Understanding the interior dynamics of gas giants like Jupiter and Saturn has long been a focus of planetary science. These planets exhibit prominent east-west banded structures, called zonal winds, clearly visible through telescopes and spacecraft observations. A central question is whether these wind patterns extend into the deep interior and, if so, to what extent and strength. Recent satellite missions to Jupiter (Juno, 2016) and Saturn (Cassini Grand Finale, 2017) measured the gravitational fields of these planets to shed light on mass transport caused by internal winds. This gravitational data enables indirect study of atmospheric dynamics. However, inferring internal mass distribution from gravitational fields, known as inverse gravimetry, is an ill-posed problem with non-unique solutions. This book explores the mathematical foundations and methods for studying gas-giant atmospheres via gravitational data, focusing on function spaces, properties of the gravitational potential, and solutions to the Helmholtz equation. The goal is to rigorously formulate and analyze the inverse problem linking zonal winds to gravitational fields.
