 Solution

For an ideal gas of fixed composition, the central temperature satisfies $T_c\propto p_c/{\rho }_c=K{\rho }_c^{1/n}$. Keeping $T_c$ fixed therefore requires $K\propto {\rho }_c^{-1/n}$, rather than keeping the polytropic constant fixed. Substitution in the Lane-Emden equation scalings givesConsequently the homologous polytropes at fixed central temperature obeyThe polytropic index and mean particle mass must be common to this homologous sequence: varying either changes the dimensionless structural coefficient. The fixed-central-temperature assumption is the idealization being used here, rather than a universal property of real stars.