Download Chapter 13 – Behavior of Spectral Lines

Absorption Coefficient and Optical Depth
• Gas absorbs photons passing through it
– Photons are converted to thermal energy or
– Re-radiated isotropically
• Radiation lost is proportional to
–
–
–
–
Absorption coefficient (per gram)
dI   I dx
Density
Intensity
d   dx
Pathlength
dI   I d



• Optical depth is the integral of the absorption
coefficient times the density along the path
L
    dx
0
I (  )  I (0)e

Radiative Equilibrium
• To satisfy conservation of energy,
the total flux must be constant at all
depths of the photosphere

F ( x)  F0   F d
0
• Two other radiative equibrium
equations are obtained by integrating
the transfer equation over solid angle
and over frequency
The Transfer Equation
• For radiation passing through gas, the
change in intensity I is equal to:
dI = intensity emitted – intensity absorbed
dI = jdx – I dx
dI /d = -I + j/ = -I + S
• This is the basic radiation transfer equation
which must be solved to compute the
spectrum emerging from or passing through
a gas.
Solving the Gray Atmosphere
• Integrating the transfer equation over
frequency:
dI
cos 
 I S
d
• The radiative equilibrium equations give us:
F=F0, J=S, and dK/d = F0/4p
• LTE says S = B (the Planck function)
• Eddington Approximation (I independent of
direction)
3
2 14
T ( )  ( (  )) Teff
4
3
Monochromatic Absorption
Coefficient
• Recall d = dx. We need to calculate
, the absorption coefficient per gram of
material
• First calculate the atomic absorption
coefficient a (per absorbing atom or ion)
• Multiply by number of absorbing atoms or
ions per gram of stellar material (this
depends on temperature and pressure)
Physical Processes
• Bound-Bound Transitions – absorption or emission of
radiation from electrons moving between bound energy levels.
• Bound-Free Transitions – the energy of the higher level
electron state lies in the continuum or is unbound.
• Free-Free Transitions – change the motion of an electron
from one free state to another.
• Scattering – deflection of a photon from its original path by a
particle, without changing wavelength
– Rayleigh scattering if the photon’s wavelength is greater
than the particle’s resonant wavelength. (Varies as l-4)
– Thomson scattering if the photon’s wavelength is much
less than the particle’s resonant wavelength. (Independent
of wavelength)
– Electron scattering is Thomson scattering off an electron
• Photodissociation may occur for molecules
Hydrogen Bound-Free Absorption Coefficient
a (cm-2 per atom) x 10^6
3.5E-14
3E-14
2.5E-14
2E-14
Balmer
Absorption
1.5E-14
1E-14
5E-15
Lyman
Absorption n=1
Paschen
Absorption
n=3
n=2
0
100
600
1200 2200 3200 4200 5200 6200 7200 8200 9200
Wavelength (A)
Neutral hydrogen (bf and ff) is the dominant
Source of opacity in stars of B, A, and F
spectral type
Opacity from the H- Ion
• Only one known bound state for bound-free
absorption
• 0.754 eV binding energy
• So l < hc/h = 16,500A
• Requires a source of free electrons (ionized metals)
• Major source of opacity in the Sun’s photosphere
• Not a source of opacity at higher temperatures
because H- becomes too ionized (average e- energy
too high)
Dominant Opacity vs. Spectra Type
Low
Electron scattering
(H and He are too
highly ionized)
He+ He
Low pressure –
less H-
Neutral H H-
H-
High
(high pressure forces more H-)
O
B
A
F
G
K
M
The T() Relation
• In the Sun, we can get the T() relation from
– Limb darkening or
– The variation of I with wavelength
– Use a gray atmosphere and the Eddington approximation
• In other stars, use a scaled solar model:
Teff Star
T ( ) 
T ( ) Sun
Teff Sun
– Or scale from published grid models
– Comparison to T(t) relations iterated through the
equation of radiative equilibrium for flux constancy
suggests scaled models are close
Hydrostatic Equilibrium
• Since d= dx
• dP/dx=  dP/d=g or
dP/d  = g/
The Paschen Continuum vs. Temperature
1.00E-02
50,000 K
Flux Distributions
Log Flux
1.00E-03
1.00E-04
1.00E-05
4000 K
1.00E-06
1.00E-07
300
400
500
600
700
Wavelength (nm)
800
900
1000
Calculating Fl from V
• Best estimate for Fl at V=0 at 5556A is
Fl = 3.54 x 10-9 erg s-1 cm-2 A-1
Fl = 990 photon s-1 cm-2 A-1
Fl = 3.54 x 10-12 W m-2 A-1
• We can convert V magnitude to Fl:
Log Fl = -0.400V – 8.451 (erg s-1 cm-2 A-1)
Log F = -0.400V – 19.438 (erg s-1 cm-2 A-1)
• With color correction for 5556 > 5480 A:
Log Fl =-0.400V –8.451 – 0.018(B-V) (erg s-1 cm-2 A-1)
Line Profiles
Note - Figure obtained from www.physics.utoledo.edu/~lsa.atnos/SAElp05.htm
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
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
The Line Transfer Equation
• We can add the continuous absorption coefficient and the line
absorption coefficient to get the total absorption coefficient:
d = (l+)dx
• 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  j
S  
l   
l
c
• Or define the line and continuum source functions separately:
– Sl=jl/l
– Sc=jc/
S 
( l /  ) Sl  S c
1 l / 
• In either case, we still have the basic transfer equation:
dI
  I  S
d

I (0)   S e sec secd
0
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 B(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,
S ( )  43p F (0)  23 
• So, at  = (4p-2)/3, S() = F(0)
• This is about =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 
• 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
formed
• 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
coefficient
– 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 dominate
– 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() 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
How do different kinds of lines behave
with temperature?
– Lines from a neutral
neutral element
– Lines from a neutral
ionized element
– Lines from an ion of
– Lines from an ion of
species of a mostly
species of a mostly
a mostly neutral element
a mostly ionized element
• Consider gas with H- as the dominant
opacity
5

  T Pe e
2
0.75 / kT
The Effect of Temperature
•
•
•
•
Temperature is the
main factor
affecting line
strength
Exponential and
power of T in
excitation and
ionization
Line strength
increases with T due
to increase in
excitation
Decrease beyond
maximum
– an increase in the
opacity
– drop in population
from ionization
The Effect of
Temperature
on Weak
Lines
1.
2.
3.
4.
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 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
DEQW
 2.5 c  0.75

 DT  T 
 0.001
2
EQW
kT


Neutral Lines of a Neutral Species
 2.5 1.16 x104 ( c  0.75)

DEQW
 DT 

 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?
Neutral Lines of an Ionized Species
 1.16 x104 ( c  0.75  I ) 
DEQW

 DT 
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
Ionized Lines of a Neutral Element
 5 1.16 x104 ( c  0.75  I )

DEQW
 DT  
 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 DT=100K)
• EQW4600=1.5EQW4500
Ionic Lines of Ionized Species
 2.5 1.16 x104 ( c  0.75)

DEQW
 DT 

 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?
H-g Profiles
• H lines are sensitive
to temperature
because of the Stark
effect
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,  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
•
•
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 www.physics.utoledo.edu/~lsa.atnos/SAElp05.htm
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
The Effect of Temperature on the
COG
Fc  F
l
 constant  
Fc

• Recall:
– (under the assumption that F comes from a characteristic optical
depth )
• Integrate over wavelength, and let l=Na
• Recall that the wavelength integral of the absorption coefficient
2
2is
w  constant 
• Express the number of absorbers in terms of hydrogen
Nr
g c
NA
NH
e
NE
u (T )
• Finally,
pe l
mc c
f
N

kT
 pe 2 N r N E

log  log  2
N H   log A  log gfl  c  log 
l
 mc u(T )

w
The COG for weak lines
 pe 2 N r N E

log  log  2
N H   log A  log gfl  c  log 
l
 mc u(T )

w
Changes in log A are equivalent to changes in log gfl, c,
or 
For a given star curves of growth for lines of the same
species (where A is a constant) will only be displaced
along the abcissa according to individual values of gfl,
c, or .
A curve of growth for one line can be “scaled” to be
used for other lines of the same species.
A Thought Problem
• The equivalent width of a 2.5 eV Fe I line in star A, a star in a
star cluster is 25 mA. Star A has a temperature of 5200 K.
• In star B in the same cluster, the same Fe I line has an
equivalent width of 35 mA.
• What is the temperature of star B, assuming the stars have
the same composition
• What is the iron abundance of star B if the stars have the
same temperature?
The Effect of Surface Gravity on the
COG for Weak Lines
• Both the ionization equilibrium and the
opacity depend on surface gravity
• For neutral lines of ionized species (e.g.
Fe I in the Sun) these effects cancel,
so the COG is independent of gravity
• For ionized lines of ionized species (e.g
Fe II in the Sun), the curves shift to
the right with increasing gravity,
roughly as g1/3
Effect of Pressure on the COG for
Strong Lines
• 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
The Effect of Microturbulence
• The observed equivalent widths of saturated lines
are greater than predicted by models using just
thermal and damping broadening.
• Microturbulence is defined as an isotropic,
Gaussian velocity distribution x in km/sec.
• It is an ad hoc free parameter in the analysis, with
values typically between 0.5 and 5 km/sec
• Lower luminosity stars generally have lower values
of microturbulence.
• The microturbulence is determined as the value of
x that makes the abundance independent of line
strength.
Microturbulence in the COG
-3
5 km/sec
Log w/lambda
-4
0 km/sec
-5
0 km/sec
1 km/sec
-6
2 km/sec
3 km/sec
5 km/sec
-7
-13
-12
-11
-10
-9
-8
-7
Log A + Log gf
Questions –
At what line strength do lines become sensitive to microturbulence?
Why is it hard to determine abundances from lines on the
“flat part” of the curve of growth?
-6
Microturbulence in the COG
-3
5 km/sec
Log w/lambda
-4
0 km/sec
-5
0 km/sec
1 km/sec
-6
2 km/sec
3 km/sec
5 km/sec
-7
-13
-12
-11
-10
-9
-8
-7
Log A + Log gf
Questions –
At what line strength do lines become sensitive to microturbulence?
Why is it hard to determine abundances from lines on the
“flat part” of the curve of growth?
-6