Certain Aspects of the Gravitational Field of a Disk

There are at least two reasons why one would study the gravitational field of a disk. The first is that many astronomical objects, such as spiral galaxies like the Milky Way, are disk-like. The second is that the field of a disk is interesting, particularly when compared to that of a spherical, or near-spherical, object, which is much easier to analyze because of its high degree of symmetry. It is hoped that this study will augment previous work on this subject. The aspects presented in this paper are as follows: 1) both the radial and vertical gravitational fields of a thin disk within the plane of the disk and above it;2) a comparison of some of the field results obtained by Lass and Blitzer (1983) involving elliptic integrals to those obtained by a standard numerical integration, now available online, and separately through the use of Legendre polynomials;3) the logarithmic divergence of the radial field at the edge of a thin disk;4) the fields in the plane of a disk containing a central hole, particularly within the hole, such as the rings of Saturn;5) circular orbits within the plane of a single disk and half way between two disks, and their stability;6) the escape velocity at a point within the Milky Way, particularly at the position of the solar system and without any added, or subtracted, orbital effects around the galactic center;and 7) the radial field at the circular edge of a disk of finite thickness.

The Gravitational Attraction between Hemispherical Masses

This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of Wolfram Mathematica. The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.

The Summation of One Class of Infinite Series

This paper presents closed-form expressions for the series, , where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: and , where l is an integer between –∞ and ∞. The values of λ include 1,,and 3. Examples of closed-form expressions obtained in this manner are first presented for , , , and . Additional examples are then presented for , , , and . This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.