Homology scaling of central stellar pressure (${\rho }_c\propto M/R^3,P_c\propto G{M}^2/R^4$)

In stellar homology, a fixed dimensionless density profile gives ${\rho }_c=C_{\rho }M/R^3$ through the mass integral. Integrating the stellar hydrostatic equation through the same profile gives $P_c=C_{P}G{M}^2/R^4$ when surface pressure is negligible. The constants depend on the shape, rather than the mass and radius. For an ideal gas, $T_c=(C_P/C_{\rho })\mu GM/(\mathcal{R}R)$.