Temprature of a planet

Description

Black-body radiation is the thermal electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body). It has a specific spectrum and intensity that depends only on the body’s temperature, which is assumed for the sake of calculations and theory to be uniform and constant.

The black-body law may be used to estimate the temperature of a planet orbiting the Sun. The temperature of a planet depends on several factors: Incident radiation from its star Emitted radiation of the planet, e.g., Earth’s infrared glow The albedo effect causing a fraction of light to be reflected by the planet The greenhouse effect for planets with an atmosphere Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due by cooling.

The analysis only considers the Sun’s heat for a planet in a Solar System.

The total power (energy/second) the Sun is emitting is given by the Stefan–Boltzmann law.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is given by this equation.

Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction α of this energy where α is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction 1 − α of the Sun’s light, and reflects the rest.

This temperature TE, calculated for the case of the planet acting as a black body by setting Pemt bb = Pabs is known as the effective temperature. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits e of the radiation that a black body would emit where e is the average emissivity in the IR range.

For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it.

This equation shows that the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.

Related formulas

Variables

TPTemprature of the planet (K)
TSTemprature of the sun (K)
RSradius of the sun (m)
αalbedo (dimensionless)
ϵaverage emissivity in the IR range (dimensionless)
Ddistance between the planet and the sun (m)