A Variable Polytrope Index Applied to Planet and Material Models

Abstract

We introduce a new approach to a century old assumption which enhances not only planetary interior calculations but also high pressure material physics. We show that the polytropic index is the derivative of the bulk modulus with respect to pressure. We then augment the traditional polytrope theory by including a variable polytrope index within the confines of the Lane-Emden differential equation. To investigate the possibilities of this method we create a high quality universal equation of state, transforming the traditional polytrope method to a tool with the potential for excellent predictive power. The theoretical foundation of our equation of state is the same elastic observable which we found equivalent to the polytrope index, the derivative of the bulk modulus with respect to pressure. We calculate the density-pressure of six common materials up to 10$^{18}$ Pa, mass-radius relationships for the same materials, and produce plausible density-radius models for the rocky planets of our solar system. We argue that the bulk modulus and its derivatives have been under utilized in previous planet formation methods. We constrain the material surface observables for the inner core, outer core, and mantle of planet Earth in a systematic way including pressure, bulk modulus, and the polytrope index in the analysis. We believe this variable polytrope method has the necessary apparatus to be extended further to gas giants and stars. As supplemental material we provide computer code to calculate multi-layered planets.