Analysis and implementation of in-plane stationkeeping of continuously perturbed Walker constellations

The stationkeeping of symmetric Walker constellations is analyzed by considering the perturbations arising from a high order and degree Earth gravity field and the solar radiation pressure. These perturbations act differently on each group of spacecraft flying in a given orbital plane, causing a differential drift effect that would disrupt the initial symmetry of the constellation. The analysis is based on the consideration of a fictitious set of rotating reference frames that move with the spacecraft in the mean sense, but drift at a rate equal to the average drift rate experienced by all the vehicles over an extended period. The frames are also allowed to experience the $J_2$-precession such that each vehicle is allowed to drift in 3D relative to its frame. A two-impulse rendezvous maneuver is then constructed to bring each vehicle to the center of its frame as soon as a given tolerance deadband is about to be violated. This paper illustrates the computations associated with the stationkeeping of a generic Walker constellation by maneuvering each leading spacecraft within an orbit plane and calculating the associated velocity changes required for controlling the in-plane motions in an exacting sense, at least for the first series of maneuvers. The analysis can be easily extended to lower flying constellations, which experience additional perturbations due to drag.     

Effects of stand-off bursts on rubble-pile targets: Evaluation of a hazardous asteroid mitigation strategy

We explore the aftereffects of stand-off burst mitigation on kilometer-scale rubble pile asteroids. We use a simple model of X-ray energy deposition to calculate the impulse transferred to the target, in particular to burst-facing blocks on the target surface. The impulse allows us to estimate an initial velocity field for the blocks on the outer side of the target facing the burst. We model the dynamics using an N-body polyhedron program built on the Open Dynamics Engine, a “physics engine” that integrates the dynamical equations for objects of general shapes and includes collision detection, friction, and dissipation.We tested several different models for target objects: rubble piles with different mass distributions, a “brick-pile” made of closely fitting blocks and zero void space, and a non-spherical “contact binary” rubble pile. Objects were bound together by self-gravity and friction/inelastic restitution with no other cohesive forces. Our fiducial cases involved objects of m=3.5×10¹² kg (corresponding to a radius of 0.7 km for the bulk object), an X-ray yield of 1 megaton, and stand-off burst distances of R=0.8–2.5 km from the target center of mass.Kilometer-scale rubble piles are robust to stand-off bursts of a yield ($Y\sim 1$ megaton) that would be sufficient to provide an effective velocity change ($\mathrm{\Delta }v\sim 0.05\mathrm{m}{\mathrm{s}}^{-1}$). Disaggregation involving some tens of percent of the target mass happens immediately after the impulse; the bulk of the object re-accretes on a few gravitational timescales, and the final deflected target contains over 95% (typically, 98–99%) of the original mass. Off-center components of the mitigation impulse and the target mass distribution cause a small amount of induced spin and off-axis components of velocity change. The off-axis velocity component amounts to an angular deviation of $\sim$ 0.05–0.1 radians from the nominal impulse vector, which may be important for mitigation planning.