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Astron. Astrophys. 331, 1108-1112 (1998)

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3. Caustics for the planar, circular, restricted problem

We considered the normalized planar, circular, restricted three-body problem. The mass ratio of the two primaries is [FORMULA] (which is similar to the Sun-Jupiter mass ratio), the reference frame has its origin at the center of mass of the two primaries and it rotates with angular velocity equal to one such that the two primaries always remain on the x axis. The trajectories of the third body are defined taking the values of the coordinates of position and velocity [FORMULA] at time [FORMULA].

In order to choose the starting conditions, we followed the "Atlas of the Planar, Circular, Restricted Three-Body Problem I and II" (Winter and Murray, 1994a, 1994b). As one can conclude from an examination of the surfaces of section for this problem, the trajectories in the phase space are arranged into two large sets: interior orbits and exterior orbits, being, in general, those that move around just one of the primaries and those that move around both primaries. Furthermore, these two sets are subdivided into families that circle a periodic orbit, which in most of the cases are associated to the resonances that correspond to commensurability between the period of the primaries and the third body.

In this work we present a representative sample of trajectories of each set. As it will be seen below, their characteristics are closely related to the type of resonance.

Next, we present some caustics for each set and analyse their main features. The procedure for the computation of the caustics is described as follows. To obtain the caustics, we used the ODE integrator given by Shampine and Gordon (1975) with accuracy [FORMULA]. The zeros of the determinant were computed with accuracy [FORMULA] and were generated around 20,000 points for each caustic. The first thousand points were considered as belonging to the transient stage, and therefore, were not taken into account.

Our main task is to show that the information provided by caustics complements that provided by the Poincaré surfaces of section. Therefore, we always show our results associated to the surfaces of section. Thus we present the caustics related to the various typical regular orbits that appear in the typical surfaces of section.

For the interior orbits we selected one surface of section corresponding to the value: [FORMULA] and for the Jacobi constant [FORMULA] (Fig. 1). For this surface of section there are nine main sets of islands immersed in a chaotic region, where the order of resonances varies from first to fifth orders.

[FIGURE] Fig. 1. The RTBP surface of section for [FORMULA] and [FORMULA] (interior region).

For the island located at [FORMULA], the caustics are given in Fig. 3.1. Computing the orbital period for the orbit located at the center of the island, it is found that it is in resonance 8:3 with the primaries. Note that the caustics shows 8 lobes, corresponding to the 8 turns that the particle completes while the primaries make 3. The order of the resonance, i.e., the difference between the two numbers of the commensurability, is the number of islands that appear in the surface of section around this orbit. The semi-major axis, a, and the eccentricity, e, of the osculating orbits of the torus represented by the caustics can be evaluated bearing in mind that the smallest distance from the torus to the massive primary corresponds to the pericenter and the largest distance corresponds to the apocenter. This can be seen in Fig. 2, where the caustics in the rotated frame are presented together with the corresponding orbit in the inertial frame. This evaluation improves when the orbit is closer to the periodic orbit. However, a and e can be estimated by using the equations of the two-body problem approximation (Hénon, 1997), i.e. for a [FORMULA] resonance, [FORMULA] and [FORMULA]. In the present case, this gives [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Orbit in the inertial frame and its caustics in the rotated frame. The initial values in the rotated frame are: [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Caustics for [FORMULA] (interior region). The values for the resonances and the initial conditions are given for each caustic.

Note also that due to the resonance the particle's apocenter near the less massive primary occurs always at its greatest distance (see for instance Figs. 3.1 and 3.2). This fact makes the orbit stable. Any other orbit with the same values of Cj, a and e, but without this property will get closer to the less massive primary becoming unstable. This is equivalent to the well known pericentric position on the line of the primaries for the stable internal orbits (see for example Winter and Murray, 1997a).

The following set of islands is around the resonance 5:2 and the associated caustics (Fig. 3.2) show similar characteristics to the one in Fig. 3.1. Next, we have Figs. 3.3 - 3.8, corresponding to resonances 7:3, 9:4, 2:1, 9:5, 7:4 and 5:3.

For the resonances 8:3, 5:2 and 7:3, the pericenter distance is less than 0.3, what, in the solar system, would correspond to the location of Mars orbit if we consider the less massive primary as Jupiter.

Note that the caustics associated to the resonances 7:4 and 5:3 have smaller eccentricities and therefore, the lobes are less noticeable, while for the 8:3 and 5:2 resonances the orbits have bigger eccentricities and the lobes are very noticeable. We also have inner circulation (Fig. 3.9), where the particle's orbit has a small eccentricity.

For the exterior orbits, we also selected one surface of section corresponding to the value: [FORMULA] and for the Jacobi constant [FORMULA] (Fig. 4)

[FIGURE] Fig. 4. The RTBP surface of section for [FORMULA] and [FORMULA] (exterior region).

The caustics of the exterior orbits show similar characteristics to the interior orbits, but the lobes are internal instead of external. For the island located at [FORMULA] (Fig. 5.1), the orbit on the invariant torus moves just around the less massive primary. This case looks like a simple circulation, but in fact it corresponds to the resonance 1:1 (non-coorbital).

[FIGURE] Fig. 5. Caustics for [FORMULA] (exterior region). The values for the resonances and the initial conditions are given for each caustic.

A particularity of this region is the existence of asymmetric orbits associated with [FORMULA] resonances. This can be seen in Figs. 5.4 and 5.6 corresponding to 1:2 and 1:3 asymmetric resonances. A particularity of this kind of resonance is that the symmetric island that appears in the surface of section corresponds to two families of trajectories (Winter and Murray, 1997b). Otherwise, symmetric islands have symmetric lobes.

Notice that to exterior non asymmetric orbits the particle's pericenter is more distant from the less massive primary (see for instance Figs. 5.2 and 5.3). We can also note that the lobes are not in opposition to the less massive primary and they are symmetric for the symmetric islands.

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Online publication: March 3, 1998