Analytical representations for simple and composite polytropes and their moments of inertia

Polytropes are widely applied in astrophysics. To facilitate their use, we derive analytical formulae for the moment of inertia as a function of polytropic index. We also provide 1- and 3-parameter equations that replicate the density variations in polytropic bodies to varying degrees of accuracy, determined by numerical calculations and analytical results for polytropic indices between 0 and 5. As an example, we construct a composite polytrope, suitable for gas giants, exoplanets, or tiny sub-solar dwarfs, wherein an inner sphere is modeled by constant density, which represents the density jump associated with production of a relatively incompressible solid, and an outer envelope is modeled as having a polytropic index near 2.5, which corresponds to a diatomic gas. Envelope sizes are constrained by the moment of inertia.