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So the uniform density collapse problem has a self-similar solution where \frac{t}{t_{ff}} = \frac{2}{\pi} \left ( \eta+\frac{1}{2} \sin(2\eta) \right )

and

\cos(\eta)=\left( \frac{\rho}{\rho_0} \right )^{-1/6}

Since \frac{R}{R_0} = \left ( \frac{\rho}{\rho_0} \right )^{-1/3}

\cos(\eta)^2=\frac{R}{R_0}

If we plot \frac{R}{R_0} as a function of \frac{t}{t_ff} we get the following.

Since the curve looks like a circle we try a polynomial expansion y=a+bu+cu^2 \mbox{ where } u=\sqrt{1-x^2}

The best fit (plotted above) has coefficients a=.7178 \mbox{, } b=.3068 \mbox{, } c=-.01842

If we then use this to scale the size of the window, the apparent radius of the cloud remains fixed as it collapses. At a finite time the radius would go to zero, so once the size of the window becomes of order 32 finest level cells, the box stops shrinking and the cloud finally appears to collapse. The dynamical time also decreases as the cloud shrinks, so in the following movie, the frame rate increases 10 fold and the collapse appears to proceed more slowly. This happens before the box stops shrinking (before the cloud appears to collapse).

movie

I also restarted the run but with sink particle formation suppressed. The results are similar…

movie