## 4. Collision probabilitiesWith the number of close encounters recorded for each population (given in Table 2), we can obtain a first glimpse of the relative importance of the different populations to the total collision probability of the Hilda asteroids. Collisions experienced by Hilda asteroids are dominated by HM collisions (74%), whereas the low velocity HH collisions only contribute 7% to the number of collisions involving Hilda asteroids. Similar contributions come from the HC and HT collisions, which add respectively, 10% and 7% to the number of collisions involving Hilda objects. The results for the other groups show that the Trojan asteroids collide with themselves (in their respective two clouds) and to a small extent (5%) with Hilda asteroids. Collisions involving objects from the Cybele group are mostly from the MC collisions (91%) with small contributions from CC (5%) and HC (4%) populations. Collisions involving main-belt objects have small contributions from the CH (5%) and the HM (2%) populations added to their total number of collisions. ## 4.1. Intrinsic collision probabilityTo obtain more quantitative results the close encounter data have been used to calculate the intrinsic collision probability between objects (Wetherill 1967). The intrinsic collision probability describes the probability for collision between two objects, and is the probability of collision between two objects with = 1 km, where and are their respective radii. The intrinsic collision probability is only dependent on the orbital elements of the two objects, and when two orbits do not intersect the intrinsic collision probability is zero. The number of close encounters
Using the dependence, where where T is the time interval used to obtain The intrinsic collision probability of each object has been calculated with Eq. 1, and the resulting mean intrinsic collision probability of the different populations are given in Table 4, where the given dispersions are the standard error of the mean. Also the calculated for the main-belt, Cybele, Hilda, and Trojan groups are given in Table 4.
Among the collisional populations involving Hilda asteroids, the
highest was found for HH collisions. However,
the small Hilda population results in a small contribution to the
total collision probability of Hilda asteroids. The determined
of HH collisions is 14% lower than the value
obtained by Marzari et al. (1996). This difference can be due to the
different (larger) Hilda population which has slightly different
orbital parameters used by Marzari and co-workers. However, a more
plausible explanation comes from the fact that their numerical
integrations were carried out for about 1.1
10 The for HM collisions are much lower than for HH collisions, this is mainly due to that HM collisions only are possible when Hilda asteroids are close to their perihelia. An even lower was found for HT collisions, this is because the collisions can only occur during the small fraction of the orbit when Hilda objects are close to one of the Trojan clouds. The of HT collisions is not affected by the incomplete Trojan population, due to the normalisation () of . The found for HC collisions is relatively high, reflecting that objects from the two populations can collide during large fractions of their orbits. The obtained for MM collisions are higher than the main-belt value = 2.86 obtained by Bottke et al. (1994). The difference is probably due to the different `main-belt' used by Bottke and co-workers, they also included Cybele asteroids in their main-belt. Therefore a more reasonable comparison can be made with the result for the main-belt group, which also includes collisions from objects in the Cybele and Hilda populations, but excludes CC collisions. The obtained for the HT and TT populations are also consistent with the mean values obtained by Marzari et al. (1996). ## 4.2. Collision probabilities of Hilda asteroidsDue to the normalisation of the intrinsic collision probabilities, the relative importance of the entries in Table 4 are not obvious, since is the collision probability per object in the different populations. In order to obtain a better view of the importance of the collisional populations involving Hilda objects, the following collision probability was derived where is the number of Hilda objects. However, this normalisation is dependent on the numbers of objects in the main-belt, Cybele, Hilda and Trojan groups. When extrapolating the results down to smaller sizes ( 50 km), will change if the relative numbers of objects in the groups are different down to the considered size. This dependence of is also discussed below in connection to the incomplete Trojan sample. The distributions of for the 39 Hilda objects with their interacting populations (HH, HM, HC, and HT) are given in Fig. 4, and mean values are listed in Table 5. The dominating contribution from HM collisions is evident, while the contributions from the HH, HC, and HT populations are quite equal, and about a factor ten smaller then the HM collisions. The distributions of HH and HM are quite narrow, indicating similar collision probabilities between most Hilda and Cybele objects, whereas HT collisions have a wide range of collision probabilities. However, the very low of HT collisions found for some of the Hilda asteroids are based on few encounters and should be interpreted with some caution.
The collision probabilities of the 909
asteroids included in the asteroid sample versus their semi-major axis
are given in Fig. 5. The main-belt objects ()
have approximately between 40-85
with peak values at about 2.6 AU, and
decreasing towards the inner and outer parts of the main-belt. The
Cybele asteroids () have decreasing
through the group, ranging from about 45 at 3.3
AU down to 10 at 3.6 AU. The Hilda asteroids
(
The mean collision probability for Hilda asteroids is about 3.5 times lower than for objects in the main-belt, and about 2 times lower than in the Cybele population. The determined of the Trojan asteroids are somewhat lower than for the Hilda asteroids, however, of the Trojan asteroids are lower limits due to undiscovered Trojan asteroids with 50 km. Increasing the population of Trojan asteroids larger than 50 km with 15% will make for the Trojan and Hilda populations about equal. But as noted above the Trojan population may be incomplete by as much as a factor of two. This will increase the collision probabilities among the Trojans up to about a factor of six. This makes is very likely that the Hilda asteroids have the lowest collision probabilities of the asteroids considered in this paper. The main reason for the low of the Hilda asteroids are that they are in `void regions' during a large part of their orbit, out of reach for both main-belt and Trojan asteroids. Most objects in the main-belt and Trojan clouds can however, collide with objects anywhere in their orbits. © European Southern Observatory (ESO) 1998 Online publication: July 27, 1998 |