Two-Body Optimization for Deflecting Earth-Crossing Asteroids

Abstract

We present a formulation of a e nite-dimensional optimization problem associated with the dee ection of Earthcrossing asteroids. The performance measure is minimizing the delta-V requirement for achieving a minimum target separation distance. A number of astrodynamical constraints are identie ed and modeled. The constrained optimization problem is numerically solved using a sequential quadratic programming method. Our numerical analysis indicates that the minimum delta-V requirement is not a monotonically decreasing function of warning time; rather, there is a e ner structure associated with the orbital period of the colliding asteroid. The analysis tool presented here may be used for optimizing the interceptor mission for impact mitigation. HE crash of the comet Shoemaker-Levy 9 into Jupiter in July 1994 was a remarkable event. Watching the results 1;2 from the impact, we felt a chilling warning of the possibility of a similar eventhappeningtotheEarth. 3 Thereareatleast1000Earth-crossing asteroids capable of global environmental catastrophe upon Earthimpact, and e ve new ones are discovered every year. 4 Crude historical data suggest that we can expect an impact greater than the equivalent of 10 Mtons of TNTabout once per century on average. 3 About 50,000 years ago the Barringer crater (1.5 km in diameter ) in Arizona was formed by an explosion estimated to be equivalent to about20 Mtons of TNT. 5 Arecentlargeatmospheric explosion was the great Tunguska bolide (»20 Mtons ofTNT )of 1908, which was caused by about a 60-m-diam comet or asteroid and exploded with the force of 1000 Hiroshima bombs. (One Hiroshima bomb has an explosive yield equivalent to about 20 ktons of TNT. ) 6 Asa resultof thepossibility of an asteroidorcometimpacting the Earth, several workshops have been held to study the fundamentals of the impact and the impact mitigation problem. The concentration of the workshops has been related primarily on the detection problem, assessing the magnitude of the threat, and impact effects and hazards to Earth, as well as the political implications of developing an impact mitigation capability. Two spacecraft exploration missions, Near Earth Asteroid Rendezvous (NEAR)7 and Clementine, 8 have included intercepts of asteroids as a major part oftheirmission to study the nature of asteroids. Despite an increase in interest on the impact mitigation problem, 9i12 little astrodynamical analysis has been performed; in particular, the mathematical optimization problem has received scant attention. 13;14 A number of dynamics and control problems in hazard mitigation are dee ned in Ref. 14. Here, we further one of these problems in formulating and solving the astrodynamical optimization problem. The dynamics of the problem is based on a two-dimensional, two-body approximation. The analysis centers on how optimal impulses applied to an Earth-crossing asteroid (ECA)at various points on its orbit affect the outcome when there is a presumption of collisionotherwise.Theperformance measureisminimizingthe delta-V required for achieving a minimum target separation distance. The constrained optimization problem is numerically solved using the sequential quadratic programming (SQP) method that is available in the MATLAB TM optimization toolbox. 15 Our numerical analysis indicates that the minimum delta-V requirement is not a monotonically decreasing function of warning time; rather, there is a e ner structure associated with the orbital period of the colliding asteroid.