Ionization Equilibrium and Line Diagnostics

Transcription

1 D SG D X D X D 7 SG O D O D 7 3 onization quilibrium, onization quilibrium and ine Diagnostics (average quasar spectrum rancis et al., 99, ig. 7) Typical spectra of G (and planetary nebulae) are dominated by Hydrogen lines, plus emission from O at 57 Å ( nebulium ). Physics: gas is in photoionization equilibrium with radiation of the vicinity of the central black hole. onization quilibrium X D D O D 7 SG ntroduction Strength of emission lines characterized by their equivalent width, defined by Z fobs(λ) fcont(λ) dλ W = fcont(λ) (7.) units of W: Å. (average quasar spectrum; rancis et al., 99, ig. 6) Similar definitions also also exist for - or ν -space! eminder: The average optical spectrum of G is dominated by broad and narrow lines. ntroduction (rancis et al., 99, Tab. )

2 D D O O D D O O D D G S D X ate quations, 7 5 G S D X, 7 7 onization structure of gas in G determined from the rate equations: toms can be ionized and can recombine = number density of ions can change with time. Define n Z (z): number density of species Z in ionization stage z (units: cm 3 ). λ(z,z + ): transition rate from stage z to z + (units: s ). then dn Z (z) dt = n Z (z )λ(z,z) n equilibrium: dn Z /dt = and thus n Z (z)(λ(z,z + ) + λ(z,z )) n Z (z + ) n Z (z) = λ(z,z + ) λ(z +,z) + n Z (z + )λ(z +,z) (7.) n q. (7.) only adjacent ionization stages are connected, calculation gets (much) more complicated if also z, z +, etc. are connected. onization quilibrium 3 (7.3) γ e / mv : onization of an ion by a photon. eaction equation: (Z,z) + γ (Z,z + ) + e Photon needs energy hν > ionization energy =: hν thresh, remaining energy, hν, goes into kinetic energy of electron and is thermalized rate: ν γ γ (z,z + ) = Γ Z,z = νthresh hν σ bf(ν)dν (7.4) where σ bf : photoionization cross-section ( bf : bound-free). onization quilibrium 5 D D G S D X ate quations, 7 6 G S D X, 7 8 The rate equations are determined from all physical processes with result in ionization or recombination. ost important processes for ionization: ollisional onization ost important processes for recombination: adiative ecombination Dielectronic ecombination We will now look at the physics of these processes in greater detail. σ bf is determined by quantum mechanics. or Hydrogen, for absorption from the nth level (enzel & Pekeris, 935): ( ) 64π4 m e e σ bf = 3 3ch 6 n 5 ν 3g bf(n, ν) (7.5) ν 3 where the Gaunt-factor, g bf, is tabulated, e.g., by Karzas & atter (96) and is ν / away from threshold. or the ground state of hydrogen: g bf;,ν = 8π 3 ν e 4z cot z ν e πz (7.6) where z = ν /(ν ν ) and where hν is the binding energy of Hydrogen. Useful fitting formulae for all elements and ions with Z 3 have been published by erner & Yakovlev 995 and erner et al. 996, detailed calculations have been performed by the opacity project (TOP, Seaton et al.). onization quilibrium 4 onization quilibrium 6

3 D D O O D D O D O D G S D X, 7 9 G S D X, 7-3 s. 3s e -4. σ bf [barn] -5. 3p 8 4 [ke] [e] [e] 3d [e] cross sections for several subshells of e (cross sections from erner et al., 996). barn = 4 cm. σ bf [barn] nergy [e] TOP cross-section for e onization quilibrium 7 onization quilibrium 9 D D G S D X, 7 G S D X, 7 σ [barn] e - total e [e] σ bf per H-atom for material of solar composition from the optical to the X-ray regime. σ 3 ke [ -4 cm ] σ [barn/h] H... [ke] H O e e- g l Si S r a r H+He... [ke] σ bf 3 in the U and X-rays (dashed: influence of dust), Wilms, llen & cray (). e i ote strong 3 dependency above the absorption edges! n the X-rays, most of the absorption is not from hydrogen, although absorbing columns are still given in terms of an equivalent hydrogen column, H. esonance structure close to ionization threshold: σ bf is influenced by autoionization resonances, where more than one electron is involved. onization quilibrium 8 onization quilibrium

4 D D O O D D O O D D G S D X ollisional onization, 7 3 G S D X adiative ecombination, 7 5 / mv e e ollisional onization rate depends on the electron velocity distribution: γ coll. (z,z + ) = n e Z (z,t e ) = n e ollisional onization: onization of an ion by a collision with an electron. eaction equation: (Z,z)+e (Z,z+)+e +e onization quilibrium vthresh σ i (v)vf(v)d 3 v =: n e vσ i (7.7) where σ i is the collisional ionization cross-section, and Z is the collisional ionization rate coefficient (units cm 3 s). n G one typically assumes f(v) to be a axwell distribution. ecombination rate: adiative ecombination: apture of an electron into the excited state of an ion with subsequent radiative cascade to ground state. eaction equation: (Z,z+)+e (Z,z)+γ+γ+... λ fb =: n e α Z,z (T) = n e σ fb (v)vf(v)d 3 v =: n e σ fb v (7.9) onization quilibrium 3 D D G S D X ollisional onization, 7 4 G S D X adiative ecombination, 7 6 σ i (v) for r (rnaud & othenflug, 985, ig. 8) Z is normally presented in tabulated form, a typical fitting formula is: / exp( /kt) Z (z,t) = z T + a z (T/T Z ) (7.8) where T Z = /kt. See, e.g., rnaud & othenflug (985) or Shull & an Steenburg (98). (ahar, Pradhan & Zhang,, ig. 3) The recombination cross-section, σ fb, can be obtained from the photoabsorption cross-section, σ bf using the ilne relation (see handout), σ fb (v) = g z,n g z+, h ν m ec v σ bf(ν) (7.) onization quilibrium onization quilibrium 4

5 D O D D O O D The cross-section for recombination, σfb, can be easily derived using the principle of detailed balance. The derivation given here follows Osterbrock (989). The microphysical processes that are balanced are photoionization by photons in the energy range from hν to h(ν + dν) on the one hand, and (spontaneous or induced) recombinations from electrons in the velocity range from v to v + dv on the other hand. Thus, v and ν are related by 7 6 mev + hνthresh = hν (7.) mevdv = hdν (7.) n thermodynamical equilibrium, the rate of induced recombinations is exp( hν/kte) times the rate of induced ionizations (this is the detailed balance, such that 4πBν(Te) nenz,z+vσfb(v)f(v)dv = ( exp( hν/kte)) nz,z σbf(ν)dν (7.3) hν Because we are in thermodynamical equilibrium, the radiation field is a Planckian, Bν, and the electron distribution, f(v), is given by the axwell-boltzmann distribution, f(v) = 4 π ( me kte ) 3/ v e mev /kte (7.4) s is shown in many introductory books to astrophysics, in thermodynamical equilibrium the ionization structure is given by the Saha equation, nz,z+ne nz,z where the gi are the statistical weights of the two ionization stages. ( ) = gz+ πmekte e hνthresh/kte zi h (7.5) nserting everything gives the ilne relation σfb(v) = gz,n h ν gz+, m ec vσbf(ν) (7.) for the recombination cross section σfb into the nth level of the ion (Z,z). Here, we ve explicitly written down the statistical weight of this level as gz,n and assumed that the recombining ion, (Z,z + ) is in its ground state (n = ). n alternative derivation using quantum mechanics uses symmetry arguments for the relevant matrix elements z H z +. G S D X ssume: cloud irradiated by photons Simplification: only source for ionization: photoionization quilibrium: number ionizations = number of recombinations = where ν ion 7 8 n(z z )σ bf (ν) ν hν dν = α(t)n en(z z+ ) (7.6) σ bf (ν): photoionization cross section (cm ; 3 ) α(t e ): total recombination coefficent (cm 3 s ) n i : particle density (cm 3 ) ν : local photon flux (erg cm s ke ) where ν is related to the source luminosity via ν = ν (7.7) 4πD quilibrium D D G S D X Dielectronic ecombination 7 7 G S D X 7 9 Dielectronic ecombination: apture of electron into excited ydberg state (followed by radiative stabilization). xcess electron energy is not radiated but transfered to another bound electron. eaction equation: (Z,z + ) + e (Z,z + ) (Z,z) + γ +... Since two electrons are excited = dielectronic recombination leads to emission of satellite lines, important, e.g., in solar corona and in photoionized gases around X-ray binaries, less so in G. Since σ bf (ν) is a strongly peaked function, we can write q. (7.6) as n(z z )σ bf (ν ion ) ν ion hν ion α(t)n e n(z z+ ) (7.8) and therefore n(z z+ ) n(z z σ bf(ν ion ) (7.9) ) α(t) 4πD n e hν ion i.e., ionization equilibrium mainly depends on U = /4πD hν ion # ionizing photons/cm3 = n e c # electrons/cm 3 (7.) where U is called the ionization parameter many other definitions available! xample: or the B: D light days, /hν 5 photons, and n = cm 3 gives U.. onization quilibrium 5 quilibrium

6 D D O O D D O O D D G S D X 7 G S D X 7 n reality, as shown before many radiative processes need to be considered: onization: collisional onization uger-onization ecombination: radiative recombination dielectric recombination ontinuum Processes: Bremsstrahlung ompton-scattering eal life: Solution using advanced radiation codes such as loudy or XST (it is not worthwhile to develop your own code... ). quilibrium 3 Strömgren sphere like structure of a cloud irradiated with a typical G spectrum (athews & erland, 987) with U =., and n H = cm 3 (typical for B). Distance is into cloud. (Hamann et al.,, ig. ) quilibrium 5 D D G S D X 7 G S D X 7 3 e X e X e X e T ( ) O y alpha ine ratios of prominent lines with respect to 549Å as a function of the ionization parameter for parameters appropriate to the B in G 45. O ] P 4 (549): Total power emitted in -line. (Korista & erland, 989, ig. 5) e ionization structure and temperature profile of a cloud with n H = cm 3 as a function of distance from a central source with a typical G continuum. Halpha ] ] g ote that line carries 5% of the total flux! = can be used to estimate bolometric flux! (erland & ushotzky, 98, ig. 5) quilibrium 4 quilibrium 6

7 D D O O D D O O D D G S D X 7 4 G S D X ine Diagnostics, 7 6 T 4 (K) /yα /yα W(yα) temperature power law continuum slope power law continuum slope quilibrium 7 / (Hamann & erland, 999, ig. 5) O/He /He ]/] ]/O] / Dependence of line ratios and cloud equilibrium temperature onto the slope of the irradiating power law. log line ratio B B ine diagnostics makes use of (de)excitation mechanisms for line emission: ollisional xcitation, adiative Deexcitation, ollisional Deexcitation, oefficients for stimulated emission, B, and for radiative excitation (=absorption), B, can generally be ignored ine Diagnostics D D G S D X ine Diagnostics, 7 5 G S D X ollisional (De)xcitation, 7 7 Before performing detailed spectral analysis of G spectra we need to understand how the physical properties of the emitting gas are determined. Density Temperature ass To get a first estimate for these parameters, full blown photoionization computations are not necessary = ine diagnostics. omputation of ij : or excitation, overall upwards rate is given by where σ : collisional cross section f(): electron velocity distribution = n e n = n e n σ ()f()d (7.) σ varies roughly. t can be parameterized as ( h σ () = 8πm e ) (Ω g ) (7.) where Ω is called the collision strength and obtained from quantum mechanics (Seaton, 958): ( ) 8π gfij Ω ij = G(T) (7.3) 3 ij where G(T) is a Gaunt factor. ine Diagnostics ine Diagnostics 3

8 D D O O D D O O D D G S D X ollisional (De)xcitation, 7 8 G S D X ine Diagnostics: Density, 7 3 Using the information from the previous slide and assuming a axwell- Boltzmann distribution for f, the upwards rate is ( ) π 4 / ( ) ( = n e n T / Ω exp ) k B m 3 e g kt (7.4) naloguously, the rate for collisional de-excitation is = n e n σ ()f()d (7.5) ( ) π 4 / ( ) = n e n T / Ω (7.6) k B m 3 e g as for de-excitation the energy threshold is zero. 3/ / 376 O 379 D Density diagnostics: hoose atom with two levels with almost same excitation energy. xcited ions then deexcite either radiatively or collisionally. ine ratio between lines depends on rate of collisional deexcitations and thus is density dependent. or n e cm 3 use [O] 379/376, for higher densities: 4 S 3/ ine Diagnostics 4 ine Diagnostics 6 D D G S D X ollisional (De)xcitation, 7 9 G S D X ine Diagnostics: Density, 7 3 To derive Ω in terms of Ω, make use of microreversibility: n equilibrium, we know that the population densities are given by the Boltzmann distribution: n = g ( exp ) (7.7) n g kt where g, g : statistical weights (g = l + ). But in equilibrium, by definition the upwards and downwards rate are the same: such that and therefore n n = ( )( Ω g g = (7.8) Ω ) exp ( kt ) (7.9) ate equations in equilibrium: such that n n e = = n + n e n 3 n e 3 = = n 3 + n e 3 n n e = n + n n e (7.3) n n e 3 = n n 3 n e 3 (7.3) n e + n e g exp( /kt) g (7.33) n e g 3 3 exp( 3 /kt) 3 + n e 3 g (7.34) ssuming the cloud is optically thin (i.e., absorption is negligable), the intensity of an emitted line is 4π = n hν (7.35) Ω = Ω (7.3) ine Diagnostics 5 ine Diagnostics 7

9 D D O O D D O O D D G S D X ine Diagnostics: Density, 7 3 G S D X ine Diagnostics: Density, 7 34 Using 4π = n hν, the line ratio is since ν ν = n hν /4π 3 n 3 hν 3 /4π insert n /n 3 from qs. (7.33) and (7.34) where the critical densities are defined by (7.36) = n 3 n 3 (7.37) = g 3 + n e 3 exp( 3 /kt) (7.38) 3 g n e = g g n e /n r,3 + n e /n r, exp( 3 /kt) (7.39) n r,i = i / i (7.4) ritical densities for T = 4 K used in G work (Hamann et al., ; Peterson, 997). Transition n cr (cm 3 ) [e ] λ [e ] λ ] λ λ ] λ99.3 λ [ ] λ599 3 ] λ [ ] λ [ ] λ λ ] λ75.9 ] λ λ Transition n cr (cm 3 ) [O ] λ O λ [O ] λ [O ] λ [O ] λ O ] λ O λ O ] λ4. O λ Si ] λ89 3. ine Diagnostics 8 ine Diagnostics D D G S D X ine Diagnostics: Density, 7 33 G S D X ine Diagnostics: Temperature K [O] (379)/(376) O S (379)/(376)..5 K K D To obtain temperature use two levels with different excitation energy and make use of different collisional excitation probabilities for levels at different energies. or T K, mainly O and n e (cm 3 ) ote: Typical temperatures in G are 4 K 3 P ine Diagnostics 9 ine Diagnostics

10 D O D G S D X ine Diagnostics: Temperature 7 36 ((4959)+(57))/(4363) n e = cm 3 n e = 5 cm 3 [O] ((4959)+(57))/(4363) T [K] ( ) = 7.7 exp(3.9 4 /T) n e T / ine Diagnostics 7 36 rnaud,., & othenflug,., 985, &S, 6, 45 erland, G. J., & ushotzky,.., 98, pj, 6, 564 rancis, P. J., Hewett, P.., oltz,. B., haffee,. H., Weymann,. J., & orris, S.., 99, pj, 373, 465 Hamann,., & erland, G., 999, &, 37, 487 Hamann,., Korista, K. T., erland, G. J., Warner,., & Baldwin, J.,, pj, 564, 59 Karzas, W. J., & atter,., 96, pjs, 6, 67 Korista, K. T., & erland, G. J., 989, pj, 343, 678 athews, W. G., & erland, G. J., 987, pj, 33, 456 enzel, D. H., & Pekeris,.., 935, S, 96, 77 ahar, S.., Pradhan,. K., & Zhang, H..,, pjs, 33, 55 Osterbrock, D.., 989, strophysics of gaseous nebulae and active galactic nuclei, (ill alley, : University Science Books) Peterson, B.., 997, n ntroduction to ctive Galactic uclei, (ambridge: ambridge Univ. Press) Seaton,. J., 958, ev. od. Phys., 3, 979 Shull, J.., & an Steenburg,., 98, pjs, 48, 95 erner, D.., erland, G. J., Korista, K. T., & Yakovlev, D. G., 996, pj, 465, 487 erner, D.., & Yakovlev, D. G., 995, &S, 9, 5 Wilms, J., llen,., & cray,.,, pj, 54, 94