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.