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Except for ice, acetic acid, bismuth and gallium and a few other things materials generally shrink when they cool and solidify, so I'm pretty sure Earth has as well.

It probably wouldn't be measurable over a period of years, but models of the Earth's current and historic rates of heat flow can probably be used to estimate a rate of change of Earth's average size and possibly oblateness.

Right now WGS84 uses 6378137.0 meters for Earth's equatorial radius and a flattening at the poles of about 1/298.257222.

How fast might those change over any given million years?

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    $\begingroup$ Perhaps relevant question: earthscience.stackexchange.com/q/18534/15419 $\endgroup$ – BMF Mar 1 '20 at 5:01
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    $\begingroup$ keep in mind the earth is slowly gaining mass due to meteors. Its a small effect but on the scale involved likely bigger than thermal changes. $\endgroup$ – John Mar 2 '20 at 2:55
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    $\begingroup$ @a_donda I guess at 10k reputation deleted posts are still displayed. In Stack Exchange nothing is ever really deleted, it just "fades from view" for most users. When I saw "linked article" in your comment and didn't see a linked article in my question, I scrolled down and saw it. i.stack.imgur.com/b8Lvq.jpg $\endgroup$ – uhoh Jul 29 '20 at 23:38
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    $\begingroup$ Did you factor space dust and meteorites arriving? that's 100 metric tons of this cosmic dust enters Earth's atmosphere every single day. also there is some loss from the top of the atmosphere. $\endgroup$ – aliential Dec 3 '20 at 9:23
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    $\begingroup$ @aliential I've asked the question, not answered it. However we can estimate that at that rate it would take several years to build up a 1 atom thick layer, something like a million years would be 100 microns, and that's ignoring any compression due to the added weight. $\endgroup$ – uhoh Dec 3 '20 at 9:51
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(Related to answers to the question Thermal expansion of Earth, which I only realized after having saved this answer. So, in principle, a double post, but I was asked to undelete :-))

Earth's thermal history may give some hints -- not sure if this is an actual answer to the question. It seems like Earth's radius depends on the layering of mantle convection, and that though there were significant changes during the Archean Eon and earlier, the current and recent (since the late Archean) shrinking rate is slow (a reduction of 12km in radius in 2.5Gy).

Source: Expanding-contracting Earth by Tsuchiya et al. 2013 in Geoscience Frontiers


Questions that remain:

What about the core ?

And is mass loss or gain from astronomical processes significant enough ?

I would like to add that WGS 84 is a reference ellipsoid, and height values can be (are) given as positive or negative numbers. They can be updated every now and then by satellite. Also, I would assume that tectonic processes or local anomalies may by far outweigh any signal from thermal shrinking over the course of next centuries or even millennia (speculation).

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