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these vehicles is reviewed by Bartz and Horsewood (1969). This review includes an extensive bibliography.
The remainder of this paper deals with the rendezvous by means of a low-thrust, solar electrically propelled vehicle because (1) this approach is generally better from a payload and flight-time basis compared to the high-thrust method and (2) it has a large degree of flexibility and can be applied to a variety of missions. It should be emphasized that here we are concentrating on rendezvous missions only, which are necessary precursors to the more complex landing or sample-return missions. The remainder of the paper deals with our results to date.
In choosing an asteroid for a mission target, one would first assume that size would be the major criterion and pick the largest one because it would present the largest surface for study. Thus a mission to Ceres would be studied at the outset as well as missions to the other asteroids if they are significantly less demanding in terms of energy and/or flight time. Such other possible targets may traverse the main asteroid belt near Ceres but have smaller inclinations, or else they may have smaller orbits passing inside that of Mars. In addition to Ceres, we have considered here missions to three bright asteroids with inclinations successively smaller than Ceres and to one with a significantly smaller orbit than Ceres as illustrated by the orbital elements of table I. Pallas and Juno were not considered because of their high inclinations of 34°8 and 13°0, respectively.
To estimate flight times for transfers to be accomplished in less than one revolution about the Sun, we indicate the value of the trip time To for a Hohmann transfer from Earth to a circular orbit at the radius of the semimajor axis. This serves as a guide because the Hohmann transfer, which uses a heliocentric central angle of 180°, is an optimal impulsive transfer between two circular orbits. It generally occurs, however, that optimal low-thrust rendezvous trajectories utilize central angles considerably greater than 180° so that flight times can be expected to be 20 to 50 percent greater than Th.
TABLE I.—Orbital Properties of Selected Asteroids
Asteroid a, TH, Synodic no. Name i AU e dāys period, yr
! . . . . . . . . . . . Ceres 10°6 2.77 0.076 473 1.28 4 . . . . . . . . . . . Vesta 7.1 2.36 .089 398 1.40 10. . . . . . . . . . Hygiea 3.8 3.15 ..100 546 1.22 20 . . . . . . . . . . Massalia .7 2.41 . 143 407 1.37 433 . . . . . . . . . Eros 10.8 1.46 .223 249 2.29
Fortunately, for solar electric propulsion, the choice of flight time is not particularly critical for determining mission feasibility.
RESULTS Launch Date Selection
Ranges of possible launch dates to be used as initial estimates of detailed searches may be found using a simple procedure that is described below. We first replace the orbits of Earth and the target asteroid by circular coplanar orbits, and we choose the time of flight and central angle to be traversed. The procedure is illustrated in figure 2, which shows launch date possibilities for Vesta from 1975 to 1977. We plot the mean anomaly of Vesta M2 versus the mean anomaly of Earth M1 as it actually occurs for the launch years we wish to consider. That is
Using the time of flight to and the central angle A0 to be traversed on the transfer orbit, we plot the mean anomaly on the target orbit M2 at the time of departure versus the mean anomaly on Earth orbit M1 at the time of departure. The relationship to be plotted is obtained by expressing the longitude of the arrival point in terms of the motion of the spacecraft along the transfer orbit and the motion of the target. Thus we obtain
where is the longitude of perihelion and n is the mean orbital rate. Intersections of lines for equations (1) and (3) indicate values of M1 for which launches meeting the requirements chosen for to and A0 selected are possible. The corresponding calendar date can be read from the abscissa. We may choose a different value for A6 and plot another curve of M2 versus M1 thus obtaining a range of launch dates for the chosen range of central angles. Having found a reasonable range of launch dates, we make a search over that range by means of a computer program that determines payloads and accurate trajectory characteristics for the particular type of mission under consideration. The optimum launch dates found for the solar electricpropulsion missions are indicated by an O.
Equations (2) and (3) can be modified to be more representative of the true situation by including the effect of the eccentricities of the two orbits. True anomaly intervals are used to obtain the longitude at arrival and thus M2 versus M1. The relation is now a curve that oscillates about the straight line and must be computed for each case of A0 and to desired. Such a graph is shown for Eros for 1974 to 1977 in figure 3.
For each asteroid chosen, the low-thrust program CHEBYTOP (Han, Johnson, and Itzen, 1969) was used in a launch-day scan mode to locate the departure data that resulted in greatest payload for a given flight time. The launch vehicle chosen was the Titan IIID/Centaur, except for Eros in which the Atlas/Centaur was used. The remaining spacecraft and launch parameters required as input for Ceres, Vesta, Massalia, and Hygiea rendezvous are as follows:
(1) Hyperbolic departure velocity at Earth, Vh1 = 5 km/s
(2) Total mass injected (at whl = 5 km/s), 3450 kg
(3) Thrustor specific impulse, Isp = 3500 s (exhaust velocity c = 34.3 km/s)
(4) Tankage factor, 6 percent (propellant and tank mass = 1.06 X mass of propellant needed)
(5) Engine efficiency, 0.667
(6) Propulsion system specific mass, 30 kg/kW
The power level at 1 AU Po is calculated by the program, but the payload results quoted have been scaled to correspond to a power level of 16 kW. The optimum hyperbolic escape velocity depends on the size of the target orbit, and for Eros it is very close to 2 km/s. With a Titan IIID/Centaur launch vehicle, payloads for Eros missions would be more than 2000 kg, which is considered to be more than necessary for preliminary missions. For this reason, the smaller Atlas/Centaur launch vehicle, which results in a total injected mass of 1000 kg at whl = 2 km/s, was used. Otherwise the values listed pertain also to Eros missions. Mission data results are presented in table II. Here flight time, payload, and encounter geometry are shown for at least two launch dates for each asteroid in the period from 1974 to 1977. It is interesting to note that inclination does
TABLE II.-Solar Electric Asteroid Rendezvous Missions
*Atlas/Centaur launch vehicle (fully optimized trajectory data) (Po 10 kW).