Astron. Nachr. / AN000, No. 00, 1 – 4 (2007) /DOI please set DOI!

The structure of the extended phase space of the Sitnikov problem

T. Kovacs⋆ andB. Erdi

Department of Astronomy, Eotvos University, Budapest, Hungary

Received 15 February 2007, accepted laterPublished online later

Key words celestial mechanics – methods: numerical

The extended phase space of the Sitnikov problem is studied by using a stroboscopic map and computing escape times.Comparisons of phase portraits and plots of escape times reveal the intrinsic connection between the geometry of the phasespace and the dynamical behaviour of the system. Propertiesof the phase space are analysed both in the central regularregion and far from it.

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1 Introduction

The Sitnikov problem (Sitnikov 1960) is a special case ofthe restricted three-body problem, which has many appli-cations in celestial mechanics and dynamical astronomy. Inthis particular case, two point-like bodies of equal masses(called primaries) orbit around their common center of massdue to their mutual Newtonian gravitational forces, and athird body of negligible mass moves along a line, perpen-dicular to the orbital plane of the primaries, going throughtheir barycenter (Fig. 1). The motion of the primaries can beeither circular or elliptic, and the problem is to determinethe motion of the third body along the perpendicular lineunder the Newtonian gravitational forces of the primaries.The problem has been known for a long time, and alreadyEuler studied it as a special case of the two fixed centersproblem. For the circular motion of the primaries, the prob-lem is integrable and MacMillan (1913) gave a closed formanalytical solution with elliptic integrals. Sitnikov’s (1960)paper, in which the existence of oscillating motion for thethree-body problem was proved, renewed the interest in theproblem. The reason is that the elliptic case of the Sitnikovproblem is one of the simplest non-integrable systems, andit serves as an example for studying chaotic motions (Moser1973).

Many recent papers deal with the Sitnikov problem. Liu& Sun (1990) derived a mapping model to investigate theproblem. Wodnar (1991) introduced a new formulation forthe equation of motion by using the true anomaly of theprimaries as independent variable. By using perturbationmethods, approximate analytical solutions were derived byHagel (1992) and Faruque (2003), valid for small oscilla-tions around the barycenter and for moderate eccentricities.Hagel & Lhotka (2005) extended the analytical approxi-mations up to very high orders by using extensive com-

⋆ Corresponding author: e-mail: T.Kovacs@astro.elte.hu

Fig. 1 The configuration of the Sitnikov problem.

puter algebra. Dvorak (1993) showed by numerical com-putations that invariant curves exist for small oscillationsaround the barycenter. Alfaro& Chiralt (1993) determinedinvariant rotational curves by applying the Birkhoff normalform of an area-preserving mapping. Belbruno, Llibre&Olle (1994) derived analytical expressions for the circularcase (MacMillan problem) using elliptic functions. Periodicsolutions were studied by Perdios& Markellos (1988), Bel-bruno et al. (1994), Jalali& Pourtakdoust (1997), Kallrath,Dvorak& Schloder (1997), Olle& Pacha (1999), and Cor-bera& Llibre (2000). Dvorak (2007) studied the completephase space numerically.

In this paper, we investigate the Sitnikov problem by nu-merical methods. Since in the elliptic case the driving forceis periodic, we study the structure of the extended phasespace by using a stroboscopic map. This approach is dif-ferent from earlier works, where surfaces of sections werealways made at the pericenter passage of the primaries. Byusing a stroboscopic map, the phase space can be studiedalong its third dimension. This allows us to get new insightinto the problem. By computing escape times, we also studythe global flow of the Sitnikov problem.

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2 T. Kovacs & B.Erdi: The Sitnikov problem

2 Equations of motion

In non-dimensional variables (taking the semi-major axisof the orbit and the total mass of the primaries as distanceand mass units, and one period of the primaries equal to2πtime units, so that the constant of gravity becomes 1), theequations of motion of the Sitnikov problem are (Alfaro&Chiralt 1993):

dz

dE= 2rv,

dv

dE= −

2rz

(z2 + r2)3

2

,(1)

wherez is the distance of the third body from the plane ofthe primaries,v is its velocity,r is the distance of the pri-maries from the barycenter, andE is the eccentric anomalyof the primaries, serving as independent variable. It is re-lated to the timet andr through the equations

t = E − e sinE, r =1

2(1 − e cosE) =

1

2

dt

dE, (2)

wheree is the eccentricity of the orbits of the primaries.When the primaries move in circular orbits,e = 0,

r = 1/2, and the problem is integrable. Whene > 0, theright hand side of Eqs (1), the driving force, depends peri-odically onE. Then the phase space can be considered asan extended 3 dimensional space of the variables(z, v, E),E taken mod2π. Systems of this kind can be studied by astroboscopic map, where the phase space is cut by a surfaceat selected values of the angular variable (mod 2π). Beforewe make a detailed study of the phase space, first we inves-tigate a particular case.

3 A phase portrait

In Fig. 2, we show a phase portrait, obtained by the methodof the surface of section fore = 0.1 andE = 0. As iswell known, periodic solutions of a system appear as fixedpoints on a surface of section. Invariant curves (or islands)around fixed points correspond to quasi-periodic motions,while scattered points on the surface of section representchaotic trajectories. In Fig. 2, several fixed points can beseen, corresponding to resonant periodic solutions, e.g. the1:1, 4:3, and 2:1 resonances. The 2:1 resonance is repre-sented by two fixed points, since we get two cuts on thesurface of section during one oscillation of the third bodyand two revolutions of the primaries. These fixed points aresurrounded by islands, corresponding to quasi-periodic mo-tions. Around these islands fixed points of a 10:1 secondaryresonance with a chain of small islands appear.

There is also a fixed point in the origin, correspondingto a stable equilibrium of the third body. Invariant curvesaround it represent oscillating motions. Interestingly, withthe increase of the eccentricity, the stable equilibrium pointcan become unstable. Bifurcation is a typical phenomenon

Fig. 2 A phase portrait fore = 0.1, made at the pericenter pas-sage of the primaries (E = 0). The initial conditions for the tra-jectories, whose intersections with the surface of sectionform thephase protrait, were taken along the section0 ≤ z ≤ 2.5, v = 0.There are invariant curves around the stable origin(z = 0, v = 0).The fixed points on thev = 0 axis correspond to resonances: 1:1 atz = 0.95, 4:3 atz = 1.19, and 2:1 atz = 1.84 and−1.84. Thereare invariant curves around the 2:1 resonance, and small islandsaround the 10:1 secondary resonance. Scattered points outside theinvariant curves originate from chaotic trajectories.

in chaotic systems. There are many values of the eccentric-ity at which bifurcation appears (Alfaro& Chiralt 1993).For example, fore = 0.8558625 the origin becomes unsta-ble and simultaneously two stable fixed points appear aboveand below it.

A phase portrait shows a slice of the real phase space.We can have other characterizations of the phase space, forexample, if we plot the stability properties of orbits on theparameter plane(z, e), evolving from points of this plane.Todetermine the stability of orbits, a suitable method is thecomputation of the Lyapunov characteristic exponents. Fig.3 shows the results of such a computation, for initial con-ditions taken along the positivez axis of Fig. 2 for differ-ent eccentricities. Thus, the stability properties of orbits,originating from points of thez > 0 axis of Fig. 2, wheree = 0.1, can be seen along the linee = 0.1 of Fig. 3. Ini-tial conditions taken from the light area of the figure resultin chaotic motions, while regular orbits start from points ofthe dark region. It can well be seen that with the increase ofthe eccentricity, the size of the islands (i.e. the size of theislands along thez axis) is changing, and there are manyvalues ofe, where the islands disappear or reappear. We canalso find this phenomenon in Dvorak (2007).

4 The extended 3D phase space

Now we study the phase space in general. In the case ofe 6= 0, the equations of motion contain the independentvariable explicitly. Our idea is to study the phase space atdifferent positions of the primaries. So instead of just onesurface of section at the pericenter, we make several cutsthrough the phase space by moving the position of the sur-

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Astron. Nachr. / AN (2007) 3

Fig. 3 Stability structure on the parameter plane(z, e) accordingto the Lyapunov characteristic exponents (LCE). The initial condi-tions were0 ≤ z ≤ 4, ∆z = 0.01, 0 ≤ e ≤ 0.995, ∆e = 0.005,v = 0, E = 0. The dark domains represent initial conditions ofstable motion, while light regions correspond to unstable orbits.

face of section of the stroboscopic map along the orbit ofthe primaries. It means that we cut the phase space at dif-ferent positions along its third dimensionE (mod 2π). Thethird body had the same initial conditions in computing allsurfaces of sections, and the primaries started always fromthe pericenter.

In Fig. 4, we show several phase portraits, obtained atdifferent positions of the primaries. One can observe themain feature of the maps, the turning of the fixed points andthe islands round the origin on the consecutive portraits. Therotation angle depends on the particular resonance. For in-stance, the fixed points of the 2:1 resonance makes a halfrotation during one revolution of the primaries (E = 2π),but the fixed point of the 1:1 resonance comes back to itsoriginal position during this time. We can imagine the phasespace in general: there are ’tubes’ that turn like a helix . Therotation rate of these ’tubes’ depends on the resonance that’sits’ in the middle of the tube.

Fig. 4 gives insight into the phase space, at different po-sitions of the primaries, for motions of the third body start-ing at the time when the primaries are at their pericenters.Since the primaries move in elliptic orbits, their positionsat the start of the motion of the third body affect the re-sulting motion. Even if the third body starts from the sameposition with the same velocity, the resulting motion willnot be the same, if the initial positions of the primaries arechanged along their orbits. It is interesting to study how theinitial positions of the primaries affect the motion of thethird body. We investigated this problem by computing theescape times.

Fig. 5 shows the escape times for motions of the thirdbody for two initial positions of the primaries,E = 0 andE = π. Dark shades represent short escape times, whilein the bright regions the escape times are longer. There is asimilarity between Figs 4 and 5 for the corresponding valuesof E. This is expected forE = 0, since in both figures thethird body begins its motion when the primaries are at peri-centers. Fig. 4(a) shows the structure of the phase plane fora

Fig. 5 Escape times fore = 0.1, for different initial phases ofthe primaries: (a)E = 0; (b) E = π. The computations weremade for 100 revolutions of the primaries by assigning the escapetimes to each initial condition on the phase plane(z, v).

Fig. 6 Escape times for a larger region of the phase plane, fore = 0.1, E = 0, and 1000 revolutions of the primaries.

small number of orbits which are either regular (periodic orquasi-periodic) or chaotic. This already gives an indicationfor where we can expect bounded or escaping orbits. Fig.5(a) confirms this expectation for a large number of orbits.Fig. 4(d) shows the structure of the phase plane atE = π fororbits started atE = 0. On the other hand, in Fig. 5(b), onecan see the bounded and escape regions for orbits started atE = π. The similarity of Fig. 4(d) and Fig. 5(b) indicatesthat in the extended phase space of the Sitnikov problem,for different initial positions of the primaries, the one andsame flow exists and we only see it at different positions ofthe primaries. This is an intrinsic connection between thegeometry of the phase space and the dynamical behaviourof the system. True, if we consider only a particular initialpoint of the third body in the phase space, the resulting tra-jectory will depend on the initial positions of the primaries.However, the flow of the trajectories is the same regardlessthe positions of the primaries.

Studying the escape times in a larger region of the phaseplane (Fig. 6), it can be seen that far from the central regularregion there exist initial conditions from which the resultingmotion remains bounded (at least for the studied 1000 rev-olutions). However, on the stroboscopic maps, these initialconditions result in chaotically scattered points (not shownhere). Thus, the corresponding orbits are chaotic which re-main stable for a long time (sticky orbits).

Considering the escape times in more detail, for exam-ple along thez = 8 line of Fig. 6, we obtain a structure of

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4 T. Kovacs & B.Erdi: The Sitnikov problem

Fig. 4 Phase portraits at different positions of the primaries: (a) E = 0; (b) E = π/4; (c) E = π/2; (d) E = π; (e) E = 3π/2; (f)E = 2π. Due to the2π periodicity ofE, the panels (a) and (f) are the same. (For clarity of the figures, invariant curves of the centralregion are not shown.)

Fig. 7 Escape times along the velocity axis forz = 8, e = 0.1,E = 0, andt = 1000 revolutions.

spikes (Fig. 7), characteristic for escape times of sticky or-bits (Dvorak et al 1998). Selecting an arbitrary∆v intervalin Fig. 7, and computing the escape times on a finer grid,we can observe the self-similarity of the distribution of thespikes. This means that the holder of the initial conditionsof the escaping orbits along the axisv is a Cantor-type set.Thus, in the Sitnikov problem, outside the well known cen-tral regular regions of the phase space, we can also find spe-cial initial conditions which result in bounded motion forlong times.

5 Conclusions

We investigated the Sitnikov problem by using a strobo-scopic map, shifting the surface of section along the orbitof the primaries. In this way, we could scan the third di-mension of the phase space. Our results show that the fixedpoints and their surrounding islands turn like a helix alongthe third direction of the phase space. The rate of the ro-

tation depends on the resonance ’sitting’ in the middle ofthe islands. The comparison of the phase portraits and plotsof escape times shows that there is only one flow in thephase space regardless the initial positions of the primaries.Computations of escape times indicate that far from the wellknown central regular regions there exist initial conditionsresulting in bounded motions for long times. However, thisdoes not mean that these motions are regular.

Acknowledgements. Helpful discussions with Prof. T. Tel, Prof.R. Dvorak, and C. Lhotka are gratefully acknowledged.

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