Dissertation Title: LAGRANJE POINTS AND THEIR CONSIDERATION IN EXOPLANETARIAN SYSTEMS
Author: AZEM HYSA
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Institution: University of Tirana, Faculty of Natural Sciences, Department of Physics
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Field of study: Physics
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Online publication date: 08.04.2026
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The dissertation is published in Albanian.
© Copyright: AZEM HYSA
Published by the University of Tirana
Based on legal acts, regulations and policies of the UT
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Briefing
The aim of this paper is to study mainly the motion of astrophysical test objects in exoplanetary systems such as Kepler 22, Kepler 16, HD 126053, HD 28185 as well as in Sun-Jupiter, Sun-Earth and Earth-Moon systems. A combination of analytical and numerical methods is used, which includes the three-body gravitational problem, the Lyapunov exponent and several numerical integration schemes such as RKF45, Euler, Verlet, Leapfrog, RK2, Ruth3 etc. In this study we found different possible orbits around a Lagrangian point, a star, a binary star system and around a planet. In the exoplanetary system Kepler 22, two Trojan configurations were identified around the L4 and L5 Lagrangian points, along with a quasi-satellite configuration near the Super-Earth Kepler 22b. Furthermore, the presence of an exomoon orbiting this Super-Earth is predicted. By calculating the mass parameter for more than 100 binary star systems, only six of them were found to have values smaller than the critical one. We found that only one of them has a mass parameter very close to the critical value. This is the exoplanetary system HD 126053. By analyzing this system, the study shows that an astrophysical object with a mass of up to several Jupiter masses can remain in stable orbit around the Lagrangian points within the system. We have also examined the possibility of the existence of a Super-Earth in the exoplanetary system HD 28185. The orbit of HD 28185b remains entirely within the habitable zone for 100% of the numerical integration time. In contrast, the test Super-Earth spends only 22% of the total integration time in the habitable zone. The study provides a detailed investigation of the dynamics of several exoplanetary systems and interesting findings on the behavior of astrophysical objects within the aforementioned exoplanetary systems. Important topics related to habitability, the existence of exomoons, undiscovered exoplanets, as yet undiscovered Trojan bodies, and the stability of exoplanetary systems are addressed here.
Keywords: Lagrange points, Exoplanets, Super-Earths, Exomoons, Lyapunov exponent, Trojan bodies.
Extended
The purpose of this paper is primarily to study the motion of astrophysical test objects inside exoplanetary systems such as Kepler 22, Kepler 16, HD 126053, HD 28185, as well as in the Sun–Jupiter, Sun–Earth, and Earth–Moon systems. A combination of analytical and numerical methods is used, including the gravitational three-body problem, the Lyapunov exponent, and several numerical integration schemes such as RKF45, Euler, Verlet, Leapfrog, RK2, Ruth3, etc.
In this study, we found various possible orbits around a Lagrange point, a star, a binary star system, and a planet. In the exoplanetary system Kepler 22–Kepler 22b, two Trojan configurations were identified around the L4 and L5 Lagrange points, along with a quasi-satellite configuration near the Super-Earth Kepler 22b.
By calculating the mass parameter for more than 100 binary star systems, it was found that only six have values smaller than the critical one. Only one of them has a mass parameter very close to the critical value: the exoplanetary system HD 126053. Analysis of this system indicates that an astronomical object with a mass up to several Jupiter masses can maintain a stable orbit around the Lagrange points within the system.
We also examined the possibility of the existence of a Super-Earth in the exoplanetary system HD 28185. The orbit of HD 28185b remains entirely within the habitable zone for 100% of the numerical integration period. In contrast, the Super-Earth test spends only 22.0% of the total integration time in the habitable zone.
The study provides a detailed investigation of the dynamics of several exoplanetary systems and presents important findings on the behavior of astrophysical objects within them. Key topics such as habitability, the existence of exomoons, undiscovered exoplanets, Trojan bodies, and system stability are addressed.
Keywords: Lagrange points, Exoplanet, Super-Earth, Exomoon, Lyapunov exponent, Trojan bodies.