Lecture #3b-- Photon emission in non-LTE
1) Sample problem
A sample problem is worked where we find the necessary dilution factor
W for a 3-eV quasi-thermal radiation spectrum, to yield 50% ionization,
in a gas where the density is 10^-16 particles per elementary state volume
(the latter is the cube of the deBroglie wavelength of thermal electrons).
We find W must be about 10^-14, so we need to be about 10^7 solar radii,
or about 100 pc, from the O-star. It is pointed out that HII regions are
rarely so large-- because we did not worry about exhausting all possible
ionizing photons.
2) Concept of Emission Measure
If we have an optically thin box, then all the light emitted in the box
will escape to be seen. Since the densities are far below critical, all
the de-excitation will be radiative, so we need only worry about the
excitation rates. Unless we are looking at flourescence, all the excitation
will either be collisional (so proportional to density-squared)
excitation, or it
will be radiative recombination (also proportional to density-squared),
so the excitation rate in the box is proportional to the density squared.
The emission just depends on inherent radiative rates, so the total rate
that light emerges will be proportional to the density-squared times the
volume. This latter quantity is therefore called the "emission measure", and
is elevated by clumping even if you have a fixed amount of stuff in the box.
3) Electronic vs. vibrational vs. rotational transitions
For a system to emit a photon, there must be something inside the system
that is responding at the frequency of the photon, so that it can resonate
with the photon it is making over many many cycles (each cycle is itself a
very weak interaction, that's the flavor of "perturbation theory"). That
also puts a constraint on the characteristic frequency the system can emit.
For quantum systems of fixed size L, the uncertainty principle dictates that
the emitted frequency be proportional to the mass of the charged particle
doing the emitting (say an electron in a hydrogen atom). For classical
oscillators, the frequency is inversely proportional to the square root
of the mass, because the restoring force limits the amplitude of the
oscillation in a way such that increasing the acceleration by reducing
the mass will shorten the period, but not proportionally because the
amplitude is also enhanced. For classical rotators, the frequency is
inversely proportional to the mass, because the amplitude of the oscillation
is fixed by the size of the molecule that is rotating.
When you go from the oscillations of the electron in an atom, to the
vibrations of the nuclei themselves, you can treat the nuclei classically
and say that their mass is up by a factor of 1843 so their frequency is
down by the square root of that-- putting it in the near-infrared.
If you look instead at the rotations of the nuclei around each other,
you are down in frequency by the full 1843, so you are in the submillimeter
regime.