Neutrino trapping in extremely compact objects: I. Efficiency of trapping in the internal Schwarzschild spacetimes

Extremely compact objects (9GM/4c2 < R < 3GM/c2) contain trapped null geodesics. When such objects enter the evolution period corresponding to the geodetical motion of neutrinos, a certain part of neutrinos produced in their interior will be trapped influencing thus neutrino luminosity of the objects and consequently their thermal evolution. The existence of trapped neutrinos indicates possibility of 'two-temperature' cooling regime of the extremely compact objects. We present upper estimates on the efficiency of the neutrino trapping effects obtained in the framework of the simplest model of the internal Schwarzschild spacetime with uniform distribution of energy density, assuming uniform distribution of neutrino emissivity. We introduce a 'global' luminosity trapping coefficient representing influence of the trapping effect on the total neutrino luminosity of the extremely compact objects and cooling trapping coefficients of both 'local' and 'global' kinds characterizing influence of the trapping on the cooling process. It is shown that the trapping of neutrinos can be relevant to moderately or even slightly extremely compact objects.