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Exoplanet transit: light curve, depth and durations

Light curve of an exoplanet transit: depth, T14 and T23 durations and shape of the floor, with limb darkening.

System

The geometry of the transit
Rp/Rs, strictly between 0 and 1. Example: 0.12086 for HD 209458b.
In stellar radii; grazing beyond 1 − k. cos i = b / (a/Rs).
Must exceed 1 + k. Example: 8.76 for HD 209458b.
dExample: 3.52474859 d for HD 209458b. Circular orbit assumed.
The star
I(μ) = 1 − u₁(1−μ) − u₂(1−μ)²; u₁ = u₂ = 0 gives a uniform disk. Values per band and stellar type: Claret & Bloemen (2011).
Domain: u₁ > 0, u₁ + u₂ < 1, u₁ + 2u₂ > 0 (Kipping 2013).
The observed time
hCounted from mid-transit, negative before.

Transit

the sky and the record
k 0.121 · b 0.503 · a/Rs 8.76t = 0 ht₁t₂t₃t₄b = 0.503transit window only, P = 3.52 d−2−10120.9850.990.9951h from mid-transitrelative fluxT14 = 3.09 ht₁t₂t₃t₄depth 1.46%T23 = 2.22 hF = 0.985 393
t = 0 h

Playback lasts twelve screen seconds for a 4.0186 h window: 0.33488 h of transit per second.

Projected separation
z = (a/Rs)·√(sin²φ + cos²i·cos²φ) = 0.50273
Flux, closed form
F = 1 − λ(k, z) = 0.985 393
Winn’s durations
T14 = (P/π)·arcsin[√((1+k)² − b²) / ((a/Rs)·sin i)] = 3.0912 h

the geometry and the curvethe four contacts, t₁ to t₄the depth δ, the answer

Result

continuous computation
The curve
δ1.4607% of the fluxbasin with a flat bottom
Total transit duration
3.0912 h
Flat-bottom duration
2.2231 h
Ingress, then egress duration
26.045 min
Flux at the bottom of the transit
0.985 393
At time t
Relative flux
0.985 393
Projected separation
0.50273 Rs

At t = 0 h, the planet is entirely in front of the star.

What the geometry fixes
Derived inclination
86.71°
Uniform-disk depth
1.4607%
Share of the orbit spent in transit
3.6542%

Calculation, assumptions and sources

Provisional · Physical model · 4 sources · reviewed on 26 August 2026

OverviewThe radius ratio k, the impact parameter b, the scaled semi-major axis a/Rs, the period and two limb-darkening coefficients give the transit light curve, its depth and its T14 and T23 durations.

The radius ratio k, the impact parameter b, the scaled semi-major axis a/Rs, the period and two limb-darkening coefficients give the transit light curve, its depth and its T14 and T23 durations.

Method & formulasz(t) = (a/Rs) √(sin²(2πt/P) + cos²i · cos²(2πt/P))
z(t) = (a/Rs) √(sin²(2πt/P) + cos²i · cos²(2πt/P))b = (a/Rs) cos i, hence z(0) = bF = 1 − λ(p, z), λ = occulted area / disk area (uniform disk)I(μ) = 1 − u₁(1 − μ) − u₂(1 − μ)², μ = cos θT14 = (P/π) arcsin[ √((1+k)² − b²) / ((a/Rs) sin i) ]T23 = (P/π) arcsin[ √((1−k)² − b²) / ((a/Rs) sin i) ]

The projected separation z, the distance between the centres of the star and the planet in stellar radii, is b at mid-transit, 1 − k at the second and third contacts and 1 + k at the first and fourth. For a uniform disk, the flux is the closed form of Mandel & Agol, a ratio of areas of two overlapping disks; with limb darkening, it is integrated by radial quadrature.

Radius ratio (k = Rp/Rs)
the radius of the planet relative to that of the star.
Impact parameter (b)
the distance from the stellar centre to the chord followed by the planet, in stellar radii: b = (a/Rs) cos i.
T14 and T23
durations from the first to the fourth contact (T14) and from the second to the third (T23, planetary disk entirely in front of the star). The ingress and egress durations are (T14 − T23)/2.
Limb darkening
the decrease of intensity from the centre of the stellar disk toward the limb, where one sees higher, cooler layers of the atmosphere.
Grazing transit
a transit where b exceeds 1 − k: the planetary disk does not move fully onto the star. T23 does not exist, nor does the flat bottom, and the curve makes a V.
AssumptionsCircular orbit: no eccentricity, no precession, no transit-timing variations.
  • Circular orbit: no eccentricity, no precession, no transit-timing variations. An eccentric orbit changes the durations, by up to a factor (1 − e²)^(1/2)/(1 ± e sin ω) on T14.
  • Spherical star with no spots or pulsations, opaque planet with no atmosphere or rings; the depth does not depend on wavelength.
  • Quadratic limb darkening integrated by radial quadrature, not by the elliptic-integral closed form of Mandel & Agol. Departure from the uniform-disk closed form: 5.1 × 10⁻⁸ at k = 0.12086, 1.4 × 10⁻⁷ at most over k ∈ [0.05, 0.2].
  • u₁ and u₂ are user inputs, within the physical triangle of Kipping (2013); the tables of Claret & Bloemen (2011) are not built in.
Validity domain2 validity limits; the calculation refuses values outside 2 of them

  • 0 ≤ k ≤ 1physical · refusal

    The radius ratio k = Rp/Rs lies strictly between 0 and 1: at k ≥ 1 the occulter is larger than the star.

    Instead · A gas giant in front of a dwarf has k < 0.3. Beyond 1 the system is an eclipsing binary, outside this model.

  • b ≥ 0physical · refusal

    The impact parameter is counted from the centre of the star: b ≥ 0. Beyond 1 + k the two disks do not overlap and there is no transit.

    Instead · Between 1 − k and 1 + k the transit is grazing and is still computed.

Depth and k²For a uniform disk, the depth is k² whatever b, as long as the planet moves entirely in front of the star.

For a uniform disk, the depth is k² whatever b, as long as the planet moves entirely in front of the star. With limb darkening, it exceeds k² when the planet crosses the centre, the brightest region, and decreases near the darker limb. The planetary radius follows from the measured depth only with a limb-darkening model.

Example: HD 209458bElements of Torres, Winn & Holman (2008) for HD 209458b, the example values of the form: P = 3.52474859 d, k = 0.12086, a/Rs = 8.76 and i = 86.71°, hence b = (a/Rs) cos i = 0.5027.

Elements of Torres, Winn & Holman (2008) for HD 209458b, the example values of the form: P = 3.52474859 d, k = 0.12086, a/Rs = 8.76 and i = 86.71°, hence b = (a/Rs) cos i = 0.5027. The model gives T14 = 3.091 h and a depth k² = 1.4607%. The published duration is about 3.07 h; the 0.7% gap comes from the circular model.

SourcesMandel, K. & Agol, E. · Winn, J. N. · Torres, G., Winn, J. N. & Holman, M. J. + 1 more

  1. Mandel, K. & Agol, E., Analytic Light Curves for Planetary Transit Searches, 2002.

    Location: ApJ 580, L171, eq. 1: occulted flux of a uniform source · accessed 2026-08-26 · DOI 10.1086/345520

  2. Winn, J. N., Transits and Occultations, 2010.

    Location: arXiv:1001.2010, eq. 3-7 (projected separation), eq. 14-15 (T14, T23) · accessed 2026-08-26

  3. Torres, G., Winn, J. N. & Holman, M. J., Improved Parameters for Extrasolar Transiting Planets, 2008.

    Location: ApJ 677, 1324: HD 209458b, P = 3.52474859 d, Rp/Rs = 0.12086, a/Rs = 8.76, i = 86.71 deg · accessed 2026-08-26 · DOI 10.1086/529429

  4. Kipping, D. M., Efficient, uninformative sampling of limb darkening coefficients for two-parameter laws, 2013.

    Location: MNRAS 435, 2152, section 2: physical domain of the quadratic law · accessed 2026-08-26 · DOI 10.1093/mnras/stt1435

Scientific validationProvisional, reviewed on 26 August 2026

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 1 · reviewed on 26 August 2026

Revision log

  1. revision 1 · 2026-08-26 · scope change

    Tool creation: transit on a circular orbit, uniform disk in closed form (Mandel & Agol 2002, eq. 1), quadratic limb darkening by radial quadrature validated to 5.1 × 10⁻⁸ against that closed form, durations T14 and T23 (Winn 2010, eq. 14-15), grazing transit.

    First published result of the tool, with no earlier value to compare against.