Intuitive explanation of why orbits are closed?

Yesterday my brother asked me how orbits work. Suppose for the sake of the question that you are trying to put a rocket in orbit around the Earth. I explained that orbiting is essentially being in free fall while going very fast sideways, so that by the time you fall, there's no ground anymore and you keep going.

He said he understands that. His question was about why, after doing a full revolution around the Earth, you end up in the same place with the same velocity instead of doing, say, an inwards spiral. I admit I was stumped by this. Obviously, conservation of angular momentum prevents you from spiraling down to the center, but I see no reason why, at least in principle, you couldn't have a spiral orbit with an amplitude that increases and decreases periodically.

Is there an explanation for this that doesn't involve actually doing the math? Of course, the equations say this doesn't happen, but that doesn't help me understand any better, nor does it help me explain it to my brother.

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–  Johannes Sep 30 '13 at 15:18
–  Emilio Pisanty Oct 1 '13 at 13:11

That is because the gravitation force is proportionnal to $1/r^2$.

As a counter-example : if you look carefully at the orbit of mercury, it actually doesn't exactly come back to the same point with the same velocity after 1 turn. Instead, the major axis of the orbit rotates a little (about 0.15° per century) mostly because of the additional force from other planets (which is obviously not proportional to $1/r^2$).

Similarly, if you imagine a force (towards a fixed point) that is proportional to $1/r^\alpha$ with $\alpha \neq 2$, you would get a similar "orbit".

I don't think I could explain it further without some maths, however.

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For anybody interested in the math, this goes by the name of "Bertrand's theorem" en.wikipedia.org/wiki/Bertrand%27s_theorem –  Jonas Sep 30 '13 at 13:20
@Nicolas, why, intuitively speaking, does a 1/r^2 law lead to a closed orbit? –  Mew Sep 30 '13 at 13:30
@Jonas, I didn't know that name, thanks! –  Nicolas Sep 30 '13 at 14:05
@Chris, that's what I don't see how to explain intuitively –  Nicolas Sep 30 '13 at 14:10