3: Interstellar Absorption Lines:Radiative Transfer in the
Interstellar Medium
James R. Graham
University of California, Berkeley
AY 216 2
Interstellar Absorption Lines
Example of atomic absorption lines• Structure of multielectron atoms & Grotrian diagrams
Radiative transfer (see Lecture 2)• Review the equation of transfer and simple solutions• Relate jν and κν to Einstein A & B, f-values, etc.• Line broadening & line shape function, φν
Optical and UV absorption lines• Variation of absorption profile with optical depth• Equivalent width vs. column density
Examples• NH/E(B-V) for the local ISM & depletion of heavy elements• Homework 1
Spitzer Ch. 3; Tielens Ch. 2; Dopita & Sutherland Ch. 2& 4
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Atomic Optical Absorption Lines
Initial evidence for a pervasive ISM came from atomicabsorption lines at visible wavelengths
• Principal ISM probe prior to radio & space-borne studies
Strong optical absorption lines:
Resonance lines• Electric dipole transitions (ΔL=±1, ΔS=0) from the ground state
• Other, weaker, optical lines (discovered 1930s - 40s) includeTi II, Ca I, K I, Li I, CH, NH, CN, CH+ & C2
NameWavelength (Å)Transition
Calcium H & K3933, 3968Ca II4s 2S1/2—4p 2Po
1/2,3/2
Sodium D1 D25890, 5896Na I3s 2S1/2—3p 2Po
1/2,3/2
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Ca II Grotrian Diagram
Multielectron atomsare labeled 2S+1LJ
• L total orbitalangular momentum
• S total spin angularmomentum and2S+1 is the spinmultiplicitiy
• J total angularmomentum J=L+S,L+S-1…|L-S|
http://nedwww.ipac.caltech.edu/level5/Ewald/Grotrian/grotrian.html
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Isoelectronic Sequence
Atoms or ions with the same number ofelectrons have similar electronic structure• Li I, Be II, C IV, N V, O VI
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Common UV Absorption Lines
Rocket & satellite observationsshow strong UV absorptionlines from the ISM
• Typical excitation energies ofresonance lines are ~ eV
• Many important atomic resonancelines are in the near-UV
+ H I+ C I - IV, O I - O VII+ MgII
• Many rare elements, e.g., Kr, Ga, Ge,As, Se, Sn, Te, Tl, Pb, Cu, Co, Mn,Zn, & Al can be traced by weak UVabsorption lines
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Observations of Absorption Lines
At spectral resolution R =λ/∆λ ≥ 104 (30 km/s)absorption lines break upinto resolved components,Doppler shifted relative toone another
Interstellar lines through thehalo towards HD93521• Hubble/GHRS data reveal
velocity structure spanning~ 90 km/s
• High SNR permits the studyof abundances & physicalconditions in individualclouds along the sight line
Individual lines withinmultiplets can be recorded
Fitz
gera
ld &
Spi
tzer
SII
2P02-
2D4
2P02-
2S2
2P02-
2S2
4S04-
4P2
4S04-
4P4
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Radiative Transfer Review
The transfer equation is
• The absorption term
energy absorbed s/cm2/sr/Hz
• Emission term
Optical depth
Source function
Equation of transfer becomes
Integrate through a slab:
€
Iν (τν ) = Iν (0) e−τν + Sν (τν ′) e
−(τν −τν ′ ) dτν ′0
τν∫€
dIνdτν
= −Iν + Sν
€
Sν ≡ jν /κν
€
dτν =κν ds
€
jν€
−κν Iν
€
dIνds
= −κν Iν + jν
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Line Emission Coefficient (jjk)
The line emission coefficient, jjk, describesradiative transitions from the kth
excited state tothe lower jth state
Usually expressed as the line emissivity
in units of erg/s/cm3
• Factor of 4# indicates total emission in alldirections. A value is in units of s-1
€
j jk = jν dνline∫
€
4π j jk = nkhν jkAkj
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Line Absorption Coefficient (κjk)
• κjk describes the total radiative excitation between thelower jth level and the excited kth state
sν = κν/nj is the atomic absorption cross-section fromlower level j at frequency ν
Line absorption coefficient, κjk has two components
• Bjk gives the rate of absorption out of level j into level k
• Bkj gives the stimulated emission from level k down to level j€
κ jk =hν jk
cn jB jk − nkBkj( )
€
κ jk = κνline∫ dν = n j sνline∫ dν = n js jk
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Einstein A & B
The B’s are the Einstein coefficients for absorption &stimulated emission• In equilibrium
• Assumption of TE shows that
• Where
• fkj is the emission oscillator strength
€
g j B jk = gk Bkj and Bkj =c 3
8π hν jk3 Akj
€
gk fkj = g j f jk€
Akj =8π 2e2ν 2
mec3 fkj
€
uν n jB jk − nkBkj( ) = nkAkj
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Optical Absorption Lines
Traditional way of studying H I clouds• If UV is accessible then HI Lyα (1216 Å), Lyβ (1026
Å, Lyγ (972 Å), etc. can used to measure HI column
• If only visible observations are possible then Na I Dand Ca II H & K lines are often the strongest lines
hν >> kT for T ≈ 80 K and ν = 3 × 1015 Hz• Neglect stimulated emission for a slab of optical
depth τν
€
Iν = Iν (0)e−τν , τν = Nl sν
€
Iν (0)
€
Iν
€
τν = Nl sν
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Departure Coefficients
Departure coefficients, bj, relate actual levelpopulations (n) with the TE populations (n*)
bj = nj / n*j
For example departure coefficients quantify the non-TEconditions of the ISM and can be used to compute sjk:
€
s jk = sνline∫ dν =κνn j
line∫ dν =hν jk
cB jk −
nkBkj
n j
=hν jk
cB jk 1−
nkgin jgk
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Departure Coefficients
The TE level populations are related by a Boltzmannfactor
In terms of the integrated atomic cross-section
su ≡ (hνjk /c) Bjk = (! e2 /me c) fjk
sjk can be defined as the total cross-section for pure
absorption, su, modified by a stimulated emissioncorrection
€
nk*
n j* =
gkg j
e−hν jk / kT
€
s jk = su 1−nkgin jgk
= su 1−
bkb j
e−hν / kT
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Special Cases
hv >> kT
• sjk ≈ su
• Pure absorption dominates because stimulated emission isnegligible, population of excited states is insignificant
• E.g., UV absorption line studies of cold gas
hv << kT• Stimulated emission is important. To first order in the exponent
or in terms of fjk
€
s jk = su 1−bkb j1−hν /kT( )
€
s jk =πe2
mecf jkhνkT
bkb j
−kThν
bkb j
−1
€
s jk = su 1−bkb je−hν / kT
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Special Cases: hν/kT << 1
When hν/kT << 1
Local Thermal Equilibrium (LTE), bk=bj=1
correction for stimulated emission reduces the crosssection by a factor of hν/kT• e.g. HI 21 cm at T = 80 K, hν/kT ≈ 8 x10-4
“Extreme” non-LTE• Absorption term is emissive, corresponding to a maser
• Level populations have been driven so far out of TE that theyare inverted (nk>nj)
€
s jk = suhνkT
=πe2
mecf jkhνkT€
s jk = su 1−bkb j1− hν
kT
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Line Shape & Doppler Shift
The cross section sν = s φ(ν) such that
Let φ1(ν) be the absorption profile of one atomφ1 = φ1(ν - ν’0 )
= φ1(ν - ν0 [1+ w/c]) = φ1(∆ν - ν0 w/c)
w = line of sight velocity of the atomν0 = rest frequencyν’0 = ν0 (1+ w/c) = non-relativistic Doppler shifted frequency∆ν = ν - ν0
€
φ(ν )dν =1line∫
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Broadening Mechanisms
For an ensemble of atoms the line of sightvelocity distribution of gas P(w) gives φ(ν)
Line broadening due to the uncertainty principle• Finite lifetime of an upper state implies an atom can
absorb at ν ≠ ν0
Lorentzian or natural profile for an atom at rest withwidth
€
φ(ν) = P(w)φ1(Δν −ν 0w /c)dw−∞
∞
∫
€
φ1(ν) =1π
γ kγ k2 + Δν 2( )
€
γ k =14π
Akik> i∑
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Natural Broadening
• Natural line widths are very small• HI Lyα, A21 = 6 x 108 s-1, ν = 2 x 1015 Hz
• γk/ ν = 3 x 10-8 or ∆w = (∆ν/ν) c = 9 m s-1
• Forbidden lines are even narrower
• Other broadening mechanisms• Stark & Zeeman effects
• Lines can be broadened by collisions• At low ISM densities, pressure broadening is
only significant for radio recombination lines
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Gaussian Line Profile
Gaussian velocity distribution
For a Maxwellian at temperature T
where m is the molecular weight and σTrepresents a turbulent component
b ≈ 0.129 (T/A)1/2 km s-1
FWHM = 2 √(2 ln 2) σ ≈ 2.355 σ
€
dP(w) = P(w)dw =1π b
e−(w / b )2
dw
€
b2 =2kTm
+ 2σT2
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Voigt Profile
The combination or convolution of the natural& Doppler profile yields the Voigt profile
Define the Doppler with ∆νD = b ν0 /c = b / λ0€
φ(ν) = P(w)φ1(Δν −ν 0w /c)dw−∞
∞
∫
€
φ(ν) =1π b
e−(w / b )2 1π
γ kγ k2 + (Δν −ν 0w /c)
2 dw−∞
∞
∫
€
φ(ν) =1
π 3 / 2be−(w / b )
2 γ kγ k2 + (Δν −ΔνDw /b)
2 dw−∞
∞
∫
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Voigt Profile
The Voigt profile varies with relative width of• Natural broadening Γ
• Doppler width ΔνD
a = γ/ΔνD
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Limiting Cases of the Voigt Profile
Case 1: Doppler core–natural line width φ1(ν) isapproximated by a δ-function for ∆ν ≈ ∆νD
• Case 2: Damping wings–good for large ∆ν >> ∆νD€
φ(ν) =1
π ΔνD
e−(Δν /Δν D )2
€
φ(ν) =γ k
πΔν 2
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UV/Visible Absorption Line Formation
Neglect stimulated emission for UV/visible interstellarabsorption lines (hv >> kT).
Lines are pure absorption, and the equation ofradiative transfer has the solution
or in terms of wavelength
Ideally, observation of an absorption-line profile can beturned into a measurement τλ• Finite spectral resolution compared to the intrinsic width, limits
on SNR, etc.signal-to-noise, makes it convenient to expressthe line strength in terms of an integrated observable, theEquivalent Width
€
Iν = Iν ,0e−τν
€
Iλ = Iλ,0e−τλ
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Absorption Line Profiles
τ0
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Doppler Cores & Damping Wings
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Equivalent Width of Spectral Lines
• Equivalent width of line:
• Wν_is the width of a rectangular profile from 0 to Iν(0)that has the same area as actual line• Wν measures line strength, units are Hz
• Similarly, in terms of wavelength
with Wλ typically measured in Å or mÅ
Wλ / λ = Wν / ν
€
Wν ≡Iν (0)−IνIν (0)
dν = 1− e−τν( )dν−∞
∞
∫−∞
∞
∫
€
Wλ ≡I λ (0)−I λI λ (0)
dλ−∞
∞
∫ = 1− e−τλ( )dλ−∞
∞
∫
λ
I/I0
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Schematic Equivalent Width
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Curve of Growth
Optically thin limit
Doppler broadened line
τ0 is the optical depth at line center
€
Wλ = 1− e−τν( )dν−∞
∞
∫ ≈ τλdλ−∞
∞
∫
= N j sφ(ν )λdνν
=−∞
∞
∫ N j sλc
2
€
Wλ ≈ 1− exp − N j s
π Δν De−(Δν /Δν D )
2[ ]{ }dλ−∞
∞
∫
=2bλ0c
ln(τ 0)
€
τ 0= N j sφν (0) =λ0πb
N j s = N jπ e2
ΔνDmecf jk
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Saturated Lines
For strong lines the width is given roughly by thepoint where τ ≈ 1 is achieved
e-1 2Δντ=1
τ0
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Saturated Lines
When lines are strong the width is given roughlyby the frequency where τ ≈ 1 is achieved• For a Doppler broadened line
€
τ 0 exp − Δντν =1 ΔνD( )2
≈1
Hence
Δντν =1 ≈ ΔνD log τ 0( ) =bλ0
log τ 0( )
Wν τ 0( ) ≈ 2Δντν =1 = 2 bλ0
log τ 0( )
Wλ =Wν
λ0ν 0
≈2bλ0c
log τ 0( )
€
τν = τ 0 exp − Δν ΔνD( )2[ ]
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Curve of Growth
Damped line
Transitions• Linear breaks down when τ0 ~ 1
• Doppler to damping when€
Wλ ≈ 1− exp −N j sγ k
πΔν 2
dλ
−∞
∞
∫
=2λc
N j sλ2γ k
€
b ln(τ 0) ≈ N j sλ2γ k
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Curve of Growth: Linear Regime
Weak lines, τ0 << 1
Linear regime:
The column is 1017 N17 cm-2
Wavelength is 1000 λ-5 Å
• Expect sensitivity limit Wλ/ λ ≈ R/SNR ~ 104/102 ~ 106
€
Wν = τν dν−∞
∞
∫ = N j σ(ν)dν = N jπe 2
mecflu−∞
∞
∫
€
Wν ∝N j
€
Wλ
λ= 0.885N17 j f λ−5
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Curve of Growth: Flat Regime
• Large τ0: all background light near line centeris absorbed, line is “saturated”• Far from line center there is partial absorption
• Wλ grows very slowly with Nj• Flat part of the curve of growth
• Onset of deviation from linear depends onDoppler parameter• Broader Doppler line will remain on the linear part of
the curve of growth for higher column
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Schematic Curve of Growth
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Curve of Growth Analysis
Goal: relate equivalent width Wν or tocolumn density Nj
Relation is monotonic, but non-linear Classical theory developed in context of stellar
atmospheres, but works for the ISM Three regimes, depending on τ at line center:
• τ0 << 1, linear regime• τ0 > 1, large, flat regime• τ0 >> 1, square-root (damping) regime
νλ
λ WW c
2=
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Interstellar Na I D Absorption
Linear - flatregime
Flat regime
Square-rootregime
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UV Absorption Lines towards ς Oph
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Optical Absorption Line Observations
Technique limited to bright backgroundsources
Mostly local (< 1 kpc), mostly AV < 1 mag.corresponding to N(H) < 5×1020 cm–2
• Strong Na I lines in every direction• Same clouds as seen in H I emission and
absorption,• Also seen in IRAS 100 µm cirrus ⇒ CNM
• Column densities from Lyα observations
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Gas Phase Abundances
Absorption line studies yield information on the gasphase abundance of a range of astrophysicallyabundant elements
Crucial for cosmological studies, e.g., D/H
Depletion D• log D = log10(abundancemeas) - log10(abundancecosmic)• Ca: log D ≈ -4 ⇒ 10,000 times less Ca than in the solar
photosphere
• Plot log D vs. condensation temperature• Shows a strong correlation?
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Depletions (log10 D) vs. Condensation Temperature
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Depletion on Grains
• In diffuse clouds many abundances are muchsmaller than solar• Strong correlation between low condensation
temperature and high depletion suggests formationof grains in circumstellar envelopes
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