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Radiance as function of the radiative flow
Equation

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Radiance is the derivative of radiative flux at the angle and projected surface section S\cos\theta

$L_i(\vec{x},t)=\displaystyle\frac{\partial^2\Phi_i(\vec{x},t)}{\partial\Omega\partial S\cos\theta}$

ID:(8486, 0)


Radiance function of the spectral radiance
Equation

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The spectral radiance L_{\Omega,
u} is the energy per area of the photons of frequency
u emitted at a solid angle d\Omega.

If the spectral radiance is integrated in the frequency, the total radiance is obtained:

$L_i(\vec{x},\hat{n},t)=\displaystyle\int d\nu L_{i,\nu}(\vec{x},\hat{n},t)$

ID:(8482, 0)


Radiant flow
Equation

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The integration of the radiance L on the solid angle d\Omega gives us the radiative flux \Phi

$\Phi(\vec{x},t)=\displaystyle\int_{4\pi} L(\vec{x},\hat{n},t)d\Omega=\sum_iL_i(\vec{x},\hat{n},t)$

ID:(8483, 0)


Radiant flow in function of the energy
Equation

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The radiative flux is the radiative energy that by time is irradiated:

$\Phi(\vec{x},t)=\displaystyle\frac{\partial Q}{\partial t}$

ID:(8485, 0)


Radiative intensity
Equation

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The radiative intensity is the radiative flux per element of solid angle:

$I_{\Omega}=\displaystyle\frac{\partial\Phi}{\partial\Omega}$

ID:(8484, 0)


Radiative transport equation (RTE)
Equation

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The photon transport equation is

$\displaystyle\frac{1}{c}\displaystyle\frac{\partial}{\partial t}L(\vec{x},\hat{n},t)+\hat{n}\cdot\nabla L(\vec{x},\hat{n},t)=-\mu_tL(\vec{x},\hat{n},t)+\mu_s\int_{4\pi}L(\vec{x},\hat{n}_h,t)P(\hat{n}_h,\hat{n})d\Omega_h+S(\vec{x},\hat{n},t)$

where \mu_t is the absorption coefficient and scattering, c the velocity of light, P(\hat{n}',\hat{n}) is the phase function that gives the probability that a photon traveling in the direction \hat{n} is deflected in the direction \hat{n}'.

ID:(8487, 0)


Thermal Photons
Equation

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For the case in which they are considered uniformly distributed thermal photons their number per cell will be according to the distribution of Bose-Einstein

\displaystyle\frac{1}{e^{\hbar\omega/kT} -1}

where \hbar is the Planck constant divided by 2\pi, \omega is the angular velocity, k the Boltzmann constant, and T the temperature.

If the flow is isotropic it will be necessary that the $ m components will be equal and therefore:

$f_i^{eq}=\displaystyle\frac{1}{e^{\hbar\omega/kT}-1}$

ID:(8561, 0)