3: Interstellar Absorption Lines: Radiative Transfer in the Interstellar Medium. James R. Graham University of California, Berkeley

Transcription

1 3: Interstellar Absorption Lines: Radiative Transfer in the Interstellar Medium James R. Graham University of California, Berkeley

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 j ν and κ ν to Einstein A & B, f-values, etc. Line broadening & line shape function, φ ν Optical and UV absorption lines Variation of absorption profile with optical depth Equivalent width vs. column density Examples N H /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 2

3 Atomic Optical Absorption Lines Initial evidence for a pervasive ISM came from atomic absorption lines at visible wavelengths Principal ISM probe prior to radio & space-borne studies Strong optical absorption lines: Transition Wavelength (Å) Name Na I 3s 2 S 1/2 3p 2 P o 1/2,3/2 5890, 5896 Sodium D 1 D 2 Ca II 3933, 3968 Calcium H & K 4s 2 S 1/2 4p 2 P o 1/2,3/2 Resonance lines Electric dipole transitions (ΔL=±1, ΔS=0) from the ground state Other, weaker, optical lines (discovered 1930s - 40s) include Ti II, Ca I, K I, Li I, CH, NH, CN, CH + & C 2 AY 216 3

4 Ca II Grotrian Diagram Multielectron atoms are labeled 2S+1 L J L total orbital angular momentum S total spin angular momentum and 2S+1 is the spin multiplicitiy J total angular momentum J=L+S, L+S-1 L-S AY 216 4

5 Isoelectronic Sequence Atoms or ions with the same number of electrons have similar electronic structure Li I, Be II, C IV, N V, O VI AY 216 5

6 Common UV Absorption Lines Rocket & satellite observations show strong UV absorption lines from the ISM Typical excitation energies of resonance lines are ~ ev Many important atomic resonance lines 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 UV absorption lines AY 216 6

7 Observations of Absorption Lines At spectral resolution R = λ/ λ 10 4 (30 km/s) absorption lines break up into resolved components, Doppler shifted relative to one another 4 S P 2 4 S P 4 SII Interstellar lines through the halo towards HD93521 Hubble/GHRS data reveal velocity structure spanning ~ 90 km/s High SNR permits the study of abundances & physical conditions in individual clouds along the sight line Individual lines within multiplets can be recorded Fitzgerald & Spitzer 2 P D 4 2 P S 2 2 P S 2 AY 216 7

8 Radiative Transfer Review The transfer equation is The absorption term energy absorbed s/cm 2 /sr/hz Emission term Optical depth Source function S ν j ν /κ ν Equation of transfer becomes j ν κ ν I ν dτ ν = κ ν ds Integrate through a slab: di ν ds = κ ν I ν + j ν di ν dτ ν = I ν + S ν I ν (τ ν ) = I ν (0) e τ ν + τ ν S ν (τ ν ) e (τ ν τ ν ) dτ 0 ν AY 216 8

9 Line Emission Coefficient (j jk ) The line emission coefficient, j jk, describes radiative transitions from the k th excited state to the lower j th state j jk = Usually expressed as the line emissivity line j ν dν 4π j jk = n k hν jk A kj in units of erg/s/cm 3 Factor of 4# indicates total emission in all directions. A value is in units of s -1 AY 216 9

10 Line Absorption Coefficient (κ jk ) κ jk describes the total radiative excitation between the lower j th level and the excited k th state κ jk = κ line ν dν = n j s line ν dν = n j s jk s ν = κ ν /n j is the atomic absorption cross-section from lower level j at frequency ν Line absorption coefficient, κ jk has two components κ jk = hν jk c ( n j B jk n k B ) kj B jk gives the rate of absorption out of level j into level k B kj gives the stimulated emission from level k down to level j AY

11 Einstein A & B The B s are the Einstein coefficients for absorption & stimulated emission In equilibrium u ν ( n j B jk n k B ) kj = n k A kj Assumption of TE shows that g j B jk = g k B kj and B kj = c 3 A 3 kj 8π hν jk Where A kj = 8π 2 e 2 ν 2 m e c 3 f kj is the emission oscillator strength g k f kj = g j f jk AY f kj

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 D and Ca II H & K lines are often the strongest lines hν >> kt for T 80 K and ν = Hz Neglect stimulated emission for a slab of optical depth τ ν I ν = I ν (0)e τ ν, τ ν = N l s ν I ν (0) I AY 216 ν τ ν = N l s ν 12

13 Departure Coefficients Departure coefficients, b j, relate actual level populations (n) with the TE populations (n * ) b j = n j / n * j For example departure coefficients quantify the non-te conditions of the ISM and can be used to compute s jk : s jk = dν = s line ν line κ ν n j dν = hν jk c = hν jk c B jk n kb kj n j B jk 1 n g k i n j g k AY

14 Departure Coefficients The TE level populations are related by a Boltzmann factor * n k n = g k e hν jk / kt * j g j In terms of the integrated atomic cross-section s u (hν jk /c) B jk = (! e 2 /m e c) f jk s jk = s u 1 n kg i n j g k = s u 1 b k e hν / kt b j s jk can be defined as the total cross-section for pure absorption, s u, modified by a stimulated emission correction AY

15 Special Cases hv >> kt s jk s u Pure absorption dominates because stimulated emission is negligible, 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 s jk = s u 1 or in terms of f jk s jk = s u 1 b k e hν / kt b j s jk = πe2 m e c f jk b k b j ( 1 hν /kt) hν b k kt b k 1 kt b j hν b j AY

16 Special Cases: hν/kt << 1 When hν/kt << 1 Local Thermal Equilibrium (LTE), b k =b j =1 s jk = s u 1 b k b j 1 hν kt s jk = s u hν kt = πe2 m e c f jk correction for stimulated emission reduces the cross section 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 they are inverted (n k >n j ) hν kt AY

17 Line Shape & Doppler Shift The cross section s ν = s φ(ν) such that 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) = non-relativistic Doppler shifted frequency ν = ν - ν 0 AY

18 Broadening Mechanisms For an ensemble of atoms the line of sight velocity distribution of gas P(w) gives φ(ν) φ(ν) = P(w)φ 1 (Δν ν 0 w /c) dw Line broadening due to the uncertainty principle Finite lifetime of an upper state implies an atom can absorb at ν ν 0 φ 1 (ν) = 1 π γ k γ k 2 + Δν 2 ( ) Lorentzian or natural profile for an atom at rest with width γ k = 1 A ki 4π AY 216 k>i 18

19 Natural Broadening Natural line widths are very small HI Ly α, A 21 = 6 x 10 8 s -1, ν = 2 x Hz γ k / 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

20 Gaussian Line Profile Gaussian velocity distribution dp(w) = P(w) dw = 1 π b e (w / b )2 dw For a Maxwellian at temperature T b 2 = 2kT m + 2σ 2 T where m is the molecular weight and σ T represents a turbulent component b (T/A) 1/2 km s -1 FWHM = 2 (2 ln 2) σ σ AY

21 Voigt Profile The combination or convolution of the natural & Doppler profile yields the Voigt profile φ(ν) = φ(ν) = 1 π b P(w)φ 1 (Δν ν 0 w /c) dw e (w / b )2 1 π γ k γ k 2 + (Δν ν 0 w /c) 2 dw Define the Doppler with ν D = b ν 0 /c = b / λ 0 φ(ν) = 1 π 3 / 2 b (w / b )2 γ e k γ k 2 + (Δν Δν D w /b) 2 dw AY

22 Voigt Profile The Voigt profile varies with relative width of Natural broadening Γ Doppler width Δν D a = γ/δν D AY

23 Limiting Cases of the Voigt Profile Case 1: Doppler core natural line width φ 1 (ν) is approximated by a δ-function for ν ν D φ(ν) = 1 e (Δν 2 / Δν D ) π Δν D Case 2: Damping wings good for large ν >> ν D φ(ν) = γ k πδν 2 AY

24 UV/Visible Absorption Line Formation Neglect stimulated emission for UV/visible interstellar absorption lines (hv >> kt). Lines are pure absorption, and the equation of radiative transfer has the solution or in terms of wavelength I ν = I ν,0 e τ ν I λ = I λ,0 e τ λ Ideally, observation of an absorption-line profile can be turned into a measurement τ λ Finite spectral resolution compared to the intrinsic width, limits on SNR, etc.signal-to-noise, makes it convenient to express the line strength in terms of an integrated observable, the Equivalent Width AY

25 Absorption Line Profiles τ 0 AY

26 Doppler Cores & Damping Wings AY

27 Equivalent Width of Spectral Lines Equivalent width of line: W ν I ν (0) I ν I ν (0) dν = 1 e τ ν W ν_ is the width of a rectangular profile from 0 to I ν (0) that has the same area as actual line I/I 0 W ν measures line strength, units are Hz Similarly, in terms of wavelength λ W λ I λ (0) I λ I λ (0) with W λ typically measured in Å or må dλ = 1 e τ λ ( ) dν ( ) dλ W λ / λ = W ν / ν AY

28 Schematic Equivalent Width AY

29 Curve of Growth Optically thin limit W λ = Doppler broadened line W λ ( 1 e τ ) ν dν τ λ dλ = N j s φ(ν) λ dν ν = N j s λ c [ ] { 1 exp N j s e (Δν / Δν D ) 2 π Δν D }dλ 2 = 2bλ 0 c ln(τ 0 ) τ 0 is the optical depth at line center τ 0 = N j sφ ν (0) = λ 0 πb N s = N π e 2 j j Δν D m e c f jk AY

30 Saturated Lines For strong lines the width is given roughly by the point where τ 1 is achieved τ 0 e -1 2Δν τ=1 AY

31 Saturated Lines When lines are strong the width is given roughly by the frequency where τ 1 is achieved For a Doppler broadened line [ ] τ ν = τ 0 exp ( Δν Δν D ) 2 τ 0 exp ( Δν τν =1 Δν ) 2 D 1 Hence Δν τν =1 Δν D log( τ 0 ) = b log( τ 0 ) λ 0 W ν ( τ 0 ) 2Δν = 2 b τν =1 log( τ 0 ) λ 0 λ W λ = W 0 ν 2bλ 0 ν 0 c log( τ 0 ) AY

32 Curve of Growth Damped line W λ = 2λ c 1 exp N j s Transitions Linear breaks down when τ 0 ~ 1 Doppler to damping when N j sλ 2 γ k γ k πδν 2 dλ b ln(τ 0 ) N j sλ 2 γ k AY

33 Curve of Growth: Linear Regime Weak lines, τ 0 << 1 W ν = Linear regime: τ ν dν = N πe j σ(ν)dν = N 2 j W ν N j m e c f lu The column is N 17 cm -2 Wavelength is 1000 λ -5 Å W λ λ = 0.885N 17 j f λ 5 Expect sensitivity limit W λ / λ R/SNR ~ 10 4 /10 2 ~ 10 6 AY

34 Curve of Growth: Flat Regime Large τ 0 : all background light near line center is absorbed, line is saturated Far from line center there is partial absorption W λ grows very slowly with N j Flat part of the curve of growth Onset of deviation from linear depends on Doppler parameter Broader Doppler line will remain on the linear part of the curve of growth for higher column AY

35 Schematic Curve of Growth AY

36 Curve of Growth Analysis Goal: relate equivalent width W ν or W λ = W c ν to column density N j 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: τ 0 << 1, linear regime τ 0 > 1, large, flat regime τ 0 >> 1, square-root (damping) regime λ 2 AY

37 Interstellar Na I D Absorption Linear - flat regime Flat regime Square-root regime AY

38 UV Absorption Lines towards ς Oph AY

39 Optical Absorption Line Observations Technique limited to bright background sources Mostly local (< 1 kpc), mostly A V < 1 mag. corresponding to N(H) < 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 Ly α observations AY

40 Gas Phase Abundances Absorption line studies yield information on the gas phase abundance of a range of astrophysically abundant elements Crucial for cosmological studies, e.g., D/H Depletion D log D = log 10 (abundance meas ) - log 10 (abundance cosmic ) 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

41 Depletions (log 10 D) vs. Condensation Temperature AY

42 Depletion on Grains In diffuse clouds many abundances are much smaller than solar Strong correlation between low condensation temperature and high depletion suggests formation of grains in circumstellar envelopes AY

43 AY