Hong Life CalculatorsUnderstanding the planet equilibrium temperature calculator — concept and principle
The equilibrium temperature is the temperature a planet would have if the sunlight it absorbs were exactly balanced by the heat it radiates, with no greenhouse effect. This calculator works it out from the star's luminosity, the planet's distance and its albedo (reflectivity), and also gives the stellar flux at that distance. It is a standard first estimate for planets in the Solar System and for exoplanets.
How it is calculated
- Stellar flux: S = 1361 × L ÷ d² W/m², with L in solar luminosities and d in AU. 1361 W/m² is the solar flux at 1 AU.
- Equilibrium temperature: T = [S × (1 − A) ÷ (4σ)]^(1/4), with σ = 5.670374419 × 10⁻⁸ W m⁻² K⁻⁴ and A the Bond albedo (0 to 1).
- °C = T − 273.15.
The principle
A planet intercepts sunlight over its cross-section, πR², and absorbs the fraction (1 − A). It radiates as a blackbody over its whole surface, 4πR², at σT⁴. Setting absorbed power equal to emitted power, S(1 − A)πR² = 4πR²σT⁴, the radius cancels and T = [S(1 − A)/(4σ)]^(1/4). The factor 4 assumes heat is spread evenly over the whole sphere, which suits a fast-rotating planet with an atmosphere. Flux falls with the square of distance, so T falls with √d: doubling the distance lowers T by a factor of √2.
Worked example
- Defaults (Earth): L = 1 L☉, d = 1 AU, A = 0.3.
- Flux S = 1361 × 1 ÷ 1² = 1,361 W/m².
- Absorbed per square metre of surface: 1361 × 0.7 ÷ 4 = 238.175 W/m².
- T⁴ = 238.175 ÷ 5.670374419e-8 = 4.2003 × 10⁹; T = 254.5781401 K.
- In Celsius: 254.578 − 273.15 = −18.57185988 °C.
| Distance (AU) | Equilibrium temperature (K) |
|---|---|
| 0.25 | 509.1562802 |
| 0.5 | 360.0278584 |
| 0.75 | 293.9615155 |
| 1 | 254.5781401 |
| 1.25 | 227.7016108 |
| 1.5 | 207.862181 |
| 2 | 180.0139292 |
How to read the result
Earth's actual average surface temperature is about 288 K (15 °C), roughly 33 K warmer than the equilibrium value; that gap is the natural greenhouse effect. Venus, with an albedo near 0.76, has an equilibrium temperature of only about 230 K, yet its thick carbon dioxide atmosphere makes the surface around 737 K. So T_eq describes the radiating balance, not what a thermometer on the ground would read.
Common mistakes
- Using visual (geometric) albedo instead of Bond albedo.
- Forgetting the factor 4, which gives a temperature about 41% too high.
- Entering distance in km or luminosity in watts.
Frequently asked questions
What if a planet is tidally locked?
If heat is not shared with the night side, the dayside temperature is often estimated with a factor 2 instead of 4, giving about 19% higher T.
Why does albedo matter so much?
It sets how much light is absorbed; raising Earth's albedo from 0.3 to 0.4 would lower T_eq by about 10 K.
What is the 1361 W/m² figure?
It is the nominal total solar irradiance at 1 AU adopted by the IAU, sometimes still called the solar constant.
Sources: IAU 2015 Resolution B3 on nominal solar and planetary values · Pierrehumbert R.T., Principles of Planetary Climate, Cambridge University Press, 2010 · CODATA 2018 recommended values of the fundamental physical constants (NIST)