= Jeans length and gravitational time conventions
{c}
{title2=$\ell_J/c_s=\sqrt{32/3}\,t_{\rm ff}$}
The physical <Jeans length> is $\ell_J=c_s\sqrt{\pi/(G\rho_0)}$. A pressureless uniform sphere of initial density $\rho_0$ has <free-fall time of a uniform sphere> $t_{\rm ff}=\sqrt{3\pi/(32G\rho_0)}$, giving the title relation. Thus a full-wavelength <sound-crossing time> and this exact free-fall convention agree in scaling, not numerical value. Using the scale $k_J^{-1}$ instead gives $1/(c_sk_J)=1/\sqrt{4\pi G\rho_0}$, the inverse long-wave <static Jeans growth-rate spectrum> limit. At the threshold itself pressure balances gravity and the net growth rate vanishes; the pressureless free-fall time must not be confused with that marginal mode's infinite growth time.
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