AY216_08 1

Lecture 2

Interstellar Absorption Lines:Line Radiative Transfer

1. Atomic absorption lines

2. Application of radiative transfer

to absorption & emission

3. Line broadening & curve of growth

4. Optical/UV line formation

5. Optical/UV line observations

References: Spitzer Secs. 3.1-3.4Ay216_06 (JRG)Dopita & Sutherland Sec. 2.1

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1. Optical & UV Absorption Lines

Familiar solar optical doublets, e.g., Fraunhofer Ca II Kand Na I D provided early evidence for a pervasive ISM.

These are “resonance” (allowed electric-dipole) transitions, that startfrom the ground state with an electron going from an s to a p orbital.Similar transitions occur across the sub-μm band; those below 0.3 μmrequire space observations. Some important examples are:

H I [1s] 1216 ÅC IV [{1s2} 2s] 1548, 1551 ÅNa I [{1s22s22p6}3s] 5890, 5896 ÅMg II [{1s22s22p6}3s] 2796, 2803 ÅK I [{1s22s22p63s23p6}4s] 7665, 7645 ÅCa II [{1s22s22p63s23p6}4s] 3934, 3968 Å

The [{ …}…] notation gives the electronic configuration, e.g., for

MgII, [{1s22s22p6}3s] indicates that it is a “one-electron” ion with a 3s electron outside neon-like {1s22s22p6} closed shell.

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Energy Levels of Atomic Hydrogen

--------------- n =

--------------- n = 5

--------------- n = 4

--------------- n = 3

--------------- n = 2

--------------- n = 1

Energies depend only on theprinciple quantum number n:

E =- R/n2, R = 13.6 eV,

The levels are 2n2-fold degenerate. The orbital quantum number l ranges From l = 0 to n-1 & the spin quantum number is = 1/2

The degenerate levels are spreadout according to orbital angularmomentum (at top), in a formatknown as a Grotrian Diagram.

The solid line indicates the maintransitions that follow from aparticular recombination event.

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Sample Grotrian Diagrams

Mg II [{1s22s22p6}3s] (2S) O I [1s22s22p4] (3P)

Spectroscopic notation for multi-electron atoms: 2S+1LJ;

S, L, & J (J = L + S) are total spin, total orbital, and totalangular momentum quantum numbers, all conserved formotion in a central potential.

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White Dwarf Hz 43A Krut et al.( 2002)

Halo star HD 93521Spitzer & Fitzpatrick (1993)

Observations of Absorption Lines

At resolution, R = > 104 (30km/s), absorption lines break upinto Doppler shifted components.

Interstellar lines through the halotowards HD 93521 revealvelocities spanning ~ 90 km/s.

FUSE operates below 1100 Åand can detect H & D Ly- lines,as discussed in Problem 1.

FUSEHST GHRS

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2. Application of Radiative Transfer

to Emission and Absorption

We assume some knowledge of radiative transfer, which is founded on the specific intensity I , defined as the energy passing through a unit area at an angle in unit time, area, solid angle and frequency:

dE = I dAcos d d dt

Standard treatments of radiative transfer are widely available:

Rybicki & Lightman Ch.1Spitzer Ch. 3Shu I Chs. 1-4McKee AY216_06, Lec. 2

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The Equation of Transfer

The equation of transfer now assumes the form,

and its solution for a slab is:

dI

ds= I + j

d = ds S j /

dI

d= I + S

I ( ) = I (0) e + S ( ) e ( ) d0

The intensity satisfies the equation of transfer

Shu shows that I may be considered a Boltzmann-like distributionfunction for photons, The equation of transfer corresponds to theBoltzmann transport equation; the source term is the emissivity j

and the absorption rate - I is the loss term. We next define the

optical depth through the differential absorption in a distance ds

and also define the “source function” S ,

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The Emission Coefficient

The emissivity j is the rate at which energy is emitted perunit volume, solid angle, frequency and time. For line emission from an upper level k to lower level j, weintegrate over frequency to get the line emission coefficient,

j jk = jline

d

This Is usually expressed in terms of the Einstein Akj

4 j jk = nkh kjAkj

The factor 4 arises from integrating over all angles.The units are erg cm-3 s-1, since the units of Ak over s-1

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The Absorption Coefficient

The absorption coefficient for radiative excitation from a lower level j to an upper level k is determined by the atomic absorption cross-section (units cm2) defined by = nj s . The frequency integrated absorption coefficient is

gives absorption rate

from level j up o level k

gives the rate ofstimulated emission fromlevel k down to level j

jk =line

d = n j sline

d n js jk

jk =h jk

cn jB jk nkBkj( )

It is related to the Einstein B-coefficients:

Note that the integrated quantity sjk has the units of cm2 s-1, and thatit can be related to the B-coefficients, as will be done in slide 11.

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Review of the Einstein A & B Coefficients

The Einstein A-coefficient gives the rate of spontaneous decay.The relations between the A and B coefficients,

can be derived from the equation of radiative balance for levels k and j for a radiation field with spectral energy distribution u

by assuming complete thermodynamic equilibrium for both the levels j and k and the radiation field. They can also be related to emission and absorption oscillator strengths, which are essentially QM matrix elements,

g j B jk = gk Bkj Bkj =c 3

8 h jk3 Akj

u n jB jk nkBkj( ) = nkAkj

Akj =8 2e2 2

mec3 fkj gk fkj = g j f jk

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The Integrated Cross Section

With these facts on the B-coefficients, the integrated cross section is:

Thermal equilibrium (TE) level populations are now labeled by an asterix,

and departure coefficients bj relate the true level populations (n) to thermal equilibrium populations (n*), as in bj = nj / n*j. Then sjk is

s jk = sline

d =n j

lined =

h jk

cB jk

nkBkj

n j

=h jk

cB jk 1

nkgin jgk

nk*

n j* =

gkg j

e h jk / kT

s jk = su 1nkgin jgk

= su 1

bkb j

e h / kT

where

su (hvjk /c) Bjk = ( e2 /me c) fjk

is the integrated absorption cross-section without stimulated emission.

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Integrated Cross Section: Limiting Cases

hv << kT -- Stimulated emission is important. Expand theexponential and introduce the oscillator strengths to find

hv >> kT -- Stimulated emission is unimportant; pureabsorption dominates

For HI 21 cm at 80 K, the correction is huge: hv/kT 8 x10-4.

s jk su 1bkb j1 h /kT( )

=

e2

mecf jk

h

kT

bkb j

kT

h

bkb j

1

s jk = su 1bkb je h / kT

su

NB Masers are an example of extreme non-LTE. The absorption term is emissive because the level populations are inverted (nk>nj).

s jke2

mecf jkh

kT

If the levels are in TE,

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3. Line Shape & Broadening

The absorption profile 1 for an atom depends on - ’0,

1 = 1( - ’0 )

= 1( - 0 [1- w/c])

= 1( + 0 w/c)

( )d =1line

Rewrite the absorption cross section as s = s ( )

(dropping the line index jk) and introduce the line shape

function ( ) with unit normalization,

Assume that the absorber is in a system moving witha line of sight velocity w, and introduce the notation:

0 = rest frequency’0 = 0 (1- w/c) = Doppler shifted frequency

= - 0

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1. Natural Line Broadening

Natural line broadening is due to the finite lifetimes of the upper and lower states (Heisenberg Uncertainty

Principle), i.e., atoms absorb and emit over a range offrequencies near 0 , and generate a “Lorentzian” lineshape

Start with the general formula for the line-shape function or a collection of atoms

The width for an atom at rest (w = 0) is

( ) = P(w) 1( + 0w /c)dw

where P(w) is the velocity distribution of the gas

1( ) =1 k

k2 + 2( )

k =1

4Aki

k> i

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Natural Line Broadening (cont’d)

Natural line widths are very small, e.g., for theelectric-dipole HI Ly transition,

A21 = 6 x 108 s-1, = 2 x 1015 Hz, k/ = 3 x 10-8

For comparison with measured line widths, typically oforder km/s, this corresponds to a velocity width,

w = ( k/ ) c = 9 m/s

Forbidden lines are even narrower.

Other broadening mechanisms are caused by:Stark & Zeeman effectsCollisions (“pressure broadening”)

At low ISM densities, “pressure broadening” is only relevantfor radio recombination lines.

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Case 2. Doppler Broadening

The velocity distribution is Gaussian

For a Maxwellian at temperature T

where m is the mean molecular mass and turb is the

dispersion of the turbulent velocities.

The total dispersion is related to the Gaussian parameterby = b/21/2 .

bth 0.129 (T/A)1/2 km s-1 (A = atomic mass)

FWHM = 2 (2 ln 2) 2.355 .

P(w) =1

be (w / b )2

b2 =2kT

m+ 2 turb

2

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Case 3. Voigt Profile

yields the Voigt profile

Convolving Doppler and natural broadening,

Define the Doppler width, D = ( 0 /c) b = b / 0, to get

( ) = P(w) 1( + 0w /c)dw

( ) =1

be (w / b )2 1 k

k2

+ ( + 0w /c)2 dw

( ) =13/ 2b

e (w / b )2 k

k2

+ ( + Dw /b)2 dw

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The Voigt profile (evaluated numerically) depends on the ratio of the natural to the Doppler widths

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Limiting Forms of the Voigt Profile

2. For large ( >> D) the slowly-decreasing dampingwings dominate (and come out of the generalconvolution integral) to yield the characteristic

Lorentzian profile wings:

1. Approximating the natural line function by a delta-function recovers the Doppler profile.

( ) =1

D

e ( / D )2

( ) = k2

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Doppler Cores and Damping Wings

QSO Absorption Spectrum

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4. UV/Visible Absorption Line Formation

Neglect stimulated emission (hv >> kT), assume pureabsorption and a uniform slab. The equation of radiativetransfer without the emissivity term has the solution,

Ideally, measuring the frequency-dependent line profile determines .

Finite spectral resolution, signal to noise (S/N), etc., often make an integrated

observable called equivalent width more useful.

I = I (0)e , = Nl s

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Equivalent Width

W is the width of the equivalent

rectangular profile from 0 to I(0)

with the same area as the actual

line. The units are Hz.

I/I0

W I (0) II (0) d = 1 e( )d

More common is the quantity in terms of wavelength W , measured in Å or mÅ, related as:

W / = W /

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Curve of Growth: Linear Regime

Optically thin limit: 0 << 1

Units: for column density, 1017N17 cm-2, for wavelength ,1000 -5 Å

Since , this is known as the linear regime for the curve of growth:

0 is the optical depth at line center,

W = d = N j ( )d = N je 2

mecflu

0=0

bN j s = N j

e2

Dmecf jk

W N j

W= 0.885N j,17 f 5

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Nonlinear Portion of the Curve of Growth

For moderately large, I.e., not too large optical depthsDoppler broadening suffices and its limit for large opticaldepths is

For large 0, the light from the source near line center isabsorbed, i.e., the absorption is “saturated”. Far fromline center there is partial absorption, and W growsslowly with Nj; this is the flat portion of the curve of growth.

For very large 0, the Lorentzian wings take over, and generate the square-root portion of the curve of growth

Wl / = (2/c) ( 2Nj s k)1/2

W 1 expN j s

De ( / D )

2[ ]{ }d 0

2b

cln( 0)

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Schematic Curve of Growth

The departure from linear depends on the Doppler parameter; broader Doppler lines remain longer on the linear part of the curve of growth.

NB The origin of this figure has been lost. The abscissa label is log Nf .

log Nf

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Summary of Curve of Growth Analysis

The goal is to determine column density Nj fromthe measured equivalent width, which increasesmonotonically but non-linearly with column Nj .

There are three regimes, depending on opticaldepth at line center:

0 << 1, linear

0 > 1, large flat

0 >> 1, square-root (“damping”)

Unfortunately, many observed lines fall on the insensitive flat portion of the curve of growth.

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Illustrating the Three Parts of the Curve of Growth

for Interstellar Na I Absorption Lines

Linear - flat

Flat

Square-root

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5. Optical/UV Absorption Line Observations

Require bright background sources with littleobscuration by dust:d < 1 kpc, AV < 1 mag or NH < 5x1020 cm–2

Strong Na I lines are observed in every direction:- same clouds are seen in H I emission and

absorption,- also seen in IRAS 100 m cirrus CNM

But the H column densities (and abundances)require UV observations of Ly .

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Absorption Lines for an Extragalactic Source

Absorption line spectra of an interstellarcloud seen towards a quasar.

Note the mm absorption lines of HCO+ at the bottom.

Molecular lines are common at radioAs well as optical and UV wavelengths.

Figure from Snow & McCall,ARAA 44 367 2006

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UV Absorption Lines towards Oph(probably the best studied diffuse interstellar cloud)

Curves are empirical curves of growth for several UV lines, but with

the same Doppler parameter.

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Results of Abundance Measurements

• Absorption line studies yield gas phase abundances of theastrophysically important elements that determine thephysical & chemical properties of the ISM, including the

D/H ratio important for cosmology.• Refractory elements are “depleted” relative to solar; the depletion factor D(X) of species X is D(X) = measured x(X) /solar x(X) e.g., log D(Ca) = - 4 10,000 times less Ca in inter-

stellar clouds the solar photosphere. The following plot ofdepletion vs. condensation temperature suggests that the“missing” elements are in interstellar dust.

• For more results, see recent review, “Diffuse Atomic andMolecular Clouds” Snow & McCall, ARAA 44 367 2006

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Depletions vs. Condensation Temperature

For more details, see Sembach & Savage, ARAA 34, 279, 1996,“Interstellar Abundances from Absorption-Line Observations with HST”

volatiles refractories

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Variations in Depletion in the ISM

warm halowarm disk + halowarm disk

cool disk