gravity - At what size will self-gravitation contribute more to stability than surface tension?

The governments of Earth have embarked on an experiment to place a massive ball of water in orbit. (umm... special water that doesn't freeze)

Imagine this to be a fluid with a given density, $\rho$ ($kg/m^3$), surface tension, $\sigma$ ($J/m^2$), and formed in a sphere of radius $R$ ($m$). I think that the viscosity $\mu$ is not needed for this question, but correct me if I'm wrong.

At what size will the restorative forces from gravity (after some small perturbation) become more significant than that from surface tension? Would the type of perturbation make a difference?

Just for fun, here's a video of a ball of water stabilized by surface tension.

Answer

Let us do a quick estimation.

Let $R_{cr}$ be a critical radius of the ball so that the condition of stability for the ball is expressed as $$R

$$R_{cr}=C\left(\frac{\sigma}{\rho^2 G}\right)^{1/3}$$ where $C$ is a dimensionless constant of orders of magnitude close to 1.

For water, $\sigma=0.07\frac{J}{m^2}$, $\rho=10^3\frac{kg}{m^3}$ and also $G=6.67\cdot10^{-11}\frac{Nm^2}{kg^2}$

So, a rough estimation:

$$R_{cr}\approx\left(\frac{\sigma}{\rho^2 G}\right)^{1/3}=10m$$