Planetary equilibrium temperature

The planetary equilibrium temperature is a theoretical temperature that a planet would be 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 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,^[1] or the effective radiation emission temperature of the planet.^[2] 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.^[3][4]

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, ${\displaystyle I_{o}}$ . 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. 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.^[4][5][6]

Derivation

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:

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

Calculation for extrasolar planets

For a planet around another star, ${\displaystyle I_{o}}$ (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:

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

The luminosity (${\displaystyle L}$ ) of the star, which can be measured from observations of the star's apparent brightness,^[7] can then be written as:

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

To find the incident stellar flux on the planet, ${\displaystyle I_{x}}$ , at some orbital distance from the star, ${\displaystyle a}$ , one can divide by the surface area of a sphere with radius ${\displaystyle a}$ :^[8]

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

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

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

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,^[9] and may in future be obtainable from direct imaging of exoplanets and a conversion from geometric albedo.^[10] Orbital properties of the planet such as the orbital distance can be measured through radial velocity and transit period measurements.^[11][12]

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

${\displaystyle {T}_{\rm {eq}}=T_{\rm {star}}{\sqrt {\frac {R}{2a}}}\left(1-A_{B}\right)^{1/4}}$ 

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 226 K (−47 °C; −53 °F), but a surface temperature of 740 K (467 °C; 872 °F).^[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 °C; 59 °F)^[15] due to the greenhouse effect in our lower atmosphere.^[5][16] The equilibrium temperatures of such planets are more accurately estimated using simple energy-balance models which include one or more levels of thermal radiation transport through the atmosphere.^[17]

Airless bodies

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 (−63 °C; −82 °F) and the mean surface temperature of emission is 215 K (−58 °C; −73 °F).^[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.^[18][19] 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][18] The same process would be necessary when considering the surface temperature of the Moon, which has an equilibrium temperature of 271 K (−2 °C; 28 °F),^[20] but can have temperatures of 373 K (100 °C; 212 °F) in the daytime and 100 K (−173 °C; −280 °F) at night.^[21] 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,^[22] geothermal energy which is driven by radioactive decay in the core of the planet,^[23] or accretional heating.^[24] 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.^[25][26] This corresponds to a ratio between power emitted and solar power received of ~2.4, indicating a significant internal energy source.^[26] Jupiter and Neptune have ratios of power emitted to solar power received of 2.5 and 2.7, respectively.^[27] 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.^[27]

See also

• Effective temperature
• Thermal equilibrium
• Earth's energy budget

References

1. Wallace & Hobbs (2006), pp. 119–120.
2. Stull, R. (2000). Meteorology For Scientists and Engineers. A technical companion book with Ahrens' Meteorology Today, Brooks/Cole, Belmont CA, ISBN 978-0-534-37214-9., p. 400.
3. Jin, Menglin; Dickinson, Robert E (2010-10-01). "Land surface skin temperature climatology: benefitting from the strengths of satellite observations". Environmental Research Letters. 5 (4): 044004. Bibcode:2010ERL.....5d4004J. doi:10.1088/1748-9326/5/4/044004. ISSN 1748-9326.
4. Lissauer, Jack Jonathan. (2013-09-16). Fundamental planetary science : physics, chemistry, and habitability. De Pater, Imke, 1952-. New York, NY, USA. p. 90. ISBN 9780521853309. OCLC 808009225.
5. Goody, Richard M. (1972). Atmospheres. Walker, James C. G. Englewood Cliffs, N.J.: Prentice-Hall. pp. 46, 49. ISBN 0130500968. OCLC 482175.
6. Catling, David C. (2017). Atmospheric Evolution on Inhabited and Lifeless Worlds. Kasting, James F. Cambridge: Cambridge University Press. p. 34. ISBN 9780521844123. OCLC 956434982.
7. "Absolute Magnitude". csep10.phys.utk.edu. Retrieved 2019-06-12.
8. "Flux, Luminosity, and Brightness". www.austincc.edu. Retrieved 2019-06-12.
9. Cowan, Nicolas B.; Agol, Eric (2011-03-01). "The statistics of albedo and heat recirculation on hot exoplanets". The Astrophysical Journal. 729 (1): 54. arXiv:1001.0012. Bibcode:2011ApJ...729...54C. doi:10.1088/0004-637X/729/1/54. ISSN 0004-637X.
10. Cahoy, Kerri L.; Marley, Mark S.; Fortney, Jonathan J. (2010-11-20). "Exoplanet albedo spectra and colors as a function of planet phase, separation, and metallicity". The Astrophysical Journal. 724 (1): 189–214. arXiv:1009.3071. Bibcode:2010ApJ...724..189C. doi:10.1088/0004-637X/724/1/189. ISSN 0004-637X.
11. Chatelain, Joey. "Exoplanets" (PDF). Georgia State University Physics and Astronomy.
12. "Exploring Exoplanets with Kepler" (PDF). NASA Jet Propulsion Laboratory.
13. "Venus Fact Sheet". nssdc.gsfc.nasa.gov. December 23, 2016. Retrieved 2017-02-01.
14. "Equilibrium Temperatures of Planets". burro.astr.cwru.edu. Retrieved 2013-08-01.
15. Science, Tim Sharp 2018-04-23T19:26:00Z; Astronomy. "What Is Earth's Average Temperature?". Space.com. Retrieved 2019-06-12.
16. Lissauer, Jack Jonathan. (2013-09-16). Fundamental planetary science : physics, chemistry, and habitability. De Pater, Imke, 1952-. New York, NY, USA. ISBN 9780521853309. OCLC 808009225.
17. "ACS Climate Science Toolkit - How Atmospheric Warming Works". American Chemical Society. Retrieved 3 October 2022.
18. Haberle, Robert M. (2013). "Estimating the power of Mars' greenhouse effect". Icarus. 223 (1): 619–620. Bibcode:2013Icar..223..619H. doi:10.1016/j.icarus.2012.12.022.
19. "Mars: Temperature overview". www-k12.atmos.washington.edu. Retrieved 2019-06-12.
20. "Moon Fact Sheet". nssdc.gsfc.nasa.gov. July 1, 2013. Retrieved 2013-08-01.
21. "What's the Temperature on the Moon?". Space.com. March 1, 2012. Retrieved 2013-08-01.
22. Strobel, Nick (March 12, 2013) [Last updated: December 12, 2018]. "Jupiter's Large Moons". Planetary Science. Retrieved 2019-03-29 – via Astronomynotes.com.
23. Anuta, Joe (March 27, 2006). "Probing Question: What heats the earth's core?". Penn State News.
24. "accretional heating". A Dictionary of Earth Sciences. Encyclopedia.com. Retrieved 2013-08-01.
25. Fortney, Jonathan J.; Nettelmann, Nadine (2010). "The interior structure, composition, and evolution of giant planets". Space Science Reviews. 152 (1–4): 423–447. arXiv:0912.0533. Bibcode:2010SSRv..152..423F. doi:10.1007/s11214-009-9582-x. ISSN 0038-6308.
26. Aumann, H. H.; Gillespie, C. M., Jr.; Low, F. J. (1969). "The internal powers and effective temperatures of Jupiter and Saturn". The Astrophysical Journal. 157: L69. Bibcode:1969ApJ...157L..69A. doi:10.1086/180388. ISSN 0004-637X.
27. "6 - Equilibrium Temperature". lasp.colorado.edu. Retrieved 2019-06-12.

Sources

• Fressin F, Torres G, Rowe JF, Charbonneau D, Rogers LA, Ballard S, Batalha NM, Borucki WJ, Bryson ST, Buchhave LA, Ciardi DR, Désert JM, Dressing CD, Fabrycky DC, Ford EB, Gautier TN 3rd, Henze CE, Holman MJ, Howard A, Howell SB, Jenkins JM, Koch DG, Latham DW, Lissauer JJ, Marcy GW, Quinn SN, Ragozzine D, Sasselov DD, Seager S, Barclay T, Mullally F, Seader SE, Still M, Twicken JD, Thompson SE, Uddin K (2012). "Two Earth-sized planets orbiting Kepler-20". Nature. 482 (7384): 195–198. arXiv:1112.4550. Bibcode:2012Natur.482..195F. doi:10.1038/nature10780. PMID 22186831.
• Wallace, J.M.; Hobbs, P.V. (2006). Atmospheric Science. An Introductory Survey (2nd ed.). Amsterdam: Elsevier. ISBN 978-0-12-732951-2.

External links

• Equilibrium Temperature at the Laboratory for Atmospheric and Space Physics, University of Colorado
• Energy balance: the simplest climate model
• HEC: Exoplanets Calculator Features a user friendly calculator to calculate the Planet Equilibrium Temperature.