Bad Astronomy: Misconceptions and Misuses Revealed, from ...

68 FROM THE EARTH TO THE MOON
a point 1,600 kilometers beneath its surface, once every month. In
a very real sense, the center of mass of the Earth (which is basically
the center of the Earth itself) is orbiting the center of mass of
the Earth-Moon system, making that little circle once a month.
This has some interesting implications. To see this, think about
the astronauts on board the space station. They float freely, as if
there is no gravity. In fact, they feel gravity almost as strongly as
we do here on the surface of the Earth; after all, they are only a
few hundred kilometers high, not much compared to the 6,400kilometer
radius of the Earth. The astronauts float because they are
in free fall; the Earth is pulling them down, so they fall. But they
have so much sideways velocity that they basically keep missing
the Earth. Their orbit carries them along a curve that has the same
curvature of the Earth, so they continuously fall but never get any
closer to the ground.
An astronaut standing on a scale in the space station would
measure her weight as zero because she is falling around the center
of the Earth. Gravity affects her, but she cannot feel it. This is
always true for an orbiting object.
But remember, the center of the Earth is orbiting the Earth-
Moon barycenter, too. So even though the center of the Earth is
affected by gravity from the Moon, someone standing there would
not actually feel that force. They would be in free fall!
But someone standing under the Moon on the Earth’s surface
would feel the Moon’s pull. Someone standing on the opposite side
would, too, but more weakly. But since the force felt from the
Moon’s gravity is zero at the Earth’s center, we can measure the
Moon’s gravity relative to the center of the Earth. To someone on
the side of the Earth nearest the Moon, there would be a force felt
toward the Moon. Someone in the center of the Earth feels no
force (remember, they are in free fall). But the person on the far
side of the Earth feels less force toward the Moon than the person
at the center of the Earth. But what’s smaller than zero force? A
negative force; in other words, a positive force in the other direction,
away from the Moon.
It seems paradoxical that gravity can act in such a way as to
make something feel a force away from an object, but in this case