The influence of moon craters on the gravity field of the moon
Loading...
Date
1964-04
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ohio State University. Division of Geodetic Science
Abstract
For the theory of satellite orbits in the neighborhood of the moon, approximate values of the influence of lunar craters on the moon's gravity field are desired. Therefore, formulas are derived for the potential V, the selenoid undulation u = V/go, the gravity components - ∂V/∂z, - ∂V/∂r, and the deflection of the plumb line ∈r, z and r being the vertical and radical coordinates, and go the normal gravity. The center of z-r system is lying in the center of the crater. First, a circular mountain such as the crater of Copernicus was approximated by a circular mass line lying 1 km above the ground plane, radius 50 km. The formulas, containing elliptic integrals of first and second kind, could numerically be evaluated for the heights 5, 10, 20, 50, 100, 200, 500, 1000, and 2000 km and the distances r = 0, 10, 20, 30, 40, 50, 60, 80, 100, 150, 200, 300, 500 km from the symmetry (z-) axis by using a desk calculator. One may imagine that the masses of such a circular mountain are, by any event, thrown out from the interior of the crater. Therefore they are missed and can be approximated by a circular mass plate of negative surface density. Such a circular plate mass can be replaced by a great number of concentric circular mass lines. It would no longer be possible to perform the calculation with a desk calculator because too much time would be needed. Thus, following the advice of Dr. Moritz, the order of magnitude of the effects on gravity was estimated by taking square flat mass for the circular mass plate. After this, Mr. R. Rapp programmed the formulas for the circular mass line and computed the gravity field of the circular mass plate, approximating it by 5, 10, 20, 30, 40, concentric mass circles. The result shows that for small values of z no less than 30 mass circles are needed, but for z ≥ 50 km the approximation by 5 mass circles is good enough. The approximation of the circular wall by a single mass line is very rough. Thus, a better approximation by circular mass lines, conveniently arrayed, was computed using a high-speed calculator. For z ≥ 50 km there is no remarkable difference between the rough and the better approximation. But for z = 5 km and r ≈ 50 km the approximation by a single circular mass line leads to numerical values which are far too high. In moon craters, central mounts of nearly parabolic or conic shape are known. If their height is h, they can be approximated by vertical mass lines on their symmetry axes for z ≥ 3 h. If z is smaller, better approximations must be taken, for instance, by an array of circular mass lines. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]
Description
Prepared for National Aeronautics and Space Administration, George C. Marshall Space Flight Center, Huntsville, Alabama: Contract No. NAS8-2493, OSURF Project 1377
Theory of the shape and gravity field of the moon, Part I
Theory of the shape and gravity field of the moon, Part I