Rings around irregular bodies. II. Numerical simulations of the 1/3 spin-orbit resonance confinement and applications to Chariklo

TL;DR

This work demonstrates that the 1/3 spin-orbit resonance can confine collisional ring material around irregular bodies like Chariklo by transferring forced resonant energy into free Lindblad modes, provided the perturbation strength μ overcomes viscous spreading. Using full 3D collisional simulations, the authors derive a confinement criterion $k\mu^2 \gtrsim \tau R^2$ (with $k\sim 4\times10^{-5}$) and show that Chariklo-like rings require $\mu \gtrsim 10^{-3}$ for confinement when $\tau\sim1$ and $R$ is meter-scale. They also demonstrate that a small outer satellite with $\mu_s \sim 10^{-6}-10^{-7}$ can prevent long-term leakage, and that self-gravity can enhance viscosity, increasing the required μ by a factor of a few to ten depending on $r_h$ and $\tau$. The findings suggest the observed rings around Chariklo, Haumea, and Quaoar can be maintained by 1/3 SOR confinement, with implications for the presence of small shepherds and for ring dynamics around other irregular bodies.

Abstract

Rings have been found around Chariklo, Haumea and Quaoar, three small objects of the Solar System. All these rings are observed near the second-order spin-orbit resonances (SORs) 1/3 or 5/7 with the central body, suggesting an active confinement mechanism by these resonances. Our goal is to understand how collisional rings can be confined near second-order SORs in spite of the fact that they force self-intersecting streamlines.We use full 3D numerical simulations that treat rings of inelastically colliding particles orbiting non-axisymmetric central bodies, characterized by a dimensionless mass anomaly parameter mu. While most of our simulations ignore self-gravity, a few runs include gravitational interactions between particles, providing preliminary results on the effect of self-gravity on the ring confinement. The 1/3 SOR can confine ring material, by transferring the forced resonant mode into free Lindblad modes. We derive a criterion ensuring that the 1/3 SOR counteracts viscous spreading. Assuming meter-sized ring particles, and tau~1, this requires a threshold value mu > 1e-3 in Chariklo's case. The confinement is not permanent as a slow outward leakage of particles is observed in our simulations. This leakage can be halted by an outside moonlet with a mass of ~1e-7 - 1e-6 relative to Chariklo, corresponding to subkilometer-sized objects. With self-gravity, the ring viscosity nu increases by a factor of few in low-tau rings due to gravitational encounters. For large tau, self-gravity wakes enhance nu by a factor of ~100 compared to a non-gravitating ring, requiring ~10-fold larger mu since the threshold value increases proportional to square-root of nu.

Figures (26)

• Figure 1: The comparison between the numerical and theoretical responses of test particles to various SORs. Numerical integrations have followed the motion of 10,000 test particles initially distributed on circular orbits, during 10,000 revolutions of the central body. The maximum eccentricities $e_{\rm max}$ reached by these particles are plotted in black as a function of $\Omega_{\rm B}/n =(a/a_{\rm cor})^{3/2}$, and are compared with the analytical estimates of Paper I (red or green curves). Upper panel: the case of a spherical body with a mass anomaly $\mu=10^{-4}$. Lower panel: the case of a homogeneous triaxial ellipsoid with elongation $\epsilon_{\rm elon}=0.01$ and oblateness $f=0$. In the case of mass anomaly the strongest responses are at the outer first-order Lindblad resonances corresponding to commensurabilities $n/\Omega_{\rm B}= m/(m-1)$, with $m=-1,-2,-3$... The insert in the upper panel is a zoom on the $\Omega_{\rm B}/n= 3$ region, using a 100 times larger mass anomaly of $\mu=0.01$. The response to the second-order 1/3 resonance is now visible and compared to the analytical estimate in green. In a case of an elongated body, the $\pi$-symmetry of the perturbation imposes even values $m=-2,-4,-6$... for the first-order resonances. In this case the strongest second-order resonance ($m=-2$ and $j=2$) is also visible, with the analytical response plotted in green. The insert in the lower panel is a zoom on the $\Omega_{\rm B}/n= 3$ region: the fourth-order 2/6 resonance has no signature even when using a large value $\epsilon_{\rm elon}=0.20$.
• Figure 2: The responses of test particles to the first-order 2/3 and second-order 1/3 SORs. Left column: the symbols represent the maximum eccentricities $e_{\rm max}$ reached by particles initially on circular orbits for three values of the mass anomaly $\mu$, near the 2/3 SOR (upper panel) and the 1/3 SOR (lower panel). The gray dashed curves are the analytical estimates of $e_{\rm max}$ displayed in Figs. 3 and 4 of Paper I. The points are plotted against the normalized distance to the resonance, $\Delta a/W_{\rm res}= (\overline{a} - a_0)/W_{\rm res}$, where $a_0$ is the radius of the circular orbit at exact resonance, $\overline{a}$ is the modified semimajor axis (Eq. \ref{['eq_modified_a']}) and $W_{\rm res}$ is the width of the resonance (Eq. \ref{['eq_peak']}). Right column: the same with the timescales $T_{\rm res}$ necessary to reach the maximum eccentricities $e_{\rm max}$. This figure shows that our numerical integrations reproduce satisfactorily the calculated distribution of $e_{\rm max}$, as well as the $\mu$-scaling expected from Eqs. \ref{['eq_peak']} and \ref{['eq_tres']}.
• Figure 3: Comparison of eccentricity evolution at the first-order 2/3 and second-order 1/3 SORs. The black curves follow the evolution of test particles released near exact resonance, while the red dashed lines display the theoretical prediction, that is a linear growth rate for the first-order SOR and an exponential growth rate for the second-order SOR.
• Figure 4: Comparison of the maximum eccentricity $e_{\rm max}$ reached by particles at the second-order 1/3 SOR (blue) compared with the first-order 2/3 (red) and 1/2 (green) SORs. Three values $\mu=0.1,0.01,0.001$ are compared. For $\mu=0.1$, the theoretical $e_{\rm max}$ associated with the 2/3 and 1/2 resonances at the location of $1/3$ resonance are $30\%$ and $8\%$ of that due 1/3 resonance, respectively. For other $\mu$'s the ratios at the 1/3 SOR location scale proportional to $\mu^{1/2}$.
• Figure 5: Three cases showing the combined effects of collisions and the 1/3 SOR on the ring confinement. The left frames show the time evolution of the vertical angular momentum ($L_z$) distribution of the particles around the 1/3 SOR at $L_z = 1.44$. The magenta lines are the average value of $L_z$ and the red lines show the RMS of the particle eccentricities; the full y-range of the frame corresponds to $e=0.1$. The inserts are polar plots of the particle positions, shown again in the right column in cartesian coordinates. For better viewing, in the cartesian projections both the width of the ring around its center position at each azimuth and the deviations of this center position from the overall mean distance have been expanded by a factor of five. We compare three cases: (i) a ring of collisionless test particles perturbed by a $\mu=0.1$ mass anomaly on an otherwise spherical central body (upper row); (ii) colliding particles around a spherical central body without perturbation ($\mu=0$, middle row); and (iii) a ring of colliding particles with $\mu=0.1$ (lower row). All simulations use the same initial conditions with 30,000 particles placed initially in an annulus $r=2.02-2.14$ straddling the $1/3$ SOR at semimajor axis $a =2.08$. In the collisional simulations the particle radius $R=10^{-3}$ corresponds to about 200 m for Chariklo's ring particles and yields an initial geometric optical depth $\tau_0=0.06$. In the case of a mass anomaly, the perturbation is turned on linearly during the first 50 revolutions of the central body.