Vol. 6, No. 1, 2018

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The effect of rotation on the flattening of celestial bodies: a journey through four centuries

Wolfgang H. Müller

Vol. 6 (2018), No. 1, 1–40

This paper presents an overview and comments on various continuum models used for predicting the deformation of celestial objects under their own rotation, also known as “flattening” — in particular from a historical perspective. Initially we shall discuss the chronology of events leading to models for fluids, solids, and gases. Our review will range from Newton’s famous Principia, Thomson and Tait’s Treatise on natural philosophy, and the treatise of the spinning top by Klein and Sommerfeld to the modern literature, which accounts for quantum mechanics and relativistic effects in exotic spinning celestial objects, such as neutron stars and white dwarfs. Then, based on previously published results by Müller and Lofink (2014) and Müller and Weiss (2016), we will present a modern treatment of the fluid model according to Newton. It will be applied not only to the Earth but also to other celestial bodies. We will compare the results to actual measurements and discuss reasons for discrepancies. Finally, we turn to a model for a solid based on Hookean linear elasticity, which we shall also state and solve in modern terminology. In particular, we will not only compute the flattening but also present closed-form solutions for the stresses in a gravitating and stationary spinning, linear-elastic sphere.

flattening, history of planet flattening, stresses in celestial bodies, fluid, degenerate matter, barytropic equations of state, linear-elastic Hookean model of Earth, linear elasticity, hydrostatics, self-gravitation
Mathematical Subject Classification 2010
Primary: 85-03, 85A99, 31-00, 74B05, 83-03
Received: 31 May 2017
Revised: 23 October 2017
Accepted: 2 December 2017
Published: 21 March 2018

Communicated by Francesco dell'Isola
Wolfgang H. Müller
Institut für Mechanik
Technische Universität Berlin