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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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Relevant: en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector –  Johannes Sep 30 '13 at 15:18
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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
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Way back in 1609, Kepler came out with Kepler's laws of planetary motion. It basically says that all orbits are an ellipse (slightly revised to a rotating ellipse caused by precession as noted by Nicolas). In an intuitive sense, perhaps this: when the planet or comet is far from the sun it slowly slows down and starts to fall towards the sun. When the planet or comet is close to the sun it quickly speeds up and shoots out farther from the sun. The act of falling closer to the sun, speeds up the planet or comet, hence it cannot spiral into the sun unless there is something else (some sort of drag, friction or crash) that slows it down.

enter image description here

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The fact that when the comet gets closer to the sun it picks up speed and goes away is clear. But I don't see any clear reason why it has to return to exactly the same place it came from and with the same velocity. In other words, if I hadn't solved the equations, I could say that in principle the comet could shoot off in a different direction of that in which it came in, and then go back and repeat the process, so its orbit would be a sort of flower, if you will. Is there any way I can explain why that doesn't happen to a 16-year old? –  Javier Badia Sep 30 '13 at 17:51
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@Javier Nadia, assuming that gravity would be none relativistic and only two point-like bodies are atracting each other due to gravity. In this case you can say that after one orbit they would be at the exact same position as before (also assuming that your reference frame is not moving relative to the center of mass of the two). You can explain this with the fact that gravity is a conservative force, so going back in time will do follow the same path as going forward, only in the reversed direction. –  fibonatic Oct 1 '13 at 6:07
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@fibonatic, even if you postulate that going back in time will trace out the same path, how do you know it doesn't take 3 or 4 orbits before it reaches its initial starting position? Why does it reach its initial position after exactly 1 orbit? –  Mew Oct 1 '13 at 8:22
    
@JavierBadia, because of the symmetry of the problem. Since in a stable orbit there will always be two (or infinite for circular motions) situations where the radial velocities relative to their center of mass of both objects is zero, so there motion is entirely tangential. –  fibonatic Oct 1 '13 at 12:47
    
The flower is exactly what you see with the orbit of Mercury. It took until Einstein to explain it and there are a lot of petals on that flower. Increasing the gravitational field will get you as many or as few petals as you wish. A great article and picture (towards the bottom) are at mathpages.com. –  AnimatedPhysics Oct 1 '13 at 15:51
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