Citation

Maccone C. Acta Astronaut. 2006; 58(12): 662-670.

Copyright

(Copyright © 2006, Elsevier Publishing)

DOI

10.1016/j.actaastro.2006.01.016

PMID

unavailable

Abstract

A system of two space bases housing missiles for an efficient Planetary Defense of the Earth from asteroids and comets was firstly proposed by this author in 2002. It was then shown that the five Lagrangian points of the Earth-Moon system lead naturally to only two unmistakable locations of these two space bases within the sphere of influence of the Earth. These locations are the two Lagrangian points L1 (in between the Earth and the Moon) and L3 (in the direction opposite to the Moon from the Earth). In fact, placing missiles based at L1 and L3 would enable the missiles to deflect the trajectory of incoming asteroids by hitting them orthogonally to their impact trajectory toward the Earth, thus maximizing the deflection at best. It was also shown that confocal conics are the only class of missile trajectories fulfilling this "best orthogonal deflection" requirement. The mathematical theory developed by the author in the years 2002-2004 was just the beginning of a more expanded research program about the Planetary Defense. In fact, while those papers developed the formal Keplerian theory of the Optimal Planetary Defense achievable from the Earth-Moon Lagrangian points L1 and L3, this paper is devoted to the proof of a simple "(small) asteroid deflection law" relating directly the following variables to each other: (1) the speed of the arriving asteroid with respect to the Earth (known from the astrometric observations); (2) the asteroid's size and density (also supposed to be known from astronomical observations of various types); (3) the "security radius" of the Earth, that is, the minimal sphere around the Earth outside which we must force the asteroid to fly if we want to be safe on Earth. Typically, we assume the security radius to equal about 10,000 km from the Earth center, but this number might be changed by more refined analyses, especially in the case of "rubble pile" asteroids; (4) the distance from the Earth of the two Lagrangian points L1 and L3 where the defense missiles are to be housed; (5) the deflecting missile's data, namely its mass and especially its "extra-boost", that is, the extra-energy by which the missile must hit the asteroid to achieve the requested minimal deflection outside the security radius around the Earth.This discovery of the simple "asteroid deflection law" presented in this paper was possible because: (1) In the vicinity of the Earth, the hyperbola of the arriving asteroid is nearly the same as its own asymptote, namely, the asteroid's hyperbola is very much like a straight line. We call this approximation the line/circle approximation. Although "rough" compared to the ordinary Keplerian theory, this approximation simplifies the mathematical problem to such an extent that two simple, final equations can be derived. (2) The confocal missile trajectory, orthogonal to this straight line, ceases then to be an ellipse to become just a circle centered at the Earth. This fact also simplifies things greatly. Our results are thus to be regarded as a good engineering approximation, valid for a preliminary astronautical design of the missiles and bases at L1 and L3.Still, many more sophisticated refinements would be needed for a complete Planetary Defense System: (1) taking into account many perturbation forces of all kinds acting on both the asteroids and missiles shot from L1 and L3; (2) adding more (non-optimal) trajectories of missiles shot from either the Lagrangian points L4 and L5 of the Earth-Moon system or from the surface of the Moon itself; (3) encompassing the full range of missiles currently available to the USA (and possibly other countries) so as to really see "which missiles could divert which asteroids", even just within the very simplified scheme proposed in this paper.In summary: outlined for the first time in February 2002, our Confocal Planetary Defense concept is a simplified Keplerian Theory that already proved simple enough to catch the attention of scholars, popular writers, and representatives of the US Military. These developments would hopefully mark the beginning of a general mathematical vision for building an efficient Planetary Defense System in space and in the vicinity of the Earth, although not on the surface of the Earth itself! We must make a real progress beyond academic papers, Hollywood movies and secret military plans, before asteroids like 99942 Apophis get close enough to destroy us in 2029 or a little later.