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

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

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…

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

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

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

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

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

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

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

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?

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

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

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