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πr2 n H f He (1 + f He )α B (He 0 , T ) = Q(He 0 ), 3
where fHe is the He:H ratio by number, αB(He0,T) is the Case B recombination coefficient for He+He0 (i.e. to all excited states of He), and we have used the fact that the densities of He+ and electrons in t€ He+ zone are nHfHe and nH(1+fHe), respectively. he There should be a correction to the above equations because some of the helium
ionizing photons will get absorbed by the small amount of neutral hydrogen present within the He+ zone. But usually this is a small correction. At 24.6 eV, the cross sections for photoionization of He0 and H0 are 8✕10−18 cm2 and 1.1×10−18 cm2, respectively, so any photon capable of ionizing helium usually does (exceptions below). The existence of the helium ionization zone also modifies the hydrogen ionization zone. This is for two reasons. One is that photons above 24.6 eV are absorbed by helium and so are not available to ionize hydrogen. However, when the ionized helium recombines, it radiates helium spectral lines and two
photon continuum radiation, some of which emerges at hν>13.6 eV. The probability p of emitting such a photon can be computed by taking the recombination coefficients to each excited level and following their decay chains with the appropriate branching probabilities, giving p = 0.96. (In high
density e
regions, n > few ✕ 103 cm−3, the metastable 1s2s 3 S1 level can collide with an electron and transition to 1s2s 1Se . In the case of this decay path, we find p = 0.66.) In this case, if we 0
compute the total number of hydrogen recombinations and set this equal to the number of ionizations, we get €
€ 4
4
2
2
π ( r13 − r23 ) n Hα B (H 0 , T ) + πr23 n H (1 + f He )α B (He 0 , T ) = Q(H 0 ) − (1 − p)Q(He 0 ) . 3
3 If we approximate p≈1, then the usual hydrogen Strömgren sphere radius is unchanged. €
The fraction of the volume of the ionized nebula in which helium is also ionized is then: −1
Ⱥ Q(H 0 )
r23
α B (He 0 , T )
α B (He 0 , T ) Ⱥ
=
−1+ p −
Ⱥ
Ⱥ . r13 α B (H 0 , T ) f He (1 + f He ) ȺQ(He 0 )
α B (H 0 , T )(1 + f He ) Ⱥ
In the limit p1, fHe<<1, and the two recombination coefficients being equal (typically good to ~10%, since Case B recombination coefficients only weakly distinguish between €
recombination on a proton versus an He+), this is r23
Q(He 0 )
≈
. r13 f HeQ(H 0 )
For harder spectra, in which the Q(He0):Q(H0) ratio is larger, more of the helium in the nebula is ionized, as we expect. €
For very hard sources, roughly a central star temperature exceeding ~4✕104 K (O7), w...
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 Winter '08
 Sargent,A

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