Radiance as function of the radiative flow
Equation
Radiance is the derivative of radiative flux at the angle and projected surface section
$L_i(\vec{x},t)=\displaystyle\frac{\partial^2\Phi_i(\vec{x},t)}{\partial\Omega\partial S\cos\theta}$ |
ID:(8486, 0)
Radiance function of the spectral radiance
Equation
The spectral radiance
u}
u
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
The integration of the radiance
$\Phi(\vec{x},t)=\displaystyle\int_{4\pi} L(\vec{x},\hat{n},t)d\Omega=\sum_iL_i(\vec{x},\hat{n},t)$ |
ID:(8483, 0)
Radiant flow in function of the energy
Equation
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
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
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
ID:(8487, 0)
Thermal Photons
Equation
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
where
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)