Nuclear fusion in dense matter: Reaction rate and carbon burning

In this paper we analyze the nuclear fusion rates among equal nuclei for all five different nuclear burning regimes in dense matter (two thermonuclear regimes, two pycnonuclear ones, and the intermediate regime). The rate is determined by Coulomb barrier penetration in dense environments and by the astrophysical S factor at low energies. We evaluate previous studies of the Coulomb barrier problem and propose a simple phenomenological formula for the reaction rate that covers all cases. The parameters of this formula can be varied to take into account current theoretical uncertainties in the reaction rate. The results are illustrated for the example of the $^{12}\mathrm{C}$+$^{12}\mathrm{C}$ fusion reaction. This reaction is important for the understanding of nuclear burning in evolved stars, in exploding white dwarfs producing type Ia supernovas, and in accreting neutron stars. The S factor at stellar energies depends on a reliable fit and extrapolation of the experimental data. We calculate the energy dependence of the S factor by using a recently developed parameter-free model for the nuclear interaction, taking into account the effects of the Pauli nonlocality. For illustration, we analyze the efficiency of carbon burning in a wide range of densities and temperatures of stellar matter with the emphasis on carbon ignition at densities $\ensuremath{\rho}\ensuremath{\gtrsim}{10}^{9}$ g cm${}^{\ensuremath{-}3}$.