Hiroshi Nakai

Symplectic integrators and their application to dynamical astronomy

AbstractSymplectic integrators have many merits compared with traditional integrators:- the numerical solutions have a property of area preserving,- the discretization error in the energy integral does not have a secular term, which means that the accumulated truncation errors in angle variables increase linearly with the time instead of quadratic growth,- the symplectic integrators can integrate an orbit with high eccentricity without change of step-size. The symplectic integrators discussed in this paper have the following merits in addition to the previous merits:- the angular momentum vector of the nbody problem is exactly conserved,- the numerical solution has a property of time reversibility,- the truncation errors, especially the secular error in the angle variables, can easily be estimated by an usual perturbation method,- when a Hamiltonian has a disturbed part with a small parameter c as a factor, the step size of an nth order symplectic integrator can be lengthened by a factor ε−1/n with use of two canonical sets of variables,- the number of evaluation of the force function by the 4th order symplectic integrator is smaller than the classical Runge-Kutta integrator method of the same order. The symplectic integrators are well suited to integrate a Hamiltonian system over a very long time span.

Analytical Solution of the Kozai Resonance and its Application

When Kozai (1962) studied the secular resonance of asteroids, he found the so-called Kozai resonance and expressed the analytical solution with the use of Weierstrass ℘. Here we discuss the case where the disturber is outside a disturbed body and give the analytical solution of the eccentricity, the inclination and the argument of pericenter with the use of the Jacobi elliptic functions, which are more familiar than the Weierstrass ℘. Then we derive the Fourier expansion of the longitude of node and the mean anomaly. The analytical expressions obtained here can be used for any value of the eccentricity and the inclination. Finally we applied these analytical expressions to several dynamical systems – Nereid, that is a highly eccentric satellite of Neptune, and newly discovered retrograde satellites of Uranus.

The Librating Companions in HD 37124, HD 12661, HD 82943, 47 Ursa Majoris, and GJ 876: Alignment or Antialignment?

We investigated the apsidal motion for the multiplanet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 UMa, and HD 82943 separately undergo apsidal alignment or antialignment. However, the companions of GJ 876 and v And are in apsidal lock only about 0degrees. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets for a certain system are in libration or circulation.

Secular perturbations of fictitious satellites of uranus

Secular perturbations of fictitious satellites that are initially circular and in the equatorial plane of Uranus are discussed. Satellites located in the region where the solar perturbation is dominant become highly eccentric and inclined with respect to the equator, and have a possibility to collide with Uranus. Satellites located in the region where the oblateness perturbation is dominant keep the original eccentricity and the inclination. A scenario of a possible extinction of outer satellites of Uranus is also discussed.

Motions of the perihelions of Neptune and Pluto

Five outer planets are numerically integrated over five million years in the Newtonian frame. The argument of Plutos perihelion librates about 90 degrees with an amplitude of about 23 degrees. The period of the libration depends on the mass of Pluto: 4.0×106 years forMpluto=2.78×10−6Msun and 3.8×106 years forMpluto=7.69×10−9Msun, which is the newly determined mass. The motion of Neptunes perihelion is more sensitive to the mass of Pluto. ForMpluto=7.69×10−9Msun, the perihelion of Neptune does circulate counter-clockwise and forMpluto=2.78×10−6Msun, it does not circulate and the Neptunes eccentricity does not have a minimum. With the initial conditions which do not lie in the resonance region between Neptune and Pluto, a close approach between them takes place frequently and the orbit of Pluto becomes unstable and irregular.

Secular perturbations of asteroids in secular resonance

When the precessional rate of the orbital plane of an asteroid is nearly equal to that of Jupiter, the orbital inclination of the asteroid changes quite largely due to this near equality of their precessional rates, which is called a secular resonance. In the vicinity of the exact resonance the difference of their longitudes of nodes librates with quite a long period of order of 1×106 yr. In this paper we treat this secular resonance by a method of semianalytical secular perturbations with use of numerical averaging for both non-resonant and resonant asteroids and show that the results by the semi-analytical treatment agrees qualitatively with those obtained by direct numerical integrations of asteroids orbits.

The Apsidal Antialignment of the HD 82943 System

We perform numerical simulations to explore the dynamical evolution of the HD 82943 planetary system. By simulating diverse planetary configurations, we find two mechanisms of stabilizing the system: the 2:1 mean motion resonance (MMR) between the two planets can act as the first mechanism for all stable orbits. The second mechanism is a dynamical antialignment of the apsidal lines of the orbiting planets, which implies that the difference of the periastron longitudes Θ3 librates about 180° in the simulations. We also use a semi-analytical model to explain the numerical results for the system under study.

The stabilising mechanism of the HD 82943 planetary system

We carry out simulations to investigate the dynamics of the HD 82943 planetary system with two resonant Jupiter-like planets, and to reveal possible stabilizing mechanism for the system. By following different coplanar configurations in the neighborhood of the best-fit orbits, we find that all the stable cases are involved in the 2:1 mean motion resonance and that the alignment of the periastra of the two planets also plays important part in the secular orbital evolution, indicating that these two kinds of mechanisms could be responsible for the dynamics of the system under study.

Note on secular perturbations between a retrograde body and a prograde body

Hirayama (1927) studied secular perturbations between a retrograde body and a prograde body by considering that the mean motion of the retrograde body is negative. In this paper we discuss the same problem by measuring angle variables from the departure point and keeping the mean motions positive for both the retrograde body and the prograde body, and compare the analytical solutions with numerically integrated orbits.

Secular perturbations between a retrograde body and a prograde body

Abstract With use of secular perturbation theory, we can discuss a long-periodic behaviour of a system of a central body and prograde bodies that have small eccentricities and inclinations (Brouwer and Clemence, 1961). Classical secular perturbation theory, however, cannot treat a system of one retrograde body and one prograde body by simply interpreting the inclination of the retrograde body as nearly 180 degrees for the following reasons: 1. 1)the classical development of the disturbing function is not applicable to the retrograde case, 2. 2)the main term in the secular part between the retrograde body and the prograde body is different from that between two prograde bodies, 3. 3)the linearized equations of motion for the retrograde body have a different form from that for the prograde body. At first we study a system of a central body and one retrograde body and one prograde body, and extend the secular perturbation theory for a system of a central body and two bodies to a system of a central body and an arbitrary number of bodies that consist of both retrograde and prograde bodies. Our theory is compared with numerically integrated orbits. The theory presented here is applicable to the system of Jupiter and Saturns satellites, and the system of outer planets which includes a hypothetical retrograde planet X.