Photoionized Gas Ionization Equilibrium Ionization Recombination H nebulae - case A and B Strömgren spheres H + He nebulae Heavy elements, dielectronic recombination Ionization structure 1
Ionization Equilibrium Equation Start with a pure H nebula # photoionizations/vol/sec = # recombinations/vol/sec: n H 4πJ ν hν a ν (H ) dν = n e n p α(h,t) (hν = 13.6 ev) ν where : J ν = 1 4π I dω = mean intensity (ergs s -1 cm 2 Hz 1 sr 1 ) ν a ν = ionization cross sec tion α = recombination coefficient for H (cm 3 s 1 ) Ex) For a pure H nebula: n p = n e, n H = n H + n p At a distance of 5 pc from an O7 star (T = 4, K) If n H = 1 cm -3 n H n H 4 x 1 4 2
Using the same parameters, when the hydrogen starts to go neutral, the thickness of the H II region is: d 1 n H a ν.1 pc So the H II region is almost completely ionized until it reaches a transition region, which is very thin Such an idealized case is called a Strömgren sphere Above calculations depends on the atomic parameters: a ν, α and the details of atomic structure: Ex) Hydrogen ground state: 1 2 S (or generally 1s 2 S) n = 1 (principal quantum number) L = (total angular momentum number) L =, 1, 2, 3 às, P, D, F m = 2 = 2S+1 (where S=spin = ½) (m =multiplicity when L > S) 3
Energy-Level Diagram for H I Permitted transitions (electric dipole) forbidden (Osterbrock & Ferland, p. 19) 4
Atomic Parameters: Transition Probabilities For permitted transitions of H I (ΔL = ± 1) : A nl,n ' L ' = 1 4 1 8 s 1, so lifetimes of states τ 1-4 1 8 sec o Ex) 2 2 P 1 2 S gives Lα photon at 1216A Using previous example (O7 star at 5 pc): # ionizations / H = 4πJ ν hν a ν (H ) dν 1 8 s 1 ν (τ 1 8 sec) So there are 1 8 sec between ionizations of an H atom all H atoms are essentially in ground state Ex) 2 2 S 1 2 S gives two - photon continuum : τ.12 sec Forbidden transitions are common in nebulae 5
Photoionization Cross Sections For a hydrogenic ion, with charge Z: a ν (Z) ~ Z 2 ν 3 (approximately) for hν > hν 1 (= 13.6 Z 2 ev) Ionization Potentials: IP (H ) = 13.6 ev IP (He ) = 24.6 ev IP (He + ) = 54.4eV (Osterbrock & Ferland, p. 21) 6
Recombination Coefficients Ejected electrons set up a Maxwell-Boltzman distribution: 3/ 2 f(v) = 4 m π 2kT v 2 e mv2 Recombination to level n,l : 2kT α nl (H,T) = v σ nl ( H,v) f(v) dv α nl T 1 2 (approximately, Osterbrock p. 21) i Total recombination coefficient: α A = nl α nl (H,T) At T = 1, K: α A = 4.18 x 1-13 cm 3 sec 1 (Osterbrock, p. 22) Recombination time: τ rec = 1 n e α A 15 n e yrs 7
Radiation Field Ex) Consider a pure H nebula around a single star: J ν = J ν stellar + J ν diffuse 4π J ν stellar = L ν 4πr 2 e-τ ν r ( ) a ν dr ' where τ ν = n H r ' at location r For a pure H nebula, 4πJ ν diffuse = ionizing flux from recombination Case A) Optically thin nebula: J ν diffuse (radiation escapes) n H L ν 4πr 2 hν a ν e-τ ν dν = n e n p α A (H,T ) ν where α A = (H,T ) n=1 α n 8
Case B) Optically thick nebula: Use "on the spot" approximation: recombinations to ground generate photons that are absorbed locally n H 4π J ν diffuse a hν ν dν = n e n p α 1 (H,T ) ν Thus: n H L ν 4πr 2 hν a ν e-τ ν dν = n e n p α B (H,T ) ν α B = (H,T ) n=2 α n Nearly all nebulae are Case B: optically thick to ionizing radiation Since τ is a function of (n H,r), ne = n p, and n H = n p + n H, the ionization equilibrium equation above can be integrated outward for a given n H (r) and T(r) to get n p /n H (r) This is a crude model, since T(r) is not known a priori. 9
Structure of Pure H Nebula:Models (Osterbrock & Ferland, p. 26) - assuming n H = 1 cm -3 and T = 75 K, constants as function of radius 1
Consider a Stromgren sphere (pure H nebula around a single star): Q(H ) = # ionizing photons emitted by star per second Q(H ) = ν L ν hν dν Global Ionization Balance = 4 3 πr 1 3 n H 2 α B r 1 = radius of Stromgren sphere (Osterbrock & Ferland, p. 27) 11
Adding He: Energy Levels for He He is a two-electron system: S = or 1 (m = 1 or 3) - ¾ of the recombinations go to triplet state (J = L-1, L, L+1) - ¼ go to singlet state (J = L) 23S is a metastable state: - can be collisionally excited to 21S or 21P (at ne 14 cm-3) - can decay to 11S by forbidden photon at 19.8 ev Transition to 11S results in H-ionizing photon: ionization equilibrium equations get more complicated (Osterbrock, p. 3) 12
H + He Nebula Models Large # of photons with hυ > 24.6 ev Small # of photons with hυ > 24.6 ev (Osterbrock & Ferland, p. 32) - Due to its abundance, H controls the full extent of the ionized zone - For very hot (O) stars, H + and He + zones are coincident - For an H II region, few photons have hν > 54.4 ev à no He +2 zone 13
Planetary Nebula Model - ionization of He + important (Osterbrock and Ferland, p. 36) 14
Photoionization of Heavy Elements # ionizations of X i = # recombinations of X i+1 n(x) = n(x ) + n(x + ) + + n(x +n ) where n(x)/n(h) = fractional abundance of element X Heavy element contribution to diffuse field a ν, α(t) not well known for many ions a ν can show complicated structure, due to resonances. There can also be ionizations of inner shell electrons from X-rays (important in AGN). Vacancies are filled with outer electrons: 1) Producing X-ray emission lines and/or 2) Auger effect: one or more electrons move down, one or more are ejected from the atom. 15
Recombination of Heavy Elements Dielectronic recombination: - Larger than normal recombination for some ions - Relative importance tends to increase with Temperature - Captured free electron excites bound electron in atom, leading to a double excited state - Both electrons eventually decay to ground state Charge exchange - Occurs when two ions have about the same I.P. - Example: O + H + O + + H (works in reverse as well) n(o ) ( ) n(h ) n( H + ) n O + (O ionization locked to H) 16
Real Photoionization Models The input parameters are typically: 1) Input energy spectrum (type of star, AGN, etc.) 2) Ionizing flux or luminosity plus distance 3) Density of hydrogen (ionized + neutral) 4) Abundances of the elements - most simple models assume plane-parallel geometry (slab) and constant density in slab. The output parameters include: 1) Ionization structure into the slab. 2) Temperature structure of the slab. 3) Emission line fluxes or ratios 17