Implementation of the velocity scaling method for elliptic restricted three-body problems

Abstract

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. 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 coordinate-dependent 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.