This Chapter is not essential for anyone willing to take on trust the slowness of the first stellar nuclear reaction, namely the fusion of two protons to form a deuteron, a positron and a neutrino. Under central solar conditions a proton typically survives the order of 10 billion years before fusing with another proton.
My main motivation for deriving this reaction rate was to confirm exactly why this reaction is so slow compared with the reaction which follows it. The formation of helium-3 from deuterium takes just a few seconds under central solar conditions. Many sources are guilty of giving the impression that the deuteron formation reaction is so slow because of the need for the protons to penetrate their mutual Coulomb barrier. This cannot be the reason because a similar Coulomb barrier applies in the case of the helium-3 formation reaction. Actually, the 17 orders of magnitude difference in the reaction rates is due solely to the weakness of the weak nuclear force compared with the electromagnetic force at the energies in question.
In this Chapter the rate of the deuteron formation reaction is derived approximately and in a rather heuristic manner. The method proceeds by analogy with that for the electro-strong reactions, and is based on Schrodinger matrix elements and an assumed - if not strictly appropriate - form of interaction Hamiltonian. The weak mediated reaction rate is obtained simply by substituting a weak vertex function in place of the electromagnetic fine structure constant.
The combined effect of the Coulomb barrier and the Maxwell distribution of particle energies at a given temperature is evaluated in the usual way, leading to the Gamow peak.
Perhaps surprisingly in view of the crude approximation used for the matrix element, the resulting expression for the reaction rate is found to agree with published data to within a factor of 2 over a wide range of temperatures.
The timescale of the deuteron formation reaction, of order 10 billion years under central solar conditions, is essentially the same as the lifetime of a solar mass star (since such stars spend the bulk of their lives burning hydrogen). It is both true and misleading to observe that the rate of this reaction determines the lifetime of a solar mass star. The reaction rate is extremely sensitive to temperature. The central temperature results from the need to balance the rate of heat production against the available heat transport mechanisms, as well as maintaining hydrodynamic equilibrium. It is more indicative to regard the rate of heat transport as determining the temperature, which in turn determines the reaction rate. Thus, the reaction rate is merely obliged to be consistent with a lifetime which is, broadly speaking, determined by the heat transport properties of the stellar medium.
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Artist's impression of a recurrent nova in the binary star system RS Ophiuch. Every 20 years or so, the red giant star dumps enough hydrogen onto its companion white dwarf star to set off a brilliant thermonuclear explosion (a nova). In the next 100,000 years enough matter will have accumulated on the white dwarf to push it over the Chandrasekhar Limit, which will result in a supernova (David A. Hardy, PPARC)