Line Broadening and Opacity

1. Line Broadening and Opacity

2. Line Broadening and Opacity Total Absorption Coefficient =κc + κl • Absorption Processes: Simplest Model • Photon absorbed from forward beam and reemitted in arbitrary direction • BUT: this could be scattering! • Absorption Processes: Better Model • The reemitted photon is part of an energy distribution characteristic of the local temperature • Bνν(Tc) > Bν(Tl) • “Some” of Bν(Tc) has been removed and Bν(Tl) is less than Bν(Tc) and has arbitrary direction. Line Broadening and Opacity

3. Doppler Broadening Non Relativistic: ν/ν0 = v/c • The emitted frequency of an atom moving at v will be ν′: ν′ = ν0 + ν = ν0 + (v/c)ν0 • The absorption coefficient will be: • Where ν = Frequency of Interest • ν′ = Emitted Frequency • ν0 = Rest Frequency Line Broadening and Opacity

4. Total Absorption Per atom in the unit frequency interval at ν • Multiply by the fraction of atoms with velocity v to v + dv: The Maxwell Boltzman distribution gives this: • Where M = AM0 = mass of the atom (A = atomic weight and M0 = 1 AMU) • Now integrate over velocity Line Broadening and Opacity

5. This is a Rather Messy Integral Doppler Equation Gamma is the effective  Line Broadening and Opacity

6. Simplify! Line Broadening and Opacity

7. Our Equation is Now • Note that ν0 is a constant = (ν0/c) (2kT/M) so pull it out • What about dv: dv = (c/ν0)d(ν) • y = ν/ν0 • dy = (1/ν0) d(ν) • dv = (cν0/ν0)dy • So put that in Line Broadening and Opacity

8. Getting Closer Constants: First the constant in the integral So then: Line Broadening and Opacity

9. Continuing On • Note the α0 does not depend on frequency (except for ν0 = constant for any line; ν0 = (ν0/c)(2kT/M)). It is called the absorption coefficient at line center. • The normalized integral H(a,u) is called the Voigt function. It carries the frequency dependence of the line. Line Broadening and Opacity

10. Pulling It Together The Line Opacity αl Be Careful about the 1/√π as it can be taken up in the normalization of H(a,u). Line Broadening and Opacity

11. The Terms • a is the broadening term (natural, etc) • a = δ′/ν0 =(Γ/4π)/((ν0/c)(2kT/M)) • u is the Doppler term • u = ν-ν0/((ν0/c)(2kT/M)) • At line center u = 0 and α0 is the absorption coefficient at line center so H(a,0) = 1 in this treatment. Note that different treatments give different normalizations. Line Broadening and Opacity

12. Wavelength Forms • Γ′s are broadenings due to various mechanisms • ε is any additional microscopic motion which may be needed. Line Broadening and Opacity

13. Simple Line Profiles • First the emergent continuum flux is: • The source function Sλ(τλ) =Bλ(τλ). • E2(τλ) = the second exponential integral. • The total emergent continuum flux is: • τλ refers to the continuous opacities and τl to the line opacity Line Broadening and Opacity

14. The Residual Flux R(λ) • This is the ratioed output of the star • 0 = No Light • 1 = Continuum • We often prefer to use depths: D(λ) = 1 - R(λ) Line Broadening and Opacity

15. The Equivalent Width • Often if the lines are not closely spaced one works with the equivalent width: • Wλ is usually expressed in mÅ • Wλ = 1.06(λ)D(λ) for a gaussian line. λ is the FWHM. Line Broadening and Opacity

16. Curve of Growth Damping/ Log (W/) Saturation Linear Log Ngf Line Broadening and Opacity

17. The Cookbook To Compute a Line You Need • Model Atmosphere: (τR, T, Pgas, Ne, ΚR) • Atomic Data • Wavelength/Frequency of Line • Excitation Potential • gf • Species (this specifies U(T) and ionization potential) • Abundance of Element (Initial) Line Broadening and Opacity

18. What Do You Do Now • τR,ΚR: Optical depth and opacity are specified at some reference wavelength or may be Rosseland values. • The wavelength of interest is somewhere else: λ • So you need αλ : You need T, Ne, and Pg and how to calculate f-f, b-f, and b-b but in computing lines one does not add b-b to the continuous opacity. Line Broadening and Opacity

19. Next • Use the Saha and Boltzmann Equations to get populations • You now have τλ • Now compute τl by first computing αl and using (a,u,ν0) as defined before. Note we are using wavelength as our variable. Line Broadening and Opacity