Emden-Chandrasekhar axisymmetric, solid-body rotating polytropes. I - Exact solutions for the special cases n = 0, 1 and 5

The basic theory on polytorpes is revisited and EC polytropes are defined. The first-order approximation theory of Chandrasekhar (1933a, b, c) and Chandrasekhar and Lebovitz (1962) is reviewed, refined and extended in such a way that better results are obtained without involving hard analytical or numerical techniques. A more precise equation is given in defining non-outer equipotential surfaces, and a new method is adopted in determining the explicit expression of the gravitational potential. This method essentially consists in equating the expression of the gravitational potential and its first radial derivative determined by accounting for the equilibrium condition, with the corresponding expression of the gravitational potential and its first radial derivative determined by accounting for mass distribution. Such expressions are to be calculated at convenient points - for instance, at the centre and at the pole of the system. In this way, an infinity of exact solutions is derived for the special cases n=0 and n=1, and we then have the problem: 'Which of the infinite number of solutions available leads to the most stable configuration?' The simplest of these solutions is taken into account in detail for both n=0 and n=1; results relative to the latter case allow us to solve the Kopal (1937) problem. EC polytropes with n=5 are found to consist of an inner massive non-rotating component and an outer zero-density rotating atmosphere. It is seen that they are equivalent in some respects to Roche systems, and the corresponding exact solution is derived. Explicit expressions for characteristic physical parameters are also determened in the general case, relative to sequences of equilibrium states characterized by constant masses and angular momenta. Detailed results are given for the special cases n=0, 1 and 5. Finally, some properties of both EC polytropes and R polytropes with n=0 (i.e., generalized Roche systems) are reported and discussed. The conclusions of this paper make it highly desirable to have an extension of the method used here to general values of n. © 1980 D. Reidel Publishing Co.