Tidally Perturbed, Rotating Stellar Systems: Asynchronous Equilibria

TL;DR

This work provides an analytic framework for tidally perturbed, rotating stellar systems with asynchronous internal rotation by extending King-model equilibria through a Jacobi-energy-based distribution function and a first-order perturbation treated with matched asymptotic expansion. The authors introduce an asynchronicity parameter $\zeta$ that couples tidal and rotational effects, yielding a three-parameter family parameterized by $\Psi$, $\zeta$, and $\epsilon$ and a global solution to the coupled Poisson-Laplace equations across an unknown boundary surface. Key contributions include the explicit internal and external solutions, a boundary-layer treatment near the truncation radius, and a thorough characterization of how triaxial morphology and kinematics depend on the perturbation regime, with two principal regimes identified: near-critical, strongly distorted models and sub-critical, nearly spherical ones. The models offer analytic equilibria for interpreting globular cluster observations and providing physically consistent initial conditions for N-body simulations, while setting the stage for extensions to misaligned configurations and higher-order perturbations. Overall, the framework bridges tidal and rotational dynamics in a tractable, analytically tractable manner, enabling quantitative exploration of their combined impact on cluster structure and dynamics.

Abstract

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.

Figures (7)

• Figure 1: The ($\Psi\!,\!\epsilon$) parameter space for first-order models with $\nu=3$. Solid lines denote the critical values of the tidal strength parameter $\epsilon_\text{cr}$, as a function of the concentration parameter $\Psi\in[2,7]$, computed for representative values of the asynchronicity parameter $\zeta\in\{-1,0,1,2,3,4,5,10\}$. The shaded area illustrates the maximum extent of the allowed parameter space (see Section \ref{['sec:regimes']} for details).
• Figure 2: The ($\epsilon\!,\!\zeta$) parameter space for first-order models with $\nu=3$. Solid lines denote the critical values of $(\epsilon,\zeta)$, computed for representative values of the concentration parameter $\Psi\in\{2,3,4,5,6,7\}$. The shaded areas show the parameter space that is accessible at each value of concentration.
• Figure 3: Intrinsic density profiles (normalized to the central value) along the $\hat{x}$-, $\hat{y}$- and $\hat{z}$-axes for a critical first-order model with $\Psi=4$, $\zeta=2$ and $\epsilon=1.211\times10^{-4}$. Also shown are profiles for the corresponding critical synchronous ($\zeta=0$, $\epsilon=1.972\times10^{-4}$) and unperturbed King models. The galactic potential is Keplerian ($\nu=3$) in all cases. Note the pale blue $\hat{x}$-axis profile of the synchronous model, the extension of which beyond the corresponding asynchronous profile is narrowly distinguishable.
• Figure 4: Intrinsic velocity dispersion profiles (normalized to the central value) along the $\hat{x}$-, $\hat{y}$- and $\hat{z}$-axes for the same models as in Fig. \ref{['fig:density']}. The $\hat{x}$-axis profiles are almost coincident, with the pale blue line of the synchronous profile just discernible beyond that of the asynchronous model.
• Figure 5: Intrinsic density profiles (normalized to the central value) along the $\hat{y}$-axis for critical first-order models with $\Psi=4$ and $\zeta\in\{-1,0,1,2,3,4,5\}$ (from left to right). The galactic potential is Keplerian ($\nu=3$).