Line Broadening and Opacity

2 Absorption Processes: Simplest Model Absorption Processes: Simplest Model –Photon absorbed from forward beam and reemitted in arbitrary direction  BUT: this could be scattering! Absorption Processes: Better Model Absorption Processes: Better Model –The reemitted photon is part of an energy distribution characteristic of the local temperature  Bν ν (T c ) > B ν (T l ) “Some” of B ν (T c ) has been removed and B ν (T l ) is less than B ν (T c ) and has arbitrary direction. “Some” of B ν (T c ) has been removed and B ν (T l ) is less than B ν (T c ) and has arbitrary direction. Total Absorption Coefficient =κ c + κ l

Line Broadening and Opacity3 Doppler Broadening The emitted frequency of an atom moving at v will be ν′: ν′ = ν 0 + = ν 0 + (v/c)ν 0 The emitted frequency of an atom moving at v will be ν′: ν′ = ν 0 +  ν = ν 0 + (v/c)ν 0 The absorption coefficient will be: The absorption coefficient will be: Whereν=Frequency of Interest Whereν=Frequency of Interest ν′=Emitted Frequency ν′=Emitted Frequency ν 0 =Rest Frequency ν 0 =Rest Frequency Non Relativistic:  ν/ν 0 = v/c

Line Broadening and Opacity4 Total Absorption Multiply by the fraction of atoms with velocity v to v + dv: The Maxwell Boltzman distribution gives this: 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) Now integrate over velocity Now integrate over velocity Per atom in the unit frequency interval at ν

Line Broadening and Opacity5 This is a Rather Messy Integral Doppler Equation Gamma is the effective 

Line Broadening and Opacity6 Simplify!

7 Our Equation is Now Note that  ν 0 is a constant = (ν 0 /c) (  2kT/M) so pull it out Note that  ν 0 is a constant = (ν 0 /c) (  2kT/M) so pull it out What about dv:dv=(c/ν 0 )d(  ν) What about dv:dv=(c/ν 0 )d(  ν) y=  ν/  ν 0 y=  ν/  ν 0 dy=(1/  ν 0 ) d(  ν) dy=(1/  ν 0 ) d(  ν) dv=(c  ν 0 /ν 0 )dy dv=(c  ν 0 /ν 0 )dy So put that in So put that in

Line Broadening and Opacity8 Getting Closer Constants: First the constant in the integral So then:

Line Broadening and Opacity9 Continuing On Note the α 0 does not depend on frequency (except for ν 0 = constant for any line;  ν 0 = (ν 0 /c)  (2kT/M)). It is called the absorption coefficient at line center. Note the α 0 does not depend on frequency (except for ν 0 = constant for any line;  ν 0 = (ν 0 /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. The normalized integral H(a,u) is called the Voigt function. It carries the frequency dependence of the line.

Line Broadening and Opacity10 Pulling It Together The Line Opacity α l Be Careful about the 1/√π as it can be taken up in the normalization of H(a,u).

Line Broadening and Opacity11 The Terms a is the broadening term (natural, etc) a is the broadening term (natural, etc) –a = δ′/  ν 0 =(Γ/4π)/((ν 0 /c)  (2kT/M)) u is the Doppler term u is the Doppler term –u = ν-ν 0 /((ν 0 /c)  (2kT/M)) 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. 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 Opacity12 Wavelength Forms Γ′s are broadenings due to various mechanisms ε is any additional microscopic motion which may be needed.

Line Broadening and Opacity13 Simple Line Profiles First the emergent continuum flux is: First the emergent continuum flux is: The source function S λ (τ λ ) =B λ (τ λ ). The source function S λ (τ λ ) =B λ (τ λ ). E 2 (τ λ ) = the second exponential integral. E 2 (τ λ ) = the second exponential integral. The total emergent continuum flux is: The total emergent continuum flux is: τ λ refers to the continuous opacities and τ l to the line opacity τ λ refers to the continuous opacities and τ l to the line opacity

Line Broadening and Opacity14 The Residual Flux R(  λ) This is the ratioed output of the star This is the ratioed output of the star –0 = No Light –1 = Continuum We often prefer to use depths: D(  λ) = 1 - R(  λ) We often prefer to use depths: D(  λ) = 1 - R(  λ)

Line Broadening and Opacity15 The Equivalent Width Often if the lines are not closely spaced one works with the equivalent width: Often if the lines are not closely spaced one works with the equivalent width: W λ is usually expressed in mÅ W λ = 1.06(  λ)D(  λ) for a gaussian line.  λ is the FWHM.

Line Broadening and Opacity16 Curve of Growth Damping/  Saturation Linear Log Ngf Log (W/)

Line Broadening and Opacity17 The Cookbook Model Atmosphere: (τ R, T, P gas, N e, Κ R ) Model Atmosphere: (τ R, T, P gas, N e, Κ R ) Atomic Data Atomic Data –Wavelength/Frequency of Line –Excitation Potential –gf –Species (this specifies U(T) and ionization potential) Abundance of Element (Initial) Abundance of Element (Initial) To Compute a Line You Need

Line Broadening and Opacity18 What Do You Do Now τ R,Κ R : 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 α λ : You need T, N e, and P g 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 Opacity19 Next Use the Saha and Boltzmann Equations to get populations Use the Saha and Boltzmann Equations to get populations You now have τ You now have τ λ Now compute τ l by first computing α l and using (a,u,  ν 0 ) as defined before. Note we are using wavelength as our variable. Now compute τ l by first computing α l and using (a,u,  ν 0 ) as defined before. Note we are using wavelength as our variable.