Planetary equilibrium temperature
The planetary equilibrium temperature is a theoretical temperature that a planet would be if it was in radiative equilibrium, typically under the assumption that it radiates as a black body being heated only by its parent star. In this model, the presence or absence of an atmosphere (and therefore any greenhouse effect) is irrelevant, as the equilibrium temperature is calculated purely from a balance with incident stellar energy. Other authors use different names for this concept, such as equivalent blackbody temperature of a planet. The effective radiation emission temperature is a related concept, but focuses on the actual power radiated rather than on the power being received, and so may have a different value if the planet has an internal energy source or when the planet is not in radiative equilibrium. Planetary equilibrium temperature differs from the global mean temperature and surface air temperature, which are measured observationally by satellites or surface-based instruments, and may be warmer than the equilibrium temperature due to the greenhouse effect. Calculation of equilibrium temperature Consider a planet orbiting its host star. The star emits radiation isotropically, and some fraction of this radiation reaches the planet. The amount of radiation arriving at the planet is referred to as the incident solar radiation, . The planet has an albedo that depends on the characteristics of its surface and atmosphere, and therefore only absorbs a fraction of radiation. The planet absorbs the radiation that isn't reflected by the albedo, and heats up. One may assume that the planet radiates energy like a blackbody at some temperature according to the Stefan–Boltzmann law. Thermal equilibrium exists when the power supplied by the star is equal to the power emitted by the planet. The temperature at which this balance occurs is the planetary equilibrium temperature. Derivation The solar flux absorbed by the planet from the star is equal to the flux emitted by the planet: Assuming a fraction of the incident sunlight is reflected according to the planet's Bond albedo, : where represents the area- and time-averaged incident solar flux, and may be expressed as: 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). Assuming the planet radiates as a blackbody according to the Stefan–Boltzmann law at some equilibrium temperature , a balance of the absorbed and outgoing fluxes produces: where is the Stefan-Boltzmann constant. Rearranging the above equation to find the equilibrium temperature leads to: Calculation for extrasolar planets For a planet around another star, (the incident stellar flux on the planet) is not a readily measurable quantity. To find the equilibrium temperature of such a planet, it may be useful to approximate the host star's radiation as a blackbody as well, such that: The luminosity () of the star, which can be measured from observations of the star's apparent brightness, can