Emden-Chandrasekhar, axisymmetric, rigidly rotating polytropes. VI. Self-consistency and continuity

TL;DR

This paper examines the self-consistency and parameter continuity of axisymmetric, rigidly rotating polytropes using Chandrasekhar's approximation methods, addressing boundary density issues and parameter behavior across polytropic indices.

Contribution

It clarifies the self-consistency of EC functions within different approximations and establishes the continuity of key parameters as functions of the polytropic index.

Findings

EC functions are well-defined internally despite boundary issues

Continuity of parameters is confirmed for some boundary conditions

Fitting curves for parameters involve exponential and polynomial segments

Abstract

Axisymmetric, rigidly rotating polytropes are considered in the framework of both the original Chandrasekhar (C33) approximation and a different version (extended C33 approximation). Special effort is devoted to two specific points, namely (i) a contradiction between the binomial series evaluation and the vanishing density on the boundary, which affects the self-consistency of the above mentioned approximations, and (ii) the continuity of selected parameters as a function of the polytropic index. Concerning (i), it is shown the Emdem-Chandrasekhar (EC) associated functions are defined at any internal point even if related EC associated equations hold only for a particular subvolume, in the framework of both the C33 and the extended C33 approximation. Concerning (ii), the continuity may safely be established, on the boundary of the domain, for part of the parameters, while additional data are needed for the remaining part. Simple fitting curves, valid to a good extent for a wide range of polytropic index, involve exponential functions and, in a single case, two straight lines joined by a parabolic segment. The expression of physical parameters in terms of the polytropic index can be used in building up sequences of configurations with changing density profile for assigned mass and angular momentum.