Planetary equilibrium temperature

The solar flux absorbed by the planet from the star is equal to the flux emitted by the planet:[4][5][6]

${\displaystyle {F}_{\rm {abs}}={F}_{\rm {emit}}}$

Assuming a fraction of the incident sunlight is reflected according to the planet's Bond albedo, ${\displaystyle A_{B}}$:

${\displaystyle (1-A_{B}){F}_{\rm {solar}}={F}_{\rm {emit}}}$

where ${\displaystyle {F}_{\rm {solar}}}$ represents the area- and time-averaged incident solar flux, and may be expressed as:

${\displaystyle F_{\rm {solar}}=I_{o}/4}$

The factor of 1/4 in the above formula comes from the fact that only a single hemisphere is lit at any moment in time (creates a factor of 1/2), and from integrating over angles of incident sunlight on the lit hemisphere (creating another factor of 1/2).[6]

Assuming the planet radiates as a blackbody according to the Stefan–Boltzmann law at some equilibrium temperature ${\displaystyle {T}_{eq}}$, a balance of the absorbed and outgoing fluxes produces:

${\displaystyle (1-A_{B})\left({\frac {I_{o}}{4}}\right)=\sigma T_{\rm {eq}}^{4}}$ where ${\displaystyle \sigma }$ is the Stefan-Boltzmann constant.

Rearranging the above equation to find the equilibrium temperature leads to:

${\displaystyle {T}_{\rm {eq}}={\left({\frac {I_{o}\left(1-A_{B}\right)}{4\sigma }}\right)}^{1/4}}$

Because of the greenhouse effect, wherein long wave radiation emitted by the planet is absorbed and re-emitted to the surface by certain gases in the atmosphere, planets with substantial greenhouse atmospheres will have surface temperatures higher than the equilibrium temperature. For example, Venus has an equilibrium temperature of approximately 260 K, but a surface temperature of 740 K.[13][14] Similarly, Earth has an equilibrium temperature of 255 K (−18 °C; −1 °F),[14] but a surface temperature of about 288 K[15] due to the greenhouse effect in our lower atmosphere.[5][16]

On airless bodies, the lack of any significant greenhouse effect allows equilibrium temperatures to approach mean surface temperatures, as on Mars,[5] where the equilibrium temperature is 210 K and the mean surface temperature of emission is 215 K.[6] There are large variations in surface temperature over space and time on airless or near-airless bodies like Mars, which has daily surface temperature variations of 50-60 K.[17][18] Because of a relative lack of air to transport or retain heat, significant variations in temperature develop. Assuming the planet radiates as a blackbody (i.e. according to the Stefan-Boltzmann law), temperature variations propagate into emission variations, this time to the power of 4. This is significant because our understanding of planetary temperatures comes not from direct measurement of the temperatures, but from measurements of the fluxes. Consequently, in order to derive a meaningful mean surface temperature on an airless body (to compare with an equilibrium temperature), a global average surface emission flux is considered, and then an 'effective temperature of emission' that would produce such a flux is calculated.[6][17] The same process would be necessary when considering the surface temperature of the Moon, which has an equilibrium temperature of 271 K,[19] but can have temperatures of 373 K in the daytime and 100 K at night.[20] Again, these temperature variations result from poor heat transport and retention in the absence of an atmosphere.

Orbiting bodies can also be heated by tidal heating,[21] geothermal energy which is driven by radioactive decay in the core of the planet,[22] or accretional heating.[23] These internal processes will cause the effective temperature (a blackbody temperature that produces the observed radiation from a planet) to be warmer than the equilibrium temperature (the blackbody temperature that one would expect from solar heating alone).[6][16] For example, on Saturn, the effective temperature is approximately 95 K, compared to an equilibrium temperature of about 63 K.[24][25] This corresponds to a ratio between power emitted and solar power received of ~2.4, indicating a significant internal energy source.[25] Jupiter and Neptune have ratios of power emitted to solar power received of 2.5 and 2.7, respectively.[26] Close correlation between the effective temperature and equilibrium temperature of Uranus can be taken as evidence that processes producing an internal flux are negligible on Uranus compared to the other giant planets.[26]