### Mercury’s Mercurial Orbit

The tiny deviation of the orbit of Mercury from a pure ellipse might seem to be of no consequence. Yet the minute precession of this planet was one of the factors leading to a revolution in our world view. Attempts to explain the anomaly in the context of Newtonian mechanics were unsatisfactory. It was only with the emergence of general relativity that we were able to understand the observed phenomenon.

Precise observations could not be reconciled with a Kepler Orbit

Kepler analysed the orbit of Mars and concluded that this planet followed an elliptic orbit. Later, Newton showed that for an attractive gravitational force varying inversely with the square of distance, all the planets would have elliptic orbits. For about 300 years, this was believed to be the case.

The elliptic orbit of Mercury according to Kepler and Newton.The perihelion and aphelion are denoted “P” and “A”.

Later, astronomers observed departures from a pure ellipse. These were explained by allowing for the perturbations of other planets on the particular planet being studied. This was first considered by Newton, and it explained most of the irregularities in the orbits. There seemed to be no reason to doubt the inverse square law of gravity as formulated by Newton,

$\displaystyle F = G Mm \left[ \frac{1}{r^2} \right] \ \ \ \ \ (1)$

The precessing orbit of Mercury according to observations. The rate of precession is greatly exaggerated in the figure.

However, a small discrepancy remained for the orbit of Mercury, the smallest planet and the closest to the Sun. It was found that the rotation of the “apsidal line” between the perihelion and aphelion was some 565 arc minutes per century. Based on perturbation analysis, the expected value was 527 arc seconds (${527^{\prime\prime}}$). This left a small additional angle of ${38^{\prime\prime}}$ per century unaccounted for.

Around 1850, Leverrier checked all the earlier calculations and found no problems with them. The idea was floated that there might be another planet, tentatively called Vulcan, which might explain the disturbance. However, observations failed to find any such object. By 1900, the anomaly in the rotation was reckoned to be ${43^{\prime\prime}}$ per century.

Changing the Law of Gravity

American astronomer Asaph Hall (1829–1907).

Astronomers considered whether Newton’s law of gravity required revision. The American astronomer Asaph Hall, best known for his discovery in 1877 of Deimos and Phobos, the two moons of Mars, made an attempt to modify the inverse square law. He found that if he changed the inverse power of the radius vector from 2 to 2.000,000,157, it would result in an extra rotation of the orbit by ${43^{\prime\prime}}$ per century, according with the best observations. Moreover, this would have an insignificant impact on the other planets.

However, this was an ad hoc solution to the problem and did not provide any improvement in our understanding of celestial dynamics [see Pask, 2015].

In 1915, Albert Einstein published his theory of general relativity, showing how mass distorts spacetime and spacetime curvature determines the motion of mass. In his paper “The Foundation of the General Theory of Relativity” the following year, he showed how the motion agrees with Newton’s laws when the curvature is small, and he derived a force law for small perturbations, which may be written

$\displaystyle F = G Mm \left[\frac{1}{r^2} + \left(\frac{3 h^2}{c^2}\right) \frac{1}{r^4} \right] \ \ \ \ \ (2)$

The first term is the usual Newtonian inverse square law of attraction. The second term depends on the specific angular momentum ${h}$ and the speed of light ${c}$, and varies as the fourth power of ${r}$. It is very small, but it means that the force does not obey a purely inverse square law, and the orbits are no longer pure Kepler ellipses, but precess from one revolution to the next. The additional term results in the missing ${43^{\prime\prime}}$ per century in the precession of the orbit of Mercury.

The explanation of the observed orbit of Mercury was one of the key confirmations of general relativity. In the century that has passed since the publication of the theory, it has withstood many challenges and has led to some dramatic predictions.

Sources

${\bullet}$ Pask, Colin, 2015: Great Calculations: A Surprising Look Behind 50 Scientific Inquiries. Prometheus Books, New York, 414pp. ISBN: 978-1-6338-8028-3 [See §6.6].