(c) (a) 3kT/2. Cascade

Transcription

1 1 AY30-HIITemp IV. Temperature of HII Regions A. Motivations B. History In star-forming galaxies, most of the heating + cooling occurs within HII regions Heating occurs via the UV photons from O and B stars Cooling occurs via dust and line-emission Epochal series of papers: Spitzer 1948, 1949ab, 1954 Summary: Burbidge, Gould, & Pottasch 1963 Ostriker, Chapter 3 C. Heating: Basics Heating occurs by photoionization of incident (stellar) radiation in the Lyman continuum Photoionization only occurs if there is an H 0 atom in the region Ultimately, we require recombination (c) n=! (a) 3kT/ n=3 n= Cascade h!* > h!0 Figure (b)

2 AY30-HIITemp Because σn rec (v) 1/v, electrons with energy f 3 kt are more likely to recombine (f 0.5). This releases a cascade of photons. The neutral HI atom is ionized by a hard photon hν > h The heating is recombination driven (a) > (b) > (c) The gas is heated by If hν < h + f 3 kt, the gas is cooled! Are the electrons in kinematic equilibrium? E = E f E i = (hν h ) f 3 kt (1) Consider the e e relaxation time Strong collisions occur when e r 1 mv () The cross-section for such a collision is ( ) e σ ee πr π (3) mv Mean free path Relaxation time λ 1 σ ee n e (4) t ee λ e m 1 e ( kt ) T 3 s (5) v e n e e 4 n e Therefore, the electrons form a Maxwellian distribution How about the protons? t pp ( mp m e Are the protons and electrons coupled? ) 1 tee 43t ee (6) Equipartition time Only a fraction of energy (m e /m p ) is exchanged during e p collisions All of these time-scales are short compared to t rec 10 5 yrs D. Heating of Hydrogen G(T ) :: Heating rate (erg cm 3 /s) t ep 1836t ee (7)

3 3 AY30-HIITemp Start by ignoring the diffuse radiation { n e n p G(T ) = n e n p < σ n (v)v > Max [ < hν > h ] } < σ n v 1 mv > Max The latter term is related to f 3 kt Define: <E> = [ < hν > h ] (9) Now consider the diffuse ionizing radiation Let s try an on-the-spot assumption n e n p G(T ) = < σ n v > Max <E> < σ n v 1 mv > Max + n= n= { < σ 1 v > Max <E> d < σ 1 v 1 mv > Max } (8) But, the On-the-spot approximation implies {} = 0 And this ignores the fact that some electrons that cascade to the ground state are ionized by the diffuse radiation field where < hν > d < hν >! We need a different approach A Self-consistent solution (or close to it): Consider the photoionization point of view Energy in: n HI Γ <E> +n HI Γ d <E> d (10) Now we need to solve for Γ and Γ d Of course, we have (from our on-the-spot approx): Consider Γ Photoionization by starlight/s Assume τ ν 1 Local density of ionizing photons is n HI Γ = n eα B (T ) {Stromgren} (11) n HI Γ d = n eα 1 (T ) {on the spot} (1) F ν = πb ν4πr 4πr e τν = L ν 4πr e τν (13) Therefore n HI Γ = 0 n γ = F ν chν n HI L ν (14) 1 dν 4πr c hν c σph (ν) (15)

4 4 AY30-HIITemp Finally, L ν L ν e τν with τ ν defined as always But, we can actually estimate the heating rate without calculating Γ explicitly Replace Γ with the recombination rate n HI Γ <E> +n HI Γ d <E> d = n eα B <E> +n eα 1 <E> d (16) For a highly ionized gas n p = n e We infer the total heating rate { Define <E> n eg(t ) = n e α B <E> +α 1 <E> d < } σ n (v)v 1 mv > Max We are left to evaluate <E>, <E> d, < σ n v 1 mv > Max <E>=< hν h >= <E>= Calculate <E> for a star (blackbody) total K.E. of ejected electrons total number of photoionizations n HI (hν h ) uνdν hν n HI u ν B ν u νdν hν σph ν ν 3 e hν/kt 1 σph ν Assume τ ν = 0 (no attenuation of stellar radiation) Approximate ( ) 3 ν = σ 0 σ ph ν (17) (18) (19) (0) (1) Substitute variables Express <E> = ( h 1 ) ν 3 dν ν e hν/kt 1 ν 3 ν3 1 e hν/kt 1 ν () dν ν 3 y hν kt β = h kt = K T (3) <E> = kt β β dy e y 1 dy y(e y 1) β kt ψ(β ) (4)

5 5 AY30-HIITemp To evaluate the integrals: Expand the denominator 1 e y 1 = e y 1 e = ( y e y 1 + e y + e y +... ) (5) = e ny (6) Uniformly convergent series can be integrated term by term ψ(β ) = = e ny dy β β (7) e ny dy/y β 1 n e nβ β (8) E 1 (nβ ) Leading terms ψ(β ) 1 1 β +... (9) Table 1: Evaluation of ψ(β ) T /10 4 β ψ(β ) Now calculate <E> d for the diffuse heating (hν h )j ph νd σph ν <E> d = 0 ν1dν j ph νd σph 1ν dν (30) Photon emissivity :: j ph d j ph d 1 ν e hν/kt (31) σ ph 1 is the photoionization cross-section to n = 1

6 6 AY30-HIITemp Substitute: β h /kt <E> d = kt β E 3(β) E 4 (β) E 4 (β) kt ξ(β) (3) ξ(β) 1 4 β +... (β > 1) (33) Finally, the loss of energy by electron recombination < σ n (v)v 1 mv > Max = m el 3 π 0 σ n (v)v 5 e Lv dv (34) L = m/kt f(v) = (4/ π)l 3 v e Lv Define β = h /kt, A = ( 5 /3 3/ )α 3 πa B < σ n (v)v 1 mv > Max = ma β { β 1 βn ( ) } βn πl 3 n 3 eβ/n E 1 The sum is written as χ(β) Using the E-M sum rule: (e.g. BGP63) χ(β) 1 [ ln β + 1 ] (3β) (35) (36) Table : Evaluation of χ(β) T/10 4 β χ(β) Altogether now! G(T ) = α B <E> +α 1 <E> d < σ n (v)v 1 mv > Max (erg cm 3 /s) (37) ( ) hν0 = α B (T )kt ψ + α 1 (T )kt ξ kt ( hν0 kt ) Am ( ) e βχ hν0 πl 3 kt (38)

7 7 AY30-HIITemp Approximations of the recombination coefficients And the integrals α B ( T 10 4 ) 0.8 cm 3 ( ) 0.55 T α cm s s (39) (40) ψ(β ) 1 1 β (41) ξ(β) 1 4/β (4) χ(β) 1 [ ln β 1/3β ] (43) Figure 10 G(T) (10 5 ergs cm 3 /s) T * = 100,000K T * = 30,000K T * = 10,000K T HII (10 4 K) E. Cooling: Collisional Excitation of Hydrogen Key excitation process Also 1 S P (44) Followed by emission of Lyα photon with λ = Å(10.eV) Dominant coolant for gas with T K Emission of photons A 1 S,1 S = 8.3 s 1 (vs. A Lyα 10 9 s 1 ) hν + hν = 10. ev Cross-section 1 S S (45)

8 8 AY30-HIITemp σhi ex is not simply proportional to v It is complicated, in particular, by wiggles due to resonances Fig Collision strengths: See Table 3.1 in Osterbrock Amazingly, accurate cross-sections are not available for n > 3 Line emissivity (see RadProc notes) F. Cooling: Collisional Excitation of Heavy Elements Definitions n (m) k f (m) k,j = Number density of element m in k th ionization state = Fraction of the ion in level j Cooling from element m in k th ionization state from level j i [ ] L (m) k,ij = n en (m) k f (m) ki q ij f (m) kj q ji (46) Total cooling rate L line (T ) = m L (m) k,ij (47) We will return to line emission from heavy elements as a means to probe the physical conditions within HII regions G. Cooling: Brehmstrahlung (a.k.a. Free-free emission) For a (nearly) full quantum treatment, see xavier/ay04b/lectures/ay04b phtrec.ps k j>i

9 9 AY30-HIITemp A semi-classical, heuristic treatment is given in the RadProc notes Astrophysics Dominant cooling process at high T (e.g. cluster gas) Means of diagnosing the T in HII regions Quantum result ḡ ff is the Gaunt factor ḡ ff = 1 for most frequencies Finally, ḡ ff = I = 4 π 3ḡff(ν)e hν/kt (48) { ( 3 π ln 4kT ) hν kt hν 1 hν kt (49) j ff ν = i n i n e ( me 3πkT ) 1 [ 3π Z i e 6 3m ec 3 ] g ff (ν)e hν/kt (50) Radio Frequency Brehmstrahlung (i.e. HII Regions) The log term in j ν introduces a weak, power-law frequency dependence ( ) j ν = ne n i Zi T 0.35 ν 0.1 (51) For fully ionized H, He the sum is 1.4n e Contrast with X-ray Brehmstrahlung Total Bolometric free-free emission ε ff = 4π HII region with T 10 4 K j ν = n et 1 e hν/kt g x (5) g x = { ( x) ln(.5/x) x 1 x 0.4 x 1 (53) x hν/kt (54) j ff ν dν (55) = T 1 n e (56) ε ff 10 5 n e (57)

10 10 AY30-HIITemp Compare with H line emission (or even [OIII]) Diagnosing HII Regions ε H 10 4 n e ε ff (58) Specific flux from a thermal radio source at distance R F ν = 1 4πj 4πR ν dv (59) = R pc.1 GHz n e dv pc T 0.35 Jy (60) Observe the radio flux to infer: (a) n edv =<n e> if T is known (or assumed) (b) Ionizing photon density φ = n eα B (T )dv α B (T ) n edv (61) α B (T )F ν R T 0.35 (6) (c) Interstellar reddening: Ratio of free-free flux to Hβ is independent of R and <n e> and insensitive to T H. Temperature Profile Equate heating with cooling Balance and solve for T Approximate value G(T ) = L R (T ) + L line (T ) + L ff (63) x = 0.9, solar abundances Figure Heating curve is G(T ) L R (T ) Free-free, collision excitation of HI is small Line radiation Peaks when kt χ Decreases slowly for kt > χ Equilibrium T is where the combined curve (not labeled) crosses the heating curve

11 11 AY30-HIITemp Temperature profile Need to iteratively solve for ionization balance and the T together And perform Radiative Transfer! CLOUDY: Gary Ferland and Associates CLOUDY Example Input file c hii_typical.in title typical HII region sphere c 05 star with temperature 4000=10**4.63 K radius 10**11.9cm blackbody 4.63, radius=11.9 c gas density in log of number density of all protons hden = c log of starting radius r_cm radius 17 stop temperature 3 plot continuum range 0.1 iterations = 3 print last iteration punch overview last file= hii_typical.ovr Solution

12 1 AY30-HIITemp Discussion Red curve: Solar metallicity Blue curve: Zero metallicity We note T as r This is because σν ph ν 3 Therefore the radiation field becomes harder as it propogates out This leads to more efficient heating (similiar to a hotter star) log T e (1 x) log depth=(r ) 10 8