Explicit Green's Function of a Boundary Value Problem for a Sphere and Trapped Flux Analysis in Gravity Probe B Experiment

TL;DR

This paper derives an explicit Green's function for a boundary value problem on a sphere, enabling analysis of trapped magnetic flux signals in the Gravity Probe B experiment, with applications in error analysis and data simulation.

Contribution

It provides a closed-form Green's function solution for a sphere boundary value problem and applies it to model trapped flux signals in the GP-B experiment.

Findings

Derived explicit Green's function for a sphere boundary problem.

Analyzed flux signal spectrum due to rotor motion.

Developed a program for trapped flux signal simulation.

Abstract

Magnetic flux trapped on the surface of superconducting rotors of the Gravity Probe B (GP-B) experiment produces some signal in the SQUID readout. For the needs of GP-B error analysis and simulation of data reduction, this signal is calculated and analyzed in the paper. We first solve a magnetostatic problem for a point source (fluxon) 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 fluxon position. Next, the time dependence of a fluxon position, caused by rotor motion according to a symmetric top model, and thus the time signature of the flux are determined, and the spectrum of the trapped flux signal is analyzed. Finally, a multi-purpose 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.