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Ok, here is some setup.

This fictional world which I have yet to name — let's just call it Earth to make it easier — has two moons, with one bigger than the other. There was then a cataclysmic event that saw the smaller of the two moons fracture and fall to the planet below.

The smaller moon is half the size of the bigger moon, which is about the same size as our moon.

My main question is, how much damage the falling moon would do to the world? And more specifically, what would be left standing in its wake?

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    $\begingroup$ Quick summary: "Rocks fall, everyone dies". It'll be the kind of event that makes biblical catastrophes look like minor inconveniences. I'll see about making a proper answer later, if someone hasn't beaten me to it already ;-) $\endgroup$
    – Starfish Prime
    Mar 22, 2021 at 16:47
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    $\begingroup$ I concur. Even with half the size of the moon, the crust would probably liquify and destroy all but (maybe) some bacteria. I'm not giving odds on the bacteria surviving. $\endgroup$
    – DWKraus
    Mar 22, 2021 at 16:59
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    $\begingroup$ This is similar to Stephenson's Seveneves. In the novel, moon fragments fall for roughly 5000 years and render the Earth's surface uninhabitable during this time. It's going to be tough to make this scenario not "rocks fall, everyone dies" without dramatically decreasing the volume of falling rock or increasing the timespan over which it does so $\endgroup$
    – Punintended
    Mar 22, 2021 at 18:00
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    $\begingroup$ @Punintended even that was a fairly benign moonfall apocalypse; much of the moon remained in orbit. The scenario the OP is suggesting is worse by a few orders of magnitude. $\endgroup$
    – Starfish Prime
    Mar 22, 2021 at 18:05
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    $\begingroup$ @Locksmith this kind of question is never dumb. You don't know these things by heart until you've ran lots of scenarios. $\endgroup$
    – The Square-Cube Law
    Mar 22, 2021 at 18:10

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So, you didn't specify exactly what "half the size" means, so I'll go with the most conservative reading of the term, which would be "half the radius". This is smaller than "half the mass" or "half the apparent area" and the like. It won't make much difference though.

Our Moon has a radius of 1737km, which will be the diameter of our new moon'o'doom. If the new m'o'd has the same density as the Moon, it will mass ~7.35x1022kg. This is fairly close to the size and mass of Io, for reference. Lets imagine it falls at ~11km/s. Given the sheer mass of the m'o'd, it is possible that it could collide with the planet at below the planet's escape velocity, if there's not enough time for it to accelerate (which depends on how it got deorbited in the first place), but it doesn't really matter either way.

Run it through the old Earth Impact Effects Calculator and you can see that the impact delivers ~5x1029J of energy. That's enough to vapourise the oceans and blow the atmosphere and crust into space, where it will form a debris cloud that might one day condense back into a new moon to replace the missing one (or it may just fall back onto the planet or be flung into interplanetary space by gravitational interactions with the existing moon). Compare and contrast with the much larger hypothesised Theia impact that may have created our own moon.

The planet will be resurfaced, and if you're prepared wait, life might (re)appear. On Earth, the Hadean era lasted about half a billion years. When the Earth's crust finally solidified at the end of the Hadean, it took another half a billion years for the first cyanobacteria to be identified.

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    $\begingroup$ I was going to say you ninja'ed me but you actually explained it far better than I could. +1 $\endgroup$
    – The Square-Cube Law
    Mar 22, 2021 at 18:09
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    $\begingroup$ Xavon actually made a very good point in the comments on the question. You seem to assume the moon comes crashing down almost in whole. What changes when there's more chunks? I assume there's quite a difference between two half-Io impacts, one half-Io impact with another half Io's worth of debris burning in the atmosphere, and a whole Io's worth of debris burning in the atmosphere? $\endgroup$
    – Egor Hans
    Mar 23, 2021 at 7:58
  • $\begingroup$ @EgorHans it makes no difference, because of the sheer amount of mass. There's so much that it doesn't "burn up" in the atmosphere, it blows the atmosphere off. The heat of that mass burning up is part of what boils the oceans away, for example. Even if all the mass were very carefully reduced to powder and evently allowed to re-enter all over the Earth, the resultant heat would still remove the atmosphere, oceans, and melt the crust. $\endgroup$
    – Starfish Prime
    Mar 23, 2021 at 9:26
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Starfish Prime's answer addresses what happens if a full lunar mass falls onto an Earth like planet. I'd like to reality-check the event itself.

In order for the Moon's mass as a whole to fall on the planet, you have to deorbit it. The amount of energy you would need to do so lies within the range of the kinetic energy you get from the impact, give or take one order of magnitude.

For the Earth this would mean resurfacing and remodeling but for our own Moon, which is two orders of magnitude less massive, this could mean total obliteration. A quick search on the internet and I found out that the gravitational binding energy of our own Moon is around 1.2 $\times$ 1029 joules - 20% of the energy you'd get from the final impact on Starfish Prime's answer. I understand that the falling moon in the question would be less massive so it would have even less binding energy.

So... The moon will become quickly expanding plasma way before its mass deorbits onto the host planet.

Much of this plasma will probably be made of particles going much over the escape velocity at the moon's altitude, and the bulk of the remaining mass would just settle in new orbits around the planet as rings. The bigger moon may get somewhat scorched, but the planet might keep its atmosphere mostly intact. Over weeks or months that plasma would cool down and become ionized gas, which would deorbit over eons so no danger there.

Edit: Starfish Prime put one some calculations here. A lot of that plasma will hit the planet - around 10 dino-killing asteroids worthy of plasma, considering the upper margin estimates for the asteroids mass. Though the impact would be spread across the side of the planet facing the plasma, the effect would be much the same. So... Mega extinction event. Life might still exist in the oceans, but the surface would probably be [redacted].

This also means a large mass that was astronomically close the planet has been removed from the system. There might be some earthquakes as the crust of the planet readjusts. Moons and planets stretch each other like a drawn bow; As the mass of the moon leaves the system the planet will contract like the same bow being slowly undrawn. That is true even if the bigger moon stays.

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    $\begingroup$ Yeah, I thought I'd gloss over the handwaving in the OP ;-) If you blew up the moon vigorously (enough to throw the debris out of Earth's SOI) you'd end up with about 7 millionths of its mass hitting the Earth directly... that's about $5\times10^{18}$kg, or 10 times the upper margins of estimates of the Chixulub impactor's mass. It'd landscape the moon-facing hemisphere pretty effectively. $\endgroup$
    – Starfish Prime
    Mar 22, 2021 at 18:36
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    $\begingroup$ Even at ~6e26J, spread out across the Earth's atmosphere, that is only about 3J per cubic meter of air facing the blast. The affected atmosphere would heat up by an average of ~0.01 degree fahrenheit. What makes something like Chixulub so devastating on a global scale is not the impact energy, but the dust and greenhouse gases it floods the atmosphere with. All that dust in the air, would cause a global winter but it would pass comparably quickly since you would not have the wildfires caused by a direct impact, and your dust would mostly be heavy particles which should settle quickly. $\endgroup$
    – Nosajimiki
    Mar 22, 2021 at 19:29
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    $\begingroup$ @Nosajimiki radiant heat from the debris entering the atmosphere will absolutely set everything on fire. Remember that the cloud of crud hitting the earth here weighs as much as the whole of earth's amosphere, and its travelling at escape velocity. If the wildfires are reduced, its because the oxygen needed to fuel them went missing. $\endgroup$
    – Starfish Prime
    Mar 23, 2021 at 9:34
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    $\begingroup$ @Nosajimiki and whilst I think of it, "3J per cubic meter of air facing the blast" is wrong, wrong. wrong. 6e26J shared out equally over 2.5e18kg of atmosphere is 240MJ per kilo of air. Forget "0.01 fahrenheit", that's more than enough energy to disassociate every molecule in that hemisphere's atmosphere and ionise it. Not gentle warming, its a colossal fireball of doom. $\endgroup$
    – Starfish Prime
    Mar 23, 2021 at 9:53
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    $\begingroup$ @StarfishPrime Hmmm.... I have no clue how I messed that calculation up so badly, but I boy did I. You are correct. $\endgroup$
    – Nosajimiki
    Mar 23, 2021 at 13:47
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The Chicxulub Impactor was an asteroid about 10 miles in diameter that struck what is now the Yucatan peninsula about 66 million years ago. This impact was powerful enough to cause a mass extinction event that wiped out 75% of all life on the planet (including the dinosaurs). The shock of the impact momentarily punched a hole in the ground that nearly extended all the way to the Earth's mantle.

Your moon impacting the planet would make Chicxulub look like a pebble.

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