Angular-momentum pairs in spherical systems: applications to the Galactic centre

TL;DR

This paper investigates vector resonant relaxation (VRR) in spherical systems where orbital planes precess and the angular-momentum directions diffuse while energies stay nearly fixed. It develops a double orbit-averaged VRR Hamiltonian and derives a Hill-radius–like stability criterion for angular-momentum pairs to remain bound under an external perturber, validating the criterion with N-ring simulations and direct N-body runs. The authors apply the framework to the Milky Way's Galactic Centre, modeling the young stellar disc as coupled two-ring fragments and showing that a putative intermediate-mass black hole (IMBH) of a few thousand solar masses at ~0.15 pc could induce retrograde structures consistent with observations, depending on fragment mass and orbital parameters. The results provide order-of-magnitude constraints on perturber properties, guide future simulations, and highlight the role of VRR in shaping angular-momentum distributions in galactic nuclei and related systems.

Abstract

Consider a system of point masses in a spherical potential. In such systems objects execute planar orbits covering two-dimensional rings or annuli, represented by the angular-momentum vectors, which slowly reorient due to the persistent weak gravitational interaction between different rings. This process, called vector resonant relaxation, is much faster than other processes which change the size/shape of the rings. The interaction is stron9gest between objects with closely aligned angular-momentum vectors. In this paper, we show that nearly parallel angular-momentum vectors may form stable bound pairs in angular-momentum space. We examine the stability of such pairs against an external massive perturber, and determine the critical separation analogous to the Hill radius or tidal radius in the three-body problem, where the angular-momentum pairs are marginally disrupted, as a function of the perturber's mass, the orbital inclination, and the radial distance. Angular-momentum pairs or multiples closer than the critical inclination will remain bound and evolve together in angular-momentum-direction space under any external influence, such as anisotropic density fluctuations, or massive perturbers. This study has applications in various astrophysical contexts, including galactic nuclei, in particular the Milky Way's Galactic centre, globular clusters, or planetary systems. In nuclear star clusters with a central super-massive black hole, we apply this criterion to the disc of young, massive stars, and show that clusters in angular-momentum space may be used to constrain the presence of intermediate-mass black holes or the mass of the nearby gaseous torus.

Figures (10)

• Figure 1: Angular-momentum vectors on the unit sphere.
• Figure 2: A comparison of analytical prediction by Equation \ref{['eq:synchronous']} with the results of N-Ring simulations. Top panels: magnitudes of the pair's internal torque (blue) and of the relative torque from the IMBH ($\theta_\bullet$; red) as functions of the pair’s inclination with respect to the IMBH, for mutual inclinations of $\theta_b=4^\circ$, $7^\circ$, and $10^\circ$, computed using Equation \ref{['eq:synchronous']} for circular orbits. Bottom panels: the mutual inclination of every pair versus $\theta_\bullet$ at $t=0$ (blue), 1 Myr (grey), 5 Myr (orange), and 10 Myr (green) obtained from N-Ring integrations of the same configurations. The simulations confirm the analytic criterion: all pairs remain bound for $\theta_b=4^\circ$, whereas those entering the unstable (red-above-blue) region are disrupted for $\theta_b=7^\circ$ and $10^\circ$. In the quadrupole-dominated regime (panel a) the torque difference exhibits a single clear maximum around $\theta_{\bullet}=90^\circ$, whereas in the general case (panel b) higher-order multipoles introduce second prominent peaks. In both configurations the analytic disruption criterion (red curve exceeding blue) accurately predicts which angular momentum binaries become unbound by 10 Myr.
• Figure 3: Average mutual inclination angle between pairs as a function of cosine of inclination with respect to the perturber and distance ratio to the perturber for initial mutual inclination of $10^\circ$. Top panels: the equal mass case, bottom panels: hierarchical case ($m_i\gg m_j$). Each column corresponds to 5,10 and 20 Myr of the evolution. The dashed lines on each of the panels show the region where the average inclination of the pair does not exceed $15^\circ$. The plot shows results for N-Ring simulations with an IMBH mass of $m_\bullet=10^5\,{\rm M}_{\odot}$ at a distance $r_\bullet=1.5$ pc from the SMBH of mass $4\times10^6 \,{\rm M}_{\odot}$.
• Figure 4: Contour lines show the analytical boundary of stability for a pair of circular orbits against disruption by an eccentric IMBH for initial mutual inclination of $7^\circ$, i.e. regions where the torque between a pair of bodies equals the right hand side of the Eq. \ref{['eq:synchronous']} for eccentric orbits, calculated from the analytical expressions in section \ref{['subsec:elliptical orbits']}. Vertical lines show the pericentre, the semi-major axis, and the apocentre of the IMBH. The IMBH mass, eccentricity, and $\kappa$ parameter are indicated in the panels.
• Figure 5: Average inclination angle between pairs after $5$ Myr (two top rows) and $10$ Myr (two bottom rows) of evolution as a function of relative distance from the IMBH and relative inclination with respect to the IMBH. In all models we used $a_\bullet=0.15$ pc and $m_\bullet=2\times10^3\,{\rm M}_{\odot}$. The colour coding represents the average mutual inclination. Odd rows -- N-Ring, even rows -- $\phi$-CPU, for the identical parameters. The dashed lines show the region where $\theta_b<10^\circ$. The dotted lines in the third row show the disruption threshold computed using Eq. \ref{['eq:synchronous']}.