Perihelion Precession

Perihelion precession is the gradual rotation of the major axis of an elliptical orbit within its orbital plane. The perihelion is the point of closest approach to the central body (the aphelion is the farthest); the line joining them is the line of apsides. In an idealized two-body system governed by a pure inverse-square force, the orbit is a closed ellipse and this line stays fixed forever. Any departure from the exact inverse-square law — the gravitational pull of other bodies, a non-spherical mass distribution, or the curvature of spacetime itself — prevents the orbit from closing, so each revolution the body returns to perihelion at a slightly rotated angle and the line of apsides slowly sweeps around the central mass.

Most of the Solar System's observed perihelion precession is Newtonian: the mutual gravitational tugs of the planets perturb one another's orbits. For Mercury, these planetary perturbations account for roughly 531 arcseconds per century of advance. But the measured dynamical advance is about 574 arcseconds per century, leaving a residual of about 43 arcseconds per century that no Newtonian accounting could supply. Identified by Urbain Le Verrier in 1859, this anomaly resisted explanation for half a century and even prompted the hypothesis of an undiscovered inner planet, Vulcan, that never existed.

General relativity resolves the residual exactly. In the Schwarzschild geometry outside a spherical mass, the effective potential governing orbits carries an additional attractive term that falls off as the inverse cube of distance, steeper than Newtonian gravity. This term un-closes the ellipse and advances the perihelion by Δϖ = 6πGM/[c²a(1−e²)] radians per orbit, where M is the central mass, a the semi-major axis, e the eccentricity, and c the speed of light. For Mercury this evaluates to about 43 arcseconds per century — the number Einstein computed on 18 November 1915, a result he said gave him heart palpitations. Because the effect scales steeply with closeness to the Sun, Mercury shows it most strongly; it also appears, far more dramatically, in the periastron advance of binary pulsars.

The anomalous residual was isolated by Urbain Le Verrier in 1859 and refined by Simon Newcomb. After decades of failed searches for a perturbing planet, Einstein derived the relativistic 43 arcseconds per century in November 1915, days before presenting the final field equations — the first quantitative confirmation of general relativity, and a retrodiction with no adjustable parameters.