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ASTEROID MASS DISTRIBUTION MEASUREMENT
WITH GRAVITY GRADIOMETERS

ROBERT L. FORWARD
Hughes Research Laboratories

A determination of the internal structure and composition of the asteroids will give us important information concerning the origin of the solar system and the formation of the planets. We can obtain this information by measuring the total mass and the internal mass distribution through the use of spacecraft missions to flyby or to rendezvous, orbit, and land on the asteroids. The Doppler tracking technique used for mass measurement on planetary probes is suitable only for a flyby mission of the larger asteroids (>20 km) because the gravity force field of the smaller asteroids is not strong enough to appreciably affect the trajectory of the probe during a flyby. If a rendezvous mission is used and the spacecraft is placed in orbit about an asteroid, the mass can be determined from the orbital period, but the effect on the spacecraft orbit due to the mass anomalies under the surface will not be easily seen unless the anomaly is very large because again the gravity force field decreases rapidly with decrease in anomaly size. The magnitude of the gradient of the gravity force field is independent of the asteroid or anomaly size, however, and a gravity gradient sensing technique for sensing of the gravity field of the asteroid will perform equally well on all except the very smallest (< 1 km) asteroids. Thus, if we desire to obtain mass measurements of the smaller asteroids during a flyby or to obtain detailed mass anomaly maps of the asteroids from an orbital survey prior to landing, it would be desirable to include a gravity gradiometer as part of the spacecraft instrument package.

GRAVITY GRADIOMETER INSTRUMENTATION

The various gravity gradient instruments that could be used to measure the gravity gradient field of an asteroid from a spacecraft all work on the principle of measuring the differential tensions, compressions, or torques induced in the sensor by the gradient of the gravity force field (the tidal effect) of the asteroid. There are many techniques and configurations possible (a fairly complete bibliography is available in (Bell, Forward, and Williams, 1970), but the instruments presently under serious consideration for spacecraft missions are those that measure the gravity gradient torque. These torque gradiometers use the same physical principle for their operation as the gravity gradient stabilized satellites. The gravity gradient satellites we are familiar with have an elongated dumbbell shape and are very large and massive in an attempt to obtain engineeringly significant torque levels out of the gravity gradient field. The gradiometers are smaller so that they can be enclosed for protection from nongravitational forces, and therefore must have high sensitivity. The typical signal levels encountered by these instruments range from 10-9 to 3X 10-9 st” (1 to 3000 Eötvös units (EU) and the present designs have a noise level of 10-9 st? at 10 s integration time (Bell, Forward, and Williams, 1970). One type of torque gradiometer is a single dumbbell in a cylindrical or spherical case that is floated at neutral buoyancy using the techniques developed for high-precision floated gyros (Trageser, 1970). Highly sensitive torque pickoffs and electromagnetic servodrivers are used to keep the float balanced and to read out the gravity gradient torques. (See fig. 1.) A second technique, based on free-fall modifications of the old Eötvös torsion balance gradiometer, would use one or more dumbbells connected by fine quartz torsion fibers with capacitive pickoffs and electrostatic feedback. The gravity gradient signal would be that of the relative torque between the two sensing arms. (See fig. 2.) A third variation also uses two opposed dumbbells, but they are connected by a stiff torsion spring that forms a resonant mechanical structure. (See fig. 3.) The entire device is then rotated at 60 to 1800 rpm so that the gravity gradient field is “chopped” by the rotating arms (Bell, Forward, Williams, 1970; Forward, Pilcher, and Norwood, 1967). The resonant frequency of the sensor structure is chosen at twice the rotation frequency, and the gravity gradient field induces vibrations into the rotating sensor structure at its

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Figure 1.-Schematic of spherical floated torque gradiometer (from MIT Charles Stark Draper Laboratory, Cambridge, Mass.).

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Figure 2.-Differential gravity gradient torques on a two-dumbbell torque gradiometer sensor structure. (The rotation axis of the rotating version of this type of gradiometer structure would be out of the page.)

Figure 3.-Rotating gravity gradiometer.

resonant frequency. Piezoelectric transducers on the torsion spring convert the resonant mechanical vibrations into ac voltages whose amplitude and phase give the strength and direction of the gravity gradient field. If the rotating gradiometer is installed in a spin-stabilized spacecraft and the sensor resonance is tuned to twice the spacecraft rotation frequency, the spacecraft then provides the rotation needed for the sensor operation. This mode of operation has the advantage that because the spacecraft is rotating along with the sensor, its gravity field is fixed with respect to the sensor arms and the sensor does not see the spacecraft gravity field, only the asteroid field. The various possible versions of the gravity gradiometer are just leaving the laboratory and are far from being tested flight hardware, but we can expect that by the time the asteroid missions begin, the instrumentation will be available. However, the gradiometer instrumentation adds significantly to the cost, weight, and power budgets of the spacecraft, whereas Doppler tracking is practically free. Therefore, the gravity gradiometer instrumentation should only be included on those missions for which the Doppler tracking data are not adequate for determination of the mass or mass distribution. In the following sections we will try to give some general guidelines that show when one technique is preferred over the other. This hopefully will help those who are planning the missions to obtain the maximum scientific return from each

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flight.
MASS MEASUREMENT DURING A FLYBY

In figure 4 we have plotted the flyby altitude at which we can expect to obtain a 1 percent measurement of the mass of asteroids of various radii using both the Doppler velocity tracking technique and the gravity gradient sensing technique. We assumed that the accuracy limit was set by the present sensitivity of the two systems, 10-9 st” (1 EU) at 10 s for the gradiometer system and 0.5 mm/s at 60 s for the Doppler tracking system. The purpose of

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Figure 4.—Flyby altitude for 1 percent mass measurement (flyby velocity = 1 km/s) using Doppler tracking (broken curve) and gravity gradiometer techniques (solid curve).

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