Long-wavelength topographic relaxation for self-gravitating planets and implications for the time-dependent compensation of surface topography

The support of planetary surface topography is controlled by the thickness of the elastic lithosphere and thus has implications for the thermomechanical structure of shallow planetary interiors. Previous analyses of the support of long-wavelength topography have generally utilized elastic formulations, which preclude consideration of the time evolution of lithospheric stresses and thermal state. Here we formulate a viscoelastic model for the support of topography on a spherical planet. Our stress relaxation model employs a multilayer linear viscoelastic rheology and includes the effect of self-gravitation. In these models we approximate internal rheology with an olivine flow law, and we estimate the internal thermal structure assuming simple conductive cooling. We analyze how relief at the surface and internal density interfaces evolves with time in response to a surface or internal load. Our analysis demonstrates that crustal compensation is strongly dependent on planetary radius, with a tendency for a large terrestrial planet like Earth to reach complete Airy isostasy at long wavelengths within a time scale controlled by the mantle viscosity. The correlation of degree of compensation with planetary radius is consistent with previous work [Turcotte et al., 1981], which showed that membrane stress can support significant long-wavelength gravity anomalies. Applied to the Moon, the degree of compensation associated with loading of 100-Ma lithosphere is less than 0.3, and the load Love numbers are greater than -0.6 for all harmonics including degree 2, even after 10(9) years of relaxation. This conclusion does not change significantly even with a moderately low viscosity for the crust. The tendency for a small body like the Moon to support long wavelength stresses may be relevant in understanding the anomalously large degree 2 terms in the lunar shape. The results also suggest that major impact basins would not be compensated throughout lunar geological history (> 4 b.y.), even if the surfaces on which the basins were initially formed were relatively young (i.e., 100 - 300 Ma). The fact that the South Pole-Aitken basin is largely compensated suggests that the lateral variations in thermal and/or mechanical structure induced by impact processes play significant roles in the compensation of this structure.