# Isostatic equilibrium in spherical coordinates and implications for crustal thickness on the Moon, Mars, Enceladus, and elsewhere

## TL;DR

This paper revises the understanding of isostatic equilibrium in spherical coordinates, showing that the common 'equal masses' assumption leads to disequilibrium and inaccuracies in estimating crustal thickness on planetary bodies.

## Contribution

It introduces a new model based on 'equal pressures' for isostatic equilibrium in spherical geometry, correcting previous assumptions and providing more accurate crustal thickness estimates.

## Key findings

- 'Equal masses' assumption overestimates compensation depth by ~27% for lunar highlands.
- 'Equal masses' assumption overestimates compensation depth by nearly a factor of two for Enceladus.
- The new 'equal pressures' model aligns better with physical equilibrium conditions.

## Abstract

Isostatic equilibrium is commonly defined as the state achieved when there are no lateral gradients in hydrostatic pressure, and thus no lateral flow, at depth within the lower viscosity mantle that underlies a planetary body's outer crust. In a constant-gravity Cartesian framework, this definition is equivalent to the requirement that columns of equal width contain equal masses. Here we show, however, that this equivalence breaks down when the spherical geometry of the problem is taken into account. Imposing the "equal masses" requirement in a spherical geometry, as is commonly done in the literature, leads to significant lateral pressure gradients along internal equipotential surfaces, and thus corresponds to a state of disequilibrium. Compared with the "equal pressures" model we present here, the "equal masses" model always overestimates the compensation depth--by ~27% in the case of the lunar highlands and by nearly a factor of two in the case of Enceladus.