# Do 2-body elliptic orbits precess in special relativity?

Einstein famously explained the anomalous precession of Mercury by showing that in general relativity elliptic orbits precess even in the 2-body problem. But apparently in the early days of quantum mechanics Sommerfeld refined Bohr's model of the hydrogen atom with circular orbits by not just introducing elliptic orbits, but also assuming that they precess based on special relativity only. "Sommerfeld postulated that only certain eccentricities of ellipse are possible trajectories for the electron... But Sommerfeld does not stop here. His next move is to apply the laws of special relativity to the different possible electron trajectories. In essence, Sommerfeld makes use of the equation for a precessing elliptical orbit, but introduces relativity by making a change to the equation for angular momentum..." (p.16).

If I understand this correctly he solves the 2-body problem in special relativity kinematics but under the classical inverse square law. Is it enough to make ellipses precess? Does this effect explain the anomalous precession of Mercury even without general relativity?

## 1 Answer

The answers are yes and no. Special relativity does make ellipses precess, but it only accounts for 7" out of 43" per century of Mercury's anomalous precession. I wonder if Einstein and/or Sommerfeld knew that.

To first order, incorporating special relativity results in a small inverse cube correction to the gravitational force, which is well known to cause precession of orbits. An elementary derivation is given by Lemmon and Mondragon, who write:"This orbit equation clearly describes three corrections to a Keplerian orbit due to special relativity: precession of perihelion; reduced radius of circular orbit; and increased eccentricity. The predicted rate of precession of perihelion of Mercury is identical to established calculations using only special relativity. Each of these corrections is exactly one-sixth of the corresponding correction described by general relativity in the Keplerian limit". Another elementary derivation derivation is given in the answer to Mercury's Orbital Precession in Special Relativity.