Mean Gravitational Potential Energy of Neutrons in Isothermal Neutron-Star Atmospheres: A Statistical-Mechanical Approach

Neutron-star atmospheres provide an important environment for investigating the interplay between gravity, thermodynamics, and quantum statistics under extreme physical conditions. This study presents a statistical-mechanical formulation for determining the mean gravitational potential energy per neutron in an infinitely high isothermal neutron-star atmosphere. Starting from the canonical and grand canonical ensembles, the single-particle partition function is derived by separating the momentum and configurational contributions in a uniform gravitational field. Evaluation of the configurational partition function yields the atmospheric scale height,
H=(k_B T)/(m_n g), and the exact analytical result for the mean gravitational potential energy per neutron, ⟨U⟩=k_B T, demonstrating that the average gravitational potential energy is equal to one unit of thermal energy in classical equilibrium. The formulation is subsequently extended to neutron-star physics by incorporating Fermi–Dirac statistics, neutron degeneracy pressure, and the relativistic equation of state governing dense nuclear matter. Representative calculations for a canonical neutron star ((M=1.4M_), (R=12" km)," (T = 10^6 K)) predict a surface gravitational acceleration of (1.29×10^12 " " 〖"m s" 〗^(-2)), an atmospheric scale height of approximately (6.4mm), and a mean gravitational potential energy of ((1.3〖8×10〗^17J) (approximately (86 eV)). Comparison with a representative neutron Fermi energy of approximately (60 MeV) shows that quantum degeneracy overwhelmingly dominates the thermodynamic behavior of the stellar interior, whereas the classical Boltzmann approximation remains applicable only to the dilute atmospheric layers. The study also discusses the application of the Tolman–Oppenheimer–Volkoff equations to neutron-star structure and the importance of the mass–radius relation as a probe of the dense-matter equation of state. The analytical derivations and numerical results provide a benchmark for validating neutron-star atmosphere models and establish a foundation for future investigations incorporating general relativity, realistic nuclear interactions, strong magnetic fields, radiative transfer, and constraints from X-ray and gravitational-wave observations.

Neutron stars; Statistical mechanics; Gravitational potential energy; Tolman-Oppenheimer-Volkoff equations; Fermi-Dirac statistics; equation of state.

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