AY2161Lec 3. Radiative Processes and HII Regions¨1. Photoionization2. Recombination3. Photoionization-Recombination Equilibrium4. Heating & Cooling of HII Regions5. Strömgren Theory (for Hydrogen)6. The Role of Helium References Spitzer Secs. 5.1 & 6.1Tielens Ch. 7Dopita & Sutherland Ch.9Osterbrock & Ferland, Ch. 3AY2162Introductory SummaryThe far ultraviolet radiation (FUV) from an O or B star Ionizes its immediate neighborhood and produces an HII region. Strömgren developed the theory in 1939 for a spherical model where the HII region slowly expands into uniform HI.HII regions illustrate basic processes that operate in allphotoionized regions of the ISM: super-Lyman radiation( < 916.6 Å - FUV or X-rays) photoionize sH: h + H  H+ + e.The H+ ions recombine radiatively H+ + e  H + hv.• The balance between these two reactions determinesthe ionization fraction.• Any excess photon energy above the ionization potential(IP = 13.6 eV) is given to the ejected electron and is thenequilibrated in collisions with ambient electrons, therebyheating the HII region.AY2163NGC 3603 RosetteNorth American Shapley’s Planetary NebulaReal HII regions are rarely spherical. Nonetheless, Strömgren’stheory Illustrates the basic roles of photoionization and recombination.AY21641. PhotoionizationAbsorption cross section per H nucleus(1-2000 Å, averaged over abundances)X-raysFUVThe ISM is opaque at 911 Å(the H Lyman edge) & partially transparent in the FUV and inthe X-ray band above 1 keV. Ionizing photons come from1. Massive young stars*2. Hot white dwarfs3. Planetary nebula stars4. SNR shocks* Table 2.3 of Osterbrock &Ferland give T(O5V )  46,000 K(BB peak at 3kT or 12 eV) & 3x1049 ionizing photons/sAY2165Hot Stars as FUV SourcesBlackbody (smooth curve)is a poor approximation inthe FUV and EUV.State of the art modelatmospheres disagree.See Lejuene et al.,A&AS 125, 229 1997and NASA ADS:1997yCat..41250229L,for an extensive libraryof stellar energy distributions.Note that the spectrum of this very hot star cuts off beyond 50 eV.AY2166Photoionization of Hydrogenic Ionswhere h1 = 13.6 Z2 eV and gbf  1 is the QM Gaunt factorfor bound-free transitions from n = 1. Compare this withKramers’ semi-classical formula:Both give an inverse-cube dependence on frequency.The mean free path at 912 Å is very small: =7.911018Z2 1       3gbfcm2 =1 1       3 with 1=6.33 1018Z2cm2lmfp(1) =1nHI 1=1.58 1017nHIcm2Quantum theory predictsAY2167Photoionization Cross Section of HeIonization potentialsH 13.6 eV (912Å)He 24.6 eV (504Å)He+ 54.4 eV (228Å)• At high frequencies, the largerHe cross section more thancompensates for its lowerabundance relative to H.• Very hot stars are needed to ionize He+ (T* > 50,000 K). He++ does not occur in H II regions except in planetary nebulae, AGN and in fast shocks.• Continuum radiation gets “harder” with increasing depthdue to the rapid decreaseof the cross section with .NB For analytic fits for heavy atomphotoionization cross sections, seeVerner et al., ApJ, 465, 487 1996AY2168Calculation of the Photoionization RateThe photoionization rate for one H atom iswhere the mean intensity J , photon number, and energydensity are derived from the specific intensity by= dv 4Jh1 = cdv n1 n=1c4Jhd1 =4J1hcI1,u=1cd I4cJ= nhThe total photon number density,In JJ11      nd11where I1 is the first inverse moment of J , defined byAY2169Calculation of the Photoionization Rate (cont’d)  n1c (I4I1)=4Jhd1 =4J1hJJ11      11 1      3d1 =4J1h1I4Because the cross section varies as -3, the ionizationrate can also be expressed in terms of these moments.Replacing the mean intensity by the photon number, this isThe moments depend on the spectrum. A typical ratio appropriate for HII regions is I4/I1  .AY21610Numerical Estimate of the Photoionization RateNB In the 2nd line, the speed of light is given in m/s, whereas cgs units are used everywhere else, especially in the 1st line.AY216112. Radiative RecombinationThe cross section for capture to level n ise-H+ + e  H + hRadiative recombination is the inverseof photoionization. Milne calculated thecross section from Kramer’s crosssection using detailed balance(Spitzer Eq. 5-9, Rybicki & Lightman Sec.10.5)It depends on electron speed as w-2. The rate-coefficient,n = <w(w)>th, is small and decreases as 1/T, e.g., thetotal rate coefficient (next slide), summed over n at 10,000Kis (104K) ~ 4x10-13 cm3 s-1.fb(w) = 2n2hmecw      2bf(v) =h1mec2h112mew211n3AY21612Rate Coefficient for Radiative Recombination n(h/kT) is a slowly varying function of T, introducedand tabulated by Spitzer (p. 107). The values for n = 1& 2 at 8000K are 2.09 & 1.34, respectively, so that(1)= 5x10-13 and (2) = 3x10-13 cm3s-1nenin(w)w = nenin(n )=mm= n= 2.06 10 11Z2T 1/ 2n()cm3s 1The rate coefficient (n) (units, cm3s-1) gives the rate ofdirect recombinations per unit volume to level n; ni isthe ion density (in this case H+),Summing the rates for all levels m  n, gives the totalrecombination rate coefficient to level n:AY21613On The Spot Approximation for HThe rate coefficient (1) includes recombination to theground state, but that process produces another ionizingphoton that is easily absorbed locally at high density, as ifthe recombination had not occurred.In this on the spot approximation, the effective recombinationrate omits recombination to the ground stateThe corresponding recombination time isCompared with the ionization time (slide 10), rec >>  inmost cases, and the gas around hot stars is highly ionized.(2)=mm= 2 2.60 1013Z2T40.8cm3s1trec=1ne(2) 3.85 1012ne1T40.8secAY216143. Photoionization Equilibrium for HHowever, the time scales for dynamical changes forgas and star are much longer than trec and tion.Thus theionization fraction is in a quasi-steady-state calledphotoionization equilibrium where the rate of ionization out of ionization state i-1 = rate of recombination intostate i:= (1) or (2), dependingon