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We consider a lattice gas in spaces of dimensionality D = 1; 2; 3. The particles are subject to a hardcore exclusion interaction and an attractive pair interaction that satisfies Gauss' law as do Newtonian gravity in D = 3, a logarithmic potential in D = 2, and a distance-independent force in D = 1. Under mild additional assumptions regarding symmetry and fluctuations we investigate equilibrium states of self-gravitating material clusters, in particular radial density profiles for closed and open systems. We present exact analytic results in several instances and high-precision numerical data in others. The density profile of a cluster with finite mass is found to exhibit exponential decay in D = 1 and power-law decay in D = 2 with temperature-dependent exponents in both cases. In D = 2 the gas evaporates in a continuous transition at a nonzero critical temperature. We describe clusters of in_nite mass in D = 3 with a density pro_le consisting of three layers (core, shell, halo) and an algebraic large-distance asymptotic decay. In D = 3 a cluster of finite mass can be stabilized at T > 0 via con_nement to a sphere of finite radius. In some parameter regime, the gas thus enclosed undergoes a discontinuous transition between distinct density profiles. For the free energy needed to identify the equilibrium state we introduce a construction of gravitational self-energy that works in all D for the lattice gas. The decay rate of the density profile of an open cluster is shown to transform via a stretched exponential for 1 < D < 2 whereas it crosses over from one power-law at intermediate distances to a different power-law at larger distances for 2 < D < 3.