An effective method in investigating structures of polytropic protoplanets formed via gravitational instability

In this article, we have reinvestigated the initial distribution of thermodynamic variables inside the protoplanets formed via gravitational instability having mass range $$ ( $$ ) by an embedded RKACeM(4,4) method assuming that the polytropic gas law holds in the protoplanets. The findings attained by our numerical experiments are recognized to be consistent with the results acquired through other notable investigations in this regard. Furthermore, the model is easily computable. The used method is found to be efficient in investing the structures of polytropic protoplanets in their initial stages in terms of accuracy, stability, computational cost, and solving endpoint constraints.