Demo for Moments of RTE (Radiative transfer equation)

[fab13]

Hello,

I would like to demonstrate the 2 below equations ($eq(1)$ and $eq(2)$) concerning the first and second moment of Radiative Transfer Equation (RTE) (where $I_{\nu}$ represents the specific intensity) :

Starting from the definitions :

$J_{\nu}=\dfrac{1}{2} \int_{-1}^{1} I_{\nu}(\mu) \text{d}\mu$

with

$\mu \dfrac{\text{d}I_{\nu}}{\text{d}\tau_{\nu}} = I_{\nu} - S_{\nu}$

and

$H_{\nu}=\dfrac{1}{2} \int_{-1}^{1} I_{\nu}(\mu) \mu \text{d}\mu$

also :

$K_{\nu}=\dfrac{1}{2} \int_{-1}^{1} I_{\nu}(\mu) \mu^{2} \text{d}\mu$

Finally, I would like to get :

$dH/d\tau = J - S\quad(1)$

and

$dK/d\tau = H\quad(2)$

with S the source function :

Maybe also, I do confusions between $\tau$ and $\tau_{\nu}$ : could you help me to grasp these subtilities and clarify the definition of these two parameters ?

Regards

 

[AndyW]

Hello,

Thank you for your question and for your interest in the radiative transfer equation (RTE). I am happy to help clarify the definitions and equations you have mentioned.

First, let's define some terms. The specific intensity, $I_{\nu}$, represents the amount of radiation at a given frequency, $\nu$, in a specific direction. The source function, $S_{\nu}$, is a measure of the amount of radiation emitted by a source at a given frequency, and it depends on the temperature and other properties of the source.

Now, let's look at the equations you have mentioned. The first equation, $J_{\nu}=\dfrac{1}{2} \int_{-1}^{1} I_{\nu}(\mu) \text{d}\mu$, is the definition of the mean intensity, $J_{\nu}$, which is the average intensity of radiation in all directions at a given frequency. The integral includes all possible directions, from -1 to 1, and the factor of 1/2 is included to account for the symmetry of the radiation field.

The second equation, $\mu \dfrac{\text{d}I_{\nu}}{\text{d}\tau_{\nu}} = I_{\nu} - S_{\nu}$, is known as the radiative transfer equation. It describes the change in intensity along a ray of radiation as it travels through a medium with optical depth, $\tau_{\nu}$. The left side of the equation represents the change in intensity with respect to optical depth, and the right side represents the balance between the incoming intensity, $I_{\nu}$, and the emitted radiation from the source, $S_{\nu}$.

The third equation, $H_{\nu}=\dfrac{1}{2} \int_{-1}^{1} I_{\nu}(\mu) \mu \text{d}\mu$, is the definition of the net flux, $H_{\nu}$, which is the amount of radiation passing through a unit area in a given direction at a given frequency. The integral again includes all possible directions and the factor of 1/2 is included for symmetry.

Finally, the fourth equation, ##K_{\nu}=\dfrac{1}{2} \int_{-1