Scale-free equilibria of isopedic polytropic clouds

We investigate the equilibrium properties of self-gravitating magnetized clouds with polytropic equations of state with negative index n. In particular, we consider scale-free isopedic configurations that have constant dimensionless spherical mass-to-flux ratio λ_r and that may constitute "pivotal" states for subsequent dynamical collapse to form groups or clusters of stars. For given Γ = 1 + 1/n, equilibria with smaller values of λ_r are more flattened, ranging from spherical configurations with λ_r = ∞ to completely flattened states for λ_r = 1. For a given amount of support provided by the magnetic field as measured by the dimensionless parameter H₀, equilibria with smaller values of Γ are more flattened. However, logatropic (defined by Γ = 0) disks do not exist. The only possible scale-free isopedic equilibria with logatropic equation of state are spherical uniformly magnetized clouds.