Stability of a spherical star system

The stability of a spherically symmetric aggregate of point gravitating particles relative to arbitrary small perturbations is studied. It is assumed that in the absence of perturbations the particles move along circular trajectories chaotically oriented in space so that the total moment of the aggregate is zero. Dimensions of the aggregate are large in comparison to the gravitational radius, and particle velocities are nonrelativistic. It is shown that there exist initial mass-density distributions unstable relative to any perturbations with the exception of radial and dipole perturbations. A general stability criterion is formulated, with the form dΩ2/dr > 0, where\(\Omega ^2 = (4\pi G / r^3 )\int\limits_0^r {p_0 r^2 dr, \rho _0 (r)} \) is the aggregate mass density, and G is the gravitational constant. The dependence of the increment onl, the perturbation harmonic number, is studied. In the case of weak inhomogeneity r(dΩ2/dr)/Ω2 ≪ 1 the increment is maximum for quadrupole perturbations (l = 2) and decreases monotonically with increase inl. In the opposite case of high inhomogeneity r(dΩ2/dr)/Ω2 ≫l the increment increases with increase inl. In the case of weak inhomogeneity the increment may be as small as desired. For high inhomogeneity, instability develops over a time period smaller than the period of revolution of an individual particle. For dΩ2/dr < 0 the system is stable. Consideration of system microstructure in this case leads to damping of macrooscillations (system “heating”).