INVESTIGATION OF THE ANGULAR DISTRIBUTION OF LAGRANGE POINTS

Abstract

This work presents a theoretical investigation of the Lagrange points for the restricted three-body problem using the polar coordinate system. Employing the effective gravitational potential, relationships were derived to describe the angular and radial positions of the Lagrange points as functions of the characteristic parameters of the system. To obtain analytical expressions, perturbation theory was applied, with the small parameter being the mass ratio of the primary components of the system. The use of polar coordinates allowed the explicit description of the angular distribution of the equilibrium points. It was shown that the angular position of the Lagrange points L4 and L5, depending on the mass ratio of the two primary components, varies within the range from π/3 to π/2. The obtained result was confirmed with examples of the stellar systems Groombridge 34, HD 155358, and HD 69830, which exhibit different mass ratios in the range from 0,443 ⋅ 10-5 to 0,383. The relationship describing the angular distribution of the Lagrange points derived in this work was applied to the study of the exoplanetary system PDS 70. Based on the analysis of images of the PDS 70 system, the angular position of a gas–dust cloud located on the orbit of the exoplanet PDS 70b was determined. Additionally, the angular position of this cloud was calculated using the known mass of the central star in the PDS 70 system and the mass of the exoplanet PDS 70b. The results show that the dust cloud in which a new planet is forming is located at the L5 Lagrange point of the PDS 70 – PDS 70b system. This finding supports the hypothesis that another “Trojan” exoplanet is forming in the orbit of PDS 70b. It also allows us to assert that co-orbital configurations, known in the Solar System (e.g., Jupiter’s Trojan asteroids), are a universal phenomenon capable of arising in exoplanetary systems.