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Ultra Scientist Vol. 24(3)B, 539-545 (2012). Polytropes and their property SUDHANSHU PANDEY 1 and K. L. PANDEY 2 (Acceptance Date 21st November, 2012) Abstract Polytropes has been analytical studied under the concept of a sphere of uniform density by the polytrophic indices n tending to zero and n tending one. In this paper we have demonstrated that the temperature of the stars keeps on decreasing from its centre to surface. Approximate analytic solutions to the equilibrium equations have been presented in phase planes such as (U,V), Transformations connecting solutions in this phase plane have been obtained and discussed. plane. Key word : Polytropes, Temperature , Analytical Study, Phase Introduction The structure of a self-gravitating equilibrium polytropic sphere is described by Lane-Emden equation which have been used to model several phenomena in mathematical physics, thermodynamics, astrophysics such as the theory of stellar structure and thermal behavior of a spherical cloud of gas, for the value of polytropic index n between 0 and 3, the equation approximates to a reasonable accuracy the structure of a variety of the realistic stellar models. Though modern texts no longer give them through treatment that classical works of Emden and Chandrasekhar do, Closed form analytical solution, has been studied by Fowler 1 , Hopf 2 and Chandrasekhar for n3, respectively. It is well known so far from some of these studies that the polytropic index n=0 and 1 represent, the liquid and gaseous states of a polytrope of uniform density respectively. The origin and the behavior of Lane-Emden equations were reported same whatever be the index of a polytrope 3-14 . The Miline 3 was able to determine the maximum limiting density 15 , whereas the structure of planet was also reported 16,17 for the same values of n, and the maximum value of mass of a star 18 for n0 and n general relativity neutron star 19 were also reported for the same values of n, and very massive stellar models in Ni’s theory of gravity 20 , relativistic stellar structures and X-ray transients in Ni’s theory of gravity 21 , Further thermo dynamical equilibrium of stars clusters embedded in an isothermal

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