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

## 1. Introduction

Objects in the asteroid belt collide with each other due to the simple fact that they have elliptic orbits. Collisions of asteroids is believed to be an essential process behind the physical parameters of asteroids that we observe today, such as rotational properties, shapes, surface morphologies, and size distribution. Collision velocities and impact rates are also key factors to determine the lifetime of asteroids. In this paper, which is the third in a series about Hilda asteroids, their collisional properties will be investigated. See Dahlgren & Lagerkvist (1995), and Dahlgren et al. (1997) for the previous papers.

The first contributions to our understanding of collision velocities and impact rates in the main-belt have been made by Öpik (1951) and Wetherill (1967). Later studies by Farinella & Davis (1992) and Bottke et al. (1994) also used Wetherill's theory, which only needs the values of semi-major axes, eccentricities and inclinations of two objects to calculate the collision velocities and impact rate between them. Wetherill's theory was refined by Bottke et al. (1994) by also including the relative probability of all possible collision geometries between each asteroid pair, which earlier was assumed to be equal for all geometries. A different approach was taken by Vedder (1996) using a purely statistical method to derive the collisional properties of asteroids.

The collisional properties of the Trojan asteroids were studied by Marzari et al. (1996), performing numerical integrations of Trojan orbits, and analysing the close encounter data obtained during the numerical integration. They also included Hilda asteroids in their study because their orbits may intersect the orbits of Trojan asteroids.

The collisional properties of the Hilda asteroids have as yet not been studied as they were not included in the main-belt studies mentioned above. In the study by Marzari et al. (1996) main-belt objects were not included and therefore their results concerning the collision probabilities of the Hilda asteroids are of minor importance. Since the orbits of Hilda asteroids intersect orbits of both main-belt and Trojan asteroids, objects from the whole asteroid belt have to be accounted for to correctly investigate the collisional properties of Hilda asteroids. The aim of this paper is to determine the collisional properties of the Hilda population since these quantities are crucial when trying to improve our understanding of the rotational properties, shapes, and lifetimes of the Hilda asteroids. Extrapolating the collision probabilities and collision velocities from the main-belt studies or using the incomplete results from the Trojan study can lead to serious errors.

To study the collisional properties of the Hilda asteroids the equations of motion for a representative population of asteroids, from the inner parts of the main-belt to the Trojan clouds, have been numerically integrated and close encounters occurring between the asteroids have been detected. The close encounter data will be analysed to infer collision velocities and collision probabilities of objects in the asteroid belt. The analysis relies on the assumptions that there are no significant differences between the close encounter data and the objects that actually collide.

The advantage of the numerical method used in this study, compared to Wetherill's, is that the close encounter data will give accurate result even when the angular orbital elements of the asteroids have non-random distributions or couplings between them. This is the case for Hilda and Trojan asteroids due to their residence in the 3:2 and 1:1 mean motion resonances with Jupiter. Ideally, the integrations should be carried out for a sufficiently long time span so that any temporary fluctuations in the osculating elements will be averaged out. In this study the numerical integrations were carried out for 55 000 years which is about equal to the longest period in the osculating elements of the Hilda asteroids. This is likely to average out any temporary fluctuations in the derived collisional properties due to the ad hoc starting time of the numerical integrations (see also Farinella & Davis 1992; Marzari et al. 1996).

In the analysis the asteroid population has been divided into four groups: main-belt, Cybele, Hilda and Trojan asteroids. The four groups gave ten `collisional populations', namely: main-belt-main-belt (MM), Cybele-Cybele (CC), Hilda-Hilda (HH), L4 Trojan-L4 Trojan (TT4), L5 Trojan-L5 Trojan (TT5), Cybele-main-belt (CM), Hilda-Cybele (HC), Hilda-main-belt (HM), Hilda-L4 Trojan (HT4), and Hilda-L5 Trojan (HT5). These populations will be analysed separately, but the results will be discussed with emphasis on the Hilda asteroids since they are the primary concern of this paper. In the discussion of the collision velocities and collision probabilities the Trojan clouds have been treated as one unit, giving the Trojan-Trojan (TT) and the Hilda-Trojan (HT) collisional populations. Note that in the case of collision probabilities it has been taken into account that the Trojan clouds are two separate populations.

© European Southern Observatory (ESO) 1998

Online publication: July 27, 1998