Black body properties from thermodynamical considerations
Anthony Brown
brown@strw.leidenuniv.nl
25.02.2008
Abstract. These notes provide some extra material on the subject of black bodies. The problem of
black body radiation is treated with classical physics, specifically thermodynamics, by considering
it as a photon gas in order to show how global properties of this radiation can be derived without
resorting to quantum physics. The material was extracted from the books by Adkins (1983) and
Rybicki & Lightman (1979).
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Contents
1
Thermal and black body radiation
1.1 Equilibrium radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
3
2
The Stefan-Boltzmann law
3
3
The limits of the classical treatment of black body radiation
4
References
Adkins C.J., 1983, Equilibrium Thermodynamics, third edition, Cambridge University Press
Carroll B.W., Ostlie D.A., 2007, An Introduction to Modern Astrophysics, Second Edition, Addison-Wesley,
San Francisco
Rybicki G.B., Lightman A.P., 1979, Radiative processes in astrophysics, John Wiley & Sons inc
1
1.1
1
Equilibrium radiation
2
Thermal and black body radiation
The material in this section is largely extracted from Adkins (1983, section 8.9). We first define a couple of
physical quantities that we will need in the discussion of thermal radiation. In order to discuss how the energy
of radiation is distributed with wavelength we define u to be the total energy density of the radiation and we
define the:
Spectral energy density uλ , such that uλ dλ equals the energy density contained in the radiation in the wavelength interval [λ, λ + dλ].
Spectral absorptivity for a surface, αλ , which is the fraction of incident radiation at wavelength λ which is
absorbed.
spectral emissive power of a surface, eλ , such that eλ dλ is the power emitted per unit area of the surface in
the wavelength interval [λ, λ + dλ].
1.1
Equilibrium radiation
Consider now an equal temperature enclosure for which all parts of the walls are at the same temperature. The
radiation in this vessel will after some time be in equilibrium with the walls surrounding it. This means that the
spectral energy density will be constant in time.
We now consider two equal temperature enclosures A and B which are initially at the same temperature
but whose walls are not the same. We bring them into contact via a narrow tube into which we insert a filter F
which only passes radiation in a narrow wavelength interval centred on λ (see figure 1). Now suppose that the
spectral energy density in A is larger than that in B:
B
uA
λ > uλ .
This would mean that energy would start to flow from A to B. This will cause the temperature of B to rise and of
A to fall. However this is not possible according to the laws of thermodynamics which forbid the spontaneous
divergence in temperature for systems that are in thermal contact. So we must have that:
B
uA
λ = uλ .
This means that the nature of the radiation is independent of the nature of the walls of the vessels. The radiation
also has to be isotropic otherwise a net flow of energy towards (part of) one of the walls would cause it to rise
in temperature thus violating the assumption that all walls are at the same temperature. This means that the
properties of radiation in equilibrium with the equal temperature enclosure can only depend on wavelength and
the temperature of the system:
uλ = f (λ, T ) .
(1)
This kind of radiation is named thermal radiation or black body radiation.
For isotropic radiation the energy incident per second on a unit area in the wavelength range [λ, λ + dλ] is
equal to:
1
cuλ dλ .
(2)
4
Of this incident radiation a fraction αλ will be absorbed and the rest will be reflected. The radiation in the equal
temperature enclosure can only be in thermal equilibrium if the walls of the vessel on balance do not absorb
any radiation. This implies that:
1
eλ dλ = αλ cuλ dλ .
(3)
4
We already know that uλ is a function of wavelength and temperature only so we have:
eλ
1
= cuλ = g(λ, T ) .
αλ
4
(4)
This result is known as Kirchoff’s law: the ratio of the spectral emissive power to the spectral absorptivity for
all bodies is a universal function of wavelength and temperature only.
2
The Stefan-Boltzmann law
3
B
A
F
Figure 1: Two equal temperature enclosures A and B in thermal contact through a very narrow tube. In the tube
a filter F is placed which only transmits radiation in a narrow wavelength interval centred on λ.
1.2
Black body radiation
A black body is defined as a body that absorbs all incident radiation, that is αλ = 1. For a black body we have:
1
eλ = cuλ .
4
(5)
This means that black body radiation has a wavelength and temperature dependence which is the same as for
thermal radiation. The latter is therefore also called black body radiation, however this is strictly speaking not
correct. For a black body the intensity is equal to the Planck function whereas for thermal radiation the source
function is equal to the Planck function. The intensity and source function will be treated in the discussion on
the formation of spectral lines in stellar atmospheres (chapter 9 in Carroll & Ostlie, 2007).
We can now define the spectral emissivity λ as the ratio of the spectral emissive power of a surface to that
of a black body. For a black body λ = αλ holds while for other bodies the following relation holds:
eλ
λ
λ 1
=
eλ,BB =
cuλ .
αλ
αλ
αλ 4
2
(6)
The Stefan-Boltzmann law
Black body radiation can be analysed further by applying thermodynamics, in particular the first and second
laws, because we are discussing a system in thermodynamical equilibrium. The material below is extracted
from Rybicki & Lightman (1979, section 1.5).
Take the equal temperature enclosure from the previous section and replace one of the walls by a piston.
The vessel has a volume V , a temperature T , and the radiation (a photon ‘gas’) has an energy density u and a
corresponding pressure p. According to the first law of thermodynamics:
dQ = dU + pdV ,
(7)
where Q is the heat en U the total energy. According to the second law:
dS =
dQ
,
T
(8)
where S is the entropy. We have that U = uV and for isotropic radiation the pressure is given by p = u/3
(pressure is one-third of the energy-density for a classical ultra-relativistic gas), where in this case u depends
3
The limits of the classical treatment of black body radiation
4
on temperature only. Hence:
dS =
V du
u
1u
dT + dV +
dV
T dT
T
3T
V du
4u
=
dT +
dV .
T dT
3T
(9)
The quantity dS is a perfect differential so we can identify the coefficients of dT and dV with the partial
derivatives of S:
∂S
∂S
V du
4u
.
(10)
=
=
∂T V
T dT
∂V T
3T
Differentiating these expressions (with respect to V and T for the left and right expressions, respectively) and
equating the new derivatives leads to:
1 du
4u
4 du
∂2S
=
=− 2 +
,
∂T ∂V
T dT
3T
3T dT
(11)
which means that:
du
4u
=
,
dT
T
du
dT
=4
,
u
T
ln u = 4 ln T + ln a ,
where ln a is an integration constant. This leads to:
Z
u(T ) = uλ dλ = aT 4 .
The total emissive power for a surface in the case of a black body is given by:
Z
Z
1
1
cuλ dλ = cu .
e = eλ dλ =
4
4
(12)
(13)
(14)
Equating this to the relation between u and T we obtain:
1
e = caT 4 = σT 4 ,
4
(15)
where σ = ac/4. This is of course the Stefan-Boltzmann law for the flux emitted from a surface radiating as a
black body.
3
The limits of the classical treatment of black body radiation
One can take this analysis of black body radiation further and derive Wien’s displacement law (see Adkins,
1983). In addition one can show that uλ = λ−5 g(λT ) from which the Wien approximation can be derived by
empirically finding an expression for g(λT ). However this is as far as classical physics can go. The derivation
of the correct functional form of uλ (T ) for black body radiation requires quantum physics.