Physics of Photon Dominated Regions. PDR Models SS 2007 M. Röllig

Transcription

1 Physics of Photon Dominated Regions PDR Models SS 2007 M. Röllig

2 The standard view from planeparallel models So far: all necessary components of a PDR model introduced: energy balance: important heating and cooling processes PE heating H 2 vibrational deexctiation H 2 formation heating H 2 photodissociation heating CR heating fine-structure emission ( [CII], [CI], [OI]) line emission (CO, H 2 O, OH, Ly α,...) gas-grain collision (heating and cooling)

3 The standard view from planeparallel models chemical network: N species (e.g. 99) L reactions (e.g. 1548) M elements ( e.g. 7 ) system of chemical rate equations N+M+1 equations for N unknowns numerical complexity scales N 2...N 3

4 The standard view from planeparallel models radiative transfer incoming radiation (radiation hitting the PDR surface from outside) outgoing radiation (radiation leaving the PDR) or emission of radiation absorption of radiation these two descriptions are not fully exchangeable, since outgoing radiation means emitted radiation, which has partly been re-absorbed in the cloud. or FUV radiation IR and FIR radiation

5 The standard view from planeparallel models n i, χ, A v T Energy Balance T(r), n(r) 1 I( ν, r, θ) dθ 4π Chemical Network n i Gas, n i Dust n dust χ, A v, n i, n dust I( ν, r, θ ) T Radiative Transfer

6 Energy Balance PE heating Γ PE = ε 0 Γ denotes heating rates ε: PE efficiency, e.g.: Gn [erg s cm ] 2 = 3 10 < ( ) >= f O G 0 T 1 2 n e <f(o)>: neutral fraction (Bakes&Tielens, 1994) ionization rate recombination rate GT 0 n e 1 2

7 (Bakes&Tielens, 1994)

8 G 0 vs. χ G 0 : FUV flux normalized to the Habing field for the solar neighborhood (Habing 1968) integrated over ev G 0 = erg cm -2 s -1 χ: FUV flux normalized to the Draine field (Draine 1978) G 0 =1.71 χ (isotropic radiation from 4π)

9 (Draine, 1978)

10 G 0 vs. χ Draine intensity: I( λ) = π λ λ λ λ in [Å], I(λ) in erg s -1 cm -2 ster -1 Å -1 1 u( λ) = I( λ) dω c 2400Å Å G = u( λ) dλ

11 Draine vs. Habing

12 Draine vs. Habing

13 Cooling [CII] 158µm fine structure emission 2 P 3/2 2 P 1/2 2-level system (collision + spont. emission) u nnγ = nnγ + n A l lu u ul u ul n cr = γ lu γ ul A ul Eul g u kt l γlu = γule gl A g ul critical density: u e γ n>n cr : collisions ul n g dominate u l = n n l 1+ n E ul kt cr detailed balance

14 Escape Probability now including absorption γ lu B lu J ul γ ul A ul B ul J ul u J ul : mean intensity B: Einstein coefficients for absorption and stimulated emission nnγ + nb J = nnγ + n A + n B J l lu l lu ul u ul u ul u ul ul Attention: local population depends on J J depends on population everywhere new concept: escape probability β(τ ul ) l

15 Escape Probability Assumptions: photons produced locally can only be absorbed locally in calculating τ ul we assume that the local population holds globally STRONG ASSUMPTIONS! now: photon balance net absorption = emitted photons that do not escape ( ) = 1 β( τ ) ( ) nb nb J n A l lu u ul ul u ul ul net # of photons used up for absorptions net # of photons made available for local absorptions

16 Escape Probability β(τ ul ): probability that a photon formed at opt. depth τ escapes through the surface nnγ + nb J = nnγ + n A + n B J l lu l lu ul u ul u ul u ul ul ( ) nnγ + nb n B J = nnγ + n A l lu l lu u ul ul u ul u ul ( 1 ( )) nnγ + n β τ A = nnγ + n A l lu u ul ul u ul u ul nnγ = nnγ + n β( τ ) A l lu u ul u ul ul simple 2-lev corrected with β factor nnγ = nnγ + n A l lu u ul u ul

17 Cooling Line cooling rate n 2 Λ=n u A ul E ul β(τ ul ) (erg s -1 cm -3 ) n u : number of atoms in upper level n n u l = g g u l 1+ nu + nl = ntot E ul kt e ncrβ n n u = n tot Eul g l kt ncrβ 1+ e 1+ g u n

18 Escape Probability β=?? depends e.g. on geometry turbulent, homogeneous, semi-infinite slab line-averaged opt. depth: 3 Ac ul n u nl g u = b πν ul ng u l Δz b: Doppler broadening parameter Δz: distance from the surface instead of thermal motion: velocity gradient b/δz dv/dz once a photon has traveled a distance Δv D /(dv/dz), with Δv D its Doppler width of the line, it will be shifted to the line-wings, where the opt. depth is small, and the photon will escape τ ul

19 Escape Probability For various geometries, β can be given analytically. For a semi-infinite, plane-parallel slab: 1 exp( 2.34 τ ) βτ ( ) = τ<7 4.68τ 1 2 τ β ( τ) = 4τ ln τ>7 π small τ: β 1/2 (photons escape through half the hemisphere) large τ: β τ -1-1

20 Emergent Intensities z 1 2 I = n Λ( z) dz 2π 0 2π: photons only escape through the front surface In thermodynamic equilibrium (n>>n cr β) νb I = B( T) f( τ ) c τ f( τ) 2 β( τ) dτ E (2.34 τ) ln(2.34 τ) [ ] = = for τ<<1 : f(τ)=τ I = A Nhν ul u ul 4π

21 Cooling [CII] 158µm n Λ= erg cm s 92K β T 1+ e 1+ 2 for n=10 3,10 4,10 5 n nβ D radiation Λ@ C I I D IHτL T@KD τ

22 3-Level System [OI]!! β=0.5, Z: metallicity

23 Cooling [CII] vs. [OI] Heating and Cooling Curves Hn=10 3,n=10 5 L erg s e c 1 c m T@KD

24 Energy Balance Heating and Cooling Curves Hn=10 3,χ=10 6 L Γ PE Λ [CII] + Λ [OI] erg s e c 1 c m Λ [OI] Λ [CII] T@KD

25 Heating and Cooling H 2 vib. deexcitation H 2 dissociation H 2 formation CR heating PE heating gas-grain collisions [OI] 63, 146, 44µm CO rot. lines [CII]158µm [CI] 610, 370, 230µm Si + 13 CO rot. lines OH H 2 O gas-grain collisions

26 1E-14 n=10 6, χ=10 6, Z=1 rate [erg s -1 cm -3 ] 1E-15 1E-16 1E-17 1E-18 1E-19 1E-20 1E T [K] COOLING HEATING H 2 vib.deexc. H 2 dissoc. H 2 form. O 63µ O 44 µ O 146 µ CR heating PE heating CO rot.lin. C + CI 610µ CI 230µ CI 370µ Si + 13 CO rot.lin. OH approx. H 2 O approx. gas grain cooling OH N-lev. H 2 O N-lev.

27 1E-14 n=10 6, χ=10 0, Z=1 rate [erg s -1 cm -3 ] 1E-15 1E-16 1E-17 1E-18 1E-19 1E-20 1E T [K] COOLING HEATING H 2 vib.deexc. H 2 dissoc. H 2 form. O 63µ O 44 µ O 146 µ CR heating PE heating CO rot.lin. C + CI 610µ CI 230µ CI 370µ Si + 13 CO rot.lin. OH approx. H 2 O approx. gas grain cooling OH N-lev. H 2 O N-lev.

28 1E-20 n=10 3, χ=10 0, Z=1 rate [erg s -1 cm -3 ] 1E-21 1E-22 1E-23 1E-24 1E-25 1E-26 1E T [K] COOLING HEATING H 2 vib.deexc. H 2 dissoc. H 2 form. O 63µ O 44 µ O 146 µ CR heating PE heating CO rot.lin. C + CI 610µ CI 230µ CI 370µ Si + 13 CO rot.lin. OH approx. H 2 O approx. gas grain cooling OH N-lev. H 2 O N-lev.

29 1E-20 n=10 3, χ=10 6, Z=1 rate [erg s -1 cm -3 ] 1E-21 1E-22 1E-23 1E-24 1E-25 1E-26 1E T [K] COOLING HEATING H 2 vib.deexc. H 2 dissoc. H 2 form. O 63µ O 44 µ O 146 µ CR heating PE heating CO rot.lin. C + CI 610µ CI 230µ CI 370µ Si + 13 CO rot.lin. OH approx. H 2 O approx. gas grain cooling OH N-lev. H 2 O N-lev.

30 Chemistry N species, M elements one rate equation per species n i = n ζ + n k + k n n + n ζ t i i q qi rs r s t ti q r s t, i=1,...,n ζ = ζ + ζ k diss ion i i i qi krs ζ ti = ζ diss ti destruction reactions formation reactions rate coefficient for photo-destruction of i rate coefficient for X +X... rate coefficient for X +X X +... ion + ζ rate coefficient for X +h ν X +... ti i r t q s i i

31 Chemistry particle conservation n = nc M M i i i e.g.: n=n +2n H H c 2 M i : number of atoms of element M in species i charge conservation n = nc charge e i i i e.g.: n = n e C + c charge i : number of charges in species i N+M+1 quations for N unknown quantities

32 Chemistry e.g.: H/H 2 balance nh ( 2) τ = Rf fshielde Idiss( τ = 0) χnh 2 t R f nnh cm s (observation) advection ( n v) H 2 z 1 Rf = S ( T, TD ) η ( TD ) ndnhσdvh (theory) I ( τ = 0) s diss

33 Chemistry Assumptions: steady-state chemistry τ=0 : cloud surface n = t 0 Rnn = I (0) χn H diss H n= n + 2n H H α nr nh = n, nh = n, α = α α χ I(0)

34 Chemistry H, H 2 densities for χ=10 0, n= ni R H 2 formation rate for χ=10 0,n=10 6 H 2 formation rate

35 Chemistry H H 2 Transition for χ=10 3, n=10 4,10 6 0: (0) ni τ α = I I e nr χ I(0) e τ τ UV UV τ connection to RT A V τ UV =k A v (e.g. k=3.02) A V = N Htot A V =1 entspricht N H = cm -2

36 Chemistry dust shielding important G 0 /n cm 3 self shielding important G 0 /n cm 3 self shielding factor approximation β N SS ( ) N H 2 = N ( ) = 10 N H 10 cm Draine & Bertoldi, 1996, Ap.J., 468

37 Chemistry H2 and CO are photodissociated via line absorption, hence they are both subject to line shielding effects (see. e.g. van Dishoeck & Black, 1988) Lee et al. 1996

38 Chemistry τ, A V, N are exchangeable measures of the amount of matter passed by radiation. Pay attention when you exchange them!

39 Radiative Transfer pure absorption di ν ds = κ ν I ν I ν ds absorption +emission with extinction di ν ds = κ I + ε ν ν ν d ds, di ν εν τ = κ I I S dτ = + κ = + ν ν ν ν ν ν ν k = κ + σ dτ = k ds ν ν ν ν ν τν τν τ ν ν ( τν) = ν(0) + ( τ ν) τ ν 0 ν I I e e S d

40 Radiative Transfer τν τν τ ν ν ( τν) = ν(0) + ( τ ν) τ ν 0 ν I I e e S d background radiation radiation emitted inside the cloud and partly re-absorbed within the cloud I ν (0) I ν (τ ν ) τ ν

41 Radiative Transfer in case of LTE (local thermodynamic equilibrium) und e.g. constant temperature S ν B ν (T) (black body) Problem: u place A hν u place B hν l l T A emission depends on n u /n l and T A T B emission depends on n u /n l and T at A and n u /n l and T at B

42 Radiative Transfer radiative properties at plave B depends on conditions at place A! non-local problem To solve the RT problem at B it is necessary to have it already solved at all other places. Same argument holds for all positions. Analytical solution only for special cases possible. numerical, iterative solution special simplifications to de-couple the RT escape probability LVG (large velocity gradient) (Sobolov approx.)

43 Radiative Transfer Through RT geometry enters the stage Since RT couples distant mass elements to each other it becomes necessary to define the model geometry

44 Geometry Cloud plane-paralle (semi-infinite or finite), spherical, disk spatial size structure (density/velocity gradient or fluctucations) Environment FUV field (isotropic?, strength) other radiation background (IR?) density, presure, temperature of the ambient medium Some configurations more advantageous due to their symmetry. pp-models with directed or isotropic FUV spherical clouds with isotropic FUV

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46

47

48 Geometry If we put a model cloud in an isotropic FUV field with G 0 =1, the flux at the cloud s edge is erg s -1 cm -2 (1/2 G 0 )

49 Geometry If we put a model cloud in an isotropic FUV field with G 0 =1, the flux at the cloud s edge is erg s -1 cm -2 (1/2 G 0 )

50 Geometry irradiation with inclination angle θ μ = cosθ diν ( μ, x) μ = κν( x) Iν( μ, x) + εν( x) dx for pure absorption: x J exp( τμ) = E ( τ ) = dμ J μ 0 E2: second order elliptical integral J: mean intensity, integral over 4π

51 Geometry

52 Geometry Flannery et al (Legendre polynomial expansion) g: mean cosine of scattering angle (1=forward) ω: scattering albedo

53 Flannery et al ω: single scattering albedo 0 ω 1 the fraction of light that is actually absorbed is 1- ω

54 Geometry uni-directional: A V is a function of depth isotropic: A V depends on the depth and the angle it is difficult to directly compare the output of isotropic models with uni-directional models. Solution: definition of an effective A V A Veff, e = E ( A ) 2 Veff, 2 ( ) A = ln E ( A ) V V

55 Geometry

56 Solution scheme Solve chemical network i n = ni ς i + nq kqi( T ) + krs( T ) nr ns + nt ς ti t q r s t destruction formation I ν (τ(r)) n i (r) τ T(r) ni (r) τ T(r) Solve radiative transfer τν τν τ ν ν ( τν) = ν(0) + ν( τ ν) τ ν 0 I I e e S d S(r) I ν (τ(r)) i Solve energy balance Γ ( r, n, χ, T) = Λ ( r, n, χ, T) i heating cooling i i

57 Standard Setup plane-parallel slab, semi-infinite FUV radiation hits surface perpendicular extinction exp(-a V ) total gas density n=n(h)+2n(h 2 )=const. A V N H z steady-state chemistry n / t =0 FUV H 0 1 N= H 2 z A V

58 Standard Setup Example: C + /C C + hν C + + e - C + + e C + hν C + + H 2 CH 2+ + hν χiexp( A k) n = a n + n + k n n C V C C C e C + C H n + n = X n C C C ( ) + n = X n n n + C C C + = n e 2 small A V larger A V α = I = k = k = , 3 10, C 7 10, 1.8

59 A V ni Standard Setup ( ) C χiexp( A k) X n n = a n,n=10,x =10 V C C C χiexp( A k) χiexp( A k) + = 0 2 V V n + n + X C C Cn ac ac n C + = 6000 exp( 1.8 A ) + V exp( 3.6 AV) 3 10 exp( 1.8 AV) + + C + C Transition for χ=10 3, n=10 4, C layers: C + outsinde, C inside next step C CO C + OH CO + H C + O 2 CO + O classical C + -C-CO stratification

60 Standard Setup χ=10 3, C/H=10-4, n=10 4 cm -3 (solid), n=10 6 cm -3 (dashed) H (red), H 2 (blue), C + (green), C (black)

61 Standard Setup 2 nd C bump charge exchange C + + S C+ S + as C + < S +, more C produced via charge exchange with S than via recombination (C + + e - ). Reason: more abundant S + recombines quicker than C +, and each so produced S then produces one C!

62 Basic structure: 1-D steady-state model, escape probability method. H 2 self-shielding and dust attenuation vs. grain surface formation. Controlling parameters: FUV ev 1) cloud density/pressure 2) FUV intensity 3) grain scattering properties 4) H 2 formation rate coefficient 5) geometry/clumpiness 6) gas phase abundances 7) magnetic field...

63 Hollenbach&Tielens, 1999

64 nr ( )

65 T, ( r ) dust gas

66 Resulting Model Cloud

67 PDR model for Orion n=2.3*10 5 cm -3 G 0 =10 5 Tielens & Hollenbach 1985

68 CII 158 μm fine-structure line cooling: Tielens & Hollenbach 1985 ApJ Expected theoretically - e.g. Dalgarno & McCray dense PDRs First far-ir (Lear jet) detections by Russell et al in NGC 2024 and Orion. Widely detected since with KAO, ISO... CNM Wolfire et al WNM

69 Fine-Structure Line Emission: line intensity (erg cm -2 s -1 sr -1 ) line intensity (erg cm -2 s -1 sr -1 ) [CII] 158μm [OI] 63μm Hollenbach & Tielens 1997 ARAA, 35, 179 Hollenbach & Tielens 1999 Rev. Mod. Phys., 71, 173 FUV intensity Kaufman et al ApJ, 527, 795 [OI] 63 μm / [CII] 158 μm log hydrogen density (cm -3 ) FUV intensity [OI] 63μm can become optically thick.

70 Hollenbach et al. 1991

71 Hollenbach et al. 1990

72 Stacey et al. 1991

73 OH driven chemistry in warm H/H 2 transition zone: T = K H 2 CO + HCO + Prediction: CO + /HCO + large in PDR NO C + CO + /HCO + small in dark core N OH Si + SiO + O S + O 2 SO +

74 For example, CO + / HCO + NGC 7023 (reflection nebula) Fuente et al. 2003, AA, 899, 913 Molecular Peak HCO + CO + CO + /HCO + Molecular Peak < PDR Peak PDR Peak ~ 0.1 HCO + illuminating star (Herbig B3Ve) CO + At interface: G 0 = 2.4 x 10 3 n = 10 4 cm -3 Plateau de Bure & 30m

75 Polycyclic Aromatic Hydrocarbons (PAHs): NGC 7023 (reflection nebula) Optical internal conversion IR

76 PAHs in PDRs: Bakes & Tielens 1998 ApJ 499, 258 PAH+ PAH PAHdashed without PAHs solid with PAHs (2 x 10-7 ) mutual neutralization X + + PAH- X + PAH competes with or dominates radiative recombination X + + e X + hν Lepp & Dalgarno 1988, ApJ, 335, 769

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79 Resulting Model Cloud

80 Resulting Model Cloud