Radiant exitance

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 (W/m2), 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 (W·m−2·nm−1). The CGS unit erg per square centimeter per second (erg·cm−2·s−1) 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 Me ("e" for "energetic", to avoid confusion with photometric quantities), is defined as M e = ∂ Φ e ∂ A , {\displaystyle M_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},} where ∂ is the partial derivative symbol, Φe is the radiant flux emitted, and A 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 e ∘ = σ T 4 , {\displaystyle M_{\mathrm {e} }^{\circ }=\sigma T^{4},} where σ is the Stefan–Boltzmann constant, and T is the temperature of that surface. For a real surface, the radiant exitance is equal to: M e = ε M e ∘ = ε σ T 4 , {\displaystyle M_{\mathrm {e} }=\varepsilon M_{\mathrm {e} }^{\circ }=\varepsilon \sigma T^{4},} where ε is the emissivity of that surface.

Spectral exitance

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

M e , ν = ∂ M e ∂ ν , {\displaystyle M_{\mathrm {e} ,\nu }={\frac {\partial M_{\mathrm {e} }}{\partial \nu }},}

where ν is the frequency.

Spectral exitance in wavelength of a surface, denoted Me,λ, is defined as M e , λ = ∂ M e ∂ λ , {\displaystyle M_{\mathrm {e} ,\lambda }={\frac {\partial M_{\mathrm {e} }}{\partial \lambda }},} where λ 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:

M e , ν ∘ = π L e , Ω , ν ∘ = 2 π h ν 3 c 2 1 e h ν k T − 1 , M e , λ ∘ = π L e , Ω , λ ∘ = 2 π h c 2 λ 5 1 e h c λ k T − 1 , {\displaystyle {\begin{aligned}M_{\mathrm {e} ,\nu }^{\circ }&=\pi L_{\mathrm {e} ,\Omega ,\nu }^{\circ }={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }^{\circ }&=\pi L_{\mathrm {e} ,\Omega ,\lambda }^{\circ }={\frac {2\pi hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}},\end{aligned}}}

where h is the Planck constant, ν is the frequency, λ is the wavelength, k is the Boltzmann constant, c is the speed of light in vacuum, T is the temperature of that surface. For a real surface, the spectral exitance is equal to: M e , ν = ε M e , ν ∘ = 2 π h ε ν 3 c 2 1 e h ν k T − 1 , M e , λ = ε M e , λ ∘ = 2 π h ε c 2 λ 5 1 e h c λ k T − 1 . {\displaystyle {\begin{aligned}M_{\mathrm {e} ,\nu }&=\varepsilon M_{\mathrm {e} ,\nu }^{\circ }={\frac {2\pi h\varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }&=\varepsilon M_{\mathrm {e} ,\lambda }^{\circ }={\frac {2\pi h\varepsilon c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}.\end{aligned}}} where ε {\displaystyle \varepsilon } is the emittance of the surface.

SI radiometry units

See also

• Radiosity