LAUSR

Abstract The basis of the famous equation attributed to Maclaurin (1742), which relates the ellipticity of homogeneous spheroids to their rotation rate, is reviewed and Todhunter's (1873) neglected but important contribution is recognized. The relationship is then re-derived in a new manner based on potentials and energy conservation rather than on force balance, yielding exact analytical equations that are valid for both homogeneous objects and inhomogeneous objects with any spheroidal mass distribution. The new equations are applied to the Earth considering three density distributions: the homogeneous case, several two layer models, and an internal density that varies according to the PREM seismic model. Earth's sidereal rotation rate plus the polar and equatorial values of the gravitational acceleration g are all accurately computed from the PREM density distribution, validating the analysis while using no fits, approximations or free parameters. Earth's core must be more spherical than Earth's outer surface. The potentially measurable difference of 0 to 5 km from a congruent core shape is inversely related to the magnitude of core super-rotation, and bears on many additional matters including the core's volume, and the magnitude and direction of its internal and external gravity field.