Estimating π Using the Topography of the Terrestrial Planets

Abstract

AbstractIt is π‐day 2024 (March 14th) and to celebrate, we ask the question: can π be estimated accurately using the shape of the terrestrial planets and the Moon? We proceed by using models for the planet topographies in slices of constant latitude and computing π as the ratio of the circumference to the diameter in each slice. We define the circumference as the topographically undulating perimeter of the slice (which deviates from the circular section of the reference ellipsoid). We define the diameter as the distance connecting two opposite points on the topography model through the center of the slice. This gives us a distribution of π estimates for each latitude slice and overall for each planetary body. The distribution and accuracy of the π estimates we compute are a representation of how smooth and circular latitude slices of the terrestrial planets are. We find that Mercury and Venus are particularly good at predicting π because they are relatively smooth and the topographic deviations from the reference ellipsoid are small compared with Mars, Earth, and the Moon. In the spirit of π‐day, this mathematical exercise motivates an interest in planetary geophysics and shape.