= Constant-opacity homologous contraction to CNO ignition
{title2=$L\propto M^3,\quad R_{\rm MS}\propto M^{15/19},\quad M_{\rm arrival}\propto t^{-19/34}$}
For radiative, gas-pressure-supported homologous stars with constant <opacity>, write $\rho=M b/(4\pi R^3)$, $P=GM^2p/(4\pi R^4)$ and $T=\mu GMp/(\mathcal R Rb)$. The <radiative diffusion in a star> equation contains the <dimensionless> coefficient $D=3\kappa_0\mathcal R^4L/(64\pi^2ac\mu^4G^4M^3)$. A common fixed profile fixes $D$, so $L\propto M^3$ and the <luminosity> of a fixed-mass member is constant. <Gravitational energy generation in a homologously contracting ideal-gas star> gives $\dot R=-2ELR^2/(3GM^2)$ for a positive profile constant $E$. Thus $R^{-1}=R_0^{-1}+2ELt/(3GM^2)$. For <CNO cycle> heating $\epsilon\propto\rho T^{16}$, the <stellar nuclear fusion> <luminosity> scales as $M^{18}R^{-19}$. Equating this with $M^3$ gives $R_{\rm MS}\propto M^{15/19}$. Neglecting $R_0^{-1}$, the arrival time is $t_{\rm MS}\propto M^{-34/19}$, hence a coeval cluster's newly arriving <mass> scales as $t^{-19/34}$. All comparisons hold <stellar composition>, <opacity> normalization and <dimensionless> profiles fixed.
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