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Relativistic polytropic spheres embedded in a chameleon scalar field

In this paper we investigate gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (polytropic) matter nonminimally coupled to an external chameleon scalar field. We show that this system has static, regular, asymptotically flat general relativistic solutions. The properties of these spherical configurations, such as total mass, distribution of matter, and size, depend strongly on the surrounding scalar field. The mass is found in terms of the parameter $σ$, which is proportional to the central mass density of the ordinary matter. We perform a stability analysis of these spherical solutions and find an upper bound for $σ$ where dynamical instability occurs.