= Solution
Efficient <convection> keeps the interior at nearly uniform <specific entropy>. For the fully ionized <monatomic gas>, the <heat capacity ratio> is $5/3$, so the <adiabatic process> gives $P=K_\rho\rho^{5/3}$, with $K_\rho$ spatially constant. The <ideal gas> relation $P=\mathcal R\rho T/\mu$ then implies
$$
\boxed{P=KT^{5/2}},\qquad K=\left(\frac{\mathcal R}{\mu}\right)^{5/2}K_\rho^{-3/2}.
$$
The constant is spatial, not temporal: loss of <specific entropy> during contraction changes it. This is a <stellar polytrope> with <polytropic index> $n=3/2$. We neglect <radiation pressure> and take the thin <photosphere> as an atmospheric matching layer rather than a significant fraction of the stellar mass.
For a fixed dimensionless profile of the <Lane-Emden equation>, the <stellar homology> relations have the form
$$
\rho_c=C_\rho\frac{M}{R^3},\qquad P_c=C_P\frac{GM^2}{R^4},\qquad T_c=\frac{C_P}{C_\rho}\frac{\mu GM}{\mathcal R R},
$$
where $C_\rho,C_P$ are positive dimensionless structure constants. More explicitly, write $\rho(r)=\rho_cf(x)$ and $m(r)=Mh(x)$ with $x=r/R$ and fixed dimensionless profiles. Mass conservation gives $M=4\pi\rho_cR^3\int_0^1f(x)x^2\,dx$. Integrating the <hydrostatic pressure support equation>, with negligible surface pressure on the interior scale, gives $P_c=(GM\rho_c/R)\int_0^1h(x)f(x)x^{-2}\,dx$. Both integrals are fixed positive numbers, proving the two central scalings. Substituting these relations into $K=P_c/T_c^{5/2}$ gives
$$
\boxed{K=K_0\mu^{-5/2}M^{-1/2}R^{-3/2}},\qquad K_0=C_\rho^{5/2}C_P^{-3/2}\mathcal R^{5/2}G^{-3/2}.
$$
Thus the same <polytropic index> fixes $K_0$ throughout this homologous sequence, while the dimensional <stellar polytrope> constant changes.
At the <photosphere>, whose <optical depth> is $2/3$, identify the temperature with the <effective temperature> $T_e$. The <ideal gas> relation and the interior adiabat give $\rho_{\rm ph}=\mu K T_e^{3/2}/\mathcal R$. With the prescribed <opacity> and <stellar surface boundary condition>,
$$
\kappa P=\frac{\kappa_0\mu}{\mathcal R}K^2T_e^8=\frac{2GM}{3R^2}.
$$
Eliminating $K$ therefore proves
$$
\boxed{T_e^8=\frac{2G\mathcal R}{3\kappa_0K_0^2}\mu^4M^2R}.
$$
Using the <Stefan–Boltzmann law>, with $\sigma=ac/4$, gives the <luminosity>
$$
L=4\pi R^2\sigma T_e^4=\pi ac\left(\frac{2G\mathcal R}{3\kappa_0}\right)^{1/2}K_0^{-1}\mu^2MR^{5/2}.
$$
The <gravitational energy of a stellar polytrope> is $\Omega=-6GM^2/(7R)$ for $n=3/2$. The <stellar virial theorem> for a <monatomic gas> gives $2U+\Omega=0$, and hence the total energy is $E=U+\Omega=-3GM^2/(7R)$. With no appreciable <stellar nuclear fusion> or <accretion>, energy conservation gives
$$
L=-\frac{dE}{dt}=-\frac{3GM^2}{7R^2}\dot R.
$$
Combining the two expressions for $L$ yields
$$
\dot R=-\frac{7\pi ac}{3}\left(\frac{2\mathcal R}{3\kappa_0G}\right)^{1/2}K_0^{-1}\mu^2M^{-1}R^{9/2}.
$$
Multiplication by $-7R^{-9/2}/2$ and integration proves the <contraction law with density-linear fourth-power stellar opacity>:
$$
\boxed{R^{-7/2}-R_0^{-7/2}=\frac{49\pi ac}{6}\left(\frac{2\mathcal R}{3\kappa_0G}\right)^{1/2}K_0^{-1}\mu^2M^{-1}(t-t_0)}.
$$
When the accumulated contraction term dominates the initial-radius term, $R\propto\mu^{-4/7}M^{2/7}(t-t_0)^{-2/7}$. Substitution into the central <ideal gas> relation gives
$$
\boxed{T_c\propto\mu^{11/7}M^{5/7}(t-t_0)^{2/7}},
$$
which has the stated $t^{2/7}$ behavior when the time origin is negligible. This late-time approximation still requires the assumed <fully convective star>, fixed composition and <opacity> law to remain applicable.
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