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Angular resolution describes the smallest angular separation or structure an observation can distinguish. A physical scale $r$ at distance $D$ subtends approximately $r/D$ radians when $r\ll D$. Resolving a black-hole sphere of influence therefore involves distance as well as its physical radius, and useful kinematics also require adequate surface brightness.

For identical self-conjugate dark matter particles, the pair-annihilation rate per volume is $⟨\sigma v⟩n_{\chi }^2/2$. If each annihilation releases radiated energy $E_a$, volume emissivity is $j=E_a⟨\sigma v⟩{\rho }^2/(2m_{\chi }^2)$. Optically thin isotropic surface brightness is $(4\pi {)}^{-1}\int jdz$. The squared-density dependence assumes a spatially constant rate coefficient and a smooth density; unresolved substructure changes the average of the square.

If a dark matter particle of mass $m_{\chi }$ radiates power $ϵ$ independently of its neighbours, optically thin isotropic emission has volume emissivity $j=ϵ\rho /m_{\chi }$. Its surface brightness is $I=(4\pi {)}^{-1}\int jdz$ and therefore traces the mass column rather than its square.

The Faber-Jackson relation is an empirical correlation between luminosity and stellar velocity dispersion for elliptical galaxies, approximately $L\propto {\sigma }^4$ in its classical form. Slope, scatter, sample and band matter. A heuristic combines $M\propto {\sigma }^{2}R$, common mass-to-light ratio, and common characteristic surface brightness $L/R^2$. The luminosity-dispersion correlation alone does not imply those auxiliary assumptions, and extrapolation to a dwarf spheroidal galaxy is not guaranteed.

The fundamental plane relates effective radius, stellar velocity dispersion, and mean surface brightness of elliptical galaxies. Homologous virial equilibrium with constant mass-to-light ratio predicts $R_e\propto {\sigma }^2/I_e$; empirical exponents differ. The two-variable Faber-Jackson relation is a projection, not the entire plane.

The signed magnification is the reciprocal determinant of the Jacobian matrix of the thin gravitational lens equation. Its absolute value gives the flux and solid-angle amplification; its sign gives image parity. Gravitational lensing preserves surface brightness in the geometric-optics approximation.

The luminosity per unit spatial volume. For a spherical source, its enclosed luminosity is $L(r)=4\pi {\int }_0^r\lambda (s)s^{2}ds$, and its projected surface brightness is $\mu (R)=2{\int }_R^{\infty }\lambda (r)rdr/\sqrt{r^2-R^2}$.

An optically thin spherical source with volume emissivity $j(r)=j_0{r}^{-p}$ has surface brightnessSubstitute $z=Ru$ in the line-of-sight integral and evaluate ${\int }_{-\infty }^{\infty }(1+u^2{)}^{-p/2}du$ using the beta function. The condition $p>1$ is convergence at large distance. A core or finite outer boundary changes the projection near that boundary; the stated scaling applies between those scales.