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In-band Radiance: Integrating the Planck Equation Over a Finite Range

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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[2], 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[3]:

This gives

(23)

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

(24)

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 References Blackbody Calculator Print Version

[2] 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

[3] CRC Handbook of Chemisry and Physics, 56th edition #521