Mean Gravitational potential energy of neutrons in isothermal neutron-star atmospheres: a statistical-mechanical approach

Abstract

We develop a statistical-mechanical derivation of the mean gravitational potential energy per particle in an infinitely extended isothermal atmosphere and establish the precise domain in which the result ⟨𝑈⟩ = 𝑘𝐵𝑇 can be used as a benchmark for compact-star surface layers. The classical model assumes a dilute, non-degenerate, non-interacting gas in plane-parallel geometry, constant temperature, and a locally uniform gravitational acceleration. Direct integration of the Boltzmann distribution and the canonical partition function both yield ⟨𝑈⟩ = 𝑘𝐵𝑇 and a scale height 𝐻 = 𝑘𝐵𝑇/(𝑚𝑔). For a canonical 1.4 M⊙, 12 km neutron star at 106 K, the Newtonian and local general-relativistic surface gravities are 1.290e+12 and 1.594e+12 ms⁻², respectively, giving free-neutron benchmark scale heights of 6.388 and 5.172 mm. Because H/R ≈ 4.31e-07, the local constant- gravity and plane-parallel approximations are geometrically well motivated, although the absolute surface gravity requires relativistic correction. The Maxwell–Boltzmann validity domain is quantified using the thermal de Broglie wavelength and neutron degeneracy parameter 𝑛𝜆𝑇³ ; at 106 K the conservative 𝑛𝜆𝑇³ = 0.1 boundary occurs near 3.2 × 104 𝑔 𝑐𝑚⁻³ for an ideal free-neutron gas. We then separate this dilute benchmark from the strongly degenerate crust and core, introduce relativistic Fermi energies, and formulate the Tolman– Oppenheimer–Volkoff equations required for global stellar structure. Comparison with established neutron-star atmosphere calculations shows that the analytical result is best interpreted as a limiting hydrostatic/statistical- mechanical benchmark, not as a complete photospheric model. The formulation therefore provides a transparent bridge from elementary statistical mechanics to relativistic compact-star physics.