# Blackbody Calculator

Enter parameters to calculate blackbody radiance or exitance in watt or photon space over the specified wave band as follows:

Paramter Photon space Watt space
Spectral Radiance $L_{q,\lambda }(\lambda ,T) = \frac{2c}{\lambda ^{4}(e^{\frac{hc}{\lambda kT}}-1)}$ $L_{e,\lambda }(\lambda ,T) = \frac{2hc^{2}}{\lambda ^{5}(e^{\frac{hc}{\lambda kT}}-1)}$
In-Band Radiance $\int_{\lambda _{1}}^{\lambda _{2}}\varepsilon L_{q,\lambda }(\lambda ,T)d\lambda$ $\int_{\lambda _{1}}^{\lambda _{2}}\varepsilon L_{e,\lambda }(\lambda ,T)d\lambda$
Total Radiance $L_{q} = \frac{\varepsilon \sigma_{q} T^{3}}{\pi }$ $L_{e} = \frac{\varepsilon \sigma_{e} T^{4}}{\pi }$
$\sigma$ (Stefan-Boltzmann Constant) $\sigma_{q} = \frac{4 \zeta (3) k^3 \pi}{h^3 c^2}$ $\sigma_{e} = \frac{2 \pi^5 k^4}{15 h^3 c^2}$
Peak emission wavelength $\lambda _{peak} = \frac{b_{q}}{T}$ $\lambda _{peak} = \frac{b_{e}}{T}$
$b$ (Wien's Displacement Constant) $b_{q} = \frac{10^6 h c}{a_4 k}$ $b_{e} = \frac{10^6 h c}{a_5 k}$
Spectral Exitance $M_q=\pi L_q$ $M_e=\pi L_e$
The equations above use the following constant values:

Symbol Name Value Reference
$c$ Speed of Light $299792458$ NIST
$h$ Planck Constant $6.62607015 \times 10^{-34}$ NIST [*]
$k$ Boltzmann Constant $1.380649 \times 10^{-23}$ NIST
$a_4$ Solution to $4 \left ( 1-e^{-x} \right ) - x = 0$ $3.92069039487$ WA
$a_5$ Solution to $5 \left ( 1-e^{-x} \right ) - x = 0$ $4.96511423174$ WA
$\zeta (3)$ Apery's Constant $1.202056903159594$ OEIS

[*] The Planck Constant has undergone many redefinitions because historically, its value was determined experimentally against a kilogram standard. However, as of 2018, the Planck Constant is the standard, and is fixed to the duration of the cesium-133 ground state transition. The use of any other value for Planck's Constant is incorrect.

References:
1. Spectral Calc Calculation of Blackbody Radiance

## Photons / sec

Value Quantity Units
Peak Wavelength Microns