Rings around irregular bodies I. Structure of the resonance mesh, applications to Chariklo, Haumea and Quaoar

TL;DR

This work analyzes how second-order Spin-Orbit Resonances shape dense rings around irregular solar system bodies using a Hamiltonian framework for resonances up to order five. It evaluates two non-axisymmetric sources—a triaxial ellipsoid with $C_{22}$ and a mass anomaly with $\,\mu$—to map resonance topology, widths, maximum eccentricities, and excitation timescales, and then applies these results to Chariklo, Haumea and Quaoar with $\mu \lesssim 0.001$. The key finding is that the $1/3$ SOR and $5/7$ SOR offer the most favorable regions for ring confinement, while corotation points can be unstable for some bodies; higher-order resonances generally have limited impact on a dense collisional disk. The analysis provides a theoretical baseline for interpreting forthcoming N-body simulations (Paper II) and supports a dynamical mechanism for ring confinement around irregular bodies that complements observational constraints.

Abstract

Three ring systems have been discovered to date around small irregular objects of the solar system (Chariklo, Haumea and Quaoar). For the three bodies, material is observed near the second-order 1/3 Spin-Orbit Resonance (SOR) with the central object, and in the case of Quaoar, a ring is also observed near the second-order resonance 5/7 SOR. This suggests that second-order SORs may play a central role in ring confinement. This paper aims at better understanding this role from a theoretical point of view. It also provides a basis to better interpret the results obtained from N-body simulations and presented in a companion paper. A Hamiltonian approach yields the topological structure of phase portraits for SORs of orders from one to five. Two cases of non-axisymmetric potentials are examined: a triaxial ellipsoid characterized by an elongation parameter C22 and a body with mass anomaly mu, a dimensionless parameter that measures the dipole component of the body's gravitational field. The estimated triaxial shape of Chariklo shows that its corotation points are marginally unstable, those of Haumea are largely unstable, while those of Quaoar are safely stable. The topologies of the phase portraits show that only first- (aka Lindblad) and second-order SORs can significantly perturb a dissipative collisional ring. We calculate the widths, the maximum eccentricities and excitation time scales associated with first- and second-order SORs, as a function of C22 and mu. Applications to Chariklo, Haumea and Quaoar using mu ~ 0.001 show that the first- and second-order SORs caused by their triaxial shapes excite large (>~ 0.1) orbital eccentricities on the particles, making the regions inside the 1/2 SOR inhospitable for rings. Conversely, the 1/3 and 5/7 SORs caused by mass anomalies excite moderate eccentricities (<~ 0.01), and are thus a more favorable place for the presence of a ring.

Figures (9)

• Figure 1: Representative phase portraits of resonances with orders $j$=1, 2, 3, 4 and 5, from top to bottom, respectively. Each phase portrait shows level curves of the Hamiltonians ${\cal H} (X,Y)$ given in Eq. \ref{['eq_H_X_Y']}, where the mixed variables $X$ and $Y$ (Eq. \ref{['eq_def_X_Y']}) define the eccentricity vector ${\bf e}$ (Eq. \ref{['eq_ecc_vector']}). The fixed elliptic points away from the origins correspond to maxima of ${\cal H} (X,Y)$. For each resonance, four representative values of $\Delta J$ decreasing from left to right have been considered to illustrate the varying topologies of the phase portraits. The homoclinic trajectories are drawn in red. In all the plots, the value of the parameter $\epsilon$ appearing in Eq. \ref{['eq_H_X_Y']} is taken as negative, as is the case for outer resonances. The topology of resonances with orders $j > 5$ are similar to the case $j$=5, except that there are $j$ islands instead of five, with widths that decrease as $j$ increases.
• Figure 2: Fixed points given by Eq. \ref{['eq_e_fixed_E']} as a function of $\Delta J$ for resonances of various orders. They are the solutions of Eqs. \ref{['eq_roots_j_1']}, \ref{['eq_roots_j_2']}, \ref{['eq_roots_j_3']}, \ref{['eq_roots_j_4']} and \ref{['eq_roots_j_geq_5']}. In all the plots, the values of $\epsilon'$ are taken as negative, as is the case for outer resonances. The cubic root $\epsilon'^{1/3}$ is then understood as the real root, i.e. ignoring the complex roots. The blue (resp. red) branches corresponding to stable elliptic (resp. unstable hyperbolic) points. Similarly, blue (resp. red) dots at the origin indicate a stable (resp. unstable) point. The units of all the plots are arbitrary. Panel (a): first-order resonances. The positions of two particular points are specified: the solution corresponding to $\Delta J=0$ and the pitchfork bifurcation point at the lower right. Panel (b): second-order resonances. The parabolic branches are the solutions of Eq. \ref{['eq_roots_j_2']}. The positions of three particular points are specified. Panel (c): third-order resonances. The parabolic branches are the solutions of Eq. \ref{['eq_roots_j_3']}. The positions of three particular points are specified. The origin of the phase portrait ($X_{\rm f} = E_{\rm f}$ =0) is stable everywhere, except for the value $\Delta J = 0$ (red dot), where it is hyperbolic. Panel (d): fourth-order resonances. The parabolic branches are the solutions of Eq. \ref{['eq_roots_j_4']}. The origin of the phase portrait ($X_{\rm f} = E_{\rm f}$ =0) is stable everywhere, except for large values of $\epsilon$, see Eq. \ref{['eq_range_epsilon_stability_origin_j_4']}. Panel (e): resonances of orders $j \geq 5$. In this case, $X_{\rm f} = E_{\rm f}$ (Eq. \ref{['eq_roots_j_geq_5']}), so that the stability of the points corresponding to each branch cannot be indicated on the plot, hence the black color used here. This plot is now indistinguishable from the unperturbed case ($\epsilon'=0$).
• Figure 3: Response to a 1$^{\rm st}$-order resonance. Left panel: the maximum eccentricity $e_{\rm max}$ reached by a particle initially on a circular orbit with modified semi-major axis $\Delta \overline{a}/a_0$, with $m < 0$ and $\epsilon' < 0$. The right (resp. left) branch of the function is given by Eq. \ref{['eq_emax_j_1_branch_1']} (resp. \ref{['eq_emax_j_1_branch_2']}). The value of $e_{\rm max}$ suffers a discontinuity at $\Delta \overline{a}/a_0 = (3/2)(m-1)|2\epsilon'|^{2/3}$, where $e_{\rm max}$ jumps from $|2\epsilon'|^{1/3}$ to $|16\epsilon'|^{1/3}$, the maximum possible eccentricity $e_{\rm peak}$. Right panel: the phase portrait corresponding to the discontinuity, with the homoclinic trajectory going through the origin. The red and blue points correspond to their counterparts shown in the left panel.
• Figure 4: The same as Fig. \ref{['fig_DJ_emax_1st_order']} for a 2$^{\rm nd}$-order resonance. Left panel: the function plotted here is given by Eq. \ref{['eq_emax_j_2']}. The value of $e_{\rm max}$ suffers a discontinuity at $\Delta \overline{a}/a_0 = -|(m-2)\epsilon'/2|$, where $e_{\rm max}$ jumps from zero to is maximum value $e_{\rm peak}= |4\epsilon'|^{1/2}$. Right panel: the phase portrait corresponding to the discontinuity. The red points correspond to their counterpart of the left panel.
• Figure 5: The same as Fig. \ref{['fig_DJ_emax_2nd_order']} for a third-order resonance. Left panel: the function plotted here is given by Eq. \ref{['eq_emax_j_3']}. Right panel: the phase portrait corresponding to the discontinuity at $\Delta \overline{a}/a_0= 0$. The red points correspond to their counterpart of the left panel.