J. R. Donnison

The Hill stability of triple minor planets in the Solar system

The triple asteroids and triple Kuiper belt objects (collectively called triple minor planets) in the Solar system are of particular interest to the scientific community since the discovery of the first triple asteroid system in 2004. In this paper, the Hill stability of the nine known triple minor planets in the Solar system is investigated. Seven of the systems are of large size ratio, that is, they consist of a larger primary and two moonlets, while the other two systems have components of comparable size. Each case is treated separately. For the triple minor planets that have large size ratio, the sufficient condition for Hill stability is expressed in a closed form. This is not possible for the systems with comparable size components, for which the Hill stability is assessed by a combination of analytical and numerical means. It is found that all the known triple minor planets are Hill stable, except 3749 Balam, for which the incomplete orbital parameters make the Hill stability of the system uncertain. This suggests that there might be more such stable triple minor planets in the Solar system yet to be observed. It is also shown that the Hill stability regions increase as the mutual inclination between the inner and outer orbits decreases, the semimajor axis ratio of the inner orbit with respect to the outer orbit decreases, and the mass ratio of the outer satellite with respect to the inner satellite increases. This study therefore provides useful information about dynamical properties of the triple minor planets in the Solar system.

A Bayesian approach to trends in cometary magnitude data

Abstract A Pareto distribution was used by Donnison (1990) to model the brightness of long- and short-period comets. The data was subsequently analysed for trends using a classical approach, including the “pool adjacent violaters algorithm” and likelihood ratio tests. In this new investigation the problems of variation in the brightness indices of long-period comets for a number of parameter ranges, including perihelion distance and discovery date, are examined from a Bayesian viewpoint using Gibbs sampling. The data set analysed is greatly enhanced and includes all those comets observed up to 1988. It is found that there is no evidence that the brightness index shows any ordering with discovery date using data from 584 ad to 1988 ad for comets with H 10 H 10

Statistical trends in cometary magnitude data

The brightness distributions of long-period comets with P > 200 years, and those short-period comets with P < 15 years (Jupiter family of comets) were modelled by Pareto power-law distributions. The models were fitted using maximum likelihood methods and analysed by a new trend analysis approach. The long-period comets were divided into a number of physical parameter ranges, notably by perihelion, inverse semi-major axis and discovery date, and examined for appropriate physical trends. Similarly, the short-period comets were examined for trends and compared with the long-period comets. Overwhelming evidence was found that the brightness indices increase with discovery date for the long-period comets. There was no statistical evidence that the brightness distribution of the long-period comets significantly altered as the 1/a values decreased. No trends were apparent in the brightness of long-period comets in terms of their perihelion distance for perihelia less than 2.5 a.u.

The distribution of short-period comet magnitudes

Short-period comets with P ≤ 15 yr represent one of the most complete comet samples. The magnitude distribution of these comets was analysed using a maximum likelihood method. The brightness (magnitude) index for the comets with H10 ≤ 11 mag was estimated together with the large sample errors and found to be 0.62 ± 0.09. It was clear that many faint comets with H10 > 11 mag remain to be discovered. Some of the faint, smaller comets have probably been removed from the distribution altogether.Observational selection was also apparent for the sample of comets with perihelia q < 1.5 AU. It was found that comets satisfying the combined criteria P ≤ 15 yr, H10 ≤ 11 mag, q < 1.5 AU probably represent the most complete set of comets available. The brightness index of this sample estimated by maximum likelihood was 0.69 ± 0.14. This translates into a mass distribution index s of 1.69 ± 0.14 indicating that most of the mass is contained in a few of the larger comets rather than spread throughout the smaller ones. This distribution, although modified by mass loss, is most likely to have been produced by a process of particle accretion.

The mass and position of possible past companions of the sun

Stars usually form as members of binary or multiple star systems, and it is likely that the Sun was no exception. The mass and position of possible past companions of the Sun is determined by considering the orbital stability of the Solar System. This is achieved by considering the stability of critical three-body subsets comprising the sun-planet-companion star which must be stable if the Solar System is to remain stable as a whole.