Scale-Invariant Mode in Collisionless Spherical Stellar Systems

An analytical solution for the perturbed equations, applicable to all ergodic models of collisionless spherical stellar systems with a single length parameter, has been derived. This solution corresponds to variations in this parameter, i.e., the expansion or contraction of the sphere while conserving total mass. During this process, the system maintains an equilibrium state. The simplicity of the solution allows for the explicit expression of the distribution function, potential, and density across all orders of perturbation theory. This, in turn, aids in clarifying the concept of perturbation energy, which, being of second order in amplitude, cannot be calculated using linear theory. It is demonstrated that the correct expression for perturbation energy, accounting for second-order perturbations, does not align with the well-known expression for perturbation energy via a quadratic form, derived from first-order perturbations within linear theory. However, both these energies are integrals of motion and differ only by a constant. The derived solution can be utilized to verify the correctness of codes and the precision of calculations in the numerical study of collisionless stellar models.