Boltzmann-Poisson equation with a central body: Analytical solutions in one and two dimensions

Area of Science:

• Astrophysics
• Statistical Mechanics
• Mathematical Physics

Background:

• Isothermal self-gravitating systems are fundamental models in astrophysics and statistical mechanics.
• Previous analytical solutions for these systems typically did not include a central body.
• The Boltzmann-Poisson equation describes the equilibrium state of such systems.

Purpose of the Study:

• To derive analytical solutions for the density profile of isothermal self-gravitating systems with a central body.
• To generalize existing analytical solutions by incorporating a central mass.
• To explore the applicability of these solutions to diverse physical phenomena.

Main Methods:

• Analytical solution of the Boltzmann-Poisson equation in one and two dimensions.
• Generalization of classical solutions (Camm, 1950; Ostriker, 1964).

Main Results:

• Explicit analytical expressions for the density profile of isothermal self-gravitating systems surrounding a central body.
• Demonstration that analytical solutions are achievable in 1D and 2D, unlike the 3D case requiring numerical methods.
• Generalization of known analytical solutions.

Conclusions:

• The derived analytical solutions provide a powerful tool for understanding systems with a central mass.
• These findings have broad implications for modeling galaxies, protoplanetary atmospheres, Brownian particle systems, bacterial chemotaxis, and 2D hydrodynamics.
• The study highlights the importance of dimensionality in the analytical tractability of the Boltzmann-Poisson equation.