Allowed regions of galaxy-like and cluster-like polytropes in the nonrelativistic regime. The figure shows the polytrope extension, R, and mass, M, at the edges of the selected intervals R = 100–200 kpc and M = 1–5 × 10¹² M_⊙ (top figure) and R = 1–5 Mpc & M = 1–5 × 10¹⁵ M_⊙ (bottom figure). We are drawing only the extensions R and M for the polytropic index n = 0.5 (4 lines). However, the “polygon shape” intersections are demonstrated for n = 0.5 up to n = 4.5 with step size 0.5. The intersection for n = 0.5 is the leftmost one, for n = 1.0 the intersection region is located to the right of the previous one, and so on. Also, the non-integers n are filled with single hue color (increasing the hue for higher polytropic index) for a better overview. The two vertical lines correspond to the critical value ρ_crit(n; σ), i.e., the λ_crit(n; σ) (see Section 4) for two polytropic index (4.5,3.0). The configurations for the particular n can only exists on the right side of the curve. The dots represent the selected points for the polytropic spheres whose profiles are given in the next section.