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

In connection with the basic theory reported in a previous paper (Paper I) for EC1 (rigidly rotating) polytropes, we define exact configurations 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. Then we restrict to the special case n=5 and divide 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 te centre. It is found a single exact configuration exists, which under some assumptions may be related to Roche systems. In the special case n=0 it is shown a particular configuration, the spheroidal one, is an exact configuration and evidence is given that spheroidal configurations are the stablest 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. In the special case n=1 it is shown each allowed configuration is expressible by a convenient series development, which reduces to the related n=0 configuration by maintaining only the first two or the first one terms of the sum. It is also deduced, by analogy with the case n=0, that pseudospheroidal configurations are exact and the stablest among all the allowed (axisymmetric) configurations. © 1987 D. Reidel Publishing Company.