Explicit Green’s function of a boundary value problem for a sphere and trapped flux analysis in Gravity Probe B experiment

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Abstract
Magnetic flux trapped on the surface of superconducting rotors of the Gravity Probe B (GP-B) experiment produces some signal in the superconducting quantum interference device readout. For the needs of GP-B error analysis and simulation of data reduction, this signal is calculated and analyzed in this article. We first solve a magnetostatic problem for a point source on the surface of a sphere, finding the closed form elementary expression for the corresponding Green’s function. Second, we calculate the flux through the pick-up loop as a function of the source position. Next, the time dependence of a source position, caused by rotor motion according to a symmetric top model, and thus the time signature of its flux are determined, and the spectrum of the trapped flux signal is analyzed. Finally, a multipurpose program of trapped flux signal generation based on the above results is described, various examples of the signal obtained by means of this program are given, and their features are discussed. Signals of up to 100 fluxons, i.e., 100 pairs of positive and negative point sources, are examined.
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