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

AY 216 3

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

AY 216 4

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

AY 216 5

Isoelectronic Sequence

Atoms or ions with the same number ofelectrons have similar electronic structure• Li I, Be II, C IV, N V, O VI

AY 216 6

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

AY 216 7

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

AY 216 8

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

AY 216 9

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

AY 216 10

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

AY 216 11

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

AY 216 12

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

AY 216 13

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

AY 216 14

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

AY 216 15

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

AY 216 16

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

AY 216 17

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∫

AY 216 18

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∑

AY 216 19

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

AY 216 20

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

AY 216 21

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−∞

∞

∫

AY 216 22

Voigt Profile

The Voigt profile varies with relative width of• Natural broadening Γ

• Doppler width ΔνD

a = γ/ΔνD

AY 216 23

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

AY 216 24

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

AY 216 25

Absorption Line Profiles

τ0

AY 216 26

Doppler Cores & Damping Wings

AY 216 27

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

AY 216 28

Schematic Equivalent Width

AY 216 29

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

AY 216 30

Saturated Lines

For strong lines the width is given roughly by thepoint where τ ≈ 1 is achieved

e-1 2Δντ=1

τ0

AY 216 31

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[ ]

AY 216 32

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

AY 216 33

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

AY 216 34

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

AY 216 35

Schematic Curve of Growth

AY 216 36

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=

AY 216 37

Interstellar Na I D Absorption

Linear - flatregime

Flat regime

Square-rootregime

AY 216 38

UV Absorption Lines towards ς Oph

AY 216 39

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

AY 216 40

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?

AY 216 41

Depletions (log10 D) vs. Condensation Temperature

AY 216 42

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

AY 216 43