= Solution
Assume <virial equilibrium>, structurally homologous <early-type galaxies>, a fixed relation between the measured central <velocity dispersion> $\sigma_c$ and the global random stellar motions, and a constant total <mass-to-light ratio> $\Upsilon$. The <virial theorem> gives $M=k_v\sigma_c^2R_e/G$, where the dimensionless structure factor $k_v$ is common to the sample. With $\mathcal I_e$ denoting mean \b[linear] <surface brightness> inside the <effective radius>, $L=2\pi R_e^2\mathcal I_e$. Thus
$$
\boxed{R_e=\frac{k_v}{2\pi G\Upsilon}\frac{\sigma_c^2}{\mathcal I_e},\qquad \log R_e=2\log\sigma_c-\log\mathcal I_e+\mathrm{constant}.}
$$
The usual observed <fundamental plane of elliptical galaxies> has a tilt relative to these exponents because <mass-to-light ratio> varies with <stellar population>, chemical composition, and <dark matter> fraction; structural and dynamical nonhomology can also change $k_v$.
\b[Printed convention.] The source places the logarithm of a value expressed in $\mathrm{mag\,arcsec^{-2}}$ in the empirical fit. This differs from the usual logarithm of linear <surface brightness>. If $\mu_e$ is the quoted magnitude <surface brightness>,
$$
\log_{10}\frac{\mathcal I_e}{L_{V,\odot}\,\mathrm{pc^{-2}}}=-0.4(\mu_e-M_{V,\odot}-21.572).
$$
A coefficient $-0.8$ on $\log\mathcal I_e$ therefore becomes $+0.32$ on $\mu_e$, with an adjusted intercept; it does \b[not] become $-0.8\log\mu_e$. For a concrete distinction, changing $\mu_e$ from 20 to 25 decreases linear <surface brightness> by a factor 100, so the linear-intensity fit changes $R_e$ by $100^{0.8}\simeq39.8$, whereas the literal printed logarithm changes it by $(25/20)^{-0.8}\simeq0.837$.
The following part evaluates the printed numerical fit literally and labels that convention. If the fit was intended for linear <surface brightness>, its calibration must be specified; a conditional result for using the same coefficients with linear units is also given. No equality between these two conventions is assumed.
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