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So the moon adds to the total mass of the Earth-Moon system so together there is more mass so then there should be more gravity due to the total system and the moon is small relative to how much gravity it exerts, and it doesn't really protect the Earth because its volume is small. So doesn't the moon not actually help the Earth and instead just deflects asteroids to Earth?

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A reasonable line of thinking, but one that doesn't quite work out in practice for non-obvious reasons.

The first thing to note is that Moon is about 80 times less massive than the Earth, so it really doesn't do much to increase the mass (hence size) of the joint Earth-Moon gravity well.

As the question hints at, it's not the volume of the Moon that matters. The chance of an asteroid on a collision course with the Earth being on a trajectory going through a Moon-sized volume of space is tiny. If you consider a ball of space centred on the Earth and extending as far as the Moon, the Moon itself occupies only one ten millionth of it.

What does matter is that the Moon's gravity pulls towards the Moon, not towards the Earth. From very far away, (eg, Mars) the two do pull in the same direction, but this breaks apart once you are close to the Earth. In fact, for an asteroid between the Earth and the Moon, lunar gravity pulls directly away from the Earth. So once asteroids are within the Earth-Moon gravity well, the Moon acts to perturb the path of asteroids that would otherwise be in a Keplerian hyperbolic trajectory around the Earth (or directly into the Earth). Now this lunar perturbation could change the trajectory of an asteroid that was on a collision course with the Earth into missing it, or turn a near miss into a hit. But the former is far more likely.

The reason for that is complicated and non-intuitive. This is a case of a 3-body problem (in fact, 4-body: Sun, Earth, Moon, asteroid) which is notoriously difficult in physics. One way of thinking about it that might help is to consider a special, but common, case. Many of the asteroids that are most likely to hit the Earth are in a similar orbit around the Sun as the Earth itself, and those orbits cross. When the timing works out (which happens regularly due to the orbital periods being related), they get relatively close; close enough for the Earth's gravity to tweak the asteroid's orbit around the Sun such that the next approach will be even closer. If this happens repeatedly, a collision becomes very likely. However, it's precisely in this kind of regular scenario that lunar gravity becomes important. It messes up this "clockwork" motion by tugging at the asteroid in a different direction when it comes in the vicinity of Earth, which throws some chaos into the system. This chaos breaks the pattern and acts to make the Earth's and asteroid's orbits around the Sun more different, and hence a collision very unlikely. It's the same kind of reasoning as to why the Moon makes geocentric orbits with large diameter unstable.

For things like comets, however, which come from the outer system very fast in a very elliptical orbit, there are no such resonance effects (sometimes with Jupiter, but not with the Earth) and hitting the Earth is little more than blind luck. In such cases, I believe lunar gravity is simply too feeble to have a discernible impact one way or another. It would be like blowing on a car from the pavement to prevent (or cause) a car crash.

Sadly, this hand wavy, qualitative answer is the best I can do. At the end of the day, to see whether the Moon makes terrestrial asteroid impacts more or less likely you need to do lots of numerical simulation of the full 4 body system for lots of different initial asteroid starting positions and velocities, and see what happens. But that's also a rather unsatisfying "computer says no."

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    $\begingroup$ So it is more about the moon deflecting rather than taking the hit itself then?Also even if asteroids hit the moon, that might still be an issue right especially if those asteroids are Kuiper Belt comets and they hit the moon, that could mess with the orbit because those objects have very high kinetic energy due to their huge gravitational potential energy with the Sun and whatever object is in its way just gets hit. Is this true? $\endgroup$ Mar 12 at 18:12
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    $\begingroup$ Yes, it's far more about the Moon deflecting them than acting as a physical impact shield. Same thing with Jupiter protecting the Earth. While high energy comets hitting the moon would leave a big crater, and some of the dust/debris generated will be ejected fast enough to leave the Moon, that's not much worry on Earth. A cloud of dust in lunar orbit might be inconvenient for our satellites in orbit there, but that's about the extent of the impact (no pun intended). $\endgroup$ Mar 12 at 21:21

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