Download Chapter 13 – Behavior of Spectral Lines

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Line Profiles
Note - Figure obtained from
Chapter 13 – Behavior of Spectral Lines
We began with line absorption coefficients which give
the shapes of spectral lines. Now we move into the
calculation of line strength from a stellar atmosphere.
• Formalism of radiative transfer in spectral
– 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
The Line Transfer Equation
• We can add the continuous absorption coefficient and the line
absorption coefficient to get the total absorption coefficient:
dtn = (ln+kn)rdx
• And the source function is the sum of the line and continuous
emission coefficients divided by the sum of the line and
continuous absorption coefficients.
j  jn
Sn  n
ln  k n
• Or define the line and continuum source functions separately:
– Sl=jnl/ln
– Sc=jnc/kn
Sn 
( ln / kn ) Sl  S c
1 ln / kn
• In either case, we still have the basic transfer equation:
  In  Sn
In (0)   Sn etn sec secdtn
The Line Source Function
• The basic problem is still how to obtain the source function to
solve the transfer equation.
• But the line source function depends on the atomic level
populations, which themselves depend on the continuum
intensity and the continuum source function. This coupling
complicates the solution of the transfer equation for lines.
• Recall that in the case of LTE the continuum source function is
just Bn(T), the Planck Function.
• The assumption of LTE simplifies the line case in the same way,
and allows us to describe the energy level populations strictly
by the temperature without coupling to the radiation field.
• This approximation works when the excitation states of the
gas are defined primarily by collisions and not radiative
excitation or de-excitation.
The Gray Atmosphere
• Recall for the gray atmosphere,
Sn (tn )  43p Fn (0)tn  23 
• So, at tn = (4p-2)/3, Sn(t) = Fn(0)
• This is about t=3.5 – and gives us a
“mapping” between the source function and
the line profile
• The center of a line is formed higher in the
atmosphere than the wings because the
opacity is higher in the center
Mapping the Line Source Function
• The line source function with depth maps into the
line profile
• The center of the line is formed at shallower
optical depth, and maps to the source function at
smaller t
• The wings of the line are formed in progressively
deeper layers
Depth of Formation
• It’s straightforward to determine
approximately where in the atmosphere (in
terms of the optical depth of the
continuum) each part of the line profile is
• But even at a specific Dl, a range of optical
depths contributes to the absorption at
that wavelength
• It’s not straightforward to characterize
the depth of formation of an entire line
• The cores of strong lines are formed at
very shallow optical depths.
The Strength of Spectral Lines
• The strengths of spectral lines depend on
– The number of absorbers
• Temperature
• Electron pressure or luminosity
• Atomic constants
– The line absorption coefficient
– The ratio of the line/continuous absorption
– Thermal and microturbulent velocities
– In strong lines – collisional line broadening
affected by the gas and electron pressures
Computing the Line Profile
The line profile results from the solution of the transfer equation
at each Dl through the line.
The line profile will depend on the number of absorbers at each
depth in the atmosphere
The simplifying assumptions are
– LTE, collisions dominage
– Pure absorption (no scattering)
How well does this work?
To know for sure we must compute the line profile in the general
case and compare it to what we get with simplifying assumptions
Generally, it’s pretty good
Start with the assumed T(t) relation and model atmosphere
Recompute the flux using the line+continuous opacity at each
wavelength around the line
For blended lines, just add the line absorption coefficients
appropriate at each wavelength
The Effect of Temperature
Temperature is the
main factor
affecting line
Exponential and
power of T in
excitation and
Line strength
increases with T due
to increase in
Decrease beyond
– an increase in the
– drop in population
from ionization
The Effect of
on Weak
neutral line, mostly neutral species
neutral line, mostly ionized species
ionic line, mostly neutral species
ionic line, mostly ionized element
Neutral lines from a neutral species
• Number of absorbers proportional to N0exp(-c/kT)
• Number of neutrals independent of temperature (why?)
• If H- is the dominant opacity, the ratio of line to continuous
absorption coefficient is given by
T5 2 
( c  0.75)
• But Pe is ~ proportional to exp(T/1000), so… (Ch. 9)
1 dR 2.5 c  0.75
 0.001
R dT
 2.5 c  0.75
 DT  T 
Neutral Lines of a Neutral Species
 2.5 1.16 x104 ( c  0.75)
 DT 
 0.001
 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?
Neutral Lines of an Ionized Species
 1.16 x104 ( c  0.75  I ) 
 DT 
• 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
Ionic Lines of a Neutral Element
 5 1.16 x104 ( c  0.75  I )
 DT  
 0.002 
• 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
Ionic Lines of Ionized Species
 2.5 1.16 x104 ( c  0.75)
 DT 
 0.001
 T
• How strong is a Ba II line (at 0 eV) in
a 6000K star compared to a 5000K
• 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
– 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
– Hence the ratio line/continuous opacity is independent of
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
• 7770 O I triplet lines in solar-type
• [O I] in K giants
• Fe I and Fe II lines in solar-type
• 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
H-g Profiles
• H lines are sensitive
to temperature
because of the Stark
The high excitation of the Balmer
series (10.2 eV) means excitation
continues to increase to high
temperature (max at ~ 9000K).
Most metal lines have disappeared
by this temperature. Why?
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, kn is independent of
Pe, while the line absorption coefficient is
proportional to Pe, so line strength is too
• In hotter stars (with electron scattering) kn 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
• 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
When abundance is small - "linear part," line strength increases linearly
When abundance is mid-range - "flat part," absorption begins to saturate
When abundance is large - "damping part," optical depth in the wings
becomes large
Note - Figures obtained from
Curves of Growth
Traditionally, curves of growth
are described in three sections
• The linear part:
– The width is set by the thermal
– 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 kn
– 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