Transit depth (%) = 100δ
For example, a depth of 0.012 means a 1.2% decrease in measured brightness.
3. Estimate the radius ratio
Take the square root of the fractional depth:
k = Rp/R★ = √δ
The square root matters because the blocked light depends on area, and circular area scales with radius squared. A 1% transit does not imply a planet that is 1% as wide as its star; it implies a radius ratio of about √0.01 = 0.10.
4. Convert to a planet radius
Multiply the radius ratio by the adopted stellar radius:
Rp = kR★
Keep units consistent. If the star’s radius is expressed in kilometres, the planet radius will be in kilometres. If the star is given in solar radii, the result will be in solar radii unless you convert it afterward.
Formula reference
| Quantity | Formula | Purpose |
|---|---|---|
| Transit depth | δ = (Fbase − Fmin)/Fbase | Fraction of light removed |
| Depth as a percentage | 100δ | Readable percentage |
| Radius ratio | k = √δ | Planet radius divided by stellar radius |
| Planet radius | Rp = R★√δ | Estimated physical radius |
| Depth uncertainty | σδ ≈ √[(σmin/Fbase)² + (Fminσbase/Fbase²)²] | Simple independent-error estimate |
5. Estimate simple uncertainty
If the baseline and minimum each have estimated uncertainties, use σbase and σmin in the depth formula above. This assumes the two measurements are independent. For the radius ratio, propagate the depth uncertainty as:
σk ≈ σδ/(2√δ)
If the stellar radius has uncertainty σ★, a simple combined estimate for the planet-radius uncertainty is:
σRp ≈ Rp√[(σk/k)² + (σ★/R★)²]
For classroom work, it is acceptable to report the measurement uncertainty separately from the uncertainty in the adopted stellar radius. That distinction makes clear whether the limitation comes from the light curve or from knowledge of the host star.
Inputs and assumptions
Before calculating, record the following inputs:
- Baseline flux: the average out-of-transit brightness.
- Minimum flux: the representative brightness during transit.
- Flux uncertainties: an estimated spread or standard error for each measurement.
- Stellar radius: the value and units used for the host star.
- Stellar-radius uncertainty: include it if a source provides one.
- Flux scale: confirm whether the data are normalized flux, raw counts, or another measurement.
Do not mix magnitudes directly with the flux formula. Magnitudes are logarithmic, while the transit-depth equation uses a linear brightness scale. Convert magnitudes to flux first, or use a light curve already supplied in flux units.
The basic estimate also assumes that the dip is caused by a planet crossing the star and that the baseline is stable. A nearby eclipsing binary, changing weather, detector trends, star spots, or an incomplete transit can produce a misleading depth. A grazing transit may also require a more detailed model because the planet does not cover the same projected stellar area as it would in a central crossing.
For calculations involving very small values, retain extra digits during the intermediate steps and round only the final result. A guide to scientific notation and significant figures can help you choose an appropriate final precision.
Worked examples
Example 1: normalized light curve
Suppose a normalized light curve has:
- Fbase = 1.000
- Fmin = 0.988
- R★ = 1.00 solar radius
First calculate the depth:
δ = (1.000 − 0.988)/1.000 = 0.012
The transit depth is therefore 1.2%. The radius ratio is:
k = √0.012 ≈ 0.1095
The planet’s estimated radius is:
Rp ≈ 0.1095 × 1.00 = 0.1095 solar radii
The result should be reported with precision supported by the input data, for example approximately 0.11 stellar radii, rather than with many unsupported decimal places.
Example 2: a non-normalized light curve
Now suppose the measured brightness is:
- Fbase = 24,500 counts
- Fmin = 24,180 counts
- R★ = 0.80 solar radii
The depth is:
δ = (24,500 − 24,180)/24,500 ≈ 0.01306
That is approximately 1.31%. The radius ratio is:
k = √0.01306 ≈ 0.1143
The estimated planet radius is:
Rp ≈ 0.1143 × 0.80 = 0.0914 solar radii
The important point is that the baseline need not equal 1. The fractional change is what matters.
Printable calculation template
Copy this checklist into a worksheet or spreadsheet:
- Target name: __________________________
- Baseline flux, Fbase: __________________________
- Minimum flux, Fmin: __________________________
- Transit depth, δ = (baseline − minimum)/baseline: __________________________
- Transit depth as a percentage: __________________________
- Stellar radius, R★, with units: __________________________
- Radius ratio, √δ: __________________________
- Planet radius, Rp: __________________________
- Uncertainty or data-quality notes: __________________________
When to recalculate
Recalculate whenever you process a different light curve, revise the baseline or minimum, adopt a new stellar-radius estimate, or change the data-cleaning method. The result should also be checked when a source dataset is updated, a classroom activity is refreshed, or a learner compares a manual estimate with software output.
Use this error-checking list before accepting an answer:
- Is the light curve expressed in linear flux rather than magnitudes?
- Were baseline and minimum values measured from several points rather than one noisy point?
- Is the transit depth positive and smaller than 1?
- Was the square root applied to the fractional depth, not the percentage?
- Are the stellar-radius units consistent with the reported planet-radius units?
- Does the number of significant figures match the quality of the data?
- Could trends, star spots, an incomplete transit, or an eclipsing companion affect the dip?
For a repeatable workflow, save the original light curve, mark the baseline and transit windows, record every input, and keep the calculation beside the result. This makes it easier to compare targets, identify changed assumptions, and revisit the estimate as better measurements of the host star become available.
