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Astron. Astrophys. 336, 1056-1064 (1998)

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4. Collision probabilities

With 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 probability

To obtain more quantitative results the close encounter data have been used to calculate the intrinsic collision probability [FORMULA] between objects (Wetherill 1967). The intrinsic collision probability describes the probability for collision between two objects, and [FORMULA] is the probability of collision between two objects with [FORMULA] = 1 km, where [FORMULA] and [FORMULA] 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 N within a specific distance R to an object is expected to be a function of the form [FORMULA], because the cross section of a close encounter is [FORMULA]. To verify this the number of close encounters as a function of distance were calculated for some of the asteroids, and the best fitting [FORMULA] function were determined. The results for three Hilda asteroids are given in Fig. 3. The encounter data of these Hilda objects were selected to validate that the [FORMULA] approximation is acceptable for wide ranges in N, orbital eccentricity and inclination of the objects.

[FIGURE] Fig. 3. Number of encounter as a function of distance for three Hilda asteroids and the best fitting [FORMULA] functions.

Using the [FORMULA] dependence, where N is the number of close encounters within a specific distance (in this case R = 0.02 AU), the intrinsic collision probability of an object is

[EQUATION]

where T is the time interval used to obtain N close encounters, and [FORMULA] is the number of possible collision pairs in the interaction population(s). In the case of the MM, CC, HH, TT4, and TT5 populations, each object is a possible target for all the other [FORMULA] objects in the population, giving [FORMULA] possible pairs. In the case of the other collisional populations (CM, HM, HC, and HT) the number of possible pairs is [FORMULA] where [FORMULA] and [FORMULA] are the number of objects in the respective population. When calculating the [FORMULA] for objects with respect to all interacting populations the number of pairs were calculated as [FORMULA] where [FORMULA] is the number of internal pairs and [FORMULA] is the number of external pairs between the [FORMULA] interacting populations. Since [FORMULA] is the collision probability per cross section unit ([FORMULA]) of the object, the total collision probability of an object can be obtained by multiplying with the actual cross section of the object in question.

The intrinsic collision probability of each object has been calculated with Eq. 1, and the resulting mean intrinsic collision probability [FORMULA] of the different populations are given in Table 4, where the given dispersions are the standard error of the mean. Also the calculated [FORMULA] for the main-belt, Cybele, Hilda, and Trojan groups are given in Table 4.


[TABLE]

Table 4. Mean intrinsic collision probability [FORMULA] for the collisional populations and for the main-belt, Cybele, Hilda, and Trojan groups.


Among the collisional populations involving Hilda asteroids, the highest [FORMULA] 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 [FORMULA] 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[FORMULA] 104 years, thereby not allowing the longest period in the osculating orbital eccentricity of the Hilda orbits to complete a whole cycle. This can give a biased [FORMULA] since the collision probability of the Hilda objects varies with time as the shape of their orbits varies with time (around a stable mean value).

The [FORMULA] 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 [FORMULA] 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 [FORMULA] of HT collisions is not affected by the incomplete Trojan population, due to the normalisation ([FORMULA]) of [FORMULA]. The [FORMULA] found for HC collisions is relatively high, reflecting that objects from the two populations can collide during large fractions of their orbits.

The obtained [FORMULA] for MM collisions are higher than the main-belt value [FORMULA] = 2.86 [FORMULA] 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 [FORMULA] 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 asteroids

Due to the normalisation of the intrinsic collision probabilities, the relative importance of the entries in Table 4 are not obvious, since [FORMULA] 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

[EQUATION]

where [FORMULA] 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 ([FORMULA] 50 km), [FORMULA] will change if the relative numbers of objects in the groups are different down to the considered size. This dependence of [FORMULA] is also discussed below in connection to the incomplete Trojan sample. The distributions of [FORMULA] 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 [FORMULA] 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 [FORMULA] of HT collisions found for some of the Hilda asteroids are based on few encounters and should be interpreted with some caution.


[TABLE]

Table 5. Mean collision probability [FORMULA] for the collisional populations involving Hilda asteroids, and for the main-belt, Cybele, Hilda, and Trojan groups.


[FIGURE] Fig. 4. Histograms of collision probabilities [FORMULA] for the 39 Hilda asteroids versus different populations: HH) Hilda-Hilda, HM) Hilda-main-belt, HC) Hilda-Cybele, and HT) Hilda-Trojan collisions.

The collision probabilities [FORMULA] of the 909 asteroids included in the asteroid sample versus their semi-major axis are given in Fig. 5. The main-belt objects ([FORMULA]) have [FORMULA] approximately between 40-85 [FORMULA] with peak values at about 2.6 AU, and decreasing towards the inner and outer parts of the main-belt. The Cybele asteroids ([FORMULA]) have decreasing [FORMULA] through the group, ranging from about 45 at 3.3 AU down to 10 [FORMULA] at 3.6 AU. The Hilda asteroids (o) at 4.0 AU have [FORMULA] from about 3 to 30 [FORMULA]. At 5.2 AU the Trojan asteroids have [FORMULA] from 2 to 20 [FORMULA]. The mean values and standard error of [FORMULA] for the main-belt, Cybele, Hilda, and Trojan groups are listed in Table 5. There is considerable scatter around [FORMULA] for individual objects in the populations, therefore [FORMULA] of the population is a bad approximation for most individual objects.

[FIGURE] Fig. 5. Collision probability [FORMULA] versus average semi-major axis during time T for main-belt ([FORMULA]), Cybele ([FORMULA]), Hilda (o), and Trojan ([FORMULA]) asteroids. Note that [FORMULA] for the Trojan asteroids are lower limits due to undiscovered objects with [FORMULA] 50 km.

The mean collision probability [FORMULA] 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 [FORMULA] of the Trojan asteroids are somewhat lower than for the Hilda asteroids, however, [FORMULA] of the Trojan asteroids are lower limits due to undiscovered Trojan asteroids with [FORMULA] 50 km. Increasing the population of Trojan asteroids larger than 50 km with [FORMULA] 15% will make [FORMULA] 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 [FORMULA] 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.

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© European Southern Observatory (ESO) 1998

Online publication: July 27, 1998
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