Positron beta decay in stellar core plasma According to introduction level textbooks $\beta^+$ decay, that is conversion of protons into neutrons only occurs in atomic nuclei.
$$p \xrightarrow{\beta^+} n + e^+ + \nu_e$$
I understand that it takes energy that free protons don't have. But what about inside stellar cores? Is there a similar, perhaps slightly more complex reaction with the same outcome, going on in there? If so, what would be the ratio, compared to better known fusion reactions?
I know there are $\beta^+$ reactions going on in bigger nuclei in the stars as part of the PP and CNO cycles. I'm asking about free protons bumping around.
 A: "Neutronisation" or inverse beta decay is endothermic, in the sense that the neutron mass is greater than the proton by 1.29 MeV, so even neglecting the energy of any lepton number-balancing neutrino emission, you need to find kinetic energy from somewhere and convert it into rest mass.
The reaction you suggest also creates a positron, which needs even more proton kinetic energy. A slightly easier way is 
$$ p + e \rightarrow n + \nu_e$$
but the combined kinetic energy of the proton and electron would still need to exceed 0.785 MeV.
Temperatures in the core of a star are of order 10-100 million K, which corresponds to particle kinetic energies of only 1-10 keV.
If you work out what fraction of the tail of a Maxwell-Boltzmann distribution has energies of hundreds of times the mean energy, it is vanishingly small (as in you wouldn't expect any particle to have that kinetic energy in the whole star).
Where this reaction can happen is in the core of a collapsing massive star. Here the temperatures approach $10^{11}$ K ($\sim 10$ MeV) and free protons and electrons can combine to form a... neutron star.
The reactions can also happen in "cold", but very dense degenerate matter. For example at the centres of very massive white dwarfs, although it would likely occur for protons in nuclei in that case. Neutronisation (or inverse beta decay as it is more normally known in this context) can happen because the degenerate electrons can have much higher kinetic energies (by factors of 1000 or more) than in a perfect gas of a similar temperature. Their kinetic energies increase at higher densities and can be sufficiently high to overcome the energy threshold for neutronisation. This may lead to instability as free electrons, that provide the majority of the pressure support, are removed from the gas.
