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Astron. Astrophys. 320, 672-680 (1997)

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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 [FORMULA] 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.

[FIGURE] Fig. 2. The secondary resonances calculated by the semi-numerical method

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 [FORMULA] for the set of initial conditions [FORMULA], [FORMULA], [FORMULA] and a between 0.624 and [FORMULA] (3.2467 and 3.2727 AU for [FORMULA] 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 [FORMULA] appear at separatrices of the secondary resonances. We identified by evaluation of [FORMULA] 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 [FORMULA] and [FORMULA] 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 [FORMULA]. Other initial values of e, [FORMULA] for instance, lead to the hyperbolic point crossing at 8/1.

[FIGURE] Fig. 3. The second derivative singularities of the frequency [FORMULA] at separatrix crossings and the ratio [FORMULA]

The frequency [FORMULA] for the estimate of the derivative in (a) was calculated from only [FORMULA] yr, what led to a slight distortion of the frequency curve. In fact, if T is longer, as it was done in (b) where [FORMULA] 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 ([FORMULA]) 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 [FORMULA] yr for a fine net of initial conditions inside the shown rectangles with spacing 0.0005 in eccentricity and 0.0002 [FORMULA] in semi-major axis (zero initial angles) and, similarly as in Fig. 3, we have computed the value of [FORMULA]. 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.

[FIGURE] Fig. 4. Position of the secondary resonances in three models: circular a, elliptic b and with Saturn c

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 [FORMULA] (3.25188 AU), a large island of the 8/1 resonance at [FORMULA] (3.26228 AU) and weaker high-order resonances further to the right.

[FIGURE] Fig. 5. The secondary resonances in the four-body planar model
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998