Features of drop model of protoplanetary disk

The drop model of a protoplanetary disk postulates a cluster of spherical bodies moving in a single plane in circular orbits around an attracting center, whereby the bodies are united during collision. The evolution of this model is discussed within the framework of the coalescence theory, based on the ideas and results of T. M. Eneyev and N. N. Kozlov (1977, 1980). Eneyev and Kozlov discovered density waves in studying the evolution of the drop model of a protoplanetary disk. A numerical study made by them of a model containing tens of thousands of bodies demonstrated the pattern of the formation of alternating annular regions of higher and lower density in the initially uniform disk. A single planet ultimately forms in each region of higher density. The radii of the boundaries of these regions form on average a geometrical progression consistent with the positioning of the planets in the solar system. Eneyev and Kozlov suggested an approach from the viewpoint of the coalescence theory for studying the evolution of the drop model with a very large number of bodies. The coalescence equation is derived and an analytical solution is given. It is demonstrated that the drop model permits a complete analytical study within the framework of the coalescence theory.