18: Lecture Solution of Equation of Radiative Transfer
The equation looks fairly simple
dI = (j - I )ds
If we knew the state of matter, we should be able to calculate j and and solve this. Problem: to know state of matter (i.e. ionization, excitation, T), we need to know I, but to know I we need to know the state of matter.
i.e. a real problem
Typlically approached iteratively
Guess one of the quantities, solve for the other refine the guess
etcetera
In Thermodynamic Equilibrium
divide by .
Solutions
giving:
1 dI j = I ds = S
ds = d where is the "optical depth"
j the "source function"
In T.E., Kirchoff showed the emissivity of an object must equal its absorptivity, and therefore
S = B (T), the Planck function i.e. j = B (T),
Note that if S > I, dI/ds > 0, i.e. I increases
and vice-versa until I = S
Solution, given a known intensity I0 at s = 0
show on board,
Take case S = const., consider emission from optically box
thin and optically thick
Formal Solution for intensity I
Equation becomes, in terms of source function and optical depth
dI + I = S d multiply both sides by e dI e + e I = S e d consider that d dI e I = e + e I d d
(
)
so the equation of radiative transfer becomes d e I = S e d
(
)
this can be formally integrated along the path starting at s = 0 where I = I0 and = 0 to the new position s = s
I0
s, = 0
I
s = s, =
Solution given by integral of source function S attenuated by exp(-) along path
The solution is
d ' ( e
d
0
'
...