Confirming Einstein’s Theory of General Relativity With Calculus, Part 7e: Computing Precession

In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.

We have shown that under general relativity, the motion of a planet around the Sun precesses by

,

where is the semi-major axis of the planet’s orbit, is the orbit’s eccentricity, is the gravitational constant of the universe, is the mass of the Sun, and is the speed of light.

Notice that for to be as observable as possible, we’d like to be as small as possible and to be as large as possible. By a fortunate coincidence, the orbit of Mercury — the closest planet to the sun — has the most elliptical orbit of the eight planets.

Here are the values of the constants for Mercury’s orbit in the SI system:

• 
• 
• 
• 
• 
• 

The last constant, , is the time for Mercury to complete one orbit. This isn’t in the SI system, but using Earth years as the unit of time will prove useful later in this calculation.

Using these numbers, and recalling that , we find that

.

Notice that all of the units cancel out perfectly; this bit of dimensional analysis is a useful check against careless mistakes.

Again, the units of are in radians per Mercury orbit, or radians per 0.2408 years. We now convert this to arc seconds per century:

.

This indeed matches the observed precession in Mercury’s orbit, thus confirming Einstein’s theory of relativity.

This same computation can be made for other planets. For Venus, we have the new values of , , and . Repeating this calculation, we predict the precession in Venus’s orbit to be 8.65” per century. Einstein made this prediction in 1915, when the telescopes of the time were not good enough to measure the precession in Venus’s orbit. This only happened in 1960, 45 years later and 5 years after Einstein died. Not surprisingly, the precession in Venus’s orbit also agrees with general relativity.