Astrophysical S-factor

The steep dependence of P on energy leads to the use of the astrophysical S-factor defined by... 

The thermally averaged reaction rate per pair is, utilizing the astrophysical S-factor and the energy dependence of the Gamow-factor ... 

Fig. 2. Cross-section and astrophysical S-factor for charged particle reactions as a function of beam energy. The effective range of energy in stellar interiors is usually far less than the Coulomb barrier energy Ec or the lower limit El of laboratory measurements. The y-scale is logarithmic for cross-section but linear for S-factor thus the cross section drops sharply in regions of astrophysical interest, while the change is much less severe for the S-factor. The extrapolation of laboratory data to lower energies relevant for astrophysical situations is more reliable for S-factor...

The reaction rate, expressed as the number of reactions per volume and per time, is proportional to the astrophysical S-factor. At the temperatures and densities relevant to the stellar environments the interacting nuclei have a Maxwell distribution of speeds. This distribution has also to be taken into account when determining the reaction rate. An introduction to astrophysical S-factors and reaction rates can be found in many textbooks on nuclear astrophysics, e.g., Arnett (1996), Rolfs and Rodney (1988), Iliadis (2007), Boyd (2008). 

Table 1. Bare nucleus astrophysical S(E)-factor at zero energy for the reactions discussed in the text.

Quantum mechanics shows that the fusion reaction probability is also proportional to a geometrical factor nX oc HE, where X in this context is the de Broglie wavelength. The strong energy dependence of this factor and the barrier penetrability have prompted the introduction of the astrophysical S-function (Burbidge et al. 1957), defined by writing the cross section as a product of three factors ... 

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