Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria.
Department of Mathematical Sciences, Faculty of Physical Sciences, Federal University Dutsin-Ma, Dutsin-Ma, Katsina State, Nigeria.
Sci Rep. 2023 Mar 15;13(1):4286. doi: 10.1038/s41598-023-30671-3.
We have examined the effects of Albedo in the Elliptic Restricted Three-Body Problem (ER3BP) with an oblate primary, a triaxial secondary, and potential due to belt for the Earth-Moon system. We have found that as the perturbed parameters increases, the possible boundary regions of the primary come closer to one other, allowing particles to travel from one region to the next freely and possibly merge the permissible regions. Our study has revealed that the formation of triangular libration points depends on the Albedo effects, semi-major axis, the Eccentricity of the orbits, triaxiality, and the potential due to the belt. As the parameters mentioned above increase, the triangular positions [Formula: see text] and [Formula: see text] move towards the center of origin in cases 1, 2, 3, and 4 and away from the center of the origin in cases 5, 6, and 7. Considering the range of a stable and unstable libration point for the problem under study given as [Formula: see text] for stable libration points and [Formula: see text] for unstable libration points, our study has established that the triangular libration points are respectively stable and unstable for cases 1, 2, and 6 and cases 3, 4, 5, and 7. Our study has also revealed that each set of values has at least one characteristic complex root with a positive real part. Hence, the triangular libration points for the Earth-Moon system are unstable in the sense of Lyapunov. The Earth-Moon system's Poincare Surface of Section (PSS) has demonstrated that a slight change in the initial conditions, the semi-major axis, and the Eccentricity of the orbits have affected the system's behavior dramatically. Further, it is seen that a chaotic dynamical behavior of the system results into either regular or irregular orbits.
我们研究了具有扁长形主星、三轴次星和地球-月球系统带引力势的椭圆限制性三体问题（ER3BP）中反照率的影响。我们发现，随着摄动参数的增加，主星的可能边界区域相互靠近，允许粒子从一个区域自由地移动到另一个区域，并可能合并允许的区域。我们的研究表明，三角形平动点的形成取决于反照率效应、半长轴、轨道偏心率、三轴性和带引力势。随着上述参数的增加，在情况 1、2、3 和 4 中，三角形位置 [Formula: see text] 和 [Formula: see text] 向原点中心移动，而在情况 5、6 和 7 中则远离原点中心。考虑到所研究问题稳定和不稳定平动点的范围分别为 [Formula: see text] 为稳定平动点和 [Formula: see text] 为不稳定平动点，我们的研究表明，在情况 1、2 和 6 以及情况 3、4、5 和 7 中，三角形平动点分别是稳定和不稳定的。我们的研究还表明，每组值都至少有一个具有正实部的特征复根。因此，地球-月球系统的三角形平动点在 Lyapunov 意义上是不稳定的。地球-月球系统的 Poincaré 截面（PSS）表明，初始条件、半长轴和轨道偏心率的微小变化都会对系统的行为产生巨大影响。此外，还可以看出，系统的混沌动力学行为会导致规则或不规则轨道。
