1. Chapter 13 – Behavior of Spectral Lines
We began with line absorption coefficients which give
the shapes of spectral lines. Now we move into the
calculation of line strength from a stellar atmosphere.
Formalism of radiative transfer in spectral lines
Transfer equation for lines
The line source function
Computing the line profile in LTE
Depth of formation
Temperature and pressure dependence of line strength
The curve of growth

2. The Line Transfer Equation
We can add the continuous absorption coefficient and the line absorption coefficient to get the total absorption coefficient:
dtn = (ln+kn)rdx
And the source function as the sum of the line and continuous emission coefficients divided by the sum of the line and continuous emission coefficients.
Or define the line and continuum source functions separately:
Sl=jnl/ln
Sc=jnc/kn
In either case, we still have the basic transfer equation:

3. The Line Source Function
The basic problem is still how to obtain the source function to solve the transfer equation.
But the line source function depends on the atomic level populations, which themselves depend on the continuum intensity and the continuum source function. This coupling complicates the solution of the transfer equation for lines.
Recall that in the case of LTE the continuum source function is just the Bn(T), the Planck Function.
The assumption of LTE simplifies the line case in the same way, and allows us to describe the energy level populations strictly by the temperature without coupling to the radiation field.
This approximation works when the excitation states of the gas are defined primarily by collisions and not radiative excitation or de-excitation.

4. Mapping the Line Source Function
The line source function with depth maps into the line profile
The center of the line is formed at shallower optical depth, and maps to the source function at smaller t
The wings of the line are formed in progressively deeper layers

5. Computing the Line Profile
The line profile results from the solution of the transfer equation at each Dl through the line.
The line profile will depend on the number of absorbers at each depth in the atmosphere
The simplifying assumptions are
LTE
Pure absorption (no scattering)
How well does this work?
To know for sure we must compute the line profile in the general case and compare it to what we get with simplifying assumptions
Generally, it’s pretty good
Start with the assumed T(t) relation and model atmosphere
Recompute the flux using the line+continuous opacity at each wavelength around the line
For blended lines, just add the line absorption coefficients appropriate at each wavelength

6. Depth of Formation
It’s straightforward to determine approximately where in the atmosphere (in terms of the optical depth of the continuum) each part of the line profile is formed
But even at a specific Dl, a range of optical depths contributes to the absorption at that wavelength
It’s not straightforward to characterize the depth of formation of an entire line
The cores of strong lines are formed at very shallow optical depths.

7. The Ca II K 3933A
Why does the line have an emission peak (a reversal) in the center?
In the Sun, the central emission peak is self-absorbed. Why?

8. The Behavior of Line Strength
The strengths of spectral lines depend on
The width of the absorption coefficient
Thermal and microturbulent velocities
The number of absorbers
Temperature
Electron pressure or luminosity
Atomic constants
In strong lines – collisional line broadening affected by the gas and electron pressures

9. The Effect of Temperature
Temperature is the main factor affecting line strength
Exponential and power of T in excitation and ionization
Increase with T due to increase in excitation
Decrease beyond maximum an increase in the opacity or from ionization

10. H-g Profiles
H lines are sensitive to temperature because of the Stark effect
The high excitation of the Balmer series (10.2 eV) means excitation continues to increase to high temperature (max at ~ 9000K).
Most metal lines have disappeared by this temperature. Why?

11. How do different kinds of lines behave with temperature?
Lines from a neutral species of a mostly neutral element
Lines from a neutral species of a mostly ionized element
Lines from an ion of a mostly neutral element
Lines from an ion of a mostly ionized element
Consider gas with H- as the dominant opacity

12. Neutral lines from a neutral species
Number of absorbers proportional to exp(-c/kT)
Number of neutrals independent of temperature (why?)
Ratio of line to continuous absorption coefficient
But Pe is ~ proportional to exp(T/1000), so…