## 1 Introduction
The equations obtained, when disregarding the SRP, eliminate the spurious Poisson terms from the theory of [PERSON] and [PERSON] [18; 19]. In the case of this work, the mathematical model considers only the main perturbing forces acting on the space debris in a GEO regime, such as the forces associated with the direct solar radiation pressure (denoted already as SRP), the third-body perturbations from the Sun and Moon (TBP), and the flattening of the Earth, expressed as the dominant term of its potential, namely the second-degree zonal harmonic coefficient \(J_{2}\). The goal is to use the changes over the orbital eccentricity along with the increase in the perigee altitude of these objects until they reach the outside of the borders of the GEO region. In the present study, we adopted averaged dynamics to study the orbital motion, since it has been successfully applied to several studies seeking a qualitative approach for the long-term dynamical evolution of a system (see, for instance, the applications of this type of dynamics presented in references [20; 21; 22; 23; 24; 25]). In this study, a model for the SRP acceleration is used that can be associated with a potential, because it depends only on the radial position of the solar sail with respect to the Sun [15], which generates the photon emission. Such a consideration allows the disturbing potential associated to the SRP to be written in a series of polynomials of Legendre. In a recent work, [PERSON] et al. [26] developed the SRP equation using the double-averaged model considering the Earth in an elliptical and inclined orbit around the Sun. It is assumed that the solar sail maintains an orientation perpendicular to the line of the Sun and the sail is located at a distance from the Sun such that the SRP model can be expressed by a potential function. Neglecting the effect of the Earth's shadow, the system is Hamiltonian. Thus, the equations are developed using Legendre polynomial expansions up to the second order. In the double-averaged model, the first-order term of the expansion of the Legendre polynomial does not contain the terms of the orbital elements of the satellite (or debris), and therefore, when it is replaced in the Lagrange planetary equations, it does not affect its orbital elements. Thus, in [26] the first-order term is neglected and only the second-order term of the Legendre polynomial is used in the dynamical system. In [27], the authors remake the modeling presentedin [26], but, in this case, it is taking into account the single-averaged model. With this approach, the first-order term of the expansion of the Legendre polynomial contains the terms of the orbital elements of the satellite (debris), and thus affect the orbital elements when it is substituted in the Lagrange planetary equations. The authors showed that the first-order SRP model is dominant compared to the second-order expansion term; that is, the first-order equation coming from the SRP is the main perturbation of the model and its effect is much greater than the term of second order. It is shown that the SRP acts strongly on the debris in the GEO region, allowing an increase in their eccentricity. This increase is proportional to the area/mass ratio of the debris. Several authors have taken into account the solar radiation pressure models to explore the dynamics of artificial satellites, asteroids, and space debris; for example, see references [9; 10; 11; 27; 12; 28; 29; 30; 31; 32; 33; 34; 35; 36] (and references therein). For the equation of the perturbing potential due to a third body, we use a single-averaged model [37] including the perturbations coming from the Sun and Moon and assuming that these perturbing bodies are moving in elliptical and inclined orbits. The equation of the disturbing potential due to the third body can be seen in Appendix B of reference [37]. It is important to mention that, in the present work, the effect of the shadow of the Earth is not included in the mathematical model, because, as presented in [32], it is not very important to the dynamics.