On the upper limit for surface temperature of a static and spherical body

Abstract

An upper limit for surface temperature of a static and spherical body in steady state is determined by considering the gravitational temperature drop (GTD). For this aim, a body consisting of black body radiation (BBR) only is considered. Thus, it is assumed that body has minimum mass and minimum GTD. By solving the Oppenheimer–Volkoff equation, density distribution of self-gravitating thermal photon sphere with infinite radius is obtained. Surface temperature is defined as the temperature at distance of R from centre of this photon sphere. By means of the density-temperature relation of BBR, surface temperature is expressed as a function of central temperature and radius R. Variation of surface temperature with central temperature is examined. It is shown that surface temperature has a maximum for a finite value of central temperature. For this maximum, an analytical expression depending on only the radius is obtained. Since a real static and stable body with finite radius has much more mass and much more GTD than their values considered here, obtained maximum constitutes an upper limit for surface temperature of a real body. This limitation on surface temperature also limits the radiative energy lose from a body. It is shown that this limit for radiative energy lose is a constant independently from body radius and central temperature. Variation of the minimum mass with central temperature is also examined. It is seen that the surface temperature and minimum mass approach some limit values, which are less than their maximums, by making damping oscillations when central temperature goes to infinity.