Lecture 2Line Radiative Transfer for the ISM

• Absorption lines in the optical & UV • Equation of transfer• Absorption & emission coefficients• Line broadening• Equivalent width and curve of growth• Observations of optical & UV lines• Abundances, depletion & dust

References: Spitzer Ch. 3, Dopita & Sutherland Chs. 2 & 4

Optical & UV Absorption LinesOptical lines familiar from the SUN, e.g., the Fraunhofer doublets Ca II K 3933, 3968 Å and NaI D 5890, 5896 Å,provided the first evidence for a pervasive ISM.

These are “resonance” (allowed electric-dipole) transitions starting from 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 [{1s22s22p63s23s6}4s] 7665, 7645 ÅCa II [{1s22s22p63s23s6}4s] 3934, 3968Å

What does the detection of these resonance transitions tell us about the physical conditions of the absorbing material?

Sample Grotrian Diagrams for UV Lines

O I [1s22s22p4] (4S)Mg II [{1s22s22p6}3s] (2S)

Observations of Absorption Lines

Halo star HD 93521 Spitzer & Fitzpatrick (1993)

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

At high resolution, R = λ/∆λ =104 – 105

individual components can be resolved (at different velocities) and their abundances and physical conditions determined, as for HD 93521. FUSE operates below 1100 Å and can detect H & D Ly-α lines

FUSEHST GHRS

To deduce the physical conditions, we need to apply the equation of radiative transfer to this case.

Radiative Transfer Equation

dτν = κν ds

dIν

ds= −κν Iν + jν

differential optical depth = fractional absorption (per unit projected area, solid angle, frequency, time) in distance dssource function = ratio of emission & absorption coefficients= Planck function for complete thermodynamic equilibrium Sν ≡ jν /κν

dIν

dτν

= −Iν + Sν (new form of eq. of transfer)

Formal solution easy to use for a slab geometry:

Iν (τν ) = Iν (0) e−τν + Sν (τν′) e−(τν −τν

′ ) dτν′

0

τν∫

The Absorption Coefficient(Following Spitzer Ch. 3)

κ jk = κνline∫ dν = n j sνline∫ dν = n js jk

The absorption coefficient for the radiative excitation from thelower j th level and the upper k th level is determined by the atomic absorption cross-section from the lower level j, sν = κν/nj . The frequency integrated absorption coefficient is related to the Einstein B-coefficients:

κ jk =hν jk

cn jB jk − nkBkj( )

↑absorption from level j into level k

↑stimulated emissionfrom level k down into level j

What are the units of sjk and the B’s?

Einstein A & B Coefficients

g j B jk = gk Bkj Bkj =c 3

8π hν jk3 Akj

uν n jB jk − nkBkj( )= nk Akj

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 levelsk and j for a radiation field with spectral energy distribution uν

by assuming complete thermodynamic equilibrium.

They can also be related to emission and absorption oscillator strengths (essentially QM matrix elements),

Akj =8π 2e2ν 2

mec3 fkj

gk fkj = g j f jk

The Integrated Cross Section

s jk = sνline∫ dν =κν

n jline∫ dν =

hν jk

cB jk −

nkBkj

n j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

=hν jk

cB jk 1−

nkgi

n jgk

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Rearrange the definition:

nk*

n j* =

gk

g j

e−hν jk / kT

Thermal equilibrium level populations are given by the Boltzmann factor

Departure coefficients bj are used to relate true level populations (n) to thermal equilibrium populations (n*)

bj = nj / n*j

The Integrated Cross-section (cont’d)

Using the departure coefficients, the final result is

s jk = su 1−nkgi

n jgk

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = su 1−

bk

b j

e−hν / kT⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

where su ≡ (hvjk /c) Bjk = (π e2 /me c) fjk

is the absorption cross-section without stimulated emission.Departure coefficients are a formal device that signal non-LTE populations.At high frequencies, stimulated emission and departures from thermal equilibrium are unimportant for absorption.

Limiting Caseshv >> kT -- Pure absorption dominates because stimulated emission & excited state populations are both insignificant

hv << kT -- Stimulated emission is important, so expand the exponential to get

s jk = su 1−bk

b j

1−hν /kT( )⎡

⎣ ⎢

⎤

⎦ ⎥

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= 1

2

j

k

j

kjk

ejk b

bhkT

bb

kThf

cmes

ννπ

or, in terms of fjk,

Stimulated emission reduces the cross section by a factor of hv/kT, e.g., for HI 21 cm at T = 80 K hv/kT ≈ 8 x10-4.

Example of extreme non-LTE: masers, where the absorption term is emissive because level populations are inverted (nk>nj).

Line Emission CoefficientThe frequency-integrated line emission coefficient, jjk, is the rate at which downward radiative transitions occur (per unit volume, time, frequency, & solid angle) from the upper k th to the lower j th level.

A useful form expresses the total emission integrated over all directions (in units of ergs/s cm3) in terms of the atomic properties of the line and the density of emitters in the upper state:

j jk = jν dνline∫

4π j jk = nkhν jk Akj

Line Shape & BroadeningWrite the cross section as sν = s φ(ν) with

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) = Doppler shifted frequency∆ν = ν - ν0

φ(ν )dν =1line∫

What is the origin of the frequency dependence of the underlying absorption and emission coefficients?

Natural Line BroadeningStart with the general formula for the line-shape function

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

The natural line width is a consequence of the finite lifetimes of the upper and lower states (Heisenberg Uncertainty Principle), i.e., atoms absorb and emit over a range of frequencies nearν0

φ(ν ) = P(w)φ1(∆ν − ν 0w /c)dw−∞

∞∫

φ1(ν) =1π

γ k

γ k2 + ∆ν 2( )

γ k =1

4πAki

k> i∑

“Lorentzian” line shape

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

Natural Line Broadening (cont’d)

Natural line widths are very small, e.g., for an electric-dipole transition like HI Lyα:

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

For comparison with measured line widths, measured in km/s, this corresponds to a velocity,

∆w = ( γk/ ν) c = 9 m s-1

Forbidden lines have A-values a million or more smaller.Lines can also be broadened by collisions, but at low ISM densities, “pressure broadening” is only significant for radio recombination lines

Gaussian Line ProfileGaussian velocity distribution

For a Maxwellian at temperature T

where m is the mass of the line carrier, σT represents a turbulent component, σ = b/21/2 is the total variance, &

FWHM = 2 √(2 ln 2) σ ≈ 2.355 σ.

For only thermal broadening and A = atomic mass,b ≈ 0.129 (T/A)1/2 km s-1

P(w) =1π b

e−(w / b )2

b2 =2kTm

+ 2σT2

Voigt Profile

substituting the Gaussian velocity distribution and the natural line-shape yields the Voigt profile

φ(ν) = P(w)φ1(∆ν −ν 0w /c)dw−∞

∞

∫Starting again from the general line-shape function

φ(ν) =1π b

e−(w / b )2 1π

γ k

γ k2 + (∆ν −ν 0w /c)2 dw

−∞

∞

∫Defining the Doppler width, ∆νD = (ν0 /c) b = b / λ0, this becomes

φ(ν) =1

π 3 / 2be−(w / b )2 γ k

γ k2 + (∆ν − ∆ν Dw /b)2 dw

−∞

∞

∫

This cannot be reduced further, and we need to use tables or approximate limiting forms.

Voigt Profile Limiting Forms

2. For large ∆ν the slowly-decreasing damping wings dominate (and come out of the convolution integral:

φ(ν) =1

π ∆ν D

e−(∆ν / ∆ν D )2

φ(ν) =γ k

π∆ν 2

1. Natural line width approximated by a delta-function, recovering the Doppler profile.

Damping Wings & Doppler Cores

UV/Visible Absorption Line Formation

Neglect stimulated emission (hv >> kT), assume pure absorption, and (usually) assume a uniform slab.

The equation of radiative transfer

without the emissivity term has the solution

dIν

ds= −κν Iν + jν

Ideally, a measurement of the frequency-dependent line profile can be turned into a determination of τν . Finite spectral resolution, signal to noise (S/N) limits, etc. often make an integrated observable useful, the Equivalent Width.

Iν = Iν (0)e−τν , τν = Nl sν

Equivalent WidthWν ≡ Iν (0)−Iν

Iν (0) dν = 1− e−τν( )dν−∞

∞∫−∞

∞∫Wν is the width of a rectangular profile from 0 to Iν(0) with the same area as the actual line. It measures line strength in Hz. More common is the quantity in terms of wavelength Wλ, measured in Å or mÅ, related as:

Wλ / λ = Wν / ν

λ

I/I0

As for the Voigt profile itself, the equivalent width cannot be evaluatedIn closed form, and we have to rely on tables or limiting cases. From the formula for optical depth, we identify the column density Nj in the lower level as the basic parameter for a given transition.

Schematic Equivalent Width

Curve of Growth: Linear RegimeOptically thin limit: τ0 << 1

Wν = τν dν−∞

∞∫ = N j σ(ν)dν = N jπe 2

mec flu−∞

∞∫

τ 0=λ0

πbN j s = N j

π e2

∆ν D mecf jk

τ0 is the optical depth at line center,

Wν ∝ N j

Wλ

λ= 0.885N17 j f λ−5

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

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

Nonlinear Parts of the Curve of GrowthFor large but not too large optical depths, Doppler broadening suffices:

The 2nd line is the asymptotic limit for Doppler broadening,

Wλ ≈ 1− exp − N j s

π ∆ν De−(∆ν / ∆ν D )2[ ]{ }dλ

−∞

∞∫

=2bλc

ln(τ 0)

For large τ0, the light from the source near line center is absorbed, i.e., the absorption is “saturated”. Far from line center there is partial absorption, and Wλ grows slowly with Nj; this is the flat portion of the curve of growth.

For very large τ0, the Lorentian wings take over, & give the square-root portion of the curve of growth

Wλ / λ = (2/c) (λ 2Nj sγk)1/2

Schematic Curve of Growth

The departure from linear depends on the Doppler parameter; broader Doppler lines remain on the linear part of the curve of growth for higher column densities.

Curve of Growth AnalysisThe goal is to determine column density Nj from the measured equivalent width, which increases monotonically but non-linearly with column Nj.,

There are three regimes, depending on optical depth at line center:

τ0 << 1, linear τ0 > 1, large flat τ0 >> 1, square-root (damping)

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

Interstellar Na I AbsorptionLinear - flat

regime

Flat regime

Square-root regime

Optical Absorption Line ObservationsRequire bright background sources not too

obscured by dust extinction: < 1 kpc & AV < 1 mag, or NH < 5x1020 cm–2

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

absorption,- also seen in IRAS 100 µm cirrus ⇒ CNM

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

UV Absorption Lines towards ς Oph

Empirical curve of growth (same Doppler parameter) for several UV lines.

Gas Phase Abundance MeasurementsAbsorption line studies yield gas phase abundances of the astrophysically important elements that determine the physical & chemical properties of the ISM.

Abundances are important for cosmology, e.g., D/H

Refractory elements are “depleted” relative to cosmic; the depletion factor d is

d(X) = measured X /cosmic Xlog d(Ca) ≈ -4 ⇒ 10,000 times less Ca than in the solar photosphere

Correlation between depletion & condensation temperature suggests that the “missing” elements are in interstellar dust.

Depletions vs. Condensation Temperature

volatiles refractories

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

Variations in Depletion