Does The Moon Orbit The Earth?

Question posed by Jennie, after I baffled her in the pub. She’d been watching this video, and said that if Jupiter were to replace the Moon, then surely Earth would end up in orbit around Jupiter?

No.

Well, not really*. The common view of the Earth-Moon system (and of any two-body system) is that the Earth sits there on the spot (ignoring its motion around the Sun, for now) and the Moon makes its stately way around it in a big circle. The same view is generally thought of with Jupiter and the Sun, or any other situation in which an ‘orbit’ is involved. This isn’t quite what happens…
 

Remember a time when you were younger; orbital mechanics are less of an issue and Big Uncle Bob, who’s six-feet tall and built like some kind of unmentionable outhouse, picks you up and swings you around. You are very definitely flying around in a scary yet enjoyable circle, but at the same time, even though you’re so light compared to him, he still has to lean back as he spins you so as not to fall over. This is pretty much what’s happening with the Earth and Moon.

The Earth is a lot more massive than the Moon, so it seems like the Moon is orbiting the Earth, but in reality they’re both orbiting a point between the centres of the two bodies. This common centre is actually under the surface of the Earth, but it is not at the centre of the Earth. The animation to the right gives a pretty good demonstration of this (the Earth and Moon and their distance apart are not to scale): the red cross in the centre is the point they’re both orbiting around.

The fact is that they’re both in orbit around each other, or more precisely, around their common centre of mass**. This point is called the ‘barycentre’ of a system.

In situations where one object is bigger than the other*** the centre of mass moves closer to the heavier object. If you adjust the model in your head to an extreme (Uncle Bob’s morbidly obese, and you’re a newborn slip of a thing), the effect is less obvious****. Can you think what might happen if both objects have the same mass?

For purposes of clarity, however, we usually say that one object (the lighter one) is ‘in orbit’ around the other (heavier) one: the Moon orbits Earth; Jupiter orbits the Sun. In Jennie’s original comment, only the biggest pedant (me) would say that her statement was incorrect- in the animation above, just imagine the big circle is Jupiter, and the small one is Earth.

Now, you’re probably thinking ill of me right now; “what an annoying, pedantic fool!” But wait! This fact is actually quite useful: the same thing happens with any two bodies that are in orbit around each other, including distant stars. This means that some stars appear to ‘wobble’. For some stars, we can detect this wobble and work out that there’s a planet there even though we can’t see it. We can even work out things like the mass of the planet and its orbital period!

* I’m being facetious for the purposes of introducing a concept that I’d like to talk about. Of course, in the situation in which Jupiter replaced our Earth there would be significant changes in the dynamics of the system, and Jupiter would indeed dominate it.
** Or ‘centre of gravity’ if you’re more comfortable with that phrase.
*** Strictly speaking, we should say ‘more massive‘ rather than ‘bigger’ because, in this case, size doesn’t matter. It’s how much stuff it’s made of that counts.
**** It is important to note, however, that the only way you can get the centre of mass to be in the centre of the heavier ball is if the lighter ball has zero mass (i.e. there’s nothing there).

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