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Astron. Astrophys. 329, 451-481 (1998)

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3. The 3D logarithmic system

The wide interest for the study of logarithmic potentials in galactic dynamics springs from their ability to fit with the observed flat rotation curves, outside the core of many galaxies. This latter observation was the most direct indication for the existence of "dark" matter in galactic halos (for a review see Binney & Tremaine 1987, p. 598). In our study, we will consider the general form of the logarithmic system (Binney 1981), whose Hamiltonian is written as follows:

[EQUATION]

where ([FORMULA]) are the Cartesian coordinates and ([FORMULA]) their conjugate momenta. It is the 3-dimensional extension of the scale-free disk potential proposed by Richstone (1980, 1982, 1984), softened by the constant [FORMULA], which represents the radius of a central core. Miralda-Escudé & Schwarzschild (1989) showed that, the regularisation of the logarithmic singularity changes substantially the dynamics of the model, thereby extending the ideas of Gerhard & Binney (1985) about galactic central cusps, which seem to be a generic feature for the great majority elliptical galaxies (e.g. see Merritt 1997). Roughly speaking, whereas the singular logarithmic potential can be thought as a model of massive halos (Kuijken 1993; Schwarzschild 1993), its softened version is more adequate for the main body of galaxies (Levison & Richstone 1987).

Poisson's equation [FORMULA] gives directly the smooth density function which reproduces the corresponding potential:

[EQUATION]

where

[EQUATION]

with [FORMULA] the symbol of Kronecker and letting [FORMULA], [FORMULA]. Hence, the density profile of the logarithmic potential is proportional to [FORMULA] at large radii ([FORMULA]), a special case of the power-law profile core models with equipotentials in stratified coeccentric ellipsoids (Evans 1994; Evans and de Zeeuw 1994; de Zeeuw 1996), which may fit well with the observational data corresponding to the M32 nearby galaxy. In a previous work, de Zeeuw & Pfenniger (1988) evidenced that the logarithmic potential, as well as its scale free limit, is a member of a wide family of potential-density pairs which share a number of interesting properties with the Stäckel potentials (Stäckel 1890, 1893; Kuzmin 1956; de Zeeuw 1985). Moreover, Evans (1993) demonstrated that the model, in its axisymmetric limit, presents a simple distribution function involving the two integrals of motion (see also Hunter & Qian (1993)).

The parameters [FORMULA] and [FORMULA] represent the axial ratios of the equipotential ellipsoids. It can be shown, though, that the isodensity surfaces are rather flatter and deviate from the ellipsoidal configuration, as soon as the axial ratios are different from 1 (Binney 1981). Indeed, we can determine approximate axial ratios of an isodensity by calculating the ratios of the values [FORMULA] and [FORMULA] of the spatial variables, where the isodensity cuts the corresponding axis. These latter are parameterized by the density (see Appendix A). We may give, though, rough estimates, for [FORMULA]

[EQUATION]

and for [FORMULA]

[EQUATION]

where

[EQUATION]

using the same convention for [FORMULA], as before. The flatness of the isodensities is one of the main deficiencies of the model preventing it from being very realistic. Nevertheless, the logarithmic potential is known to be very efficient for the representation of the dynamics of core models with ellipsoidal isopotentials (Gerhard 1994). Throughout this work we will choose an arbitrary value for the energy ([FORMULA]) and the axial ratios [FORMULA] and [FORMULA] will play the role of the perturbation parameters, as in Paper I. Following previous studies (e.g. Binney & Spergel 1982) the softening parameter [FORMULA] is set to 0.1.

Even if in the great majority of studies it is considered that [FORMULA] we prefer the liberty to give to the isopotential axial ratios any values [FORMULA]. In fact, these parameters respect certain conditions in order for the density function of Eq. (12) to be positive everywhere in [FORMULA] and conserve the physical significance of the model. In general, it is easily verified from Eq. (12) (de Zeeuw & Pfenniger 1988) that, for [FORMULA]:

[EQUATION]

The last equation is exact only in the scale-free case.

The general logarithmic system preserves all the main dynamical characteristics of the restricted 2D system (Paper I). Indeed, its phase space remains a compact (every variable attains a maximum value when all the other vanish). Moreover, the Hamiltonian of the system continues to present the symmetry by reflection with respect to all the variables. For the minimum permitted energy value [FORMULA], there exists a fixed point at [FORMULA], which is linearly elliptic (with six pure imaginary eigenvalues), as for the 2D system.

For general values of the parameters, the 2D system is non-integrable. This should be also the case for the 3-dimensional case, considering the fact that the addition of the third degree of freedom in the Hamiltonian of the 2D system does not induce a new symmetry. However, as in the 2D case, there exist special values of the variables and/or the parameters for which the system is integrable. These special cases will be the guide for the understanding of the complicated dynamics of the full system. They will be studied in detail in the following subsections.

3.1. Rectilinear case

As for the 2D system (Paper I), the planes [FORMULA], [FORMULA] and [FORMULA] are invariant by the Hamiltonian flow. The restriction of the system on these planes provides integrable 1 degree of freedom Hamiltonians. By performing a transformation in the positions and rescaling the time variable (Scuflaire 1995) we may express the three systems by the same Hamiltonian:

[EQUATION]

where the [FORMULA] correspond to [FORMULA], [FORMULA] and [FORMULA], for each case, respectively. These systems preserve all the main characteristics of the general Hamiltonian (compactness of the phase space, mirror symmetries, etc.). Their phase space is covered with periodic orbits parameterized by the corresponding energy value. In the configuration space, these orbits oscillate on straight lines along the corresponding p -axis.

Linear stability analysis:

In order to unravel the behaviour of the periodic orbits in the general system, we will undertake a linear stability analysis, when the two perturbing parameters vary. A semi-analytical study, using a perturbative Poincaré-Lindstedt approach was already applied by Scuflaire (1995) to the 2D problem. Here, instead, we rely on a numerical application of the Floquet-Lyapounov theory (Floquet 1883, Lyapounov 1892), as in Paper I. A similar study with the present was performed by Magnenat (1982), using specific values of the axial ratios. In our study, we will provide a global image of the behaviour of the periodic orbits for a wide number of axial ratios. In the following paragraphs we outline briefly the application of this theory in a Hamiltonian context, which dates back to the work of Poincaré (1892). Many of the mathematical propositions stated here can be found grouped in the introductory book of Meyer and Hall (1992) or in other more advanced works (e.g. Yakubovich & Stazhinskii 1975).

By linearizing the complete system in the vicinity of the periodic orbit we construct a system of linear differential equations with periodic coefficients. In order to study the stability of these equations, we should search for the eigenvalues of the monodromy matrix [FORMULA]. For this reason, we integrate numerically 6 linearly independent initial conditions, corresponding usually to the columns of the identity matrix I, for one period of the periodic orbit in question (we use the elegant method proposed by Hénon (1982), in order to find its exact period). The numerical integration is carried out through a Runge-Kutta 7/8th order integrator based on the formulas of Prince & Dormand (1981) (the routine Dopri8 is given by Hairer et al. (1981)), which assures a precision in the calculation of the energy of the order of [FORMULA].

For autonomous systems, one eigenvalue of the monodromy matrix is always equal to 1, as the direction along the periodic orbit is degenerate. Moreover, due to the symplectic structure of the system, the algebraic multiplicity of the eigenvalue [FORMULA] would be even and, hence, at least 2. This is simply due to the fact that an autonomous Hamiltonian system has always an integral3. Furthermore, the characteristic polynomial should be symmetric. Thus, for 3 degrees of freedom systems, the reduced polynomial has the form (Broucke 1969): [FORMULA], where [FORMULA] and [FORMULA]. The characteristic polynomial can be factored to the product of two quadratic polynomials of the form [FORMULA] and [FORMULA], where [FORMULA] are the roots of the equation [FORMULA] with [FORMULA]. The linear stability of the orbit is thereby determined completely by the roots [FORMULA], [FORMULA] and the discriminant [FORMULA] of the equation involving mu. If [FORMULA], all the eigenvalues are complex and not on the unit circle, defined by [FORMULA] and the orbit is complex unstable. On the other hand, if [FORMULA] and [FORMULA] or/and [FORMULA] the orbit is semi/doubly unstable. The only case leading to linear stability is when [FORMULA] and [FORMULA].

The case of complex instability is interesting, as it can appear only in systems with 3 or more degrees of freedom (for a review see Contopoulos (1994)). The Krein - Moser theorem (Krein 1950, 1955; Moser 1958) states the necessary conditions, in order for the eigenvalues of the linearized system to leave the unit circle. Nevertheless, in the case of the rectilinear orbits, complex instability does not occur (Magnenat 1982). We can check out that the linearized systems are composed by three independent pairs of linear equations with periodic coefficients. Hence, the monodromy matrix reduced on a Poincaré map transversal to the periodic orbit can be always written in the block form:

[EQUATION]

Consequently, the linearized system is decomposed in two 2-dimensional uncoupled systems whose eigenvalues can never become complex. Indeed, in that case the two stability indexes [FORMULA] and [FORMULA] are identified as the traces of the submatrixes [FORMULA] and [FORMULA], respectively. They designate the parametric stability, following the two directions perpendicular to the periodic orbit.

By means of the numerical approach exposed in the previous paragraphs we have calculated the stability indexes [FORMULA] and [FORMULA], for various values of the parameters [FORMULA] and [FORMULA] (not only the "physical" ones). In Fig. 1, we display the complete stability diagrams for each rectilinear orbit, on the plane [FORMULA]. The latter are often called "existence diagrams" (Contopoulos & Magnenat 1985; Patsis & Zachilas 1990), for the obvious reason that they also display the parameter space for which the orbits exist. In our case, though, the orbits exist for all [FORMULA], [FORMULA] [FORMULA].

[FIGURE] Fig. 1a-c. Complete stability diagram on the plane formed by the perturbation parameters [FORMULA] a x -axial periodic orbit b y -axial periodic orbit and c z -axial periodic orbit.

These diagrams separate the [FORMULA] -plane in 4 regions, depending on the values of the stability indexes: the bold points denote the double instability, the semi-instabilities are depicted by two kind of stars and the linear stability by the fine points. The solid lines represent the cases for which the orbit is transient, in the sense that its stability changes, i.e. when one of the stability indices crosses the (in)stability thresholds [FORMULA]. As in our case the stability indices never become less than -2, the transition lines represent all the values of [FORMULA] and [FORMULA] for which the orbit is degenerate in one or both directions (when the lines intersect). These lines delineate the exact values of the axial ratios for which we have bifurcations of the corresponding periodic orbit.

Due to the invariance of the linearized equations with an interchange of the perturbation parameters the stability diagram of the x -axial orbit is symmetric with respect to the diagonal [FORMULA], and the two other diagrams are identical with an interchange of their axes. Furthermore, in the case of the x -axial orbit, the boundaries of transition are vertical and horizontal lines in the [FORMULA] -plane, as the linearized equations defining each stability index depend only on one of the perturbation parameters. In the other two cases, only the linearized equations involving [FORMULA] depend on one perturbation parameter. For this reason, the transition lines are vertical to the [FORMULA] -axis for the y -axial orbit and to the [FORMULA] -axis for the z -axial orbit. Finally, for [FORMULA] or [FORMULA] or [FORMULA], the orbits are degenerate. In these cases the system preserves another integral of motion (a component of the angular momentum) and can be reduced to a 2 degrees of freedom system, as the one studied in Paper I.

3.2. Spherical case

By setting both [FORMULA] and [FORMULA] equal to 1, the system becomes integrable. This can be easily checked through a canonical transformation to spherical coordinates [FORMULA], [FORMULA] and [FORMULA] with [FORMULA], [FORMULA], [FORMULA] and their conjugate momenta [FORMULA]. The Hamiltonian of the system is written as a function of the new variables:

[EQUATION]

This Hamiltonian represents a classical case of a system with a spherical symmetry, where there exist two more integrals of motion in addition to the energy. Indeed, it can be shown by the equations of motion that the component of the angular momentum [FORMULA], parallel to the z -direction, represented by the variable [FORMULA], is an integral of motion. Eventually, the quantity [FORMULA], representing the square total amplitude of the angular momentum vector [FORMULA], completes the triplet of independent integrals. Due to this conservation of the angular momentum, the configuration space of the axisymmetric system is covered with planar orbits centered at [FORMULA] - the well known rosettes.

An interesting characteristic of this type of systems is that we can consider the integrals as parameters and thereby, reduce our study to the space [FORMULA] of a 1 degree of freedom system. This was the case for the 2 degrees of freedom axisymmetric system studied in Paper I. For the reasons that we explained in that work the singularity at [FORMULA] is not restrictive: the family of periodic orbits passing through the center, which are excluded from this representation, can be produced by rotations of the rectilinear orbits generated by the integrable systems discussed in the previous subsection. These orbits are completely degenerate in that case (see Fig. 1). On the other hand, there exist two more singularities at [FORMULA] and [FORMULA] which will not influence our study. In that case [FORMULA] and therefore, the singularities correspond to the restriction of the general 3D system to a family of 2D centrally symmetric systems, discussed in Paper I (see also next subsection).

As for the general system, many of the properties of the 2D axisymmetric system are preserved in its 3-dimensional extension. We can always find on the plane [FORMULA], for a fixed value of the energy [FORMULA], an elliptic fixed point [FORMULA]:

[EQUATION]

with a constant value of the total angular momentum [FORMULA]. For the considered energy value [FORMULA], the periodic orbit is defined by [FORMULA] and corresponds to a value of the total angular momentum [FORMULA]. This fixed point corresponds to a circular orbit. We can resolve explicitly the equations of motion for [FORMULA] and [FORMULA], which yield:

[EQUATION]

where [FORMULA] are the values of [FORMULA] and [FORMULA], for [FORMULA]. The parameters A, B and C are constants depending on the fixed values of [FORMULA] and [FORMULA]:

[EQUATION]

Thus, the circular orbit is periodic, with period:

[EQUATION]

In fact, Eqs. (19) and (20) define a family of periodic orbits on a sphere [FORMULA] of the configuration space, parameterized by the values of [FORMULA] and [FORMULA]. It is sufficient to choose any initial condition on this surface, with an appropriate value of the total angular momentum [FORMULA], and construct a circular periodic orbit. All these periodic orbits are linearly elliptic. When the systems is perturbed the fate of this family of periodic orbits is predicted by the Poincaré-Birkhoff fixed point theorem (Poincaré 1892; Birkhoff 1927). Indeed, only a finite number of them survive and it happens to be the orbits restricted on the perturbations of the 2D central systems, studied in the next subsection.

A first test for the precision on the determination of the fundamental frequencies provided by the NAFF algorithm can be obtained through the comparison of the numerical value given by the application of the method to the numerical solutions of a member of the family of periodic orbits and the theoretical frequency [FORMULA] given by Eq. (22). The numerical frequency is [FORMULA] and its difference with the theoretical one is of the order of [FORMULA], for an integration time span of only 800 periods, which agrees with the rigorous result of Laskar (1996) proving thus the high accuracy of the method (see also Sect. 2).

3.3. 2D centrally symmetric cases

As noted in the previous paragraphs, the planes [FORMULA], [FORMULA] and [FORMULA] are invariant by the Hamiltonian flow. If we set in the general system of Eq. (11) one pair of conjugate variables equal to 0, we restrict the system in one of the 4-dimensional hypersurfaces generated by two of these planes, which are still invariant by the Hamiltonian flow. These restricted systems have 2 degrees of freedom and their Hamiltonians are:

[EQUATION]

Let us set, now, [FORMULA] and [FORMULA] for the Hamiltonians [FORMULA] and [FORMULA], respectively, and [FORMULA], [FORMULA], [FORMULA] (or [FORMULA], [FORMULA], [FORMULA]) for the third Hamiltonian. The latter systems become equivalent to the 2D logarithmic system studied in Paper I. For [FORMULA], these systems are integrable. This can be verified by a canonical transformation to the usual polar coordinates: in that case, the corresponding component of the angular momentum is a second integral of motion, in addition to the energy.

Indeed, the 2D axisymmetric systems can be thought as restrictions of the 3D axisymmetric system on the surfaces [FORMULA], or [FORMULA], or finally, [FORMULA]. These systems preserve the characteristics of the general axisymmetric system. Their phase space is covered by the well-known loop orbits. As for the circular periodic orbit, around which the loops rotate in the phase space, it respects the same equations of motion (Eqs. (19), (20)), modified, by taking into account the restrictions mentioned above. What is important about these periodic orbits is that they are the representatives of the family of circular orbits of the integrable 3D system which persist when the perturbation parameters are different from one.

3.4. General system - types of orbits

For the general system, we use the numerical integrator mentioned above, in order to integrate the equations of motion and explore its dynamics. For the energy value used in the previous subsections ([FORMULA]), we take two values of the axial ratios which are close to 1 ([FORMULA] and [FORMULA]), in order to study the departure of our system from the integrable case. As before, the parameter [FORMULA] is set equal to 0.1.

The main drawback for the visualization of the dynamics of a higher-dimensional system (with 3 or more degrees of freedom) is that its Poincaré map is at least 4-dimensional. Nevertheless, we may try to have a hint about the systems behaviour by displaying the most representative orbits in the configuration space. The axes labeled by x, y or z are the middle, long and short. The latter designation stems from the comparison of the maximum values of the variables x, y and z and, therefore, depends on the axial ratios.

Before proceeding further to the study of the orbital morphology of the system, let us comment on the periodic orbits studied in the previous subsections. It seems clear that these orbits play a fundamental role in the structure of the phase space. As already mentioned, the rectilinear orbits exist for all values of the perturbation parameters. Indeed, they are tracing the boundaries of the phase space, as they pass from the maximum values of the variables defining them. A quick glance at Fig. 1 should convince us that the (middle) x -axial orbit is elliptic in the (long) y -direction and hyperbolic in the (short) z -direction, the (long) y -axial orbit is elliptic-elliptic, whereas the (short) z -axial orbit is hyperbolic in both directions.

Contrarily to the rectilinear periodic orbits, it is not at all obvious that the periodic orbits springing from the axisymmetric systems exist for these values of the perturbation parameters. In effect, as we do not know explicitly the flow defining these periodic orbits as soon as the axial ratios are different from 1, we are not able to prove their existence rigorously through the implicit function theorem. However, a numerical approach using a Newton's method (see Press et al. 1988) may help us to locate these periodic orbits. A further application of a numerical linear stability analysis shows that the orbits on the planes which are tangent to the middle axis ([FORMULA] and [FORMULA]) are elliptic-elliptic. Yet, the orbit restricted on the plane formed by the long and short axes [FORMULA] is elliptic-hyperbolic.

As stated in numerous previous studies (e.g. Schwarzschild 1993) the principal categories of orbits of the 3D logarithmic system are the boxes and 3 families of tubes (Fig. 2). These orbits do not only characterise the logarithmic system, but they also appear in other galactic models with ellipsoidal isopotentials, as for example the Stäckel systems (de Zeeuw 1985; Statler 1987).

The boxes are the straightforward 3-dimensional generalisations of the 2D box orbits. They have the general characteristic of forming with their edges, in the configuration space, the 8 corners of a parallelepiped and always fill the part near its center (Fig. 2a). They oscillate along the three dimensions of the configuration space. Hence, these orbits can be thought as the effect of the coupling between the three integrable systems studied in Sect. 3.1.

[FIGURE] Fig. 2a-d. Typical orbits of the logarithmic system on the configuration space [FORMULA]: a Box orbit [FORMULA] b short axis tube [FORMULA] and c,d inner and outer long axis tube orbits ([FORMULA] and [FORMULA] respectively).

The tubes, on the other hand, are the 3D extension of the loop orbits. They move around the principal axes of the configuration space. However, we may find only the long axis and the short axis tubes (here, the ones rotating around the y and z axis, respectively). The latter is due to the hyperbolicity of the periodic orbit moving around the middle axis, which, in our case, is the planar [FORMULA] orbit (see Binney (1981) for some theoretical arguments confirming this fact). We should quote here, that the long axis tubes are separated to the inner and the outer ones (Fig. 2c and 2d, respectively). We may consider that the tubes are the product of the coupling of a perturbed 2D axisymmetric system (setting the axial ratio different from 1), and the addition of a third non-rotationally symmetric dimension. Thus, in the case of tubes, a 1:1 resonance should appear between two degrees of freedom, when the system is written in the rectangular variables. However, due to the variety of their shape in the configuration space, we may anticipate that they present some fundamental differences between them, as in the case of the tubes produced by the Stäckel system (de Zeeuw 1985; Statler 1987).

Indeed, in this level of our study, many questions may be raised regarding the dynamics of these orbits. There is no doubt that the projections to the configuration space are poor guides, especially, when we have to deal with multidimensional systems as this one. Another important aspect, which we cannot account by this cursory study is the nature of the rest of the phase space which separates these presumably regular orbits. In order to clarify our ideas about these issues, we will rely on the fundamental aspect of the frequency map analysis method, i.e. the quasi-periodic approximation of a numerically determined function.

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Online publication: December 8, 1997
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