Multidimensional extension of the continuity equation method for debris clouds evolution

Abstract

As the debris spatial density increases due to recent collisions and inoperative spacecraft, the probability of collisions in space grows. Even a collision involving small objects may produce thousands of fragments due to the high orbital velocity and the high energy released. The propagation of the trajectories of all the objects produced by a breakup would be prohibitive in terms of computational time; therefore, simplified models are needed to describe the consequences of a collision with a reasonable computational effort. The continuity approach can be applied to this purpose as it allows switching the point of view from the analysis of each single fragment to the study of the evolution of the debris cloud distribution of area-to-mass ratio and eccentricity among the fragments. Results for these three applications are shown and discussed in terms of accuracy compared to the numerical propagation and to the one-dimensional approach.

Closed-Loop Control of the Orbit Evolution of “Smart Dust” Swarms

“…Colombo [10], and debris by Letizia et al [11]. However the smart dust devices considered in this paper are assumed to be actively controlled through modulation of their lightness number  and pitch angle  defined by Eq.…”

“…Similar to modelling the evolution of interplanetary dust [7,8], nano-satellite constellations [9] and high area-to-mass spacecraft [12,13], the continuum evolution of the swarm can be obtained from a continuity equation linking the swarm density and the velocity vector field of the flow of 'smart dust' devices, as [11]   -n n n t…”

“…If both the source and sink terms are ignored in this equation, it is referred as the homogeneous evolution model. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Solutions to the continuum evolution equation can be achieved by propagating the initial data for the swarm forward to describe the evolution of n as a function of both  and . The initial data function is defined as…”

“…[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Moreover, the device deposition strategies proposed in Section III.II are considered to compensate for on-orbit failures, which act as a boundary controller in the classical PDE control problem. Compared with the active device controllers for individual devices in Section III.I, the boundary control is provided by the macroscopic deposition of new devices.…”

Wavelike Patterns in Precessing Elliptical Rings for Swarming Systems

“…In principle, the resulting density peaks could provide enhanced coverage [2]. Letizia et al [12] summarized applications of the continuity equation approach, including interplanetary dust [9], nanosatellites [11] and high area-to-mass SpaceChips [13] and developed a standard procedure for the analytic solution to the continuity equation in orbital element space. McInnes [14] studied the evolution of the density of swarms of self-propelled devices in heliocentric orbit with closed-form solutions developed analytically for several scenarios, such as the evolution of an infinite sheet or a finite disk, with on-orbit failures, and with the constant disposition of new devices at one boundary.…”