1 AY230-HIIReg II. HII Regions (Ionization State) A. Motivations Theoretical: HII regions are intamitely linked with past, current and future starforming regions in galaxies. To build theories of star-formation it is vital to have a working knowledge of HII regions. Observational: The spectra of emission-line galaxies are dominated by its HII regions. To derive basic properties about the galaxy (e.g. its dynamics, metallicity, age, etc.), one must understand the physical nature of HII regions B. Ionization balance (Hydrogen) Balance Equilibrium e + H + H 0 + hν (1) Photoionization rate equals the recombination rate This balance holds locally, i.e. in every cm 3 of the gas Photionization of H Photoionization rate per H atom :: Γ Therefore, the photoionization rate per cm 3 is n HI Γ Define 4πJν n HI Γ = n HI hν σph ν dν (2) J ν :: Mean intensity of ionizing radiation (see radproc notes) σν ph :: Cross-section of H to ionizing photons ( ) σν ph = 29 π 2 4 IH 3 αa2 0 f(η) (3) hν f(η) exp [ 4η cot 1 η ] 1 exp [ 2πη ] ( ) 1 ( IH 2 IH η = E f ω I H ) 1 2 (4) (5) See http://www.ucolick.org/ xavier/ay204b/ay204b phtrec.ps I H = h σ ph ν 0 ω < I H ν 8/3 ω I H ν 3 ω I H ν 7/2 ω I H (6)
2 AY230-HIIReg 10 18 2.5 σ ph (ν) 10 20 10 22 10 24 dlog σ(ν) / dlog ν 3.0 3.5 10 26 1 10 100 hν/i H 4.0 1 10 100 hν/i H σ ph ( ) = 6 10 18 cm 2 Overall, ( ) 3 ν σ ph (ν > ) σ ph ( ) (7) Recombination of H Total recombination rate n e n p α(t ) Total recombination rate per proton and per electron :: α(t ) α(t ) = n α n (T ) (n specifies the energy level) (8) Recombination rate to a given energy level n :: α n α n (T ) < σ rec n (v)v > (9) σn rec (v) is the cross-section for recombination to level n. It is convolved with the velocity distribution f(v) of electrons. The functional form of f(v) is generally a Maxwellian described by the electron temperature T. See the radproc notes on Milne s relation to derive: σ rec n (v) = 2 C. Stromgren Radius (Simplified case) ( hν Crude estimate of the size for an HII region Concept cm e ) 2 n 2 v 2 σph n (ν) (10) Globally balance the photoionizing rate by a star with the recombination rate of the gas surrounding it Ignore radiative transfer, dust, etc. Specific assumptions Single O or B star Embedded within a cloud of HI gas (no dust) Constant density
3 AY230-HIIReg Ionizing photon luminosity of a star Total number of ionizing photons emitted per second φ φ = B ν is the blackbody spectrum h = 1 Ryd = 13.6 ev Recall Astro 101 πb ν hν 4πR2 dν (photons/s) (11) F = σ is the Stefan-Boltzman constant Black body Similiarly For kt > h, Expect φ L/h Note T 0 = h /k 150, 000 K Substituting x hν/kt 0 πb ν dν = σt 4 (12) σ = 2π5 k 4 15c 2 h 3 = 5.67 10 5 (cgs) (13) B ν (T ) 2hν3 c 2 1 (e hν/kt 1) (14) L = 4πR 2 F (15) φ = 15 π 4 L h x 0 x 2 dx (e x 1) (16) Plotting f(x 0 ) equal to the x integral 2.5 2.0 f(x 0 ) 1.5 1.0 0.5 0.0 10 20 30 40 50 T/(10 4 K) Recombination rate of the surrounding medium Key: Ignrore recombinations to the ground-state because these will produce an ionizing photon (i.e. no net change)
4 AY230-HIIReg Recombination rate per cm 3 Where we have defined n e n p < σ n 2 v > (17) σ n 2 Note n p = n e for pure Hydrogen Define Case B recombination coefficient σ n (18) n=2 α B (T ) < σ n 2 (v) v > (19) Integrated over a Maxwellian T is the temperature of the electrons For Hydrogen α B (T ) 2.60 10 13 cm 3 (T/10 4 K) 0.8 s (20) Recombination rate of a constant density cloud with radius R S n 2 eα B (T ) 4 3 πr3 S (21) Stromgren Radius: R S Adopt ionization equilibrium: Total number of recombinations equals the total number of ionizations Physics φ = n 2 eα B (T ) 4 3 πr3 S (s 1 ) (22) Rearrange ( ) 1/3 3φ R S = (23) 4πn 2 eα B (T ) Consider real stars (TABLE) Table 1: Stromgren Radii Spectral type T /10 4 log φ log(rs 3 n2 e) R S (pc) [n e = 10] B0 3.0 47.67 4.1 4.9 O9 3.2 48.24 4.6 7.3 O7 3.5 48.84 5.2 12 O5 4.8 49.67 6.1 23 What happens at r R S?
5 AY230-HIIReg Consider the mean free path of a photon λ From equation 7 and with ν λ mfp 1 σ ph ( ) n H (24) λ mfp 1 ( nh ) 1 0.1 σ ph ( )n H cm 3 pc (25) Therefore, the transition from HII to HI is very sharp! For harder spectra, σ < σ( ) and λ mfp is larger D. Ionization structure:: On the spot Next simplest model is to approximate the radiative transfer and emissivity within the HII region Assumptions Pure Hydrogen Allow for partial ionization, radiative transfer On-the-spot approximation Radiative transfer (see radproc notes) Ionization equilibrium In steady state, ionizations balance recombinations n HI 4πJν hν σph ν Define: Case A recombination coefficient Equation of transfer (for ionizing photons ν ) Opacity: Ionization of Hydrogen dν = n p n e < σ rec n 1v > (26) α A (T ) = < σ rec n 1(v)v > maxwellian (27) di ν ds = κ νi ν + j ν (28) κ ν = n HI σ ph ν (29) Emissivity: Reemission of photons with ν due to recombinations to n = 1. We will refer to this as the diffuse radiation field Express the intensity as two terms I ν = I ν + I νd (30)
6 AY230-HIIReg I ν : Stellar radiation field I νd : Diffuse radiation field Stellar radiation field I ν (r) = I ν (R) R2 e τν (31) r 2 1/r 2 term is standard dilution τ ν is the optical depth due to the HI atoms in the HII region Diffuse radiation field τ ν (r) = di νd ds r 0 n HI (r ) σ ph ν dr (32) = n HIσ ph ν I νd + j ν (33) Recombinations give j ν We will discuss an exact expression later Consider the total number of photons generated per cm 3 Text books frequently define On-the-spot (OTS) approximation j ν 4π hν dν = n pn e < σ n=1 v > (34) = n p n e [ α A (T ) α B (T ) ] (35) α 1 (T ) α A α B =< σ rec n=1(v)v > max (36) In an optically thick cloud where no ionizing photon escapes, every diffuse photon generated is absorbed somewhere in the cloud jν J νd 4π dv = 4π n HI hν hν σph ν dv (37) J νd is the mean intensity of the diffuse radiation field J νd n HI σν ph is the rate of absorption by HI atoms in the HII region If this relation holds locally, we have the on-the-spot approximation J νd = j ν n HI σ ph ν (38) This implies a very small mean free path Only valid in regions with large n HI This is not the case for most of the HII region, yet the approximation still works well!
7 AY230-HIIReg Ionization structure assuming OTS Approx Original equation Substitute n HI 4πJ ν hν σph ν dν = n p n e α A (39) J ν = J ν + J νd (40) = J ν + j ν n HI σ ph ν (41) Rearrange n HI Substituting for J ν from Equation 31 n HI R 2 r 2 Solving for the ionization structure Adopt a density profile 4πJ ν hν σph ν dν = n p n e α B (42) 4πJ ν (R) σν ph e τν dν = n p n e α B (T ) (43) hν n H (r) = n HI (r) + n p (r) (44) Assume a temperature profile T (r) Can integrate Equation 43 and the optical depth (Equation 32) to find n HI (r) and n p (r) Rough calculation Recall φ(r ) = 4π Introduce the intensity weighted ionization cross-section σ J ν (R)4πR 2 dν (45) hν ph J ν σν hν Using our on-the-spot approximation, we rewrite dν (46) J ν dν hν n HI σ φ 4πr = n en 2 p α B (47)
8 AY230-HIIReg Ionization fraction x x n e = xn H for Hydrogen gas n HI = (1 x)n H for Hydrogen gas Ionization fraction in the on-the-spot approximation At radii r R S, the gas is highly ionized: x 1 n HII n HII + n HI = n HII n H (48) x 2 1 x = σ φ 1 (49) α B 4πr 2 n H 1 1 x = σ φ 1 (50) α B 4πr 2 n H 1 x α B σ Example: O7 star (see Table 1) Evaluate at r = R S /2 = 6 pc 1 x = HI density profile If x 1, n HI n H Example: O7 star again 4πr 2 φ n H (51) 2.6 10 13 6 10 18 4π(6pc) 2 10 48.8 10 = 2.7 10 4 1 (52) n HI = (1 x)n H r 2 (53) small optical depth τ = n H τ = r 0 = n H r 0 n HI σdr (54) r 0 (1 x) σdr (55) r 2 (R S /2) 2 1.6 10 21 dr (56)
9 AY230-HIIReg 10 0 10 1 10 2 τ(r) 10 3 10 4 10 5 0.0 0.2 0.4 0.6 0.8 1.0 r/r S This plot breaks down at r R S where x is no longer very close to 1 At r < R S /2, the ionizing spectrum is largely unattenuated Consider the mean free path at R S /2 J ν (r) = R 2 J ν r 2 e τν R2 r 2 J ν (R) (57) φ(r) φ(r)e τ φ(r) (58) λ 1 n HI σ 1 n H (1 x) σ 20 pc (59) This is greater than R S!! Ionizing radiation is not absorbed on-the-spot Need to include an extra term for diffuse n 1 ionizing radiation On the other hand, in most conditions J νd J ν Interesting aside: Life cycle of an electron in an HII region Time scales (a) Recombination time (b) Cascade time (spontaneous emission) (c) Time in ground state t rec = 1 n H α B = 105 n H yrs (60) t cascade = 1/A n l n=1 10 7 s (61) t n=1 = 4πr2 φ σ ( ) 2 r 3yrs (62) R S /2
10 AY230-HIIReg Electrons spend the majority of their time as free particles E. Ionization structure:: Helium, heavy elements and dust Helium Physics Extra opacity for hν > 24.6 ev Extra source of electrons He + region Follows HII region for stars with h ν > 24.6 ev Much smaller region for h ν < 24.6 ev Heavy elements Dust Mainly contribute electrons Important for cooling in the HII region Offer useful observational diagnostics of HII regions Optical depth Rough estimate dτ d = κ d ds (63) τ d = κ d N d 2 10 21 cm 2 N H (64) N H = n H R S Evaluating with Equation 23 for R S ( nh ) ( ) 1/3 1/3 τ d = 0.19 T 0.26 φ cm 3 4 (65) 10 48 Therefore, dust can be very important for dense HII regions