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.