3. Secondary resonances
Following the semi-numerical method of Lemaître & Henrard (1990) and using the Ferraz-Mello & Sato (1989) evaluation of the perturbing function, as it was done in Moons & Morbidelli (1993), we have computed a position of several secondary resonances in the planar and circular three-body model. The lowest and the highest-order secondary resonances are denoted by a corresponding ratio in Fig. 2. The high-order secondary resonances reach the moderate eccentricity region and, in the forthcoming analysis, we will be interested in their exact position in the elliptic and four-body planar models.
The behaviour of frequencies at the secondary resonances is shown in Fig. 3a and b. In Fig. 3a we have plotted a numerical estimate of for the set of initial conditions , , and a between 0.624 and (3.2467 and 3.2727 AU for AU), computed in the planar elliptic three-body model by the technique described in Sect. 2. The three discontinuities at approximately 0.6249, 0.6258 and 0.6278 appear at separatrices of the secondary resonances. We identified by evaluation of that the first two correspond to the 8/1 and the later to the 9/1 secondary resonances. In between the first and the second discontinuity, the frequencies and are locked in a precise ratio of 8/1 (Fig. 3b). This plateau, which is typical for the elliptic point crossing, as well as the hyperbolic point crossing at 9/1, is associated with chosen . Other initial values of e, for instance, lead to the hyperbolic point crossing at 8/1.
The frequency for the estimate of the derivative in (a) was calculated from only yr, what led to a slight distortion of the frequency curve. In fact, if T is longer, as it was done in (b) where yr, the second derivative is equal to zero inside the libration island.
Fig. 4a-c is a comparison of the secondary resonances position in three planar models: (a) is the circular three-body model, (b) is the elliptic three-body model () and in (c), the effect of Saturn was included. Each rectangle is a narrow strip centered at the eccentricity 0.217. (a) is in fact a small part of Fig. 2, the result of the semi-numerical method. The bright strips in (b) and (c) are the places of the separatrix crossings seen in Fig. 3a. We have made integrations spanning yr for a fine net of initial conditions inside the shown rectangles with spacing 0.0005 in eccentricity and 0.0002 in semi-major axis (zero initial angles) and, similarly as in Fig. 3, we have computed the value of . Now, the bright areas in (b) and (c) are the places where this quantity has a high value, which shows an approximate position of the separatrices. In the elliptic and four-body models the 8/1 and 9/1 secondary resonances are shifted to the left in comparison to the circular model, which means to lower values of the semi-major axis. They are close together and the 8/1 secondary resonance apparently broadens in the four-body model. Moreover, the 10/1 secondary resonance appears on the right. The 8/1 secular resonance has an elliptic island on the studied surface while the 9/1 resonance has a hyperbolic point there. It is clearly visible in Fig. 4b but less clear in Fig. 4c where the resonant borders get fuzzy.
The same computation was extended in the planar four-body model to a larger area of initial conditions in eccentricities. The result is shown in Fig. 5. We see there, going along a diagonal from bottom-left to top-right, the overlapping resonances 6/1 and 7/1 at (3.25188 AU), a large island of the 8/1 resonance at (3.26228 AU) and weaker high-order resonances further to the right.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998