Manifold Corrections Of N-Body Simulations In The Solar System And Their Applications

It is very clear that a reliable numerical method is the basic tool for investigating nonlinear dynamics. Because traditional numerical methods are full of artificial dissipation, and symplectic integrator is limited to its application, it is an essential task to develop the traditional numerical methods at present.Extending Nacozy's manifold correction scheme and using the integral invariant relation, we propose three new manifold correction methods to keep slow-varying quantities or quasi-integrals in n-body problems in the solar system. Our aim is to improve the accuracy of the related elements.First, as variants of the velocity-position single scaling method of Fukushima, a new velocity-scaling method and a new position-scaling method for correcting the varying Keplerian energy of each body are presented. We use one scale factor for the integrated velocity or position so as to satisfy a rigorous equation related the Keplerian energy. It is easy to get the value of scale factor from the equation. The value of the after-corrected velocity or position is regarded as the initial condition at next step. From numerical tests for a pure Keplerian model and the three-body problem consisting of the Sun, Jupiter and Saturn, we find that like other existing methods including the method of Fukushima and that of Wu et al., the two new methods are almost the same effectiveness in significantly improving the orbital semi-major axis or mean anomaly. In particular, compared with the position method, the new velocity method is more convenient in application.Second, for each body of the solar system, there are two slowly varying quantities or quasi-integrals, Keplerian energy and Laplace integral, which are closely associated with the orbital semi-major axis and eccentricity, respectively. To correct numerical errors of them, we develop a new velocity manifold correction method, in which a 2×3 matrix is made up of partial derivatives of Keplerian energy and Laplace integral. In terms of Nacozy's manifold correction ideal, it is easy to get the control term, which is added to the numerical solution. As a result, the new scheme and the rigorous dual scaling method of Fukushima are almost equivalently in raising the precision of the semi-major axis and the eccentricity. Especially, our new method is superior in the correction of eccentricity if the adopted integrator provides a sufficient precision.Third, unlike the second work, in this time the correction we consider is to all orbital elements. For each object of an n-body problem in the solar system, orbital elements except the mean anomaly are directly determined by five independently integrals or quasi-integrals, which include the Keplerian energy, the three components of the angular momentum vector, and the z-component of the Laplace vector. It should be emphasized that the mean anomaly depends on the mean motion specified by the Keplerian energy. Compared with the second work, the new technique is more complicated in construction. Numerical tests show that the new method is almost the same as the linear transformation method of Fukushima. Both of them are good devices to correct all orbital elements of each object.Finally, the velocity-scaling method is used to study the nonlinear evolution of a closed Friedman Robertson Walker (FRW) universe with a conformally coupled scalar field. Our numerical scheme conserves almost the level of energy, and provides higher precision to numerical solution. By the numerical tool, we investigate the dependence of chaos on two parameters of cosmological constant and self-interacting coefficient in FRW model. It is found that Poincare plots for the two parameters less than 1 are almost the same as those in the absence of the cosmological constant and self-interacting terms. For energies below the energy threshold 0.5, an abrupt transition to chaos occurs when at least one of the two parameters is 1. However, for energies larger than the energy threshold, the strength of chaos does not increase in this sense. For other situations, chaos is weaker, and even disappears as the two parameters grow.