In-band Radiance: Integrating the Planck Equation Over a Finite Range(click on equations to view enlarged)
Above we analytically integrated the spectral radiance over the entire spectral range. The result, Eq. (20), is the well-known Stefan-Boltzmann law. Similarly, Eq. (22) gives the integrated photon radiance. As useful as the Stefan-Boltzmann law is, for many applications a finite spectral range is needed. To facilitate this, we compute the one-sided integral of the spectral radiance. We follow the method described by Widger and Woodall, using units of wavenumber. Note that using other spectral units produces the same result, because it represents the same physical quantity.
Noting that , we get .
The remaining integral can be integrated by parts:
Testing shows that carrying the summation up to n = min(2+20/x, 512) provides convergence to at least 10 digits.
Any finite range can be computed using two one-sided integrals:
Further, the complimentary integral is easily evaluated using (19):
A similar formula can be derived for the in-band photon radiance:
Again using , we get .
Integrating by parts so
Equations (23) and (24)
provide efficient formulas for computing in-band radiance. Example C++ computer source code is provided
in Appendix A.
|Calculation of a Blackbody Radiance|
|Units of Frequency|
|Units of Wavelength|
|Units of Wavenumbers|
|Radiance: Integrating the Planck Equation|
|In-band Radiance: Integrating the Planck Equation over a Finite Range|
|Appendix A: Algorithms for Computing In-band Radiance|
|Appendix B: The Doppler Effect|
|Appendix C: Summary of Formulas|
 Widger, W. K. and Woodall, M. P., Integration of the Planck blackbody radiation function, Bulletin of the Am. Meteorological Society, 57, 10, 1217-1219, Oct. 1976
 CRC Handbook of Chemisry and Physics, 56th edition #521