• Journal of Astronomy and Space Sciences
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• v.22 no.3
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• pp.249-262
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• 2005

Optimization problems are formulated to calculate optimal impulses for deflecting Earth-Crossing Objects using a Nonlinear Programming. This formulation allows us to analyze the velocity changes in normal direction to the celestial body's orbital plane, which is neglected in many previous studies. The constrained optimization in the three-dimensional space is based on a patched conic method including the Earth's gravitational effects, and yields impulsive ${\Delta}V$ to deflect the target's orbit. The optimal solution is dependent on relative positions and velocities between the Earth and the Earth-crossing objects, and can be represented by optimal magnitude and angle of ${\Delta}V $ as a functions of a impulse time. The perpendicular component of ${\Delta}V $ to the orbit plane can sometimes play un-negligible role as the impulse time approaches the impact time. The optimal ${\Delta}V $ is increased when the original orbit of Earth-crossing object is more similar to the Earth's orbit, and is also exponentially increased as the impulse time reaches to the impact time. The analyses performed in present paper can be used to the deflection missions in the future.