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In the framework of a chiral hadronic model it has been numerically proven that the parabolic expansion withrespect to δ2 at finite density and temperature is a good approximation in the relativistic mean field theory. It isalso shown that the proton fraction in the modified Urca process is considerably restrained in comparison with thatof the direct Urca process. Then, we discuss the influences of the density-dependent and temperature-dependentasymmetry energy on the proton fraction in hot neutron star matter.