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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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