Crossing-time cloud support mass ($M_{\mathrm{cross}}=\frac{4\pi }{3}(\frac{3\pi }{32}{)}^{3/2}\frac{c_s^3}{G^{3/2}{\rho }^{1/2}}$)

Equate the free-fall time of a uniform sphere $\sqrt{3\pi /(32G\rho )}$ to a radius-based sound-crossing time $R/c_s$, then substitute $R$ in $M=4\pi \rho R^3/3$. This gives the displayed mass. It has the scaling of a Jeans mass, but its coefficient differs from the convention using a sphere of diameter equal to the Jeans length. Equality of these two times is an approximate support comparison, not the exact marginal mode of an infinite-medium Jeans instability.