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this graph is not to say that one technique is better than another, but to bring out the general features of the advantages and disadvantages of the two complementary techniques. This graph is for a flyby velocity of 1 km/s. Other curves with flyby velocities from 0.1 to 10 km/s show the same general result. As we can see, the Doppler tracking technique is preferable for the larger asteroids, but becomes quite insensitive for the smaller asteroids, even for very close flybys. This general conclusion that the Doppler tracking technique is not suitable for flyby missions to the smaller asteroids is discussed in greater detail in the paper by John D. Anderson in these proceedings." (The Doppler tracking curve in fig. 4 is taken from fig. 3 in his paper.) The gravity gradiometer technique will give results for all asteroids above 1 km, but its measurement range for the larger asteroids is poorer than the Doppler tracking technique. Future increases in the accuracy of either system will not change these general conclusions significantly because of the rapid falloff of both curves. A change in sensitivity of an order of magnitude will only shift the curves a factor of 2 in asteroid radius or in flyby altitude.
ASTEROID RENDEZVOUS MISSIONS
When we investigate techniques for mass measurement and mass anomaly measurement that are applicable to a rendezvous mission to an asteroid, we find four techniques that can be considered: orbital velocity tracking, orbital period measurement, gravity gradient measurement, and acceleration measurement (after landing). All of these techniques can give accurate measurement of the mass of the larger asteroids, although the orbital velocity and the accelerometer techniques become less accurate for the smaller asteroids.
The orbital velocity of a spacecraft about an asteroid
ranges from 300 m/s for Ceres to 9 mm/s for a 10 m radius asteroid. The velocity about the larger asteroids is high enough that present Doppler velocity tracking techniques are more than adequate for an accurate mass measurement. For the smaller asteroids, however, it is better to measure the time for one orbital period rather than the orbital velocity directly. The orbital period for a close orbit is independent of the mass of the asteroid and is a function of the average asteroid density
The period ranges from about 3 hr for an iceball to 1 hr for a very dense asteroid.
If the rendezvous vehicle lands on the asteroid, then we can use gravimeters or the spacecraft navigation accelerometers to measure the gravity force to obtain an estimate of the mass. The acceleration field
ranges from 0.03 g for Ceres to 10-6 g for a 10 m radius asteroid. Although quite small, these acceleration levels can be measured to high accuracy by any number of available accelerometers and gravimeters. Both the accelerometer technique and the orbital period technique, however, are limited to obtaining an estimate of the total mass or average density of the asteroid. If we are interested in obtaining data on the internal density distribution of the asteroid, the use of the Doppler velocity tracking and the gravity gradiometer techniques from orbit are most suitable. The horizontal gravity gradient of an asteroid is
T GM 4 G
The data that can be obtained on the internal mass distribution of an asteroid from Doppler tracking and gravity gradiometer measurements using an orbiting vehicle are compared in the following figures. Figure 5 shows a schematic of the hypothetical asteroid that was used in the computer simulations. The asteroid is 100 km in radius and has an average density of 3.5 g/cm3. Embedded in this asteroid are spherical mass anomaly regions with radii of 1, 3, 10, and 30 km and a density difference of 0.5 g/cm3. If the orbiting vehicle is 1 km above the surface, then the output of the Doppler tracking system and the gravity gradiometer system during the passage over the anomalies is as shown in figure 6. At this altitude we can see that the gradiometer system gives significantly improved resolution and signal level for the smaller anomalies. If the altitude is raised to 3 km (to possibly avoid collision with the surface features), then we obtain the comparative plots shown in figure 7. The advantage of the gradiometer data is now not so
significant, and the advantages of the slight improvement in data must be weighed against the costs. The significant advantage of the gradiometer technique is shown in figure 8 where we have assumed a decrease in scale of the simulation by a factor of 10. Instead of a spacecraft in an orbit 3 km above a 100 km asteroid with 1 to 30 km sized anomalies, we have simulated a spacecraft in an orbit 300 m above a 10 km asteroid with 100 to 3000 m sized anomalies. The orbital period has not changed, because the asteroid density is assumed to be the same, so the time required for the measurement is the same. The gravity gradient signal has the same magnitude and resolution for the 10 km asteroid as it had for the 100 km asteroid, but the Doppler velocity signal has decreased by an order of magnitude and the accuracy of this technique for mass anomaly measurement has decreased in the same proportion.
As a general rule, our studies show that the average density of an asteroid can best be obtained by Doppler tracking techniques if the mission is a flyby mission to one of the larger asteroids. If the mission involves a flyby of a smaller asteroid, or a rendezvous and orbit of any asteroid, the addition of a gravity gradiometer to the spacecraft instrument package will give a significant improvement in the quality of the gravity data and should be seriously considered for such missions.
Bell, C. C., Forward, R. L., and Williams, H. P. 1970, Simulated Terrain Mapping With the Rotating Gravity Gradiometer. Proc. Invitational Symp. Dyn. Gravimetry (Fort Worth), Mar. 16-17, pp. 45-60.
Forward, R. L., Pilcher, L. S., and Norwood, Virginia T. 1967, Asteroid Belt Investigation Using Small, Spin-Stabilized Fly-By Probes. Proc. AAS Symp. Planet. Geol. Geophys. (Boston), May 25-27, pp. 327–347.
Trageser, M. B. 1970, A Gradiometer System for Gravity Anomaly Surveying. Proc. Invitational Symp. Dyn. Gravimetry (Fort Worth), Mar. 16-17, pp. 1-43.
HARRIS: With a gravity gradiometer is there an ambiguity in the density distribution?
FORWARD: There is always a mathematical ambiguity in the details of the internal density distribution of an object obtained from external gravity data alone. The ambiguity will have to be resolved with additional data obtained from magnetic, acoustic, or borehole surveys along with reasonable assumptions for the types of materials (rock, iron, ice, etc.)