Calculator
Choose a mode, enter radii or ratio, and optionally refine with geometry, limb darkening, and dilution.
Example Data
These examples assume b=0, u1=u2=0, and no dilution.
| Scenario | Star radius | Planet radius | Depth (%) | Depth (ppm) |
|---|---|---|---|---|
| Earth transiting Sun | 1.000 R☉ | 1.000 R⊕ | 0.008386 | 83.86 |
| Neptune transiting Sun | 1.000 R☉ | 3.865 R⊕ | 0.125257 | 1,252.57 |
| Jupiter transiting Sun | 1.000 R☉ | 1.000 Rj | 1.009827 | 10,098.27 |
| Hot Jupiter, 0.8 R☉ star | 0.800 R☉ | 1.000 Rj | 1.577855 | 15,778.55 |
Formula Used
The geometric transit depth for a small planet is: δ = (Rp / Rs)², where Rp is planet radius and Rs is star radius.
With quadratic limb darkening, intensity is evaluated at mid-transit: I(μ) = 1 − u1(1−μ) − u2(1−μ)², with μ = √(1−b²).
Average disk intensity for this law is: Ī = 1 − u1/3 − u2/6. The limb-darkened depth is: δLD = δ × I(μ)/Ī.
If extra light dilutes the signal, observed depth becomes: δobs = δLD / (1 + D), where D is contaminant-to-target flux ratio.
How to Use This Calculator
- Select the calculation mode you need.
- For depth mode, enter radii or the ratio.
- For inverse modes, enter observed transit depth.
- Add impact parameter, limb darkening, and dilution.
- Press Calculate to show results above the form.
- Use the buttons to export CSV or print PDF.
Transit Depth Guide
1) What transit depth measures
Transit depth is the fractional loss of starlight during a planet crossing. In the simplest case, depth equals the blocked area ratio. A depth of 1% means the flux drops by 0.01 of baseline. Observers also report depth in parts per million (ppm). For example, 1% equals 10,000 ppm and 10 ppt.
2) Geometric depth from radii
The geometric estimate is δ = (Rp/Rs)². If Rp/Rs = 0.10, then δ = 0.01, or 1%. An Earth-sized planet around a Sun-sized star gives Rp/Rs ≈ 0.0092. That produces δ ≈ 8.4×10−5, or about 84 ppm. These values set the scale for detection difficulty.
3) Impact parameter and mid-transit chord
The impact parameter b describes how centrally the planet crosses. b = 0 is a central transit, while b near 1 is grazing. Higher b moves the transit chord toward the stellar limb. Since limb regions are dimmer, the same planet can yield a different depth. This calculator uses μ = √(1−b²) to evaluate limb effects.
4) Quadratic limb darkening in practice
Quadratic limb darkening uses coefficients u1 and u2. The local intensity follows I(μ) = 1 − u1(1−μ) − u2(1−μ)². Disk-averaged intensity is Ī = 1 − u1/3 − u2/6. At mid-transit, depth scales by I(μ)/Ī. Typical optical coefficients often fall between 0 and 0.6.
5) Dilution from blended light
Nearby stars or background light dilute the measured dip. If contaminant-to-target flux ratio is D, observed depth is δobs = δLD/(1+D). With D = 0.20, a true 1% limb-darkened dip appears as 0.833%. In ppm, 10,000 ppm becomes about 8,333 ppm. Always check aperture contamination for precise radii.
6) Inverse solutions for planet size
When you measure depth, you can solve for Rp/Rs as √δ after corrections. Combine that ratio with an independent stellar radius. If Rs = 1 R☉ and corrected δ = 1%, then Rp ≈ 0.10 R☉. That is about 1.0 Rj, close to Jupiter’s size.
7) Photometric noise and sampling
Depth must be compared to noise per cadence. If per-point scatter is 300 ppm, an 84 ppm Earth-like dip is subtle. Binning improves precision roughly with √N, when noise is uncorrelated. Systematics, stellar variability, and instrument drift can dominate. Use consistent detrending before interpreting small depths.
8) Interpreting depth alongside other observables
Depth mainly constrains Rp/Rs, not mass or composition. Combine depth with radial velocity mass for density. Add transit duration and impact parameter to constrain inclination. Multi-band depths can hint at atmospheric scattering or spots. Report your coefficients, bandpass, and dilution assumptions for clarity.
FAQs
1) What is a typical exoplanet transit depth?
Hot Jupiters often produce ~0.5% to 2% dips (5,000–20,000 ppm). Neptune-size planets can be a few hundred to a few thousand ppm. Earth-size planets around Sun-like stars are roughly ~80–100 ppm.
2) Why does impact parameter change the depth?
Limb regions are dimmer than the stellar center. A higher impact parameter samples lower intensity. With limb darkening enabled, the same area block can yield a different mid-transit depth. Setting u1=u2=0 removes this effect.
3) What dilution value should I enter?
Use D = contaminant flux divided by target flux in your aperture. If a nearby star contributes 10% of the target flux, enter D = 0.10. If you have no blending estimate, start with D = 0 and compare later.
4) Why is the “observed depth” smaller than geometric depth?
Dilution reduces depth because extra light raises the baseline. If D is positive, δobs = δLD/(1+D) must be smaller. Limb darkening can also shift δLD relative to δ, depending on b and coefficients.
5) Can I use this for grazing transits?
It supports b up to 1, which approximates grazing geometry. However, very grazing cases can violate the small-planet mid-transit assumption. Treat results as an estimate and use a full light-curve model for high precision.
6) Which limb-darkening coefficients should I choose?
Coefficients depend on stellar temperature, gravity, metallicity, and bandpass. Use values from your light-curve pipeline, stellar models, or literature tables. If uncertain, explore a plausible range and see how Rp/Rs changes.
7) What units should I use for depth?
Use ppm for small dips and percent for large dips. Remember 1% = 10,000 ppm and 1 ppt = 1,000 ppm. The calculator converts and reports fraction, percent, and ppm together.