Effective temperature

The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation.^[1] Effective temperature is often used as an estimate of a body's temperature when the body's emissivity curve (as a function of wavelength) is not known.

When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. The net emissivity may be low due to surface or atmospheric properties, including greenhouse effect.

Star

The effective temperature of the Sun (5777 K) is the temperature a black body of the same size must have to yield the same total emissive power.

The effective temperature of a star is the temperature of a black body with the same luminosity per surface area ( $\mathcal{F}_{\rm Bol}$ ) as the star and is defined according to the Stefan–Boltzmann law $\mathcal{F}_{\rm Bol}=\sigma T_{\rm eff}^4$ . Notice that the total (bolometric) luminosity of a star is then $L=4 \pi R^2 \sigma T_{\rm eff}^4$ , where $R$ is the stellar radius.^[2] The definition of the stellar radius is obviously not straightforward. More rigorously the effective temperature corresponds to the temperature at the radius that is defined by a certain value of the Rosseland optical depth (usually 1).^[3]^[4] The effective temperature and the bolometric luminosity are the two fundamental physical parameters needed to place a star on the Hertzsprung–Russell diagram. Both effective temperature and bolometric luminosity actually depend on the chemical composition of a star.

The effective temperature of our Sun is around 5780 kelvin (K).^[5]^[6] Stars actually have a temperature gradient, going from their central core up to the atmosphere. The "core temperature" of the sun—the temperature at the centre of the sun where nuclear reactions take place—is estimated to be 15 000 000 K.

The color index of a star indicates its temperature from the very cool—by stellar standards, that is—red M stars that radiate heavily in the infrared to the very blue O stars that radiate largely in the ultraviolet. The effective temperature of a star indicates the amount of heat that the star radiates per unit of surface area. From the warmest surfaces to the coolest is the sequence of star types known as O, B, A, F, G, K, and M.

A red star could be a tiny red dwarf, a star of feeble energy production and a small surface or a bloated giant or even supergiant star such as Antares or Betelgeuse, either of which generates far greater energy but passes it through a surface so large that the star radiates little per unit of surface area. A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more heat per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel.

Planet

The effective temperature of a planet can be calculated by equating the power received by the planet with the power emitted by a blackbody of temperature T.

Take the case of a planet at a distance D from the star, of luminosity L.

Assuming the star radiates isotropically and that the planet is a long way from the star, the power absorbed by the planet is given by treating the planet as a disc of radius r, which intercepts some of the power which is spread over the surface of a sphere of radius D (the distance of the planet from the star). We also allow the planet to reflect some of the incoming radiation by incorporating a parameter called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed. The expression for absorbed power is then:

$P_{\rm abs} = \frac {L r^2 (1-A)}{4 D^2}$

The next assumption we can make is that the entire planet is at the same temperature T, and that the planet radiates as a blackbody. The Stefan–Boltzmann law gives an expression for the power radiated by the planet:

$P_{\rm rad} = 4 \pi r^2 \sigma T^4$

Equating these two expressions and rearranging gives an expression for the effective temperature:

$T = \left (\frac{L (1-A)}{16 \pi \sigma D^2} \right )^{\tfrac{1}{4}}$

Note that the planet's radius has cancelled out of the final expression.

The effective temperature for Jupiter from this calculation is 112 K and 51 Pegasi b (Bellerophon) is 1258 K. A better estimate of effective temperature for some planets, such as Jupiter, would need to include the internal heating as a power input. The actual temperature depends on albedo and atmosphere effects. The actual temperature from spectroscopic analysis for HD 209458 b (Osiris) is 1130 K, but the effective temperature is 1359 K. The internal heating within Jupiter raises the effective temperature to about 152 K.

Surface temperature of a planet

The surface temperature of a planet can be estimated by modifying the effective-temperature calculation to account for emissivity and temperature variation.

The area of the planet that absorbs the power from the star is A_abs which is some fraction of the total surface area $A_{\rm total} = 4 \pi r^2$ , where r is the radius of the planet. This area intercepts some of the power which is spread over the surface of a sphere of radius D. We also allow the planet to reflect some of the incoming radiation by incorporating a parameter a called the albedo. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed. The expression for absorbed power is then:

$P_{\rm abs} = \frac {L A_{\rm abs} (1-a)}{4 \pi D^2}$

The next assumption we can make is that although the entire planet is not at the same temperature, it will radiate as if it had a temperature T over an area A_rad which is again some fraction of the total area of the planet. There is also a factor ε, which is the emissivity and represents atmospheric effects. ε ranges from 1 to 0 with 1 meaning the planet is a perfect blackbody and emits all the incident power. The Stefan–Boltzmann law gives an expression for the power radiated by the planet:

$P_{\rm rad} = A_{\rm rad} \varepsilon \sigma T^4$

Equating these two expressions and rearranging gives an expression for the surface temperature:

$T = \left (\frac{A_{\rm abs}}{A_{\rm rad}} \frac{L (1-a)}{4 \pi \sigma \varepsilon D^2} \right )^{\tfrac{1}{4}}$

Note the ratio of the two areas. Common assumptions for this ratio are 1/4 for a rapidly rotating body and 1/2 for a slowly rotating body. This ratio would be 1 for the subsolar point, the point on the planet directly below the sun and gives the maximum temperature of the planet.^[7]

Let's look at the Earth. The Earth has an albedo of about 0.367.^[8] The emissivity is dependent on the type of surface and many climate models set the value of the Earth's emissivity to 1. However, a more realistic value is 0.96.^[9] The Earth is a fairly fast rotator so the area ratio can be estimated as 1/4. The other variables are constant. This calculation gives us an effective temperature of the Earth of 252K or -21 °C. The average temperature of the Earth is 288K or 15 °C. One reason for the difference between the two values is due to the Greenhouse effect, which increases the average temperature of the Earth's surface.

Also note here that this equation does not take into account any effects from internal heating of the planet, which can arise directly from sources such as radioactive decay and also be produced from frictions resulting from tidal forces.

References

↑ Archie E. Roy, David Clarke (2003). Astronomy. CRC Press. ISBN 978-0-7503-0917-2.
↑ Tayler, Roger John (1994). The Stars: Their Structure and Evolution. Cambridge University Press. p. 16. ISBN 0-521-45885-4.
↑ Böhm-Vitense, Erika. Introduction to Stellar Astrophysics, Volume 3, Stellar structure and evolution. Cambridge University Press. p. 14.
↑ Baschek (June 1991). "The parameters R and Teff in stellar models and observations". Astronomy and Astrophysics 246 (2): 374–382. Bibcode:1991A&A...246..374B.
↑ "Section 14: Geophysics, Astronomy, and Acousticse". Handbook of Chemistry and Physics; section=14-18: Solar Spectral Irradiance (88 ed.). CRC Press.
↑ Jones, Barrie William (2004). Life in the Solar System and Beyond. Springer. p. 7. ISBN 1-85233-101-1.
↑ Swihart, Thomas. "Quantitative Astronomy". Prentice Hall, 1992, Chapter 5, Section 1.
↑ http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
↑ Jin, Menglin and Shunlin Liang, (2006) “An Improved Land Surface Emissivity Parameter for Land Surface Models Using Global Remote Sensing Observations” Journal of Climate, 19 2867-81. (www.glue.umd.edu/~sliang/papers/Jin2006.emissivity.pdf)

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