A new spectral calculation scheme for long‐term deformation of Maxwellian planetary bodies

Viscoelasticity plays an important role in long‐term deformation of large‐scale topographies on small planetary bodies, such as the Moon. Although we should consider increases in viscosity due to secular cooling during long‐term deformation, previous spectral calculation schemes for a Maxwellian viscoelastic body have problems in calculating deformation with temporal and radial variation in viscosity. To resolve these problems, we propose a new formulation of the constitutive equation of a Maxwell body. Compared with conventional time domain spectral schemes, our scheme, with its second‐order approximation in time, demonstrates significant improvement in both accuracy and stability for long time steps. As a result, the computational costs for time‐consuming calculations, such as the long‐term evolution of surface topography, are greatly reduced. Using this scheme, we investigate the effect of planetary cooling on the deformation of an initially hot but monotonically cooling Moon. Our calculation results show that the effective elastic thickness depends on the wavelength of deformation for loads emplaced at very early times (<20 Myr). The results of our calculations also show that without additional heat sources other than initial accretion heat, hardening due to secular cooling prevents substantial isostatic compensation and crustal lateral flow underneath large impact basins. This result indicates that small planetary bodies, such as the Moon, do not retain the memory of their initial state during the accretion phase for very long and that their current viscoelastic states reflect sustained heat production and/or transport from their deep interior.