Planetary Surface Temperature Calculator

This planetary surface temperature calculator estimates how hot the surface of a planet is, whether it's a planet in our solar system or an exoplanet orbiting a distant star. It uses a simple energy balance climate model and assumes that the stellar radiation absorbed by a planet is equal to the thermal energy radiated out into space.

Please enter the luminosity of the star, the distance of the planet from its orbit, the albedo (reflectivity) of the planet and the thickness of the atmosphere. The tool will give you the equilibrium temperature of a planet with no atmosphere and an upper surface temperature if there is a greenhouse atmosphere. If you leave any field blank then the tool can calculate that value to answer questions such as "how far away from its star does a planet need to be to have a particular temperature?"

What is a planet's equilibrium temperature?

A planet continues to absorb heat until the energy it absorbs from its star is equal to the energy it radiates as infrared light. This equilibrium state determines the temperature.

A planet receives starlight on a circular surface and radiates heat from its entire surface. In this energy balance equation, the radius is considered on both sides and cancels out, so that the final temperature ultimately depends only on the incoming stellar radiation and the reflectivity of the planet, not its size.

Here's how to use this calculator.

First we start with four input fields describing stars and planets. The brightness of a star is given as a value relative to the sun, where the sun has a value of 1, a dark red dwarf would have a value of 0.01 and a bright star could have a value of 10 or more. The distance is given by the orbital radius, with Earth being at a distance of 1 AU (Astronomical Unit). Albedo is the percentage of light that a planet reflects, ranging from 0 which would be a completely dark planet to values close to 1 for a bright snow-covered planet. Optical thickness gives the strength with which an atmosphere traps heat in the form of infrared radiation; 0 means there is no atmosphere.

If you leave the surface temperature blank, the calculator will fill in the result. To run the model in reverse, enter a target temperature and leave one of the other four fields blank. If you leave the distance field blank, it will give an answer to the classic question: At what orbital distance would this planet have that temperature? This is at the heart of the concept of the habitable zone.

Formula for energy balance

If the star's luminosity is given as a value relative to that of the Sun, and its distance in Astronomical Units, then the flux received by the planet will be equal to the Solar constant, multiplied by the luminosity, and reduced by the square of the distance. The equilibrium temperature without an atmosphere can then be calculated from the Stefan-Boltzmann law.

S sub 0 is the solar constant and it's about 1361 watts per square meter. L is the luminosity relative to the sun, a is the albedo and d is the distance in AU. Sigma is the Stefan-Boltzmann constant and that has the value of 5.67 times 10 to the minus 8. If there's a greenhouse atmosphere then the surface temperature will be above this, where the enhancement is determined by the optical depth of infrared radiation, which is denoted by the Greek letter tau; so the larger tau is, the greater the enhancement.

For example, if the luminosity is 1, the distance is 1 AU, the albedo is 0.29 and the optical thickness is 0.78, then the equilibrium temperature without an atmosphere would be about 255 K, or about minus 18 degrees Celsius. This temperature is well below freezing and corresponds to the approximate temperature of Earth if greenhouse gases were removed. If an atmosphere is added, the surface temperature rises to about 287 K, which approximates the actual global average temperature of 288 K, or 15 degrees Celsius. This difference of 33 degrees is the greenhouse effect that keeps our oceans liquid.

Examples of calculations in the solar system:

The same model reproduces the temperature of rocky planets without atmospheres very well, and the difference between predicted values and actual surface temperatures directly shows the greenhouse effect on each planet.

Mars and Mercury have almost no atmosphere so the equilibrium temperature is nearly the same as the surface temperature. Earth has a moderate warming due to a 33 degree greenhouse effect. Venus is an extreme example, with a very thick carbon dioxide atmosphere causing runaway greenhouse effects that cannot be captured by a simple model; thus a very large optical depth is required to approach its actual temperature of 737 K.

Optical thickness and greenhouse effect.

The optical depth for infrared radiation indicates how transparent the atmosphere is to heat radiated by a planet. Starlight heats the surface, which then radiates infrared light. Greenhouse gases absorb some of this outgoing energy and re-radiate it back down. The larger the optical depth, the more heat is trapped, and the warmer the surface becomes.

Earth's atmosphere has a moderate optical thickness of about 0.8. Venus has an intolerably thick carbon dioxide atmosphere with an effective optical thickness of several tens or hundreds. This explains why Venus is hotter than Mercury, even though it is farther from the Sun. Setting the optical thickness to zero in a calculation tool removes the atmosphere and allows you to find the equilibrium temperature of a bare planet.

Finding the Habitable Zone

The habitable zone is the range of orbital distances in which liquid water can exist on a planet's surface. It roughly extends from the freezing point (273 K) to the boiling point (373 K). As temperature decreases with increasing distance, there is such a band-like region for each star. For red dwarfs, the habitable zone is closer to the star, while it is farther away for brighter stars.

To find the habitable zone, enter the luminosity of the star and the albedo and optical depth of the planet, input a target temperature (e.g. 288 K) and leave the distance field blank. The calculator will then return the orbital distance that produces this temperature. A graph shows distances compared to temperature curves with freezing point and boiling point lines for visualizing this concept.

Model limitations

This is not a full climate simulation but an educational model. Planets are treated as one object with the same temperature everywhere, ignoring day and night, seasons, oceans, and heat transport by winds and ocean currents. It cannot be applied to planets close to a star that have tidal locking so that one side always faces the star. The greenhouse effect used here is very simplified on purpose and underestimates the effects of a thick atmosphere like Venus has. These numbers are rough estimates to give an overall picture, not meant as accurate predictions.