Exoplanet Discovery Calculator: Transit, Radial Velocity, Habitable Zone
This calculator covers five methods astronomers use to find and characterize exoplanets: the transit method (planet radius from brightness dip), radial velocity (stellar wobble speed), Kepler orbital period, equilibrium surface temperature, and habitable zone boundaries. Choose a detection mode, enter the star and planet parameters, and get instant results with the step-by-step math shown. Defaults are set to a Jupiter-sized planet around a Sun-like star so a sensible result appears on first load.
Formula
Worked example
Transit example: a 1% dip around a 1.0 R☉ star gives R_p = 1.0 × 695,700 km × √0.01 = 6,957 km ≈ 1.09 R⊕. That is roughly an Earth-sized planet. In ppm: 0.01 × 10^6 = 10,000 ppm, easily detectable by Kepler.
How the transit method works
When a planet passes directly between its host star and our line of sight, it blocks a fraction of the star's light. That fractional dimming is the transit depth: δ = (R_p / R★)². Measuring δ with a photometer and knowing R★ from spectroscopy immediately gives the planet radius. The transit method produced the majority of confirmed exoplanet discoveries - more than 3,800 planets detected by Kepler and TESS alone as of 2026. It works best for large planets close to their stars (where transits are more likely and more frequent) and requires the orbital plane to be nearly edge-on from our vantage point.
The transit depth ranges from about 10 ppm for a 1.3 R⊕ super-Earth around a Sun-like star to roughly 10,000-15,000 ppm (1-1.5%) for a hot Jupiter. Kepler could detect depths as low as ~20 ppm in quiet stars; TESS, with its wider field and shorter baselines, typically needs at least 1,000 ppm to confirm a signal.
Radial velocity: measuring the stellar wobble
A planet does not orbit its star; both objects orbit their common center of mass. The star wobbles, and that wobble produces a periodic Doppler shift in its spectral lines. The radial velocity semi-amplitude K quantifies the maximum speed of that wobble:
K = (2πG/P)^(1/3) × (Mp sin i) / (M★ + Mp)^(2/3) / √(1 - e²)
K depends on the planet mass, orbital period, eccentricity, and the inclination angle i between the orbital plane and the sky. Because i is unknown from RV alone, radial velocity gives only a minimum planet mass: M_p sin i. Earth induces a wobble of only 0.09 m/s in the Sun; Jupiter causes 12.5 m/s. Modern instruments like HARPS on the 3.6 m La Silla telescope and ESPRESSO on the VLT can routinely measure velocities below 1 m/s, making rocky-planet detection possible around quiet stars.
Kepler's Third Law and orbital periods
Kepler's Third Law (1619) states that the square of an orbital period is proportional to the cube of the semi-major axis: P² ∝ a³ / M★. In SI units, P = 2π × √(a³ / GM★). This is one of the most powerful tools in exoplanet science: observing multiple transits gives the period P directly; knowing the stellar mass from spectroscopy then gives the orbital distance a, which sets the planet's irradiation environment. Planets with periods of just 1-5 days (hot Jupiters) orbit at roughly 0.03-0.05 AU, while an Earth-period planet at 365 days orbits at 1 AU. Future missions like PLATO (ESA, launch 2026+) are designed to detect year-long periods in Solar-type stars.
Equilibrium temperature and the habitable zone
A planet's equilibrium temperature is the blackbody temperature it would reach if it absorbed and re-emitted all incoming stellar radiation, with no greenhouse effect or internal heat:
T_eq = T★ × (R★ / 2a)^(1/2) × (1 - A)^(1/4)
where A is the Bond albedo (Earth ≈ 0.30, Venus ≈ 0.76). Earth's equilibrium temperature is about 255 K; the greenhouse effect adds ~33 K to reach the observed mean of ~288 K. The habitable zone (HZ) is the annular region around a star where T_eq could support liquid surface water on a rocky planet with a CO₂-rich atmosphere. The conservative HZ (Kopparapu 2013) spans roughly 0.95-1.67 AU for the Sun (inner: runaway greenhouse; outer: maximum greenhouse). For low-luminosity red dwarfs the HZ shrinks to 0.1-0.3 AU, putting habitable planets in range of ground-based RV and JWST transmission spectroscopy.
Planet classification by radius
| Radius range (R⊕) | Planet class | Likely composition |
|---|---|---|
| 0 - 1.5 | Rocky / terrestrial | Iron, silicate rock (Earth-like) |
| 1.5 - 3.5 | Super-Earth / mini-Neptune | Rock + thick H/He envelope or deep ocean |
| 3.5 - 10 | Neptune-class | Gas/ice dominant, rocky core |
| 10 - 15 | Saturn-class | Hydrogen-helium gas giant |
| > 15 | Jupiter-class or brown dwarf | Fully gaseous, possible companion star |
Standard size categories based on transit-derived radius from Fulton et al. (2017) and NASA Exoplanet Archive conventions.
Frequently asked questions
What is transit depth and why is it given in ppm?
Transit depth is the fractional decrease in a star's brightness when a planet crosses its disk. It equals (R_p / R★)². Parts per million (ppm) is used because space telescopes measure stellar brightness to extraordinary precision and the dips are tiny: a Jupiter-sized planet blocks about 10,000 ppm (1%) of Solar light, while an Earth-sized planet blocks only about 84 ppm. Expressing depth in ppm avoids leading zeros and makes instrument sensitivity comparisons straightforward.
Why does radial velocity give only a minimum planet mass?
The radial velocity signal depends on Mp sin i, where i is the inclination of the orbit. Only if i = 90° (edge-on) does sin i = 1 and the measured mass equal the true mass. For any other inclination the true planet mass is larger: M_true = M_RV / sin i. Without an independent inclination measurement (e.g., from a transit), we can only report the lower bound M_p sin i.
What is the difference between the transit and radial velocity methods?
Transit photometry detects the drop in stellar brightness when a planet passes in front of its star, giving planet radius but only a mass lower bound. Radial velocity spectroscopy measures the Doppler wobble of the star caused by the planet's gravity, giving a minimum planet mass. Used together, they yield both radius and true mass, which gives bulk density and hints at internal structure (rocky, icy, or gaseous). Most confirmed exoplanets from Kepler and TESS are confirmed via follow-up RV.
How do astronomers use Kepler's Third Law to find orbital distances?
Kepler's Third Law links the orbital period P (measured directly from repeated transits or RV cycles) to the semi-major axis a via a³ = GM★ P² / (4π²). Stellar mass M★ is determined from spectroscopy. Plugging in P (observed) and M★ (spectroscopic) solves for a, which sets the orbital distance and therefore how much stellar energy the planet receives - the fundamental input to habitability calculations.
What is an equilibrium temperature and does it predict whether a planet is habitable?
Equilibrium temperature is the theoretical blackbody temperature assuming the planet absorbs and instantly re-emits all incoming starlight. It is a first estimate of thermal environment, not a surface temperature. Atmospheric greenhouse effects, tidal heating, and internal radioactivity all raise the real surface temperature above T_eq. Earth's T_eq is 255 K but its average surface is 288 K. A planet with T_eq of 260 K could be habitable with a CO₂ greenhouse effect, or frozen if it has none - T_eq alone does not determine habitability.
What are the limits of the habitable zone concept?
The habitable zone assumes a rocky planet with an Earth-like water-rich, CO₂-buffered atmosphere. Planets with oceans of different depth, different atmospheric compositions (e.g., H₂ can push the HZ outward), subsurface oceans (Europa), or extreme geothermal activity (Io-like) can support liquid water well outside the classical HZ. The HZ is a useful observational target, not a strict boundary on life.
Can I use this calculator to verify real exoplanet data?
Yes. The NASA Exoplanet Archive provides transit depth, stellar radius, orbital period, stellar mass, and stellar luminosity for thousands of confirmed systems. Enter those values here and compare the derived planet radius, RV amplitude, orbital period, or equilibrium temperature against the archive values. Small differences arise from limb darkening, stellar model uncertainties, and eccentricity effects not captured in simplified formulas.