In this paper, we consider the nonrelativistic limit of Chandrasekhar variational model for neutron stars. We show that the minimizer $\rho_{ c}$ of Chandrasekhar energy $E_c(N)$ converges strongly to the minimizer $\rho_{ \infty}$ of limit energy $E_{ \infty}(N)$ in $L^1\cap L^{ \frac{ 5}{ 3}}(\mathbb{ R}^3)$ as the speed of light $c\rightarrow\infty$, this is a limit between two free boundary problems. Moreover, we develop a novel approach to obtain the convergence rates, we show that the above nonrelativistic limit has the optimal convergence rate $\frac{ 1}{ c^2}$. For the radius $R_c$ of the compact support of $\rho_c(x)$ and the radius $R_\infty$ of the compact support of $\rho_\infty(x)$, we also get the optimal convergence rate $\frac{ 1}{ c^2}$, this means that $R_\infty-R_c=O(\frac{ 1}{ c^2})$ as $c\rightarrow\infty$. Moreover, we also obtain the optimal uniform bounds of $R_c$ and $L^\infty$-norm of $\rho_c$ with respect to $N$ as $c\rightarrow \infty$.
Categories: math.AP math-ph math.MP