PADEU. The 1 : 2 resonance in the Sitnikov problem. 1 Introduction. T. Kovács 1

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1 PADEU PADEU 7, (6) ISBN , ISSN c Pulished y the Astron. Dept. of the Eötvös Univ. The : resonance in the Sitnikov prolem T. Kovács Eötvös University, Department of Astronomy, H-58 Budapest, Pf. 3, Hungary t.kovacs@astro.elte.hu Astract In simple dynamical systems of two degrees of freedom chaotic ehaviour appears. There is a special case of the restricted three-ody prolem, the Sitnikov prolem, which has even less than two degrees of freedom, ut in the phase space of this system typical forms of chaotic ehaviour can e well studied. In the Sitnikov prolem a massles ody moves along a line under periodic force. Regular and chaotic orits can appear in this configuration depending on the initial conditions. In this paper we study the dependence of resonant motions of the Sitnikov prolem on the parameter of the prolem which is the eccentricity of the primaries orit. We show that there is a ifurcation when the eccentricity reaches a critical value. Keywords: Sitnikov prolem, resonances, chaos Introduction It is well known that the gravitational three-ody prolem can not e analytically integrated. The evolution of computers in the past decades permits the use of computers with great efficiency to solve prolems of celestial mechanics. Due to this progress we came to know that in simple dynamical systems various types of motion appear depending on the different initial conditions. These motions can produce complex ut ordered structures in the phase space, as well as chaotic formations. In this paper we investigate a special case of

2 T. Kovács the restricted three-ody prolem, namely the Sitnikov prolem (SP). We know that in this very simple configuration there appears chaotic ehaviour. In the Sitnikov prolem a massles ody moves along a line perpendicular to the plane of the equal-mass primaries going through the arycenter of the system. Mac Millan (93) showed first that in the circular Sitnikov prolem (the primaries revolve in circular orit around each other) the motion of the third massles ody can e descried y elliptic integrals. There are various trajectories if we allow eccentric motion of the primaries. This prolem was investigated y Sitnikov (96). He showed that there are periodic orits in the elliptic restricted three-ody prolem. After that other authors demonstrated the existence of periodic motions in the Sitnikov prolem. Dvorak (993) investigated numerically the phase space of this prolem via Poincaré s surfaces of sections. In the same year Alfaro & Chiralt (993) showed that the =, ż = point in the middle of the phase space ecomes unstale for several values of the eccentricity. Kallrath et al. (997) studied the periodic orits (PO) of the SP and gave a summary of low order resonances in the phase space. The ifurcations and staility of families of periodic orits was studied y Perdios and Markellos (998). Hagel (99) and Faruque (3) used a perturation theory to approximate the solution for the third ody s motion. This solution is valid only when the eccentricity is small enough. In this paper we investigate numerically (using strooscopic map) the staility of the : resonance in the SP depending on the increase of the eccentricity. Description of the system In the Sitnikov prolem two equal masses m and m revolve in Keplerian orits around each other. A massles ody m 3 moves perpendicularly to the plane of the primaries through their arycenter (Fig. ). The motion has one degree of freedom, ut in the eccentric prolem (when the orit of the primaries is eccentric) the equation of motion contains explicitly the time or the true anomaly. Therefore the energy of the third ody does not remain constant.. Equation of motion The equation of motion of the third ody is very simple, when we introduce suitale units. Choosing the total mass of the primaries as mass unit, the period of the primaries to e π, the semi-major axis of the orit of the primaries as

3 The : resonance in the Sitnikov prolem 3 Figure : The Sitnikov prolem. distance unit, the Gaussian constant of gravity ecomes. Then the equation of motion of the massles ody is where ( m = r 3 + m ) r 3, r = r = R +, R = e cose = p + e cosv, R is the distance etween the primaries, is the distance of the massles ody from the plane of the primaries, e is the eccentricity, p is the parameter of the elliptic orit of the primaries (semilatus rectum), and E is the eccentric anomaly, which depends on the time according to Kepler s equation: t τ = E e sin E. The τ phase constant corresponds to the pericenter passage.. Periodic orits In a dynamical system periodic orits constitute the ackone of the structure of the phase space. Periodic orits are elliptic fixed points on a map, the quai-

4 4 T. Kovács periodic motion appears as a closed curve around the PO. For want of friction these invariant curves remain finite distance to the fixed point. The trajectories lie on n-dimensional invariant tori (-dimensional in the SP) in the phase space. Small perturations can destroy the tori with resonant frequencies. Non-resonant and sufficiently,,irrational tori can survive the perturations and remain in distorted forms in the phase space, these are called KAM tori. The KAM tori are important, ecause in their neighourhood chaotic motion appears. This phenomenon was studied y Dvorak (997)..3 The phase space As desried aove, periodic orits play an important role in dynamical systems. These trajectories appear as elliptic fixed points on a map. Closed curves around the fixed points on the map correspond to invariant tori in the phase space on which the motion is quasi-periodic. Fig. shows the phase space of the Sitnikov prolem, when the eccentricity of the primaries is e =.4. There are closed curves around the central = ż = point. Some resonances are visile in this resolution, namely the :, :, and 3: resonances. These are the centers of the little islands in Fig.. It is important that in the Sitnikov prolem we can have periodicity only for resonant motions etween the primaries and the third ody. When the trajectories are started from initial conditions taken from the,,chaotic sea etween the islands, the test particle m 3 escape the system. The location of the individual resonances was studied y Kallrath et al. (997). In Fig. it is well visile that the centers of the small island are in the locations given y Kallrath et al. (997). Tale : List of the visile resonances in Fig.. (y Kallrath et al. (997)) Resonance :.6947 : :.479 We study in detail the : resonance in the next section. We have investigated what happens with the elliptic fixed point when the eccentricity of the

5 The : resonance in the Sitnikov prolem 5.5 e = Figure : The phase spase of the Sitnikov prolem (e =.4). There are invariant curves around the fixed pont in the middle. The small islands correspond to the :, :, and 3: resonances. The test particle starting from other initial conditions escapes the system. primaries is increased. 3 The : resonance There is one parameter in the Sitnikov prolem, the eccentricity of the primaries. In the circular SP it is possile to find periodic orits, i. e. initial conditions, for any period T. When the primaries revolve in eccentric orits, other types of motion appear in the system, ut there remain orits with multiple periods of π, these periodic orits are the resonances. The : resonance means a whole period of m 3 aout the arycenter during which the primaries cover two revolutions. We have investigated the : resonance numerically and found a value of the eccentricity when a ifurcation appears. Figure 3 shows the phase space of the SP around this value of the parameter. We can see one fixed point efore the critical value of the eccentricity e c. When the eccentricity gets larger than e c, the fixed point ecomes unstale and simultaneously two stale trajectories (periodic orits) appear past the original stale fixed point. In the figure there are 3 initial conditions in each panel, [.;.6], =.5, the integration time was period of the primaries. The initial velocity ż was ero in all computations. In our simulation the initial position of the primaries was v =. (For more details see Kovács, T. (5).)

6 6 T. Kova cs e= e= e= e= e= e= e= e=.53.4 e= e= Figure 3: The : resonance in the Sitnikov prolem. The eccentricity is changing with the figures from.5 to.6, e =.. See details in the text. It is well visile that for larger eccentricities the closed curves around the PO ecome more and more prolonged. This corresponds to the decrease of the frequency of the motion started from these initial conditions. This phenomenon is the critical deceleration. Nem rtem a elnevest. Deceleration lassulst jelent. We determined numerically the critical value of the eccentricity, where the disjunction of the two (new) stale fixed points is appreciale, and it is ec = It is well visile in Fig. 4 that around the PO (.7958) the invariant curves corset and close to it two little loops appear. In fact, for e =.6 these two fixed points correspond to the : mean motion resonance. We checked the two islands with high resolution and allocated the location of the initial conditions of periodic orits. The values are lef t =.43 for the island on the left hand side, and right =.375 for the

7 7 The : resonance in the Sitnikov prolem e= e-8 5e-9-5e-9 -e Figure 4: For the critical eccentricity (ec = ) the stale PO ecomes unstale and two new stale trajectories appear. These new POs gets more farther from the formation place with increasing the eccentricity. ( [.7955;.796], = 5 5, Tint = 4 5 period.) [the distance from the primaries plane] island on the right hand side. We plotted the motions started from these initial positions. It is ovious that the period of the motions is 4π (two periods of the primaries), see Fig left right π 4π 6π 8π time [in period of primaries] Figure 5: The distance from the plane of the primaries versus the true anomaly (time) for e =.6. The test ody m3 started from the initial points corresponding to the two fixed points in Fig. 3 (th panel). The period in oth cases are 4π. The left curve corresponds to the periodic orit in the middle of the island on left side, the right curve to the PO on right side.

8 8 T. Kovács 4 Conclusions In this paper we investigated the Sitnikov prolem which is a special case of the restricted three-ody prolem. We found that in this system the phenomenon of ifurction appears, when the parameter of the system (the eccentricity of the primaries) reaches a critical value. We studied the : resonance and determined the critical value of the eccentricity, where the stale fixed point ecomes unstale. Simultaneously, two stale fixed points evolve near the original periodic orit. These trajectories correspond again to the : mean motion resonance. It is possile that there are other critical values of the eccentricity of the primaries where ifurcation appears. We are going to study this prolem in the future. References Alfaro, J. M., Chiralt C. 993, Celestial Mechanics and Dynamical Astronomy, 55, pp Dvorak, R. 993, Celestial Mechanics and Dynamical Astronomy, 56, 7 Dvorak, R., Contopoulos, G., Ch. Efthymiopoulos, Voglis, N. 997, Planet. Space Sci., 46 Faruque, S. B. 3, Celestial Mechanics and Dynamical Astronomy 87, Hagel, J. 99, Celestial Mechanics and Dynamical Astronomy 53, Kallrath J., Dvorak, R., Schödler, J. 997,in The Dynamical Behaviour of our Planetary System (ed. R. Dvorak and J. Henrard) Kluwer Academic Pulishers, The Netherlands, pp Kovács, T. 5, in Proceedings of the 4th Hungarian Austrian Workshop on Trojans and related topics (ed. Á Süli, A. Pál, F. Freistetter) Eötvös Univ. Press, Budapest Mac Millan, W.D. 93, Astron. J. 7,. Perdios, E., Markellos, V.V. 998, Celestial Mechanics and Dynamical Astronomy, 4, pp Sitnikov, K. 96, Dokl. Akad. Nauk. USSR, 33, pp