Black-body Radiation - Temperature Relation Between A Planet and Its Star

Temperature Relation Between A Planet and Its Star

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 sun
• 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 Stefan–Boltzmann law gives the total power (energy/second) the Sun is emitting:

where

is the Stefan–Boltzmann constant,

is the effective temperature of the Sun, and

is the radius of the Sun.

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:

where

is the radius of the planet and

is the astronomical unit, the distance between the Sun and the planet.

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 of this energy where is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction of the sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

Even though the planet only absorbs as a circular area, it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law

where is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting, 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 of the radiation that a black body would emit where is the average emissivity in the IR range. The power emitted by the planet is then:

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:

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, T_P:

In other words, given the assumptions made, 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.