TITLE:
Trapped Vlasov-Poisson-Λ Equilibria and Tidal Response
AUTHORS:
Nikolay N. Fimin
KEYWORDS:
Vlasov-Poisson Equation, Cosmological Constant, Bounded Equilibria, Jeans Distribution, Tidal Response
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
27,
2026
ABSTRACT: A positive cosmological constant makes the Newtonian particle potential decrease without bound at large radius. The same energy and angular momentum may describe both bounded motion and escape. We prove that a nontrivial global Jeans distribution, which assigns a non-negative density to all states with energy and squared angular momentum, has infinite mass in a class of spherical potentials decreasing as a power of radius. The positive cosmological constant gives the quadratic case, with at least logarithmic mass divergence. We construct instead a smooth distribution of finite mass that occupies only the bounded inner component of each admissible invariant level. The generated gravitational field satisfies the same equations that define particle motion. A smooth radial cutoff removes the exterior component without changing any occupied trajectory. The occupied energies lie a positive distance below the outer radial separatrix, the barrier separating bounded radial motion from escape, and this separation persists under small parameter changes. Linearization about the equilibrium yields a Volterra equation: the density perturbation depends through a time integral on its previous gravitational field and on the imposed tide. On every bounded time interval, we prove existence, uniqueness, convergence of the repeated time-integral series, and continuous dependence on the initial perturbation and tidal history. Rotational symmetry isolates the quadrupole, the trace-free part of the second spatial moment, and reduces its tidal response to a scalar kernel with an exact initial derivative. Orbit formulas express the projected kernels through unperturbed trajectories and their dependence on initial momentum. An explicit equilibrium of uniform density and solutions with one scale factor for all directions provide exact comparisons; deformation about a fixed axis supplies a separate approximation. These results distinguish the global restriction imposed by cosmological repulsion from the local response sustained by self-gravity, without assuming stability or decay over arbitrarily long times.