The pertinent conclusion to be drawn from the foregoing discussion is that a spacecraft must be maneuvered to result in close encounters with the numbered asteroids because the radius of the sphere in which encounters are likely to take place (0.1 AU) is large compared to the distance required for a measurement (107* AU). It is a good approximation to assume that a spacecraft must be guided to hit the target asteroid. MISSION ANALYSIS Multiple asteroid flyby missions that involve maneuvering a spacecraft away from a nominal trajectory with onboard propulsion may be categorized as follows: (1) Opportunities resulting from randomly generated trajectories that combine flybys of several asteroids, none of which are preselected, into one mission requiring a small total Av for maneuvering (2) Multiple flyby missions that are required to include particular asteroids of known scientific interest; for example, Ceres (3) Missions to major planets that include favorable opportunities for maneuvering close to one or more asteroids Additional categories might involve circularizing the spacecraft orbit in the asteroid belt or rendezvous with a major asteroid. Such maneuvers require large values of Av compared to those resulting from the present study of multiple flybys, hence they belong to a different type of mission that involves much larger spacecraft and an advanced propulsion system such as solar electric-ion engine propulsion. The basic logic for examining multiple flyby missions is contained in a computer program that compares cartesian position coordinates of a spacecraft with the positions, at the same time, of the 1748 numbered asteroids. The asteroid positions are obtained by using the osculating elements in the 1971 Ephemeris volume as Kepler elements and by advancing the listed mean anomalies along unperturbed ellipses to the desired time. The positions so generated are considered to be exact for the purposes of this study. The spacecraft trajectory may be a Keplerian ellipse in the ecliptic, generated internally by the computer program, or an externally generated trajectory, as in the case of missions involving major planets. As the positions of spacecraft and asteroids are computed, their relative separations are compared against a preselected search radius, and all asteroids passing within the search radius are counted as possible targets for that particular mission. For a search radius of 0.1 AU, a sizable number of encounters can be expected on the average—about six, according to equation (1), for a trajectory to 3 AU. To determine impulsive Av requirements for multiple flyby missions, various asteroid flyby sequences are examined after possible targets are identified. First, the spacecraft is retargeted at Earth launch to intercept one of the asteroids. Then, an algorithm that solves Lambert's problem is utilized to determine the impulsive Av required for the spacecraft to encounter the second target at the time of closest approach to the second target. This process is then repeated as necessary to complete the desired sequence. The asteroids are assumed to be massless points that are intercepted exactly by the spacecraft. Figure 1 illustrates how the Av maneuvers are performed for a three-asteroid sequence. The Av required at Earth for the spacecraft to intercept the first target is assumed to be provided by the launch vehicle and is not charged against the capability of the spacecraft. RESULTS Random Search Trajectories For spacecraft trajectories that are not constrained to intercept a major planet or some particular asteroid, a Keplerian ellipse in the ecliptic can be specified by the launch date and aphelion; the perihelion is assumed to be at the launch point. Table I shows a listing of asteroid encounters resulting from 11 such trajectories, launched between Julian dates 2444500 (late 1980) and 2445000 (early 1982) at 50 day intervals to an aphelion of 3 AU. The search radius is 0.1 AU. The number of encounters ranges from 8 to 15, and the average number is 10 per mission, as compared to the 6 per mission predicted by equation (1). The agreement between these two numbers is acceptable, considering the simplifying assumptions made in deriving equation (1) and the fact that the listings in table I are not sufficient to constitute a statistically significant sample. Additional details concerning the seven closest encounters during the trajectory launched on Julian date 2444800 (mid-1981) are given in table II; this launch date yielded the largest number of different targets of the 11 launch dates shown in table I. The encounters take place over a time of 659 days, representing a range in spacecraft true anomaly of 115°. Certainly, it will not be possible, in general, to maneuver to all asteroids passing within 0.1 AU of a spacecraft. For example, consecutive flybys of all the asteroids listed in table II would require an impulsive Av of 41.6 km/s, computed as indicated in TABLE I.—Numbered Asteroids Passing Within 0.1 AU of a Spacecraft in a 1 by 3 AU Orbit Launched on the Julian Date Shown the previous section. When encounters with the spacecraft occur in relatively rapid succession (for example, 1515 – 149 and 1740 - 561 in table II), it will generally be required to make a choice between two targets. There are, however, several opportunities for flying close to three asteroids for a total impulsive Av of less than 1 km/s; such missions are listed in table III. It is always assumed that the spacecraft is launched from Earth to intercept the first target in a multiple flyby sequence. All seven of the asteroids in table II are TABLE II.-Data for the Seven Closest Asteroid Encounters With a Spacecraft Launched on Julian Date 2444800 Into a 1 by 3 AU Orbit Asteroid Julian Radius, True Minimum Relative no. date, AU anomaly" separation, velocity, 244XXXX 10° km km/s 4971 1.97 118° 5.3 11.4 4980 2.02 121 5.4 11.1 5134 2.72 154 6.7 8.3 5422 2.91 194 7.6 5.1 5508 2.69 207 7.1 9.2 5528 2.63 211 4.0 10.5 5630 2.16 233 2.8 12.3 *Of the spacecraft. included in one or more of the triple flybys listed in table III. A four-asteroid mission, 1515 - 1674 – 561 - 1720, is possible with a total impulsive Av of only 0.8 km/s. Trajectories to a Preselected Asteroid For missions that are constrained to include a close flyby of a particular asteroid, a Keplerian ellipse in the ecliptic is still used for the spacecraft trajectory; but the launch date and aphelion are adjusted to include an encounter with the desired asteroid when it passes through the ecliptic. Data for generating trajectories to Ceres and Vesta are readily available in the NASA Planetary Flight Handbook (Lockheed Missiles & Space Co., 1966). As an example for this study, the trajectory resulting from a late 1975 launch to TABLE IV.-Data for the Five Closest Asteroid Encounters With a Space. craft Constrained to Pass Near 1 Ceres During a 1975 Launch Opportunity Asteroid Julian Radius, True Minimum Relative no. date, AU anomaly" separation, velocity, 244XXXX 10° km km/s Ceres" was examined for other close asteroid approach opportunities. Data for the five closest approaches are summarized in table IV. They take place over a period of 612 days and a spacecraft true anomaly range of 81°. For the trajectory studied here, 1 Ceres was encountered last, but this would not be generally true. Of the several theoretically possible multiple asteroid missions for this trajectory, two triple flybys can be performed with an impulsive Av of less than 1 km/s: 632 – 946 -> 1 with Av = 0.73 km/s and 632 – 947-> 1 with Av = 0.24 km/s. Asteroid Encounters on Trajectories to Jupiter The possibility of passing close to an asteroid enroute to a major planet provides extra motivation for performing the planetary mission if the required maneuvering can be assumed to have no significant effect on the primary mission goal. As an example of such a mission, a 1975 trajectory to Jupiter has been examined for close approaches to asteroids, and those passing within 0.1 AU are tabulated in table V. Of course, because of the relatively short time spent in the asteroid belt on this trajectory, there are not as many close approach opportunities as for the previous two mission types. Nevertheless, the encounter sequence 666 – 396 – Jupiter can be performed with an impulsive Av of only 0.52 km/s. It should be noted that the flyby velocities for this type of trajectory are necessarily higher than for trajectories having an aphelion in the asteroid belt. TABLE V.—Asteroid Encounters on a Trajectory to Jupiter Launched During a 1975 Opportunity Asteroid Julian Radius, Minimum Relative no. date, AU separation, velocity, 244XXXX 10° km km/s 2734 1.97 12.9 16.7 2736 1.99 9.7 16.2 2751 2.12 8.6 18.0 2875 3.06 7.1 10.2 DISCUSSION OF RESULTS It is clear that the Av results presented above should be taken only as representative of what can be expected for multiple asteroid flyby missions. Most important, computing asteroid positions from Kepler's equations does "Ceres has an aphelion of 2.9776 AU, a perihelion of 2.5574 AU, and an orbital inclination of 10°6. |