On the Instabilities of Small-Scale Modes of Oscillations Against the Background of a Collapsing Galaxy Model

In this work, the gravitational instability of small-scale perturbations with an azimuthal wave number $$m = 2$$ in disk-like self-gravitating systems is considered. Calculations of horizontal small-scale oscillation modes $$(m;N) = (2;10)$$ and (2; 20) against the background of a nonlinearly non-equilibrium anisotropic model of a self-gravitating disk are performed. Critical diagrams of the relationship between the virial parameter and the degree of rotation for these modes are constructed, and the increments of instability for different values of the rotation parameter are calculated. The results show that the instability for the oscillation mode (2; 10) begins at a virial parameter value of $${{(2T{\text{/}}\left| U \right|{\kern 1pt} )}_{0}} \approx 0.217$$ at $$\Omega = 0$$ and reaches 0.413 at $$\Omega = 1$$ . For the oscillation mode (2; 20), the instability starts at a virial parameter value of $${{(2T{\text{/}}\left| U \right|{\kern 1pt} )}_{0}} \approx 0.128$$ at $$\Omega = 0$$ and reaches 0.146 at $$\Omega = 1$$ . It is found that with an increase in the rotation parameter, the instability region also increases, while with an increase in the degree of small-scale structure, the instability region significantly decreases. The work is partially based on a talk presented at the Modern Stellar Astronomy 2024 conference.