A planet going round the Sun traces an ellipse, with the Sun at a focus rather than at the centre. The point where the planet comes closest to the Sun is called the perihelion.
Under a force inversely proportional to the square of the distance, the ellipse closes exactly after one lap and the direction of the perihelion does not change.
Only two forces always give closed orbits: the inverse-square force and a spring force proportional to distance (Bertrand's theorem). If the law of force differs even slightly, the orbit no longer closes and the perihelion slowly turns.
Mercury's perihelion turns by arcseconds a century, about degrees. One arcsecond is a th of a degree.
Most of this is explained by the pull of Venus, the Earth and Jupiter. Newtonian gravity gives arcseconds, leaving arcseconds over.
In Le Verrier proposed an unknown planet inside Mercury's orbit to account for the remainder. The planet, later called Vulcan, was searched for but never found.
Working out planetary motion in general relativity gives Newton's inverse-square force plus a small term inversely proportional to the fourth power of the distance.
Far away this term hardly matters; it grows only close to the Sun. The figure draws the added term much larger than it really is.
With this term the orbit does not close, and each lap the perihelion advances by radians. Here is the gravitational constant, the mass of the Sun, the semi-major axis of the orbit and its eccentricity.
The shift per lap is larger closer to the Sun, and a shorter period means more laps per century.
Mercury is the closest planet to the Sun, with a period of days. Its ellipse is also elongated (), so the position of its perihelion is well defined and the shift is easy to measure.
Mercury's orbit has m and . Putting these into the formula together with the Sun's m³/s² and the speed of light m/s gives radians per lap.
One radian is about arcseconds, so a lap turns the perihelion by arcseconds. With laps in a century, that comes to arcseconds a century.
The arcseconds left over and the arcseconds calculated agreed. Einstein obtained the result in November and wrote in a letter the following year that for a few days he was beside himself with joyous excitement.
These arcseconds had stood unexplained for more than years since . They were explained not by adding a planet but by rewriting the law of gravity.
The turning of the closest point is larger where gravity is stronger. In the binary pulsar PSR B1913+16, where two neutron stars orbit each other, the point where the two come closest (the periastron) turns degrees a year, covering Mercury's century of arcseconds in about a day.
The orbital period is hours, so the pair goes round times a year. Per orbit that is arcseconds, about times Mercury's .
For the star S2, which orbits Sagittarius A*, the black hole at the centre of our galaxy, every years, a turn of about arcminutes ( degrees) per orbit was confirmed by observation in .
In both cases the turning agrees with the value calculated from general relativity.