Planet Equilibrium Temperature Calculator

Planet Equilibrium Temperature

The equilibrium temperature is the temperature a planet would have if it were a uniformly-emitting blackbody balancing absorbed starlight against thermal re-emission. It is the simplest first-order estimate of an exoplanet’s surface conditions and a key parameter in habitable-zone studies.

Formula

T_eq = T_star × √(R_star / 2a) × (1 − A)^(1/4)

where:

• T_star = star’s effective surface temperature (K)
• R_star = star’s radius
• a = orbital semi-major axis (same units as R_star)
• A = bond albedo (fraction of light reflected, 0 to 1)

This calculator works in solar units: stellar luminosity L (in L_sun) and orbital distance a (in AU). The equivalent expression is:

T_eq = 278.5 × (L × (1 − A) / a²)^(1/4) K

The constant 278.5 K comes from Earth at 1 AU around a 1 L_sun star with A = 0.

Worked Examples

For Earth: T_eq = 278.5 × (1 × 0.69 / 1)^(1/4) ≈ 254 K (-19°C). The actual surface average is 288 K, and the 33 K difference is due to the greenhouse effect.

The Greenhouse Adjustment

A real planetary surface is warmer than T_eq because the atmosphere traps outgoing infrared. Mars (thin CO₂) gains about 5 K from greenhouse warming. Earth (water vapor + CO₂) gains about 33 K. Venus (massive CO₂) gains about 500 K, a runaway greenhouse.

Enter one of those figures in the optional greenhouse box and the calculator adds it to T_eq to give a surface estimate. That step is where the honesty ends and the guessing begins: greenhouse warming depends on atmospheric mass and composition, neither of which appears anywhere in the equilibrium formula. Venus and Earth sit two-thirds of an AU apart and differ by 500 K at the surface, and T_eq alone predicts Venus to be the colder of the two because its clouds are so reflective. Any exoplanet paper quoting a T_eq is quoting the number before this correction, not after it.

Habitable Zone Definition

A common simple criterion: T_eq must allow liquid water with reasonable greenhouse warming. Using roughly 190 K < T_eq < 270 K with Earth’s albedo, this formula gives a generous 0.88 to 1.79 AU around the Sun. Full climate models narrow the real habitable zone to roughly 0.95 to 1.4 AU, so treat the T_eq band as a first cut. This calculator lets you find that range for any star and albedo.

Caveats

The model assumes a fast-rotating planet with uniform temperature. Slowly-rotating tide-locked worlds have huge day-night temperature gradients that make global T_eq misleading. Also, the bond albedo can change with phase: ice ages, cloud cover, and atmospheric chemistry all shift A and thus T_eq.

Quick Reference Albedos