Line Broadening and Opacity Line Broadening and Opacity

Line Broadening and Opacity

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

Doppler Broadening Non Relativistic: ν/ν 0 = v/c The emitted frequency of an atom moving at v will be ν′: ν′ = ν 0 + ν = ν 0 + (v/c)ν 0 n The absorption coefficient will be: n Where n ν′ n ν 0 n ν = = = Frequency of Interest Emitted Frequency Rest Frequency Line Broadening and Opacity 3

Total Absorption Per atom in the unit frequency interval at ν n Multiply by the fraction of atoms with velocity v to v + dv: The Maxwell Boltzman distribution gives this: – Where M = AM 0 = mass of the atom (A = atomic weight and M 0 = 1 AMU) n Now integrate over velocity Line Broadening and Opacity 4

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

Simplify! Line Broadening and Opacity 6

Our Equation is Now n n n Note that ν 0 is a constant = (ν 0/c) ( 2 k. T/M) so pull it out What about dv: dv = (c/ν 0)d( ν) y = ν/ ν 0 dy = (1/ ν 0) d( ν) dv = (c ν 0/ν 0)dy So put that in Line Broadening and Opacity 7

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

Continuing On Note the α 0 does not depend on frequency (except for ν 0 = constant for any line; ν 0 = (ν 0/c) (2 k. T/M)). It is called the absorption coefficient at line center. n The normalized integral H(a, u) is called the Voigt function. It carries the frequency dependence of the line. n Line Broadening and Opacity 9

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 10

The Terms n a is the broadening term (natural, etc) – a = δ′/ ν 0 =(Γ/4π)/((ν 0/c) (2 k. T/M)) n u is the Doppler term – u = ν-ν 0/((ν 0/c) (2 k. T/M)) n At line center u = 0 and α 0 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 11

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

Simple Line Profiles n First the emergent continuum flux is: The source function Sλ(τλ) =Bλ(τλ). n E 2(τλ) = the second exponential integral. n The total emergent continuum flux is: n n τλ refers to the continuous opacities and τl to the line opacity Line Broadening and Opacity 13

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

The Equivalent Width n • Wλ is usually expressed in mÅ Often if the lines are not • Wλ = 1. 06( λ)D( λ) for a closely spaced one gaussian line. λ is the FWHM. works with the equivalent width: Line Broadening and Opacity 15

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

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

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, bf, and b-b but in computing lines one does not add b-b to the continuous opacity. Line Broadening and Opacity 18

Next Use the Saha and Boltzmann Equations to get populations n You now have τλ n Now compute τl by first computing αl and using (a, u, ν 0) as defined before. Note we are using wavelength as our variable. n Line Broadening and Opacity 19