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 H bf opacity dominates,  is independent of P e, while the line absorption coefficient is proportional to P e, 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 P e 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 P e –Degree of ionization depends on 1/P e

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

Simple approach Lines are formed in a cool non- emitting gas placed above the sourece of continuous emission Surface flux in the line is    = optical depth, A = N E /N H = flux in cntinuum

Line strength Weak lines:  << 1 Line depth is proportional to   or A  each point in the weak-line profile proportional to A  EW is proportional to A Strong lines : wings dominate  gaussian componet acts like a  -function 

Strong lines Then, : depth average damping constant EW = U 2 =  2 / Afh  EW for strong line :proportional to A 1/2

Be careful! The whole chemical composition of the model is changing along with A 1,ionization equilibrium for the species may change, making the relation between N and A a function of A 2. continuum absorption may change with the change in electric donors 3. collisional damping may be different because of the changes in Pe and Pg Measurement of A for one element is coupled to what the remainder of the elements are doing.

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  Recall  that the wavelength integral of the absorption coefficient is Express the number of absorbers in terms of hydrogen Finally,

The COG for weak lines Changes in log A are equivalent to changes in log gf, , or  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 gf, , or . A curve of growth for one line can be “scaled” to be used 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 g 1/3

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 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