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

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2. Asteroid sample and calculations

In an ideal case the asteroid sample should include all asteroids there is down to a relevant size limit. For instance, to obtain a complete picture of the collision properties of asteroids larger than 50 km in diameter, all asteroids larger than [FORMULA] 1 km (or even smaller) should be included in the sample. This ideal case is, however, for several reasons not obtainable. The number of asteroids larger then 1 km is not known, but is certainly to large to be handled easily computationally. To overcome this problem a representative sample of asteroids should be selected down to a size where the sample is reasonably complete and large enough that most types of orbits are represted in the sample. When these criteria are satisfied the collision probabilities and collision velocities determined should be close to the `real' values obtained with a much larger population. This assumes that there are no significant differences between the orbits of the used population and the real population down to smaller sizes. Whether or not this is a valid assumption is difficult to assess, but it is a reasonable assumption to use.

In order to obtain a population as free as possible from observational biases, and representative of asteroids even in the outer-belt groups, all asteroids with [FORMULA] 50 km were included in the asteroid sample. Most certainly, only a few asteroids with [FORMULA] 50 km are still not discovered in the main-belt to Hilda region of the asteroid belt. (Cellino et al. 1991; Farinella & Davis 1992; Lagerkvist et al. 1996). In the Trojan region, however, there may be up to a factor of two more asteroids with [FORMULA] 50 km than presently known (C.-I. Lagerkvist, private communication). The orbital elements are from Bowell (1997) and, when available, the asteroid diameters were taken from IRAS data (Tedesco & Veeder 1992). The diameters for asteroids with no IRAS diameter was estimated from the absolute magnitude H (Bowell et al. 1989) by assigning asteroids with a semi-major axis [FORMULA] 2.7 AU an albedo [FORMULA] = 0.15, representative of S-type albedos, and when [FORMULA] 2.7 AU, the albedos were set to [FORMULA] = 0.04 to mimic typical albedos of C-type surfaces. The change in [FORMULA] was made to approximate the compositional change with heliocentric distance seen in the asteroid belt. Tests showed that the obtained population is not sensitive to the exact value of [FORMULA] or the limit in semi-major axis. An asteroid sample of 909 asteroids was obtained (see Table 1 for details of their orbital properties).


Table 1. Orbital parameters for the asteroid sample. The number of objects N, range in semi-major axis a, mean and 1-[FORMULA] of eccentricity e and inclination i are given for the four groups.

The equations of motion of the 909 asteroids were numerically integrated with the RADAU integrator (Everhart 1985) and with Jupiter and Saturn as perturbing planets. For each integration time step (set to 3 days) the distance between all asteroids (and planets) were computed. When the distance between two asteroids was less than 0.03 AU the position and velocity vectors for both asteroids were saved. However, for reasons explained below, only encounters with separation [FORMULA] 0.02 AU were used in the analysis. The recorded close encounters have to be further processed to calculate the final values of the position and velocity vectors at the (true) minimum distance of each encounter. A more accurate minimum distance was searched for in the time interval [FORMULA] 1.5 days, with a time step of 10 minutes with keplerian orbits of the two objects to obtain the final position and velocity vectors. The close encounter data will be biased towards low-velocity-deep encounters because the time spent by two asteroids within 0.03 AU from each other have an increasing probability to be less than the integration time step for high-velocity-shallow encounters. Therefore a significant part of these encounters will be lost. To get statistically unbiased close encounter data only encounters with distance [FORMULA] 0.02 AU will be used in the analysis. Together with the time step of the numerical integration (3 days), this ensures that close encounters with relative velocities [FORMULA] 26 [FORMULA] will be detected. This velocity is high enough to ensure that all close encounters occurring with the used asteroid sample will detected.

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© European Southern Observatory (ESO) 1998

Online publication: July 27, 1998