8.4: The Gravity Gradient

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This strange rotation of the moon is maintained because the moon is slightly elongated along the axis which points towards earth. To understand the effect we look at the motion of a body with a much more pronounced elongation --- an artificial satellite with the shape of a symmetric dumbbell (see Figure \(\PageIndex{1}\) below).

It can be shown that if the forces on the dumbbell (or indeed on a satellite of any shape) are unbalanced, it rotates around its center of gravity. That point will be defined here http://www.phy6.org/stargaze/Srocket.htm, but in a symmetric dumbbell with two equal masses marked A and B, the center of gravity is right in the middle between them.

Both masses A and B are attracted to the Earth, and if the attracting forces were equal, their tendencies to rotate the satellite (“rotation moments" or “torques") are equal and cancel each other, so that no rotation occurs. If however A starts closer to the center of Earth, the force on it is just a little stronger. Therefore the satellite will rotate until A is as close to Earth as it can be, which is a possible position of equilibrium. Of course, it may then overshoot its equilibrium position, and end up swinging back-and-forth like a pendulum, only slowly (like a pendulum) losing energy and settling down. The elongated moon acts like a dumbbell too.

The rotating force which lines up the moon or an orbiting dumbbell is the difference between the pull on A and on B. It depends not on how strongly gravity pulls these masses, but on how rapidly the pull of gravity changes with distance --- on the “gravity gradient." Near Earth that is a gentle force, though still strong enough to line up elongated satellites. Among those was Triad, (http://www.phy6.org/Education/wtriad.html) deliberately shaped like a long dumbbell with an additional payload in the middle, the first satellite to map the electrical currents associated with the polar aurora.

Near a black hole or pulsar, though, the gravity-gradient force can be fierce enough to rip a spacecraft apart.

Actually, the long axis of the Moon does not always point exactly to the center of the Earth, but swings back and forth around that direction, a motion known as liberation. Most of this is caused because the Moon rotates around its axis with a fixed period, while its motion around its orbit slows down far from Earth and speeds up close to it. This speeding up and slowing down is the result of Kepler's 2^nd law, discussed in section 12, and is a rather small effect, since the moon's orbit is very close to circular.

Because of liberation, even though at any time only half the Moon is visible, over time 59% can be seen, since it lets astronomers look at the Moon from slightly different viewing directions. Liberations are a rather specialized subject --- but if you want to know more about it, go to http://www.phy6.org/stargaze/Smoon4.htm