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# Radiative equilibrium

In physics, **radiative equilibrium** is the condition where a steady state system is in dynamic equilibrium, with equal incoming and outgoing radiative heat flux and negligible heat transfer by conduction and convection.^{[1]} In atmospheric physics, under conditions of radiative equilibrium, total flux is constant with depth.^{[clarification needed]}^{[2]} In astrophysics, radiative equilibrium is used to determine atmospheric radiation of stars. In climate science, the net change in the tropopause after temperatures readjust to radiative equilibrium in the stratosphere, is used to determine the radiative forcing, as part of an assessment of natural and anthropogenic climate change.^{[3]}

## Contents

## Definitions

### History

In 1791 Pierre Prevost showed that all bodies radiate heat and concluded, *Radiation will exactly compensate absorption*. He used the terms absolute and relative equilibrium to describe changes.^{[4]} Prevost considered that what is nowadays called the photon gas or electromagnetic radiation was a fluid that he called "free heat" (French: *le feu*). Prevost proposed that free radiant heat is a very rare fluid, rays of which, like light rays, pass through each other without detectable disturbance of their passage. Prevost's called his theory *movable equilibrium of heat*, now designated as the theory of exchanges, which stated that each body radiates to, and receives radiation from, other bodies. The radiation from each body is emitted regardless of the presence or absence of other bodies.^{[5]}^{[6]}^{[7]}

In 1906 Karl Schwarzschild postulated the radiative equilibrium (German: *Strahlungsgleichgewicht*) dependent on Kirchhoff's law of thermal radiation, when he studied the sun.^{[8]}

### Pointwise radiative equilibrium

Following Planck (1914),^{[9]} a radiative field is often described in terms of specific radiative intensity, which is a function of each geometrical point in a space region, at an instant of time.^{[10]}^{[11]} A detailed definition is given by Goody and Yung (1989).^{[11]} They think of the interconversion between thermal radiation and heat in matter. From the specific radiative intensity they derive <math>\mathbf{F}_\nu</math>, the monochromatic vector flux density of radiation at each point in a region of space, which is equal to the time averaged monochromatic Poynting vector at that point (Mihalas 1978^{[12]} on pages 9–11). They define the monochromatic volume-specific rate of gain of heat by matter from radiation as the negative of the divergence of the monochromatic flux density vector; it is a scalar function of the position of the point:

- <math>h_\nu = - \nabla \cdot \mathbf{F}_\nu</math>.

They define **(pointwise) monochromatic radiative equilibrium** by

- <math>\nabla \cdot \mathbf{F}_\nu = 0 </math> at every point of the region that is in radiative equilibrium.

They define **(pointwise) radiative equilibrium** by

- <math>h = \int_0^{\infty} h_\nu d\nu = 0</math> at every point of the region that is in radiative equilibrium.

This means that, at every point of the region of space that is in (pointwise) radiative equilibrium, the total, for all frequencies of radiation, interconversion of energy between thermal radiation and energy content in matter is nil.

#### Approximate pointwise radiative equilibrium

Chandrasekhar (1950, p. 290)^{[13]} writes of a model of a stellar atmosphere in which "there are no mechanisms, other than radiation, for transporting heat within the atmosphere ... [and] there are no sources of heat in the atmosphere." This is hardly different from Schwarzschild's 1906 approximate concept, but is more precisely stated.

### Exchange equilibrium between systems

Radiative exchange equilibrium occurs with thermodynamic systems. Planck (1914)^{[9]} refers to a condition of thermodynamic equilibrium, in which "any two bodies or elements of bodies selected at random exchange by radiation equal amounts of heat with each other."

The term **radiative exchange equilibrium** can also be used to refer to two specified regions of space that exchange equal amounts of radiation by emission and absorption (even when the steady state is not one of thermodynamic equilibrium, but is one in which some sub-processes include net transport of matter or energy including radiation).

## Radiative equilibrium in astrophysics

Radiative equilibrium for a star, is taken as a whole and not confining attention only to its atmosphere. And when the rate of heat transfer from nuclear reactions plus viscosity to the microscopic motions of the material particles of the star is *balanced* by the transfer of energy by electromagnetic radiation from the star to space.

A star that is radiating energy to space cannot be in a steady state of temperature distribution unless there is a steady supply of energy from nuclear reactions within the star, to support the radiation to space.^{[14]}

Planetary equilibrium temperature is the theoretical temperature for a blackbody, which does not consider a radiative atmosphere.

## Radiative equilibrium in Earth sciences

**Radiative equilibrium** or **radiative balance** or just **energy balance**, describe a steady state net change of infrared radiation from Earth and shortwave radiation from outer space.

## See also

- Earth's energy budget
- Non-equilibrium thermodynamics
- Planetary equilibrium temperature
- Thermal equilibrium
- Thermodynamic equilibrium

## References

**^**Modest, Michael F. (2013).*Radiative heat transfer*(3rd ed. ed.). Oxford: Academic. p. 298. ISBN 9780123869906.**^**de Pater, Imke; Lissauer, Jack J. (2010).*Planetary sciences*(2nd ed. ed.). New York: Cambridge University Press. p. 68. ISBN 9780521853712.**^**IPCC (2007). "2.2 Concept of Radiative Forcing".*Climate Change 2007: Working Group I: The Physical Science Basis*. Cambridge University Press.**^**Prevost, P. (1791). Mémoire sur l'equilibre du feu.*Journal de Physique*(Paris), vol 38 pp. 314–322.**^**Maxwell, J.C. (1871). [AbstractPDF*Theory of Heat*] (PDF). Longmans, Green and Co, London. pp. 221–222.**^**[AbstractFull Text*The Laws of Radiation and Absorption: Memoirs by Prévost, Stewart, Kirchhoff, Bunsen*]. American Book Company. 1901.**^**Partington, J.R. (1949).*An Advanced Treatise on Physical Chemistry*, volume 1,*Fundamental Principles. The Properties of Gases*, Longmans, Green and Co, London, p. 467.**^**Schwarzschild, K. (1906). Über das Gleichgewicht der Sonnenatmosphäre.Download PDF- ^
^{a}^{b}Planck, M. (1914).*The Theory of Heat Radiation*p. 40 **^**Mihalas, D., Weibel-Mihalas, B. (1984).*Foundations of Radiation Hydrodynamics*, Oxford University Press, New York, ISBN 0-19-503437-6.- ^
^{a}^{b}Goody, R.M., Yung, Y.L. (1989).*Atmospheric Radiation. Theoretical Basis*, second edition, Oxford University Press, New York, 1989, ISBN 0-19-505134-3. **^**Mihalas, D. (1978).*Stellar Atmospheres*, 2nd edition, Freeman, San Francisco, ISBN 0-7167-0359-9.**^**Chandrasekhar, S. (1950).*Radiative Transfer*, Oxford University Press, Oxford, 1950.**^**Cox, J.P. with Giuli, R.T. (1968, reprint 1984).*Principles of Stellar Structure*, Gordon and Breach, New York, ISBN 0-677-01950-5, p. 134.