= Solution
For approximately circular motion in a spherically dominated <Newtonian gravitational field>, $v_c^2(r)=GM(<r)/r$. If most gravitating mass followed the concentrated luminous <galaxies>, then beyond their main light distribution the enclosed mass would nearly saturate, giving a declining Keplerian <galaxy rotation curve>, $v_c\propto r^{-1/2}$.
Instead, extended <galaxy rotation curves> commonly remain roughly flat. The <spherical mass profile for a flat rotation curve> then requires
$$
\boxed{M(<r)\simeq\frac{v_0^2r}{G},\qquad
\rho(r)\simeq\frac{v_0^2}{4\pi Gr^2}.}
$$
The inferred mass continues growing where the luminous contribution is small, so the <mass-to-light ratio> increases outward. Under the assumed gravitational dynamics this is evidence for an extended <dark matter halo>. Measurements of neutral-gas motion can probe radii beyond the bright stellar disk. A realistic disk contribution must be computed using its flattened geometry; the spherical formula describes a halo-dominated region and does not turn every disk <rotation curve> into a spherical mass profile.
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