Homologous isothermal mass-radius relation ($M/R=(c_T^2/G){\xi }_b{\Psi }^{\prime }({\xi }_b)$)

Use the isothermal Lane-Emden equation variables $r=a\xi$, $a^2=c_T^2/(4\pi G{\rho }_c)$ and $\rho ={\rho }_c{e}^{-\Psi }$. Truncation at fixed dimensionless radius ${\xi }_b$ gives $R=a{\xi }_b$ and $M=4\pi {\rho }_c{a}^3{\xi }_b^2{\Psi }^{\prime }({\xi }_b)$. Division proves the displayed relation. A common temperature, composition and dimensionless truncation therefore imply $M\propto R$, while the external pressure generally varies along that homologous sequence. Arbitrary changes of truncation at fixed temperature do not retain a common coefficient. A finite regular cloud needs an outer boundary such as that of a pressure-confined isothermal sphere.