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"Appear" to wobble is an important part of the idea. This is an apparent motion of the moon and not a true motion. It is called east-west libration (from Libra, the scale). It happens because of the combination of these essential facts:

  1. The moon's orbit around the earth is an ellipse and not a circle.
  2. Objects in elliptical orbits move in their orbital paths more quickly when they are closer to the objects they are orbiting, and less quickly when they are farther away. The moon's maximum orbital velocity happens when the moon is at perigee (closest to earth).
  3. The rate of the moon's rotation (turning on its axis) is for our purposes a constant velocity.
  4. The moon is tidally locked with earth, meaning that it rotates exactly once during every sidereal [absolute] orbit around earth. This is why the same face of the moon, for the most part, is always oriented toward earth.

Now if you view the moon from the earth in a time lapsed version of the moon's orbit, something interesting happens. When the moon is getting nearer and nearer to perigee, it picks up speed in its orbit. But its rotational speed is almost exactly constant. So the moon moves in its orbit 'ahead' of its rotation, in a sense. We get a glimpse of the leading edge, or limb, of the moon. Then, when it is past perigee and starting to slow down in its orbit, its rotational speed moves ahead of its orbital speed, and we get a glimpse of the trailing edge or limb of the moon. Observing this with software, you can see the obvious apparent wobble of the moon. There is also north-south libration which happens for different reasons.

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