Application Of Analytical Solutions Of Two-body Problems In Numerical Improvement Of Multi-body Problems

The results of the conventional algorithms are unreliable in the long-term orbital integration because of the artificial dissipation.Based on Nacozy's idea of manifold correction,we proposed a new manifold correction scheme by slightly modifying a Kepler solver.The difference between the new method and the Kepler solver is that the eccentric anomaly is not calculated by iterating the Kepler equation,but is calculated by using the geometric relationship between the unit position vector obtained by numerical solutions,the Laplace vector and the angular momentum vector.When the system is perturbed,the five orbital elements,which are constants in the two-body problem,vary slowly with time.The seven quasi-integrals obtained by combining the invariant relations and the equations of motion are more accurate than the corresponding values obtained by positions and velocities.Taking the quasi-integrals given by the integral invariant relations as the reference values of the correction,the implementation of the new manifold correction method is not hindered.We test the numerical performance of the new manifold correction method in three different quasi-Keplerian problems.For the post-Newtonian two-body problem,compared with the basic integrator fourth-order Runge-Kutta method(RK4),the new method not only significantly improves the accuracy of the five time-varying orbital elements except for the mean anomaly,but also suppresses the errors of the mean anomaly and the position that increase with the square of time to a linear increase,which are similar to Fukushima's method.The method we used to calculate the eccentric anomaly has the best numerical performance when different speed of light magnitude is used to obtain different magnitude of the post-Newtonian effect.For the dissipative two-body problem,the new method not only achieves better accuracy than the basic integrator and the fourth-order implicit midpoint rule,but also has the highest calculation efficiency.The position errors calculated by the new method increase linearly with time,which is consistent with the performance of the fourth-order implicit midpoint rule(IM4).For the five-body problem of the outer solar system,the fourth-order Runge-Kutta method,as an artificial dissipative algorithm,has obvious secular changes in the semi-major axis,z-direction angular momentum and eccentricity of Jupiter over 10~7years of integration time.The corresponding orbital elements and quasi-integrals remain bounded even if the integration time is 100 million years.When the basic integrator used is RKF5(6)with fixed step size,the accuracy of the new method M1 is better than that of the original algorithm and the second-order symplectic Wisdom-Holman(WH)method.However,the error slope of M1 is smaller than that of RKF5(6)and larger than that of WH method.Since the accuracy of the positions and velocities of each celestial body have been improved after correction,the errors of the total energy and total angular momentum calculated by M1 is only one ten thousandth of the uncorrected solutions.In this paper,we propose a new form of manifold correction method for the perturbed two-body problem.This method is simple to operate and is suitable for various one-step integrators,and only requires a small amount of extra computation time.The new method can deal with the two-body problem of various perturbation types,whether they are conservative or dissipative systems.