Advances in Astronomy

Volume 2016, Article ID 8945090, 23 pages

http://dx.doi.org/10.1155/2016/8945090

## High-Order Analytic Expansion of Disturbing Function for Doubly Averaged Circular Restricted Three-Body Problem

National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka, Tokyo 181-8588, Japan

Received 3 June 2016; Revised 18 August 2016; Accepted 19 September 2016

Academic Editor: Elbaz I. Abouelmagd

Copyright © 2016 Takashi Ito. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio between perturbed and perturbing bodies in the inner case , and up to the fifteenth order in the outer case . The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the doubly averaged disturbing function yields a result that agrees well with the numerical quadrature and with the numerical integration. Local extremums of the doubly averaged disturbing function are quantitatively reproduced by the high-order analytic expansion even when is large. Although the analytic expansion is not applicable in some circumstances such as when orbits of perturbed and perturbing bodies cross or when strong mean motion resonance is at work, our expansion result will be useful for analytically understanding the long-term dynamical behavior of perturbed bodies in circular restricted three-body systems.

#### 1. Introduction

In the long tradition of celestial mechanics, the restricted three-body problem has occupied a fundamental role. In this problem, the mass of one of the three bodies is assumed to be small enough so that it does not affect the motion of the other two bodies. The restricted three-body problem is often considered on a rotating coordinate where central body and perturbing body are always located on the -axis. See Szebehely [1] for more general characteristics of the restricted three-body problem.

Among many variants of the restricted three-body problem, its circular version called the circular restricted three-body problem (hereafter referred to as CR3BP) has been studied particularly well, and it makes a basis for understanding solar system dynamics and many other fields in celestial mechanics. In this system, the perturbing body lies on a circular orbit around central body. As is well known, the degree of freedom of CR3BP becomes unity and the system turns into integrable once we average the disturbing function of the system by mean anomalies of perturbed and perturbing bodies. Theories of the so-called classical Lidov–Kozai cycle have been developed based on the integrable characteristics of the doubly averaged CR3BP [2–4], where stationary points of argument of pericenter around appear when the vertical component of the angular momentum of perturbed body is smaller than a certain value. The present paper deals with the doubly averaged CR3BP.

In the classical theory of the Lidov–Kozai cycle, the doubly averaged disturbing function is expanded using the Legendre polynomials of even orders. Putting , where is the semimajor axis of perturbed body and is that of perturbing body, only the lowest-order terms up to are considered in many of the studies along this line (e.g., [5–9]); it is the quadruple-order approximation. Recent studies of the so-called eccentric Lidov–Kozai mechanism (e.g., [10–13]) that deal with the eccentric restricted three-body problem (ER3BP), where the orbit of perturbing body has a finite eccentricity, are based on the octupole-order approximation of disturbing function up to . It is now getting better known that the inclusion of octupole terms in the disturbing function substantially changes the dynamical behavior of ER3BP. Even in CR3BP, the quadruple-order approximation is not accurate enough when is large. To eliminate this shortcoming, in the early days, Kozai [3] expanded the doubly averaged disturbing function of the inner CR3BP up to . More recently, Laskar and Boué [14] calculated analytic expansions of the general three-body disturbing function together with a practical method to compute the Hansen coefficients. Laskar and Boué [14] explicitly showed expressions of secular disturbing function up to for planar problems and up to for spatial problems.

In the present paper we will show specific expressions of the analytic expansion of the doubly averaged spatial disturbing function up to for the inner CR3BP and up to for the outer CR3BP using the Legendre polynomials. As most readers are aware, very wide varieties of studies have been already done on the analytic expansion of the doubly averaged disturbing function of CR3BP. Compared with previous literature, the present paper intentionally aims to be rather expository. The major purpose of this paper is to explicitly show expressions of the high-order analytic expansion of the doubly averaged disturbing function for CR3BP so that readers with interest can consult the high-order terms without going through algebraic manipulation by themselves. We also aim at analytically reproducing local extremums that secular disturbing function of CR3BP intrinsically has, particularly when or is large. Thus, even on a very basic subject like this, we have felt it advisable to give more details than would otherwise be necessary.

In Section 2 we give a brief description of the disturbing function of the three-body problem that we consider in the present paper. Section 3 goes to double averaging and analytic expansion of the disturbing function for CR3BP: general procedure (Section 3.1) and specific forms (from Sections 3.2 to 3.8). In Section 4 we show a comparison between the results obtained by the analytic expansion and by numerical quadrature. We also carried out direct numerical integration of equations of motion for comparison and show its result in Section 5. Section 6 is devoted to summary and discussion.

For readers’ convenience, before getting into the main sections let us quickly write down the basic equations of motion of the system that we deal with in the present paper. The differential equation that we will consider is the simple classical Newtonian equation of motionwhere indicates the position vector of the perturbed body and relates to the central mass. The disturbing function that plays a central role in this paper is denoted as . As for a literal definition of the doubly averaged disturbing function, particularly its direct part, we use the following one:where is related to the mass of perturbing body, and denote mean anomaly of perturbed and perturbing bodies, respectively, and is the osculating distance of the two bodies in space. Consult later sections for detailed definitions of the variables in the above equations. Needless to say, the considered system contains only three bodies.

#### 2. Disturbing Function

In the present paper we categorize CR3BP in two cases: (i) the inner case where the orbit of the perturbed body is located inside that of a perturbing body , such as the Sun-asteroid-Jupiter system, and (ii) the outer case where the orbit of the perturbed body is located outside that of a perturbing body such as the Sun-Neptune-TNO system (TNO = Trans-Neptunian Object). The coorbital case is out of the scope of the present paper.

Following the long-term convention of celestial mechanics, in the present paper we express the disturbing function of CR3BP in relative coordinates where the origin is located on the primary body (Figure 1(a)). In this coordinate system, the disturbing function that describes the perturbation on the motion of an object with mass due to the motion of another mass has the following general form (e.g., [15, p. 228]):where with the gravitational constant , is the position vector of the mass with respect to the central body, is the position vector of the mass with respect to the central body, and . In what follows we will consider only the first term of the right-hand side of (3) which is often referred to as the direct part. The second term is called the indirect part. As is well known, the indirect part makes no contribution to long-term dynamics of the system because it vanishes after the double averaging procedure, unless nonnegligible mean motion resonances are at work and we cannot simply employ the double averaging procedure.