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Lec 3. Radiative Processes and HII Regions

¨

1. Photoionization2. Recombination3. Photoionization-Recombination Equilibrium4. Heating & Cooling of HII Regions5. Strömgren Theory (for Hydrogen)6. The Role of Helium

References Spitzer Secs. 5.1 & 6.1Tielens Ch. 7Dopita & Sutherland Ch.9Osterbrock & Ferland, Ch. 3

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Introductory Summary

The far ultraviolet radiation (FUV) from an O or B star Ionizes its immediate neighborhood and produces an HII region. Strömgren developed the theory in 1939 for a spherical model where the HII region slowly expands into uniform HI.

HII regions illustrate basic processes that operate in allphotoionized regions of the ISM: super-Lyman radiation( < 916.6 Å - FUV or X-rays) photoionize sH:

h + H H+ + e.The H+ ions recombine radiatively

H+ + e H + hv.• The balance between these two reactions determinesthe ionization fraction.• Any excess photon energy above the ionization potential(IP = 13.6 eV) is given to the ejected electron and is thenequilibrated in collisions with ambient electrons, therebyheating the HII region.

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NGC 3603 Rosette

North American Shapley’s Planetary Nebula

Real HII regions are rarely spherical. Nonetheless, Strömgren’stheory Illustrates the basic roles of photoionization and recombination.

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1. Photoionization

Absorption cross section per H nucleus(1-2000 Å, averaged over abundances)

X-raysFUVThe ISM is opaque at 911 Å(the H Lyman edge) & partially transparent in the FUV and inthe X-ray band above 1 keV.

Ionizing photons come from1. Massive young stars*2. Hot white dwarfs3. Planetary nebula stars4. SNR shocks* Table 2.3 of Osterbrock &Ferland give T(O5V ) 46,000 K(BB peak at 3kT or 12 eV) & 3x1049 ionizing photons/s

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Hot Stars as FUV Sources

Blackbody (smooth curve)is a poor approximation inthe FUV and EUV.

State of the art modelatmospheres disagree.

See Lejuene et al.,A&AS 125, 229 1997and NASA ADS:1997yCat..41250229L,for an extensive libraryof stellar energy distributions.

Note that the spectrum of this very hot star cuts off beyond 50 eV.

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Photoionization of Hydrogenic Ions

where h 1 = 13.6 Z2 eV and gbf 1 is the QM Gaunt factor

for bound-free transitions from n = 1. Compare this withKramers’ semi-classical formula:

Both give an inverse-cube dependence on frequency.The mean free path at 912 Å is very small:

=7.91 10 18

Z 21

3

gbf cm2

=1

1

3

with 1 =6.33 10 18

Z 2 cm2

lmfp ( 1) =1

nHI 1

=1.58 1017

nHIcm2

Quantum theory predicts

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Photoionization Cross Section of He

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

• At high frequencies, the largerHe cross section more thancompensates for its lowerabundance relative to H.

• Very hot stars are needed to ionize He+ (T* > 50,000 K). He++ does not occur in H II regions except in planetary nebulae, AGN and in fast shocks.

• Continuum radiation gets “harder” with increasing depthdue to the rapid decreaseof the cross section with .

NB For analytic fits for heavy atomphotoionization cross sections, seeVerner et al., ApJ, 465, 487 1996

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Calculation of the Photoionization Rate

The photoionization rate for one H atom is

where the mean intensity J , photon number, and energydensity are derived from the specific intensity by

= dv 4 J

h1

= c dv n1

n =1

c

4 J

hd

1

=4 J

1

hcI1,

u =1

cd I

4

cJ = n h

The total photon number density,

InJ

J1

1

nd

11

where I1 is the first inverse moment of J , defined by

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Calculation of the Photoionization Rate (cont’d)

n 1c (I4I 1

)

=4 J

hd

1

=4 J

1

h

J

J1

1

1

1

1

3d

1

=4 J

1

h 1I4

Because the cross section varies as -3, the ionizationrate can also be expressed in terms of these moments.

Replacing the mean intensity by the photon number, this is

The moments depend on the spectrum. A typical ratio appropriate for HII regions is I4/I1 .

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Numerical Estimate of the Photoionization Rate

NB In the 2nd line, the speed of light is given in m/s, whereas cgs units are used everywhere else, especially in the 1st line.

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2. Radiative Recombination

The cross section for capture to level n is

e-

H+ + e H + hRadiative recombination is the inverseof photoionization. Milne calculated thecross section from Kramer’s crosssection using detailed balance

(Spitzer Eq. 5-9, Rybicki & Lightman Sec.10.5)

It depends on electron speed as w-2. The rate-coefficient,

n = <w (w)>th, is small and decreases as 1/T, e.g., thetotal rate coefficient (next slide), summed over n at 10,000Kis (104K) ~ 4x10-13 cm3 s-1.

fb (w) = 2n2h

mecw

2

bf (v) =h 1

mec2

h 112mew

21 1

n3

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Rate Coefficient for Radiative Recombination

n(h /kT) is a slowly varying function of T, introduced

and tabulated by Spitzer (p. 107). The values for n = 1

& 2 at 8000K are 2.09 & 1.34, respectively, so that(1)= 5x10-13

and (2) = 3x10-13 cm3s-1

neni n (w)w = neni n

(n )= m

m= n

= 2.06 10 11Z 2T 1/ 2n ( ) cm3s 1

The rate coefficient (n) (units, cm3s-1) gives the rate of

direct recombinations per unit volume to level n; ni isthe ion density (in this case H+),

Summing the rates for all levels m n, gives the totalrecombination rate coefficient to level n:

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On The Spot Approximation for H

The rate coefficient (1) includes recombination to theground state, but that process produces another ionizingphoton that is easily absorbed locally at high density, as ifthe recombination had not occurred.

In this on the spot approximation, the effective recombination

rate omits recombination to the ground state

The corresponding recombination time is

Compared with the ionization time (slide 10), rec >> inmost cases, and the gas around hot stars is highly ionized.

(2)= m

m= 2

2.60 10 13Z 2T40.8

cm3s 1

trec =1

ne(2) 3.85 1012ne

1T40.8 sec

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3. Photoionization Equilibrium for H

However, the time scales for dynamical changes forgas and star are much longer than trec and tion.Thus theionization fraction is in a quasi-steady-state calledphotoionization equilibrium where the rate of ionization

out of ionization state i-1 = rate of recombination into

state i:

= (1) or (2), dependingon optical depth

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

ni 1 = nenini /ni 1 = / ne = trec / t

nH+

nH0

= (2)ne

=n c 1 I4 /I1

(2)xenH

The ISM is not in LTEThe Saha Equation (chemical equilibrium) is not valid.

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Ionization Parameter

The H+/H ratio depends on the ratio of photon density(strength of the radiation field) to the particle density.This fact is a general property of external energysources and is recognized by the ionization parameterU n / nH. The H+/H ratio can now be rewritten as

A typical value in an HII region is U ~ 10-2.5 >> UH

Since xe = ne / nH 1.2 (<10% He), hydrogen in HIIregions is fully ionized.

nH +

nH 0

=U

UH

UH

(2)xe

1c (I4 /I1)

=1.37 10 6 xeT40.7(I4 /I1)

Similar equations apply to all elements using appropriate values of n and UZ (replacing UH).

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4. Temperature of Photoionized Gas

This section follows closely the treatment of the heatingand cooling of HII regions given in Spitzer Sec. 6.1

The one important heating mechanism involves thedissipation of the excess energy of the photoelectrons(generated by the absorption of stellar UV photons) inCoulomb collisions with ambient electrons:

Ee= hv - hv1 ~ kT

The mean energy of the photoelectrons is

where Jv is the mean intensity of the radiation field.

E 2 =

h( 1)4 J

hd

1

4 J

hd

1

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Magnitude of the Photoelectric Heating

0 - value near star, ‹ 0› is averaged over an HII region; the 1st

decreases and the 2nd increases with T*, as does the ratio.

1.6551.3801.1991.1011.051‹ 0›

0.7750.8640.9220.9590.9770

64,00032,00016,00080004000Te(K)

Spitzer Sec. 6.1, expresses the mean photoelectronenergy in terms of the stellar effective temperature

With nH the photoionization rate per unit volume,the volumetric heating rate is

Spitzer, Table 6.1

= E 2 /kT*

= nH kT

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Hardening of the Ionizing Radiation

The mean photoelectron energy increases as the softerphotons get preferentially absorbed. This is why theentries of the previous table, and especially the ratio‹ 0›/ 0 ,,increase with stellar effective temperature.

Jv

1

Near star

Jv

“Low” energy photonsabsorbed first

The spectrum hardens because of -3 absorption close tothe star.

Into the HI region

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Recombination Cooling

In HII regions, radiation from recombination provides a

minimum amount of cooling: each recombination drains

thermal energy mew2 from the gas. The total cooling

rate per ion is

The recombination cross section varies as the inverse

square of the speed w. Therefore the rate of cooling

by recombination is determined by the thermal average

of w3xw-2= w, or T1/2 . This is borne out roughly by the

exact calculations described by Spitzer:

1

2m w3

j

j= k

Radiation is the main way that interstellar matter cools.

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Recombination Cooling Rate

= recombination function (Spitzer Table 5-2, p. 106

= energy gain function - which come from Spitzer’s

calculation of the recombination cooling (Eq. 6-8, p. 135)

Roughly 2/3 kT of electron thermal energy is lost in

each recombination in an HII region.

0.510.630.690.73(2)/ (2)

64,00016,00080004000T/K

The volumetric cooling, neglecting recombinations tothe ground state (on-the-spot rate approximation) is

rec =(2)nen(H

+)kT((2)

(2) )

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Preliminary Thermal Balance for Pure H

NB Spitzer’s formula for photo-electric heating subtractsrecombination cooling. HII regions, where the on-the-spotapproximation applies, require the use of (2) and (2).

For recombination cooling to balance with photoelectricheating requires ep = 0, or

T/T* = ( / ) ‹ ›)

Both / and ‹ › increase with T, so / ) ‹ › is typically~2-3. Thus the predicted temperature is much greater thanobserved, e.g., for a B0.5 star with T* = 32,000, T would be~ 100,000K. There must be other coolants - not surprisingsince many emission lines are observed in HII regions.

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5. The Strömgren Sphere

Real HII regions are inhomogeneous. Their properties aredetermined by the local ionization parameter. Modeling HIIregions requires a good calculation of the stellar FUVradiation field (usually Monte Carlo). Here we avoid all suchcomplications by using Strömgren’s simple but very usefultheory of a uniform spherical region of radius Rst.

Rst is determined by equating the total rates of ionizationand recombination inside a sphere of radius Rs :

SH =4

3RSt3 ne

2 (2)

Here SH = 1049 s-1 SH,49 is the rate at which the centralstar produces photons that ionize H.

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The Strömgren Radius

The numerical value assumesT=7000 K.

As mentioned above, HII regionsare neither uniform nor spherical.Rather, the dynamics determinen, e,g., expansion into the non-uniform surrounding ISM.

RSt =3

4

SHxen

2 (2)

1/ 3

61.7SH 49n2

1/ 3

pc

Numerical parameters are given in the next slide from Osterbrock’s book.

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Properties of Strömgren Spheres

Osterbrock p. 22

RSt 61.7SH 49n2

1/ 3

pc

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Characteristics of Strmgren Spheres

1. Ionization parameter and radial column density

Consider a location just inside Rst where xe = 1; ignore attenuation of the spectrum, and apply ionization equilibrium:

USt =n

n=

S

4 RSt2cn

=(4 /3)RSt

3n2 (2)

4 RSt2cn

=

(2)

3cnRSt

nRSt = n3

4S

n2 (2)

1/ 3

=3

4nS

(2)

1/ 3

=3

4 (2)

1/ 3

(nS)1/ 3

and finally

U =(2)

3c(

34 (2) )

1/ 3 (nS)1/ 3

Both Ust and nRst are proportional to (nS)1/3.

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2. Numerical values of Ust and nRst

N =43 nRSt

3

RSt2 =

4

3nRSt 1.15 1021 S49n2( )

1/ 3cm 2

Substitute the numerical values, = 3.65x10-13 cm3s-1,n = 100 cm-3 n2 and S = 1049 S49 s

-1

Ust = 3.49x10 3(n2S49)1/ 3=10 2.46(n2S49)1/ 3,

nRst = 8.6x1020(n2S49)1/ 3 cm2

The radial column density nRst is related to the “averagecolumn density:

As n increases for fixed S, the column increasesas n1/3 : small dense HII regions can have largecolumns, as in ultra compact (UC) HII regions.

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nH +

nH 0

=1

3 1nRSt (I4 /I1) =

1

4 1N (I4 /I1)

1=

1N = (6.33 10 18) (1.15 1021) S49n2( )

1/ 3

= 7280 S49n2( )1/ 3

3. The H+/H Ratio

nH +

nH 0

=U

UH

UH

(2)xe

1c (I4 /I1)

Start from previous results for the ionization parameter

USt =

(2)

3cnRSt

to find

This expresses the H+/H ratio in terms of the opticaldepth at the Lyman edge. Recalling 1 = 6.33x10-18 cm2.

The H+/H ratio is about 1/8 of this value, or about 900.

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4. Transition from H+ to H

How thick is the region in

which x(H) goes from 0 to 1?Roughly the distance for an ionizingphoton to be absorbed:

Neglect hardening of the spectrum anddefine the transition where n(H) = 0.5 n,

n(H0)

R

R

1= Rn

H 01

=1

R

RSt

=1

12 n

H 01RSt

=1

38 1

N =2

3

UH

USt

I4I1

3.5 10 4 S49n2( )1/ 3

This property, as well as the three previous ones,all depend on the Strömgren parameter (Sn)1/3.

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5. The Role of He in HII Regions

1. High IP H - 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).O stars won’t do, so their HII regions have no He++.Need planetary nebula stars or AGN.

2. The radiation that ionizes He also ionizes H.3. He recombination radiation photoionizes H.4. He has a dual set of energy levels arising from thetwo total spin states: S=0 (singlet) and S=1 (triplet).See JRG 2006 Lec-03 on HII regions for further discussion.

5. The He threshold photoionization cross section islarger than that for H, largely compensating for itssmaller abundance

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Ionization of He by OB Stars

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

• Many photons in 13.6 - 24.6 eV range• 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 beyond 24.6 eV

• Lots of photons with hv > 24.6 eV

– Single Strömgren sphere• H+ and He+ zones coincide

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He+ Zones in Model H II Regions

Osterbrock Figures 2.4 & 2.5

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

He

He

H

O6

B0

For an O6 star, the abundant supply of He-ionizing photons keepsboth H and He ionized, whereas the smaller number generated bya BO star are absorbed close to the star.