Lec 3. HII Regions: Radiative Processes

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1. Photoionization2. Radiative Recombination3. Photoionization Equilibrium4. Strömgren Models5. The Role of He in HII RegionsAppendix: Introduction to Thermal Properties

Basic Strömgren model (1939): spherical HII region slowly expanding into uniform HI

References: Spitzer; Chs. 3 & 5 Osterbrock, Ch. 3

NGC 3603 Rosette

North American Shapley’s Planetary Nebula

Real HII regions look nothing like this, in part because they form in clusters. The simple model helps Illustrate basic physical processes.

1. Photoionization

Absorption cross section per H nucleus for IS matter (average over abundances)

X-raysFUVISM is quite opaque at 911Å(the H Lyman edge) & partially transparent in the FUV and X-rays (c.f. heavy elements).

Sources of ionizing photons:1. Massive young stars,

e.g.T(O5V ) ≈ 46,000 KBB peaks at hν ≈ 3 kT ~ 12 eV Q ≈ 1049.53 γ s-1 hν > 13.6 eV

2. Hot white dwarfs & PN star 3. SNR shocks

Hot Star Atmospheres

State-of-the-art model atmospheres disagree with blackbody approximation, especially above 13.6 eV.

Photoionization Cross Sectionfor Hydrogenic ions, charge Z

where hν1 = 13.6 Z2 eV & gbf ≤ 1 is the QM Gaunt factor for bound-free transitions from n = 1.

Kramers’ semi-classical approximation s:

Mean free path at 912 Å

σν =7.91×10−18

Z 2ν1

ν⎛ ⎝ ⎜

⎞ ⎠ ⎟

3

gbf cm2

λ =1

nHIσν

=1.58 ×1017

nHI

cm2 =0.051

nHI

pc

σν = σν 1

ν1

ν⎛ ⎝ ⎜

⎞ ⎠ ⎟

3

and σν =6.33 ×10−18

Z 2 cm2

ν

σ

Photoionization Cross Sections for H, He, & He+

Ionization potentialsH 13.6 eV (912Å)He 24.6 eV (504Å)He+ 54.4 eV (228Å)

Very hot stars are needed to ionize He+ (T* > 50,000 K), and He++

does not occur in H II regions except in planetary nebulae & AGN (also fast shocks).

Continuum radiation will become harder” with increasing depthbecause of the rapid decreaseof the cross section with ν.

At high frequencies, He has a larger cross section than H that more than compensates forits lower abundance (~10%).

Photoionization RatePhotoionization rate per H atom

The mean intensity Jν enters, as expected fromthe kinetic theory formulawhere nπ is the number density of ionizing photons,

N.B. π is used here a label for ionizing photons, not pions, etc.

ζ π =4π Jν

hνσν dν

ν 1

∞

∫

ζ π ≈ nπσν 1c

nπ =1c

4π Jν

hνdν

ν 1

∞

∫ =4π Jν 1

hcI1

In =Jν

Jν 1

ν1

ν⎛ ⎝ ⎜

⎞ ⎠ ⎟

n dννν 1

∞

∫

Photoionization Rate (cont’d)I4 / I1 depends on the spectrum; a typical value is ≈ 1/2

For example, 1 pc from an O5 star

The ionization time tπ = 1/ζπ = 1.1 x 106 s ~ two weeks

N.B. γ is used here as a symbol for photons, not specifically γ-rays

nπ =Nπ

•

4π R2c=

1049.53s−1

4π (3×1018cm)2 3×1010cms−1 = 9.46γ cm−3

ζ π =1.79 ×10−6 × 12 = 8.96 ×10−7 s−1

2. Radiative Recombination

e-

γγ

γH+ + e → H + hν

is the inverse of recombination.Milne calculated the cross sectionfrom Kramer’s photoionizationcross section using detailed balance.

The (small) cross section for capture to level n is:

σ fb (w) = 2n 2 hνmecw

⎛

⎝ ⎜

⎞

⎠ ⎟

2

σ bf (v) =hν1

mec2

hν112 mew

2ν1

νσ1

n3 ~ 3×10−21T4−1 cm2

Note the inverse-square dependence on electron speed w.

Radiative Recombination Rates

Recombination rate coefficient αn (cm3s-1) to level n

Recombination summed over all higher levels to n

φ = φ(hν/kT) is tabulated by Spitzer (p. 107). The values

for n = 1 & 2 at 8000K are 2.09 & 1.34, respectively, so that

α(1) = 5x10-13 & α(2) = 3x10-13 cm3s-1

neni σ recw = neniαn [cm3s−1]

α (n ) = αmm= n

∞

∑ = 2.06 ×10−11Z 2T−1/ 2φ (n )(β) cm3s−1

On The Spot Approximation for HThe rate coefficient α(1) includes recombination to the ground state, but that produces another ionizing photon that is easily absorbed locally at high density, as if the recombination had not occurred.

In this on the spot approximation, the effective recombination rate omits recombination to the ground state

α (2) = αmm= 2

∞

∑ ≈ 2.60 ×10−13 Z 2T4

−0.8

cm3s−1

The recombination time is

trec =1

neα(2) ≈1.22 ×105 ne

−1T40.8 yr

If τrec >> τπ, the gas near on O5 star, for example, will be highly ionized.

3. Photoionization Equilibrium for HSince the ISM is not in LTE, the Saha equation is not valid. Instead, steady-state balance applies if the time scales for changes in particle density n & photon density nπ are much longer than trec. Photoionization equilibrium is defined so that the rate of ionization ζπ out of i-1 state = rate of recombination of into state i.

ζ π ni−1 = α neni

ni /ni−1 = ζ π /αne = trec / tπ

α = α(1) or α(2)

depending on optical depth

For H, assume α = α(2) and nπ = nπ (hv > 13.6eV)

xe = ne / nH ≤ 1.2 for 10% He abundance

nH +

nH 0

=nπσν 1

α (2)xenH

(I4 /I1)

Ionization ParameterThe last formula for the H+/H ratio depends on the ratio ofof photon density (i.e., strength of the radiation field) to the particle density. This is a general property of photoionizationequilibrium, where an external ionization rate (1-body process) is balanced against recombination (2-body process).

It is helpful to recognize this fact by formally introducing the ionization parameter U ≡ nπ / nH and re-express the previous result as

nH +

nH 0

=U

UHUH =

α (2)xe

σν 1c (I4 /I1)

=1.37 ×10−6 xe

T40.7(I4 /I1)

A typical value in an HII region is U ~ 10-2.5 >> UH, i.e., HII regions are fully ionized.

4. The Strömgren SphereThe size of the ionized volume is determined by balance between ionization and recombination. If a star emits SH = 1 x 1049 SH49 hydrogen ionizing photons per sec,what is the radius of the ionized zone (xe=1) density n?

(assuming T=7000 K). RSt is the Strömgren radius & the ionized volume is the Strömgren sphere. As mentioned at the start, HII regions are not spherical. Rather, the dynamics determine n.

SH =4π3

RSt3 ne

2α (2)

RSt =3

4πSH

xen2α (2)

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

≈ 61.7 SH 49

n2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 3

pc

Properties of Strömgren Spheres

Osterbrock p. 22

Characteristics of Strömgren SpheresIonization parameter

Average column

(small, dense HII regions have large columns, e.g, UC HII regions)

nH +

nH 0

=USt

UH

=13

σν 1nRSt (I4 /I1) =

14

σν 1N (I4 /I1)

USt =nπ

n=

α H(2)xe

3cnRSt ≈10−2.5 S49 n2( )1/ 3

N =43 πnRSt

3

πRSt2 =

43

nRSt ≈1.18 ×1021 S49n2( )1/ 3 cm−2

τν 1= σν 1

N = 6.33×10−18 ×1.1×1021 S49n2( )1/ 3

= 750 S49n2( )1/ 3

Ionization parameter in terms of column

Optical depth at the Lyman edge if the HII region were neutral

(τ >> 1 at the Lyman edge)

Transition from H+ to H

How thick is the region in which x(H) goes from 0 to 1?

Roughly the distance for an ionizing photon to be absorbed:

Neglect hardening of the spectrum and taking the average density as 0.5 n,

n(H0)

R

∆R

τν 1= ∆Rn

H 0σν 1=1

∆RRSt

=1

12 n

H 0σν 1RSt

=1

38 σν 1

N =

23

UH

USt

I4

I1

≈ 3.5 ×10−4 S49n2( )−1/ 3

5. The Role of He in HII RegionsMoving away from pure H models, special considerations are required for He:

1. It is sensitive to high- radiation, i.e., to spectral type.2. The radiation that ionizes He also ionizes H (and isabsorbed by both H and He)3. Unlike the case of H, the electron density is not determined by He alone, but as

ne = n(H+) + n(He+) + n(He++)

4. He recombination radiation can ionize H.

Therefore the photoionization of He and H are coupled problems.

Ionization of He by OB Stars

Example 1: B0 star, Teff ≈ 30,000K– Spectrum peaks at ~13.6 eV

• Many photons 13.6 eV < hv < 24.6 eV• Few photons with hv > 24.6 eV

– Two Strömgren spheres• Small central He+ zone surrounded by large

H+ region

Example 2: O6 star, Teff ≈ 40,000 K– Spectrum peaks at ≥ 24.6 eV

• Lots of photons with hv > 24.6 eV– Single Strömgren sphere

• H+ and He+ zones coincide

He+ Zones in Model H II RegionsOsterbrock Figures 2.4 & 2.5

He

He

Fractional ionization vs. fractional radius He+/H+ radius ratio vs. Teff

Introduction to Thermal PropertiesThe basic hydrodynamic equations state the conservation laws for continuum problems.

After mass and momentum conservation,the equation for the entropy is the mostImportant, and is known as the heat →equation.

The volumetric rates of gain and loss of The entropy per unit volume are called the heating rate Γ and the cooling rate Λ.

In the steady state, these two rates must balance & solving Γ = Λ yields the temperature T.

Line Cooling Function

Λ can be expressed in terms of the net rate at which energy is lost in collisions. For HII regions, electrons are the most important collision partners. The collisions are described by upward and downward rate coefficients that are related by detailed balance.

NB Instead of j and k for lowerAnd upper level, we use here u and l

Steady Cooling & Escape Probability

By analyzing the steady rateequations for the level population and introducing the escape probability, the cooling can be expressed in a compact form that resembles the emissivity of the transition.

Two-Level SystemThis problem can be solved exactly once one makes a critical but very useful approximation in the radiationtransfer, i.e., uses the idea ofescape probability.

The closed form for the population of the upper level then leads to a simple form for the cooling function. Both involve an important new quantity, the critical density:

Ncr = Aul/kul,where kul is the collisional rate coefficient for the downward transition.

High densities (n >> ncr) give the TE result, whereas low densities (n<< ncr) give a characteristic quadratic density cooling.