When will the planet break apart?

A solid body will break up if the centrifugal force at a point is equal to the gravitational force. At Earth's equator, the equation becomes $$\frac{G{M}_{\oplus }}{R_e^2}={\omega }^2{R}_e$$ where $M_{\oplus }$ is the mass of Earth, $R_e$ is its equatorial radius, and $\omega$ is its angular speed. Rearranging and solving for $\omega$ gets $$\omega =\sqrt{\frac{G{M}_{\oplus }}{R_e^3}}$$ Assuming $R_e\approx {\overline{R}}_{\oplus }$, its mean radius, we have $$\omega =1.24\times {10}^{-3}{\text{ rad s}}^{-1}$$ Earth's current angular velocity is approximately $\omega =7.29\times {10}^{-5}{\text{ rad s}}^{-1}$. This is almost exactly 1/17^th of Earth's breakup speed.

Now, this assumes that $R_e$ doesn't change with $\omega$ - which it does. In fact, the equatorial bulge is $$R_e=R_p+\frac{5}{4}\frac{{\omega }^2{\overline{R}}_{\oplus }^4}{G{M}_{\oplus }}$$ At the original breakup speed, we have $$R_e=6.356\times {10}^6+\frac{5}{4}\frac{(1.24\times {10}^{-3}{)}^2(6.371\times {10}^6{)}^4}{6.673\times {10}^{-11}\times 6\times {10}^{24}}=1.43\times {10}^7\text{ m}=2.24{\overline{R}}_{\oplus }$$ This decreases the gravity by a factor of roughly 5, so the breakup speed is actually a bit lower. So at 100 or 1000 times its current rotation speed, Earth wouldn't be here.

The Coriolis force

The maximum magnitude of the Coriolis acceleration is $$|{\mathbf{a}}_c{|}_{\text{max}}=2|\mathbf{\Omega }||\mathbf{v}|$$ where $\mathbf{\Omega }$ is the rotational velocity, and $\mathbf{v}$ is the velocity of a particular particle. This means that the force scales linearly with $|\mathbf{\Omega }|$, and thus can only be 17 times as large as it is right now, at a given point on Earth.

We'd see Rossby numbers (which characterize how much a system is affected by Coriolis forces) lower by no more than a factor of 17. As Anoplexian noted, this would lead to stronger winds and likely stronger hurricanes - although nothing catastrophic; an order of magnitude change here won't be devastating. I wouldn't be too concerned.

Daily life

At 17 times the Earth's current rotation speed, each day would last about 45 minutes, and each night would last about 45 minutes. Even at 12 hours per day-night cycle, you have to worry about humans getting the proper amount of deep sleep each cycle. 45 minutes is enough for a nice catnap, but not good sleep.

I'd assume that creatures would either adapt to such a short sleep cycle, or sleep through several days and nights, and then spend several days and nights asleep. Whether or not either of these is feasible is something I'll have to look into; I'm not overly optimistic.

posted over 8 years ago

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