Semi-Analytical Solution for the Optimal Low-Thrust Deflection of Near-Earth Objects

Abstract

This paper presents a semi-analytical solution of the asteroid deviation problem when a low-thrust action, inversely proportional to the square of the distance from the sun, is applied to the asteroid. The displacement of the asteroid at the minimum orbit interception distance from the Earth's orbit is computed through proximal motion equations as a function of the variation of the orbital elements. A set of semi-analytical formulas is then derived to compute the variation of the elements: Gauss planetary equations are averaged over one orbital revolution to give the secular variation of the elements, and their periodic components are approximated through a trigonometric expansion. Two formulations of the semi-analytical formulas, latitude and time formulation, are presented along with their accuracy against a full numerical integration of Gauss equations. It is shown that the semi-analytical approach provides a significant savings in computational time while maintaining a good accuracy. Finally, some examples of deviation missions are presented as an application of the proposed semi-analytical theory. In particular, the semi-analytical formulas are used in conjunction with a multi-objective optimization algorithm to find the set of Pareto-optimal mission options that minimizes the asteroid warning time and the spacecraft mass while maximizing the orbital deviation. Nomenclature A MOID = matrix form of proximal motion equations a = acceleration vector, km=s 2 a = semimajor axis, km b = semiminor axis, km d m = diameter of the mirror, m E = incomplete elliptic integral of the second kind e = eccentricity e r = relative error F = incomplete elliptic integral of the first kind G t = matrix form of the Gauss equations h = angular momentum, km 2 =s I sp = specific impulse of the spacecraft engine, s i = inclination, deg j = integer number k = proportionality constant of the acceleration, km 3 =s 2 M = mean anomaly, deg m d = dry mass, kg m 0 = mass into space, kg n = angular velocity, s 1 p = semilatus rectum, km r = orbital radius, km T = transition matrix T NEO = asteroid nominal orbital period, s or days t = time, s t d = departure time from the Earth, s t e = time when the low-thrust arc ends, s t i = interception time, s t MOID = time at the minimum orbit interception distance point, s t w = warning time, s v = velocity vector, km=s v = orbital velocity, km=s = vector of the orbital parameters r = vector distance of the asteroid from Earth at the minimum orbit interception distance, km r = deviation vector in the Hill coordinate frame, km s = component of the deviation vector, km v = impulsive maneuver vector, km=s = orbital element difference between the perturbed and the nominal orbit = true anomaly, deg = argument of the latitude, deg = sun gravitational constant, km 3 =s 2 = argument of the ascending node, deg ! = argument of the perigee, deg Subscripts h = direction of the angular momentum h = tangential direction in the orbital plane n = normal direction in the orbital plane

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