= Solution
Put $\Delta=R-r\ll R$. The near-surface <mass density> and <mean free path> from <2/a> and <2/b> are
$$
\rho\simeq\rho_c\frac\Delta R,\qquad
\ell\simeq\frac{LR}{\Delta}.
$$
A simple local escape estimate sets the remaining distance equal to the <mean free path>, $\ell\simeq\Delta$. It gives
$$
\boxed{\Delta R_{\rm esc,local}\simeq\sqrt{LR},\qquad
\frac{\Delta R_{\rm esc,local}}R\simeq\sqrt{\frac LR}.}
$$
For the <Sun>'s parameters this is $\boxed{\Delta R_{\rm esc,local}\simeq1.94\,\mathrm{km},\quad\Delta R_{\rm esc,local}/R\simeq2.79\times10^{-6}}$.
A more precise operational definition uses the outward radial <optical depth>. In this varying <mass density>,
$$
\tau(r)=\int_r^R\frac{ds}{\ell(s)}
=\frac1{aL}\int_{ar}^{\pi}\frac{\sin x}{x}\,dx
\simeq\frac{\Delta^2}{2LR}.
$$
The chance of an outward radial <photon> escaping without another scattering is $e^{-\tau}$. Thus the conventional $\tau=1$ last-scattering depth gives
$$
\boxed{\Delta R_{\rm esc,\tau=1}\simeq\sqrt{2LR}
\simeq2.75\,\mathrm{km},\qquad
\Delta R_{\rm esc,\tau=1}/R\simeq3.95\times10^{-6}.}
$$
There is no sharp escape surface: $\tau\ll1$ gives nearly free propagation, and oblique rays have different <optical depths>. The local $\ell=\Delta$ estimate actually has $\tau\simeq1/2$ in this model, not $1$; the factor $\sqrt2$ is a convention-dependent order-one correction, not a discrepancy in the <Thomson cross-section>. Neither scattering criterion establishes a thermalization surface.
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