Radiant exitance

Template:Short description In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre (Template:Nobreak), while that of spectral exitance in frequency is the watt per square metre per hertz (W·m⁻²·Hz⁻¹) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m⁻³)—commonly the watt per square metre per nanometre (Template:Nobreak). The CGS unit erg per square centimeter per second (Template:Nobreak) is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

Mathematical definitions

Radiant exitance

Radiant exitance of a surface, denoted Template:Math ("e" for "energetic", to avoid confusion with photometric quantities), is defined as $$M_{\mathrm{e}}=\frac{\partial {\mathrm{\Phi }}_{\mathrm{e}}}{\partial A},$$ where Template:Math is the partial derivative symbol, Template:Math is the radiant flux emitted, and Template:Mvar is the surface area.

The radiant flux received by a surface is called irradiance.

The radiant exitance of a black surface, according to the Stefan–Boltzmann law, is equal to: $$M_{\mathrm{e}}^{\circ }=\sigma T^4,$$ where Template:Mvar is the Stefan–Boltzmann constant, and Template:Mvar is the temperature of that surface. For a real surface, the radiant exitance is equal to: $$M_{\mathrm{e}}=\epsilon M_{\mathrm{e}}^{\circ }=\epsilon \sigma T^4,$$ where Template:Mvar is the emissivity of that surface.

Spectral exitance

Spectral exitance in frequency of a surface, denoted M_e,ν, is defined as

$M_{\mathrm{e},\nu }=\frac{\partial M_{\mathrm{e}}}{\partial \nu },$

where Template:Mvar is the frequency.

Spectral exitance in wavelength of a surface, denoted M_e,λ, is defined as $$M_{\mathrm{e},\lambda }=\frac{\partial M_{\mathrm{e}}}{\partial \lambda },$$ where Template:Mvar is the wavelength.

The spectral exitance of a black surface around a given frequency or wavelength, according to Lambert's cosine law and Planck's law, is equal to:

$\begin{array}{cc}{M}_{\mathrm{e},\nu }^{\circ }& =\pi L_{\mathrm{e},\mathrm{\Omega },\nu }^{\circ }=\frac{2\pi h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1},\\ M_{\mathrm{e},\lambda }^{\circ }& =\pi L_{\mathrm{e},\mathrm{\Omega },\lambda }^{\circ }=\frac{2\pi h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1},\end{array}$

where Template:Mvar is the Planck constant, Template:Mvar is the frequency, Template:Mvar is the wavelength, Template:Mvar is the Boltzmann constant, Template:Mvar is the speed of light in the medium, Template:Mvar is the temperature of that surface. For a real surface, the spectral exitance is equal to: $$\begin{array}{cc}{M}_{\mathrm{e},\nu }& =\epsilon M_{\mathrm{e},\nu }^{\circ }=\frac{2\pi h\epsilon {\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1},\\ M_{\mathrm{e},\lambda }& =\epsilon M_{\mathrm{e},\lambda }^{\circ }=\frac{2\pi h\epsilon c^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}.\end{array}$$

SI radiometry units

Comparison of photometric and radiometric quantities

See also

• Radiosity