# Outcomes of earth slowing down spinning on its own axis

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?

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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.

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+1 - I was reading the other answers and planned to correct a few misconceptions, but here, you gave the correct and complete answer! –  ysap Jun 30 '11 at 14:17
This is not a problem of "misconceptions", but of severe editing of the question. To see, which part of an answer referred to which status of the question is difficult meanwhile. –  Georg Jun 30 '11 at 17:14

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'

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The topic here is outcome from slowing down of rotation, not another (and irrelevant) mechanism of slowing down. -1 –  Georg Jun 19 '11 at 16:02
the question is pertaining to earth's spinning on its own axis, not the planetary rotation round the sun. –  pongapundit Jun 21 '11 at 17:08

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.

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Whoops, should have read a bit more. Thank you for the correction, I'll edit it in. –  yatima2975 Jun 29 '11 at 13:22