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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 the Lyman continuum Photoionization only occurs if there is an H 0 atom in the region Ultimately, we require recombination Figure n=1 n=2 n=3 n=! (a) (b) (c) 3kT/2 Cascade h! * > h! 0

IV. Temperature of HII Regions - Lick Observatoryxavier/AY230/ay230_HIItemp.pdfIV. Temperature of HII Regions A. Motivations • In star-forming galaxies, most of the heating + cooling

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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 =

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

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

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

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)