The gravitational coupling between longitudinal segments of a hollow cylinder and an arbitrary gravitational source: relevance to the STEP experiment

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Abstract
The gravitational interaction is derived between a solid longitudinal segment cut from a cylinder of uniform density, and an external point mass. The derivation is expressed in terms of the associated Legendre functions , and the parametric form of the coupling coefficients is presented. This theory is applied to the gravitational interaction between a point mass and a finite hollow cylinder, where the cylinder bears a number of `flats' cut into its outer surface. The `flats' are imagined to be regularly spaced in azimuth around the cylinder, each flat being treated as the removal of a solid segment from the full cylinder. Such forms of test mass have been proposed for the satellite test of the equivalence principle (STEP) experiment, since the masses may then be prevented from rotating in azimuth - a factor which is considered to be essential for this experiment. The gravitational theory developed here is applied to such STEP test masses, and two `low gravitational susceptibility' designs for test-mass pairs are considered, having four and six `flats', respectively. An expression for the axial force on such masses is derived which is more than times faster to compute than a Monte Carlo integration of similar accuracy, by virtue of which it is shown that a design with six or more `flats' is to be preferred. This theory is shown to have much wider applicability to gravitational problems involving general segmented cylindrical bodies, including square- and hexagonal-section prisms of finite length (hollow or solid).