Icarus’s perihelion advance explored through Newtonian and relativistic gravity, and the Vulcan hypothesis

Abstract

In this work, a comparative study of the perihelion precession of the inner planets is presented, considering both general relativity and Newtonian gravity extended to include a hypothetical planet, Vulcan, modeled as a planetary-mass primordial black hole located between Mercury and Venus. Two computational approaches are examined: the conventional method based on the secular evolution rate of the perihelion longitude and an alternative approach based on the rotation of the Laplace–Runge–Lenz vector. A Vulcan with a semi-major axis of 0.545 astronomical units and a mass approximately one-third that of Mercury reproduces the observed perihelion advances of Mercury, Earth, and Mars. However, it predicts a perihelion advance for the asteroid Icarus significantly larger than the relativistic value. As a planetary mass primordial black hole, Vulcan would be electromagnetically invisible, and its presence could only be inferred through gravitational perturbations. For Mercury, Earth, and Mars, the perihelion advances computed using both methods are mutually consistent within the Newtonian-plus-Vulcan framework and in general relativity. The analysis of Venus’s perihelion advance reveals large fluctuations, and it is therefore excluded from the study. For asteroid Icarus, the two methods yield divergent results: only the Laplace–Runge–Lenz vector approach produces values consistent with general relativity. These results demonstrate that the Laplace–Runge–Lenz vector method provides a reproducible and intuitive framework for calculating perihelion advance, capturing the effect as a rotation of a vector that directly points to the perihelion position, offering a straightforward and physically intuitive alternative to the traditional perihelion-longitude approach. The orbital motion of Icarus provides a particularly sensitive test for distinguishing between classical Newtonian dynamics extended to include a hypothetical Vulcan and relativistic predictions. A deeper understanding of the orbital dynamics of solar system bodies enables refinement of gravitational models in the inner solar system and more accurate predictions for near-Earth object trajectories.