Emden-chandrasekhar axisymmetric, solid-body rotating polytropes

Abstract

According to the general results of a previous work (Caimmi, 1980; hereafter referred to as Paper I), solutions to EC equation, which expresses a necessary and sufficient condition for equilibrium of Emden-Chandresekhar axisymmetric, solid-body rotating polytropes (EC polytropes), are taken into consideration, of the type

$$\vartheta (\xi ,\mu ) = A_0 \vartheta _0 (\upsilon ,\xi ) + \sum\limits_l^\infty {_l {\rm A}_{2l} (\upsilon )\vartheta _{2l} (\xi )P_{2l} (\mu ),} $$

with ϑ _2l later defined as the EC associated function of degree 2l. Thus the EC equation, involving (ϑ, μ), is found to be equivalent to the infinite set of EC associated equations, involving ϑ _2l (μ). We approximate g (ϑ, μ) by neglecting all terms of degree higher than 2 which appear in the above expression, and then search power series solutions to EC associated equations of degree 0 and 2, corresponding to any choice ofn (polytropic index, related to density distribution) andv (related to rotational distorsion). To this aim, we extend the methods used by Seidov and Kuzakhmedov (1977), and Mohan and Al-Bayaty (1980), to construct power series of the type outlined above, related to solid-body rotating configurations and originating both inside and outside the radial boundary (defined as the first zero of ϑ₀(μ)=0). The corresponding expressions of ϑ₀ and ϑ₂ may serve to derive an approximate expression of, and future work becomes possible concerning the determination of some physical parameters (such as volume, mass, potential energy, angular momentum) related to any choice ofn andv.

Computations have been performed forn=k/4 (0≤k≤20, i.e. 0≤n≤5) andv=0,v≈v _R/2,v≈v _R, withv _R lowest value ofv leading to balance between gravitation and centrifugal force at the equator of the system. An upper limit to the error, ε^*(μ), done in computing ϑ _2l , ϑ‚ _2l , and ϑ„ _2l at any point ϑ for a given choice ofn andv, is estimated, ranging from large values (ε^*=1E-2) forn close enough to 0 and ϑ close enough or outside the radial boundary, to low values (ε^*=1E-10) forn far enough from 0 and no constraint on ϑ. Comparison between results of this paper and the accurate results by Linnell (1977, 1981) obtained using a different approach and available forn=2,v=0, andn=3,v=0, lead to a fair agreement (up to (1E−5−1E−6).

It is apparent that the method followed here continues to hold when the first EC associated functions up to degree 2l are taken into account, leading — at least in way of principle — to a more refined approximation to the EC function; this would only make the related calculations much more complicated.