= Solution
Use $\mathcal R$ for the <stellar gas constant>, reserving $R$ for the stellar <radius>. The <specific internal energy> of a <monatomic gas> obeying the <ideal gas> law is $e=P/[(\gamma-1)\rho]=3P/(2\rho)$. At fixed enclosed <mass>, the <first law of thermodynamics> gives the heat available to the outgoing <luminosity> as $\epsilon=-\partial_te-P\partial_t(1/\rho)$ when there is no <stellar nuclear fusion> heating. Therefore
$$
\boxed{\epsilon=-\frac3{2\rho}\left(\frac{\partial P}{\partial t}\right)_m
+\frac{5P}{2\rho^2}\left(\frac{\partial\rho}{\partial t}\right)_m.}
$$
The second coefficient includes both the <mass density> dependence of $e$ and <pressure> work. This is <gravitational energy generation in a homologously contracting ideal-gas star>.
Write the characteristic scales as
$$
\rho_0=\frac M{4\pi R^3},\qquad P_0=\frac{GM^2}{4\pi R^4},\qquad T_0=\frac{\mu GM}{\mathcal R R}.
$$
Then $\rho=\rho_0b$, $P=P_0p$, $T=T_0p/b$, $m=Mq$ and $L_r=Ll$. Since $d/dr=R^{-1}d/dx$, direct substitution in the <stellar hydrostatic equation> and <mass conservation> gives
$$
\boxed{\frac{dp}{dx}=-\frac{bq}{x^2},\qquad\frac{dq}{dx}=x^2b.}
$$
For <radiative diffusion in a star>, dividing the <temperature> equation by $T_0/R$ gives
$$
\frac d{dx}\left(\frac pb\right)
=-\frac{3\kappa_0\rho_0L}{16\pi acRT_0^4}\frac{bl}{x^2(p/b)^3}
=-D\frac{b^4l}{x^2p^3},\qquad
\boxed{D=\frac{3\kappa_0\mathcal R^4L}{64\pi^2ac\mu^4G^4M^3}.}
$$
In particular the fourth power here is of $\mathcal R$, not of the <radius>.
Because $q(x)$ is fixed and $M$ is constant, a fixed <mass> label has fixed $x$. Thus <stellar homology> gives $\partial_t\rho=-3\rho\dot R/R$ and $\partial_tP=-4P\dot R/R$. The heating rate reduces to $\epsilon=-3P\dot R/(2\rho R)$, positive during <Kelvin-Helmholtz contraction>. Substitution in $dL_r/dr=4\pi r^2\rho\epsilon$ now yields
$$
\frac{dl}{dx}=-\frac{6\pi R^2\dot R}{L}x^2P
=Ex^2p,\qquad
\boxed{E=-\frac{3GM^2\dot R}{2R^2L}>0.}
$$
The <dimensionless> profiles fix $D$ and $E$. In a common <stellar homology> family with fixed <stellar composition> and <opacity> normalization, the expression for $D$ implies
$$
\boxed{L=\frac{64\pi^2ac\mu^4G^4D}{3\kappa_0\mathcal R^4}M^3\propto M^3.}
$$
It is independent of <radius> and constant in time for a particular fixed-mass star. The formula for $E$ gives $\dot R=-2ELR^2/(3GM^2)$, which integrates to
$$
\frac1{R(t)}=\frac1{R_0}+\frac{2ELt}{3GM^2}.
$$
When the initial-radius term is negligible,
$$
\boxed{\frac{RLt}{GM^2}=\frac3{2E}.}
$$
This is the <Kelvin-Helmholtz contraction> time law in the prescribed <stellar homology> model. Infinite initial <radius> is a limiting idealization; a finite initial <radius> retains the first term.
At <main sequence> arrival, <CNO cycle> heating with fixed <stellar composition> has <luminosity>
$$
L_{\rm nuc}=\epsilon_0M\rho_0T_0^{16}\int_0^1b(q)\left(\frac{p(q)}{b(q)}\right)^{16}dq
\propto\frac{M^{18}}{R^{19}}.
$$
The <integral> is <dimensionless> and fixed within the <stellar homology> family. The constant-<opacity> radiative scaling still gives $L\propto M^3$. Equating the <stellar nuclear fusion> and transported <luminosities> gives
$$
\boxed{R_{\rm MS}\propto M^{15/19}.}
$$
Finally the <Kelvin-Helmholtz contraction> law evaluated at $R_{\rm MS}$ gives $t_{\rm MS}\propto M^2/(LR_{\rm MS})\propto M^{-34/19}$. In a coeval <star cluster>, the <mass> just arriving at the <main sequence> consequently satisfies
$$
\boxed{M_{\rm arrival}\propto t^{-19/34}.}
$$
These results comprise <constant-opacity homologous contraction to CNO ignition>; their <mass> exponents compare fixed-composition models with common <dimensionless> profiles.
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