Astro 501: Radiative Processes Lecture 34 April 19, 2013

Transcription

1 Astro 501: Radiative Processes Lecture 34 April 19, 2013 Announcements: Problem Set 10 due 5pm today Problem Set 11 last one! due Monday April 29 Last time: absorption line formation Q: at high resolution, what qualitative and quantitative information does a line give? Q: what must be known to extract this information? 1

2 line transmission profile F ν /F ν (0) = e τ ν directly measures optical depth τ ν σ lu N l So if we assume we know the spectral shape F ν (0) of the background source across the line transmission profile then measure product σ lu τ ν but the absorption cross section is σ lu (ν) = πe 2 /m e c f lu φ lu (ν) (1) 2 oscillator strength f lu usually known (i.e., measured in lab) so at high resolution: transmission profile depth absorber column density N l transmission profile shape absorber profile function φ lu (ν) which encodes, e.g., temperature via core width b = 2kT/m, and collisional broadening via wing with Γ

3 Depth of Line Center if the absorbers have a Gaussian velocity distribution then the optical depth profile is τ ν = τ 0 e v2 /b 2 with the Doppler velocity v = (ν 0 ν)/ν 0 c, and thus τ ν is also Gaussian in ν the optical depth a the line center is τ 0 = ( ) e 2 Nl f π lu λ lu m e c b [ 1 g un u g l N l ignoring the stimulated emission term [ ], for H Lyman α ( ) ( ) ( ) N τ 0 = l flu λ ( lu 10 km/s cm Å b ] (2) ) 3 so if we can measure τ 0, we get column N l Q: in low-resolution spectra, what sets transmission profile? Q: what information is lost? what information remains?

4 Equivalent Width if instrumental resolution R = λ inst /λ low: λ inst line shape all information about true astrophysical line profile is lost! and observed profile is just instrumental artifact yet flux is still removed by the absorption line so that we still can measure integrated effect of line i.e., the total flux lost due to absorbers F line = ν line [F ν (0) F ν ] dν where ν 0 is frequency of line center 4 useful to define a dimensionless equivalent width W F line ν 0 F ν (0) = F ν (0) F ν dν (3) ν line F ν (0) ν 0 Q: what does this correspond to physically?

5 equivalent width W = ν line F ν (0) F ν F ν (0) so Wν 0 equivalent to width of 100% absorbed line i.e., saturated line with rectangular profile and W is width as fraction of ν 0 ν0 dν ν 0 F ν F ν(0) 1 W ν 0 ν note: many authors use dimensionful equivalent with W W λ λ 0 = λ line F λ (0) F λ F λ (0) so that W λ λ λ 0 W or the velocity equivalent width W v = c W dλ λ 0 (4) 5

6 Curve of Growth in terms of optical depth, equivalent width is W = ν line [ 1 F ] ν dν = F ν (0) ν 0 ν line and thus W = W(N l ) via τ ν = σ ν N l dependence of W vs N l : curve of growth ( 1 e τ ν ) dν ν 0 (5) even if line is unresolved, equivalent width still measures F = Wν 0 F ν (0) = total missing flux across the line Q: what is W if absorbers are optically thin? what do we learn? 6

7 Optically Thin Absorption: τ 0 < 1 for an optically thin line: τ 0 < 1 and thus maximal flux reduction at line center is e τ 0 > 1/e if τ ν 1 then we can put 1 e τ ν τ ν : W τ ν dν ν 0 = N l so W N l : linear regime in curve of growth line σ lu(ν) dν ν 0 (6) for Gaussian profile, good fit to second order in τ 0 is 7 W π b c τ 0 and thus when τ 0 1, 1+τ 0 /(2 2) = πe2 m e c 2 N l f lu λ lu τ 0 1+τ 0 /(2 2) N l = m ec 2 πe 2 W f lu λ lu = cm 1 (7) W f lu λ lu (8)

8 if line optically thin, then W N l width measures absorber column density sketch W vs N l Q: what happens if line is optically thick? Q: what if line is thick and we assume thin? Q: how can we use W to check if line is thick or thin? 8

9 Flat Part of Curve of Growth: 1 < τ 0 < τ damp once τ 0 > 1, line center has essentially no flux line core is totally dark and thus saturated true line profile is nearly box-shaped true line shape still has damping wings but there cross section is small, so if τ 0 < τ damp then wings only round the edges of the line box if we treat the unresolved line as a box then width is just Gaussian width W ( ν) FWHM ν 0 = ( v) FWHM c = 2 b lnτ0 c 2 (9) 9 and thus W b lnτ 0 Q: implications?

10 when 1 < τ 0 < τ damp then equivalent width W b lnτ 0 depends very weakly on N l flat part of curve of growth add to W vs N l sketch solving for column: N l ln2 π m e c e 2 column is exponentially sensitive to W b e (cw/2b)2 (10) f lu λ lu 10 Warning! if a line is in this regime: difficult to get N l from measurements of W reliable result requires very accurate measurements of W and b confidence that true line profile is Gaussian Q: what if absorber column density increases further?

11 Damped Part of Curve of Growth: τ 0 > τ damp if N l and thus τ 0 very large, then absorption very strong, then high-res profile shows Lorentzian damping wings away from line center, in wing regime ν ν 0 ν 0 /b/c: 4Γ lu τ ν πe2 m e c N lf lu 16π 2 (ν ν 0 ) 2 +Γ 2 lu full width at half-max, i.e., width at 50% transmission, is ( λ) FWHM λ 0 = ( u) FWHM c = 1 πln2 e 2 m e c N lf lu λ lu Γ lu ν lu (11) 11

12 thus equivalent width is W = π ln2 ( λ) FWHM λ 0 = e2 m e c N lf lu λ lu Γ lu ν lu = finish W vs N l sketch www: professional plot of curve of growth b c τ 0 π Γ lu λ lu c (12) 12 so the column density is N l = m ec 3 W 2 e 2 f lu Γ lu λ 2 lu transition from flat to damped when W flat W dampled : τ damp 4 π b Γ lu λ lu ln [ 4 π ln2 b Γ lu λ lu ] (13) (14)

13 Awesome Example: Quasar Absorption Lines Q: let s remind ourselves what s a quasar? quasar (QSO) rest-frame optical to UV spectra F λ (0) = F qso λ : smooth continuum, with broad peak at rest-frame Lyman-α line www: high-resolution quasar spectrum quasars generally at large redshift, typically z qso 3 distance very large: > d H 4000 Mpc observed peak at λ peak,obs = (1+z qso )λ Lyα 3600 Å: optical! QSO light passes through all intervening material at z < z qso 13 Q: what is intervening material made of? Q: effect if absorbers have uniform comoving cosmic density? Q: why can we rule out a uniform density?

14 Quasar Absorption Line Systems quasars are distant, high-redshift backlighting to all of the foreground universe but thanks to big-bang nucleosynthesis, we know: cosmic baryonic matter mostly made of hydrogen if universe uniformly filled with H in 1s ground state, then: at redshift z, Lyα 1s 2s absorption at absorber-frame λ Lyα, and observer-frame λ obs = (1+z)λ Lyα absorption should occur at all λ < (1+z qso )λ Lyα absorbers have same comoving density at each z so optical depth τ λ and hence transmission spectrum should be smooth as a function of λ 14 in cosmo-practice: a baryon = neutron or proton or combinations of them = anything made of atoms = ordinary matter dark matter

15 Observed quasar spectra: do show absorption shortwards of the quasar Lyα! but transmitted spectrum is not smooth continuum, rather, a series of many separate lines Implications: diffuse intervening neutral hydrogen exists! there is an intergalactic medium intergalactic neutral gas is not uniform but clumped into clouds of atomic hydrogen note: low-z quasars show few absorption lines high-z quasars show many: Lyman-α forest a major cosmological probe 15 Q: what information does each forest line encode?

16 The Lyman-α Forest: Observables each forest line cloud of neutral hydrogen absorber z abs gives cloud redshift absorber depth gives cloud column density N(H I) note that absorbers span wide range in column densities most common: optically thin forest systems correspond to modest overdensities δρ/ρ 1 rare: optically thick damped Lyα systems damping wings of seen in line profile N(H I) > cm 2 correspond to large overdensities: protogalaxies! 16 www: zoom into damped Lyα system