1. Chapter 13 Cont’d – Pressure Effects
When/why does line strength depend on pressure?
Mg b lines
Hydrogen lines
More curves of growth
How does the COG depend on excitation potential, ionization potential, atmospheric parameters (temperature and gravity), microturbulence

2. 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 dependence of damping constant
Pressure dependence of Stark broadening
Pressure effects are much weaker than temperature effects
The Fe II l4508 line weakens with increasing pressure because the continuous opacity decreases (less H- - WHY?)

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

4. Hydrogen lines depend on pressure
If Teff > 7500, hydrogen lines becomes sensitive to pressure (why, and why are they less sensitive at lower temperature?)
Lines get stronger with increasing pressure

5. 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?

6. 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

7. 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

8. 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

9. 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

10. 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

11. The Effect of Temperature on the COG
Recall:
(under the assumption that Fn comes from a characteristic optical depth tn)
Integrate over wavelength, and let lnr=Na
Recall that the wavelength integral of the absorption coefficient is
Express the number of absorbers in terms of hydrogen
Finally,

12. The COG for weak lines
Changes in log A are equivalent to changes in log gfl, qc,
or kn
For a given star curves of growth for lines of the same
species (where A is a constant) will only be displaced
along the abcissa according to individual values of gfl,
c, or kn.
A curve of growth for one line can be “scaled” to be
used for other lines of the same species.

13. A Thought Problem
The equivalent width of a 2.5 eV Fe I line in star A, a star in a star cluster is 25 mA. Star A has a temperature of 5200 K.
In star B in the same cluster, the same Fe I line has an equivalent width of 35 mA.
What is the temperature of star B, assuming the stars have the same composition
What is the iron abundance of star B if the stars have the same temperature?

14. The Effect of Surface Gravity on the COG for Weak Lines
Both the ionization equilibrium and the opacity depend on surface gravity
For neutral lines of ionized species (e.g. Fe I in the Sun) these effects cancel, so the COG is independent of gravity
For ionized lines of ionized species (e.g Fe II in the Sun), the curves shift to the right with increasing gravity, roughly as g1/3

15. Effect of Pressure on the COG for Strong Lines
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

16. The Effect of Microturbulence
The observed equivalent widths of saturated lines are greater than predicted by models using just thermal and damping broadening.
Microturbulence is defined as an isotropic, Gaussian velocity distribution x in km/sec.
It is an ad hoc free parameter in the analysis, with values typically between 0.5 and 5 km/sec
Lower luminosity stars generally have lower values of microturbulence.
The microturbulence is determined as the value of x that makes the abundance independent of line strength.

17. Microturbulence in the COG
5 km/sec
0 km/sec
Questions –
At what line strength do lines become sensitive to microturbulence?
Why is it hard to determine abundances from lines on the
“flat part” of the curve of growth?