Lecture #3a-- Rate and Energy Balance
1) Non-LTE rate balance:
What "non-LTE" means is essentially that the characteristics of the
radiation field are much different from what would be the case for
thermodynamic equilibrium at the same temperature as the gas.
That tends to happen when you have a nonisotropic radiation field (a
nearby star) coupled with low optical depths, and that also tends to
mean low density. Low density means that the collisional rates are
negligible compared to the radiative rates (so densities much less than
the "critical density"), so we have a radiatively-dominated gas and
the radiation field is coming from special places (the stars).
2) Critical densities
There are actually all sorts of critical densities, depending on the
processes being compared. They are always the densities where a certain
rate may be accomplished equally often by a radiative version or a collisional
version of that process (the collisional version always substitutes the role of a colliding particle, often an electron, for the photon, so always involves
an extra power of density in the rate-- ergo the critical density). If we
consider a single process and its inverse, then if the density is well above
the critical value for both the forward and reverse process, we have a
situation that is the same as you get in thermodynamic equilibrium. That
means the likelihood of finding the system (atom or molecule) in the parent
or daughter state of that process will be determined entirely by
counting the accessible states, and the Boltzmann factor will do that for
the rest of the universe. However, if the density drops until it goes
below either the upward-transition or reverse-transition critical densities,
then the population ratio for the upper over lower state will either go
above (in the former case) or below (in the latter case) the value it
would have in thermodynamic equilibrium at the gas temperature.
The former case is like the presence of a bright radiation field
(i.e., a nearby star), and the
latter case is in the absence of a bright radiation
field (i.e., the cosmic microwave background),
both at low density (like in the ISM).
3) The rate equations
The population rate equations describe the balance between upward and
downward transitions that connect two states of the system.
In LTE, the population ratio is indepedent of density for transitions
between bound states,
and inversely proportional to electron density for ionizing transitions.
In non-LTE, the radiation temperature and dilution factor control
that ratio, yet it is still independent of density for bound-state
transitions and inverse to density for ionizations.
When the density is far below critical, the value of the critical
densities no longer matters, but what does matter is the number of
accessible states per free electron, given by 2 over the density divided
by the cube of the deBroglie wavelength of thermal electrons.
The latter can be thought of as the volume per state.