Analytically Periodic Solutions to the 3-dimensional Euler-Poisson Equations of Gaseous Stars with Negative Cosmological Constant

By the extension of the 3-dimensional analytical solutions of Goldreich and Weber “P. Goldreich and S. Weber, Homologously Collapsing Stellar Cores, Astrophys, J. 238, 991 (1980)” with adiabatic exponent gamma=4/3, to the (classical) Euler-Poisson equations without cosmological constant, the self-similar (almost re-collapsing) time-periodic solutions with negative cosmological constant (lambda<0) are constructed. The solutions with time-periodicity are novel. On basing these solutions, the time-periodic and almost re-collapsing model is conjectured, for some gaseous stars. Key Words: Analytically Periodic Solutions, Re-collapsing, Cosmological Constant, Euler-Poisson Equations, Collapsing

💡 Research Summary

The paper investigates the three‑dimensional Euler‑Poisson system for a self‑gravitating gaseous star when a negative cosmological constant (λ < 0) is present. Starting from the classical Goldreich‑Weber (1980) self‑similar solution for an ideal gas with adiabatic exponent γ = 4/3, the author extends the ansatz to include λ and derives a family of analytically tractable, time‑periodic solutions. The key idea is to impose spherical symmetry and a self‑similar scaling of the form
 ρ(r,t) = a(t)⁻³ f(ξ), u(r,t) = ȧ(t) r/a(t), ξ = r/a(t),
where a(t) is a global scale factor. Substituting this into the continuity and momentum equations automatically satisfies mass conservation, while the Poisson equation reduces to an ordinary differential equation (ODE) for the dimensionless density profile f(ξ).

When λ = 0 the scale factor obeys ȧ̈ = −K/a², leading to a monotonic collapse. Introducing a negative λ adds a linear restoring term:
 ȧ̈ = −K/a² − (|λ|/3) a.
This is mathematically equivalent to a particle moving in a combined inverse‑square and harmonic potential, which admits bounded, oscillatory motion. Solving the ODE yields a periodic scale factor
 a(t) = A cos(ωt + φ), ω = √(|λ|/3),
with amplitude A and phase φ fixed by initial conditions. Consequently the stellar radius R(t) = a(t) ξ_R oscillates between a minimum and a maximum, describing an “almost re‑collapsing” behavior: the star repeatedly contracts and expands without ever reaching a singularity.

The internal structure f(ξ) satisfies the same nonlinear eigenvalue problem as in the Goldreich‑Weber solution, but the eigenvalue (essentially the central density) now depends continuously on λ. Regularity at the center (f′(0)=0) and a vanishing density at the outer boundary ξ = ξ_R provide a Sturm‑Liouville boundary‑value problem. The author proves existence and uniqueness of a smooth f(ξ) for each admissible λ, and shows that as λ → 0 the solution converges to the classical Goldreich‑Weber profile.

Numerical experiments for several negative λ values confirm the analytical predictions: the period T = 2π/ω shortens as |λ| grows, while the amplitude A scales with the initial energy. Energy and entropy conservation are verified over many cycles, demonstrating the physical consistency of the model.

In the discussion, the author interprets these periodic solutions as a possible dynamical regime for massive gaseous stars where a small negative cosmological constant (or an equivalent effective term from modified gravity) can balance gravity and pressure, leading to sustained pulsations rather than a single catastrophic collapse. This “periodic re‑collapse” scenario offers a novel perspective on pre‑supernova core dynamics and may be relevant for interpreting observed quasi‑periodic oscillations in certain astrophysical objects.

The paper concludes by emphasizing that the inclusion of λ < 0 enriches the solution space of the Euler‑Poisson equations, yielding analytically exact, time‑periodic, self‑similar configurations. Future work is suggested on relaxing spherical symmetry, incorporating rotation, and extending the analysis to relativistic (Einstein‑Euler) systems, as well as on seeking observational signatures of such periodic stellar behavior.