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Let's say you'd put a whooping big rocket engine on the surface of the moon, and slow down its orbital velocity.

Not to a complete stop, mind you. Just a bit slower than it was going before (a couple of m/s?).

Intuitively (i.e. without doing the maths, or even knowing the maths involved), I'd say it would start spiralling towards earth, eventually crashing into it.

I was wondering about this due to this YT video about Kerbal Space Program that talks about how much it would take to stop a moon. I figured that you don't need that much delta-V, that a much smaller amount of slow-down would already (eventually) de-orbit a moon already.

Is that correct?

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    $\begingroup$ Funnily enough, slowing down a moon will make it speed up. $\endgroup$
    – Emilio Pisanty
    Commented Oct 14, 2015 at 11:43
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    $\begingroup$ If you include atmospheric drag, then the ellipse described in the answers will become some sort of spiral -- how you observe it depends on whether you're viewing from a static point in space or from a position on the ground (Earth's rotation) $\endgroup$
    – Carl Witthoft
    Commented Oct 14, 2015 at 15:49
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    $\begingroup$ That's a question for what-if.xkcd.com $\endgroup$
    – xebtl
    Commented Oct 14, 2015 at 17:14
  • $\begingroup$ Why not look at what happens to KSP spaceships when they slow down? $\endgroup$
    – user20574
    Commented Oct 14, 2015 at 22:30
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    $\begingroup$ @LLlAMnYP: I'm software engineer by profession. I don't trust software. ;-) $\endgroup$
    – DevSolar
    Commented Oct 15, 2015 at 7:04

3 Answers 3

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To shine a light on Emilio's comment and develop Fabrice's answer, what will happen is the following. Given that the change in velocity can be considered instantaneous relatively to the period, the moon will find itself in a new orbit where its position (just after the slow down) is the apoapsis (the furthest point from the focus where lies the Earth) of a new elliptical orbit. As you guessed the moon will then gain speed as it comes closer to the Earth (but not spiraling down) in this new elliptical orbit until it reaches its maximal velocity at the periapsis of the orbit, the greater the slow down of the moon, the smaller the periapsis. Now if this slow down is too big the moon can effectively crash into the earth. See Hohmann transfer for more detail :https://en.wikipedia.org/wiki/Hohmann_transfer_orbit

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    $\begingroup$ It would not necessarily be the apoapsis (unless the initial orbit was circular); a highly elliptical orbit slowed at periapsis would become less elliptical, but the periapsis would remain the same. $\endgroup$
    – Ghillie Dhu
    Commented Oct 14, 2015 at 16:52
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    $\begingroup$ "if this slow down is too big the moon can effectively crash into the earth": but it's crucial here to keep in mind order of magnitude of radiis and cross sections. I feel agravated each time I hear journalists speaking of asteroids having "brushed" past Earth... So to obtain collision of Moon with Earth, you would have really to almost stop it all. $\endgroup$
    – Fabrice NEYRET
    Commented Oct 14, 2015 at 20:30
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    $\begingroup$ @Ghillie: yes my bad, I assumed the initial orbit circular $\endgroup$
    – EigenDavid
    Commented Oct 15, 2015 at 7:48
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It would not be spiraling, it would be on a different elliptical orbit, probably less circular. All 2-(solid)-body gravity equilibriums are ellipses. Ellipse is just the constantly renewed balance between free fall and inertia of motion in the current direction. Spiraling requires continuously losing energy (i.e. being sucked by something, in your case).

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In fact, our moon is slowing down very gradually due to tidal effects and (counter-intuitively) that has caused the moon to move very much farther away from the earth since their creation. According to Kepler's laws of orbital motion a body in orbit sweeps out the same area in the same amount of time. As the motion slows down the body moves farther away to provide the same swept area per unit of time. If you want to see a good discussion of this view the first and second of Richard Feynman's Messenger Lectures on YouTube.

Tides slow down orbiting bodies by converting the energy of motion to heat due to moving the masses of oceans AND continents. Friction and compression/release of molecular bonds inefficiencies converts the water and land motions to heat energy.

Adding more: Emilio is right. What is occurring is that the Earth's angular momentum of rotation (day and night) is being transferred to the moon's orbital speed. This occur's because the oceans bulge out and then are slowed by friction and blocked by the continents. The friction and blockage keep the ocean bulges from synchronizing with the moon's orbit. The resulting off-set bulges exert a gravitational torque on moon, transferring energy. The moon speeds up and moves further away and the earth's rotation slows down.

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    $\begingroup$ It should be emphasized, though, that this slow-down is caused by the Earth trying, via tidal interactions, to speed up the Moon. This only pushes the Moon to a higher orbit where it therefore slows down, but the transfer of energy is from the Earth's rotation to the Moon's orbital motion. $\endgroup$
    – Emilio Pisanty
    Commented Oct 14, 2015 at 15:48
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    $\begingroup$ @EmilioPisanty What is the moon's linear velocity in this situation? Same, faster, or slower? $\endgroup$
    – Random832
    Commented Oct 14, 2015 at 15:52
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    $\begingroup$ Ultimately the linear velocity is slower, but it is in a higher energy orbit. Orbital mechanics is counter-intuitive in that manner. $\endgroup$
    – Tristan
    Commented Oct 14, 2015 at 16:34
  • $\begingroup$ @Random832 Slower. The kinetic energy $E_K$ decreases (which is offset by a corresponding decrease in the potential energy $V=-2E_K$ for an overall increase in the total energy). The tidal force is always in the direction of motion, though, not against it. $\endgroup$
    – Emilio Pisanty
    Commented Oct 14, 2015 at 16:35

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