Page 33 - 83 basic knowledge of astronomy

Taking into account Einstein’s relations among the differential coefficients in
equations (21) and (25):

n
m
g m β m = g n β ,
n
α m = 2hν 3 ,
n
β n m c 3
we can express the emissivity–opacity ratio as

1 2hν 3
ν
= , (53)
n m g n − 1 c 2
κ ν
n n g m
using equation (39). Obviously, the right hand side of equation (53) becomes
the Planck function if the Boltzmann distribution

n n − E n
P n = = g n e kT ,
n
(equation (17)) holds, and hence

n m g n hν
= e kT .
n n g m

Therefore, we can interpret LTE as a physical situation where the Boltzmann
distribution is established among the particles in a medium due, for example,
to their mutual collisions, but in which the radiation is still not in equilibrium
with the particles.

10.5 Spectrum of the Orion Nebula

Now we are in a position to interpret qualitatively the bend in the spectrum
of the Orion Nebula, as shown in Figures 2 and 26. If we neglect the contri-
bution of the background radiation in the solution of the radiative transfer
equation in the isothermal LTE case (equation (51)), the intensity is given
by
I ν = B ν (T)(1 − e −τ ν (0) ).

Therefore, the bending can be explained if the nebula is completely opaque
(τ ν (0) 1) at low frequencies, but not at high frequencies. In the higher
frequency range, the radiation no longer shows the blackbody spectrum. But
this type of radiation is still usually included in the category of thermal
radiation, since this is caused by the thermal motion of free electrons in
the plasma gas, and tends to have the blackbody (Planck) spectrum in the
completely opaque limit.

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