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Advances in Astronomy
Volume 2016, Article ID 8945090, 23 pages
http://dx.doi.org/10.1155/2016/8945090
Research Article

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 [24], 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., [59]); it is the quadruple-order approximation. Recent studies of the so-called eccentric Lidov–Kozai mechanism (e.g., [1013]) 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.

Figure 1: A schematic illustration of the three-body configuration considered here in two kinds of coordinate. (a) Relative coordinate. Both the vectors and originates from the central mass denoted as . (b) Jacobi coordinate. The vector is originated from the primary mass , and the vector is originated from the barycenter of the primary mass and the secondary mass denoted as .

When designating as the angle between the vectors and , it is also well known that on the right-hand side of (3) can be expanded using the Legendre polynomials aswhen (i.e., the inner case), andwhen (i.e., the outer case). Once again, it is well known that the terms of and in (4) and (5) do not contribute to secular motion of the bodies, as they will vanish or become constant after the double averaging procedure. Hence in the remaining part of this paper we will just consider terms with in (4) and (5).

Readers find the expressions of the disturbing functions (3), (4), and (5) and their derivations in many textbooks such as Brouwer and Clemence [16], Danby [17], or Murray and Dermott [15]. In the inner case, the direct part of the disturbing function can be derived also in a more general way. Consider a general three-body system with three masses: primary , secondary , and tertiary . Now we use the Jacobi coordinate (e.g., [18]): measuring ’s position vector from , measuring ’s position vector from the barycenter of and , and is the angle between the vectors and . Naturally and have different origins, and the angle is different from in general (see Figure 1(b)). We assume . In this coordinate system, the equations of motion of and become (see [16, 19], for detailed derivation)whereare reduced masses used in the Jacobi coordinate system (e.g., [20, 21]), and is the common force functionwhereis the mass factor. Using the force function , this system can be written in a canonical form governed by a Hamiltonian. By expressing and as the semimajor axis of the orbits of the secondary and the tertiary (e.g., [2225]), the Hamiltonian becomes

The first and the second terms of in (10) drive the Keplerian motion of the secondary and tertiary mass, respectively. Note that the third term which represents the mutual interaction of the secondary and the tertiary does not include terms of or . This is a typical consequence of the use of the Jacobi coordinate that separates the motions of three bodies into two separate binaries and their interactions by a single infinite series.

Now, let us think about the limit where is infinitesimally small; this would correspond to the inner R3BP. In this case, we divide the force function in (8) by in (7) for normalization by mass before taking the limit. Then the third term of becomes

Now we take the limit of , and the position of in Figure 1(b) gets overlapped with the position of . Thus we can replace for , for , for , and for and will end up with an expression equivalent to the direct part of the disturbing function of the inner case written in the relative coordinate (4).

On the other hand, deriving the form of the disturbing function of the outer case written in the relative coordinate (5) by taking the mass-less limit of Hamiltonian (10) is difficult, if not impossible. In the outer case, the origin of the position vector of the tertiary is the barycenter of the primary and secondary ( in Figure 1(b)). But would not be overlapped with the position of the primary regardless of the value of the tertiary’s mass , unless . Therefore, in the present paper, we will not mention the conversion of the general three-body Hamiltonian into the outer disturbing function written in the relative coordinate. In modern celestial mechanics, more and more methods for expanding disturbing function without using the relative coordinate are becoming available (e.g., [14, 26, 27]).

3. Doubly Averaged Disturbing Function for CR3BP

3.1. General Form

From (3) and (4), the direct part of the disturbing function for the inner CR3BP where becomes as follows:where we ignore the term of . In (12) we also ignore all the terms including the odd Legendre polynomials because they all vanish after the averaging procedure using mean anomaly of the perturbing body. Note that in the remaining part of this paper we will not consider the indirect part of the disturbing function either, as they do not have any secular dynamical contributions in nonresonant systems. Therefore we just use the variable for representing the entire part of the disturbing function.

Assuming there is no major resonant relationship between the mean motions of perturbed and perturbing bodies, we now try to get the double average of (12) over mean anomalies of both the bodies. Nonexistence of a resonant relationship means that the mean anomalies of perturbed and perturbing bodies (referred to as and in what follows) are independent from each other. The procedure to carry out double averaging of is straightforward as follows: Let us pick up the th term of and name it as . We have

First we average by mean anomaly of the perturbing body , aswhere

The angle is expressed by orbital angles through a relationship (e.g., [3, Eq. () in p. 592]) where , are true anomalies of the perturbed and perturbing bodies, , are arguments of pericenter of the perturbed and perturbing bodies, and is their mutual inclination measured at the node of the two orbits. We choose the orbital plane of the perturbing body as a reference plane for the entire system, and then and can be measured from the mutual node. Note that is not actually defined in CR3BP. Therefore, in (16) we regard as a single, fast-moving variable when we carry out averaging of (15). Practically, we can simply replace for in the discussion here.

To obtain of (15), we calculate the average of by asThen we average of (14) by mean anomaly of the perturbed body , as

If we switch the integration variable from to eccentric anomaly , (18) becomes

We can obtain the doubly averaged disturbing function for the outer CR3BP in the same way as above. From (3) and (5), the direct part of the disturbing function for the outer CR3BP becomes as follows:

Note that our definition of for the outer case (5), hence also in (20), may be different from what is seen in conventional textbooks (e.g., [15, Eq. () in p. 229]): Roles of the dashed quantities may be the opposite. This difference comes from the fact that conventional textbooks always assume , while we assume for the outer problem. This is because we make it a rule to always use dash for the quantities of perturbing body, whether it is located inside or outside the perturbed body.

Similar to the procedures that we went through for the inner CR3BP, we again assume that there is no major resonant relationship between mean motions of perturbed and perturbing bodies in the outer CR3BP. We then try to get the double average of over mean anomalies of both the bodies. Let us pick the th term of in (20) and name it . We have

First we average by mean anomaly of the perturbing body , aswhere is already defined in (15).

Then we average in (22) by mean anomaly of the perturbed body , as

If we switch the integration variable from to true anomaly , (23) becomes

Note that has the order of as in (23) and (24), not .

In Sections 3.2 to 3.8 we show the expressions of in (17), in (15), the Legendre polynomial in its original form, in (19), and in (24) for . We used Maple™ for algebraic manipulation to obtain the series of expressions. Note that in what follows we use instead of because it generally makes the formulas simpler. For this reason, some of the expressions look apparently different from what was presented in the previous literature in spite of their equivalence.

3.2.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .

We just describe the resulting expressions of the expansion as follows: Let us emphasize again that the dashed quantities such as and are those of the perturbing body, whether its orbit is located inside or outside of the orbit of the perturbed body.

The expression in (28) shows the leading term of the doubly averaged disturbing function that causes the classical (circular) Lidov–Kozai cycle in the inner CR3BP that we have often seen in the previous literature. Meanwhile, the expression in (29) shows the leading term of the doubly averaged disturbing function for the outer CR3BP, but somehow we do not see it often. We should note that the leading term of the doubly averaged disturbing function for the outer problem (29) does not contain dependence on . The -dependence in the doubly averaged outer CR3BP first shows up in the next order: . This is why the Lidov–Kozai mechanism for the outer CR3BP is more subtle than the inner one, particularly when is small, and perhaps this is why we rarely see the expression in the literature.

3.3.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .

Note that we now see the -dependence of the doubly averaged disturbing function for the outer CR3BP in the expression of (34).

3.4.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .

3.5.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .

3.6.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .

3.7.

At this order, the corresponding component of the disturbing function for the inner problem is of , and that for the outer problem is of .