Behaviour of the Centrifugal Force and of Ellipticity for a Slowly Rotating Fluid Configuration with Different Equations of State

TL;DR

This study examines the behavior of centrifugal force and ellipticity in slowly rotating fluid configurations with various equations of state, revealing conditions for stability and shape deformation.

Contribution

It provides a detailed analysis of centrifugal force and ellipticity using the Hartle-Thorne solution for different equations of state, highlighting new insights into stability and shape.

Findings

Centrifugal force exhibits a maximum in all cases.

Reversal in centrifugal force sign occurs only in one case.

Configurations can become unstable depending on the equation of state.

Abstract

We have evaluated the centrifugal force acting on a fluid element and the ellipticity of the fluid configuration, which is slowly rotating, using the Hartle-Thorne solution for different equations of state. The centrifugal force shows a maximum in every case, whereas the reversal in sign could be seen in only one case, and the system becomes unstable in other cases. The ellipticity as calculated from the usual definition shows maxima, whereas the definition obtained from the equilibration of the inertial forces, shows a negative behaviour, indicating that the system is prolate and not oblate. This prolate shape of the configuration is similar to the one earlier found by Pfister and Braun for a rotating shell of matter, using the correct centrifugal force expression for the interior. The location of the centrifugal maxima gets farther away from the Schwarzschild radius as the equation of state gets softer.