## 4. Frequency map analysis of the triaxial logarithmic system: quasi-periodic approximationsIn this section, we will proceed as in Paper I, in order to verify our conjectures regarding the principal categories of orbits. In effect, we will endeavour to understand their complicated structure through the quasi-periodic approximations given by the frequency map analysis. ## 4.1. Box orbitsWe apply the NAFF algorithm to the representative box orbit (Fig. 2a), generated by the initial conditions . The orbit is integrated for about 20 orbital periods represented by the number of crossings of the surface . This time span assures a precision in the determination of the frequency vector for quasi-periodic KAM solutions of the order of (Sect. 2). We present in Table 1 the quasi-periodic approximations given
by the frequency analysis when applied to each pair of conjugate
variables, considered as complex functions of the form
, and
, respectively (long, middle and short axis
variables for this choice of the perturbation parameters). Here, the
number of terms
The three fundamental frequencies are non resonant, since the
boxes, as their 2-dimensional analogues (Paper I), are orbits
circulating in the phase space parameterized by the Cartesian
coordinates and their conjugate momenta. They can be generated by the
coupling of the 1 degree of freedom integrable systems studied in
detail in Sect. 3.1. To be more precise, let us first note that the
only axial periodic orbit which is linearly elliptic for values of
The second part of the Hamiltonian (24) can be considered as a
perturbation if ## 4.2. Tube orbits## 4.2.1. Short axis tubesWe generate a typical small axis tube orbit , whose projection in the configuration space is plotted in Fig. 2b, by integrating the equations of motion of the system with initial conditions for the same time span as for the box orbit. The quasi-periodic approximations extracted by the NAFF algorithm for the complex functions , and are represented in Table 2.
In Sect. 3.4, we inferred that the short axis tubes should be
resonant in the employed variables. This is indeed true as we may
observe from the frequencies of maximum amplitude of the
quasi-periodic approximations involving the long
and middle axis
variables. By considering ,
and as fundamental
frequencies, we manage to determine with a good precision all the
harmonics of the quasi-periodic approximations. On the other hand, the
solutions corresponding to the resonant variables are very similar to
the ones of an axisymmetric system (see Paper I). Thus, we may suspect
that the small axis tubes are generated by the perturbation of the
axisymmetric system restricted on the long-middle variables
() hypersurface by setting the perturbation
parameter ( in our case) different from 1 and a
second perturbation by adding in the argument of the logarithm of the
potential a third non-axisymmetric dimension. This perturbation may
generate quasi-periodic orbits, considering the fact that the periodic
orbit on the plane formed by the middle and long axis spatial
variables ( where . The part is
a perturbation if is close to 1 and The part of the phase space covered by these resonant orbits can be thought as an extension of a librating island (or mathematically speaking of a central manifold of dimension 4). In that part of the phase space, the ratios between the frequencies of maximum amplitude is constant, due to the fact that the variables associated with these frequencies label the tori of the circulating part of the phase space. Consequently, in order to explore the librating part corresponding to the short axis tube orbits, we should find a proper system of variables where the resonance is eliminated. The approach the most direct is the construction of some suitable normal forms which would transform the motion into products of distorted circles. However, we have observed in Paper I, that the transformation to some adapted action-angle variables is rather cumbersome for such a complex system. Alternatively, the resonant long and middle axis variables may be transformed in polar coordinates, as in Eq. 25(see also Paper I), which is a more adapted set of variables for the study of the short axis tubes. A third solution can be given by using a trick inspired by the averaging procedure in dynamical systems. In fact, as we have determined the resonant frequency given by the frequency analysis, we may integrate the equations of motion of the system using as step the associated period . Thus, we "average" in a certain way the system, as it is performed, for example, in planetary dynamics, in order to eliminate the fast motions of the orbit and recover the "real" frequencies, which are "hidden" by the resonance. Principally due to lack of space, we will not present here any example of these "averaged" quasi-periodic solutions (for more details see Papaphilippou 1997). ## 4.2.2. Outer long axis tubesThe third important family of orbits of the logarithmic potential
is the outer long axis tubes. In the case of the chosen axial ratios,
these are the orbits which are moving around the
The conjecture that these orbits present a 1:1 resonance between the middle and the short axis variables is verified by the computed quasi-periodic approximations. All the harmonics can be considered as linear combinations of three fundamental frequencies , and and this particular orbit seems quasi-periodic. We remark the prominent resemblance of these quasi-periodic approximations with the ones issued by the application of the method in the short axis tubes (see Table 2), especially in the quasi-periodic decomposition of the harmonics. More specifically, the resonant variables present approximately the same frequencies in their quasi-periodic approximations. Indeed, these orbits may be considered as the perturbation of an axisymmetric 2 degrees of freedom system by setting the axial ratio different from one, plus a second perturbation by adding a third dimension which is not rotationally symmetric with the two previous ones, as in the case of the short axis tubes. Considering the fact that the periodic orbit in the middle-short axis plane (the -plane in the present choice of the axial ratios) is linearly elliptic, the Hamiltonian of the system may be written: where . The second part
of the Hamiltonian is a perturbation if
is close to 1 and Formally, the Hamiltonian of the system can be also written as a
perturbation of the axisymmetric system restricted in the surface of
the long-short axis variables (here the ) with
a perturbation on the middle ( ## 4.2.3. Inner long axis tubesThe last principal quasi-periodic family of orbits of the logarithmic system are the inner long axis tubes. The behaviour of these orbits in the configuration space is very similar with the outer long axis tubes. In order to study more closely their dynamics, we display the quasi-periodic approximations given by the frequency analysis (Table 4) when applied to the orbit of Fig. 2d. In fact, we integrate the equations of motion of the system, with initial conditions , which are slightly different from the ones of the previously studied outer long axis tube. Table 4. Frequency analysis of the inner long axis tube orbit .
It is straightforward to check by the quasi-periodic approximations of the functions and that there exists a 1:1 resonance between the middle and short axis variables, as for the outer long axis tubes. The difference though is apparent in the quasi-periodic approximation of the function (first part of Table 4). First, for a small change of the initial conditions comparing to the ones of the outer long axis tube (see Sect. 4.2.2), the leading frequency has changed dramatically from -2.5947176 to -2.1157969. Furthermore, we observe the appearance of many uncoupled harmonics, as for the corresponding approximation of the box orbits (see Table 1), which testifies that the system is circulating in the long axis variables. In consequence, it is reasonable to consider that the inner long axis tubes can be generated by a perturbation of the long axial integrable system, as in the case of box orbits but with the fundamental difference that this perturbation is performed by a slightly perturbed axisymmetric system. We may thus write: is indeed a perturbation, if
is close to 1 and is
sufficiently small comparing to © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |