Analytical approximations for the collapse of an empty spherical bubble

The Rayleigh equation (3)/(2)\Rdot +R\Ruml +pρ^-1=0 with initial conditions R(0)=R₀, \Rdot(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density ρ. The solution for r≡R/R₀ as a function of time t≡T/T_c, where R(T_c)≡0, is independent of R₀, p, and ρ. While no closed-form expression for r(t) is known, we find that r₀(t)=(1-t²)^2/5 approximates r(t) with an error below 1%. A systematic development in orders of t² further yields the 0.001% approximation r_*(t)=r₀(t)[1-a₁Li_2.21(t²)], where a₁≈-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these approximations is demonstrated by comparison to high-precision cavitation data obtained in microgravity.