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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 ${\rm Y}$. The virial theorem gives $M=k_v{\sigma }_c^2{R}_e/G$, where the dimensionless structure factor $k_v$ is common to the sample. With ${\mathcal{I}}_e$ denoting mean linear surface brightness inside the effective radius, $L=2\pi R_e^2{\mathcal{I}}_e$. ThusThe 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 $k_v$.

Printed convention. The source places the logarithm of a value expressed in $\mathrm{mag}\mathrm{arcse}{\mathrm{c}}^{-2}$ in the empirical fit. This differs from the usual logarithm of linear surface brightness. If ${\mu }_e$ is the quoted magnitude surface brightness,A coefficient $-0.8$ on $\log {\mathcal{I}}_e$ therefore becomes $+0.32$ on ${\mu }_e$, with an adjusted intercept; it does 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 $100^{0.8}\simeq 39.8$, whereas the literal printed logarithm changes it by $(25/20{)}^{-0.8}\simeq 0.837$.

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.

Evaluate the source's printed logarithm using the numerical magnitude value 21.0, with base-ten logarithms:This is the literal printed-fit convention identified above. Convert the mean visual surface brightness to a linear luminosity per projected area. At 10 parsecs, one square arcsecond subtends $100/(206264.806{)}^2\mathrm{p}{\mathrm{c}}^2$. The definition of absolute magnitude consequently givesThe constant $21.572=5{\log }_{10}(206264.806)-5$ is the conversion between the angular and physical projected area. Since the effective radius encloses half the total luminosity,For the stipulated stellar population of Solar-type stars, approximate the stellar mass-to-light ratio by $M_{\odot }/L_{V,\odot }$. Summing 100 such galaxies then givesThis assumes negligible interstellar extinction and that the supplied surface brightness is an intrinsic or already corrected value; an observed cosmological surface brightness requires additional dimming and passband corrections before this conversion.

For comparison only, if the empirical coefficients and intercept were instead meant for ${\log }_{10}({\mathcal{I}}_e/[L_{V,\odot }\mathrm{p}{\mathrm{c}}^{-2}])$, substituting $140.9$ in that fit would give $R_e=1.09\mathrm{kpc}$, $L_V=1.06\times 10^9{L}_{V,\odot }$, and $M_{\ast ,\mathrm{cluster}}=1.06\times 10^{11}{M}_{\odot }$. Those are conditional alternative results, not an unannounced correction of the printed units.

The effective radius is one tenth of the optical outer radius: $R_e\simeq 2.13$ kpc. A flat rotation speed then gives $M_{\mathrm{dyn}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$. Half the luminosity, with the same mass-to-light ratio, implies $M_{\mathrm{luminous}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$.The concentration of light into a small fraction of the optical radius suggests a high-surface-brightness, centrally concentrated disk galaxy, plausibly an early-type spiral or lenticular with a substantial bulge. The kinematic and photometric data do not uniquely distinguish those morphologies. Disk geometry, an unknown inclination correction, or a varying stellar mass-to-light ratio would alter the exact numerical inner comparison.