Dynamical Systems: An International Journal, Vol. 20, No. 1, March 2005, 23–44
The Restricted Hill Full 4-Body Problem: application to spacecraft motion about binary asteroids D. J. SCHEERES* and J. BELLEROSE University of Michigan, 1320 Beel Avenue, 3048 FXB Bldg, Ann Arbor, MI 48109-2140, USA (Received 7 June 2004; in final form 5 August 2004)
The Restricted Hill Full 4-Body Problem (RHF4BP) models the motion of a spacecraft or particle about two mutually orbiting distributed bodies in the tidal gravity field of a larger body. The practical application of this problem is to the motion of a spacecraft or particle about a binary asteroid system. Current estimates are that up to 16% of near-Earth asteroids (NEAs) may be binary asteroids, thus this is an extremely relevant topic for future missions to NEAs. It is also an interesting topic from an academic point of view, as this problem integrates four classical problems of astrodynamics: the Hill problem, the restricted 3-body problem, the non-spherical orbiter problem, and the full 2-body problem. In this paper, we define the RHF4BP in terms of these classical models and present results that the RHF4BP inherits from these classical problems. Some initial steps towards the analysis of this problem are also given, relating to the stability of motion about the Lagrange points in the Restricted Full 3-Body Problem and the Restricted Hill 4-Body Problem, both one-stage simplifications of the RHF4BP.
1. Introduction Spacecraft motion about asteroid binaries is a topic of interest for future missions to near-Earth asteroids (NEAs). Current estimates are that up to 16% of NEAs are binaries [1], meaning that there is a good chance that future mission targets will be binaries. Additionally, the number of Main-belt asteroids has grown at an exceedingly rapid pace within the last few decades [2], raising many interesting questions of how ejecta evolve in these systems. Finally, there is a fundamental interest in sending a spacecraft to a binary, as they provide a very dynamic environment within which many important issues of asteroid science can be investigated. Solutions to the problem of orbital motion about an asteroid binary, in its most general form, are difficult and require modelling of the non-spherical mass distributions of the bodies, the mutual orbital and rotational motion of the
*Corresponding author. Email:
[email protected] Dynamical Systems: An International Journal ISSN 1468–9367 print/ISSN 1468–9375 online # 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/1468936042000281321
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bodies, and the perturbative influence of the sun on the system. Orbits in these systems can be very complex, and in general include orbits bound to either of the asteroid bodies, exchange trajectories between the bodies and with the exterior region, and orbits isolated in the exterior region about both bodies. When properly formulated, the binary environment integrates four classical problems of celestial mechanics into a single environment: the Hill problem, the restricted 3-body problem, the non-spherical orbiter problem (what we call the restricted full 2-body problem in this paper), and the full 2-body problem. In this paper, we pose the most general problem and a number of simpler, although still difficult, approximations of this problem that describe the range of motions that can exist. For these more specific problems, we can pose equations of motion and study certain aspects of their solutions. As a point of focus in this paper, we also investigate how these additional effects modify the classical stability results for the Lagrange points in the restricted 3-body problem. Investigation of the stability of these points is of prime interest for a space mission, as it would be quite attractive to park a spacecraft in the vicinity of one of these points to observe the system. We find, in general, that the effect of the solar tide perturbation and the effect of a non-spherical central body is to further restrict the stability of motion about these points.
2. The Restricted Hill Full 4-Body Problem 2.1 Problem statement In its most general form, the problem we consider is the motion of a massless particle (a spacecraft) attracted by three massive bodies: the two components of the binary asteroid and the sun (figure 1). Hence, it is a Restricted 4-Body Problem (R4BP) in general. Additional generality must be added, however, since two of the masses (the binary asteroid) may have non-spherical mass distributions which leads to coupling between their translational and rotational motion. Thus, to specify the motion of the massive bodies we must also know the translational and rotational motion of both mass distributions, which are in turn orbiting about the sun. Both the solar and the non-spherical asteroid shape perturbations can be significant, and modify the dynamics in non-trivial ways.
Figure 1.
The Restricted Hill Full 4-Body Problem.
Restricted Hill Full 4-Body Problem
25
In the following, we introduce terminology which will help us keep track of the different models we must consider. First is the well-known ‘restricted’ terminology, which indicates that one of the bodies in question (the spacecraft in general) has negligible mass, and does not influence the motion of the other bodies. Thus, whenever a problem is a Restricted n-Body Problem, we can immediately separate the problem into two parts, the motion of the n 1 bodies with mass (or the (n 1)Body Problem) and the motion of the 1 body with negligible mass attracted to the n 1 bodies as they move about each other. Another useful restriction that applies to our system is the ‘Hill approximation’ which leads, in our case, to the Restricted Hill 4-Body Problem (RH4BP). This indicates that n 1 of the bodies (including the massless body) are relatively close to each other and in orbit about a more massive body (in our system, the sun). Further simplification is found if the centre of mass of the cluster of close bodies is in a circular orbit about the massive body, although this assumption is not necessary to apply the Hill approximation [3]. Such a model was derived previously in [4] for application to the motion of a spacecraft in the Earth–Moon–Sun system. We find that this same model can be applied to motion about a binary asteroid, under certain approximations. The next term we use is ‘full-body’, which indicates that one or more of the bodies has a non-spherical mass distribution and thus has non-trivial rotational dynamics that are coupled with its translational dynamics. This appelation is not as familiar, and is introduced to account for the non-spherical shapes that asteroids and other small solar system bodies have, in general. This also greatly complicates the motion of the system, and leads to a host of interesting dynamical effects which are a current subject of study [5–7]. So, to give our model problem a name, we call it the Restricted Hill Full 4-Body Problem (RHF4BP). This indicates that it studies the relative motion of three bodies, one massless and the others with non-spherical mass distributions, which are in a mutual orbit about the sun.
2.2 Heirarchy of approximations To properly pose this problem, however, we must first solve for the motion of the massive bodies about each other, the Hill Full 3-Body Problem, i.e. the motion of non-spherical binary asteroids about each other within the tidal field of the sun. This specific problem has not been considered in the literature, to the best of our knowledge, and is a fundamental problem for future study. What has been studied are the next simplifications to this problem: the Hill 3-Body Problem (H3BP) and the Full 2-Body Problem (F2BP). The H3BP has been extensively studied in the past (c.f. [8] for a recent review and rigorous derivation of this problem), and its dynamics (although non-integrable) are understood at a relatively sophisticated level. The F2BP has not been studied at the same level, however. A comprehensive introduction to this problem was given by [9], and some recent work has uncovered basic stability results, whose implications are still being evaluated [5, 6, 10]. The general F2BP is difficult due to the large number of degrees of freedom in the system (12) and to the presence of an arbitrary mutual potential between the two bodies. To circumvent this we often consider the Sphere Restriction to the F2BP which assumes that one of the two massive bodies is a sphere. This approximation
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Figure 2.
Heirarchy of approximations.
allows for many analytical results, as the mutual potential between the two bodies then has the same basic form as the gravitational potential for the non-spherical body, a model that is well understood. Of more importance, with this restriction the system can be reduced to purely relative motion between the two mass distributions without having to solve for the attitude of either body [9]. Starting from our knowledge of motion in these two fundamental problems, we will derive the equations of motion of a particle in each of these systems, leading to the RH4BP and the Restricted Full 3-Body Problem (RF3BP). The resulting equations are both approximations of the more general RHF4BP. Each model will be properly introduced, followed by a discussion of some specific results on the stability of motion associated with the Lagrange points. Much of what we discuss here is new, and serves as a requisite for the study of the RHF4BP. Figure 2 outlines the sequence of approximations that link these different models to each other. The starred boxes represent problems not intensively studied to date. The ovals represent problems where some limited study has been performed. The rectangular boxes represent problems that are well understood and have been intensively studied in the past. The shaded enclosures are restricted problems, which are focused on the motion of a massless particle within some system. The clear enclosures are the problems that stand behind the restricted problems, and which must be solved first before we can pose the restricted problems. This diagram is useful to help keep track of the different model approximations we make and to define the existing, open problems in the literature. In [11], the implications of the three simplest models, the Restricted 3-Body Problem (R3BP), Restricted Hill 3-Body Problem (RH3BP) (equivalent to the H3BP), and Restricted Full 2-Body Problem (RF2BP), for the stability of motion about binary asteroids is discussed. Our initial goal for analysing these systems is to determine how coupling between these simplest problems can modify classical stability results. The equilibrium points of most interest are the Lagrange points in the R3BP (also known as the equilateral triangular points, or L4,5). These are of interest as they are the only equilibrium points that may be stable, and thus can potentially be used to observe the binary system. Of specific interest is how their stability conditions become modified under solar perturbations and shape effects in one of the primaries.
Restricted Hill Full 4-Body Problem
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3. The Restricted Hill 4-Body Problem A derivation and study of this problem was given previously in [4]. Even though that paper focused on motion in the Earth–Moon–Sun system, its models, applications, and results can be naturally generalized to the motion of a spacecraft about a binary asteroid in the tidal field of the sun. In our approach to the problem the binary asteroid (modelled as two point masses) evolves subject to the Hill equations of motion, and thus their motion is consistent with the tidal force field that also affects the motion of the particle. By deriving the problem in this way, the resulting equations of motion contain both the Hill Problem and the R3BP as limiting cases.
3.1 Equations of motion For the motion of the primaries we take the Hill Variation orbit, a periodic orbit known analytically which satisfies the H3BP [12]. To lowest order, the relative motion of the two primaries in the Hill Problem can be described with the relative position vector: 2 11 2 2=3 2 ^þ R¼m m sinð2Þ þ . . .Þ^y 1 m m cos ð2Þ þ . . . x ð1Þ 3 8 where m is the ratio of the sidereal period of the relative orbit of the binaries over the orbit period of the binary asteroid about the sun, or m ¼ Tbin/Thelio, and is the time ^-axis is non-dimensionalized by the sidereal orbit angular rate of the binaries. The x aligned with the sun–asteroid centre of mass, with the ^y-axis pointing in the direction of motion of the binary asteroid about the sun. This orbit serves as the basis for Hill’s theory for the motion of the moon [13]. The resulting equations of motion for a particle can be found in [4] (equation (55)) and are: x00 2ð1 þ mÞy0 ¼ Vx
ð2Þ
y00 þ 2ð1 þ mÞx0 ¼ Vy
ð3Þ
z00 ¼ Vz
ð4Þ
1 3 2 1 1 þ 2m þ m ðx2 þ y2 Þ m2 z2 Vðx, y, z, ; , mÞ ¼ 2 2 2 3 þ m2 ðx2 y2 Þ cos 2 2xy sin 2 4 m2 1 þ 3 þ a0 R1 R
ð5Þ
where is the mass parameter of the system and equals the smaller of the primary masses divided by the sum of the two masses, m is the parameter described previously, R1 and R are the distance between the attracting primaries and the spacecraft and are oscillating as a function of time, and a0 is the average radius of the
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orbit and is a function of the parameter m. Specifically: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ¼ ðx þ Rx Þ2 þ ðy þ Ry Þ2 þ z2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðx ð1 ÞRx Þ2 þ ðy ð1 ÞRy Þ2 þ z2 2 7 2 2=3 a0 ¼ m 1 m þ m þ ... 3 18
ð6Þ ð7Þ ð8Þ
where Rx and Ry are the components of the vector in equation (1). It is important to note that setting ! 0 recovers the classical H3BP (after transforming into a frame rotating with the sun-line), while setting m ! 0 recovers the R3BP. 3.2 Modified stability of the Lagrange points In [4], the dynamical environment about the Lagrange points was studied for a range of and m values. This study included detailed computation and analysis of periodic orbits in the vicinity of these points and the characterizations of the manifolds associated with these points. One results is that stable Lagrange equilibrium points will bifurcate into stable periodic orbits as m increases from 0. As m grows larger, however, a critical value is reached where these periodic orbits lose their stability. A derivation of this result is given here due to its applicability to the binary asteroid orbiter problem, an application not foreseen earlier. As detailed and studied in [4], a periodic solution about L4,5 can be found by analytic continuation in m from the R3BP. The period of this orbit is in normalized time units, and corresponds to half an orbit of the binary asteroid. The characteristic multipliers of the resulting periodic orbits can, from Floquet theory, be shown to equal: 2
ei ½ð1þmÞo1 þOðm Þ ,
ei ½ð1þmÞo2 þOðm
2
Þ
ð9Þ
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1h o1,2 ¼ 1 1 27 þ 272 ð10Þ 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1 Þ þ . . . ð11Þ o1 2 27 ð12Þ o2 1 ð1 Þ þ . . . 8 and o1, o2 are the natural frequencies of the L4,5 points in the R3BP. The classical criterion for stability of the Lagrange points in the R3BP is the Routh criterion. " rffiffiffiffiffi# 1 23 1 < 0:0385: ð13Þ 2 27 For equal density bodies, this restriction can be restated in terms of the ratio of mean radii of the binary components: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 ð23=27Þ R1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:342 . . . ð14Þ < R2 1 þ ð23=27Þ
Restricted Hill Full 4-Body Problem
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For small values of m, we see by inspection that 0 < ð1 þ mÞo1 ð1 þ mÞo2 < 1:
ð15Þ
Thus, the Floquet multipliers of these periodic orbits will not intersect with the positive real axis or with each other, which implies that stable equilibrium points will continue into stable periodic orbits. For large enough values of m, however, the Floquet multipliers may intersect with the negative real axis, i.e. a period doubling bifurcation may occur. The condition for this is eið1þmÞo2 ¼ 1, or ð1 þ mÞo2 ¼ 1:
ð16Þ
At this point, the multipliers meet on the negative-real axis and the periodic orbits become unstable. From this condition we can derive a constraint on m for stability: m
