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The optimum thrust programs for power-limited propulsion systems are used to generate rendezvous trajectories from Earth to Mars for various flight times and launch dates during the years 1968 to 1971. The manner in which the propulsion requirements vary with flight time and launch date is considered, and a comparison of vehicle performance using the variable and constant thrust programs is presented. The existence of optimum launch dates is interpreted in terms of certain transversality conditions derivable from the calculus of variations. A brief comparison of the advanced propulsion vehicle and the ballistic vehicle propulsion requirements is made for Earth-Mars rendezvous trajectories.T HE emergence of advanced propulsion for interplanetary flights has generated great interest in the application of optimization theory to advanced propulsion vehicle systems and to trajectory design. It becomes necessary to obtain fairly accurate estimates of the payload capabilities of advanced propulsion vehicles for various interplanetary missions. In Ref. 1, there appeared the results from a series of trajectories to the planets Venus and Mars. An optimum variable thrust program was used to generate these trajectories. Moreover, certain terminal conditions, such as the orientation of the terminal orbit and the terminal position on the orbit, were left unspecified, and corresponding transversality conditions derived from the calculus of variations were satisfied instead.This paper is concerned with the problem in which all end conditions, as determined by the planetary ephemerides, are specified, and the main purpose is to show the manner in which the propulsion requirements vary both with flight time and with launch date. This procedure is analogous to the problem in ballistic trajectories in which the velocity increments required for interplanetary missions are determined (2,3). 3 In advanced propulsion trajectories, however, the propulsion intervals constitute a significant portion of the trajectory, and, therefore, the thrust program employed becomes quite important in payload studies, and optimization theory as applied to the trajectory analysis is of considerable use. A comparison of vehicle performance will be made between the use of an optimum variable thrust program and the use of an optimum constant thrust program. Optimum Thrust EquationsIn order to develop an optimum thrust program that extremizes some terminal quantity indicative of vehicle performance, it is necessary to include the constraints of the system. For the power-limited propulsion system, the constraints are the equations of motion of the vehicle and an equation describing the fact that the amount of kinetic power contained in the exhaust propellant is constrained. Generally, the kinetic power depends on the efficiency of power conversion from the nuclear powerplant of the vehicle, and the efficiency, in turn, is dependent on the exhaust velocity employed. In this treatment, the kinetic power is constant, which is the case for the constant thrust pro...

This paper describes the optimization of Solar-Sail h t e r p l a n e t e r y trajectories using the Calculus of Variations. A n "ideal" plane sail is assumed and the control variables a r e the c h a racteristic thrust acceleration and the orientation of the s a i l with respect to the direction of sunlight, square force field model in three dimensions and an inverse square variation of s o l a r p r e s s u r e with heliocentric distance. The analysis employs a two-body inverseThe optimization has been applied to rendezvous trajectories for each of the t e r r e s t r i a l planets, In addition an example of an asteroid roundtrip ( e . g . , sample return mission) i s p r esented to demonstrate the capabilities of a solars a i l spacecraft.

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