03/05/16 Ionizaon & Recombinaon Hydrogen Recombinaon Free electrons recombine with protons into bound energy levels and rapidly • Photoionizaon is o9en the most important process in astronomical cascade down to the ground state, eming hydrogen series photons. objects. Hot stars (or the accre.on of material onto compact objects, as in AGN, quasars and X-ray binary systems) emit substan.al UV fluxes that ionize atoms. I.P. of H = 13.6, He = 24.6, 54.4eV. Transi.ons to the ground state (n=1) form the Lyman series with lines • Photons with E > 13.6 eV can photoionize hydrogen the dominant gas between 121.6 nm (Ly(alpha) and the series limit at 91.2nm (which species, though atoms with lower I.P. will be ionized with lower energy corresponds to the ionizaon poten.al of H). photons. i i+1 − X + hν → X + e The Balmer (n -> 2) series lies in the visible, from H(alpha) 653 nm (n= 3->2) to the series limit at 364 nm (corresponding to the ionizaon energy from n=2), and further series in the infrared. where Xi is the atom or ion that is ionized by the incident photon hν. Xi+1 is the resul.ng ion and e- the liberated electron • In an idealised model HII region, the Stromgren sphere is the region The emieed line intensi.es reflect the level populaons through within which hydrogen is ionized. It has a uniform density and a sharp € recombinaon and the transi.on probabili.es down to lower levels. boundary and may be strafied with regions of He++ and He+ if the star is sufficiently hot. Note: recombinaon lines are also seen from other species – especially He • It may be surrounded by a region where H is neutral, but low I.P species and He+ but also C,N,O however the laer are usually orders of magnitude may be ionized (Mg, Na etc). Beyond that, there may be molecular gas. weaker, reflec.ng their much lower abundances. 6E5 2E5 5E4 1E6 3E6 H Recombinaon Transi.ons 7E5 4E6 1E5 9E5 7E5 3E6 5E6 Allowed Transi.ons from n=5 With transi.on probabili.es 5E6 1E6 9E6 3E7 From Osterbrock & Ferland 2006 1 03/05/16 Hydrogen Recombinaon spectrum H Line spectrum The spontaneous emission coefficients for the H lines have values A > 106 s-1, while the collisional cross sec.ons have values Q ~ 10-10 m3 s-1 . This means that collisional transi.ons for Hydrogen are 16 -3 important only at densi.es ne ≥ A/Q = 10 m . Logarithmic representaon of the UV- Op.cal - Near-IR The energy difference between the ground state and level 2 is ~10eV. At nebular temperatures, ΔΕ < kTe and collisional 4 hydrogen recombinaon series. excitaon is unimportant at Te below 10 K. Right: high-n transi.ons converging at the Balmer series Below these values of Te and ne, the level populaon are determined primarily by the A values, and so are only limit at 365nm measured in the weakly dependent upon Ne and Te. Orion nebula (Esteban et al 2004) Recombinaon Coefficents Absorp.on probabili.es • Calculated by integrang the probability of e- capture by a proton and branching raos into all levels. Defini.ve tables by Hummer & Storey (MNRAS 224, 801. 1987). The ionizaon cross sec.on (for photons at 912 Å) is σ =6.30 × 10− 18 cm2. This • These give the recombinaon coefficient as a func.on of density and temperature and νo -13 3 -1 is orders of magnitude smaller than the absorp.on cross sec.ons of the low-n the line intensi.es that result, relave to Hβ (n=4-2): αB = 2.6 x10 cm s Lyman lines. So the bright Lyman lines are absorbed by nearby atoms, producing enormous op.cal depths, and maintaining high ionizaon frac.ons. The mean free path ~1/(σ n ) ~ 2 .10-5pc or 1 A.U. for n ~ 104 cm-3 νo H H 2 03/05/16 Line Emission Dependence of H line raos on Te Recombinaon coefficients are calculated from the pho.onizaon cross sec.ons and summed for recombinaon to all atomic levels. The total recombinaon coefficient should be used in the low density limit, Case A, which occurs where the Lyman lines are op.cally thin. The intensity of emission lines can be expressed in terms of the effec.ve line emissivity jν – usually given for Hβ. In thermal equilibrium, jν = κνBν (T ), but the recombinaon process These approximaons hold to 1% over the range : produces level populaons that are far from LTE. 100 < n < 10000 cm-3 and 5000 < T < 12000 K and so are valid for many nebulae. e e In case B, the Hβ emission line luminosity : The rao of ionizing photons to the number of Hα photons = 2.2 within these ranges, and so measurement of the Balmer lines gives a good es.mate of Ni L = 4π n n j dV providing that the Ly series is op.cally thick and the Balmer lines are op.cally H β ∫ e p ν thin. This condi.on is known as Menzel’s Case B. In this case, all of the Lyman where jν(T) is the line emissivity of Hβ. lines are mul.ply scaered in the nebula and only emerge as transi.ons to level 2 j (T) = N N α hc/(4πλ) Wm-3 (Balmer series) or higher. ν p e Metal lines arise from trace elements and are usually op.cally thin € Connuum Emission Free-free In addi.on to emission lines, the electrons in the ionized gas give rise to free-free n 2 and free-bound transi.ons Now e ds τν = ∫ 2.1 3 / 2 Free-free emission is usually most important at radio wavelengths where the ν T Rayleigh Jeans approximaon can be used for thermal emission. At high radio frequencies τ << 1 and so the flux Sν is: 4 The electrons have a thermal Maxwellian distribu.on at Te ~ 10 K. At long 2 wavelengths, we can use the Rayleigh-Jeans approximaon: 2kTν S −0.1 ν ∝ 2 τν ∝ ν 2kT j € c as jν = kνBν(T) ν Bν (T) = 2 = As the frequency decreases, the opacity increases un.l it λ kν becomes op.cally thick and the flux then falls with decreasing frequency as a R-J BB tail, ν2, with a turnover at τ=1. −1.5 −2.1 k ∝T ν n n ds ν e ∫ e p € The opacity : Note: the quan.ty ∫ ne n p dV is known as the emission measure (for a pure hydrogen nebula) € nenp k ds ds τ ν = ∫ ν ∝ ∫ 2.1 3/2 and the op.cal depth : ν T € € 3 03/05/16 Representave free- Stromgren Sphere free spectrum • The extent of the Stromgren sphere is determined by the ionizaon and Ploeed as Sν against ν as usual at radio recombinaon of H atoms. i.e. the flux of ionizing photons with E > frequencies. 13.6eV • A hot star emits Ni ionizing photons per second, and we assume that all of these are eventually absorbed by H atoms in the surrounding HII The op.cal depth region. In equilibrium, the rate of recombinaons equals N . increases with i decreasing S • With a radiave recombinaon coefficient α, the recombinaon rate per frequency un.l the ν unit volume per second: Nr = α nen H emission becomes op.cally thick, when • For a fully ionized, pure hydrogen nebula, ne = nH, and within the volume it follows a Rayleigh V the radius of the Stromgren sphere is: Jeans blackbody 3N slope. r3 = i € s 2 4πα n H At higher • This is an idealised model HII region, but gives an es.mate of the size of frequencies thermal an HII region around a par.cular star surrounded by a region of a certain dust emission density. usually becomes 3 4πrs 4π jν dominant • The Hβ line flux from an idealised sphere FH = Frequency (GHz) € β 3 4π D2 Stromgren Sphere in more detail Resul.ng HII region • In equilibrium, the recombinaon rate equals N . Because radiave i • With a recombinaon coefficient, α= 2.6 x 10-19 m3 s-1 transi.ons have very short life.mes, excited states will rapidly decay 4 to the ground state, so we can treat all H atoms as being in the (appropriate for Te~ 10 K), we can es.mate the radius of ground state. the Stromgren sphere around different types of star. 10 -3 • The photoionizaon rate from the ground state is balanced by the Adop.ng a density of 10 m . total recombinaon rate to all levels in the H atom. -1 ∞ Sp Type Ni (s ) rs (pc) 4πJ 39 n ν a (1s)dν = α (T ) n n G2V 10 6.7E-5 1s ∫ ν A e e p B0V 4x1046 1.6E-2 ν hν 1 O6V 1049 0.1 • However a recombinaon directly to the ground state emits a Lyman con.nuum photon which immediately is absorbed in ionizing Note that the HII region can be either maer bounded or another nearby H atom so we actually use the recombinaon radiaon bounded, where the Stromgren sphere extends coefficient to just the excited states (i.e. excluding to the ground € beyond the surrounding material or is contained within it. state) αB 4 -19 3 -1 3 3Ni At T=10 K, αB = 2.6 x10 m s r s = 2 A similar calculaon can be carried out for He (IP = 24.6eV) 4πα B n H € 4 03/05/16 He and H photoionizaon model Note the large difference in the extent of the He ionizaon zone (from Osterbrock) The photo-ionizaon (bound-free) cross sec.ons increase abruptly from zero at the IP and are high at energies just above that energy, falling as ν-3.