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We know that the Moon's gravity is about one-sixth that of Earth. Then I recently read that the mass of the Moon is about one-eightieth the mass of Earth. Since gravity depends on the mass of the 2 objects, shouldn't the Moon's gravity be one-eightieth the gravity of Earth? Is the Moon extra dense?

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    $\begingroup$ The moon being made from other stuff than the Earth probably factors into this. A lot fewer liquids there. $\endgroup$ – Parrotmaster Dec 12 '19 at 15:59
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    $\begingroup$ Do you know the formula for determining the gravitational force between two masses? $\endgroup$ – Eric Lippert Dec 12 '19 at 18:08
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    $\begingroup$ @Alchemista: If the Moon had its current mass but was the size of the Earth -- and was therefore enormously less dense -- then it would have 1/80th of the surface gravity. But the Moon is not the size of the Earth, and that matters. I asked if the original poster knew the formula as a nudge towards thinking about the fact that the distance between centers of the masses is just as important a factor as the masses. $\endgroup$ – Eric Lippert Dec 13 '19 at 13:43
  • $\begingroup$ I get it. If the Moon had its same mass, but with a radius the same as Earth, then the surface gravity would be 1/80th. The distances would correspond, and mass would be the only variable. The Moon's smaller radius from center to surface gives us the 1/6 instead of 1/80th. No wonder it's so hard to split an atom. $\endgroup$ – John Canon Dec 14 '19 at 2:26
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    $\begingroup$ @JohnCanon: The difficulty of splitting an atom is not due to the gravitational force holding it together, but that belongs on physics.se Gravity is truly negligible at that scale. $\endgroup$ – Ross Millikan Dec 14 '19 at 4:43
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As you said, the mass of the Moon is 1.2 percent that of the Earth. Now, if you mean the gravitational acceleration at the surface, it is calculated like this $G\frac{M}{R^2}$, where $M$ is the mass, and $R$ is the radius of the celestial body. The moon's mass is a hundred times smaller, but the radius is four times smaller, meaning its surface gravity will be $100/16 \approx 6$ times smaller. Considering the factor is mass over radius to the power of two here, density alone does not help you determine the ratio of surface accelerations.
On a side note, the moon's density is around 60 per cent that of the Earth.

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    $\begingroup$ ... and that 60% density value should derive directly from knowledge of the radius and gravitational force at that radius :-) $\endgroup$ – Carl Witthoft Dec 12 '19 at 16:36
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    $\begingroup$ Low density because Theia drops most of its iron core onto proto-Earth at collision time. Still the 2nd densest moon in the solar system; only Io is denser $\endgroup$ – David Tonhofer Dec 12 '19 at 22:00
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    $\begingroup$ @DavidTonhofer that must mean Io is worth more, since it has a higher metal content. How do we go about trading? $\endgroup$ – corsiKa Dec 14 '19 at 5:26
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    $\begingroup$ @corsiKa That's easy. I give you \$10 for a bit of the Moon and you give me \$50 for the same amount of Io. $\endgroup$ – Andrew Leach Dec 14 '19 at 12:06

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