Proto-neutron star in generalized thermo-statistics

The proto-neutron star (PNS) is investigated for the …rst time in the generalized thermo-statistics. The subextensive entropy for the power-law index q > 1 leads to much lower temperature in a core of PNS than the Clausius extensive entropy. Consequently, we can see a clear di¤erence between the mass-radius relation of PNS in the standard (q = 1:0) and generalized (q = 1:3) thermo-statistics. The result is essentially due to the e¤ect of gravitation that is consistent with the subextensivity of the generalized entropy. A proto-neutron star (PNS) [1,2] is born during 0.1-1s after the core bounce of successful supernova explosion. The PNS is a hot and lepton-rich object and so is quite di¤erent from the ordinary neutron star (NS) observed as a radio pulsar. It is composed of a shocked envelope and an un-shocked core of entropy per baryon S 1. The core is in quasi-stationary -equilibrium state because the time scale of weak interaction is much shorter than the time scale of neutrino di¤usion. We can therefore study an equation-ofstate (EOS) of PNS in the nuclear physics of dense baryon matter. So far the EOS of PNS has been investigated [3-10] within the standard Boltzmann-Gibbs thermo-statistics. To the contrary the present paper calculates the EOS for the …rst time within the generalized thermo-statistics [11-13]. We make use of the relativistic mean-…eld (RMF) model of PNS matter developed in Ref. [14]. The RMF model is reasonable because the investigation of PNS needs the general relativity that is based on the validity of the special relativity at local space-time, where the microscopic nuclear model is developed. We extend the thermodynamic potential in Ref. [14] using the q-deformed exponential and logarithm [15]: expq(x) [ 1 + (1 q)x ] 1=(1 q) ; (1) lnq(x) x q 1 1 q : (2) 1 Proto-neutron star in generalized thermo-statistics Consequently, the thermodynamic potential q in the generalized thermo-statistics is q = 1 2 m h i 2 + 1 2 m h 3i 2 + 1 2 m h i 2 1 2 m! h!0i 2 1 2 m h 03i 2 1 2 m h 0i 2 2 kB T X B=p;n; ; +; 0; ; 0; Z dk (2 ) lnq 1 + expq B E kB kBT + lnq 1 + expq B E kB kBT kB T X l=e ; ; e; l Z dk (2 ) lnq 1 + expq l ekl kBT + lnq 1 + expq l ekl kBT ; (3) where kB is the Boltzmann constant and B is given by the chemical potential B and the vector potential VB of baryon as B = B VB: (4) The spin degeneracy factor for leptons is l = f2; 2; 1g for l = fe ; ; g. M B = m BMB and E kB = (k 2 +M B ) 1=2 are the e¤ective mass and the energy of baryon while l and ekl = (k 2 +ml ) 1/2 are the chemical potential and the energy of lepton. The scalar mean …elds h i, h 3i and h i are expressed [14] in terms of three independent e¤ective masses of p, n and while the vector mean …elds h!0i, h 03i and h 0i are expressed [14] in terms of three independent vector potentials of p, n and . Then, M p , M n, M , Vp, Vn and V are determined from extremizing the thermodynamic potential q in terms of them. The results are p + X Y 6= g Y Y ! g nn + I3Y g Y Y g nn! g !=g g Y Y g nn g pp! g nn + g nn! g pp Y g nn m 2 ! h!0i+ g nn!m h 03i g !=g g nn m 2 h 0i g pp! g nn + g nn! g pp = 0; (5) n + X Y 6= g Y Y ! g pp I3Y g Y Y g pp! g !=g g Y Y g pp g pp! g nn + g nn! g pp Y g pp m 2 ! h!0i g pp!m h 03i g !=g g pp m 2 h 0i g pp! g nn + g nn! g pp = 0; (6) + X Y 6= g Y Y g Y m g !2 (V g ! h!0i) = 0; (7)