= Solution
For a <uniform-density stellar model>, $\rho=3M/(4\pi R^3)$ and $m(r)=4\pi\rho r^3/3$. With zero surface <pressure>, integrate the <stellar hydrostatic equation>:
$$
P(r)=\int_r^R\frac{4\pi G\rho^2}{3}s\,ds=\frac{2\pi G\rho^2}{3}(R^2-r^2).
$$
Using the <ideal gas> equation of state, $T(r)=P(r)/(\mathcal R\rho)$. Therefore
$$
\boxed{P_c=\frac{2\pi G\rho^2R^2}{3}=\frac12\left(\frac{4\pi}{3}\right)^{1/3}GM^{2/3}\rho^{4/3},\qquad T_c=\frac{P_c}{\mathcal R\rho}=\frac{G}{2\mathcal R}\left(\frac{4\pi}{3}\right)^{1/3}M^{2/3}\rho^{1/3}.}
$$
Equivalently $P_c=3GM^2/(8\pi R^4)$ and $T_c=GM/(2\mathcal R R)$. The constant <mass density> is imposed as a structural approximation; a <perfect gas> can realize it hydrostatically by allowing its <temperature> and <entropy> to vary with radius.
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