of Bo as $$\label{eq:Re=f(Bo)}
\mathrm{Re_1}=\left(\frac{\gamma^3 \rho_1^2}{\Delta\rho g \eta_1^4}\right)^{1/4}\mathrm{Bo}^{3/4}.$$ The factor in between parenthesis is the inverse of a Morton number [*e.g.* @clift_1978]. It only involves material properties in addition to $g$, and typically varies by a factor of 10 mainly due to uncertainties on the viscosity $\eta_1$, which is relatively small compared to Re$_{1}$ and Bo varying by orders of magnitude depending on the size of the metal. Equation \[eq:Re=f(Bo)\] (with uncertainties) thus define the part of the regime diagram which is the most relevant for metal segregation in a magma ocean. This is shown in figure \[fig:drop\_regime\] (dashed blue lines) for parameters values given in table \[tab:parameters\_magma\] and a viscosity of the metal phase in the range $10^{-3}-10^{-1}$ Pa.s [@rubie_2003; @rubie_2015]. If starting from a volume of metal on the large Re and Bo side of the regime diagram (i.e. Re$_{1}\gtrsim 10^3$ and Bo$\gtrsim 10^2$, which corresponds to a diameter of $\gtrsim 5$ cm), the falling metal would be expected to be in the stirring regime (turbulent thermal). If fragmentation produces droplets at moderate Re$_{1}$ and Bo, these droplets will most likely be in either the wobbling or spherical regimes, or possibly in the ellipsoidal-cap regime.

Equilibration times {#sec:path_equilibration}
