Unveiling protoplanetary structure equations: Semi-analytical solutions via the homotopy analysis method

This paper revisits the distribution of thermodynamic variables within initial protoplanets formed via gravitational instability (GI) across a broad mass spectrum ranging from $0.3M_J\mathrm{to}10M_J$ (where $1M_J$ denotes 1 Jupiter mass, equal to $1.8986\times 10^{30}$ g), using the Homotopy Analysis Method (HAM), a novel approach in this context. Concerning heat transfer within the protoplanets, consideration is given to the convective mode. Our findings reveal a noteworthy alignment between the results obtained via the HAM, utilizing only the first four terms (third approximation), and numerical outcomes. The HAM is found to demonstrate rapid convergence towards the exact solution, showcasing its effectiveness in obtaining such solutions to nonlinear problems. Comparative graphical illustrations of approximate series solutions obtained via HAM and the Adomian decomposition method highlight the superior accuracy and analytical depth of HAM within this framework. This establishes HAM as a powerful and insightful tool for studying astrophysical phenomena. The method offers significant advantages in accuracy and analytical depth over traditional methods, demonstrating its potential for broader applications in astrophysical research and providing robust solutions where conventional approaches may fall short.