Transit Timing Calculator
Free Transit timing Calculator for astronomy. Enter variables to compute results with formulas and detailed steps. Enter your values for instant results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Transit Timing Calculator
Calculator
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Formula: T14 = (P/pi) arcsin(sqrt((Rs+Rp)^2 - b^2 Rs^2) / (a sin i))
Worked example — Transit duration: ~3.0 hours | Depth: ~1.5% (14,950 ppm) | Impact parameter: 0.505
Formula
T14 = (P/pi) arcsin(sqrt((Rs+Rp)^2 - b^2 Rs^2) / (a sin i))
Where T14 = total transit duration, P = orbital period, Rs = stellar radius, Rp = planet radius, b = impact parameter = a cos(i)/Rs, a = semi-major axis, i = orbital inclination. Transit depth = (Rp/Rs)^2.
Worked Examples
Example 1: Hot Jupiter Transit (HD 209458 b)
Problem:Calculate transit timing for a hot Jupiter with P = 3.524 days, Rs = 1.16 solar radii, Rp = 1.38 Jupiter radii, a = 0.0475 AU, i = 86.71 degrees.
Solution:Rs = 1.16 * 6.957e8 = 8.07e8 m Rp = 1.38 * 7.149e7 = 9.87e7 m a = 0.0475 * 1.496e11 = 7.106e9 m b = (7.106e9 * cos(86.71)) / 8.07e8 = 0.505 T14 = (3.524*24/pi) * arcsin(sqrt((8.07e8+9.87e7)^2 - (0.505*8.07e8)^2) / (7.106e9*sin(86.71))) T14 = 26.93 * arcsin(0.1089) = 26.93 * 0.1096 = 2.95 hours Depth = (9.87e7/8.07e8)^2 = 0.01495 = 1.495%
Result:Transit duration: ~3.0 hours | Depth: ~1.5% (14,950 ppm) | Impact parameter: 0.505
Example 2: Earth-like Planet Transit
Problem:Calculate transit parameters for an Earth-like planet: P = 365.25 days, Rs = 1.0 solar radii, Rp = 0.0892 Jupiter radii (1 Earth radius), a = 1.0 AU, i = 89.99 degrees.
Solution:Rs = 6.957e8 m, Rp = 6.371e6 m, a = 1.496e11 m b = (1.496e11 * cos(89.99)) / 6.957e8 = 0.0375 Depth = (6.371e6/6.957e8)^2 = 8.39e-5 = 0.00839% = 83.9 ppm T14 = (365.25*24/pi) * arcsin(sqrt((6.957e8+6.371e6)^2 - (0.0375*6.957e8)^2) / (1.496e11*sin(89.99))) T14 approximately 13.1 hours Transit probability = Rs/a = 6.957e8/1.496e11 = 0.465%
Result:Transit duration: ~13.1 hours | Depth: ~84 ppm | Transit probability: 0.47%
Frequently Asked Questions
What is an exoplanet transit and how is it detected?
An exoplanet transit occurs when a planet passes directly between its host star and the observer, causing a small, periodic dip in the observed brightness of the star. This is the most prolific method for discovering exoplanets, responsible for the vast majority of confirmed planets found by missions like Kepler and TESS. The depth of the brightness dip reveals the planet size relative to its star, while the period between consecutive transits gives the orbital period. The transit method works best for large planets on short-period orbits around small stars, where the fractional brightness decrease is greatest and transits occur frequently enough to confirm the signal.
How is transit duration calculated?
Transit duration depends on the orbital period, the stellar radius, the planet radius, the semi-major axis, and the orbital inclination. The total transit duration T14 (from first to fourth contact) is given by T14 = (P/pi) * arcsin(sqrt((Rs+Rp)^2 - b^2*Rs^2) / (a*sin(i))), where P is the orbital period, Rs is stellar radius, Rp is planet radius, a is the semi-major axis, i is the inclination, and b is the impact parameter. For eccentric orbits, this is multiplied by sqrt(1-e^2). Typical hot Jupiter transits last 2-3 hours, while Earth-like planets around Sun-like stars transit for about 13 hours.
What is the impact parameter in transit observations?
The impact parameter b describes how centrally the planet crosses the stellar disk, defined as b = (a*cos(i))/Rs, where a is the semi-major axis, i is the orbital inclination, and Rs is the stellar radius. When b = 0, the planet crosses the exact center of the star (an equatorial transit), producing the longest possible transit duration and a symmetric, flat-bottomed light curve. When b approaches 1, the planet grazes the edge of the star, producing a shorter, V-shaped transit. For b greater than 1+Rp/Rs, no transit occurs at all. The impact parameter is a key observable that helps constrain the orbital inclination independently of other measurements.
What is transit depth and what does it reveal?
Transit depth is the fractional decrease in stellar brightness during a transit, equal to the square of the planet-to-star radius ratio: delta = (Rp/Rs)^2. For a Jupiter-sized planet transiting a Sun-sized star, the depth is about 1% (10,000 ppm). For an Earth-sized planet around a Sun-like star, it is only 0.008% (84 ppm), which is extremely challenging to detect from the ground. The transit depth directly measures the planet size relative to the star, making it one of the most fundamental observables in exoplanet science. When combined with radial velocity mass measurements, the planet density and bulk composition can be determined.
What are Transit Timing Variations (TTVs)?
Transit Timing Variations (TTVs) are deviations from strictly periodic transit times, caused by gravitational interactions between planets in a multi-planet system. If a transiting planet has a companion planet, the gravitational tug causes the transiting planet to arrive slightly early or late for its transits, with variations typically ranging from seconds to tens of minutes. TTVs are particularly large near mean motion resonances, where orbital periods are related by small integer ratios like 2:1 or 3:2. This technique has been used to confirm and characterize hundreds of exoplanets, including measuring planet masses without radial velocity data, and even discovering non-transiting planets through their gravitational influence.
What is the geometric probability of observing a transit?
The geometric transit probability is approximately Rs/a, where Rs is the stellar radius and a is the semi-major axis of the planet orbit. This represents the fraction of randomly oriented orbital planes that would produce observable transits from our perspective. For a hot Jupiter at 0.05 AU around a Sun-like star, the probability is about 10%, which is relatively high. For an Earth-like planet at 1 AU, the probability drops to only about 0.5%. This means that for every transiting Earth analog discovered, roughly 200 similar planets exist in non-transiting geometries. Transit surveys must therefore monitor tens of thousands of stars to discover a significant number of planets at various orbital distances.
How does eccentricity affect transit timing and duration?
Orbital eccentricity affects transit observations in several important ways. First, the transit duration is modified by a factor of sqrt(1-e^2), so eccentric orbits generally have shorter transits than circular orbits of the same period when the planet transits near apastron. Second, the transit and occultation (secondary eclipse) are no longer separated by exactly half the orbital period, and this timing offset directly measures e*cos(omega), where omega is the argument of periastron. Third, the orbital velocity varies around the orbit, so transits occurring near periastron are shorter than those near apastron. These effects allow eccentricity to be constrained purely from photometric transit observations without requiring radial velocity data.
What is the difference between T14 and T23 in transit photometry?
T14 is the total transit duration measured from first contact (when the planet disk first touches the stellar disk) to fourth contact (when the planet completely exits the stellar disk). T23 is the flat-bottom duration from second contact (planet fully inside the stellar disk) to third contact (planet begins to exit). The difference (T14 - T23)/2 gives the ingress or egress duration, which is the time for the planet to cross the stellar limb. The ratio T23/T14 depends on the impact parameter and the radius ratio, providing an independent constraint on transit geometry. For grazing transits with large impact parameters, T23 approaches zero and the light curve becomes V-shaped rather than flat-bottomed.
How do space telescopes like Kepler and TESS detect transits?
Space telescopes detect transits by continuously monitoring the brightness of thousands to millions of stars with extreme photometric precision, typically better than 100 parts per million. Kepler stared at a single field of about 150,000 stars for four years, achieving precision sufficient to detect Earth-sized planets around Sun-like stars. TESS surveys nearly the entire sky in 27-day sectors, focusing on nearby bright stars where follow-up observations are easier. Both missions use CCD detectors to measure stellar brightness every few minutes, and sophisticated algorithms remove instrumental systematics and stellar variability to isolate the transit signal. Machine learning techniques are increasingly used to identify transit candidates in the vast datasets these missions produce.
What follow-up observations confirm a transit detection?
Confirming a transit detection requires multiple lines of evidence to rule out false positives. Radial velocity measurements detect the gravitational wobble of the host star, confirming the companion is planetary mass rather than a stellar binary. High-resolution imaging checks for background eclipsing binaries blended with the target star. Multi-color photometry verifies the transit depth is achromatic (wavelength-independent), as eclipsing binaries produce color-dependent depth changes. Statistical validation techniques use the multiplicity of transit signals in the same system and the low false positive probability in certain galactic environments. For the most interesting planets, atmospheric characterization via transmission spectroscopy during transit reveals the composition of the planet atmosphere.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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