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Ionizaon & Recombinaon Hydrogen Recombinaon Free electrons recombine with protons into bound energy levels and rapidly • Photoionizaon is oen the most important process in astronomical cascade down to the ground state, eming hydrogen series photons. objects. Hot stars (or the accreon of material onto compact objects, as in AGN, quasars and X-ray binary systems) emit substanal UV fluxes that ionize atoms. I.P. of H = 13.6, He = 24.6, 54.4eV. Transions 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 potenal 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 resulng ion and e- the liberated electron • In an idealised model HII region, the Stromgren sphere is the region The emied line intensies reflect the level populaons through within which hydrogen is ionized. It has a uniform density and a sharp € recombinaon and the transion probabilies 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, reflecng their much lower abundances.

6E5 2E5 5E4 1E6 3E6 H Recombinaon Transions 7E5 4E6

1E5 9E5 7E5 3E6 5E6

Allowed Transions from n=5 With transion probabilies

5E6 1E6 9E6

3E7 From Osterbrock & Ferland 2006

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Hydrogen Recombinaon spectrum H Line spectrum

The spontaneous emission coefficients for the H lines have values A > 106 s-1, while the collisional cross secons have values Q ~ 10-10 m3 s-1 .

This means that collisional transions for Hydrogen are 16 -3 important only at densies ne ≥ A/Q = 10 m . Logarithmic representaon of the UV- Opcal - 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 transions 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 Absorpon probabilies • Calculated by integrang the probability of e- capture by a proton and branching raos into all levels. Definive tables by Hummer & Storey (MNRAS 224, 801. 1987). The ionizaon cross secon (for photons at 912 Å) is σ =6.30 × 10− 18 cm2. This • These give the recombinaon coefficient as a funcon of density and temperature and νo -13 3 -1 is orders of magnitude smaller than the absorpon cross secons of the low-n the line intensies 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 opcal depths, and maintaining high ionizaon fracons. The mean free path ~1/(σ n ) ~ 2 .10-5pc or 1 A.U. for n ~ 104 cm-3 νo H H

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Line Emission Dependence of H line raos on Te Recombinaon coefficients are calculated from the phoonizaon cross secons 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 opcally thin.

The intensity of emission lines can be expressed in terms of the effecve 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 esmate of Ni L = 4π n n j dV providing that the Ly series is opcally thick and the Balmer lines are opcally H β ∫ e p ν thin. This condion 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 mulply scaered in the nebula and only emerge as transions 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 opcally thin €

Connuum Emission Free-free

In addion to emission lines, the electrons in the ionized gas give rise to free-free n 2 and free-bound transions 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 distribuon 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 unl it

λ kν becomes opcally 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 quanty ∫ 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 opcal depth : ν T € €

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Representave free- Stromgren Sphere free spectrum • The extent of the Stromgren sphere is determined by the ionizaon and Ploed 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 opcal 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 unl the ν unit volume per second: Nr = α nen H emission becomes opcally 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 esmate of the size of frequencies thermal an HII region around a parcular 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 Resulng 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 transions have very short lifemes, 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 esmate 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 Adopng 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 connuum 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

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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 secons increase abruptly from zero at the IP and are high at energies just above that energy, falling as ν-3. Note that σ(He I) > σ(H) above the IP(He I). Figure from Osterbrock for H I, He I and He II.

The Rosee nebula M57 The Ring Nebula

A young HII region almost 1 deg Planetary nebula showing straficaon across illuminated by a cluster of Red [NII] 6584 OB stars (NGC 2244) which have The Orion nebula seen with HST. The dominant ionizing Green [OIII] 5007 1 cleared out a central cavity Blue Helium 4686 source is ϑ Ori, of spectral type O6, but there are thousands of newly formed stars in the central parsecs

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With ne = np = nH : dr 4π 4 r2 n (r) N r3 n 2 (r) Recombinaon and Ionizaon Timescales π H = i − α H dt 3 -19 3 -1 10 The recombinaon coefficient, αB= 2.6 x 10 m s , which with a density of 10 3N m-3 gives a recombinaon me of ~3 108 s ~ 10 yr. 3 3 i The expression for the Stromgren radius: r = rs = 2 The total recombinaon rate over the volume of the HII region is: 4πα n H Now define τ = t/τrec where € τrec = 1/αnH, and λ = r/rs, to express the equaons in terms of the recombinaon me and Stromgren radius, 4π 3 R = r neα B n p 3 to give: 3 2 €rs dλ 4π 3 3 2 As a hot star forms, it will start to ionize its surroundings and the ionizaon front 4πλ n H (r) = Ni − rs λ α n H (r) will advance through the surrounding medium. The mescale for advance τ rec dτ 3 depends on the density of the medium and Ni. 3 2 dλ τ rec 4π 3 The number of H atoms in a shell outside an ionized sphere is € And mulplying by τrec/rs : 4πλ n (r) = N − λ n (r) H i 3 H dτ rs 3 2 4πr n H (r) Δr € Now N τ 4π n 2 dλ 3 while the number of ionizing photons available in me Δt is N Δt, and the i rec H i 3 = ⇒ 3λ =1− λ number of ionizing photons available for ionizing the shell is the total number - And for uniform density, rs 3 dτ the number of recombinaons 2 4π 3 the soluon is €3 1/ 3 within the ionized zone 4πr n H (r) Δr = NiΔt − r neα B n pΔt λ =1− exp(−τ) ⇒ r(t) = r [1− exp(−t /t )] € 3 S rec

€ € € The ‘Pillars of Creaon’ or The Eagle Esmates of stellar temperature Nebula or M 16 The ionizing photons produce H emission lines (and an associated free-free connuum) in well-understood flux raos, with a As an ionizaon front relavely weak dependence on temperature and density. advances through a cloud, This holds for the normal range of nebular condions, but at high residual dense structures densies, collisional effects become important. remain aer The Ly connuum flux increases sharply from stars of type B0 to photoevaporaon of lower 7 O4 (Mass ~ 10 to 40Mo ), rising as ~ T . Measurement of H lines density material allows us to infer the number of ionizing photons in a nebula. If the total luminosity can be measured (integrated over the whole spectrum), then the rao of the energy in ionizing photons Dense cores are the sites of compared to Lbol gives an esmate of the effecve temperature star formaon and so newly of the excing star(s). formed stars may lie in the In pracce, some Lyα photons do escape from the nebula cores of dense globules (mulple scaering means that some have wavelengths that random walk to the edge of the line profile so that the cross secon decreases).

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Stellar Models The effects of Dust The number of high energy photons (producing highly ionized species such as He 2+) depends on the detailed shape of the Free-free stellar spectrum which emission is complex and poorly understood for many objects. Geng accurate esmates of the effecve temperature of the excing stars requires careful In the idealised HII region we have assumed that all Lyman photons are trapped and mulply scaered, comparison between populang electrons in excited levels unl they emit in the Balmer or higher series . observaons and However, if dust is mixed with the gas, some fracon of the photons will be absorbed by dust grains. The grains are heated and will emit thermal infrared photons, but this then means that Ni will be models underesmated. In this PN, dust absorbs most of the UV photons, and most of the luminosity emerges in the IR.

Collisional Ionizaon Collisional Recombinaon • Is important in hot plasmas with densies substanally higher than in typical nebulae • Here an electron is scaered as it passes an ion, giving up sufficient • Collisional recombinaon is the inverse of energy to remove a bound e- collisional ionizaon, but note that it requires a 3 X i + e− → X i+1 + e− + e− parcle interacon between two free electrons and an ion. • The collisional ionizaon rate coefficient Qi is obtained by integrang q v f(v) where q is the ionizaon cross-secon and f(v) is usually X i+1 + e− + e− → X i + e− assumed to be Maxwellian, so that Qi = qv -1 • The ionizaon rate per unit volume is then € QiNiNe (s ) and Qi can be approximated as: • This will only be important in regions of very high density such as stellar interiors and need not be ξ Q = 2 ×1014 T1/ 2e−χ / kTe m3s−1 considered in nebulae € i χ 2 e • Where ξ is the number of electrons in the subshell .

• Collisional ionizaon may be important when kTe is larger than the hydrogen ionizaon potenal, χ, i.e. at T>20,000K . €

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Collisional excitaon and autoionizaon Charge Exchange In ions that have more electrons in an inner subshell than in an • Charge exchange occurs when an electron is exchanged between an atom outer shell, excitaon of an inner electron is quite likely. and an ion. E.g. in a configuraon such as 2p63s, an inner electron may be • Here X and Y are an atom and ion of different elements. 5 excited to a higher level giving 2p 3snl (e.g. Na I, Mg II, Al III …. i j i−1 j +1 Fe XVI) X + Y → X + Y + ΔE The excited state produced can lie above the ionizaon potenal • Charge exchange is most important when the energy difference ΔE is small, of the ion (the energy required to remove the outermost and where the probability of exchange is high • The ionizaon potenals of H and O are 13.598 and 13.618 eV respecvely electron – 3s in this example). i.e. ΔE= 0.02eV 2 6 Consider Fe XVI ground state 2s 2p 3s Na I – like ion € + + Collisional excitaon of a p shell electron gives an excited state O + H → H + O 2s22p53snl which lies above the ground state of Fe XVII 2s22p6 • The reverse process is important at the edge of HII regions where the H+ fracon decreases and because of the large number of neutral H atoms, O The excited state can either decay spontaneously back to the becomes predominantly neutral. Charge exchange is the most important ground state, eming an EUV photon, or make a radiaonless recombinaon process from O II to O I transion into the connuum (analogous to Auger transions • Because of the small O abundance, this process does not alter the H from inner shells). In this case, the electron becomes unbound ionizaon structure of the HII region € and results in ionizaon to Fe XVII

Excitaon and Recombinaon Di-electronic recombinaon Processes • Dielectronic recombinaon occurs when an ion in the ground state • Different processes dominate under different condions, and so in has an electron excited by a passing electron, which is itself order to know which approach to take, we need to be able to i captured forming a doubly excited state X2 in ion X . esmate the prevailing density and temperature in the medium under study. i+1 − i X + e → X 2 • This excited state may be above the first ionizaon state of Xi and • We can do this by using pairs of transions that are sensive to the can decay by autoionizaon, producing Xi+1 or it can decay to a density, e.g. because they have very different collisional rates, and so lower level in ion Xi , eming a photon and constung a the rao of the line intensies varies as a funcon of the density. recombinaon. • € i i Most diagnosc measurements use lines in the opcal because they X 2 → X + hν have well proven and are well understood, but as spectra in other wavelength regimes have become more sensive, diagnoscs in e.g. the infrared are becoming more important, and benefit from • This process is important in hot plasmas at moderate densies reduced sensivity to the effects of interstellar reddening (e.g. in the solar corona) and in low ionizaon stages at nebular temperatures (e.g. for Planetary Nebulae) €

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