ast403_phy of fusion notes

ast403_phy of fusion notes - The physics of fusion in stars...

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Thephysicsoffusioninstars Most stars derive their luminosity from the conversion of hydrogen to helium. The rest mass of one 4 He atom is about 0 . 71% less than the combined rest masses of four hydrogen atoms (note that the electrons are included in the atomic masses here). The di ff erence, or about 26 . 7 MeV / c 2 , is released as heat, except for 0 . 6 MeV worth of neutrinos (in the pp chain). There are two paths from 4 1 H to 4 He : the pp cycle, which predominates in the Sun and cooler stars, and the CNO cycle, which predominates in stars with slightly higher central temperatures. pp CNO p + p 2 H + e + + ν e 12 C + p 13 N + γ 2 H + p 3 He 13 N 13 C + e + + ν e 3 He + 3 He 4 He + 2 p 13 C + p 14 N + γ 14 N + p 15 O + γ 15 O 15 N + e + + ν e 15 N + p 12 C + 4 He Table 1: The main channels of the pp and CNO cyles [2]. In Table 1, the isotopic designations refer to nuclei rather than whole atoms, so that 1 H would be equivalent to a proton, p . In some books, the helium nucleus is denoted by α instead of 4 He, and the deuterium nucleus by d instead of 2 H. About 0 . 4% of pp reactions in the Sun start with 2 p + e 2 H + ν e instead of the first reaction shown in the Table. About 15% involve 3 He + 4 He 7 Be + γ 7 Be + e 7 Li + ν e 7 Li + p 4 He + 4 He (1) instead of the third reaction shown. Even more rarely (0 . 02%), the second and third reactions of (1) are replaced by 7 Be + p 8 B + γ 8 B + p 8 Be + e + + ν e 8 Be 4 He + 4 He (2) in which 8 Be is a metastable state. This last side chain is energetically negligible but experimentally important because it produces an exceptionally energetic neutrino (up to 15 MeV) which, though much rarer, is easier to detect than the paltry 0 . 420 MeV neutrino resulting from the first reaction in the Table. The carbon, nitrogen, and oxygen in the CNO reactions serve as catalysts: no net production of these elements occurs. The second column of Table 1, for example, replaces the original 12 C nucleus. There is a side chain that goes through 16 O, but this also involves no net production of elements other than helium. Thus, even at high central temperatures, the CNO cycle could not have occurred in metal-free primordial high-mass stars. 1
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On a per-proton basis, the pp and CNO cycles in stars proceed extremely slowly. Fusion has reduced the central hydrogen abundance of the Sun by about a factor of two in the 4 . 6 Gyr since its formation; thus the fusion rate per proton is 5 × 10 18 s 1 . Let us compare this to a characteristic proton-proton collision rate, n p σ v th , where n 6 × 10 25 cm 3 is the central number density of protons and v th = (3 k B T c / m p ) 1 / 2 600 km s 1 is their thermal velocity. The choice of the collision cross section, σ , depends upon what one considers a collision. As will be seen later, a natural scale for cross sections is π lambdaslash 2 dB where lambdaslash dB planckover2pi1 / mv is the reduced de Broglie wavelength. If v = v th then lambdaslash dB 10 11 cm, and the collision rate n p π lambdaslash 2 dB v 10 12 s 1 . Comparing this with the fusion rate estimated above, one sees that the
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