Research Article Balancing Conditions for the Relativistic Correction Using Lorentz Acceleration M. A. Yousef , 1 M. I. El-Saftawy , 1,2 and A. Mostafa 3 1 Department of Astronomy and Space Science, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 2 National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt 3 Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt Correspondence should be addressed to A. Mostafa; ahmedmostafa@sci.asu.edu.eg Received 2 July 2023; Revised 26 July 2023; Accepted 8 August 2023; Published 19 August 2023 Academic Editor: Yue Wang Copyright © 2023 M. A. Yousef et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, the average effects of the Lorentz acceleration on the charged spacecraft’s orbit are studied encounter with relativistic correction. The relativistic correction as function of the orbital elements, and may be time, is formulated. Lagrange planetary equations are used to calculate the perturbations due to considered perturbing forces. The needed conditions to neutralize the effects of the relativistic corrections, using Lorentz acceleration, are derived. Numerical examples for different kinds of orbits are applied. 1. Introduction The perturbing forces on the artificial bodies that can be either spacecrafts or artificial satellites are substantial topics in the field of astrodynamics. By studying these forces, the orbits of artificial bodies can be controlled differently based on the purpose of the orbits. To determine the orbital design, six Keplerian orbital elements must be monitored. As time passes with any partic- ipating outer perturbing force on the artificial body, the orbital elements are significantly impacted. Indeed, the aim of the orbital theory is to minimize the influence of the per- turbing forces as much as possible. One of the perturbing forces is the relativistic correction (RC), which is referred to as post Newtonian effects. This disturbance perturbs the orbital motion of the earth’s artifi- cial body. Astronomers have paid attention to Einstein’s prediction in 1915 for precession of a planet’s perihelion around the sun which explained the discrepancy among the observed and theoretical shift of Mercury’s perihelion. Later, Lense and Thirring [1] confirmed that from the aspect of general relativity. The restricted relativistic two- body problem was also discussed in detail by Bogorodskii [2]. According to Cugusi and Proverbio [3], the RC can be divided into three components addition to secular peri- gee shift such as the effect regard to the perigee shift, the processional term of the line of nodes, both depended on rotational earth, and a variation of the time of perigee pas- sage. The general theory of relativity is applied in many astronomical applications. The RC, per unit inertial mass, in near-earth body motion can be outlined as being con- tained in the four effect components such as the Schwarzs- child solution, the geodesic precession, the Lense-Thirring precession, and the earth oblateness components (repre- sented mainly by the J 2 geopotential spherical harmonic coefficient) [4–7]. Due to perturbing forces such as RC, Lorentz accelera- tion (LA) is utilized to balance the perturbations on the arti- ficial body’s orbit [8]. It is important to mention that the space orientation of the orbits should be fixed for long time intervals. The family of orbits fulfills this status is called “fro- zen orbits.” Many studies have discussed this type of orbits such as Condoleo et al. [9], El-Salam and El-Bar [10], Khat- tab et al. [11], and El-Salam et al. [12]. Recently, many stud- ies using charge on the surface of the satellite are made to control one or more of the natural perturbing forces such as Mostafa et al. [13] and Yousef et al. [14]. The previous authors studied the usage of LA to balance the effects of solar Hindawi International Journal of Aerospace Engineering Volume 2023, Article ID 5593887, 14 pages https://doi.org/10.1155/2023/5593887