Line Broadening and Opacity. 2 Absorption Processes: Simplest Model Absorption Processes: Simplest Model –Photon absorbed from forward beam and reemitted

Line Broadening and Line Broadening and OpacityOpacity

Line Broadening and OpacityLine Broadening and Opacity 22

Line Broadening and OpacityLine Broadening and Opacity

Absorption Processes: Simplest ModelAbsorption Processes: Simplest Model– Photon absorbed from forward beam and reemitted in Photon absorbed from forward beam and reemitted in

arbitrary directionarbitrary direction BUT: this could be scattering!BUT: this could be scattering!

Absorption Processes: Better ModelAbsorption Processes: Better Model– The reemitted photon is part of an energy distribution The reemitted photon is part of an energy distribution

characteristic of the local temperaturecharacteristic of the local temperature BBνννν(T(Tcc) > B) > Bνν(T(Tll))

““Some” of BSome” of Bνν(T(Tcc) has been removed and B) has been removed and Bνν(T(Tll) is less ) is less

than Bthan Bνν(T(Tcc) and has arbitrary direction.) and has arbitrary direction.

Total Absorption Coefficient =κc + κl


Doppler BroadeningDoppler Broadening

The emitted frequency of an atom moving at v will be ν′: The emitted frequency of an atom moving at v will be ν′: ν′ = νν′ = ν00 + + ν = ν = ν00 + (v/c)ν + (v/c)ν00

The absorption coefficient will be:The absorption coefficient will be:

WhereWhere νν == Frequency of InterestFrequency of Interest ν′ν′ == Emitted FrequencyEmitted Frequency νν00 == Rest FrequencyRest Frequency

Non Relativistic: ν/ν0 = v/c

2

222

1

4( )

4e

ea f

m c


Total Absorption Total Absorption

Multiply by the fraction of atoms with velocity v to v Multiply by the fraction of atoms with velocity v to v + dv: The Maxwell Boltzman distribution gives this:+ dv: The Maxwell Boltzman distribution gives this:

– Where M = AMWhere M = AM00 = mass of the atom (A = atomic weight and = mass of the atom (A = atomic weight and

MM00 = 1 AMU) = 1 AMU)

Now integrate over velocityNow integrate over velocity

Per atom in the unit frequency interval at ν

2

2

2

M

kTdN Me d

N kT

v

v

2

22

222

0 0

24

( )4

M

kT

e

Me

e kTa f dm c

c

v

vv


This is a Rather Messy IntegralThis is a Rather Messy Integral

Doppler Equation

Gamma is the effective

00

0

0

2

( )

kT

c M

cc

d d

v

v

0

0

0

0

4

y

u

a


Simplify!Simplify!2

22

222

0 0

24

( )4

M

kT

e

Me

e kTa f dm c

c

v

vv

20

0

2 20

20

22 20

2 2 2 220

2 2 20 0

2

2

2

2

M kT

kT c M

kT

M

c

My

kT

vand

c

v

v v

c

22 2 2

0

2 2 20 0 0

4

( ) ( )

a

u yc

v


Our Equation is NowOur Equation is Now

Note that Note that νν00 is a constant = (ν is a constant = (ν00/c) (/c) (2kT/M) so 2kT/M) so

pull it outpull it out What about dv:What about dv: dvdv == (c/ν(c/ν00)d()d(ν)ν)

yy == ν/ν/νν00

dydy == (1/(1/νν00) d() d(ν)ν)

dvdv == (c(cνν00/ν/ν00)dy)dy

So put that inSo put that in

2

2 2 2 20 0 ( )

yed

a u y

v


Getting CloserGetting Closer

Constants: First the constant in the integral

So then:

2

2 20 0 ( )

yc edy

a u y

1 1

2 20

0

1 1

2 2

M M

kT kT c

2 20

2 2 20 0 0 0

0

2 2

00 0 0

1 1 1

4 4

4

1 1 1 1

e e

e e

e c ef f

m c c m c

and a so

e e a af f

m c m c


Continuing OnContinuing On Note the Note the αα00 does not depend on frequency (except does not depend on frequency (except

for νfor ν00 = constant for any line; = constant for any line; νν00 = =

(ν(ν00/c)/c)(2kT/M)). It is called the absorption (2kT/M)). It is called the absorption

coefficient at line center.coefficient at line center. The normalized integral H(a,u) is called the The normalized integral H(a,u) is called the

Voigt function. It carries the frequency Voigt function. It carries the frequency dependence of the line.dependence of the line.

2

2 2( , )

( )

ya eH a u dy

a u y


Pulling It TogetherPulling It TogetherThe Line Opacity αl

Be Careful about the 1/√π as it can be taken up in the normalization of H(a,u).

2

0

1 1( , ) 1

h

kTl

e

eN f H a u em c


The TermsThe Terms

a is the broadening term (natural, etc)a is the broadening term (natural, etc)– a = a = δδ′/′/νν00 =( =(ΓΓ/4/4ππ)/((ν)/((ν00/c)/c)(2kT/M))(2kT/M))

u is the Doppler termu is the Doppler term– u = ν-νu = ν-ν00/((ν/((ν00/c)/c)(2kT/M))(2kT/M))

At line center u = 0 and αAt line center u = 0 and α00 is the absorption is the absorption coefficient at line center so H(a,0) = 1 in this coefficient at line center so H(a,0) = 1 in this treatment. Note that different treatments give treatment. Note that different treatments give different normalizations.different normalizations.


Wavelength FormsWavelength Forms

• Γ′s are broadenings due to various mechanisms

• ε is any additional microscopic motion which may be needed.

1

22

1

21 2

2

22

n S W

ukT

c M

akTM


Simple Line ProfilesSimple Line Profiles

First the emergent continuum flux is:First the emergent continuum flux is:

The source function SThe source function Sλλ(τ(τλλ) =B) =Bλλ(τ(τλλ).).

EE22(τ(τλλ) = the second exponential integral.) = the second exponential integral.

The total emergent continuum flux is:The total emergent continuum flux is:

ττλλ refers to the continuous opacities and τ refers to the continuous opacities and τll to the line opacity to the line opacity

20(0) 2 ( ) ( )F S E d

20(0) 2 ( ) ( ) ( )l l lF S E d


The Residual Flux R(The Residual Flux R(λ)λ)

This is the ratioed output of the starThis is the ratioed output of the star– 0 = No Light0 = No Light

– 1 = Continuum1 = Continuum

We often prefer to use depths: D(We often prefer to use depths: D(λ) = 1 - R(λ) = 1 - R(λ)λ)

20

2( ) ( ) ( ) ( )

(0) l l lR S E dF


The Equivalent WidthThe Equivalent Width

Often if the lines are not Often if the lines are not closely spaced one closely spaced one works with the works with the equivalent width:equivalent width:

• Wλ is usually expressed in mÅ• Wλ = 1.06(λ)D(λ) for a gaussian

line. λ is the FWHM.

( )W D d


Curve of GrowthCurve of Growth

Damping/

Saturation

Linear

Log Ngf

Log (

W/)


The CookbookThe Cookbook

Model Atmosphere: (τModel Atmosphere: (τRR, T, P, T, Pgasgas, N, Nee, , ΚΚRR))

Atomic DataAtomic Data– Wavelength/Frequency of LineWavelength/Frequency of Line– Excitation PotentialExcitation Potential– gfgf– Species (this specifies U(T) and ionization Species (this specifies U(T) and ionization

potential)potential)

Abundance of Element (Initial)Abundance of Element (Initial)

To Compute a Line You Need


What Do You Do NowWhat 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.

0

0

0

R

l

lR R

RR

dl

dl

d


NextNext Use the Saha and Boltzmann Equations to get Use the Saha and Boltzmann Equations to get

populationspopulations You now have τYou now have τλ

Now compute τNow compute τll by first computing α by first computing αll and and

using (a,u,using (a,u,νν00) as defined before. Note we are ) as defined before. Note we are

using wavelength as our variable.using wavelength as our variable.2

0

0

( ) ( ) ( , ) 1( )

hc

kTl R R

e R

l

l l

e fN H a u e

m c

dl

Documents

Line Broadening and Opacity. 2 Absorption Processes: Simplest Model Absorption Processes: Simplest Model –Photon absorbed from forward beam and reemitted