In 1796, Laplace conjectured that a primitive interstellar gas cloud may evolve under the influence of its own gravity to form a system of isolated rings, which may in turn lead to the formation of planetary systems. In fact, such a system of rings around a protostar has been observed recently in the constellation Taurus. In this paper, we use the time dependent Euler–Poisson equations to find out under what conditions such a pattern formation can take place. To this end, we consider these equations assuming that the gas is composed of incompressible stratified fluid. Under these assumptions, the original six governing equations are then reduced to three. These equations are then solved analytically (approximately), and these solutions are used to determine the evolution of the mass density distribution within the interstellar gas cloud. We show that under proper conditions, mass density waves can emerge within the cloud.

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