Original Drag-Free Paper

B. Lange, The Drag-Free Satellite, AIAA Journal 2(9), p. 1950, 1964.

B. Lange, The Control and Use of Drag-Free Satellites, Ph.D. Thesis, SUDAER 194, June, 1964.

Summary

Outer Satellite's Self Gravity Is Biggest Disturbance!

One of the most remarkable results of this paper (and one which still holds true today) is that the largest disturbance to the proof mass comes from the self gravity of the outer satellite.

This result was calculated for the case of a spherical proof mass, a wide gap (of the order of one cm), and a charge potential of one volt or less. But it is generally true.

When the DISCOS module was built at Stanford, a considerable amount of effort was put into calculating the gravity and gravity gradient of all the masses in the satellite. This was responsible for that satellite exceeding its specification by a factor of two. Part of this success was also due to the fact that an unknown mechanism was discharging the DISCOS proof mass.

The details of calculating the gravity and gravity-gradient at the DISCOS proof mass are reported in: Fleming, Alan W. and Tasker, Michael G., Mass Attraction of Triad I / DISCOS, Stanford University Department of Aeronautics and Astronautics Report (SUDAAR) No. 445, September 1972 (Available from the Stanford Engineering Library). The report concludes that the disturbances can be held below 10-11 ge .

Drag-Free Performance Depends on the Attitude Control Choice

In addition to investigating the disturbances which could act on a Drag-Free proof mass, the AIAA paper also discusses the effect of satellite-fixed disturbances on how well the orbit is determined solely by gravity.

This depends on the attitude control of the satellite. The best case occurs in a spinning Drag-Free satellite with the spin axis perpendicular to the orbit plane. This is because any satellite-fixed disturbing forces in the orbit plane are roll averaged by about a factor of 10-5 to 10-7 depending on the attitude accuracy, the Drag-Free control accuracy, and the spin rate. Thus in-plane disturbances can be reduced to about 10-17 ge , so that a satellite would deviate in-plane from a purely gravitational orbit by about 1.5 >fdx,y t2 = 1.5 * 10-16 meters/sec2 * (3 x 107 sec)2 = 0.15 meters (15 cm!) in one year. The out-of-plane disturbances are not roll averaged, but its deviation is given by a harmonic oscillator. Thus the out-of-plane deviation is fdz (1 - cos(nt)) / n2 = 10-10 meters/sec2 * 106 sec2 = 10-4 meters for a mean orbit rate, n of 10-3 rad/sec.

The worst case is when the attitude control system keeps a Drag-Free satellite locally level such as is the case with gravity-gradient stabilization. In this case a component of the disturbing forces is always parallel to the satellite velocity vector and thus changes the orbit energy and, most importantly, the period. For example, a disturbance of 10-11 ge causes a deviation from a pure gravity orbit of 1.5 fdx,y t2 = 1.5 * 10-10 meters/sec2 * (3 x 107 sec)2 = 1.5 x 105 meters = 150 km in one year. As another example, consider two weeks = 106 seconds. For this time, the in-track error is 1.5 * 10-10 meters/sec2 * (106 sec)2 = 1.5 x 102 meters, or about 150 meters.

DISCOS Attitude Control Was the Best for Measuring Its Performance

The DISCOS satellite was gravity-gradient stabilized and thus the orbit had the worst disturbances. While locally-level attitude control is the worst for Drag-Free performance, it is the best for measuring the size of the disturbing forces by means of satellite tracking.

Because of this, it was possible to accurately measure the level of the disturbing forces in DISCOS. This has been useful in confirming the expected gyro drift in a Relativity-Gyro Experiment (PDF), calculating how well the cancellation could be done in a DC-Cancellation High-Accuracy Equivalence-Principle Experiment (PDF), and estimating the proof-mass electric charge and discharge rate in DISCOS' 750-km polar orbit.

The Unsupported Gyroscope

If a spherical Drag-Free proof mass is spun, it becomes a gyroscope. Because there are no support forces or torques deliberately applied to the rotor, it is an extraordinarily good gyro exceeding the best gyroscopes on earth by about 9 orders of magnitude. The orginal DFS paper discusses the Unsupported Gyroscope, but the calculation of the drift rates does not taken into account all of the possibilities for improving the performance such as roll averaging due to precision attitude control or choice of the Drag-Free orbit. The best calculation the drift rates of a properly designed Unsupported Gyroscope are in the PRL and PRD gyro papers (PDF).

The performance of the DISCOS satellite (5 x 10-12 ge) can be used to bound most of the sources of gyro drift in an Unsupported Gyroscope, and this gives experimental support to the calculated drift rates. This is also discussed in the PRD gyro paper.

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Copyright (c) 2001, Benjamin Lange, All rights reserved.

Benjamin Lange
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