A one-percent exoplanet transit is often translated into a planet one-tenth the radius of its host star. That shortcut assumes a uniformly bright star and a complete occultation. Impact parameter and limb darkening change where the planet crosses and how much of the star’s light is actually blocked.
This guide uses one synthetic circular orbit and a linear stellar intensity profile to show why the same planet can produce different mid-transit depths. Worked geometry, grazing-event limits, FAQs and connected observations provide a complete route from classroom formula to responsible transit modelling.
eduKateSG · The Gold Standard of Astronomy
Choose your reading route
Establish the geometry · Chapters 1–3: impact parameter, inclination and duration.
Model the actual brightness · Chapters 4–6: limb darkening, grazing transit and uncertainties.
Check and continue · Chapters 7–8: FAQs, subject links and sources.
Chapter index
Geometry · Chapters 1–3
Stellar brightness · Chapters 4–6
Questions and sources · Chapters 7–8
When a planet crosses its star, a telescope records a dip in the star’s apparent brightness. A first explanation says that if the planet blocks one percent of a uniformly bright stellar disk, its projected radius is ten percent of the star’s radius: sqrt(0.01)=0.10. The familiar relation δ≈(Rp/Rstar)² is useful precisely because its assumptions can be stated.
Real stellar disks are not generally uniform in brightness. Many stars appear brighter toward the centre and dimmer toward the edge in a particular wavelength band. That effect is called limb darkening. A planet crossing a bright central region can block a different fraction of the star’s total light from one crossing a dimmer off-centre region, even when the planet has the same radius.
The chord travelled across the star also matters. A nearly central transit usually follows a longer path across the disk than a near-edge crossing at the same projected speed. A transit grazing the stellar limb may never place the complete planetary disk in front of the star, so a simple square-law depth gives an incomplete description of the occulted area.
NASA’s Exoplanet Archive lists orbital inclination, impact parameter, scaled orbital separation, radius ratio and limb-darkening coefficients as distinct model parameters. The archive describes the squared radius ratio as the transit depth for a non-grazing transit without limb darkening, not as a universal formula for every light curve.
The useful problem is therefore not merely how deep the dip is. Ask where the planet crossed, what brightness profile it occulted, whether the planetary disk fully entered the star and whether the surrounding time-series data support the assumed geometry.
The impact parameter b is the sky-projected distance between the planet centre and stellar centre at conjunction, divided by the stellar radius. A central crossing has b=0. A larger b means the planet passes closer to the apparent stellar limb. Because b is normalised by stellar radius, it has no physical unit.
For a circular Keplerian orbit in a simple transit geometry, b=(a/Rstar)cos i, where a is the orbital semi-major axis, Rstar is the stellar radius and i is the orbital inclination. Under the standard convention, i=90 degrees is edge-on and can produce a central crossing. The formula also assumes the circular-orbit geometry appropriate to this example.
Choose a fictional system with a/Rstar=10 and a planet-to-star radius ratio k=Rp/Rstar=0.10. At i=90 degrees, cos i=0 and b=0. If b=0.60, cos i=0.60/10=0.060, so i≈86.56 degrees. If b=0.80, cos i=0.080 and i≈85.41 degrees.
A difference of only a few degrees in inclination can therefore move the transit chord substantially toward the limb when the orbit is several stellar radii wide. That is why a ‘nearly edge-on’ orbit is not automatically a central transit.
The planet can fully cross the face of the star at mid-transit when b≤1−k in the simple circular projected geometry. With k=0.10, that threshold is b≤0.90. Between b=0.90 and b=1.10 a mid-transit overlap is grazing or partial; at b≥1+k=1.10 the two disks do not overlap at conjunction in the assumed geometry.
The full detailed transit geometry varies with eccentricity, argument of periastron and the projected orbital speed. Do not transfer b=(a/Rstar)cos i unchanged into an eccentric model without including the appropriate orbit-dependent factor and reference geometry.
| Fictional orbit | b | Cosine of i | Inclination i |
|---|---|---|---|
| Central transit | 0.00 | 0.000 | 90.00° |
| Off-centre crossing | 0.60 | 0.060 | ≈86.56° |
| Further toward limb | 0.80 | 0.080 | ≈85.41° |
| Grazing example | 0.95 | 0.095 | ≈84.55° |
An off-centre chord is geometrically shorter across the stellar disk than a central chord. Ignoring the finite size of the planet for one preliminary calculation, the projected chord across the stellar disk is L=2Rstar sqrt(1−b²). This describes the path of the planet’s centre through the stellar disk, not the full first-to-fourth contact duration.
For b=0 the chord length is 2Rstar. At b=0.60 it is 2Rstar sqrt(0.64)=1.60Rstar. At b=0.80 it is 2Rstar sqrt(0.36)=1.20Rstar. In this simplified comparison, the projected path shrinks as b increases.
At a constant projected velocity, a shorter path would generally produce a shorter central crossing. However, the observational transit duration T14 spans from first contact to fourth contact of the planetary and stellar disks. It depends on planetary radius, impact parameter, orbital motion and the adopted geometry, not only the star-only chord just calculated.
For a circular orbit, a/Rstar=10, k=0.10 and orbital period P=10 days, a commonly used duration approximation is T14≈(P/π)arcsin{sqrt[(1+k)²−b²]/[(a/Rstar)sin i]}. This approximation presumes appropriate Keplerian projected geometry and no extreme orbital complications.
In the central case b=0, sin i=1 and the argument is 1.10/10=0.11. The resulting T14 is about 0.351 day, or 8.42 hours. At b=0.80, sin i≈0.9968 and the argument is sqrt(1.21−0.64)/(10×0.9968)≈0.0757, giving about 0.241 day, or 5.78 hours.
Both cases use the same orbital period, stellar radius scale and planetary size. Their different durations arise from geometry. A fitting algorithm therefore needs the light-curve shape and contact timing, not solely a single measured minimum flux point.
| Parameter | Central case b=0 | Off-centre b=0.80 |
|---|---|---|
| Planet ratio k | 0.10 | 0.10 |
| Simplified centre chord | 2.00 Rstar | 1.20 Rstar |
| Approximate T14, P=10 days | ≈8.42 h | ≈5.78 h |
| Interpretation | Longer stellar crossing | Shorter stellar crossing |
For a simple linear limb-darkening law let the stellar specific intensity be I(μ)=Icenter[1−u(1−μ)], where μ=sqrt(1−r²/Rstar²) for a projected point at radial coordinate r. The coefficient u depends on stellar properties and the observing band. This is a deliberately simplified intensity model, not a universal stellar atmosphere prescription.
Choose u=0.60 in a fictional photometric band. At the stellar centre, μ=1 and I=Icenter. At b=0.60, μ=sqrt(1−0.36)=0.80, giving I=Icenter[1−0.60×0.20]=0.88Icenter. At b=0.80, μ=0.60 and I=0.76Icenter.
The disk-averaged stellar intensity for this linear law is Iaverage=Icenter(1−u/3)=0.80Icenter. In the small-planet, non-grazing approximation, a planet blocking a small local patch at mid-transit produces δ≈k²[I(μ)/Iaverage]. This explicitly accounts for the brighter or darker part of the star being occulted.
With k=0.10 and u=0.60, a central crossing produces δ≈0.01×1/0.80=0.0125, or 1.25%. At b=0.60 the model gives 0.01×0.88/0.80=1.10%. At b=0.80 it gives 0.01×0.76/0.80=0.95%.
All three examples use the same true planet-to-star radius ratio. The measured mid-transit depth differs because the planet covers different portions of the limb-darkened star. A naive sqrt(depth) estimate would infer different planetary sizes from identical planets.
Astronomy & Astrophysics publishes the corresponding small-planet expression for a quadratic limb-darkening law, including the disk-average denominator 1−u1/3−u2/6. The conceptual point carries through: the occulted region’s brightness must be compared with the brightness of the entire star.
| Impact parameter | Local intensity / Icenter | Toy mid-transit depth | Naive sqrt(depth) |
|---|---|---|---|
| b=0.00 | 1.00 | 1.25% | ≈0.1118 |
| b=0.60 | 0.88 | 1.10% | ≈0.1049 |
| b=0.80 | 0.76 | 0.95% | ≈0.0975 |
The previous mid-transit approximation treats the occulted patch as locally small and entirely in front of the star. It is not a complete overlapping-circle calculation for a grazing event. Near b=1−k, the planet covers a range of local intensities and part of its disk may extend beyond the apparent stellar boundary.
Return to the fictional k=0.10 system. At b=0.95, the projected centre of the planet is 0.95 stellar radii from the star centre. The outer edge of its disk reaches b+k=1.05, beyond the stellar limb. The planet is therefore only partially overlapped with the star at conjunction.
A simple depth δ=k² no longer equals the occulted area fraction, even for a uniform-brightness star. The overlapping area must be calculated from the geometry of two disks. If the star is limb darkened, the overlap must also be weighted by the spatial brightness distribution.
A grazing event can have a rounded or V-shaped light curve lacking a fully in-transit flat interval. That may create degeneracies among k, b, stellar limb darkening and any third-light contribution. Very precise observations or independent constraints may be needed to distinguish a larger planet grazing the limb from a smaller planet crossing a different chord.
The key caution is to avoid extending a safe calculation outside its regime. A small-planet formula used at b=0.80 may illustrate intensity weighting; the same formula must not be stretched to b=0.95 and presented as a complete prediction of a partially overlapping disk.
At low signal-to-noise or long exposure durations, ingress and egress may be smeared. This can further weaken the geometry information and make the radius-ratio estimate more model dependent.
Limb-darkening coefficients depend on wavelength and the star’s atmosphere. A redder and a bluer observing band may sample different centre-to-limb intensity patterns. That is why a model fitted to a broadband optical light curve may not have the same limb-darkening law as one fitted to an infrared observation.
Planet radius also depends on the physical stellar radius. Even a perfectly fitted k=Rp/Rstar gives a relative size, not a size in kilometres, until Rstar is supplied from an independently supported stellar model or measurement. Errors in that host radius pass into the planet radius estimate.
Unresolved starlight creates a separate bias by diluting the observed depth. The earlier transit dilution and blended starlight guide explains why a neighbouring star can make a dip shallower even when geometry and limb darkening have been modelled correctly.
To investigate a physical radius, a researcher should therefore fit the time series with a suitable stellar limb-darkening model, orbital geometry, exposure integration, contamination model and uncertainty propagation. A single square-root calculation provides a valuable approximate cross-check but should not replace the complete analysis.
A strong analysis can vary plausible limb-darkening priors, fitting assumptions and stellar-radius values to assess sensitivity. If an apparently precise planetary radius changes materially when the stellar intensity profile changes within a supported range, the reported uncertainty should include that dependence.
Do not conflate a transmission-spectrum signal with every band-dependent measurement effect. Stellar heterogeneity, limb darkening, third light and instrumental calibration can also introduce wavelength dependence. A physical atmospheric interpretation needs tests against these alternatives.
What is impact parameter? It is the projected planet-centre-to-star-centre separation at conjunction, expressed in units of stellar radius.
Does b=0.80 mean the planet is 80% the size of the star? No. b describes the location of the transit chord, not the planetary radius. The radius ratio k is a different fitted quantity.
Is the observed transit depth always k²? No. That simple relation assumes a uniform-brightness star and a full non-grazing occultation under suitable conditions. Limb darkening and grazing geometry change the result.
Why do near-central transits often look deeper with limb darkening? They can cover a brighter part of the stellar disk relative to its disk-average brightness.
Can a shallow transit be caused by a small planet or by geometry? Yes. A smaller planet, a dim stellar limb, a grazing overlap or third-light contamination can all contribute to a shallower measured dip.
Does a grazing transit have a flat bottom? It often lacks a full-overlap interval and can be V-shaped, though precise observed shape depends on limb darkening and data quality.
Is the chord length equal to the first-to-fourth-contact duration? No. The planet’s finite disk and projected velocity affect the full duration.
Do the same limb-darkening coefficients apply to every wavelength? No. They depend on the star and bandpass and must be used within an appropriate model.
Can a ten percent radius ratio imply a measured depth above one percent? Yes. A central transit across a limb-darkened star may occult a brighter-than-average patch and exceed k² under the stated approximation.
Can I obtain the planet’s radius without knowing the star’s radius? A transit fit usually yields Rp/Rstar; a physical radius additionally requires the host radius.
The Gold Standard of transit analysis begins with the observational data and a clear model of what blocks what. Geometry determines the path, limb darkening describes the host’s brightness across that path, and the planetary radius helps determine the occulted area. The measured light curve combines those effects.
Recreate the toy example: keep k=0.10 and u=0.60, and compare b=0 with b=0.80. A small-planet approximation predicts mid-transit depths around 1.25% and 0.95% despite the same planet. Explain why the square-root shortcut estimates different k values and why the full light-curve shape helps break that ambiguity.
Continue through The Gold Standard of Astronomy, Transit Dilution and Planet Radius, Vetting an Exoplanet Candidate, Finite Exposure Averaging, and Barycentric Transit Timing.
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Authoritative reading, reviewed 26 October 2026: NASA Exoplanet Archive: EXOFAST Transit Parameter Definitions; NASA Exoplanet Archive: Impact Parameter and Radius Ratio; Astronomy & Astrophysics: Limb Darkening and Inferred Planet Radii; Astronomy & Astrophysics: Impact Parameter in Transmission Spectroscopy.
All numerical radii, inclination values, intensities and durations in this article are fictional. The equations and intensity profile are introduced to show their assumptions and are not fitted measurements of a real exoplanet.