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The Theory of Motion and Ephemerides of The Second Neptunian Satellite Nereid
en
Saad Abdel-Navy Saad
サード アビデルナビ サード
Saad Abdel-Navy Saad
総研大甲第492号
The problem of obtaining accurate ephemerides for the second Neptunian satellite Nereid has intrigued many astronomers since its discovery by Kuiper in 1949. That is because of its bizarre orbit. The satellite's orbit has unusually large eccentricity (e ～0.75) which is considered as the most eccentric known natural satellite in the solar system. This very elongated orbit renders the usage of the classical methods for expanding the disturbing function in terms of the eccentricity, that is because of the slow convergence of the power series solutions especially at higher orders.In this work we aim to study the dynamical motion of the second Neptunian satellite Nereid using both analytical and numerical methods. We construct an analytical theory of the motion of a highly eccentric Nereid which accurately represents a real satellite system, then we pose emphasis upon comparison with numerical integration of the equations of motion. The theory is elaborated by the use of Lie transformation approach advanced by Hori's device. This method enables us to express the relations between the osculating and the mean elements in an explicit form instead of the implicit form arised by Poincare'-von Zeipel's approach. By the virtue of Hori's perturbations method, we can also get the inverse transformations easily. The main perturbing forces on Nereid which come from the solar influence are only taken into account through the present theory. The disturbing function is developed in powers of the ratio of the semimajor axes of the satellite and the Sun. To avoid the slow convergence of the power series solution, the disturbing function is put in a closed form with respect to the eccentricity of Nereid. In addition, replacing functions of the true anomaly by expressions involving the mean anomaly is also avoided, and the eccentric anomaly of Nereid has been adopted as independent variable. The present theory includes secular perturbations up to the fourth order, short and long period perturbations up to the third order and small parameter e (which defines the ratio between the orbital period of Nereid and that of Neptune)～6 × 10-3. The results of the present theory satisfy the required accuracy for future observations. We intend to develop this theory to be applied on the retrograde satellites of the major planets. The dissertation is organized as following: In chapter 1 we give a general introduction which includes the advantages of the use of the analytical techniques and their expected outcome. A review on Nereid, the second Neptunian satellite, and its enigmatic according to different sources are summarized. Chapter one contains also a section about the classification of natural satellites according to their orbits and perturbing forces. Then we pose the motivation and aim of this study. Chapter 2 contains the method that we have used, equations of motion and the disturbing function. Hori's perturbation method is introduced briefly, and some of the merits and demerits of canonical methods in celestial mechanics have been shown. This chapter includes also procedures for obtaining the osculating orbital elements starting from the mean elements and conversely. We implement each procedure for digital computations by constructing a computational algorithm described by its purpose, input and its computational sequence. In chapter 3 we dealt with the circular planar restricted three-body problem. In this case, the inclination of Nereid to the orbital plane of Neptune is zero. The osculating orbital elements of the fictitious Nereid are evaluated and given in figures. The results are compared with those computed by the numerical integration of the equations of motion. The residuals are tabulated and showed also by figures. At the end of this chapter we give a short note about d'Alembert characteristics which permit the validity of the analytical expressions based on Lie transform approach.Chapter 4 is devoted to the circular nonplanar restricted three-body problem. In this case we take the inclination of Nereid into account and deal with the nonplanar solution for a real Nereid. The analytical expressions of the short, intermediate and long periodic perturbations are evaluated. After elimination of the short and intermediate terms, the Hamiltonian system equations are solved in e, I and ωusing Jacobi's elliptic function (Kinoshita and Nakai, 1999), whereas the longitude of ascending node and the mean anomaly are expressed in Fourier series expansion. By this solution we got the mean elements which are used for evaluating the osculating orbital elements and ephemerides of Nereid. All these processes are summarized in a computational algorithm and carried out by the powerful MATHEMAT- ICA software package. Moreover, the analytical expressions are transformed into FORTRAN format and programmed to be easy to handle.We compared the analytical results with those computed by the direct numerical integration of the equations of motion for short and long periodic perturbations. As a result of this comparison, the global internal accuracy of the present theory reached 0.3 km in the semimajor axis, 10 - 7 in the eccentricity and 10 - 5 degree in the angular variables over a period of several hundred years. The behaviour of the orbital motion of the satellite is exhibited in analytical expressions, tables and figures. The way of comparison is discussed briefly. Finally, we close this research by discussion and conclusions. By this end we provide to the observers an efficient analytical theory, capable of generating accurate ephemerides for the motion prediction of highly eccentric Nereid.
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