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Homologous collapse refers to a collapse of material such that the density throughout the material is uniform at each point in time. (The word homologous means the same). A perfect, non-rotating sphere of material of uniform density obeying Newton's laws would undergo such collapse from its own gravity if pressure were not a factor, i.e., there was no resistance to its collapse. Given this ideal, all the material reaches the sphere's central point at the same time. The concept is useful to serve as a simplified model (i.e., first approximation) of collapses, such as the collapse of a cloud of gas and/or dust. It has the advantage of exact solutions.
In a cloud of low-density gas (such as much of the interstellar medium) and sufficiently optically thin that it radiates away energy pretty much as soon it is unleashed by the collapse, a homologous collapse approximates much of the first portion of the collapse, and can serve as a workable model for the collapse of fairly uniform bodies, e.g., during star formation, and possibly in some core collapse supernovae.
Such an ideal collapse has an easy-to-model structure, described by a relatively straightforward equation (though computation is required to use it). At any point in time, the further from the center, the higher the acceleration, due to more matter within the same radius from the center, which is exactly enough to keep the density (ratios) the same and bring all the matter to a point. In a real world collapse, potential energy released by gravitational collapse heats the gas and the combination of heat and density raise the pressure, slowing the collapse. Other real-world factors that affect actual collapses include rotation of the cloud, its shape (undoubtedly not quite spherical), its surrounding matter, and relativity.
It is of note that though mathematics describing the ideal collapse has been well-established, this known mathematics calculates time and radii from a chosen third parameter rather than calculating radii directly from time. A newer method that actually calculates radius from time was published in 2021.