Behavior of Spectral Lines – Part II
Formalism of radiative transfer in
spectral lines
Transfer equation for lines
The line source function
Computing the line profile in LTE
Depth of formation
• Temperature and pressure
dependence of line strength
• The curve of growth
How do different kinds of lines
behave with temperature?
– Lines from a neutral species of a mostly neutral
element
– Lines from a neutral species of a mostly ionized
element
– Lines from an ion of a mostly neutral element
– Lines from an ion of a mostly ionized element
• Consider gas with H- as the dominant
opacity
5

2
  T Pe e
0.75 / kT
Neutral lines from a neutral species
• Number of absorbers proportional to exp(-c/kT)
• Number of neutrals independent of temperature
(why?)
• Ratio of line to continuous absorption coefficient
T5 2 
R

e

Pe
l
( c  0.75)
kT
• But Pe is ~ proportional to exp(T/1000), so…
1 dR 2.5 c  0.75


 0.001
2
R dT
T
kT
EQW
c  0.75


 T  2T.5 

0
.
001

2
EQW
kT


Neutral Lines of a Neutral Species
 2.5 1.16 x104 ( c  0.75)

EQW
 T 

 0.001
2
EQW
T
 T

• Oxygen triplet lines at 7770A.
– Excitation potential = 8 eV
– Ionization potential = 13.6 eV
• Oxygen resonance line [O I] at 6300A
• By what factor will each of these lines
change in strength from 5000 to 6000K?
• Factors of ~4 and 0.75
Neutral Lines of an Ionized Species
 1.16 x104 ( c  0.75  I ) 
EQW

 T 
2
EQW
T


• How much would you have to change
the temperature of a 6000K star to
decrease the equivalent width of the
Li I 6707 resonance line by a factor
of two?
• Ionization potential = 5.4 eV
• Raise T by ~333K
Ionized Lines of a Neutral Element
 5 1.16 x104 ( c  0.75  I )

EQW
 T  
 0.002 
2
EQW
T
T

• Fe II lines in giants are often used to
determine the spectroscopic gravity.
• How sensitive to temperature is a
2.5eV Fe II line (I=7.9 eV) in a star
with Teff=4500K? (Estimate for
T=100K)
• EQW4600=1.5EQW4500
Ionized Lines of Ionized Species
 2.5 1.16 x104 ( c  0.75)

EQW
 T 

 0.001
2
EQW
T
 T

• How strong is a Ba II line (at 0 eV) in
a 6000K star compared to a 5000K
star?
• How do the strengths of a 5 eV Fe II
line compare in the same two stars?
• For Ba II, EQW decreases by 25%
• For Fe II, EQW is almost x3 larger
Line Strength Depends on Pressure
• For metal lines, pressure
(gravity) affects line
strength in two ways:
– Changing the line-tocontinuous opacity ratio
(by changing the ionization
equilibrium)
– Pressure broadening
• Pressure effects are
much weaker than
temperature effects
Rules of Thumb for Weak Lines
•
•
•
When most of the atoms of an element are in the next higher state
of ionization, lines are insensitive to pressure
– When H- opacity dominates, the line and the continuous
absorption coefficients are both proportional to the electron
pressure
– Hence the ratio line/continuous opacity is independent of
pressure
When most of the atoms of an element are in the same or a lower
state of ionization, lines are sensitive to pressure
– For lines from species in the dominant ionization state, the
continuous opacity (if H-) depends on electron pressure but the
line opacity is independent of electron pressure
Lines from a higher ionization state than the dominant state are
highly pressure dependent
– H- continuous opacity depends on Pe
– Degree of ionization depends on 1/Pe
Examples of Pressure Dependence
• Sr II resonance lines in solar-type
stars
• 7770 O I triplet lines in solar-type
stars
• [O I] in K giants
• Fe I and Fe II lines in solar-type
stars
• Fe I and Fe II lines in K giants
• Li I lines in K giants
The Mg I b lines
• Why are the Mg I b lines sensitive to
pressure?
Pressure Effects on Hydrogen Lines
• When H- opacity dominates, the continuous opacity
is proportional to pressure, but so is the line abs.
coef. in the wings – so Balmer lines in cool stars
are not sensitive to pressure
• When Hbf opacity dominates,  is independent of
Pe, while the line absorption coefficient is
proportional to Pe, so line strength is too
• In hotter stars (with electron scattering)  is
nearly independent of pressure while the number
of neutral H atoms is proportional to Pe2. Balmer
profiles are very pressure dependent
What Is Equivalent Width?
• The equivalent width is a
measure of the strength of a
spectral line
• Area equal to a rectangle with
100% depth
• Triangle approximation: half
the base times the width
• Integral of a fitted line
profile (Gaussian, Voigt fn.)
• Measured in Angstroms or
milli-Angstroms
• How is equivalent width
defined for emission lines?
The Curve of Growth
•
•
The curve of growth is a mathematical relation between the chemical
abundance of an element and the line equivalent width
The equivalent width is expressed independent of wavelength as log W/l
Wrubel COG from Aller and Chamberlin 1956
Curves of Growth
Traditionally, curves of growth
are described in three sections
• The linear part:
– The width is set by the thermal
width
– Eqw is proportional to abundance
•
The “flat” part:
– The central depth approaches
its maximum value
– Line strength grows
asymptotically towards a
constant value
•
The “damping” part:
– Line width and strength depends
on the damping constant
– The line opacity in the wings is
significant compared to 
– Line strength depends
(approximately) on the square
root of the abundance
Effect of Pressure on the COG
• The higher the damping constant, the stronger the lines get
at the same abundance.
• The damping parts of the COG will look different for
different lines