Velocity Scaling Correction Method For Non-conservative And Dissipative Restricted Three-body Problems

The velocity scaling method of Ma et al.,as an extension of Nacozy’s manifold correction scheme,can frequently take the solution of a numerical integration back to a surface determined by an integral of the equations of motion.First,an elliptic restricted three-body Hamiltonian of the Sun,major and minor planets in a rotating frame is explicitly dependent on time and,therefore,is not a conserved quantity.In this case,there is no Jacobi conservative integral available but there is a Jacobi non-conservative integral.This seems to be an obstacle to applying the velocity scaling correction method.Here are two points about an effective way to overcome this obstacle.First,because of the Hamiltonian having momentum-and coordinatedependent terms associated with the contributions from the non-inertial frame,a scaling correction factor should be used to act on the velocities in the Jacobi non-conservative integral although the momenta are integration variables.Secondly,at each integration step,the value of the Hamiltonian obtained from an integral invariant relation is referred to as a more accurate reference value;the scaling factor versus the velocities is given by constraining the numerical solution to remain on the Jacobi non-conservative integral along this reference value.Numerical experiments show that a lower-order non-symplectic algorithm plus the velocity scaling scheme demonstrates good numerical performance in suppressing the rapid growth of integration errors,compared to the lower-order uncorrected algorithm.The correction scheme is powerful for eliminating spurious non-physical chaos due to integration errors.It is found that a larger eccentricity of the giant planet will increase the possibility of chaos or escape of the asteroid.Second,we survey the effect of radiation pressure,Poynting-Robertson drag and solar wind drag on the motion of negligible mass dust grains,subject to the gravities of the Sun and Jupiter moving in circular orbits.The effect of the dissi-pative parameter on the locations of_ve Lagrangian equilibrium points is estimated analytically,approximately.Instability of the triangular equilibrium point L4 is also shown analytically due to the presence of drag forces.In this case,the Jacobi constant varies with time,whereas its integral invariant relation still provides a chance for the applicability of the conventional fourth-order Runge-Kutta algorithm combined with the velocity scaling manifold correction scheme.As a result,the corrected algorithm suppresses signi_cantly the extent of arti_cial dissipation effects caused by the uncorrected one.Irrespective of whether an orbit is chaotic or not in the conservative problem,its stability time for the dissipative forces included is apparently longer in the corrected case than in the uncorrected case.In spite of the arti_cial dissipation ruled out,the drag dissipation leads to an escape of grains.Numerical evidences also support that more orbits near the triangular equilibrium point L4 escape with the integration time increasing.