Blackbody Calculator

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[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




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
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