= Homologous isothermal mass-radius relation
{title2=$M/R=(c_T^2/G)\xi_b\Psi'(\xi_b)$}
Use the <isothermal Lane-Emden equation> variables $r=a\xi$, $a^2=c_T^2/(4\pi G\rho_c)$ and $\rho=\rho_ce^{-\Psi}$. Truncation at fixed dimensionless radius $\xi_b$ gives $R=a\xi_b$ and $M=4\pi\rho_ca^3\xi_b^2\Psi'(\xi_b)$. 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>.
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