When the gas is optically thin, the
radiation field at each radius is determined simply by geometrical
dilution of the given source spectrum .
Then, as shown by [Tarter Tucker and Salpeter 1969],
the state of the gas depends only on the ionization parameter
,
where
is the (energy) luminosity of the incident radiation integrated from 1 to 1000 Ry,
is the gas density, and
is the distance from the radiation source. This
scaling law allows the results of one model calculation to be
applied to a wide variety of situations. For a given choice of
spectral shape this parameter is
proportional to the various other customary ionization parameter
definitions, i.e.
([Davidson and Netzer 1979]), where
is the incident photon number
flux above 1 Ry;
, where
is incident (energy) flux at 1 Ry; and
(e.g. [Krolik McKee and Tarter, 1981]).
In the optically thick case,
[Hatchett Buff and McCray 1976], and [Kallman 1983]
showed that the state of the gas could be
parameterized in terms of an additional parameter which is a
function of the product of and either
(the number density)
or
(the pressure), depending on which quantity is held fixed.
In the case
= constant,
this second parameter is simply
([McCray, Wright and Hatchett 1977]).
This parameter does not allow easy scaling of model results from
value of
to another, since the dependence on this parameter
is non-linear, but it does provide a useful indicator of which
combinations of parameter values are likely to yield similar results
and vice versa.
When the electron scattering optical depth, , of the cloud
becomes significant, the outward-only approximation used here breaks
down, and different methods of describing the radiative
transfer must be used (e.g. [Ross 1979]). Therefore, the range of
validity of the models presented here is restricted to
,
or electron column densities
cm
.