Outcomes of earth slowing down spinning on its own axis

Question

What are the possible outcomes if earth slows down spinning on its own axis?

To be specific: Can the decrease in the internal centrifugal (or centripetal) force due to slowing down earth's spin:
a. change earth's structure given the fact that major part of earth is not solid and the earth's oblong shape may be in dynamic equilibrium of gravity vs this internal centrifugal force.
b. alter the sense of the "vertical" as indicated by the plumb line especially when we are off equator, given the fact that the direction of centrifugal force acting on a physical body has a non-zero component on the horizontal direction except on equator and poles? If this is true, will the altered horizontal component be sufficient enough to affect high rise buildings?

Answer 1

I would say: a - yes, it will change the shape of the Earth and b - no, it will not change the vertical, because of a.

Now, this needs a clarification.

a - Yes, because the Earth is globally in hydrostatic equilibrium, and the change in rotational speed is so slow that it cannot drive the Earth out of this equilibrium. Faster excitations, like plate tectonics or glaciations, do drive some out of equilibrium behavior, but the rotation slow down is just too slow.

Just to get an idea of how slow this is: The earth flattening is now $f = 1/298$. Assuming it is proportional to the rotational speed of the Earth, and assuming a slowdown of 2 ms/d/century as suggested before, we get

$$ \frac{df}{dt} = - f \times 2.3\times 10^{-8}/\text{cy} = - 7.8\times10^{-11}/\text{cy} $$

That means that the equatorial radius is getting shorter by 0.16 mm per century, while the polar radius is increasing at twice this rate.

b - It actually depends on what you are comparing the vertical to. If you compare the new vertical with what the old vertical was at the same, fixed geocentric latitude, then yes, it will change. If you compare at the same position on the crust, it will also change, but for another reason: plate tectonics will move your reference point far from where it was originally. If you compare at a fixed geographic latitude, it will not change, simply because the geographic latitude is defined by the direction of it's vertical. In any case, the vertical will stay perpendicular to the geoid because of the definition of the vertical (gradient of potential) and of the geoid (equipotential). It will also stay roughly perpendicular to the Earth's crust because of point a. Comparing with tall buildings will not be possible because the buildings we build today will not be here in a billion years.

Answer 2

For point b: Yes, but very slightly. (I can't properly answer a, since I'm not a geophysicist). Let's do some back of the envelope calculations.

The centripetal acceleration at the equator is given by $\omega^2 r$, where $\omega$ is the angular velocity of the earth, and $r$ is the radius. $\omega$ is pretty small, somewhere around $7 \times 10^{-5} \frac{rad}{s}$, $r$ is about $6.4 \times 10^6 m$, which makes the centripetal acceleration somewhere around $4 \frac{cm}{s^2}$.

Given that:

• the gravitional acceleration is $980 \frac{cm}{s^2}$
• the slowing down is very gradual (the change in Earth's rotational period (a day in other words) is 2 milliseconds per century according to Wikipedia)

you're not going to notice anything over your lifetime, and I doubt whether it could be measured experimentally over the course of a hundred years. At the very least, high rise buildings won't notice it as much as the forces they are continually subjected to; think thermal expansion/contraction, wind (the Empire State Building sways up to 3 cm in bad storms), earthquakes and even passing traffic!.

If the earth's rotation would suddenly scream to a halt, things are different. For starters, the rotational energy is $\frac{m \omega^2 r^2}{5}$, about $2.9 \times 10^{29} J$ (if I didn't mess up too much :) which is 'equivalent' to $2 \times 10^{11}$ Japan 2011 earthquakes, or 'enough' to raise the sealevel temperature by several thousand degrees. All that energy has to go somewhere, but fortunately it also has to come from somewhere.

Answer 3

EDIT: Answer below is talking about a radical sudden slow-down of angular velocity, not a gradual one.

Question a is actually pretty interesting. We can see a very extreme example of this in the case of a neutron star, some of which rotate with surface speeds of 1/3 the speed of light due to conservation of angular momentum, making it an oblate shape. The basic structure of a neutron star is analogous to earth, it has a hard "crust" (which is significantly stronger than earth's crust) and a fluid like core. As it slows down, the crust holds firm against the inner fluid, until suddenly it ruptures and the neutron star undergoes a "stellar quake," which create small gamma ray bursts.

I imagine if for some reason or other the earth is significantly slowed down, a similar event would occur. At first there would be rather little seismic activity, but sometime the pressure would be so great that somewhat catastrophic fissures would occur in the earth's crust as it ruptures and re-shapes itself to be more spherical.

Answer 4

Yarkovsky Effect

quoting wikipedia - "The Yarkovsky effect is a force acting on a rotating body in space caused by the anisotropic emission of thermal photons, which carry momentum."

http://en.wikipedia.org/wiki/Yarkovsky_effect

Essentially, the Earth absorbs momentum from Sun photons and remits this momentum as the Earth rotates, for retrograde planetary motion this causes the planet to move Sunwards, and oppositely for prograde motion (Earth has a prograde motion).

While this effect is minimal over a short period of time, it will sum to measurable effects over a great period (to wit millions of years).

It can be easily imagined that for sufficient time frames, any rotating-orbiting body will move closer or further from the Sun until it reaches an orbit where the momentum released via the Yarkovsky effect will balance that gained from Sun.

So in answer to your question (from the Yarkovsky effect) any change in rotation will cause an eventual change in orbit, and depending on the planets retrograde or prograde rotation will cause the planet to either heat up or cool down depending on the new orbit.

Noting once more - this may take 'billions of years'