Blackbody Radiation

A blackbody is defined as an object that does not reflect or scatter radiation shining upon it, but absorbs and re-emits the radiation completely. A blackbody is a kind of an ideal radiator, which cannot exist in the real world. Yet many objects behave very much as if they were blackbodies.

The radiation of a blackbody depends only on its temperature, being perfectly independent of its shape, material and internal constitution. The wavelength distribution of the radiation follows Planck's law, which is a function of temperature only. The intensity at a frequency v of a blackbody at temperature T is

2hv3

where h = the Planck constant = 6.63 x 10-34 J s , c = the speed of light ~ 3 x 108 ms-1 , k = the Boltzmann constant = 1.38 x 10-23 JK-1 .

By definition of the intensity, the dimension of Bv is Wm-2 Hz-1 sterad-1.

Blackbody radiation can be produced in a closed cavity whose walls absorb all radiation incident upon them (and coming from inside the cavity). The walls and the radiation in the cavity are in equilibrium; both are at the same temperature, and the walls emit all the energy they receive. Since radiation energy is constantly transformed into thermal energy of the atoms of the walls and back to radiation, the blackbody radiation is also called thermal radiation.

The spectrum of a blackbody given by Planck's law (5.12) is continuous. This is true if the size of the radiator is very large compared with the dominant wavelengths. In the case of the cavity, this can be understood by considering the radiation as standing waves trapped in the cavity. The number of different wavelengths is larger, the shorter the wavelengths are compared with the size of the cavity. We already mentioned that spectra of solid bodies are continuous; very often such spectra can be quite well approximated by Planck's law.

We can also write Planck's law as a function of the wavelength. We require that Bv dv = -BA dA. The wavelength decreases with increasing frequency; hence the minus sign. Since v = c/A, we have

dv

c

dA =

-A2 '

whence

v dA

or

2hc2

The functions Bv and Bk are defined in such a way that the total intensity can be obtained in the same way using either of them:

Let us now try to find the total intensity using the first of these integrals:

2h v3 dv

We now change the integration variable to x = hv/(kT), whence dv = (kT/h)dx:

The definite integral in this expression is just a real number, independent of the temperature. Thus we find that

B(T) = AT4 , where the constant A has the value

(In order to get the value of A we have to evaluate the integral. There is no elementary way to do that. We can tell those who are familiar with all the exotic functions so beloved by theoretical physicists, that the integral can rather easily be expressed as r(4)Z(4), where Z is the Riemann zeta function and r is the gamma function. For integral values, r(n) is simply the factorial (n — 1)!. The difficult part is showing that Z(4) = n4 /90. This can be done by expanding x4 — x2 as a Fourier-series and evaluating the series at x = n.)

The flux density F for isotropic radiation of intensity B is (Sect. 4.1):

This is the Stefan-Boltzmann law, and the constant o (= nA) is the Stefan-Boltzmann constant,

In fact this defines the effective temperature of the star, discussed in more detail in the next section.

The luminosity, radius and temperature of a star are interdependent quantities, as we can see from (5.19). They are also related to the absolute bolometric magnitude of the star. Equation (4.13) gives the difference of the absolute bolometric magnitude of the star and the Sun:

But we can now use (5.19) to express the luminosities in terms of the radii and temperatures:

As we can see in Fig. 5.10, the wavelength of the maximum intensity decreases with increasing total intensity (equal to the area below the curve). We can find the wavelength Xmax corresponding to the maximum intensity by differentiating Planck's function BX(T) with o = 5'67 x 10—8Wm—2 K—4

From the Stefan-Boltzmann law we get a relation between the luminosity and temperature of a star. If the radius of the star is R, its surface area is 4nR2, and if the flux density on the surface is F, we have

If the star is assumed to radiate like a blackbody, we have F = aT4, which gives

500 nm

1,000 nm

Fig. 5.10. Intensity distributions of blackbodies at temperature 12,000 K, 9000 K and 6000 K. Since the ratios of the temperatures are 4:3:2, the wavelengths of intensity maxima given by the Wien displacement law are in the proportions 1:4, 1:3 and 1:2, or 3, 4 and 6. The actual wavelengths of the maxima are 2415 nm, 322 nm and 483 nm. The total intensities or the areas below the curves are proportional to 44, 34 and 24

500 nm

1,000 nm

Fig. 5.10. Intensity distributions of blackbodies at temperature 12,000 K, 9000 K and 6000 K. Since the ratios of the temperatures are 4:3:2, the wavelengths of intensity maxima given by the Wien displacement law are in the proportions 1:4, 1:3 and 1:2, or 3, 4 and 6. The actual wavelengths of the maxima are 2415 nm, 322 nm and 483 nm. The total intensities or the areas below the curves are proportional to 44, 34 and 24

respect to X and finding zero of the derivative. The result is the Wien displacement law:

where the Wien displacement constant b is b = 0.0028978 Km.

We can use the same procedure to find the maximum of Bv. But the frequency vmax thus obtained is different from vmax = c/Xmax given by (5.22). The reason for this is the fact that the intensities are given per unit frequency or unit wavelength, and the dependence of frequency on wavelength is nonlinear.

When the wavelength is near the maximum or much longer than Xmax Planck's function can be approximated by simpler expressions. When X « Xmax (or hc/(XkT) » 1), we have ehc/(XkT) » 1

In this case we get the Wien approximation 2hc2

When hc/(XkT) < 1 (X » Xmax), we have ehc/XkT « 1 + hc/(XkT), which gives the Rayleigh-Jeans approximation 2hc2 XkT 2ckT

X5 hc

physical phenomena, and often there is no unique 'true' temperature.

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 only on the total radiation power integrated over all frequencies, it is well defined for all energy distributions even if they deviate far from Planck's law.

In the previous section we derived the Stefan-Boltzmann law, which gives the total flux density as a function of the 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

The total flux is L = 4nR2 F, where R is the radius of the star, and the flux density at a distance r is

4nr2

This is particularly useful in radio astronomy.

Classical physics predicted only the Rayleigh-Jeans approximation. Were (5.24) true for all wavelengths, the intensity would grow beyond all limits when the wavelength approaches zero, contrary to observations. This contradiction was known as the ultraviolet catastrophe.

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