ASTEROID SECULAR DYNAMICS: CERES' FINGERPRINT IDENTIFIED

To identify which ones, if any, among the different possible dynamical mechanisms are actually at work here, we performed a set of numerical integrations. For this purpose we employed the ORBIT9 integrator embedded in the multipurpose OrbFit package.⁴ The dynamical model includes the gravitational effects of the Sun and the four outer planets, from Jupiter to Neptune. It also accounts for the Yarkovsky thermal effect, a subtle non-gravitational force due to the recoil force of anisotropically emitted thermal radiation by a rotating body (Bottke et al. 2006), causing mainly a secular drift in the semimajor axis. The indirect effect of the inner planets is accounted for by applying a barycentric correction to the initial conditions.

Our simulations follow the long-term orbital evolution of test particles initially distributed randomly inside an ellipse determined by the Gauss equations. This ellipse corresponds to the dispersion of the Hoffmeister family members immediately after the breakup event, assuming an isotropic ejection of the fragments from the parent body.

The total number of particles used is 1678, the same as the number of asteroids we currently identified as members of the family. The family membership is determined utilizing the hierarchical clustering method and standard metric as proposed by Zappala et al. (1990).

For simplicity, the Yarkovsky effect is approximated in terms of a pure along-track acceleration, inducing on average the same semimajor axis drift speed ${da}/{dt}$ as predicted from theory.⁵ Assuming an isotropic distribution of spin axes in space, to each particle we randomly assign a value from the interval $\pm {({da}/{dt})}_{\mathrm{max}}$, where ${({da}/{dt})}_{\mathrm{max}}$ is the estimated maximum of the semimajor axis drift speed due to the Yarkovsky force. The value of ${({da}/{dt})}_{\mathrm{max}}$ is determined using a model of the Yarkovsky effect developed by Vokrouhlický (1998, 1999), and assuming thermal parameters appropriate for regolith-covered C-type objects. In particular, we adopt values of ${\rho }_{s}$ = ${\rho }_{b}$ = 1300 $\mathrm{kg}\;{{\rm{m}}}^{-3}$ for the surface and bulk densities (Carry 2012), Γ = 250 ${\rm{J}}\;{{\rm{m}}}^{-2}\;{{\rm{s}}}^{-1/2}\;{{\rm{K}}}^{-1}$ for the surface thermal inertia (Delbó & Tanga 2009), and = 0.95 for the thermal emissivity parameter. In this way we found that for a body of D = 1 km in diameter ${({da}/{dt})}_{\mathrm{max}}$ is about $4\times {10}^{-4}$ AU Myr⁻¹. Next, we select sizes of the test particles equal to sizes of the Hoffmeister family members, estimated using their absolute magnitudes provided by AstDys database⁶ , and geometric albedo of p_v = 0.047 (Masiero et al. 2011). Finally, since the Yarkovsky effect scales as $\propto 1/D$, the particle sizes are then used to calculate corresponding values of ${({da}/{dt})}_{\mathrm{max}}$ for each particle by scaling from the reference value for objects of D = 1 km.

The orbits of the test particles are propagated for 300 Myr, which is a rough estimate of the age of the Hoffmeister family (Nesvorný et al. 2005; Spoto et al. 2015). Time series of mean orbital elements⁷ are produced using on-line digital filtering (Carpino et al. 1987). Then, for each particle we compute the synthetic proper elements (Knežević & Milani 2000) for consecutive intervals of 10 Myr. This allows us to study the evolution of the family in the space of proper orbital elements.

If our first dynamical model were complete we should be able to reproduce the current shape of the family. However, we found that the shape cannot be reproduced with the afore described model. In particular, in these simulations we observed only a dispersion of the semimajor axis caused by the Yarkovsky effect, but no evolution in inclination (Figure 2). This implies that neither mean-motion nor secular resonances involving the major outer planets are responsible for the strange shape of the Hoffmeister family.

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In order to clarify the situation we turned our attention to the inner planets, and their possible role in the evolution of the family. For a representative sample of about 200 test particles we repeated the above described simulations using a model with seven planets (from Venus to Neptune), which also includes the Yarkovsky thermal force. However, the seven-planet model did not give any different results, which remain practically the same as the one obtained within the model with the four outer planets only. Thus, obviously there is still something missing in the dynamical model.

Being left with almost no other option, we turn our attention to asteroid (1) Ceres. Having a proper orbital semimajor axis of 2.767 AU, and therefore being inside the range covered by the family members, it seems to be the only remaining candidate. Hence, we again numerically integrated the same test particles, using the four-planet dynamical model but this time also including Ceres as a perturbing body.⁸

These new simulations already after about 150 Myr very nearly matched the current spreading of the family in the semimajor axis versus inclination plane, clearly implicating Ceres as the culprit for what we see today. The striking feature observed in these runs is a fast dispersion of orbital inclinations for objects with semimajor axis of about 2.78 AU. After 15 Myr of evolution the spread in sine of inclination is 3 times larger than the initial one, as can be seen in the middle panel in Figure 2.

The fact that the main perturber is Ceres raises a very important question. What is the exact mechanism by which Ceres is perturbing the members of the Hoffmeister family to such a high degree? The current paradigm suggests this may be the result of close encounters, or it might be the consequence of the 1/1 mean motion resonance with Ceres. However, our simulations undoubtedly show that none of these two mechanisms could explain the evolution of the family. Actually, we found that most of the evolution is taking place within a narrow range of the semimajor axis. However, this range does not correspond to the location of the 1/1 resonance with Ceres, neither is there any reason for close encounters to affect only objects within this specific range of semimajor axes. Moreover, it is very unlikely that these two effects would primarily affect orbital inclinations. Finally, regardless of the mechanism, such a large perturbation on asteroids caused by Ceres has never been observed in the asteroid belt. This situation motivates us to test some other mechanisms which are at work in this region, despite generally being accepted to be negligible. These are the secular resonances with asteroid Ceres.

Analyzing the secular frequencies of the Hoffmeister family members we immediately notice that some of these are very close to the nodal frequency of Ceres (${s}_{{\rm{c}}}=-59.17$ arcsec yr⁻¹). This is an interesting fact because the secular resonances involving the nodal frequencies are known to affect mainly the orbital inclination.

To better understand the reasons for the dispersion in inclination, and to determine the possible role of the secular resonance with Ceres, we pick a few particles that experienced significant changes in inclination during our numerical simulations, and analyzed their behavior in more detail. This analysis identified a mechanism responsible for the evolution of orbital inclinations, revealing a completely new role of Ceres in asteroid dynamics.

As an illustration, in Figure 3 we show the evolution of one test particle. This particle was initially located at a semimajor axis of 2.788 AU, with its Yarkovsky induced drift set to be negative, forcing it to move toward the Sun. After about 65 Myr this particle enters the region where the fast dispersion in orbital inclination has been observed. During the time spent in this area, the particle's inclination has experienced a fast increase, with the average value jumping from about 44–49.

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Let us recall here that a resonance occurs when the corresponding critical angle librates.⁹ In the case of the $s-{s}_{{\rm{c}}}$ secular resonance the critical angle is $\sigma ={\rm{\Omega }}-{{\rm{\Omega }}}_{{\rm{c}}}$, where Ω and ${{\rm{\Omega }}}_{{\rm{c}}}$ are the longitude of the ascending node of an asteroid and Ceres, respectively. Clearly, as can be seen in Figure 3, the period of increase in inclination exactly corresponds to the period of libration of the critical angle of the ${\nu }_{1{\rm{c}}}=s-{s}_{{\rm{c}}}$ secular resonance. This correlation is a direct proof that this resonance is responsible for the evolution and observed spread in the orbital inclination for asteroids belonging to the Hoffmeister asteroid family. In Figure 2 we plotted the location of the ${\nu }_{1{\rm{c}}}$ resonance.

Though our numerical simulations clearly show that passing through the ${\nu }_{1{\rm{c}}}$ resonance may cause significant changes in orbital inclination, we further investigated the mechanism. A key to understanding the observed behavior are the cyclic oscillations in inclination for objects trapped inside this resonance (see Figure 4). In the scenario with the Yarkovsky effect included in the model, some of the objects are reaching the border of the ${\nu }_{1{\rm{c}}}$ while steadily drifting in semimajor axis, and enter it at random value of the inclination cycle. During the time spent inside the resonance their inclination is continuously repeating the cycles. However, as the semimajor axis is evolving due to the Yarkovsky effect, the objects must sooner or later reach the other border of the ${\nu }_{1{\rm{c}}}$, and subsequently exit from the resonance. As the exit also happens at a random value of the inclination cycle, the values of orbital inclination with which bodies enter the cycle typically differ from those with which they exit. In this way the ${\nu }_{1{\rm{c}}}$ resonance changes the inclination of objects that cross it. Certainly, this mechanism would not work without the Yarkovsky effect.

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Interestingly, this process is very similar to the one observed inside another relatively weak secular resonance, namely, the $g+2{g}_{5}-3{g}_{6}$, which affects members of the Koronis asteroid family (Bottke et al. 2001).

It is also worth mentioning that although passage across the ${\nu }_{1{\rm{c}}}$ secular resonance could result in a very fast change in orbital inclination, the total changes are limited and could not exceed the maximal variations. The amplitude of variations in orbital inclination is not the same for all objects, but it is generally similar to the one shown in Figure 4.

Moreover, we have found that our test particles typically spend 15–30 Myr inside the ${\nu }_{1{\rm{c}}}$, before being moved outside by the Yarkovsky effect. Thus, as a libration period is very long (about 40 Myr), most of the particles spend less than one librating cycle inside this resonance (Figure 3).

Finally, let us re-examine the role of ${z}_{1}$ for the members of the Hoffmeister family. The results presented above clearly indicate that without Ceres in the model, the ${z}_{1}$ does not affect the family. Still, with Ceres included in the dynamical model ${z}_{1}$ may only affect a limited number of members. Our analysis has shown that about 5% of family members could reach this resonance after their inclination is pumped-up, at least a bit, by the ${\nu }_{1{\rm{c}}}$ resonance (see Figure 2). The orbital inclination of these objects is then additionally dispersed by ${z}_{1}$, being the main reason for the slightly different distributions toward high and low inclinations. Moreover, although the analysis along this line is beyond the scope of the Letter, we noticed that the ${z}_{1}$ resonance also slightly affects the orbital eccentricity of family members.

Nevertheless, the role of the ${z}_{1}$ in the dynamical evolution of the Hoffmaister family members is minor compared to the role of the ${\nu }_{1{\rm{c}}}$ secular resonance. Thus, the evolution of the family is almost completely determined by the combined effect of the ${\nu }_{1{\rm{c}}}$ resonance and the Yarkovsky effect, and would be practically the same even if the ${z}_{1}$ is not that close.