Calculator
Formula used
The equilibrium temperature is found by balancing absorbed stellar power with emitted thermal radiation:
- Teq = [ (1 − A) · S / (ε · σ · F) ]^(1/4)
- S = L / (4πd²) (luminosity mode)
- S = σT★^4 · (R★/d)² (radius–temperature mode)
- Optional eccentricity correction: S̄ = S / √(1 − e²)
A is Bond albedo, ε emissivity, σ Stefan–Boltzmann constant, and F controls redistribution (4, 2, or 1).
How to use this calculator
- Select an input mode that matches your available data.
- Enter either flux, or stellar properties with orbital distance.
- Set albedo, emissivity, and redistribution to match conditions.
- Optionally apply eccentricity correction for average heating.
- Use greenhouse factor to estimate surface warming.
- Press Submit to see results, then export CSV or PDF.
Example data table
| World | Mode | Key inputs | Albedo | Redistribution | Teq (K) | Notes |
|---|---|---|---|---|---|---|
| Earth | Flux | Flux 1361 W/m² | 0.30 | 4 | ≈ 255 | Typical global-average equilibrium temperature. |
| Mars | Flux | Flux 590 W/m² | 0.25 | 4 | ≈ 210 | Lower flux gives colder equilibrium temperature. |
| Hot Jupiter | Luminosity | 1.0 solar, 0.05 AU | 0.10 | 2 | ≈ 1900 | Dayside averaging raises temperature estimate. |
| Exoplanet (cool star) | Radius–Temp | 0.3 R☉, 3400 K, 0.10 AU | 0.35 | 4 | ≈ 270 | Demonstrates direct use of star radius and temperature. |
Planetary equilibrium temperature guide
1) What equilibrium temperature represents
A planet’s equilibrium temperature is the blackbody temperature that would radiate the same power the planet absorbs from its star. It is a first-order climate baseline, useful for comparing worlds before adding atmospheric physics, oceans, clouds, or seasons. The calculator reports both an equilibrium value and an optional surface estimate using a greenhouse multiplier.
2) Energy balance at the top of atmosphere
Incoming stellar energy is reduced by the Bond albedo, the fraction reflected back to space across all wavelengths and angles. Absorbed power scales with incident flux and the planet’s cross-section, while emitted thermal power scales with emissivity and the Stefan–Boltzmann law. Small changes in absorbed flux matter because temperature depends on the fourth root of heating.
3) Typical reference numbers
Near Earth’s orbit the solar flux is about 1361 W/m², and Earth’s Bond albedo is roughly 0.30. With efficient redistribution (F = 4) and emissivity near 1, this produces an equilibrium temperature near 255 K. Earth’s mean surface temperature is about 288 K, illustrating how atmospheric absorption raises the surface above Teq.
4) Heat redistribution choices
The redistribution factor captures how absorbed energy is spread before infrared emission. F = 4 assumes uniform emission over the whole sphere, F = 2 approximates dayside-only averaging, and F = 1 represents the substellar point. For tidally locked planets, the difference between F = 4 and F = 2 can shift Teq by hundreds of kelvin for strongly irradiated systems.
5) Using luminosity or star properties
If you know stellar luminosity and orbital distance, the flux at the planet is computed from the inverse-square law, S = L/(4πd²). When luminosity is not available, you can derive it from a star’s radius and effective temperature. That mode uses S = σT★⁴(R★/d)², which is convenient for many exoplanet catalogs.
6) Eccentric orbits and average heating
Planets on eccentric orbits receive more energy near periapsis than apoapsis. For long-term averages, a useful correction is the mean inverse-square factor, which increases the average flux by 1/√(1−e²). This adjustment helps estimate climate forcing when only semimajor axis and eccentricity are known.
7) Emissivity and the greenhouse multiplier
Emissivity accounts for how efficiently a planet emits infrared radiation relative to a perfect blackbody. Values slightly below 1 can represent spectral windows or high-altitude emission. The greenhouse factor is a practical knob to approximate atmospheric warming; it is not a full radiative-transfer model.
8) Interpreting results for decision making
Use Teq to compare environments across planets, test sensitivity to albedo or distance, and sanity-check published parameters. Use the surface estimate to explore “what-if” scenarios, but treat it as illustrative. Exporting CSV or PDF provides reproducible records for reports, lab notes, or classroom exercises.
FAQs
1) Is equilibrium temperature the same as surface temperature?
No. Equilibrium temperature is a top-of-atmosphere radiative balance estimate. Surface temperature depends on greenhouse gases, clouds, pressure, circulation, and heat storage, which can raise or lower the surface relative to Teq.
2) Which albedo should I use?
Use Bond albedo when possible because it averages reflection over wavelengths and angles. If you only have geometric albedo, treat it as an approximation and test a plausible range to see sensitivity.
3) What does the redistribution factor mean physically?
It represents how much surface area emits the absorbed energy. Uniform global emission uses 4, dayside emission uses 2, and a substellar hotspot uses 1. Real planets fall between these limits.
4) When should I apply the eccentricity correction?
Apply it when you want time-averaged heating over many orbits and you know eccentricity. Avoid it if you are modeling instantaneous conditions at a specific orbital phase or focusing on seasonal extremes.
5) Why does temperature change slowly with flux?
Because radiative emission follows σT⁴. Doubling absorbed flux increases temperature by only the fourth root of two, about 19%. This is why albedo and redistribution tweaks can be compared systematically.
6) Can emissivity be greater than one?
Physical emissivity is bounded by 1. For quick sensitivity checks, values slightly above 1 sometimes appear in simplified fits, but they are not strictly physical. For realistic cases, keep emissivity in the 0.8–1.0 range.
7) What units does the calculator use internally?
All calculations use SI units: watts, meters, and kelvin. You can enter distance in AU, km, or meters, and the tool converts as needed. Output flux is W/m² and temperatures are in kelvin.