# Maximum rotation speed of a planet of given mass and size

What will be the limit of the rotation speed of a planet with given radius and size? will it depend on factors other than mass and size?

Neutron stars are known to rotate at very high speeds but they are very small in size, so, given a planet, how would we calculate the maximum rotation speed beyond which it might disintegrate?

I would go about defining a centrifugal pressure : $$P_{cent} = \frac {F_{cent}}{A_{surf}} = \frac {m\omega ^{2}}{4\pi r}$$

Then somehow you need to find out Ultimate tensile strength,- $$\sigma_{ts}$$ of a planet. Which may depend on a planet chemical composition, matter phase (gas, solid, etc), temperature and other factors. Then, when you'll know planet's breaking tensile strength, solve given equation for an angular speed $$\omega$$, in edge case of $$P_{cent} = \sigma_{ts}$$.

Also you can take a look into Breaking stress of neutron star crust, 2018 article by Chugunov & Horowitz on how they derive crust breaking stress formula for a rotating neutron star. Albeit it's not exactly the same problem.

Part of the planet of mass $$m$$ is attracted to the rest with a force $$\frac{GMm}{r^2}$$, where $$M$$ is the mass of the planet.

To be held on, the centripetal force acting on the mass needs to be $$m\omega^2 r$$, where $$\omega$$ is angular velocity.

So parts of the planet will begin to be flung off, if $$m\omega^2 r \gt \frac{GMm}{r^2}\tag1$$ $$\omega^2 \gt \frac{GM}{r^3}\tag2$$

For the earth that's $$\omega = 1.25\times 10^{-3}$$ or a rotation period of about $$84$$ minutes.

As parts were flung off, since $$M$$ is proportional to $$r^3$$ (for constant density), the value for the angular velocity would be approximately constant, the outer layers would be flung off, and the other layers too, at the same rate of angular rotation.