Fig. 1. Resonance sticking in the scattered disk: some individual examples (Particles 1–6: panels a–f). We used osculating orbital elements in the heliocentric frame. Left: small dots represent the orbital evolution plotted every 0.5–1 Myr for a period of 4 Gyr. Dashed vertical lines refer to the location of relevant resonances. Perihelia of 30 and 35 AU are shown by two curves (upper and lower, respectively). All particles started on Neptune-encountering orbits, and with a<50 AU. The final orbital elements of the particle are indicated by gray circles (out of range in panel a). Right: evolution of semimajor axis a, perihelion distance q and inclination i with time.

Fig. 2. Relative importance of resonances (described in the form r:s) during resonance sticking for 255 particles that remained in the scattered disk after 4 Gyr. First, we determined the ratio of the total time spent in a particular type of resonance (s=1,2,…,12) to the total time in all resonances. This quantity was later weighted by the number of individual resonances for a given s, and normalized to the value found for r:1 resonances.

Fig. 3. Distribution of the total time spent in resonances to the total dynamical lifetime of 255 particles that remained in the scattered disk after 4 Gyr. See text for discussions.

Fig. 4. Cumulative time spent in resonances during resonance sticking and semimajor axes. The time spent in resonances within 150, 200 and 250 AU represents approximately 90, 97 and 99% of that computed for all resonance captures experienced by the 255 particles over 4 Gyr, respectively.

Fig. 5. Relative resonance stickiness of relevant resonances (described as r:s) detected in the evolution of 255 particles that remained in the scattered disk after 4 Gyr. Resonance stickiness illustrates the likelihood of capture into a resonance, and the ability of that resonance to retain a captured object (i.e., timescale) (see text for details). Resonance stickiness was normalized to the largest value, found at the 6:1 resonance (a=99.4 AU). Resonances at a<47.8 AU were omitted due to their overlap with the initial conditions, not allowing a proper calculation of relative resonance stickiness for these resonances. Relative resonance stickiness is given as a function of resonance order (top) and argument s (bottom). We show resonances up to 25th order and with argument s ranging from 1 to 12, with values increasing from black to light gray.

Fig. 6. Resonance strength of relevant resonances (described as r:s) detected in the evolution of 255 particles that remained in the scattered disk after 4 Gyr. Resonance strength was calculated by solving the SR function (see the main text, and Gallardo, 2006b) and it is given as a function of resonance order (top) and argument s (bottom). We show resonances up to 25th order and with argument s ranging from 1 to 12, with values increasing from black to light gray. Compare with Fig. 5.

Fig. 7. Orbital distribution of 255 particles that remained in the scattered disk after 4 Gyr (black circles). The orbital elements (heliocentric frame) were averaged over the last 100 Myr. From the left, dashed vertical lines refer to the location of the 10:3, 6:1, 13:2, 7:1, 8:1, 21:2, 11:1, 23:2, 27:2 and 14:1 resonances. These resonances played an important role in lifting the perihelion of some objects.

Objects locked in resonances with Neptune in the scattered disk after 4 Gyr^d