As discussed in Part 2 (12/11 post) we can express the equivalent width in two ways, based on frequency (u) *or* wavelength (l):

W = ò** **^{¥}_{0 } (I _{c} – I _{u }/ I _{c} )du = ò ^{¥}_{0 } (F _{c} – F _{l }/ F _{c} ) dl

The left side defines W in terms of the intensity e.g. from the continuous spectrum outside of the spectral line where the quantity (I _{c} – I _{u}/ I _{c} ) is referred to as the “*depth of the line*”. This is the analogous quantity to (F _{c} – F _{l }/ F _{c} ) on the right side where we have radiant flux units.

Having obtained the equivalent width, the next logical step is to
generate the “curve of growth”. Recall
from the previous section that W, the equivalent width, represents the line
strength. Then the curve of growth describes how the latter increases as the
optical depth, τ increases.

To fix ideas, we use a model slab of
finite thickness ds and over which the optical depth increases by d
τ. The incident intensity is I _{c}_{
} for frequencies in the
neighborhood of the line, and the
intensity emerging from the opposite side is I _{u }which we seek to find.

Starting with the
original basic transfer equation based on wavelength:

dI(l)/ds = -k(l) I(l) + k(l) S(l)

We can derive the frequency form:

d I_{u
}/ dτ _{u
} = - I_{u} – S _{u}_{ }

We have (from the
original equation) after adjusting for frequency:

d I** _{u
}**/ ds

**= -k**

_{ }_{u}I

_{u}+ k

_{u}S

_{u}

Where: k _{u} = d τ _{u }/ ds

So:

d I_{u
}/ ds** _{ }** = -( d τ

_{u }/ ds) I

_{u}+ (d τ

_{u }/ ds) S

_{u}

_{}d I_{u
}= - d τ _{u
} I_{u} + d τ _{u } S _{u
}= - d τ _{u } (I_{u} – S _{u})_{}

whence:

d I_{u
}/ dτ** **_{u } =
- I_{u} – S _{u}_{ }

For which the solution
is found (on integration):

I** _{u}** = I

**exp (-τ**

_{ c}

_{u})

*Where: **τ *

_{u}

*= N L*

*a*

_{ u}

*= N L*

*a*

_{o}*f*

_{u}

Where ‘N’ is the number
of absorbing atoms per unit volume and we already saw that:

a
** _{o }**= a

**/**

_{u }**f**

_{ }_{u }= a

**[1 - e**

_{u}^{- h }

^{u o}

^{ / kT}] (

**p**e

^{2}/ mc) f

In terms of this newly derived information, the equivalent width of the line can then be written as:

W** _{ u}** = ò

^{¥}

_{0}

**[1 - exp (- N L a**

_{ }**f**

_{o}_{u}) ] du

For small x we may use
the approximation such that:

_{}

exp (-x) » 1 – x

So rewrite the equation for W** _{
u}**:

W** _{ u }** » ò

^{¥}

_{0}

**[1 - (1 - N L a**

_{ }**f**

_{o}_{u}) du

» ò ^{¥}_{0}_{ }** _{ }** N L a

**f**

_{o}_{u}du » N L a

_{o}**ò**

^{¥}

_{0}

**f**

_{ }_{u}du

So we find W** _{
u}** depends only on the form for
the broadening function. For very weak
lines, for example:

ò ^{¥}_{0}** _{ }**f

_{u}du » 1

so that W** _{
u}** » N L a

_{o}Or simply proportional to N, the number of
absorbing atoms. For “strong” lines the absorption near the center is very
large so we can expect:

N L a ** _{o}** f

_{u}>> 1

And for the *moderately strong* lines:

W** _{ u}** =
2

**D**u

_{D}

**{ln (N L a**

**/Ö**

_{o}**p**D u

_{D})}

^{½}

While *very
strong lines* yield:

W** _{ u}** =
1/p (N L a

_{o}g)

^{½}

Where g is the damping
constant. Assembling all the diverse W_{ }_{u} and
plotting log W** _{ u}**
vs. log (N
L a

_{o}) one gets the curve of growth shown below:

__Suggested Problem: __

How would the curve of growth appear if the broadening function:

f _{u }du = 1/ Ö**p** [exp (u - u _{o})_{ }/D u_{D} ]^{2} du / D u_{D}

Had: (u) » (u_{o})

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