Transcript PowerPoint
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 5 2 T Pe e 0.75 / kT 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 T5 2 R e Pe l ( c 0.75) kT • But Pe is ~ proportional to exp(T/1000), so… 1 dR 2.5 c 0.75 0.001 2 R dT T kT EQW c 0.75 T 2T.5 0 . 001 2 EQW kT Neutral Lines of a Neutral Species 2.5 1.16 x104 ( c 0.75) EQW T 0.001 2 EQW T T • 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 1.16 x104 ( c 0.75 I ) EQW T 2 EQW T • 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 5 1.16 x104 ( c 0.75 I ) EQW T 0.002 2 EQW T T • 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 T=100K) • EQW4600=1.5EQW4500 Ionized Lines of Ionized Species 2.5 1.16 x104 ( c 0.75) EQW T 0.001 2 EQW T T • 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-tocontinuous 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, 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 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 • • 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 – 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