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The Formation of Spectral Lines
I. Line Absorption Coefficient
II. Line Transfer Equation
Line Absorption Coefficient
Main processes
1. Natural Atomic Absorption
2. Pressure Broadening
3. Thermal Doppler Broadening
Line Absorption Coefficient
The classical model of the interaction of light with a photon is a
plane electromagnetic wave interacting with a dipole.
∂E
∂2 E
2
= v
2
∂ x2
∂t
Treat only one frequency since by Fourier composition the total
field is a sum of all sine waves.
E = E0 e–iw(x/v – t)
The wave velocity through a medium
½
e
m
0
0
v=c ( e m
e and m are the electric and magnetic
permemability in the medium and
free space. For gases m = m0
(
Line Absorption Coefficient
The total electric field is the sum of the electric field E and the
field of the separated charges which is 4pNqz where z is the
separation of the charges and N the number of dipoles per
unit volume
The ratio of e/e0 is just the ratio of the field in the medium to the
field in free space
E + 4pNqz
e
=
e0
E
We need z/E
= 1+
4pNqz
E
Line Absorption Coefficient
For a damped harmonic oscillator where z is the induced
separation between the dipole charges
d2 z
d t2
dz
+ g dt
+ w0
2
e
E0 eiwt
=
m
e,m are charge and mass of electron
g is damping constant
Solution: z = z0e–iwt
E0 eiwt
e
z=
m w20 – w2 + igw
E
e
=
m w20 – w2 + igw
Line Absorption Coefficient
2
e
4pNe
1+
=
e0
E
1
w02 – w2 + igw
For a gas e ≈ e0
The wave velocity can now be written as
2
1 4pNe
e ½
(e ≈ 1 + 2 m
0
(
c
≈
v
1
w02 – w2 + igw
Where we have performed a Taylor expansion (1 + x) = 1 + ½ x
for small x
Line Absorption Coefficient
c
≈ 1+
v
2pNe2
m
w20 – w2
(w20 – w2)2 + g2w2
–i
gw
(w20 – w2)2 + g2w2
This can be written as a complex refractive index c/v = n – ik. When it is
combined with iwx/v it produces an exponential extinction e–kwx/c . Recall
that the intensity is EE* where E* is the complex conjugate. The light
extinction can be expressed as:
I = I0 e–kwx/c = I0e–lnrx
Line Absorption Coefficient
ln r =
4pNe2
gw
mc
(w20 – w2)2 + g2w2
This function is sharply peaked giving non-zero values when w ≈ w0
2
2
w0 – w =(w0 – w)(w0 + w) ≈ (w0 – w)2w ≈ 2wDw
The basic form of the line absorption coefficient:
ln r =
Npe2
gw
mc
Dw2 + (g/2)2
This is a damping
profile or Lorentzian
profile
Line Absorption Coefficient
Consider the absorption coefficient per atom, a, where lnr = Na
2pe
a=
mc
gw
Dw2 + (g/2)2
g/4p
2pe
a=
mc
Dn2 + (g/4p)2
2pe
a=
mc
l2
c
gl2/4pc
Dl2 + (gl2/4pc)2
Line Absorption Coefficient
∞
∫ a dn
=
0
pe2
mc
This is energy per unit atom
per square radian that the
line absorbs from In
A quantum mechanical treatment
∞
∫0 a dn
= f
pe2
mc
f is the oscillator strength and is related to the transition probability Blu
∞
∫0 a dn
= Blu hn
Line Absorption Coefficient
f=
mc
pe2
Blu hn = 7.484 ×
10–7
Blu
l
gu A
ul
f=
2pe2n2 gl
mc
3
There is also an f value for emission gu fem = gl fabs
Most f values are determined from laboratory measurements and most
tables list gf values. Often the gf values are not well known. Changing the
gf value changes the line strength, which is like changing the abundance.
Standard procedure is you take a gf value for a line, fit it to the solar
spectrum, and change gf until you match the solar line. This value is then
good for other stars.
The Damping Constant for Natural Broadening
Classical dipole emission theory gives an equation of the form
dW
2 e2 w2
–g W
–
=
=
W
3
dt
3 mc
Solution of the form
W= W0e–gt
2 w2
2e
2 in cm
g =
=
0.22/l
3
3mc
The quantum mechanical radiation damping is an order of magnitude
larger which is consistent with observations. However, the observed widths
of spectral lines are dominated by other broadening mechanisms
Pressure Broadening
Pressure broadening involves an interaction between the atoms absorbing
the light and other particles (electrons, ions, atoms). The atomic levels of
the transition of interest are perturbed and the energy altered.
• Distortion is a function of separation R, between absorber and perturber
• Upper level is more strongly altered than the lower level
u
hn
E
l
1
2
3
R
1: unperturbed energy
2. Perturbed energy less than
unperturbed
3. Energy greater than unperturbed
Pressure Broadening
Energy change as a function of R:
DW = Const/Rn
n
Type
Lines affected
Perturber
2
Linear Stark
Hydrogen
Protons, electrons
4
Quadratic Stark
Most lines, especially in hot stars
Ions, electrons
6
Van der Waals
Most lines, especially in cool stars
Neutral hydrogen
Dn = Cn /Rn
Pressure Broadening: The Impact Approximation
Dtj
Photon of duration Dt is an infinite sine wave times a box
Spectrum is just the Fourier transform of box times sine which is sinc pDt(n-n0)
and indensity is sinc2pDt(n-n0). Characteristic width is Dn= 1/Dt
Pressure Broadening: The Impact Approximation
With collisions, the original box
is cut into many shorter boxes of
length Dtj < Dt
The distribution, P, of Dtj is:.
dP(Dtj) = e–Dtj/Dt0 dDtj/Dt0
Because Dtj < Dt the line is broadened with
Dnj = 1/Dtj. The Fourier transform of the sum
is the sum of the transforms.
The line absorption coefficient:
2
∞
∫
Dt2
0
a=
a= C
sinpDt(n – n0)
pDt(n – n0)
e–Dt/Dt0
dDt
Dt0
C
4p2(n – n0)2 + (1/Dt0)2
gn/4p
(n – n0)2 + (gn/4p)2
In other words this is the Lorentzian. To use this in a line profile
calculation need to evaluate gn = 2/Dt0. This is a function of depth in the
stellar atmosphere.
Evaluation of gn
Simplest approach is to assume that all encounters are in one of
two groups depending on the strength of the encounter. If phase
shift is too small ignore it. The cumulative effect of the change in
frequency is the phase shift.
∞
∫
∞
∫
f = 2p n dt = 2p Cn R–n dt
0
0
v
y
Assume perturber moves
past atom in a straight line
r = R cos q
∞
∫
f = 2p Cn cos q dtn
r
0
Atom
r
q
x
R
Perturber
Evaluation of gn
v = dy/dt = (r/cos2q) dq/dt => dt = (r/v)dq/cos2q
p/2
n–2 q dq
n
f = 2p C
cos
vrn–1
∫
n
Usually define a limiting impact parameter for a
significant phase shift f = 1 rad
r0 =
2p Cn
v
∫
∫ cosn–2 q dq
–p/2
–p/2
p/2
p/2
1/(n–1)
2
p
3
2
4
p/2
6
3p/8
cosn–2 q dq
–p/2
The number of collisions is pr0vNT where N is the number of perturbers per unit
volume, T is the interval of the collisions
gn = 2pr02vN
Evaluation of gn : Quadratic Stark
In real life you do not have to calculate gn
For quadratic Stark effect
g4 = 39v⅓C4⅔N
Values of the constant C4 has been measured only for a few lines
Na 5890 Å log C4 = –15.17
Mg 5172 Å log C4 = –14.52
Mg 5552 Å log C4 = –13.12
Evaluation of gn
For van der Waals (n=6) you only have to consider neutral hydrogen and helium
log g6 ≈ 19.6 + 0.4 log C6(H) + log Pg – 0.7 log T
log C6 = –31.7
Linear Stark in Hydrogen
Struve (1929) was the first to note that the great widths of hydrogen
lines in early type stars are due to the linear Stark effect. This is
induced by ions near the hydrogen atom. Above are the Balmer
profiles for an A0 V star.
Thermal Broadening
Thermal motion results in a component of the thermal motion along
the line of sight
Dl
Dn
=
l
n
=
vr
vr = radial velocity
c
N
We can use the Maxwell Boltzmann distribution
[(
2
[(
vr
dN
1
=
exp – v
v 0 p½
0
N
dvr
1.18s
variance v0 = 2kT/m
v
Velocity
Thermal Broadening
The Doppler wavelength shift
Dl
exp –
DlD
[(
[(
2
(
2kT ½
m
Dl
d Dl
D
(
dN
= p–½
N
(
(
v0
n
DnD =
n =
c
c
2kT ½
m
(
DlD =
l
l =
c
c
v0
(
The energy removed from the intensity is (pe2f/mc)(l2/c) times dN/N
l2
f
c
1
exp –
DlD
[(
Dl
DlD
2
[(
a dl =
p½e2
mc
dl
The Combined Absorption Coefficient
The Combined absorption coefficient is a convolution of all processes
a(total) = a(natural)*a(Stark)*a(v.d.Waals)*a(thermal)
The first three are easy as they can be defined as a single dispersion profile with g:
g= gnatural + g4 + g6
The last term is a Gaussian so we are left with the convolution of a Gaussian with
the Dispersion (Lorentzian) profile:
g/4p
pe2
f
a=
mc Dn2 + (g/4p)2 *
2
Lorentzian
1
p½
e–(Dn/DnD)
Gaussian
2
The Combined Absorption Coefficient
p½e2 f
H(u,a)
a=
mc DnD
H(u,a) is the Hjerting function
u = Dn/DnD = Dl/DlD
H(u,a) =
∫
–∞
∞
g
a=
4p
1
DnD =
g
4p
2
l0
c
1
DlD
g/4p2
–(Dn1/DnD)2
2
2
e
dn1
(Dn – Dn1) + (g/4p)
a
H(u,a) = p
∫
–∞
∞
2
e–
du
u
(u – u11)2 + a2 1
Hjerting function
tabulated in Gray
The Line Transfer Equation
dtn = (ln + kn)rdx
ln= line absorption coefficient
kn= continuum absorption coefficient
Source function:
Sn =
jln + jcn
l n + kn
jln = line emission coefficient
jcn = continuum emission coefficient
dIn
= –In + Sn
dtn
This now includes spectral
lines
Using the Eddington approximation
S(t) =
3Fn
(t + ⅔)
4p
At tn = (4p – 2)/3 = t1 , Sn(t1) = Fn(0), the surface flux and source
function are equal
Across a stellar line ln changes being larger towards the center of the line.
This means at line center the optical depth is larger, thus we see higher up
in the atmosphere. As one goes farther from line center, ln decreases and
the condition that tn = t1 is deeper in the atmosphere. An absorption line is
formed because the source function decreases outward.
Computing the Line Profile
In local thermodynamic equilibrium the source function is the Planck function
∞
∫
F = 2p Bn(T) E2(tn)dtn
0
∞
dtn
= 2p Bn(tn) E2(tn) dt dt0
0
∫
0
∞
dlog t0
l n + kn
= 2p Bn(tn) E2(tn)
t0
k0
log e
∫
–∞
Computing the Line Profile
To compute tn
log t0
tn(t0) =
∫
–∞
Fc – Fn
Fc
ln + kn dlog t0
k0 t0 log e
Sn(tc=t1) – Sn(tn = t1)
=
Sn(tc=t1)
Take the optical depth and divide it into two parts, continuum and line
t0
t0
tn =
∫
ln dt +
k0 0
0
tn =
tl + tc
∫
kn
dt
k0 0
0 t0
Computing the Line Profile
t l ≈ l n t0
k0
tc ≈ k n t 0
k0
We need Sn(tn = t1) = Sn(tl + tc = t1) = Sn(tc = t1 – tl)
We are considering only weak lines so tl << tc and evaluate Sn at
t1 – tl using a Taylor expansion around tc = t1
Sn(tn = t1) ≈ Sn(tc = tn) +
dSn (–t )
l
dtn
Computing the Line Profile
Fc – Fn
Fc
=
tl
Sn(tc=t1)
dlnSn
t
= l dt
c
t1
dSn
dtc
ln t dlnSn
≈ k 0 dt
c
n
t1
ln
= C k
n
Weak lines
• Mimic shape of ln
• Strength of spectral line can be increased either by decreasing the
continuous absorption or increasing the line strength
Contribution Functions
∞
∫
–∞
Fn = 2p
Bn(tn) E2(tn) ln + kn t0 dlog t0
k0
log e
Contribution
function
How does this behave with line strength and position in the line?
Sample Contribution Functions
Strong lines
Weak line
On average weaker lines are
formed deeper in the
atmosphere than stronger
lines.
For a given line the
contribution to the line center
comes from deeper in the
atmosphere from the wings
The fact that lines of different strength come from different depths in the
atmosphere is often useful for interpreting observations. The rapidly
oscillating Ap stars (roAp) pulsate with periods of 5–15 min. Radial velocity
measurements show that weak lines of some elements pulsate 180 degres
out-of-phase with strong lines.
In stellar
atmosphere:
+
─
z
Radial node where
amplitude =0
Conclusion: The two lines are formed on opposite sides of a
radial node where the amplitude of the pulsations is zero
Ca II line
Dl (Å)
Strong absorption lines are formed higher up in the stellar atmosphere. The core
of the lines are formed even higher up (wings are formed deeper). Ca II is formed
very high up in the atmospheres of solar type stars.
Behavior of Spectral Lines
The strength of a spectral line depends on:
• Width of the absorption coefficient which is a function of thermal and
microturbulent velocities
• Number of absorbers (i.e. abundance)
- Temperature
- Electron Pressure
- Atomic Constants
Behavior of Spectral Lines: Temperature Dependence
Temperature is the variable that most strongly controls the line strength because
of the excitation and power dependences with T on the ionization and excitation
processes
Most lines go through a maximum
•
Increase with temperature is due to increase in excitation
•
Decrease beyond maximum can be due to an increase in continous opacity
of negative hydrogen atom (increase in electron pressure)
•
With strong lines atomic absorption coefficient is proportional to g
•
Hydrogen lines have an absorption coefficient that is temperature sensitive through
the stark effect
Temperature Dependence
Example: Cool star where kn behaves line the negative hydrogen ion‘s boundfree absorption:
kn = constant T–5/2 Pee0.75/kT
Four cases
1. Weak line of a neutral species with the element mostly neutral
2.
Weak line of a neutral species with the element mostly ionized
3.
Weak line of an ion with the element mostly neutral
4.
Weak line of an ion with the element mostly ionized
Behavior of Spectral Lines: Temperature Dependence
Case #1:
The number of absorbers in level l is given by :
Nl = constant N0 e–c/kT ≈ constant e–c/kT
The number of neutrals N0 is approximately constant with temperature until
ionization occurs because the number of ions Ni is small compared to N0.
Ratio of line to continuous absorption is:
R=
ln
kn
T5/2 –(c+0.75)/kT
= constant
e
Pe
Behavior of Spectral Lines: Temperature Dependence
Recall that Pe = constant eWT
5
ln R = constant +
2
1
R
dR
dT
ln T – c + 0.75
kT
c + 0.75
2.5
– WT
=
+
2
T
kT
– WT
Behavior of Spectral Lines: Temperature Dependence
Exercise for the reader:
Case 2 (neutral line, element ionized):
1
R
dR
dT
=
c + 0.75 – I
kT2
Case 3 (ionic line, element neutral):
1
R
dR
dT
=
c + 0.75 + I
5
T
+
– 2WT
kT2
Case 4 (ionic line, element ionized):
1
R
dR
dT
2.5
=
T
c + 0.75
+
kT2
– WT
Behavior of Spectral Lines: Temperature Dependence
Exercise for the reader:
Case 2 (neutral line, element ionized):
1
R
dR
dT
=
c + 0.75 – I
kT2
Case 3 (ionic line, element neutral):
1
R
dR
dT
=
c + 0.75 + I
5
T
+
– 2WT
kT2
Case 4 (ionic line, element ionized):
1
R
dR
dT
2.5
=
T
c + 0.75
+
kT2
– WT
The Behavior of Sodium D with Temperature
The strength of Na D decreases with increasing temperature. In this
case the absorption coeffiecent is proportional to g, which is a function of
temperature
Behavior of Hydrogen lines with temperature
B3IV
B9.5V
A0 V
G0V
F0V
The atomic absorption coefficient of hydrogen is temperature sensitve through
the Stark effect. Because of the high excitation of the Balmer series (10.2 eV)
this excitation growth continues to a maximum T = 9000 K
Behavior of Spectral Lines: Pressure Dependence
Pressure effects the lines in three ways
1. Ratio of line absorbers to the continous opacity (ionization equilibrium)
2. Pressure sensitivity of g for strong lines
3. Pressure dependence of Stark Broadening for hydrogen
2
For cool stars Pg ≈ constant Pe
Pg ≈ constant
g⅔
Pe ≈ constant g⅓
In other words, for F, G, and K stars the
pressure dependencies are translated into
gravity dependencies
Gravity can influence both the line wings and
the line strength
Example of change in line strength with gravity
Example of change in wings due to gravity
Pressure dependence can be estimated by considering
the ratio of line to continuous absorption coefficients
Rules:
1. weak lines formed by any ion or atom where most of the element is in the
next higher ionization stage are insenstive to pressure changes.
2. weak lines formed by any ion or atom where most of the element is in that
same stage of ionization are presssure sensitive: lower pressure causes a
greater line strength
3. weak lines formed by any ion or atom where most of the element is in the
next lower ionization stage are very pressure sensitive: lower pressure
causes a greater line strength.
Rule #1
Ionization equation:
Nr+1
Nr
Fj(T)
=
Pe
By rule one the line is formed in the rth ionization stage, but most of
the element is in the Nr+1 ionization stage: Nr+1 ≈ Ntotal
Nr ≈ constant Pe
ln ≈ constant Nr ≈ constant Pe
The line absortion coeffiecient is proportional
to the number of absorbers
The continous opacity from the negative hydrogen ion dominates:
kn = constant T–5/2 Pee0.75/kT
ln
kn
is independent of Pe
Rule #2
If the line is formed by an element in the r ionization stage and most of
this element is in the same stage, then Nr ≈ Ntotal
ln
kn
=
constant
Pe
≈ constant g–⅓
Note: this change is not caused by a change in l, but because the continuum
opacity of H– becomes less as Pe decreases
Also note:
∂ log(ln/kn)/∂ log g = –0.33
Proof of rule #3 similar.
In solar-type stars cases 1) and 2)
are mostly encountered
Behavior of Spectral Lines: Abundance Dependence
The line strength should also depend on the abundance of the absorber, but
the change in strength (equivalent width) is not a simple proportionality as
it depends on the optical depth.
3 phases:
Weak lines: the Doppler core dominates
and the width is set by the thermal
broadening DlD. Depth of the line grows in
proportion to abundance A
Saturation: central depth approches
maximum value and line saturates towards
a constant value
Strong lines: the optical depth in the wings
become significant compared to kn. The
strength depends on g, but for constant g
the equivalent width is proportional to A½
The graph specifying the change in
equivalent width with abundance is called the
Curve of Growth
Behavior of Spectral Lines: Abundance Dependence
Assume that lines are formed in a cool gas above the source of the continuum
Fn = Fce–tn Fc is continuum flux
L
L
∫
tn = lnrdx =
0
0
∫ Na dx
L is the thickness of the cool gas.
N/r = number of absorbers per unit mass
N
r
N NE
=
NE NH
NH
r
L
tn = A
0
∫
(N/NE)Nha dx
N/NE is the fraction of element E capable of
absorbing, NE/NH is the number abundance A,
NH/r is the number of hydrogen atoms per unit
mass
tn is proportinal to the abundance A and the
flux varies exponentially with A
Behavior of Spectral Lines: Abundance Dependence
For weak lines tn << 1
Fn ≈ Fc(1 – tn)
Fc – Fn
Fc
≈ tn
→ line depth is proportional tn and thus A. The line
depth and thus the equivalent width is proportional to A
Behavior of Spectral Lines: Abundance Dependence
What about strong lines?
p½e2 f
H(u,a)
a=
mc DnD
The wings dominate so
f
pe2 g
a=
mc 4p2 DnD
L
tn = A
0
∫
Af
g
(N/N
)N
dx
(N/NE)Nha dx =
E
H
2
mc Dn2
4p
0
<g> A f h
≈
Dn2
pe2
L
∫
<g> denotes the depth average
damping constant, and h is the
constants and integral
Fc – Fn
Fc
= 1 – e–tn
The equivalent width of the line:
∞
W=
0
∫
(1 – e–tn) dn
∞
W=
0
∫
(1 – e–<g>Af h/Dn2) dn
Substituting u2 = Dn2/<g>A f h
∞
W = (<g>A f h)½
0
∫
2
(1 – e–1/u ) du
Equivalent width is proportional to the square root of the abundance
A bit of History
Cecilia Payne-Gaposchkin (1900-1979).
At Harvard in her Ph.D thesis on Stellar Atmospheres she:
• Realized that Saha‘s theory of ionization could be used to determine the
temperature and chemical composition of stars
• Identified the spectral sequence as a temperature sequence and correctly
concluded that the large variations in absorption lines seen in stars is due to
ionization and not abundances
• Found abundances of silicon, carbon, etc on sun similar to earth
• Concluded that the sun, stars, and thus most of the universe is made of
hydrogen and helium.
Otto Struve: „undoubtedly the most brilliant Ph.D thesis ever written in Astronomy“
Youngest scientist to be listed in American Men of Science !!!