Mathematical model of thermal energy in spherical celestial bodies with application to solar system planets

The internal thermal energy of celestial bodies is a fundamental driver of planetary evolution and habitability. However, a universal mathematical framework for quantifying it from first principles has yet to be established. Current approaches rely on indirect measurements in system-specific models, with no reflection of a unified theoretical foundation. This study introduces a mathematical method that defines a fundamental relationship between internal pressure and non-equilibrium temperature in spherical celestial objects. By expressing this relationship as a function of radial distance from the core, we can derive a set of three equations that can be used to calculate internal pressure, distributed temperature (non-equilibrium), and self-generated heat. A key theoretical contribution lies in identifying two coefficients for temperature–pressure proportion and heat production for rocky and gaseous celestial bodies, respectively, thereby establishing a unified classification criterion. To rigorously validate the model, we calculated the surface temperatures of all eight planets in the solar system. The results align well with comparable empirical data and successfully distinguish between the effects of stellar irradiation and the intrinsic heat of the planetary body. The three derived equations provide a novel theoretical perspective on the thermodynamic analysis of spherical celestial bodies, offering a new base to advance the tool for planetary-scale environmental studies.