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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 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?) 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-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? 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 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/l 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 kn – Line strength depends (approximately) on the square root of the abundance The Effect of Temperature on the COG • Recall: Fc Fn ln constant Fc kn – (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 e 2 l2 N w constant mc c f kn • Express the number of absorbers in terms of hydrogen • Finally, Nr g NA NH e NE u (T ) kT e 2 N r N E log log 2 N H log A log gfl log kn l mc u(T ) w The COG for weak lines e 2 N r N E log log 2 N H log A log gfl log kn l mc u(T ) w Changes in log A are equivalent to changes in log gfl, , 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, , or kn. 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 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 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. Microturbulence in the COG -3 5 km/sec Log w/lambda -4 0 km/sec -5 0 km/sec 1 km/sec -6 2 km/sec 3 km/sec 5 km/sec -7 -13 -12 -11 -10 -9 -8 -7 Log A + Log gf 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? -6