– Radial drift rate vs. misalignment phase for all four gyroscopes. 

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... ε 0 is the permittivity of free space, a is the rotor radius, Δ is the gap between the housing and the rotor, and γ is the angle between the principal axis of the gyroscope corresponding to the maximum moment of inertia and the spin axis of the gyroscope. The coefficients in the spherical harmonic expansion of the electrostatic potential on the surface of the rotor in the principal axis reference frame are R lm and coefficients in the expansion of the housing potential are H lm , in a reference frame with the satellite roll axis along the z -axis. P l (cos γ ) denotes the Legendre polynomial of order l . For small angular misalignments between the satellite roll axis and the gyroscope spin axis, the magnitude of the disturbance drift rate is proportional to the misalignment and its direction is perpendicular to the misalignment. It is proportional to the product of the coefficients of the spherical harmonic expansion of the rotor and housing potentials having the same order and increases with increasing order of the spherical harmonic. It is important to note that the torque is due to an interaction of the potential on the surface of the rotor with the potential on the surface of the housing. A patch effect potential on either surface alone produces no torque. Also, the interaction of the a.c. potential applied to the electrodes will produce no net torque since the 20 Hz modulation frequency is well averaged. As observed during the calibration phase, a d.c. potential applied to the electrodes will produce a torque on the gyroscopes, but this d.c. operating mode for the electrostatic suspension system was not used during the science data collection phase. Even though this model clearly explains the observed effects, some of the details of the interaction between the potentials on the housing and the rotor are not known. Although the effect could be explained by patch effect potentials on the surface of the rotor interacting with patch effect potentials on the surface of the housing, additional contributions may results from the patch effect potentials on the surface of the rotor interacting with holes in the gyroscope housing. This model of the interaction of the patch effect fields on the surface of the rotor and the surface of the housing also explains an additional observation. At those times when a harmonic of the rotor’s polhode frequency coincided with satellite roll frequency, offsets in the gyroscope spin axis as large as 0.1 arcsec occurred over intervals of approximately one day. As the harmonic of the slowly changing polhode frequency drifts through the roll frequency, there is a torque which has a nonzero average value [57]. In this case, the orientation of the gyroscope spin axis follows a Cornu spiral, which can be separated from the uniform drift rate because of its unique time signature. . 4 3. Data analysis in the presence of misalignment torques . – One important question is whether the gyroscope drift rate due to these misalignment torques can be clearly separated from the relativistic drift rate. At any given time, the gyroscope drift rate may be divided into two components: one component in a direction parallel to the misalignment between the gyroscope spin axis and the satellite roll axis (the radial component) and the other component in a direction perpendicular to the misalignment between the gyroscope spin axis and the satellite roll axis (the azimuthal component). The azimuthal component of the drift rate has contributions from the misalignment torque and the relativistic drift rate, but the radial component is entirely due to the relativistic drift rate. As the direction of the misalignment changes, primarily due to the annual aberration, a uniform relativistic drift rate will vary sinusoidally as a function of the misalignment phase (the direction of the misalignment in an inertial reference frame). The amplitude of the sine wave is equal to the magnitude of the relativistic drift, while the phase of the sinusoidal variation determines the direction of the uniform, relativistic drift rate. A plot of the radial component of the drift rate vs. the misalignment phase is shown in fig. 16. Data was not used at those times when a harmonic of the polhode frequency was close to the roll frequency. Although this method of analyzing the flight data shows that the relativistic drift rate may be clearly separated from the effects of the misalignment torque, other methods of analyzing the data will very likely be more accurate. To apply this method, the data is divided into segments several days long and the drift rate for each segment is determined. Dividing the data into short segments decreases the potential accuracy advantage of the drift rate estimates decreasing as 1 /T 3 / 2 where T is the measurement time. On the other hand, if the misalignment torques and the relativistic drift rate are simultaneously estimated, there is no limitation due to the short data segments. Initial efforts to simultaneously estimate these two effects have shown promising results. As of July, 2007, the data analysis gives a drift rate of − 6638 ± 97 marcsec/y in the plane of the orbit (North-South direction). This result may be compared with the predicted uniform drift rate in the North-South direction of 6571 ± 1 marcsec/y. This drift rate includes the predicted terrestrial geodetic effect for the measured orbit of the Gravity Probe B satellite, a contribution from the predicted solar geodetic, and the proper motion of the guide star. The measurement error is dominated by systematic errors as indicated by the disagreement between the measured drift rates of the four gyroscopes. At this time, the estimate for the drift rate in the West-East direction is consistent with frame- dragging effect predicted by general relativity, but additional work is needed before clear comparison between the measurements and the predicted effect may be made. There is significant room for improvement in the accuracy, and the estimated error will decrease as the data analysis progresses. Updates on the status of the data analysis at the end of 2008 may be found in refs. [57-61]. We would like to acknowledge the enormous contribution to this work by numerous people at Lockheed-Martin Space Corporation, NASA’s Marshall Space Flight center, and Stanford University. This work was supported by NASA under contract number NAS8-39225. The authors would also like to acknowledge the generous support of R. D. Fairbank ...