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.

25 pages, 7 figures, 5 tables. This version has been accepted to Monthly Notices of the Royal Astronomical Society (MNRAS) Changes from version 1: Change in the title, some extraneous text and a table were cut, Fig. 4 now only describes rocky planets, more references were added. Also a supplemental Python code to create a multi-layered planet is available with this version

References in corpus (7)

• Mass-Radius Relationships for Solid Exoplanets
• Detailed Models of super-Earths: How well can we infer bulk properties?
• Radius and Structure models for the First Super-Earth Planet
• The Kepler-10 planetary system revisited by HARPS-N: A hot rocky world and a solid Neptune-mass planet
• On the radius of habitable planets
• Electronic properties and magnetism of iron at the Earth's inner core conditions
• Semi-empirical pressure-volume-temperature equation of state; MgSiO3 perovskite is an example

Cited by in corpus (10)

• Evaporation of dark matter from celestial bodies
• Multiple Stellar Evolution: a population synthesis algorithm to model the stellar, binary, and dynamical evolution of multiple-star systems
• An analytic model for mass transfer in binaries with arbitrary eccentricity, with applications to triple-star systems
• New Perspectives on the Exoplanet Radius Gap from a Mathematica Tool and Visualized Water Equation of State
• Metric-affine gravity effects on terrestrial (exo-)planets profiles
• Catching a planet: A tidal capture origin for the exomoon candidate Kepler 1625b I
• Planetary seismology as a test of modified gravity proposals
• Comment on "The SPOCK equation of state for condensed phases under arbitrary compression" by R. Myhill
• Elastic Planetoids
• Multi-component, rigidly rotating polytropes: improved and extended theory