• Mayur Pawar

Do you know: Do Newton's laws always work right in planetary motion?

Updated: Dec 5, 2020

The ideas outlined in Newton’s laws of motion and universal gravitation stood unchallenged for nearly 220 years until Albert Einstein presented his theory of special relativity in 1905. Newton’s theory depended on the assumption that mass, time, and distance are constant regardless of where you measure them.

The theory of relativity treats time, space, and mass as fluid things, defined by an observer’s frame of reference. All of us moving through the universe on the Earth are in a single frame of reference, but an astronaut in a fast-moving spaceship would be in a different reference frame. Within a single frame of reference, the laws of classical physics, including Newton’s laws, hold true. But Newton’s laws can’t explain the differences in motion, mass, distance, and time that result when objects are observed from two very different frames of reference. To describe motion in these situations, scientists must rely on Einstein’s theory of relativity.

At slow speeds and at large scales, however, the differences in time, length, and mass predicted by the relativity are small enough that they appear to be constant, and Newton’s laws still work. In general, few things are moving at speeds fast enough for us to notice relatively. For large, slow-moving satellites, Newton’s laws still define orbits. We can still use them to launch Earth-observing satellites and predict their motion. We can use them to reach the Moon, Mars, and other places beyond Earth. For this reason, many scientists see Einstein’s laws of general and special relativity not as a replacement of Newton’s laws of motion and universal gravitation, but as the full culmination of his idea.

In Newtonian theory, the gravitational attraction is a central force, and all planets move in a constant plane around the sun. Hence in polar coordinates, the motion of this plane is dependent on the distance r of the planet from the centre, and I the angle between the line that connects the planet to the centre and a line that is chosen arbitrarily. One obtains the orbit equation and r as a function of I (the distance of the planet from the sun at any given angle). The solution of the Newtonian orbit equation is the equation of an ellipse – an orbit in the plane, and the eccentricity e determines the characteristic of the elliptic orbit.

The perihelion of the orbit is the point in which the planet is closest to the sun. This point is found on the major axis of the ellipse, its longest diameter, the line that runs through the centre and both its foci. This major axis was found to slowly turn around the sun, and the perihelion turned as well. This is the precession of the perihelion, and it is more pronounced the more the eccentricity e is larger.

In Einstein's theory, the geodesic equation leads to an orbit equation. The geodesic equation led Einstein to a relativistic equation of the orbit. Einstein found that the difference between the Newtonian orbit equation and the relativistic orbit equation was in an additional term: 2GM/c^2 r^3 that appears in the relativistic equation. He treated first the Newtonian solution to this equation as a first approximation. He then checked, what was the size of the correction that resulted from the addition of this term? He integrated the Newtonian orbit equation first. The Newtonian solution to the Newtonian orbit equation describes an ellipse of a planet, for which the direction of the major axis and the perihelion should both stay fixed.

Einstein then added the perturbation of the additional term 2GM/c^2 r ^3 to this solution in order to see whether the turning of the perihelion resulted from this additional term in the relativistic equation. If this was indeed the result, then the precession of the perihelion would turn to be a result of a relativistic effect, and this was the first triumph of Einstein's 1915 theory. Einstein concluded his scheme by saying, "The calculation yields, for the planet Mercury, an advance of the perihelion of 43'' per century, while the astronomers indicated 45'' ± 5'' as the unexplained remainder between observations and the Newtonian theory. This means full compatibility". A great triumph for Einstein's November 1915 theory.