= Solution
Let $z$ measure distance inward from the surface. While $a\ll R$, curvature is negligible and this is a planar two-phase <Stefan problem>. Use a common reference <mass density> $\rho$, <specific heat capacity> $c_p$, <thermal conductivity> $k$ and <thermal diffusivity> $\kappa=k/(\rho c_p)$; retain the <mass density> differences only in the subsequent <buoyancy> estimates. Define $\Delta T=T_m-T_s>0$, $\Theta=\overline T-T_m\geq0$, and $\vartheta=\Theta/\Delta T$. The paper's parameter $S=L/(c_p\Delta T)$ is the <latent-to-sensible heat ratio>, reciprocal to the more usual <Stefan number>.
For <superheated two-phase Stefan similarity>, solving $T_t=\kappa T_{zz}$ with $T(0)=T_s$, $T(a)=T_m$ and $T(\infty)=\overline T$ gives the <Neumann solution of the Stefan problem>. With $\eta=z/(2\sqrt{\kappa t})$,
$$
\boxed{a(t)=2\lambda\sqrt{\kappa t},\quad
T_s(z,t)=T_s+\Delta T\frac{\operatorname{erf}\eta}{\operatorname{erf}\lambda},\quad
T_l(z,t)=\overline T-\Theta\frac{\operatorname{erfc}\eta}{\operatorname{erfc}\lambda}.}
$$
The solid expression holds for $0<z<a$ and the liquid expression for $z>a$. The interface gradients inserted in the <Stefan condition> determine the positive constant $\lambda$:
$$
\boxed{S\lambda=\frac{e^{-\lambda^2}}{\sqrt\pi}\left[\frac1{\operatorname{erf}\lambda}-\frac{\vartheta}{\operatorname{erfc}\lambda}\right],\qquad
\dot a=\lambda\sqrt{\frac\kappa t}.}
$$
The right-hand bracket decreases from infinity and crosses zero, while the equivalent left-hand side $S\sqrt\pi\lambda e^{\lambda^2}$ increases, proving uniqueness. The liquid-side flux subtracts from the roof cooling: it is heat brought to the interface by the superheated interior.
For large $S$, $\lambda\ll1$. Keeping the superheat term while using $\operatorname{erf}\lambda\simeq2\lambda/\sqrt\pi$ and $\operatorname{erfc}\lambda\simeq1$ yields $S\lambda^2+(\vartheta/\sqrt\pi)\lambda\simeq1/2$. Hence \b[the <small-interface-parameter growth with liquid superheat> approximation] is
$$
\boxed{a(t)\simeq A\sqrt{\kappa t},\quad A=\frac{2}{\sqrt{\vartheta^2/\pi+2S}+\vartheta/\sqrt\pi},\quad\dot a\simeq\frac{A\sqrt\kappa}{2\sqrt t}.}
$$
The solid <temperature> is approximately linear across its thin crust, while the liquid <thermal boundary layer> has thickness of order $\sqrt{\kappa t}$. For fixed superheat and $S\gg1$, $A\simeq\sqrt{2/S}$; for $\vartheta\gg\sqrt S$, $A\simeq\sqrt\pi/\vartheta$. This retains the strong suppression of growth by a very hot interior.
To estimate overturn, take local <gravitational acceleration> $g$ at the roof. The liquid's <mass density> contrast is $\Delta\rho_l=\rho_f\alpha\Theta$, while the solid contrast is $\Delta\rho_s=\rho_s-\rho_f$. A liquid thermal layer of thickness $\delta$ has <Rayleigh number> $\Delta\rho_lg\delta^3/(\mu_l\kappa)$. As $\delta\sim\sqrt{\kappa t}$, it reaches its critical value after a time scaling as
$$
\tau_l\sim\left(\frac{\mu_l}{\Delta\rho_lg\sqrt\kappa}\right)^{2/3}.
$$
In <viscous overturn of a dense crust>, an unstably stratified viscous crust has a <Rayleigh-Taylor instability> growth time of order $\mu_s/(\Delta\rho_sga)$. Setting this equal to its age, with $a=A\sqrt{\kappa t}$, gives
$$
\boxed{\tau_s\sim\left(\frac{\mu_s}{\Delta\rho_sgA\sqrt\kappa}\right)^{2/3},\qquad
\frac{\tau_l}{\tau_s}\sim\left[\frac{\mu_l}{\mu_s}\frac{\Delta\rho_s A}{\rho_f\alpha\Theta}\right]^{2/3}.}
$$
These are formation-to-overturn times. A specified critical liquid <Rayleigh number> $\mathrm{Ra}_c$ multiplies $\tau_l$ by $\mathrm{Ra}_c^{2/3}$; logarithmic amplification and geometrical constants also affect the crust estimate. They are omitted for the requested order-of-magnitude scaling.
The given numbers yield $S\simeq2.54$, $\Delta\rho_s=500\,\mathrm{kg\,m^{-3}}$ and $\Delta\rho_l=100\vartheta\,\mathrm{kg\,m^{-3}}$. Thus
$$
\frac{\tau_l}{\tau_s}\sim\left(\frac{5A}{\vartheta}\,10^{-18}\right)^{2/3}\simeq
\begin{cases}
5.8\times10^{-11},&\vartheta=10^{-2},\\
9.2\times10^{-15},&\vartheta=10^2.
\end{cases}
$$
\b[Liquid thermal overturn is enormously faster than crustal overturn] throughout the stated range. Even a critical-Rayleigh prefactor of order $10^2$ does not change that ordering. Since $S\simeq2.54$ is not asymptotically large, low-superheat coefficients from the permitted approximation are only approximate. At the upper end $\alpha\Theta=1.25$, so the linear thermal-density approximation also ceases to be quantitatively credible; that endpoint is a formal scaling estimate under the supplied constant-property model. At a common age the instantaneous viscous instability-time ratio instead scales as $5A\,10^{-18}/\vartheta$; it should not be confused with the above formation-time ratio.
For a <boundary-layer renewal model>, the liquid layer reaches $\delta_l\sim\sqrt{\kappa\tau_l}$. Its cycle-averaged thermal flux is consequently
$$
\boxed{F_l\sim\frac{k\Theta}{\delta_l}\sim k\Theta\left(\frac{\rho_f\alpha\Theta g}{\mu_l\kappa}\right)^{1/3}.}
$$
If a critical <Rayleigh number> is retained, insert a factor $\mathrm{Ra}_c^{-1/3}$. Averaging the diffusive $t^{-1/2}$ flux over a cycle only changes an order-one coefficient. The crust reaches
$$
a_s\sim\left(\frac{\mu_sA^2\kappa}{\Delta\rho_sg}\right)^{1/3},
$$
so its average solid-<volume flux> per roof area is \b[the crust-removal flux]
$$
\boxed{J_s\sim\frac{a_s}{\tau_s}\sim\left(\frac{\Delta\rho_sg}{\mu_s}\right)^{1/3}(A^2\kappa)^{2/3}.}
$$
The corresponding <mass flux> is $\rho_sJ_s$ and latent-<heat flux> is $\rho_sLJ_s$. This independent-cycle estimate is a scale for a crust that reaches overturn thickness and is then renewed. Because the liquid renews much more rapidly, in a hot interior its average <heat flux> can limit actual crust growth: the consistent interfacial budget is $\rho L\dot a\simeq k\Delta T/a-F_l$. A stalled crust would not realize repeated solid cycles. In a superheated interior, detached solid may also remelt; permanent core deposition is most directly justified in the late regime requested next.
At late times $\Theta=0$, liquid thermal <buoyancy> vanishes and $A^2\kappa\simeq2\kappa/S=2k\Delta T/(\rho L)$. Thus
$$
\boxed{J_0\sim\left(\frac{\Delta\rho_sg}{\mu_s}\right)^{1/3}\left(\frac{2k\Delta T}{\rho L}\right)^{2/3}.}
$$
Assuming thin renewed crust, efficient sinking and constant asteroid radius, <conservation of mass> gives $4\pi b^2\dot b\simeq4\pi R^2J_0$ under the permitted equal-density approximation. Therefore \b[the <asteroidal core growth by crustal sinking> law is]
$$
\boxed{b(t)^3\simeq b_0^3+3R^2J_0(t-t_0),\qquad \frac{b(t)}R\simeq\left[\left(\frac{b_0}R\right)^3+\frac{3J_0(t-t_0)}R\right]^{1/3}.}
$$
For a self-gravitating approximately uniform sphere, $g\simeq4\pi G\rho R/3$, so $J_0\propto R^{1/3}$ and a zero-seed core scales as $b\propto R^{7/9}(t-t_0)^{1/3}$. This law applies while the thin-crust renewal model remains valid and liquid remains available; it cannot continue beyond complete solidification. The specific numerical lifetime cannot be inferred without the absolute solid <viscosity> and conductivity.
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