## 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. All these different temperatures are needed to describe different 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 X the flux density FX on the surface of the star is obtained from Planck's law, we get the brightness temperature Tb. In the isotropic case we have then FX = nBX(Tb). If the radius of the star is R and distance from the Earth r, the observed flux density is

Again Fx can be determined only if the angular diameter a is known. The brightness temperature 7b can then be solved from

Since the star does not radiate like a blackbody, its brightness temperature depends on the particular wavelength used in (5.27).

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 brightness as the observed source.

Since radio wavelengths are very long, the condition hv ^ kT of the Rayleigh-Jeans approximation is usually satisfied (except for millimetre and submillimetre bands), and we can write Planck's law as

2h v3 1

2kv2

Thus we get the following expression for the radio astronomical brightness temperature:

rIv -TT Iv

2kv2 2k

A measure of the signal registered by a radio telescope is the antenna temperature TA. After the antenna temperature is measured, we get the brightness temperature from

"A

The colour temperature Tc can be determined even if the angular diameter of the source is unknown (Fig. 5.11). We only have to know the relative energy distribution in some wavelength range [A.i,A.2]; the absolute value of the flux is not needed. The observed flux density as a function of wavelength is compared with Planck's function at different temperatures. The temperature giving the best fit is the colour temperature in the interval [X1,X2]. The colour temperature is usually different for different wavelength intervals, since the shape of the observed energy distribution may be quite different from the blackbody spectrum.

A simple method for finding a colour temperature is the following. We measure the flux density F'k at two wavelengths and A.2. If we assume that the intensity distribution follows Planck's law, the ratio of these flux densities must be the same as the ratio obtained from Planck's law:

The temperature T solved from this equation is a colour temperature.

The observed flux densities correspond to certain magnitudes m^ and m-k2. The definition of magnitudes gives

where the constant term is a consequence of the different zero points of the magnitude scales. If the temperature where n is the beam efficiency of the antenna (typically 0.4 < n ^ 0.8). Equation (5.29) holds if the source is wide enough to cover the whole beam, i. e. the solid angle £A from which the antenna receives radiation. If the solid angle subtended by the source, £S, is smaller than £2A, the observed antenna temperature is

Blackbody radiation | ||

// // |
Observed flux | |

// // / / // / / |
^ density | |

Fig. 5.11. Determination of the colour temperature. The ratio of the flux densities at wavelengths X1 and X2 gives the temperature of a blackbody with the same ratio. In general the result depends on the wavelengths chosen

Fig. 5.11. Determination of the colour temperature. The ratio of the flux densities at wavelengths X1 and X2 gives the temperature of a blackbody with the same ratio. In general the result depends on the wavelengths chosen b

is not too high, we can use the Wien approximation in the optical part of the spectrum:

where a and b are constants. This shows that there is a simple relationship between the difference of two magnitudes and the colour temperature.

Strictly speaking, the magnitudes in (5.32) are monochromatic, but the same relation can be also used with broadband magnitudes like B and V. In that case, the two wavelengths are essentially the effective wavelengths of the B and V bands. The constant is chosen so that B — V = 0 for stars of the spectral type A0 (see Chap. 8). Thus the colour index B — V also gives a colour temperature.

The kinetic temperature Tk, is related to the average speed of gas molecules. The kinetic energy of an ideal gas molecule as a function of temperature follows from the kinetic gas theory:

Solving for Tk we get

where m is the mass of the molecule, v its average velocity (or rather its r. m. s velocity, which means that v2 is the average of the squared velocities), and k, the Boltzmann constant. For ideal gases the pressure is directly proportional to the kinetic temperature (c. f. *Gas Pressure and Radiation Pressure, p. 238):

where n is the number density of the molecules (molecules per unit volume). We previously defined the excitation temperature Texc as a temperature which, if substituted into the Boltzmann distribution (5.11), gives the observed population numbers. If the distribution of atoms in different levels is a result of mutual collisions of the atoms only, the excitation temperature equals the kinetic temperature, Texc = Tk.

The ionization temperature Ti is found by comparing the number of atoms in different states of ionization. Since stars are not exactly blackbodies, the values of excitation and ionization temperatures usually vary, depending on the element whose spectral lines were used for temperature determination.

In thermodynamic equilibrium all these various temperatures are equal.

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