Mercury: Advance of the perihelion

All planets in the Solar System move in elliptic orbits with the Sun at one of the foci. Their orbits are approximately coplanar with one another. An ellipse is an elongated circle and the foci are two points equidistant from its centre. They lie on the line joining the points nearest to (perihelion) and farthest from (aphelion) the Sun. The perihelion and aphelion are collectively called apsides and the line joining them is the major axis of the ellipse.
According to Newton, the attractive gravitational force exerted by the Sun keeps the planet on its orbit. Without this force, the planet will stray from its orbit and move along a straight path. The force is proportional to the product of the masses of the Sun and the planet, and inversely proportional to the square of the distance between them. That is why it is known as the inverse-square-law force.
A purely Newtonian description of gravity requires that an elliptic orbit be exactly closed, i.e. the orbit is ‘reentrant,’ or repeats itself each time a planet completes a revolution around the Sun. Consequently, the perihelion remains fixed in space.
However, Newton showed that if the force differs slightly from the inverse-square-law, e.g. if the exponent in the inverse-square law, instead of being exactly equal to two, is equal to two plus some small fraction, the orbit will not be reentrant. Instead, the major axis of the orbit will slowly rotate in the plane of the planet’s orbit in the same direction as the revolution of the planet itself such that the angular location of the perihelion will gradually change. This phenomenon is known as the advance or precession of the perihelion.
In the Solar System where there’s more than one planet orbiting the Sun, the situation is a bit more complicated than the simple inverse-square-law. The mutual gravitational pull of the planets on each other results in perturbations. Consequently, the net force experienced by a planet does not vary exactly as inverse-square. As a result, planetary orbits do exhibit advance of the perihelion. The effect falls off rapidly for planets beyond Earth’s orbit because of their large distance from the Sun.
Mercury, being closest to the Sun with a highly elliptic orbit, shows the largest effect. Its perihelion advances by about 574 seconds of arc (arcsec) per century (3,600 arcsecs = 1 degree). This anomalous rate of advance of the perihelion of Mercury’s orbit was first recognised in 1859 by the French mathematician and astronomer Urbain Le Verrier. Although very small, the effect is cumulative so that observations of Mercury over centuries allowed even this small effect to be noticeable.
Detailed calculations made with Newton’s law of gravity and the gravitational perturbations of other planets on the motion of Mercury could account for 531 arcsecs. The discrepancy of 43 arcsecs between theory and measurement perplexed astronomers and remained an unresolved issue of the Newtonian theory for many years. To resolve the dilemma, some suggested that Newton’s law of gravity is flawed. Others speculated that the presence of a planet called Vulcan was responsible for the quandary created by Mercury’s precession. But later observations showed that no such planet exists.
After the imaginary Vulcan, some astronomers assumed that there’s probably an asteroid field or a massive field of dust near Mercury. This would add a little extra mass to the equations and explain why Mercury precessed so quickly. Years went by, and no field of asteroids or dust showed up.
The unexplained advance of Mercury’s perihelion was finally resolved by Einstein in 1915, after publication of his theory of general relativity. According to him, heavenly objects don’t just float around in an unperturbed vacuum. They shape the space around them – curves or warp it. The more the mass, the greater is the curving. And a mass experiencing the pull of gravity is simply responding to the curvature of the space around it.
Because of its proximity to the massive Sun, space around Mercury is more curved than the space around other planets. Thus, Mercury’s perihelion should precess at a different rate than Newtonian physics would predict. Surely, the modification introduced into Newton’s equations due to the curvature of space accounts for the additional 43 arcsecs.
The advance of perihelion occurs in other planets too. But the advance due to Newtonian effect gradually decreases as the distance between the planet and the Sun increases. Nevertheless, if the planet is massive or if it is in the vicinity of a massive one such as Mars near the giant planet Jupiter, relativistic effect can become appreciable. For example, the perihelion of Earth advances by 3.84 arcsecs per century due to general relativity, Venus by 8.62 arcsecs and Mars by 1.35 arcsecs.
The advance of Mercury’s perihelion changed our view of the Universe forever. We now know that space is not flat and rigid. It is malleable and could be bent or twisted by the presence of matter. And although we can’t see a twisted Universe, we do know it’s there.

The writer is Professor of Physics at Fordham University, New York.
Photos: Google Image