Lecture 2 Line Radiative Transfer for the ISM

Transcription

1 Lecture 2 Line 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

2 Optical & UV Absorption Lines Optical 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 µm require space observations. Some important examples are: H I [1s] C IV [1s 2 2s] Na I [{1s 2 2s 2 2p 6 }3s] Mg II [{1s 2 2s 2 2p 6 }3s] K I [{1s 2 2s 2 2p 6 3s 2 3s 6 }4s] Ca II [{1s22s22p63s23s6}4s] 1216 Å 1548, 1551 Å 5890, 5896 Å 2796, 2803 Å 7665, 7645 Å 3934, 3968Å What does the detection of these resonance transitions tell us about the physical conditions of the absorbing material?

3 Sample Grotrian Diagrams for UV Lines Mg II [{1s 2 2s 2 2p 6 }3s] ( 2 S) O I [1s 2 2s 2 2p 4 ] ( 4 S)

4 Observations of Absorption Lines HST GHRS At high resolution, R = λ/ λ = individual components can be resolved (at different velocities) and their abundances and physical conditions determined, as for HD FUSE operates below 1100 Å and can detect H & D Ly-α lines FUSE Halo star HD Spitzer & Fitzpatrick (1993) To deduce the physical conditions, we need to apply the equation of radiative transfer to this case. White Dwarf Hz 43A Krut et al.( 2002)

5 Radiative Transfer Equation di ν ds = κ ν I ν + j ν dτ ν = κ ν ds S ν j ν /κ ν differential optical depth = fractional absorption (per unit projected area, solid angle, frequency, time) in distance ds source function = ratio of emission & absorption coefficients = Planck function for complete thermodynamic equilibrium diν dτ ν = I ν + S ν (new form of eq. of transfer) Formal solution easy to use for a slab geometry: I ν (τ ν ) = I ν (0) e τ ν + τ ν S (τ ν ν ) e (τ ν τ ν ) dτ 0 ν

6 The Absorption Coefficient (Following Spitzer Ch. 3) The absorption coefficient for the radiative excitation from the lower j th level and the upper k th level is determined by the atomic absorption cross-section from the lower level j, s ν = κ ν /n j. The frequency integrated absorption coefficient is related to the Einstein B-coefficients: κ jk = κ line ν dν = n j s line ν dν = n j s jk κ jk = hν jk c absorption from level j into level k What are the units of s jk and the B s? ( n j B jk n k B ) kj stimulated emission from level k down into level j

7 Einstein A & B Coefficients The Einstein A-coefficient gives the rate of spontaneous decay. The relations between the A and B coefficients, g j B jk = g k B kj B kj = u ν c 3 8π hν A 3 kj jk can be derived from the equation of radiative balance for levels k and j for a radiation field with spectral energy distribution u ν ( n j B jk n k B kj )= n k A kj by assuming complete thermodynamic equilibrium. They can also be related to emission and absorption oscillator strengths (essentially QM matrix elements), A kj = 8π 2 e 2 ν 2 m e c 3 f kj g k f kj = g j f jk

8 The Integrated Cross Section Rearrange the definition: s jk = dν = s line ν κ ν line n j dν = hν jk c = hν jk c n k * n j * = g k g j e hν jk / kt B jk n kb kj n j B jk 1 n kg i n j g k 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

9 The Integrated Cross-section (cont d) Using the departure coefficients, the final result is where s jk = s u 1 n kg i n j g k = s 1 b k u e hν / kt b j s u (hv jk /c) B jk = (π e 2 /m e c) f jk 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.

10 Limiting Cases hv >> 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 = s u 1 b k ( 1 hν /kt) b j or, in terms of fjk, 2 πe hν b k kt = bk s 1 jk f jk m ec kt b j hν b j 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 (n k >n j ).

11 Line Emission Coefficient The frequency-integrated line emission coefficient, j jk, 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. j jk = line j ν dν A useful form expresses the total emission integrated over all directions (in units of ergs/s cm 3 ) in terms of the atomic properties of the line and the density of emitters in the upper state: 4π j jk = n k hν jk A kj

12 Line Shape & Broadening Write the cross section as s ν = s φ(ν) with line φ(ν) dν =1 Let φ 1 (ν) be the absorption profile of one atom φ 1 = φ 1 (ν - ν 0 ) = φ 1 (ν - ν 0 [1+ w/c]) = φ 1 ( ν - ν 0 w/c) w = line of sight velocity of the atom ν 0 = rest frequency ν 0 = ν 0 (1+ w/c) = Doppler shifted frequency ν = ν - ν 0 What is the origin of the frequency dependence of the underlying absorption and emission coefficients?

13 Natural Line Broadening Start with the general formula for the line-shape function φ(ν) = P(w)φ 1 ( ν ν 0 w /c) dw 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 nearν 0 φ 1 (ν) = 1 γ k Lorentzian line shape π γ 2 k + ν 2 ( ) The width for an atom at rest (w = 0) is γ k = 1 A ki 4π k>i

14 Natural Line Broadening (cont d) Natural line widths are very small, e.g., for an electric-dipole transition like HI Lyα: A 21 = 6 x 10 8 s -1, ν = 2 x Hz, γ k / ν = 3 x For comparison with measured line widths, measured in km/s, this corresponds to a velocity, w = ( γ k / 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

15 Gaussian Line Profile Gaussian velocity distribution P(w) = 1 π b e (w / b )2 For a Maxwellian at temperature T b 2 = 2kT m + 2σ 2 T where m is the mass of the line carrier, σ T represents a turbulent component, σ = b/2 1/2 is the total variance, & FWHM = 2 (2 ln 2) σ σ. For only thermal broadening and A = atomic mass, b (T/A) 1/2 km s -1

16 φ(ν) = Voigt Profile Starting again from the general line-shape function P(w)φ 1 ( ν ν 0 w /c) dw substituting the Gaussian velocity distribution and the natural line-shape yields the Voigt profile φ(ν) = 1 π b e (w / b )2 1 π γ k γ k 2 + ( ν ν 0 w /c) 2 dw Defining the Doppler width, ν D = (ν 0 /c) b = b / λ 0, this becomes φ(ν) = 1 π 3/2 b (w / b )2 γ e k γ k 2 + ( ν ν D w /b) 2 dw This cannot be reduced further, and we need to use tables or approximate limiting forms.

17 Voigt Profile Limiting Forms 1. Natural line width approximated by a delta-function, recovering the Doppler profile. φ(ν) = 1 e ( ν 2 / ν D ) π ν D 2. For large ν the slowly-decreasing damping wings dominate (and come out of the convolution integral: φ(ν) = γ k π ν 2

18 Damping Wings & Doppler Cores

19 UV/Visible Absorption Line Formation Neglect stimulated emission (hv >> kt), assume pure absorption, and (usually) assume a uniform slab. The equation of radiative transfer di ν ds = κ ν I ν + j ν without the emissivity term has the solution I ν = I ν (0)e τ ν, τ ν = N l s ν Ideally, a measurement of the frequency-dependent line profile can be turned into a determination of τ ν. Finite spectral resolution, signal to noise (S/N) limits, etc. often make an integrated observable useful, the Equivalent Width.

20 W ν Equivalent Width I ν (0) dν = ( 1 e τ ν )dν I ν (0) I ν W ν is the width of a rectangular profile from 0 to I ν (0) with the same area as the actual line. It measures line strength in Hz. More common is the quantity in I/I 0 terms of wavelength W λ, measured in Å or må, related as: λ W λ / λ = W ν / ν As for the Voigt profile itself, the equivalent width cannot be evaluated In closed form, and we have to rely on tables or limiting cases. From the formula for optical depth, we identify the column density N j in the lower level as the basic parameter for a given transition.

21 Schematic Equivalent Width

22 Curve of Growth: Linear Regime Optically thin limit: τ 0 << 1 W ν = τ ν dν = N πe j σ(ν)dν = N 2 j m e c f lu τ 0 is the optical depth at line center, τ 0 = λ 0 πb N j s = N j π e 2 ν D m e c f jk Since W ν N j, this is known as the linear regime for the curve of growth: W λ λ = N 17 j f λ 5 Units: for column density, 1017 N 17 cm -2, for wavelength,1000 λ -5 Å

23 Nonlinear Parts of the Curve of Growth For large but not too large optical depths, Doppler broadening suffices: W λ { 1 exp N j s e ( ν / ν } D )2 dλ π ν D = 2bλ c ln(τ 0 ) [ ] The 2 nd line is the asymptotic limit for Doppler broadening, For large τ 0, 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 W λ grows slowly with Nj; this is the flat portion of the curve of growth. For very large τ 0, the Lorentian wings take over, & give the square-root portion of the curve of growth Wλ / λ = (2/c) (λ 2N j sγ k ) 1/2

24 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.

25 Curve of Growth Analysis The goal is to determine column density N j from the measured equivalent width, which increases monotonically but non-linearly with column N j., There are three regimes, depending on optical depth at line center: τ 0 << 1, linear τ 0 > 1, large flat τ 0 >> 1, square-root (damping) Unfortunately, many observed lines fall on the insensitive flat portion of the curve of growth.

26 Interstellar Na I Absorption Linear - flat regime Flat regime Square-root regime

27 Optical Absorption Line Observations Require bright background sources not too obscured by dust extinction: < 1 kpc & A V < 1 mag, or N H < 5x10 20 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.

28 UV Absorption Lines towards ς Oph Empirical curve of growth (same Doppler parameter) for several UV lines.

29 Gas Phase Abundance Measurements Absorption 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 X log 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.

30 Depletions vs. Condensation Temperature volatiles refractories For more details, see Sembach & Savage, ARAA 34, 279, 1996, Interstellar Abundances from Absorption-Line Observations with HST

31 Variations in Depletion