Emden-Chandrasekhar axisymmetric, rigid-body rotating polytropes IV - Exact configurations for n = 5

In connection with the basic theory reported in a previous paper for EC1 (rigidly rotating) polytropes, exact configurations are defined as configurations for which the equilibrium equation has solutions which are infinitely close to some analytical function and the related gravitational potential coincides, in fact, with the gravitational potential due to mass distribution, at any point not outside the system. The author restricts to the special case n = 5 and divides the related polytropes into two components, a massive body where each mass element has a finite (polytropic) distance from the centre, and a massless atmosphere where each mass element has an infinite (polytropic) distance from the centre. In the special case n = 0 it is shown that a particular configuration, the spheroidal one, is an exact configuration and evidence is given that spheroidal configurations are the most stable among all the allowed (axisymmetric) configurations. It is also pointed out that EC1 polytropes with n = 0 and incompressible Maclaurin spheroids belong to different sequences, even if they exhibit some common features.