Normalized source function: physical interpretation

[fab13]

Hello,

In physical radiative processes, I am studying the solution of the following equation :

$S=(1-\epsilon) J + \epsilon B$

with :

1) S the source function
2) J the mean intensity ( $J=\dfrac{1}{2}\int_{-1}^{1} I_{\nu}\text{d}\tau_{\nu}$)
3) B the Boltzmann distribution
4) $\epsilon$ a coefficient
5) Below $\tau_{\nu}$ is optical depth

With Milne-Eddington approximation, we get : $J=3K$ (with K the seond moment of RTE (Radiation Transfer Equation)).

Moreover, we have : $\dfrac{\text{d}K}{\text{d}\tau_{\nu}}=H\quad(1)$ and $\dfrac{\text{d}H}{\text{d}\tau_{\nu}}=J-S\quad(2)$

Taking these two equations (1), (2) and $J=3K$, one has the following solution for J :

$J = J(\tau_{\nu}) = B - \dfrac{B}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}$

which implies :

$J = B \bigg[1-\dfrac{1}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg]$

So : $S=(1-\epsilon) J + \epsilon B = J - \epsilon J + \epsilon B = B \bigg[1-\dfrac{1}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg] - \epsilon B \bigg[1-\dfrac{1}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg] + \epsilon B$

$S= B \bigg[1-\dfrac{1}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}} + \dfrac{\epsilon}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg]$

$S=B\bigg[1-\dfrac{(1-\epsilon)}{1+\sqrt{\epsilon}}\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg]$

$S=B\bigg(1-(1-\sqrt{\epsilon})\,e^{-\sqrt{3\epsilon} \tau_{\nu}}\bigg)\quad(3)$

As a conclusion, It is asked to do a graphical representation of normalized source function S/B and to comment about non-LTE "law" that $S \approx \sqrt{\epsilon} B$ ?

I would like to know what graphical representation is expected for normalized function S/B and what can I comment about the equation I got $(1)$ and the "non-LTE" law that says : $S \approx \sqrt{\epsilon} B$ ??

I mean, I would like to correctly and physically interpret the result in (3) and do the link between "LTE" and "non-LTE" cases.

Could I say that :

1) If $\tau_{\nu}$ is small (at the surface), then I find the non-LTE case :

$S/B=\sqrt{\epsilon}\quad(4)$

2) If $\tau_{\nu}$ is large (i.e, large depth), then I find the Boltzmann case :

$S/B=1\quad(5)$

?

Any help is welcome

Thanks in advance

 

[AndyW]

Thank you for your post and interest in physical radiative processes. The equation you are studying is a common one in radiative transfer, and it is important to understand its implications and behavior. Your solution for J is correct, and it leads to a simplified expression for S. To answer your questions, I will first explain the significance of the LTE and non-LTE cases, and then provide a graphical representation of the normalized source function S/B.

LTE stands for Local Thermodynamic Equilibrium, which is a state in which the energy distribution of a system is determined only by its local temperature. In this case, the source function S is equal to the Planck function B at all depths. This is because in LTE, the radiation field is in equilibrium with the matter, and the matter is in thermal equilibrium. Therefore, the radiation field is described by the Boltzmann distribution, which is also the source function in this case. In other words, the LTE case is a special case of the non-LTE case where $\epsilon=0$.

In the non-LTE case, the source function S is not equal to the Planck function B, and it is affected by the coefficient $\epsilon$. As you correctly pointed out, when $\tau_{\nu}$ is small (i.e. at the surface), the non-LTE case approaches the LTE case, and the source function S is approximately equal to the Planck function B. This is because at the surface, the radiation field is in equilibrium with the matter, and the matter is in thermal equilibrium, resulting in an LTE state. However, as you move deeper into the medium, the radiation field becomes more and more affected by the matter, and the non-LTE case deviates from the LTE case.

To graphically represent the normalized source function S/B, you can plot it as a function of $\tau_{\nu}$. This will show you the behavior of the source function at different depths. In the LTE case, the normalized source function is equal to 1 at all depths, as S=B. In the non-LTE case, the normalized source function will initially approach 1 at the surface, but as you move deeper into the medium, it will start to deviate from 1 and approach $\sqrt{\epsilon}$. This behavior is shown in the graph below:

[Graph showing the behavior of the normalized source function S/B as a