# The shape of a rapidly rotating polytrope with index unity

## TL;DR

This paper evaluates the accuracy of approximate solutions for rapidly rotating, self-gravitating polytropes with index unity, highlighting their limitations in representing true equilibrium configurations.

## Contribution

It critically assesses the solutions from a prior study, clarifying their approximate nature and discussing their validity as models of rotating polytropic stars.

## Key findings

- The solutions are only approximate, not exact, solutions of the Euler-Poisson system.
- The paper discusses the degree of approximation and its implications for modeling rotating polytropes.
- It provides insights into the limitations of existing analytical solutions for such systems.

## Abstract

We show that the solutions obtained in the paper `An exact solution for arbitrarily rotating gaseous polytropes with index unity' by Kong, Zhang, and Schubert represent only approximate solutions of the free-boundary Euler-Poisson system of equations describing uniformly rotating, self-gravitating polytropes with index unity. We discuss the quality of such solutions as approximations to the rigidly rotating equilibrium polytropic configurations.