Powered Trajectory Studies for Low Thrust Space Vehicles

Abstract

and 0o as follows. Assume that initially r = r 0 , r = 0 and 0 = 0. Then, by Equation [46] 1 GM , . ^ 7? -= -77 + 5 o sin 0o r 0 h 2 r 0[50]and, from the first derivative of Equation [46] 0 = s 0 (l -r) cos 0o [51] It then follows that the initial position is at pericenter with 0o = TT/2 [52] So = -rr 1 53 h 2 r 0 r 0 hence, using Equation [48] so ~ v/ro [54] which in conjunction with Equation [48] implies that rj is approximately equal to the eccentricity. To get the maximum excursion of the radial distance r, it is only necessary to take the difference between r max and r m i n as obtained from Equation [46] and write n = 2r 0 s 0 = 2 77 [55]in agreement with Dobrowolski.In the case of circumferential thrust, the maximum radial distance in each cycle does not remain stationary. Its rate of growth is given by Equation [26] for the nearly circular orbit, i.e., Ar a /r a ~ 4 7rrj. This result agrees directly with that of Perkins (5) as can be seen by substituting A0 = 2 w in Equation [29] of Perkins 7 paper. ConclusionBriefly, application of the nonlinear techniques of Kryloff and Bogoliuboff has been shown to provide concise solutions to the problems of small radial thrust and small circumferential thrust previously discussed in the literature, and to the problem of intermittent thrust, which has not had previously published solutions. It is hoped that this work will inspire more extensive application of the method. Powered trajectory calculations have been carried out for missions from Earth to Venus, Mars,Jupiter and return. The procedure used was based upon an optimization which minimizes total energy requirements and maximizes payload. Data are presented which relate payload fraction, total mission time, and minimum specific power requirements. It is shown that propulsion plant weight-to-exhaust power ratios of 122 lb/kw and 22 lb/kw are adequate for missions ranging in difficulty from one-way trips to Mars with 10 per cent payload fraction to round trips to Mars with 30 per cent payload, respectively. In both cases the transfer is assumed to be between low altitude satellite orbits about the planets in question. D URING the last year a study of electrical propulsion has been carried out at the Lawrence Radiation Laboratory. In order to lay a proper foundation for such a study it was necessary to determine the relationship among mission requirements (characteristic velocity), propulsion system specific power, mission time, and payload fraction. Consequently, some theoretical studies of a simple nature were made, supplemented by some powered trajectory calculations on the IBM-704. The results, while certainly not optimal, do provide the desired results in an approximate form with minimum expenditure of effort and computer time. 28 ARS JOURNAL Downloaded by UNIVERSITY OF CALIFORNIA -DAVIS on February 5, 2015 | http://arc.aiaa.org |

A natural first approximation to low-thrust trajectories between satellite orbits of the earth and moon

“…The joining of apsides is seen to be a feature in common with the familiar Hohmann-type transfers. Some similarity can also be noted in the low-thrust interplanetary trajectories of Fox,8 which also are synthesized by comparing the osculating orbits associated with powered phases initiated at the two terminals. Fox's approximation involved only a single conic segment (with nonapsidal extremities), however, since only the sun's field was involved.…”