= Solution
Let $R$ be a characteristic radius enclosing mass $M$, and let $\sigma$ denote a one-dimensional <velocity dispersion>. Gravity and random-motion support give a relation $GM/R=\eta\sigma^2$, where the dimensionless structure coefficient $\eta$ depends on the chosen radius and density profile. A crossing-based <galactic dynamical time> is therefore
$$
t_{\rm dyn}=\frac R\sigma=\frac{GM}{\eta\sigma^3},\qquad
\boxed{t_{\rm dyn}\sim\frac{GM}{\sigma^3}}.
$$
The scaling follows by equating acceleration scales $\sigma^2/R$ and $GM/R^2$, not by identifying a complete orbital period with a crossing time. An explicit scale-free example checks the coefficient: constant <circular speed> and <spherical symmetry> give $M(<r)=v_c^2r/G$ and $\rho=v_c^2/(4\pi Gr^2)$. With isotropic constant <velocity dispersion>, the <Spherical Jeans equation> becomes
$$
\sigma^2\frac{d\rho}{dr}=-\rho\frac{v_c^2}{r},
$$
so $v_c^2=2\sigma^2$, $M(<R)=2\sigma^2R/G$, and $R/\sigma=GM/(2\sigma^3)$. This is the <singular isothermal sphere> scaling, valid over an interior radial range; a literal infinite model has no finite total mass, and a hard finite boundary needs a pressure term or an outer transition. A <free-fall time of a uniform sphere> or orbital period changes the numerical factor. The subsequent estimates use the nominal scale $GM/\sigma^3$, keeping these order-unity structural uncertainties explicit.
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