Tidally Perturbed, Rotating Stellar Systems: Asynchronous Equilibria

We present a new three-parameter family of self-consistent equilibrium models for quasi-relaxed stellar systems that are subject to the combined action of external tides and rigid internal rotation. These models provide an idealised description of globular clusters that rotate asynchronously with respect to their orbital motion around a host galaxy. Model construction proceeds by extension of the truncated King models, using a newly defined asynchronicity parameter to couple the tidal and rotational perturbations. The method of matched asymptotic expansion is used to derive a global solution to the free boundary problem posed by the corresponding set of Poisson-Laplace equations. We explore the relevant parameter space and outline the intrinsic properties of the resulting models, both structural and kinematic. Their triaxial configuration, characterised by extension in the direction of the galactic centre and flattening toward the orbital plane, is found to depart further from spherical symmetry for larger values of the asynchronicity parameter. We hope that these simplified analytical models serve as useful tools for investigating the interplay of tidal and rotational effects, providing an equilibrium description that complements, and may serve as a basis for, more realistic numerical simulations.

💡 Research Summary

The paper introduces a novel three‑parameter family of self‑consistent equilibrium models designed to describe globular clusters that experience both external tidal forces from their host galaxy and internal rigid rotation that is not synchronized with the orbital motion. Building on the classic King (1966) models, the authors replace the single‑star energy E with the Jacobi integral H defined in a co‑rotating frame, thereby incorporating the combined effects of tides and rotation into a lowered Maxwellian distribution function. A new “asynchronicity” parameter ε = (ω − Ω)/Ω quantifies the mismatch between the cluster’s internal angular velocity ω and the orbital angular velocity Ω.

The dynamical framework assumes a cluster on a circular orbit of radius R₀ in a static, spherically symmetric galactic potential Φ_G(R). The cluster’s centre of mass follows this orbit while the internal rotation axis is aligned with the orbital axis. In a locally rotating frame that eliminates the bulk rotation, the authors derive a Lagrangian that includes Coriolis, centrifugal, and tidal terms. The resulting equations of motion are those of Hill’s approximation, with a time‑dependent perturbation potential Φ_P(α,β,z,t) that varies periodically with the phase η = (ω − Ω)t. By assuming that the internal dynamical timescale is much shorter than the period of this variation, the Jacobi integral H can be treated as effectively conserved for the purpose of constructing the distribution function.

The distribution function takes the form
f(H) = A