Moon Layer
In the science fiction novel "Seveneves" by Neal Stephenson, the Moon inexplicably disintegrates into small pieces and falls to the Earth with disastrous results for mankind.
If this were to really happen, approximately how thick of a layer of moon dust would fall on the Earth's surface?
Assume that the the Moon starts out perfectly spherical with a radius of , that the Earth starts out perfectly spherical with a radius of , that all of the Moon is evenly distributed over the surface of the Earth, and that the moon dust settles with the same density as the original Moon.
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1 solution
When the lunar dust settles on Earth, the two will have combined into one large sphere, with a volume equal to the total volume of Earth and the Moon. The formula for the volume of a sphere is V = 3 4 π r 3 , which can be used to find the volumes of Earth and the Moon:
V E a r t h = 3 4 π ( 6 3 7 1 k m ) 3 ≈ 1 . 0 8 3 ∗ 1 0 1 2 k m 3 V M o o n = 3 4 π ( 1 7 3 7 k m ) 3 ≈ 2 . 2 ∗ 1 0 1 0 k m 3
Adding these gives the total volume, about 1 . 1 0 5 ∗ 1 0 1 2 k m 3 . Using this volume, we can find the new radius of the sphere by inverting the volume fomula:
V = 3 4 π r 3 ⇒ r = 3 4 π 3 ∗ V
Plugging in our total volume gives a new radius of about 6414 km.
Finally, to get the increase in radius, which is due to the thickness of the moon dust, simply subtract the original radius of the Earth:
6 4 1 4 k m − 6 3 7 1 k m = 4 3 k m ≈ 4 0 k m