Temperatures
Temperatures of astronomical objects range from almost absolute zero to millions of degrees. Temperature can be defined in a variety of ways, and its numerical value depends on the specific definition used. The temperature is welldefined only in a state o
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Temperatures of astronomical objects range from almost absolute zero to millions of degrees. Temperature can be defined in a variety of ways, and its numerical value depends on the specific definition used. The temperature is welldefined only in a state of thermodynamic equilibrium. Since most astrophysical objects are not in equilibrium, they cannot be assigned a unique temperature. For many purposes it is still useful to describe phenomena in terms of a temperature, but its value will then depend on how it is defined. Often the temperature is determined by comparing the object, a star for instance, with a blackbody. Although real stars do not radiate exactly like blackbodies, their spectra can usually be approximated by blackbody spectra after the effect of spectral lines has been eliminated. The resulting temperature depends on the exact criterion used to fit Planck's function to observations. The most important quantity describing the surface temperature of a star is the effective temperature Te. It is defined as the temperature of a blackbody which radiates with the same total flux density as the star. Since the effective temperature depends onlyon the total radiation power integrated over all frequencies, it is welldefined for all energy distributions even if they deviate far from Planck's law. In the previous chapter we derived the Stefan-Boltzmann law, which gives the total flux density as a function of temperature. If we now find a value Te of the temperature such that the Stefan-Boltzmann law gives the correct flux density F on the surface of the star, we have found the effective temperature. The flux density on the surface is (6.1)
F= aT!.
The total flux is L = 4 n R 2 F, where R is the radius of the star, and the flux density at a distance r is F'
= _L_ = 4 n r2
R2 F=
r2
(~)2 aT:, 2
(6.2)
where a = 2R / r is the observed angular diameter of the star. For direct determination of the effective temperature, we have to measure the total flux density and the angular diameter of the star. This is possible only in the few cases in which the diameter has been found by interferometry. If we assume that at some wavelength A the flux density FA on the surface of the star is obtained from Planck's law, we get the brightness temperature T b • In the isotropic case we have then, FA = nBA(Tb ). If the radius of the star is R and distance from the Earth r, the observed flux density is
H. Karttunen et al. (eds.), Fundamental Astronomy © Springer-Verlag Berlin Heidelberg 1994
6. Temperatures
126
Again, F';. can be determined only if the angular diameter a is known. The brightness temperature Tb can then be solved from (6.3) Since the star does not radiate like a blackbody, its brightness temperature depends on the particular wavelength used in (6.3). In radio astronomy, brightness temperature is used to express the intensity (or surface brightness) of the source. If the intensity at frequency v is Iv, the brightness temperature is obtained from
Tb gives the temperature of a blackbody with the same surface b
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