Reader Scott writes in with a question:

Okay, let’s assume something knocked the moon out of its orbit and it is going to crash into the earth in an arbitrary amount of time. (7 days?)

You are humanity’s last hope. You have the entire nuclear salvos of the USA and Russia at your disposal, which are being fitted to be capable of being launched into outer space and to reach the moon at another arbitrary rate. Your goal is to knock the moon back onto its original orbit by striking it from the side in such a way that will redirect its net force back to a position that will result in the moon having its original distance from the earth and angular momentum.

Is this possible? If so, under what conditions?

(The answer is sort of long, so continued below)

The short answer is that if the moon’s going to hit us in 10 days, we’re toast. There just ain’t no way around it, and the world will be sterilized as thoroughly as if you put the entire planet in a giant autoclave. For smaller and more realistic dangers like asteroids, and given more warning time, we might have a chance. Let’s run the numbers, shall we?

First, let’s talk a little bit about how an explosion can move something. Newton’s laws require that an isolated system with no external forces must have a constant momentum. Fire a rifle and the forward momentum of the bullet is matched by the equal and opposite backwards recoil of the rifle. A nuclear explosion on the lunar surface is not so different – the momentum of the blast debris leaving the moon is matched by an equal and opposite recoil of the entire moon. Momentum is mass times velocity, so the momentum of the blast is very roughly the mass of the bomb times the velocity of the fragments. The moon’s recoil momentum in the opposite direction will have the same value, but the moon’s mass is *huge*. Huge mass means the corresponding recoil velocity is small.

How big do we need the recoil velocity to be to save us? That’s an extremely difficult question in general, but as a rough estimate we might say that we need enough velocity to move the moon (or asteroid) by a distance equal to the diameter of the earth in the time before impact in order for it to miss. Given the earth’s diameter and a week to move that distance, this corresponds to a velocity of about 20 meters per second. This doesn’t sound so bad. Given a lot more time – maybe 10 years for a very faraway asteroid detected early – we only need to give it a velocity of 0.04 meters per second. This is downright promising.

So how much impulse can we get from a bomb? This too is an extremely difficult question in general. We can’t get the velocity of the fragments directly from the energy because plenty of it will be lost in the form of low-momentum electromagnetic radiation and other inefficiencies. The blast isn’t perfectly directed away from the moon either. Nonetheless we can give ourselves the benefit of a doubt. Let’s say all available nuclear energy goes into the kinetic energy of the fission fragments and it’s all directed away from the moon. Using the usual classical equations for kinetic energy and momentum, and assigning the Greek letter rho = E/m for the released-energy-per-mass of the nuclei, we can derive an equation:

Savvy readers may object to my use of the classical rather than relativistic expressions here, but we’re ok. Typical nuclear fusion energies are in the ~5 MeV per nucleon range, which is well below the mass of a proton or neutron and thus mostly classical. For electrons we’d have no choice but to use the relativistic equation, but for the heavier nuclear particles the classical expressions are still quite accurate.

All right. Taking a wild guess that the innards of a fusion bomb are about 100 kg and using the energy density estimated above, a single bomb should net us an impulse of 3 x 10^{9} Newton-seconds. Translated into less inside-baseball terms, this means each blast can give a 1 kilogram target a velocity of 3 x 10^{9} meters per second, or a 3 x 10^{9} kilogram target a velocity of 1 meter per second, or anything else with that ratio. Extrapolating to the 7.3 x 10^{22} kg mass of the moon, one blast will change the velocity of the moon by… 4 x 10^{-14} m/s. Yikes. Considering we need 10 m/s total, that’s quadrillions of bombs. As I said at the top of the post, it ain’t happening.

Asteroids are another story. The near-earth asteroid 99942 Apophis has a mass of only about 2 x 10^{10} kg. That’s only a dozen or so bombs, even with only a week’s notice. Now our estimate is very generous in terms of the effects of one bomb, but it’s certainly plausible that a few dozen or a few hundred bombs might well do the trick if we caught it reasonably early.

Let’s hope we never need to test this experimentally.

[The title for this post is ~~stolen from~~ a tribute to the immortal IMAO essay of the same name]