Approximate gravitational acceleration (left) and corresponding density for the modified Lane-Emden equation (right). The left panel shows uχ(u) (χ from Eq. (10)) compared to a Newtonian acceleration of a point mass (−1 /u²). Above u = 1, both functions coincide by construction. The right panel shows , the dimensionless density distribution from Eq. (18) associated with a softened gravitational acceleration as shown in the left panel.