from Space.com
A consequence of General Relativity is that the apside of an object orbiting another object will move as the smaller object moves in its orbit. The apside in our Solar System can be thought of as being similar to the perihelion of a planet relative to the Sun. However, apsides are the point closest to the center of mass of a two-body system, like the Sun and a planet. Since the Sun is much more massive than any planet, we can equate the apside of a planet to the perihelion.
What we mean by the precession of the apside is that as the planet moves in its orbit, the planet goes deeper into the space-time well as shown above. This causes the planet's apside to move farther along as it orbits the Sun.
This is why full moons vary in size from month to month and why solar eclipses can vary from total, annular, or partial. The apside of the Earth-Moon system precesses as the Moon orbits the Earth.
Since this post is about Mercury proving General Relativity, we should discuss how this is so. Back in 1859, French Mathematician Urbain Le Verrier discovered that Mercury's orbit had some strange anomalies. It was at first thought that the anomalies were due to a planet closer to the Sun than Mercury. This explanation helped in the discovery of Neptune by Le Verrier and two others: John Couch Adams and Johann Galle earlier in the century. But once Einstein's theory came out, the equations describing the precession of the asides calculated the anomalies present in Mercury's orbit.
There are other examples of observations proving the Theory of General Relativity. They can be found in the above Wikipedia link.
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