3.1 Hyperbolic cases

Let us begin with the most unstable orbits, that is with those that already during the 2 Myr integration became hyperbolic (see Table 1, upper part). For these 9 escaping asteroids it was not possible to determine synthetic proper elements, and they are removed from the output files. 14 additional bodies became hyperbolic in the extended 10 Myr integration (Table 1, lower part). For the sake of completeness, for these bodies udergoing hyperbolic escape within 10 Myr we have included in the output files the synthetic proper elements derived from the 2 Myr integrations, even if these values are highly unreliable and should be considered with extreme caution. In few cases (3480, 1998 OP12) the computation of the LCE failed because of overflow, due to too long interval between renormalizations; the LCE value has been set to zero manually. Note that we thus provide proper elements for a total of 10,256 asteroids.

 

Table 1: Upper part: asteroids that became hyperbolic in the 2 Myr integration. Lower part: additional bodies that escaped during the extended 10 Myr integration. Columns contain asteroid identifier, semimajor axis, eccentricity and inclination at the beginning of the integration, and comment on the relevant dynamical features that might give rise to chaotic diffusion. T_L denotes the Lyapunov time in years.

Asteroid	a0 (AU)	e0	I0 (deg)	Remark
1921	3.263	0.402	19.45	2/1
5164	3.643	0.504	19.84	outer belt
7458	3.830	0.137	1.81	outer belt
8119	2.963	0.166	0.61	7/3
1988DX1	2.828	0.175	8.13	5/2
1989TT3	2.820	0.157	3.00	5/2
1993SG13	2.958	0.228	6.85	7/3
1993TC14	3.304	0.273	0.35	2/1
2202T-1	3.122	0.329	4.14	2/1 ?
677	2.962	0.044	8.47	7/3 TL = 6060 yr
2658	3.076	0.290	9.32
3480	3.045	0.283	3.76
6626	3.025	0.388	2.12	9/4
7410	3.022	0.376	5.85	9/4
8560	2.959	0.068	13.38	7/3 TL = 6200 yr
1984LK	3.115	0.386	6.55	2/1 ?
1990OB2	3.035	0.335	11.68
1991WB	2.747	0.330	35.99	secular res.
1996EJ14	3.029	0.096	15.80	9/4
1998DD2	3.058	0.318	8.35
1998KY30	3.203	0.190	14.29	2:1 TL = 6200 yr
1998OP12	3.038	0.307	1.31
1998QO47	3.085	0.340	6.17

We have not performed a detailed analysis of the reasons for such extreme instability. However, as it can be seen from the Remark column in Table 1, most of these orbits are from the start either inside or very close to some low order mean motion resonances, such as 2/1, 5/2, 7/3 and 9/4. Most of the others have a large initial eccentricity, the only exception being one case in the most unstable portion of the outer belt, with $a\simeq 3.8$ AU. This leakage we found from the main belt to hyperbolic orbits, due to gravitational perturbations only, can be extrapolated to get an estimate of the total loss over the age of the Solar System. If we extrapolate linearly from 9 bodies lost in 2 Myr, we could have as much as 20,000 bodies lost over 4.5 Gyr. If we extrapolate from the 10 Myr integration, with 23 orbits going hyperbolic, that is $0.2\%$ of the considered sample, we could have $\simeq 10,000$ lost over the age of the solar system. Even if this oversimplified computation does not take into account the size of the original population, its collisional evolution, non gravitational perturbations, nor any other refinement, it clearly demonstrates that this is by no means a negligible phenomenon, and that the original number of asteroids must have been significantly larger than the present one.