Line Broadening and
Opacity
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
2
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:
a 



Where
e
2
f

4
1
2
  
(    )  

4



ν = Frequency of Interest
ν′ = Emitted Frequency
ν0 = Rest Frequency
m ec
2
2
Line Broadening and Opacity
3
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:
dN
M v
M

2 kT
N
e
2 kT
2
dv
– Where M = AM0 = mass of the atom (A = atomic weight and
M0 = 1 AMU)

Now integrate over velocity
M
a 
e
2
m ec
f

4
2

2  kT


( 0 
v
c
M v
2
2 kT
e
  

 4 
2
dv
0  )  
Line Broadening and Opacity
2
4
This is a Rather Messy Integral
 0 
 
0
2 kT
c
M
0
v
u 
Doppler Equation
c
dv 
c
0
y 

d ( )
Line Broadening and Opacity
a 

 0
 0
 0

4
Gamma is the
effective 

 0
5
Simplify!
M
a 
Mv
2
2 kT
an d   0 
c  0
2
0
2

2
e
2
m ec
0
2 kT
c
M
f

4
2

2  kT


( 0 

Mv
2
2 kT
2 kT
2
c  0
2
2

2
  

 4 
2
dv
0  )  
2
( 0 
  2
 0 v
 c 
 0 v
2 kT
e
  
2
2
2
    a  0
 4 


2
2
c
2
2
M

v
M v
v
c
 0   )   0 (u  y )
2
2
2
  2 
2
 


y

2
  0 
Line Broadening and Opacity
6
Our Equation is Now
 






e
2
0
y
2
a    (u  y )
2
2
0
2
dv
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
7
Getting Closer
c
 0 0
a
e
2
y
2
 (u  y )
2
dy
Constants: First the constant in the integral
1
1
 M 2

 
 2 kT 
e
So then: m c
e

e

2
f

f
1
c
 c 0  0 0
a 
and
4
2
m ec
4
0
1
2
1  M 2

 
  2 kT 
1

   0  0


 c 0
e

2
f
m ec

4
1
2
1
  0
2
so
 0
e
0
1
2
f
m ec
Line Broadening and Opacity
1
1
a
  0 
 0
a

8
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.

H (a, u ) 
a

a
e
2
y
2
 (u  y )
Line Broadening and Opacity
2
dy
9
Pulling It Together
The Line Opacity αl
l  N
e
2
m ec
f

H ( a , u ) 1  e
  0

1
1
 h
kT



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

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
11
Wavelength Forms
u 
a 

1

2

2
kT


2




 c  M






 n   S  W
1
2 
1
2
 2 kT
2

 M



• 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

First the emergent continuum flux is:

F (0 )  2  S  (  ) E 2 (  ) d  
0

The source function Sλ(τλ) =Bλ(τλ).
 E2(τλ) = the second exponential integral.
 The total emergent continuum flux is:

F (0 )  2  S  (    l ) E 2 (    l ) d (    l )
0

τλ refers to the continuous opacities and τl to the line opacity
Line Broadening and Opacity
13
The Residual Flux R(λ)
2
R ( ) 
F


(0)

0
S  (    l ) E 2 (    l ) d (    l )
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
14
The Equivalent Width

Often if the lines are not
closely spaced one
works with the
W 
equivalent width:



D (   )d  
• Wλ is usually expressed in mÅ
• Wλ = 1.06(λ)D(λ) for a
gaussian line. λ is the FWHM.
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
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
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.

R

 
Line Broadening and Opacity



l
0
l
0
   dl
 R  dl
R

0
R
d R
18
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.
 hc


 kT
 l ( R )  N ( R )
H ( a , u ) 1  e

m e c   0 ( R )


e
l 

l
0
2
f
 l  dl
Line Broadening and Opacity
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