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_{c}=λ

_{c}(n ,σ ). In the case of n >3 , the critical value λ

_{c}tends to zero for special values of σ . The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with an extension comparable to those of the dark matter halos related to galaxies, i.e., with extension ℓ>100 kpc and mass M >1 0

^{12}M

_{☉}. For such largely extended polytropes, the cosmological parameter relating the vacuum energy to the central density has to be larger than λ =ρ

_{vac}/ρ

_{c}̃10

^{-9}. We demonstrate that the extension of the static general relativistic polytropic configurations cannot exceed the so-called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on the extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.

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_{q}=-1/3 of the quintessence. In dependence on the dimensionless quintessential field parameter c, we determine the black hole silhouette and the spectral line profiles of Keplerian disks generated in this special quintessential Kerr geometry, representing an extension of the general modifications of the Kerr geometry introduced recently by Ghasemi-Nodehi and Bambi (Eur. Phys. J. C 56:#290, 2016). We demonstrate that due to the influence of the parameter c, the silhouette is almost homogeneously enlarged, and the spectral line profiles are redshifted with almost conserved shape.

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_{q}in(-1;-1/3) governing the equation of state of the dark energy, and the dimensionless quintessential field parameter tilde{c}. We focus on the acceleration of particles due to collisional processes and show how the center of mass energy depends on the quintessential field parameter tilde{c}. We also make comparison of the obtained results to the collisional energetics of quintessential static black holes demonstrating the crucial role of the rotation parameter a in the particle acceleration. Finally we study the dependence of the maximal value of the efficiency of energy extraction through Penrose process for rotating black hole with quintessential field parameter tilde{c}. It is found that quintessence field decreases the energy extraction efficiency through Penrose process and when the parameter tilde{c} vanishes one can get the standard value of the efficiency coefficient for the Kerr black hole as η̃ 21 %.

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