8) THE TRANSFER EQUATION

Text Reading: Chpt 8 (Sect 7)

A) ABSORPTION

This is any process (including true absorption as well as scattering) that removes light from the "beam".

dI_n=- c_nI_n dl

where dI_n is change in intensity, c_n is absorption (in cm^-1), I_n is the amount of original intensity, and dl is the path length travelled.

Note that mfpµ1/cm^-1»1/c

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B) THE TRANSFER EQUATION

dI_n/dl = -c_n I_n + h_n

where h_n is the "emissivity of gas", and represents that part of the emerging emission that was emitted by the gas itself, and the "-cI_n" term represents that part of the emission that was absorbed by the gas.

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We will define a convenient term called the source function, defined as:

Snºhn/cn

yielding:

®

-dI_n/dl = c_n (I_n - S_n)

Note some special cases:

• If I_n=S_n ® no change in I_n (since dI_n/dl=0)

• If I_n>S_n ® I_n decreases along the pathlength

• If I_n<S_n ® I_n increases along the pathlength

If S_n=B_n ® LTE (but note that I_n=B_n only if the optical depth (t) is very large. More about t in a moment ... )

C) OPTICAL DEPTH IN THE NORMAL (ORTHOGONAL-TO-STELLAR-SURFACE) DIRECTION



This is the parallel-plane approximation. Note that dr=cosq dl

We define:

mºcosq

and

dt_n= -c_n dr = -c_nm dl

thus, the total optical depth is:

tn=òcn dl

So, the final form of the transfer equation, using our new definitions, is:

m dIn/dtn = In - Sn

D) APPLICATIONS OF THE TRANSFER EQUATION

a) LIMB DARKENING

For simplicity, assume that the source function is linearly dependent on the optical depth:

Sn(tn)=a+btn

This is called the "Eddington-Barbier" relation.

With this assumption, and a few others (such as assuming a "semi-infinite" atmosphere, e.g. ,that the atmosphere of the star is so thick that eventually, t®¥ at some depth), the intensity can be written as:

In=a+bm

or, the intensity is the source function evaluated at an optical depth exactly equal to cosq (or m).

Given these assumptions, the emergent intensity is linearly dependent upon the viewing angle q. How? Well, consider this:

When q=0, cosq=m=1 ® I_n is at its maximum value.

When q=90°, cosq=m=0 ® I_n is minimized (at the edges, which is where 90° corresponds to).

®Limb Darkening!



another way to look at it:



Note that when m=t_n, then

In=Sn

® This LTE approximation allows you to measure the source function at different depths in the Sun.

For example, if m=1, ®q=0° (center of disk)

® t=m=1 (as deep as you can see into the Sun)

® get the source function at this depth, by measuring the intensity.

if m=0, ®q=90° (edge of disk)

® t=m=0 (right on top of the solar atmosphere)

® get source function at this depth.



b) SPECTRAL ABSORPTION FEATURES



In the line wings, ® opacity smaller ® formed lower in the atmosphere, since photons can go a long way before getting absorbed.

In the line core, ® opacity largest ® formed higher up in the atmosphere, since photons couldn't travel very far before getting absorbed.

c) GAS NEBULAE

The general solution to the transfer equation is:

I_n=I_n(0) e^{- t/m} + m^-1ò₀^t S_n e^{- t/m} dt

Assume that S_n is independant of t (e.g., S_n constant in above integral), and also consider (in this case) only thenormally-emergent (m=0) light from the slab (e.g., discard the I_n(0) term):

® can show that the solution to the transfer equation is I_n=S_n(1-e^-t)

i) OPTICALLY THICK

t>>1 ® I_n=S_n (blackbody radiation)

ii) OPTICALLY THIN

t<<1 ® I_n=S_nt_n