4 Applications

4.1 The uranian satellite system

 

Table 1: Uranian satellite system: co-orbital frequencies $\sqrt {(27/4)\mu _i} n_i$and forcing frequencies g_i-g_j and f_i-f_j (units are rads/day). Note that g_i-g_j and f_j-f_i are nearly equal; this is due to the fact that g_k and f_kare respectively the eigenvalues of matrices A and B which by definition have A_kk=-B_kk and which are also nearly diagonal (the diagonal term is mostly due to the oblateness and therefore $A_{kk}=-B_{kk} \approx \beta (a_k)$)

i	$\sqrt {(27/4)\mu _i} n_i$	g1-gi	g2-gi	g3-gi	g4-gi	g5-gi
		f1-fi	f2-fi	f3-fi	f4-fi	f5-fi
1	1.1  10-2	0	-7.0  10-4	-8.5  10-4	-9.1  10-4	-9.7  10-4
		0	7.0  10-4	8.5  10-4	9.1  10-4	9.7  10-4
2	2.9  10-2	7.0  10-4	0	-1.5  10-4	-2.1  10-4	-2.7  10-4
		-7.0  10-4	0	1.5  10-4	2.1  10-4	2.8  10-4
3	1.3  10-2	8.5  10-4	1.5  10-4	0	-6.0  10-5	-1.2  10-4
		-8.5  10-4	-1.5  10-4	0	5.4  10-5	1.2  10-4
4	1.1  10-2	9.1  10-4	2.1  10-4	6.0  10-5	0	-6.1  10-5
		-9.0  10-4	-2.1  10-4	-5.4  10-5	0	6.9  10-5
5	7.0  10-3	9.7  10-4	2.7  10-4	1.2  10-4	6.1   10-5	0
		-9.7  10-4	-2.8  10-4	-1.2  10-4	-6.9   10-5	0

We will now apply our secular theory to the case of Trojan orbits associated with the five major uranian satellites. As already noticed in Paper I, this system provides a good testing ground for our theory due to two reasons. First: there are no mean motion resonances amongst these satellites and thus the secular approximation (i.e. the average of the disturbing potential over the mean longitudes) is valid. Second: these satellites have nearly-circular and nearly-equatorial orbits thus satisfying the requirements of our secular theory.

We used the data from Malhotra et al. (1989) to calculate the parameters of the secular theory for the uranian system and then identified the modes that satisfy Eq. (87) (or Eq. (88)) inside the co-orbital regions of its major satellites. For instance, in order to locate secular resonances involving the pericentres (or nodes) inside the co-orbital region of the mass m_k, we first plot $\gamma _k$ (or $\Gamma _k$) as a function of a₀ (a parameter that characterizes the size of the tadpole or horseshoe orbit already defined in Eq. (14)) and then identify the intersection with the horizontal lines taken at $g_i-\bar{A}_k$ (or $f_i-\bar{B}_k$). We show the location of the secular resonances which occur inside the co-orbital regions of Miranda (m₁), Ariel (m₂), Umbriel (m₃), Titania (m₄) and Oberon (m₅) in Fig. 2. Note that we can condense the results for the five satellites in one single picture due to the fact that both $\gamma _k$ and $\Gamma _k$scale as $\mu _k n_k$ (see also Paper I).

We also used a Runge-Kutta-Nystron 12th order scheme (Brankin et al. 1987) to integrate the equations of motion of the system consisting of Uranus, its satellites and associated test particles in tadpole or horseshoe orbits. The parameters and initial conditions were again taken from Malhotra et al. (1989), and we incorporated an oblateness potential which takes into account only the dominant zonal harmonic.

In Paper I we showed the result of a numerical integration for a test particle located near the L₄ point of Titania (m₄), which included the gravitational interaction with Umbriel (m₃) only. We saw that in this case the long-term behaviour of $(e,\varpi)$ is affected by the proximity of a secular resonance involving the mode g₃ and is in good agreement with the secular solution (Eq. (97)). In fact, we can see from Table 1 that the adiabatic approximation made in Sect. 3.3.2 should be valid in the case of the uranian satellites, as the co-orbital frequencies are much larger than the forcing frequencies.

Figure 2: Location of secular resonances involving a) the pericentres and b) the nodes, inside the co-orbital regions of the uranian satellites Miranda (m1), Ariel (m2), Umbriel (m3), Titania (m4) and Oberon (m5). Secular resonances associated with the modes gi (or fi) occur at the locations $a_0/\sqrt {\mu _k}$ determined by the intersections of the curves $\gamma _k$ (or $\Gamma _k$) with the lines labeled Ak-gi (or Bk-fi). Note that the frequencies $\gamma _k$ and $\Gamma _k$ were divided by $\mu _k n_k$ in order to be able to condense the results for all the five satellites in one single picture. The singularity in $\gamma _k$ and $\Gamma _k$ at $a_0=\sqrt {(8/3) \mu _1}$corresponds to the tadpole-horseshoe separatrix

In Fig. 3 we show the evolution of an orbit located near the L₄ point of Oberon with a₀=0. This is very close to the secular resonance involving the mode g₄ (cf. Fig. 2) and we see that the slow periodic motion of the critical argument $\varpi-g_4 t$ is correlated with a large amplitude oscillation of the eccentricity. The small amplitude fast oscillation in the eccentricity has the same periodicity as the argument $\varpi-g_5 t$ and is indeed caused by the forced term due to the 1:1 mean motion resonance with Oberon. There is no apparent threat to the long-term stability of this orbit, as the maximum eccentricity is still very far from the threshold required for close approaches with nearby Titania (i.e. e=0.25).

In Fig. 4 we show the evolution of a horseshoe orbit associated with Oberon which has $a_0=2 \sqrt {\mu _5}$. The eccentricity exhibits very irregular behaviour which is probably due to the interaction between two nearby secular resonances (involving the modes g₃ and g₄; cf. Fig. 2) as suggested by the behaviour of the associated critical arguments. Due to the overlap of the separatrices of these two secular resonances, the eccentricity diffuses chaotically and can potentially reach stability-threatening values.

Figure 3: Evolution of eccentricity (upper figure), critical arguments associated with the modes g4 (middle figure) and g5 (lower figure) for an orbit located near Oberon's L4 point with a0=0. Time is in units of Miranda's orbital period

Figure 4: Evolution of eccentricity (upper figure), critical arguments associated with the modes g3 (middle figure) and g4 (lower figure) for a horseshoe orbit associated with Oberon with $a_0=2 \sqrt {\mu _5}$. Time is in units of Miranda's orbital period

Figure 5: Location of secular resonances involving the pericentres inside the co-orbital regions of the planets. The frequencies shown in the vertical axes are adimensional and correspond to the true frequencies divided by $\mu _k n_k$. Secular resonances associated with the modes gi occur at the locations $a_0/\sqrt {\mu _k}$ determined by the intersections of the lines labeled Ak-gi with the curves $\gamma _k$. The secular resonances involving the modes g1 and g2 (not shown here as they are likely to be very weak) also occur inside the co-orbital regions of Uranus and Neptune

Figure 6: Location of secular resonances involving the nodes inside the co-orbital regions of the planets. The frequencies shown in the vertical axes are adimensional and correspond to the true frequencies divided by $\mu _k n_k$. Secular resonances associated with the modes fi occur at the locations $a_0/\sqrt {\mu _k}$ determined by the intersections of the lines labeled Bk-fi with the curves $\Gamma _k$

We remark here that although our secular theory cannot describe accurately the true behaviour in the very close vicinity of secular resonances (due to the occurrence of singularities in Eqs. (97) and (99) which is an artifact caused by the truncation of the disturbing potential at degree two in e and I) or whenever there is overlap of adjacent secular resonances (due to the underlying chaotic nature of the phase space), it is still very useful in the sense that it allows us to obtain the location of these secular resonances, whose dynamical effect can always be subsequently investigated with numerical integrations.

4.2 Our planetary system

We also applied our secular theory to the system consisting of the eight planets, Mercury to Neptune, using values for the eigen-frequencies of the secular system from Brouwer & Van Woerkom (1950). As these were calculated taking in account the effect of the 2:5 near commensurability between the orbital periods of Jupiter and Saturn, the secular system is characterised by ten eigen-frequencies g_i (with g₉=2 g₅-g₆ and g₁₀=2 g₆-g₅) and eight eigen-frequencies f_i. We then used these in order to obtain the locations of secular resonances inside the planetary co-orbital regions condensed in the six panels of Figs. 5 and 6.

Figure 5f proves that the secular resonances involving the modes g₆ and g₁₀ can affect Saturn's tadpole orbits which is in agreement with recent numerical integrations by Marzari & Scholl (2000). In particular, these latter authors suggest that the mixed secular resonance involving the mode g₁₀ is the major factor responsible for the destabilization of tadpole orbits associated with Saturn. In Fig. 6b we see that the secular resonances involving the modes f₃ and f₄occur very close to each other inside Venus' horseshoe region, thus suggesting their possible overlap. In fact, the numerical integrations of Michel (1997) showed that a clone of asteroid (4660) Nereus becomes, at some stage in its life, a Venus' horseshoe orbit which exhibits chaotic diffusion of the inclination caused by the overlap of these two secular resonances.

From Fig. 6d and as already mentioned in Paper I, we see that the secular resonance involving the mode f₆ occurs within Jupiter's tadpole region. Recent numerical integrations by Marzari & Scholl (2000) seem to support earlier suggestions by Yoder (1979) and Milani (1994) concerning the important role played by this secular resonance in the dynamical shaping of the Trojan cloud.