Rick's Cosmology Tutorial: Appendix A4 Abstract

Derivation of the Rate of the Hypothetical Reaction p + p --> Diproton

Like proton-deuteron capture, proton-proton capture also has no magnetic dipole contribution. This is again because only one spin state is available. Alternatively, as we have seen in Appendix A2, the magnetic dipole matrix element is proportional to the difference of the magnetic moments of the two interacting particles, which is clearly zero for identical particles. Also like proton-deuteron capture, proton-proton capture involves a Coulomb barrier. However, unlike proton-deuteron capture, proton-proton capture has no electric dipole contribution either. The reason is that we are dealing with identical fermions whose wavefunction must be antisymmetric under interchange. Thus, the possible two-proton states are 1S, 3P, 1D, etc. Unlike the neutron-proton or proton-deuteron systems, there is no possible matrix element. Thus, the lowest order contribution arises from the electric quadrupole matrix element. For this reason, the pp cross section is many orders of magnitude smaller than the pn cros section.

Evaluation of this matrix element including the Coulomb barrier requires numerical calculation, but an algebraic expression is also derived for the case of no Coulomb barrier to facilitate checking the numerical program. Conversion to a cross section confirms that the pp cross section is 5 or 6 orders of magnitude smaller than the pn cross section at BBN energies. At central stellar temepartures, this is more like 14 or 15 orders of magnitude smaller.

The implications of hypothetical diproton stability have been dealt with in detail in the "Cosmic Coincidences" - see the links below.

Read Appendix A4 (pdf): Derivation of the reaction rate for Proton-Proton Capture to form deuterium

Go to the "Cosmic Coincidences" treatment of the Implications of Diproton Stability

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Comet Hale-Bopp during its 1997 visit to our vicinity, with Andromeda showing in the background. Note the two tails. This is normal. The blue tail is comprised of gas and points straight away from the Sun. The yellowish tail is dust and curves away along the comets orbit. Unusually, Hale-Bopp also had a third tail, composed of neutral sodium atoms (too faint to be seen in this photo). There was a total solar eclipse during the comet's passage (if you happened to be China/Mongolia/Siberia), allowing day-time viewing in those locations.