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spacecraft might record a complex magnetic signature as the asteroid rotated in the solar wind nearby. A magnetometer might thus serve as an important advance guide to favorable locations on the asteroid for collection of the most scientifically useful samples. Correlation of magnetic signature with optical appearance and other physical characteristics of the asteroid might yield definitive inferences on how it was formed. The question of how various combinations of asteroid features, including detectable magnetic fields and noise wakes, or even miniature magnetospheres, might relate to asteroidal formation is a suitable subject for further study. There seems to be no reason at present to believe that measurements made by a sensitive magnetometer borne to the neighborhood of an asteroid will prove less valuable than the many measurements made by similar instruments carried by spacecraft to numerous other destinations in the solar system.

CONCLUSION

The fields demanded by theory in order that the larger asteroids support a magnetic interaction with the solar wind are compatible with fields found associated with extraterrestrial objects. Steady fields detected by Apollo magnetometers on the lunar surface are comparable to equatorial fields with which the largest asteroids could support small magnetospheres. Magnetizations found in meteorites and in material from the lunar surface could, if duplicated in the bodies of the few asteroids of radius greater than 100 km, maintain magnetic cavities, or, more probably, multilobed magnetic cavities around these small planets. It would be worthwhile to conduct a theoretical investigation in advance of an asteroidal mission to determine the extent to which magnetic measurements at an asteroid might serve to distinguish among models of asteroidal origin.

REFERENCES

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Binsack, J. H., Mihalov, J. D., Sonett, C. P., and Moutsoulas, M. D. 1970, Possible Lunar
Surface Fossil Magnetism (abstract). EOS, Trans. Amer. Geophys. Union 51, 774.
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Shkarofsky, I. P. 1965, Laboratory Simulation of Disturbances Produced by Bodies Moving Through a Plasma and of the Solar Wind Magnetosphere Interaction. Astronaut. Acta 1 1, 169.

Stacey, F. D., and Lovering, J. F. 1959, Natural Magnetic Moments of Two Chondritic Meteorites. Nature 183, 529.

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FEASIBILITY OF DETERMINING THE MASS OF AN
ASTEROID FROM A SPACECRAFT FLYBY

JOHN D. A/VDERSON
Jet Propulsion Laboratory

APPROXIMATION TO DOPPLER OBSERVABLE

The orbit of a spacecraft with respect to an asteroid can be approximated to zero order by a hyperbola of zero bending angle (fig. 1). In an orbital system of

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where r is the distance between the asteroid and the spacecraft, f is the true anomaly, b is the impact parameter or miss distance, v is the constant

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REFERENCE ORBIT |

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| Figure 1.-Geometry of the flyby reference orbit in an orbit-plane coordinate system. The

actual orbit is developed to the first order in the mass m as a perturbation from this reference orbit.

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hyperbolic velocity, t is the time of observation, and T is the time of closest

approach. After this zero-order solution is substituted into the two-body

equations of motion, an approximation to the actual orbit can be obtained. The approximate equations of motion are given by

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and the results of integration of equations (3) and (4) to the first order in Gm are

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where the constants of integration have been chosen such that v is the hyperbolic velocity at infinity. It is not necessary to carry the integration further because the mass m of the asteroid will be determined from Doppler data. Of course, the expressions for x., and y, can also be obtained from the hyperbolic orbital equations by establishing approximations for large eccentricities, but the perturbational derivation presented here is slightly easier and more straightforward. For purposes of determining the mass of an asteroid from the velocity history given by equations (6) and (7), it is sufficient to consider the geocentric range rate A' to the spacecraft. If the orientation of the spacecraft orbit is referred to the plane of the sky (fig. 2), then the range rate is given by

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Note that the range rate is independent of the location of the node of the spacecraft's orbit on the plane of the sky, but it does depend on the argument of the perifocus and the inclination.

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