1. Behavior of Spectral Lines – Part II
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. 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

3. 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…

4. Neutral Lines of a Neutral Species
Oxygen triplet lines at 7770A.
Excitation potential = 8 eV
Ionization potential = 13.6 eV
Oxygen resonance line [O I] at 6300A
By what factor will each of these lines change in strength from 5000 to 6000K?
Factors of ~4 and 0.75

5. Neutral Lines of an Ionized Species
How much would you have to change the temperature of a 6000K star to decrease the equivalent width of the Li I 6707 resonance line by a factor of two?
Ionization potential = 5.4 eV
Raise T by ~333K

6. Ionized Lines of a Neutral Element
Fe II lines in giants are often used to determine the spectroscopic gravity.
How sensitive to temperature is a 2.5eV Fe II line (I=7.9 eV) in a star with Teff=4500K? (Estimate for DT=100K)
EQW4600=1.5EQW4500

7. Ionized Lines of Ionized Species
How strong is a Ba II line (at 0 eV) in a 6000K star compared to a 5000K star?
How do the strengths of a 5 eV Fe II line compare in the same two stars?
For Ba II, EQW decreases by 25%
For Fe II, EQW is almost x3 larger

8. Line Strength Depends on Pressure
For metal lines, pressure (gravity) affects line strength in two ways:
Changing the line-to-continuous opacity ratio (by changing the ionization equilibrium)
Pressure broadening
Pressure effects are much weaker than temperature effects

9. Rules of Thumb for Weak Lines
When most of the atoms of an element are in the next higher state of ionization, lines are insensitive to pressure
When H- opacity dominates, the line and the continuous absorption coefficients are both proportional to the electron pressure
Hence the ratio line/continuous opacity is independent of pressure
When most of the atoms of an element are in the same or a lower state of ionization, lines are sensitive to pressure
For lines from species in the dominant ionization state, the continuous opacity (if H-) depends on electron pressure but the line opacity is independent of electron pressure
Lines from a higher ionization state than the dominant state are highly pressure dependent
H- continuous opacity depends on Pe
Degree of ionization depends on 1/Pe

10. Examples of Pressure Dependence
Sr II resonance lines in solar-type stars
7770 O I triplet lines in solar-type stars
[O I] in K giants
Fe I and Fe II lines in solar-type stars
Fe I and Fe II lines in K giants
Li I lines in K giants

11. The Mg I b lines
Why are the Mg I b lines sensitive to pressure?

12. Pressure Effects on Hydrogen Lines
When H- opacity dominates, the continuous opacity is proportional to pressure, but so is the line abs. coef. in the wings – so Balmer lines in cool stars are not sensitive to pressure
When Hbf opacity dominates, kn is independent of Pe, while the line absorption coefficient is proportional to Pe, so line strength is too
In hotter stars (with electron scattering) kn is nearly independent of pressure while the number of neutral H atoms is proportional to Pe2. Balmer profiles are very pressure dependent

13. What Is Equivalent Width?
The equivalent width is a measure of the strength of a spectral line
Area equal to a rectangle with 100% depth
Triangle approximation: half the base times the width
Integral of a fitted line profile (Gaussian, Voigt fn.)
Measured in Angstroms or milli-Angstroms
How is equivalent width defined for emission lines?

14. The Curve of Growth Wrubel COG from Aller and Chamberlin 1956
The curve of growth is a mathematical relation between the chemical abundance of an element and the line equivalent width
The equivalent width is expressed independent of wavelength as log W/l
Wrubel COG from Aller and Chamberlin 1956

15. Curves of Growth
Traditionally, curves of growth are described in three sections
The linear part:
The width is set by the thermal width
Eqw is proportional to abundance
The “flat” part:
The central depth approaches its maximum value
Line strength grows asymptotically towards a constant value
The “damping” part:
Line width and strength depends on the damping constant
The line opacity in the wings is significant compared to kn
Line strength depends (approximately) on the square root of the abundance

16. Effect of Pressure on the COG
The higher the damping constant, the stronger the lines get at the same abundance.
The damping parts of the COG will look different for different lines