Periodic Orbit Construction Around The Triangular Libration Points Based On Nolinear Relationship

Libration points are the five equilibrium solutions in the circular restricted threebody problem(i.e.CRTBP)and there are a lot of periodic orbits and quasi-periodic orbits around them.Studies about probes moving around orbits in the vicinity of the libration points have theoretical significance.This article aim at the dynamics modeling of the periodic orbit near the triangular libration points in the circular restricted three body problem,and the analytical solutions periodic orbit by Legendre expansion method will unfold to higher order.At the same time,we appliy the polynomial expansion method to obtain high order analytical solution of planar periodic orbits and vertical periodic orbits.Also it can provide analytic relationship between each motion of the periodic orbits and the application of the traditional perturbation methods were compared for periodic orbit.And we improve constraint condition of differential correction algorithm by nonlinear relationship,and provide theoretical basis for the analysis of the orbits dynamic characteristics.Numerical method is used to analyze the accuracy of analytical solutions of higher order polynomial expansion method,analyzed the influence of different system parameters on the orbital state.The main research contents include:(1)Circular restricted three body model: using the circular restricted three body model,and the deep space activities of small bodies is simplified to two main force source of circular restricted three body problem,the motion problem when calculating will be unit normalization.Obtain the equation of the small celestial body movement and provide theoretical basis for later analysis of the solution.(2)Study of planar periodic orbit design method :Analyze planar periodic orbits from the vibrational point of view.Using earth-moon system,for example,expand analytical solution of the periodic orbits to higher-order by Legendre expansion method.The polynomial expansion is applied to construct asymptotic relationship of planar periodic orbits' two motions in the main coordinates.The dynamic characteristics of the system are analyzed from a new viewing angle,and the internal relation and physical laws of the two directions are analyzed.This relationship can be used as a constraint to numerical differential correction algorithm,looking for periodic orbit by means of iterative.Numerical simulation examples verify the validity of the method and precision.In this paper,the two-order analytic solution of the equation of the plane periodic motion equation is solved by multiscale method,and the correctness of the analytic solution is verified by comparison with the traditional perturbation method.(3)Accuracy analysis and comparison of polynomial expansion solution of planar periodic orbit:The high order polynomial expansion solution of planar periodic orbits is obtained and solving the coefficient of high order polynomial expansion.The accuracy of the solution of different order polynomials is analyzed by comparing the orbitals obtained by using different order solutions.To change the initial value,the Jacobi integral distribution and invariant surface distribution of the solution of higher order polynomials are analyzed.(4)Study of vertical periodic orbit design method: Analyze vertical periodic orbits from the vibrational point of view.We find the analytical relationship between the three direction of the vertical periodic orbits and obtain third-order analytic solution.The correctness of the analytic solution is verified by comparison with the traditional L-P method.In addition,vertical periodic orbit is designed by numerical differential correction algorithm that using the relationship.(5)Accuracy analysis and comparison of polynomial expansion solution of vertical periodic orbit: The high order polynomial expansion solution of vertical periodic orbits is obtained and solving the coefficient of high order polynomial expansion.The accuracy of the solution of different order polynomials is analyzed by comparing the orbitals obtained by using different order solutions.To change the initial value,the Jacobi integral distribution of the solution of higher order polynomials are analyzed.