# If the Earth stopped spinning, how long would it take before it had gone from oblate to roughly spherical?

I know that because of the centrifugal force of earth's spin, it is not entirely spherical. According to wikipedia, earth's equatorial bulge is roughly 43 kilometers. So how long would it take for the equatorial bulge to disappear after earth's spin (and the centrifugal force) stopped? (For the purpose of this question let's assume earth just magically stopped spinning)

I started by looking through this cool article describing where the earth's oceans would settle if the earth stopped. It got me curious about how long it would take for the crust to normalize as well. Millions or billions of years?

• More of an Earth Sciences SE question? Just a thought.
– user154420
Jul 3, 2017 at 10:50
• In essence, you are asking what the effective viscosity is? Jul 3, 2017 at 14:41
• @JonCuster Uh, yeah maybe. I guess the effective viscosity would allow one the calculate the time it'd take for the equatorial bulge to recede? Not something I'd know how to do though, even if I knew the viscosity. Jul 3, 2017 at 14:50

An answer to your question is given by equation (221) in https://arxiv.org/pdf/1105.6086.pdf

I am providing the link not to the journal version but to the arXiv version of the paper, because the latter version is more extended. The said formula is absent in the journal publication.

The said eqn renders the evolution of the additional tidal potential $U_{l}$, while your question was about the tidal elevation. These two quantities are, obviously, interrelated; and it can be shown that the relaxation law for the elevation will be the same, up to an overall coefficient.

The problem discussed in my paper emerged in the context of tides. However, the solution is applicable to your problem also, because the centrifugal force can be expanded into a purely radial part (which brings a miniscule addition into the overall deformation) and a part that looks, mathematically, like a quadrupole part of the tide-raising potential. For details, see Appendix B in https://arxiv.org/abs/1706.08999

As you can see from eqn (221), relaxation is exponential (of course).

Also, as can be seen from eqn (219), the characteristic time of relaxation is $(1+A_2) \tau_M$ where $\tau_M$ is the Maxwell time (equal to the mean viscosity of the mantle divided by the mean rigidity).

Had we been playing with a sample of viscoelastic magma, the characteristic time would be simply $\tau_M$. An extra factor $1 + A_2$ shows up because we are dealing not with a small sample, but with a self-gravitating object. For planets of about the Earth size, $A_2 \approx 2.2$ (see Table 1 in https://arxiv.org/abs/1105.3936 )
So, in the end of the day, the characteristic time is only a bit longer than the Maxwell time.

Studies of post-glacial rebound indicate that the Maxwell time for the Earth is between several hundred years and several thousand years (probably, some 200 to 500 yr for the upper mantle).

With the extra factor of $1 + A_2$ included, it will not be very wrong to say that the typical timescale of relaxation of the centrifugal bulge should be somewhere from a thousand to several thousand yr.

Please be mindful that the afore-cited eqn (221) was derived under the assumption that the mantle, at large, behaves as a Maxwell body. It probably does. However, employment of a different rheological model would furnish you a different answer. (See, e.g., eqn 227.)

Best regards,

Michael