## Abstract:

The evolution of rocky and icy bodies with relative core sizes that differ substantially from the Earth's present distinct and unresolved questions. Here we examine the effect of core size on the cooling of planetary mantles for geometries ranging that of lunar structure to thin shelled convecting mantles. Several previous studies have employed purely thermal convection studies to investigate the thermal structure of generic planets. The studies feature variable curvature three-dimensional shells and different thermal energy input and viscosity contrasts to determine the onset of stagnant-lid convection. Recently, we have employed a temperature dependent Frank-Kamenetskii rheology in a 2D spherical annulus geometry to distill the effect of curvature when viscosity contrasts and internal heating rates are varied. Contrary to convection in a plane-layer, for a geometry independent viscosity-law, as viscosity contrast is first increased convection in spherical shells exhibits a curvature dependent decrease in vigour before effective Rayleigh number monotonically increases. Furthermore, the transition to stagnant-lid convection in small core bodies shows a divergence in the characteristics found for 2D annulus and 3D spherical shell convection as the ratio of core-to-planet radius, f, is decreased. We also looked at the effect of increasing internal heating rates in convection models with different size cores. For an isothermal core mantle boundary and a fixed viscosity contrast of 10^5 when internal heating is increased, we find that heat can flow from the mantle to the core and follows a monotonic increasing trend with curvature. Furthermore, we find substantial differences in the behaviour of thin shell (f=0.9) and plane-layer (Cartesian geometry) models for a variety of internal heating rates, indicating that the latter may be poor approximations for modelling variable viscosity convection in thin shells. The internal structure of the Moon is of particular interest since there is still some disagreement on whether convection is still present today and the origin of the inferred partial melt above the core. Candidate models of the Moon must be in the stagnant-lid regime and have a sufficiently hot lower mantle to produce partial melt. In this presentation, I discuss the effect of purely thermal convection for cases representing the lunar structure from post-magma ocean solidification to possible present day internal structure. My calculations feature decaying internal heat sources and secular cooling over timescales relevant to lunar evolution.