User:Lmfifer/Equilibrium Temperature Sandbox

The planetary equilibrium temperature is a theoretical temperature that a planet would be at when considered simply as if it were 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 not considered, and one treats the theoretical black body temperature as if it came from an idealized surface of the planet.

Other authors use different names for this concept, such as equivalent blackbody temperature of a planet, or the effective radiation emission temperature of the planet. 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 an equilibrium temperature due to greenhouse effects.

Calculation of equilibrium temperature
The equilibrium temperature of a planet depends on the incident solar radiation ("insolation") at its orbital distance from the Sun, $$I_o$$, and the planet's bond albedo, $$A_B$$:

$${ T }_{ eq }={\left( { \frac {I_o \left( 1-A_B \right) }{ 4 \sigma } }\right)}^{ 1/4 } $$ where $$\sigma = $$ Stefan-Boltzmann constant

Derivation
Consider a star orbiting its host planet. 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, $$I_o$$. 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. Assume 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.

The solar flux absorbed by the planet from the star is equal to the flux emitted by the planet:

$${ F }_{ abs }={ F }_{ emit }$$

$$(1-A_B){ F }_{ solar }={ F }_{ emit }$$

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

$$F_{solar}=I_o/4$$

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).

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

$$(1-A_B)\left ( \frac{I_o}{4} \right )=\sigma T_{eq}^4$$ where $$\sigma$$ is the Stefan-Boltzmann constant.

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

$${ T }_{ eq }={\left( { \frac {I_o \left( 1-A_B \right) }{ 4 \sigma } }\right)}^{ 1/4 } $$

Calculation for extrasolar planets
For a planet around another star, $$I_o$$(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:

$$F_{star}=\sigma T_{star}^4$$

The luminosity ($$L$$) of the star, which can be measured from observations of the star's apparent brightness, can then be written as:

$$L=4\pi R_{star}^2\sigma T_{star}^4$$ where the flux has been multiplied by the surface area of the star.

To find the incident stellar flux on the planet, $$I_x$$, at some orbital distance from the star, $$a$$, one can divide by the surface area of a sphere with radius $$a $$ :

$$I_x=\left ( \frac{L}{4\pi a^2}\right )$$

Plugging this into the general equation for planetary equilibrium temperature derived earlier gives:

$${ T }_{ eq }={\left( { \frac {L \left( 1-A_B \right) }{ 16 \sigma \pi a^2} }\right)}^{ 1/4 } $$

If the luminosity of the star is known from photometric observations, the other remaining variables that must be determined are the bond albedo and orbital distance of the planet. Bond albedos of exoplanets can be constrained by flux measurements of transiting exoplanets, and may in future be obtainable from direct imaging of exoplanets and a conversion from geometric albedo. Orbital properties of the planet such as the orbital distance can be measured through radial velocity and transit period measurements.

Alternatively, the planetary equilibrium may be written in terms of the temperature and radius of the star:

$${ T }_{ eq }=T_{star}\sqrt \left( 1-A_B \right)^{ 1/4 } $$

Caveats
The equilibrium temperature is neither an upper nor lower bound on actual temperatures on a planet. There are several reasons why measured temperatures deviate from predicted equilibrium temperatures.

Greenhouse effect
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 227 K, but a surface temperature of 740 K. Similarly, Earth has an equilibrium temperature of 255 K, but a surface temperature of about 288 K due to the greenhouse effect in our lower atmosphere.

Airless bodies
On airless bodies, the lack of any significant greenhouse effect allows equilibrium temperatures to approach mean surface temperatures, as on Mars, where the equilibrium temperature is 210 K and the mean surface temperature of emission is 215 K. It should be noted that 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. 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. The same process would be necessary when considering the surface temperature of the Moon, which has a equilibrium temperature of 271 K, but can have temperatures of 373 K in the daytime and 100 K at night. Again, these temperature variations result from poor heat transport and retention in the absence of an atmosphere.

Internal energy fluxes
Orbiting bodies can also be heated by tidal heating, geothermal energy which is driven by radioactive decay in the core of the planet, or accretional heating. 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). For example, on Saturn, the effective temperature is approximately 95 K, compared to an equilibrium temperature of about 63 K. This corresponds to a ratio between power emitted and solar power received of ~2.4, indicating a significant internal energy source. Jupiter and Neptune have ratios of power emitted to solar power received of 2.5 and 2.7, respectively. Close correlation between the effective temperature and equilibrium temperature of Uranus can be taken as evidence that internal flux processes are negligible on Uranus compared to the other giant planets.