| The study on thermal evolution of neutron star is a frontier issue in the field of nuclear physics. The article studies the properties of neutron star cooling in the framework of the relativistic mean field theory (RMFT). There are four parts in the article, which are as follows.Firstly, we present the expressions of relativistic neutrino emissivity. As well known, the non-relativistic expression is applied widely for calculating neutrino emissivity. However, the threshold densities of the direct URCA processes are about several times of the normal saturate density. Above the densities, the movement of hadrons is in relativistic regime. The relativistic formulae of the direct URCA processes must be used. One can find that the theoretical calculation should be self-consistent, when the relativistic formulae of neutrino energy losses and the relativistic equation of state are used. The relativistic results express that with the increase of density for neutron star matter, the neutrino emissivity for the direct URCA processes increases. While the non-relativistic results have almost opposite trend. With the increase of density, the neutrino emissivity decreases. In fact, with the increase of density one must consider the relativistic effect. The results show that the relativistic results are more reliable, whereas the non-relativistic results have an obvious deviation with the increase of density. Furthermore, the neutron star cooling curves are presented in the non-superfluidity situation. The results express that the cooling velosity of the nucleon direct URCA processes is very fast, which does not agree with the observations.Secondly, we study the influence of the degree of freedom for hyperons and the hyperon direct URCA processes (Y-DURCA) on the nucleon direct URCA processes (N-DURCA) and the total neutrino emissivity of neutron stars. There are two parts in our reserch work. One part is that the influence of the hyperon appearance on the N-DURCA processes. The research results show that the hyperon appearance suppresses the neutrino emissivity of the N-DURCA processes. Because of the hyperon appearance, the abundances of proton and electron are suppressed, which makes the decrease of the neutrino emissivity for the N-DURCA processes. Another work is that the influence of hyperons,σ* andφmesons on the Y-DURCA processes is investigated. The investigation results express that although the order of magnitude of the Y-DURCA process is lower than the one of the N-DURCA process, the Y-DURCA processes is still a fast cooling process. The different AA interaction energy, for exampleΔBΛΛ≈1 MeV andΔBΛΛ≈5MeV can impact on the neutrino emissivity of the Y-DURCA processes sensitively and change the threshold density of the reaction D. When the AA interaction becomes weaker, the neutrino emissivity of the Y-DURCA processes is stronger and in this case the threshold density of reaction D (Ξ-→Ξ0+e+(?)) is advanced, which can accelerate the cooling speed of massive neutron stars. The occurrence of the Y-DURCA processes do not need hyperon aboundance reach a certain quantity, once hyperons appear the reaction can happen. Anyhow, the appearance of hyperon freedom of degree can speed up the cooling of small mass stars and slow the cooling speed of massive stars.Thirdly, the influence of the isovector-scalar interaction with exchangingδmeson on the N-DURCA processes is investigated in neutron star matter. Withδmeson, the effective masses of nucleons will split. This split just satisfies the production condition of the N-DURCA processes. The fraction Yp with 8 meson is higher than the one withoutδmeson. With the increase of Yp, the electron fraction increases due to the charge neutral condition andβequilibrium. In this case the momentum conservation is satisfied at lower density region. The simultaneous increase of the abundances of protons and electrons can lead to the decrease of the threshold density and the increase of the neutrino emissivity for the N-DURCA processes. Only superfluidity (SF) for nucleons or hyperons can suppress the fast cooling of neutron stars and agree with the observations. In this paper, the strong proton SF is considered and the neutron SF is neglected. With the strong proton SF, the thermal capacity and neutrino emissivity in the cores of neutron stars are suppressed effectively and the cooling speed becomes lower. In addition, with the same mass, the neutron star includingδmeson has a faster cooling speed than the one withoutδmeson. By comparison with the observations of the isolated neutron stars, PSR J0205+6449, PSR J1119-6127, etc, one finds that the theoretical cooling curves are agreement with the observation region.Finally, we study the effects of the 1S0 SF of A hyperons on the neutrino emissivity of the Y-DURCA processes for reactions B (Λ→p+e+v) and C (Ξ-→Λ+e+v) in the core of neutron star. Since the new experimental data express that∑potential is repellent, our model excludes E hyperons. After excluding∑hyperons, the most important Y-DURCA processes are B and C. We calculate the gap energies ofΛhyperons in neutron star matter. The meson-hyperon coupling constants are choosed consistant with the recent experimental data, which suggests that the effective AA interaction should be about 1 MeV(△BΛΛ=1.01±0.20-0.11 +0.18 MeV). On the other hand, we adopt theΛΛpotential based on the SU(6) quark model which is in line with the recent experimental data (△BΛΛ=1.01±0.20 -0.11 +0.18MeV). In our results, the 1S0 SF of A hyperons exist in the core within the baryon density ratio range of n= 2.19-4.33n0. While the ranges of baryon densities for processes B and C are n=2.19-5.15n0 and n=2.67-4.33n0, respectively. That is to say, the 1S0 SF of A hyperons is existing in the most density ranges of happening reaction B and C. The neutrino emissivity and thermal capacity of the neutron star in the two Y-DURCA processes are sharply suppressed below the critical temperature TCA of A SF, and the neutron star cooling rate is slowed. The cooling curves for Y-DURCA processes are presented. And one can find that with the appearance of A 1S0 SF, the cooling curves of reactions B and C are suppressed obviously, when the system temperature is lower than the critical temperature Tc. By comparsion with the observed neutron star, which are PSR J0205+6449, PSR J0538+2817, PSR J2043+2740, PSR B0656+14, etc, one find the theoretical results are agreement with the observations. We think that through comparing the results of theoretical model and the observed thermal radiation from isolated neutron stars the hyperon SF may exist.
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