single line

curve of growth

m: molecules cm-2

cm – atm g cm-2

as we increase amount of gas from m1 to m3

we see more absorption.

Note: No multiple scattering is attempted here.

Define

Equivalent width

The way Wan is defined, it is a measure of the area

Optically thin case

Note: W is linear in amount of gas a, and “integrated cross-section”

set

Then

Optically thick case

   

Note W is square root in the amount of gas a, the line width αL and integrated crossection S.

The quantities a and S are clear to understand.Square root means things are no longer very efficient.Why αL1/2 ? Reason: We are eating into the far wings, and αL tells us where to stop.

Band Models

A band consists of a large number of lines

each with intensity S , width α , and

spacing δ e.g.

Since there are a large number of lines, it is convenient to define some sort of statistical mean transmission over the entire band.

There are two types of band models *

(1) Elsasser type models

(a) vi equally spaced

(b) si all equal αi (2) Goody type models ; Godson

(a) vi random

(b) S has a distribution P(s,σ) such that s = σ (Malkmus)

(*) Analytic

Computer age “band models”

k – distributions

spectral mapping technique

Godson

Goody

Malkmus

Godson

Elssasser : Regular

Elsasser Model

Infinite band

Note that is a measureof the packing of lines

One line

(4.66)

(4.67)

Langest contribution to (4.66) is n≡0 term

Goody’s random band

LorentzDefine probability distribution

Given m, the system has a chance toSample range of weak and string lines

exponential