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Elementary Considerations

When the gas is optically thin, the radiation field at each radius is determined simply by geometrical dilution of the given source spectrum $f_\varepsilon$. Then, as shown by [Tarter Tucker and Salpeter 1969], the state of the gas depends only on the ionization parameter $\xi=L/nR^2$, where $L$ is the (energy) luminosity of the incident radiation integrated from 1 to 1000 Ry, $n$ is the gas density, and $R$ 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. $U_H=F_H/n$ ([Davidson and Netzer 1979]), where $F_H$ is the incident photon number flux above 1 Ry; $\Gamma=F_\nu(\nu_L)/(2hcn)$, where $F_\nu(\nu_L)$ is incident (energy) flux at 1 Ry; and $\Xi=L/(4\pi R^2 cnkT)$ (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 $L$ and either $n$ (the number density) or $P$ (the pressure), depending on which quantity is held fixed. In the case $n$ = constant, this second parameter is simply $(Ln)^{1/2}$ ([McCray, Wright and Hatchett 1977]). This parameter does not allow easy scaling of model results from value of $Ln$ 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, $\tau_e$, 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 $\tau_e\le0.3$, or electron column densities $\le10^{24}$ cm$^{-2}$.


next up previous contents
Next: Algorithm Up: The Physics Behind XSTAR Previous: Input   Contents
Tim Kallman 2014-09-12