Chapter 13 Cont’d – Pressure Effects

• More curves of growth• How does the COG depend on excitation potential,

ionization potential, atmospheric parameters (temperature and gravity), microturbulence

• When/why does line strength depend on pressure?

• Mg b lines• Hydrogen lines

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 4508 line weakens with increasing pressure because the continuous opacity decreases (less H- - WHY?)

The Mg I b lines

• Why are the Mg I b lines sensitive to pressure?

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

H- 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?

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, is independent of Pe, while the line absorption coefficient is proportional to Pe, so line strength is too

• In hotter stars (with electron scattering) is nearly independent of pressure while the number of neutral H atoms is proportional to Pe

2. Balmer profiles are very pressure dependent

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

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

The Curve of Growth

• 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/

Wrubel COG from Aller and Chamberlin 1956

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

– Line strength depends (approximately) on the square root of the abundance

The Effect of Temperature on the COG

• Recall:

– (under the assumption that F comes from a characteristic optical depth )

• Integrate over wavelength, and let l=N

• Recallthat the wavelength integral of the absorption coefficient is

• Express the number of absorbers in terms of hydrogen

• Finally,

l

constant

c

c

F

FF

Nf

cmc

ew

22

constant

kTH

E

r eTu

gN

N

NAN

)(

logloglog)(

loglog2

2

gfAN

Tu

NN

mc

ewH

Er

The COG for weak lines

logloglog)(

loglog2

2

gfAN

Tu

NN

mc

ewH

Er

Changes in log A are equivalent to changes in log gf, ,or

For a given star curves of growth for lines of the samespecies (where A is a constant) will only be displaced along the abcissa according to individual values of gf,, or .

A curve of growth for one line can be “scaled” to beused for other lines of the same species.

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?

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

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

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 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 that makes the abundance independent of line strength.

Microturbulence in the COG

-7

-6

-5

-4

-3

-13 -12 -11 -10 -9 -8 -7 -6

Log A + Log gf

Lo

g w

/la

mb

da

0 km/sec

1 km/sec

2 km/sec

3 km/sec

5 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?

0 km/sec

5 km/sec