6. Mechanics of a satellite cluster. Methods for estimating the probability of their maximal approach in flight

Annotation: The methods are proposed (analytical and numerical based on motion equations integration) to evaluate probability of first approaches to small distances of satellites of cluster uncontrolled in flight in long time intervals. As the number of satellites injected into area of one base orbit grows, the necessity of evaluating such probability constantly increases – already at present their number in some cases exceeds hundred units. In flight, such satellites form in limited area of space rather compact cluster; the satellite density in such cluster exceeds by many orders the density of operating space objects at their functioning altitudes. Due to somewhat different satellite orbiting periods, the distances between them in flight direction continuously change, different precession motion of orbital planes determines their angular spread – approach in flight. It was determined that maximal probability of approach of whatever pair of satellites of cluster to small distances is the case if in some neighborhood of numbers of their flight orbits, simultaneously two events are realized – the satellites approach to minimal distances in flight direction and angular spread of their orb ital planes is close to zero. The conditions are determined of separation of whatever two satellites of cluster (their separation directions and velocities) – that ensure simultaneous realization of the above events in some neighborhood of number of flight orbits. The analytical relations were obtained that allow determining the corresponding numerical values of satellite approach parameters. For particular case – satellite separation at the equator – maximal probability of approach of two satellites of cluster to small distances is the case when their relative separation velocities are equal in flight direction and in perpendicular to this direction. For the option of injecting 12 satellites to the area of one base orbit of ~ 650 km altitude and  98 inclination, the parameters of satellites separation at the equator were determined that realize their uniform dispersion in the first orbits of autonomous flight. For 2 pairs (out of 66 formed for considered injection case) the conditions of maximal probability of their first approaches to small distances are realized. Using two developed methods evaluations of such probability were obtained.