Planck and Rosseland Means

With the doMean option in the param.dat file, the Plank and Rosseland means, \(\kappa_P\) and \(\kappa_R\) respectively:

(8)\[\kappa_P = \frac{\int_0^\infty \kappa B_{\nu}(T) d\nu}{\int_0^\infty B_{\nu}(T) d\nu},\]

and

(9)\[\kappa_R = \left( \frac{\int_0^\infty \kappa^{-1} \frac{\partial B_{\nu}(T)}{\partial T} d\nu} {\int_0^\infty \frac{\partial B_{\nu}(T)}{\partial T} d\nu} \right)^{-1}\]

can be calculated, where the infinity integral is truncated to the specified wavenumber limits, and \(d\nu\) is equal to dnu set in the param.dat file.

Note that the denominators in \(\kappa_P\) and \(\kappa_R\) can be computed analytically as

(10)\[\int_0^\infty B_{\nu}(T) d\nu = \frac{\sigma T^4} {\pi}\]

and

(11)\[\int_0^\infty \frac{\partial(B_{\nu}(T))}{ \partial (T)} d\nu = \frac{4 \sigma T^3} {\pi}.\]

Therefor, it is usefull to compare those analytic results to the numerical integration

(12)\[\int_0^\infty B_{\nu}(T) d\nu = \sum_i B_{\nu}(T) dnu\]

and

(13)\[\int_0^\infty \frac{\partial(B_{\nu}(T))}{ \partial (T)} d\nu = \sum_i \frac{\partial(B_{\nu}(T))}{ \partial (T)} dnu\]

The results of the Planck and Resseland means are stored in the file Out<name>_mean.dat, together with the analytic and numerical expressions (10) to (13). If the numerical expressions deviate strongly from the analytical expression, then it is a hint that the wavenumber resolution is not set fine enough.

An example of the Planck and Rosseland means is shown in Fig. 12.

Relevant parameters for this example:

doStoreFullK = 1

doMean = 1