Role of membrane stresses in the support of planetary topography

Abstract

The principal purpose of this paper is to examine whether membrane stresses can support topographic loads on planetary elastic lithospheres. It is found that the ability of a spherical shell to support loads through membrane stresses is determined by the nondimensional parameter τ = Ed/Δρ gR2 where d is the thickness of the elastic lithosphere, Δρ is the density difference between the mantle and crust, and R is the radius of the planetary body. When this parameter is large membrane stresses can fully support topographic loads without flexure, and when it is small the influence of membrane stresses can be neglected. Solutions of the equation governing the behavior of a spherical shell are obtained for a topographic load expressed in terms of spherical harmonics. Spherical harmonic expansions of the measured gravity and topography for Mars and the moon are compared with the theory. It is found that for Mars the support of topography is primarily due to membrane stresses for n < 8 and for the moon for n < 17. For Mars the data for 4 ≤ n ≤ 7 give τ ≃ 0.5. For the moon the data have considerable scatter that is attributed to the mascons but generally correlate with τ = 0.5. If bending stresses are neglected, the governing equation for the deflection of the spherical shell is Legendre's equation. A general solution is obtained for an axisymmetric load. This solution is applied to the Tharsis region on Mars. The 60–65% compensation of this region requires that τ = 0.6. The well defined fracture pattern surrounding the Tharsis region is attributed to tensional membrane stresses.