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# At u1 the above solution breaks down there is a

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Unformatted text preview: ation coefficient (the two form a Laplace transform pair). The value at threshold is 6×10−18 cm2. C. STRUCTURE OF THE HYDROGEN NEBULA We are now ready to consider a typical H II region associated with star formation. The typical timescale associated with its ionization evolution is (αBnH)−1; for nH ~ 10 cm−3, this is ~104 yr, which is short compared to the lifetime of an O star. Therefore, we expect the ionization structure to have time to reach equilibrium, and we set ξ(r,t)=ξ(r). Then: r Q(H 0 ) a 0 2 0 = −α (H , T ) n Hξ ( r) + [1 − ξ ( r)] exp − ∫ 0 n H[1 − ξ ( r' )]a dr' . 2 4 πr It is also convenient to define the radius r1 within which the rate of recombinations at full ionization would equal the rate of emission of source photons, € 43 2 πr1 α Bn H = Q(H 0 ) . 3 If we then let u = r/r1, we have € u C ξ 2 ( u) = 2 [1 − ξ ( r)] exp −C ∫ 0 [1 − ξ ( u' )] du' , 3u where we have defined the dimensionless parameter C = nHr1a. For our typical H II region, with a photon emission rate of 1049/s, we find r1 = 4.5×1019 cm = 15 pc and C = 3000. € Indeed, the typical case is that C >> 1. This case can be solved by writing the above equation in the form. u 1d ξ 2 ( u) = − 2 exp −C ∫ 0 [1 − ξ ( u' )] du' . 3u du In the deep interior of the nebula, where the left ­hand side is approximately 1, this admits the solution: € [ [ [ [ u exp −C ∫ 0 [1 − ξ ( u' )] du' = 1 − u 3 , or 3u 2 . C (1 − u 3 ) € Thus for u<1, the interior of the nebula is almost fully ionized. At u~1 the above solution breaks down; there is a transition region of thickness Δu~C−1/3 over which the nebula becomes mostly neutral. B€ eyond this, there is an exponentially decreasing flux of photons ξ ( u) = 1 − and the outside material is neutral. Thus, an ionized bubble with a sharp edge is formed. This bubble is called a Strömgren sphere. Its radius is r1. 3. Hydrogen Helium Nebulae We are now ready to add helium in to the mix. We note here that the first ionization energy of helium (He0He+) is 24.6 eV, whereas the second ionization energy is 54.4 eV. The He:H ratio (by number of atoms) is 0.08 for primordial gas, although the modern ­day values tend to be somewhat larger. A. He+ ZONES We now consider a source that emits some number of photons capable of ionizing helium, ∞ L Q(He 0 ) = ∫ν ( He0 ) ν dν . i hν For cool sources, Q(He0) is usually quite a bit smaller than Q(H0), although this will not be true for planetary nebulae or AGN. In this case, a helium Strömgren sphere appears inside the hydrogen sphere. Its €adius is given by r...
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