When self-similarity meets mass spectrum and anisotropy

Self-similar evolution is widely used in the theory of collisional stellar dynamics, but its applicability to systems with multiple stellar masses is not well established. We investigate the structural stability of self-similar evolution in multi-mass star clusters and assess the roles of mass segregation and velocity anisotropy. Using a gaseous-model approximation, we developed a theoretical framework to describe the response of a self-similar background to mass-dependent perturbations with isotropic and anisotropic velocity distributions. We show analytically that mass-dependent relaxation leads to a separation of characteristic similarity scales and renders the single-scale solution structurally unstable. In the presence of velocity anisotropy, this similarity-breaking instability splits into distinct radial and tangential modes whose growth rates are modified in a direction-dependent manner. Radial anisotropy reduces the instability through enhanced radial kinetic support, whereas tangential anisotropy increases the effective growth rates and enables faster central evolution. In systems with a mass spectrum, this instability drives mass segregation and the emergence of a multi-scale, near-homologous evolution. Together, these results place self-similar evolution in a consistent theoretical context for collisional star clusters with multiple stellar masses and anisotropic velocity distributions.