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1 AY230-HIITemp
IV. Temperature of HII Regions
A. Motivations
• 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
B. History
• ‘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 theLyman continuum
� Photoionization only occurs if there is an H0 atom in the region
� Ultimately, we require recombination
• Figure
n=1
n=2
n=3
n=!
(a)
(b)
(c)
3kT/2
Cascad
e
h!* > h!0
2 AY230-HIITemp
� Because σrecn (v) ∝ 1/v2, electrons with energy f · 3
2kT 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∗ > hν0
� The heating is “recombination driven” (a) − > (b) − > (c)
� The gas is heated by
∆E = Ef − Ei = (hν∗ − hν0)− f · 3
2kT (1)
� If hν∗ < hν0 + f · 32kT , the gas is cooled!
• Are the electrons in kinematic equilibrium?
� Consider the e−e− relaxation time
� Strong collisions occur whene2
r∼ 1
2mv2 (2)
� The cross-section for such a collision is
σee ∼ πr2 ∼ π
(2e2
mv2
)2
(3)
� Mean free path
λ ∼ 1
σeene
(4)
� Relaxation time
tee ∼λe
ve
∼ m12e ( kT )
32
nee4∼ 0.92
T32
ne
s (5)
� Therefore, the electrons form a Maxwellian distribution
• How about the protons?
tpp ∼(mp
me
) 12
tee ∼ 43tee (6)
• Are the protons and electrons coupled?
� Equipartition time
� Only a fraction of energy (me/mp) is exchanged during e−p collisions
tep ∼ 1836tee (7)
• All of these time-scales are short compared to trec ∼ 105yrs
D. Heating of Hydrogen
• G(T ) :: Heating rate (erg cm3/s)
3 AY230-HIITemp
• Start by ignoring the diffuse radiation
nenpG(T ) = nenp
{∞∑
n=1
< σn(v)v >Max [< hν >∗ −hν0 ]−∞∑
n=1
< σnv1
2mv2 >Max
}(8)
� The latter term is related to f · 32kT
� Define:<E>∗= [< hν >∗ −hν0 ] (9)
• Now consider the diffuse ionizing radiation
� Let’s try an “on-the-spot” assumption
nenpG(T ) =∞∑
n=2
< σnv >Max<E>∗ −∞∑
n=2
< σnv1
2mv2 >Max +{
< σ1v >Max<E>d − < σ1v1
2mv2 >Max
}N But, the On-the-spot approximation implies {} = 0
N And this ignores the fact that some electrons that cascade to the groundstate are ionized by the diffuse radiation field where < hν >d 6=< hν >∗ !
N We need a different approach
• A Self-consistent solution (or close to it):
� Consider the photoionization point of view
N Energy in:nHIΓ∗ <E>∗ +nHIΓd <E>d (10)
N Now we need to solve for Γ∗ and Γd
� Of course, we have (from our on-the-spot approx):
nHIΓ∗ = n2eαB(T ) {Stromgren} (11)
nHIΓd = n2eα1(T ) {on the spot} (12)
� Consider Γ∗ ≡ Photoionization by starlight/s
Fν∗ =πBν4πR
2∗
4πr2e−τν =
Lν
4πr2e−τν (13)
N Assume τν � 1
N Local density of ionizing photons is
nγ =Fν
chν(14)
N Therefore
nHIΓ∗ =
∞∫0
nHILν
4πr2
1
c
dν
hν· c · σph(ν) (15)
4 AY230-HIITemp
N Finally, Lν → Lνe−τν with τν defined as always
� But, we can actually estimate the heating rate without calculating Γ∗ explicitly
N Replace Γ with the recombination rate
nHIΓ∗ <E>∗ +nHIΓd <E>d = n2eαB <E>∗ +n2
eα1 <E>d (16)
N For a highly ionized gas np = ne
N We infer the total heating rate
n2eG(T ) = n2
e
{αB <E>∗ +α1 <E>d − <
∞∑n=1
σn(v)v1
2mv2 >Max
}(17)
N We are left to evaluate <E>∗, <E>d, <∑σnv
12mv2 >Max
• Define <E>
<E>=< hν − hν0 >=total K.E. of ejected electrons
total number of photoionizations(18)
<E>=
nHI
∞∫ν0
(hν − hν0)uνdνhν
σphν
nHI
∞∫ν0
uνdνhν
σphν
(19)
� Calculate <E>∗ for a star (blackbody)
uν ∝ Bν ∝ν3
ehν/kT − 1(20)
N Assume τν = 0 (no attenuation of stellar radiation)
N Approximate
σphν = σ0
(ν
ν0
)−3
(21)
<E>∗=
h∞∫ν0
(1− ν0
ν
)ν3
ehν/kT∗−1dνν3
∞∫ν0
ν3
ehν/kT∗−11ν
dνν3
(22)
N Substitute variables
y ≡ hν
kT∗β∗ =
hν0
kT∗=
158000K
T∗(23)
N Express
<E>∗= kT∗
∞∫β∗
dyey−1
∞∫β∗
dyy(ey−1)
− β∗
≡ kT∗ψ(β∗) (24)
5 AY230-HIITemp
N To evaluate the integrals:
◦ Expand the denominator
1
ey − 1=
e−y
1− e−y= e−y
(1 + e−y + e−2y + . . .
)(25)
=∞∑
n=1
e−ny (26)
◦ Uniformly convergent series can be integrated term by term
ψ(β∗) =
∞∑n=1
∞∫β∗
e−nydy
∞∑n=1
∞∫β∗
e−nydy/y
− β∗ (27)
=
∞∑n=1
1ne−nβ∗
∞∑n=1
E1(nβ∗)− β∗ (28)
◦ Leading terms
ψ(β∗) ≈ 1− 1
β∗+ . . . (29)
Table 1: Evaluation of ψ(β∗)
T∗/104 β∗ ψ(β∗)
10.5 1.5 0.6865.27 3.0 0.8083.16 5.0 0.8680.8 20.0 0.95
� Now calculate <E>d for the diffuse heating
<E>d=
∞∫ν0
(hν − hν0)jphνdσ
ph1νdν
∞∫ν0
jphνdσ
phν1dν
(30)
N Photon emissivity :: jphd
jphd ∝ 1
νe−hν/kT (31)
N σph1 is the photoionization cross-section to n = 1
6 AY230-HIITemp
N Substitute: β ≡ hν0/kT
<E>d= kTβE3(β)− E4(β)
E4(β)≡ kTξ(β) (32)
ξ(β) ≈ 1− 4
β+ . . . (β > 1) (33)
� Finally, the loss of energy by electron recombination
∞∑n=1
< σn(v)v1
2mv2 >Max=
2meL32
√π
∑n=1
∞∫0
σn(v)v5e−Lv2
dv (34)
N L = m/2kT
N f(v) = (4/√π)L
32v2e−Lv2
N Define β = hν0/kT , A = (25/33/2)α3πa2B
∞∑n=1
< σn(v)v1
2mv2 >Max=
mA√πL3
β∞∑
n=1
β
n3
{1− β
n2eβ/n2
E1
(β
n2
) }(35)
N The sum is written as χ(β)
N Using the E-M sum rule: (e.g. BGP63)
χ(β) ≈ 1
2
[0.735 + ln β +
1
(3β)
](36)
Table 2: Evaluation of χ(β)
T/104 β χ(β)
15.8 1 0.537.9 2 0.803.16 5 1.211.00 10.5 1.560.8 20 1.870.31 50 2.32
• Altogether now!
G(T ) = αB <E>∗ +α1 <E>d −∞∑
n=1
< σn(v)v1
2mv2 >Max (erg cm3/s) (37)
= αB(T )kT∗ψ
(hν0
kT∗
)+ α1(T )kTξ
(hν0
kT
)− Ame√
πL3βχ
(hν0
kT
)(38)
7 AY230-HIITemp
� Approximations of the recombination coefficients
αB ≈ 2.6010−13
(T
104
)−0.8cm3
s(39)
α1 ≈ 1.5710−13
(T
104
)−0.525cm3
s(40)
� And the integrals
ψ(β∗) ≈ 1− 1
β∗(41)
ξ(β) ≈ 1− 4/β (42)
χ(β) ≈ 1
2[ 0.735 ln β − 1/3β ] (43)
� Figure
1 2 3 4 5 6THII (104 K)
−4
−2
0
2
4
6
8
10
G(T
) (1
0−25
erg
s cm
3 /s) T* = 10,000K
T* = 30,000KT* = 100,000K
E. Cooling: Collisional Excitation of Hydrogen
• Key excitation process1 2S → 2 2P (44)
� Followed by emission of Lyα photon with λ = 1215.67A(10.2eV)
� Dominant coolant for gas with T ∼ 104−5 K
• Also1 2S → 2 2S (45)
� Emission of 2 photons
� A2 1S,1 2S = 8.23 s−1 (vs. ALyα ≈ 109 s−1)
� hν ′ + hν ′′ = 10.2 eV
• Cross-section
8 AY230-HIITemp
� σexHI is not simply proportional to v−2
� It is complicated, in particular, by ‘wiggles’ due to resonances
� Fig
• Collision strengths: See Table 3.12 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 = Number density of element m in kth ionization state
� f(m)k,j = Fraction of the ion in level j
• Cooling from element m in kth ionization state from level j → i
L(m)k,ij = nen
(m)k
[f
(m)ki qij − f
(m)kj qji
](46)
• Total cooling rate
Lline(T ) =∑m
∑k
∑j>i
L(m)k,ij (47)
• We will return to line emission from heavy elements as a means to probe the physicalconditions within HII regions
G. Cooling: Brehmstrahlung (a.k.a. Free-free emission)
• For a (nearly) full quantum treatment, see
� http://www.ucolick.org/∼xavier/AY204b/Lectures/ay204b phtrec.ps
9 AY230-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
I =4
π√
3gff (ν)e
−hν/kT (48)
� gff is the Gaunt factor
� gff = 1 for most frequencies
gff =
{√3π
(ln 4kT
hν− 0.577
)hν � kT
1 hν � kT(49)
� Finally,
jffν =
∑i
nine
(2me
3πkT
) 12[
32π2Z2i e
6
3m2ec
3
]¯gff (ν)e
−hν/kT (50)
• Radio Frequency Brehmstrahlung (i.e. HII Regions)
� The log term in jν introduces a weak, power-law frequency dependence
jν = 6.51 × 10−38( ∑
neniZ2i
)T−0.35ν−0.1 (51)
N For fully ionized H, He the sum is ≈ 1.4n2e
� Contrast with X-ray Brehmstrahlung
jν = 7.6 × 10−31n2eT
− 12 e−hν/kTgx (52)
gx =
{(0.551 + 0.682x) ln(2.25/x) x ≤ 1
x−0.4 x ≥ 1(53)
x ≡ hν/kT (54)
� Total Bolometric free-free emission
εff = 4π
∫jffν dν (55)
= 2.4 × 10−27T12n2
e (56)
N HII region with T ≈ 104 K
εff ≈ 2 × 10−25n2e (57)
10 AY230-HIITemp
N Compare with H line emission (or even [OIII])
εH ≈ 10−24n2e � εff (58)
• Diagnosing HII Regions
� Specific flux from a thermal radio source at distance R
Fν =1
4πR2
∫4πjνdV (59)
=2.5 × 103
R2pc ν
0.1GHz
∫n2
edVpc
T 0.35Jy (60)
� Observe the radio flux to infer:
(a)∫n2
edV =<n2e> if T is known (or assumed)
(b) Ionizing photon density
φ =
∫n2
eαB(T )dV ≈ αB(T )
∫n2
edV (61)
∝ αB(T )FνR2T 0.35 (62)
(c) Interstellar reddening: Ratio of free-free flux to Hβ is independent of R2 and<n2
e> and insensitive to T
H. Temperature Profile
• Equate heating with cooling
G(T ) = LR(T ) + Lline(T ) + Lff (63)
• Balance and solve for T
• Approximate value
� x = 0.9, solar abundances
� Figure
� Heating curve is G(T )− LR(T )
� Free-free, collision excitation of HI is small
� Line radiation
N Peaks when kT ≈ χ
N Decreases slowly for kT > χ
� Equilibrium T is where the combined curve (not labeled) crosses the heatingcurve
11 AY230-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 42000=10**4.6232 K radius 10**11.92cm
blackbody 4.6232, radius=11.92
c gas density in log of number density of all protons
hden = 2
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 AY230-HIITemp
N Red curve: Solar metallicity
N Blue curve: Zero metallicity
• Discussion
� We note T ↑ as r ↑� This is because σph
ν ∝ ν−3
� Therefore the radiation field becomes harder as it propogates out
� This leads to more efficient heating (similiar to a hotter star)
15 16 17 18 19log depth=(r − 1017)
2.5
3.0
3.5
4.0
4.5
log
Te
15 16 17 18 1910−8
10−6
10−4
10−2
100
(1−x)