Issue 
A&A
Volume 567, July 2014



Article Number  A46  
Number of page(s)  7  
Section  Galactic structure, stellar clusters and populations  
DOI  https://doi.org/10.1051/00046361/201423813  
Published online  08 July 2014 
Conditions of consistency for multicomponent stellar systems
II. Is a pointaxial symmetric model suitable for the Galaxy?
Dept. Matemàtica Aplicada IV, Universitat Politècnica de
Catalunya,
08034 Barcelona,
Catalonia,
Spain
email:
rcubarsi@ma4.upc.edu
Received:
14
March
2014
Accepted:
14
April
2014
Under a common potential, a finite mixture of ellipsoidal velocity distributions satisfying the Boltzmann collisionless equation provides a set of integrability conditions that may constrain the population kinematics. They are referred to as conditions of consistency and were discussed in a previous paper on mixtures of axisymmetric populations. As a corollary, these conditions are now extended to pointaxial symmetry, that is, point symmetry around the rotation axis or bisymmetry, by determining which potentials are connected with a more flexible superposition of stellar populations. Under pointaxial symmetry, the potential is still axisymmetric, but the velocity and mass distributions are not necessarily. A pointaxial stellar system is, in a natural way, consistent with a flat velocity distribution of a disc population. Therefore, no additional integrability conditions are required to solve the Boltzmann collisionless equation for such a population. For other populations, if the potential is additively separable in cylindrical coordinates, the populations are not kinematically constrained, although under pointaxial symmetry, the potential is reduced to the harmonic function, which, for the Galaxy, is proven to be nonrealistic. In contrast, a nonseparable potential provides additional conditions of consistency. When mean velocities for the populations are unconstrained, the potential becomes quasistationary, being a particular case of the axisymmetric model. Then, the radial and vertical mean velocities of the populations can differ and produce an apparent vertex deviation of the whole velocity distribution. However, single population velocity ellipsoids still have no vertex deviation in the Galactic plane and no tilt in their intersection with a meridional Galactic plane. If the thick disc and halo ellipsoids actually have nonvanishing tilt, as the surveys of the solar neighbourhood that include RAdial Velocity Experiment (RAVE) data seem to show, the pointaxial model is unable to fit the local velocity distribution. Conversely, the axisymmetric model is capable of making a better approach. If, in the end, more accurate data confirm a negligible tilt of the populations, then the pointaxisymmetric model will be able to describe nonaxisymmetric mass and velocity distributions, although in the Galactic plane the velocity distribution will still be axisymmetric.
Key words: galaxies: kinematics and dynamics / solar neighborhood / galaxies: statistics
© ESO, 2014
1. Introduction
The purpose of the present work is to complete some aspects of the analysis of conditions of consistency for mixtures of axisymmetric stellar systems (Cubarsi 2014, hereafter Paper I) by studying the more general pointaxial symmetry (or bisymmetry) case, i.e., rotational symmetry of 180° for the potential and the phase space density functions.
To simplify the solution of the Boltzmann collisionless equation (BCE) it is necessary to introduce some symmetries for the mass and the velocity distributions, such as the assumptions of axisymmetry, steady state, or Galactic plane of symmetry. These hypotheses provide serious limitations for describing in a realistic way the kinematic observables of the Galaxy unless a mixture model is assumed. The conditions of consistency are integrability conditions allowing for a mixture of independent populations of generalised Schwarzschild type to share the same potential function. Since the potential may depend on the population parameters involved in the velocity distribution, the less the potential depends on them, the less kinematically constrained the populations will be.
Several kinematic analyses using the newest radial velocity data from the RAdial Velocity Experiment (RAVE) survey (Siebert et al. 2011; Zwitter et al. 2008; Steinmetz et al. 2006) confirmed that the thin disc has nonvanishing vertex deviation, the thick disc has a radial mean motion differing from that of the thin disc, and the halo velocity ellipsoid is likely to be tilted (e.g., Pasetto et al. 2012a,b; Moni Bidin et al. 2012; CasettiDinescu et al. 2011; Carollo et al. 2010; Smith et al. 2009a,b).
It was suggested (Pasetto et al. 2012b; Steinmetz 2012) that the axisymmetry assumption should be relaxed towards a model with pointaxial symmetry to account for these features. However, in Paper I we proved that an axisymmetric mixture model is able to describe the actual velocity distribution in the solar neighbourhood provided that the potential is quasistationary^{1} and the phase space density function is timedependent. This family of potentials is consistent with populations having different mean velocities producing a nonnull vertex deviation of the disc distribution. In addition, if the potential is separable in cylindrical coordinates, the velocity ellipsoids may have an arbitrary tilt.
Unlike in the axisymmetric model, in steady state pointaxial systems, nonnull radial and vertical differential motions are also possible (SanzSubirana 1987; JuanZornoza 1995). Pointaxial symmetry is indeed not a relaxation of the axial symmetry, but a more informative symmetry which may account for ellipsoidal, spiral, or bar structures, and includes axial symmetry as a degenerate case. In particular, along with a quadratic velocity distribution, a pointaxial symmetry model provides triaxial mass distributions and velocity ellipsoids with nonvanishing vertex deviation. However, a quadratic pointaxial velocity distribution is still symmetric in the peculiar velocities, so that it has null oddorder central moments. Therefore, either in axial or in pointaxial symmetric systems, a mixture model is compulsory to fit the full set of local velocity moments. Nevertheless, if each population of the mixture had a velocity ellipsoid with an arbitrary orientation, as, in principle, in the pointaxial model, a lower number of populations would likely be required to fit the overall velocity distribution.
Hereafter, this analysis is organised as follows. In the next two sections we review the solution of the BCE for a single pointaxial population. In a first step, Chandrasekhar’s system of equations provides the kinematic parameters involved in the velocity distribution function whilst, in a second step, they provide an axisymmetric potential. In the fourth section we find the general solution for the potential, in both the separable and nonseparable cases. In the fifth section we study the conditions of consistency for pointaxisymmetric mixtures. In the last section we discuss the results in contrast to the axisymmetric case.
2. Pointaxial system
For fixed position and time (r,t), a single stellar population is usually described through a Gaussian velocity distribution function, which is a particular case of a generalised quadratic velocity distribution function in terms of the peculiar velocities (u_{1},u_{2},u_{3}), that is, f(Q + σ) with Q = ∑ _{i,j}A_{ij}(r,t) u_{i}u_{j}, where A_{ij} are the elements of a symmetric, positive definite secondrank tensor. Under pointaxial symmetry these functions satisfy, in cylindrical coordinates, A_{ij}(ϖ,θ,z,t) = A_{ij}(ϖ,θ + π,z,t), likewise the function σ and the components of the mean velocity v.
For the above generalised Schwarzschild velocity distribution, the BCE yields the Chandrasekhar equations (Chandrasekhar 1960), which are equivalent to the moment equations (Cubarsi 2007, 2010). Their solution provides the tensor A, the function σ, the mean velocity v, and the potential U. The two first Chandrasekhar equations are Eqs. (1) and (2) in Paper I, that may be written, with the notation used in Paper I and using the variable Δ = A·v, as which yield the elements of the secondrank tensor A and the vector Δ. In the pointaxial model these elements have the functional form (JuanZornoza et al. 1990; JuanZornoza & SanzSubirana 1991) (3)and (4)with (5)where k_{1},k_{3},q,ϕ_{1} are time dependent functions and k_{2},k_{4},n, ϕ_{2},β constants. The condition that A is positivedefinite implies that k_{1},k_{3} are positive functions and k_{2},k_{4} nonnegative constants, with the requirements k_{1}>q ≥ 0, k_{4}>n ≥ 0. If K_{4} is null, the velocity distribution is independent from z and, in addition, if k_{2} is null, then it is independent from ϖ too, which makes no sense in a threedimensional and finite Galaxy. Thus, in general, these constants are assumed to be positive, with the exception of the limiting case K_{4} = 0 of a twodimensional disc distribution^{2}. The particular case k_{2} = 0 will not be considered here, although it is discussed at the end of the Conclusions section. It would correspond to a particular stellar component with constant angular rotation at fixed height z, similarly to the axisymmetric model.
The uppercase letter K is used for a function also depending on θ. The accents mean derivatives with respect to the angle and the dots with respect to the time. As expected, the functional form of A is similar to the axisymmetric case in Paper I, with the difference that some parameters, those written in capital letters, have an additional term depending on cos2θ and sin2θ, responsible for the rotational symmetry of order 2.
As in the axisymmetric case, the model also provides a time dependent parameter k_{5} that determines a plane of symmetry for the tensor A at z = − k_{5}/k_{4}. Without loss of generality (Camm 1941) this symmetry plane may be assumed as the Galactic plane, being fixed by taking k_{5} = 0, resulting then in a symmetric velocity distribution about this plane. Therefore, the pointaxisymmetric model, with the inherent symmetry plane, also possesses pointtopoint central symmetry.
3. Equations for the potential
The remaining Chandrasekhar equations are Eqs. (3) and (4) in Paper I, which provide the potential U and the function σ. They can be written more easily by using the variable as follows: By elimination of between Eqs. (6)and (7), with the new variables and , which are appropriate to the symmetry plane of the system, six secondorder partial differential equations for the potential are obtained. In their vector notation they can be found in Chandrasekhar (1960, Eqs. (3.448) and (3.450), p.100). After substitution of the elements of A and the components of Δ, SanzSubirana (1987) and JuanZornoza (1995) proved that continuity conditions on the function force the potential to be axisymmetric. A similar result was obtained by Vandervoort (1979) for pointaxial systems, which he called galactic bars, although the study was limited to a twodimensional disc with a steadystate potential.
3.1. The potential is axisymmetric
These equations are explicitly written in the Appendix. Their solution is tedious and long, and, unfortunately, the abovementioned thesis papers cannot be accessed easily. Since this is one of the key properties of the pointaxial model, we shall see a shorter and alternative justification to this crucial fact.
We note that in the Galactic plane ζ = 0, the three Eqs. (28)–(30)in the Appendix are reduced to Eq. (30)by providing the basic dependence of the potential on the radius and the angle variables. Hence, we focus on this equation in its complete form. First, we consider the main case , hence q ≠ 0, since if K_{1} does not depend on θ, the velocity distribution has no vertex deviation in the Galactic plane^{3}, which was one of the most important observables that justified trying a noncylindrical model.
If is nonnull, we write Eq. (30)as (8)We define the function and bear in mind that the continuity and differentiability of the potential, at least up to the second derivative, implies that V is also differentiable. In the Galactic plane, the foregoing equation becomes (9)Since K_{1} is a πperiodic function of the angle, a simple recall to the mean value theorem provides us with a value θ_{0} ∈ [ 0,π) for which . Then, if V and are nonnull functions, in order to avoid any singularity, the righthand side member of the above equation must vanish, at least for θ = θ_{0}. We see that the potential does not satisfy If so, the solution would be that of the wave equation in the new variable x = lnτ, hence the solution satisfies U = F_{1}(x + 2θ) + F_{2}(x − 2θ), but the potential is a onevalued function and a periodic function of θ with period^{4}2π; therefore, U(x + 2θ) = U(x + 2(θ + 2kπ)) = U((x + 4kπ) + 2θ) is fulfilled for all k ∈ Z. However, as the Galaxy is of finite extent, such a potential taking the same value at all points x + 4kπ is unrealistic.
On the other hand, the righthand side of Eq. (9)is nonnull. If so, it would be a linear and homogeneous differential equation in V, with solution , which is discontinuous at . In particular, when , according to Eq. (5), the singularity takes place at . It is worth noticing that for k_{2} = 0 such a singularity does not exist, so that we might have noncylindrical potentials in that degenerate case. Therefore, the only admissible, continuous, and differentiable solutions to Eq. (9)are axisymmetric potentials satisfying V = 0, otherwise, in the Galactic plane, the potential is not differentiable.
This means that, the axial symmetry is the way that the differential equations for the potential avoid the singularity produced by any root of the function . It is actually a situation similar to the axisymmetric model, where the equations for the potential, in the quasistationary case in Paper I, did avoid the singularity produced by the zero of the timedependent function by providing a solution that does not depend on .
The case where is null, consequently also vanishes, requires that be nonnull, otherwise the velocity distribution is axisymmetric. Similarly, as in the above case, there is also an angle θ_{1} for which that would produce a singularity in the solution of Eq. (8), for ζ ≠ 0, unless the potential is axisymmetric^{5}.
4. Potential
Therefore, in a rotating pointaxial system, the potential consistent with a quadratic velocity distribution is still axisymmetric. The set of partial differential equations for the potential in the Appendix generalises the ones for the axisymmetric model in Paper I. We write these equations once they are simplified by taking advantage of the potential satisfying . The first three equations, derived from Eqs. (28)–(30), are (10)
(12)The remaining three equations, obtained from Eqs. (31)–(33), are (13)
(15)However, Eqs. (11)and (12)can be simplified further. By taking the θderivative in Eq. (10)and subtracting from Eq. (11), we get (16)Also, by taking into account Eq. (12), we get (17)Similarly, Eq. (15)can be expressed in a simpler form. By taking the θderivative in Eq. (13)and subtracting from Eq. (15)we get , which does not add any new condition to the previous equation.
Therefore, the equations for the potential in the pontaxial model are the set of Eqs. (10)–(14), which are similar to the ones of the axial case (Paper I, Eqs. (7)–(9)), with the additional integrability conditions given by Eqs. (16)and (17). We note that the equations for the potential do not depend on the parameters and . Under axial symmetry, the conditions depending on the θderivatives and are identically null. Thus, in a mixture model these equations are similarly planned for each population component, and depend on the respective population parameters K_{1}(θ,t), k_{3}(t), and K_{4}(θ).
In the axisymmetric case, when applying the conditions of consistency for a flat velocity distribution, that is, for a potential independent from the population parameter K_{4}, the potential becomes dramatically simplified. In the pointaxial case, we shall see that a similar reasoning and solution are inherent to the pointaxial symmetry assumption, since the reasoning can be done in regard to the angle dependence as well as to the population dependence of the parameters. In other words, a pointaxial system is consistent with a flat velocity distribution unless it degenerates towards an axisymmetric system.
Thus, being at least one of the population parameters K_{1},K_{4} functions of the angle θ (otherwise the system is axisymmetric), Eq. (10), once divided by K_{4}, becomes separated into two parts, one independent from θ and the other depending on θ, which must be null separately^{6}, (18)These equations are equivalent to the conditions of a potential independent from K_{4} in Paper I, Eqs. (12) and (14). The latter equation leads to the two typical cases of a potential additively separable in cylindrical coordinates, or a nonseparable potential.
4.1. Separable potential
The separable potential satisfies In the pointaxial model, at least one of the parameters K_{1} or K_{4} depends on the angle. In particular, if , owing to Eq. (11), the potential must be separable, otherwise would be held, rendering the axisymmetric model.
For a separable potential, either with null or nonnull, we are led to the same equations as for the axisymmetric case in Paper I, Eq. (15), with the addition of Eqs. (16)and (17), which add the new condition , yielding a separable potential in their harmonic form (19)where continuity conditions in the Galactic plane have been applied in order to neglect the term proportional to . Therefore, the separable potential reduces to the simple case of the harmonic function, and does not depend on the population kinematic parameters except for the unique function A(t) discussed in Paper I.
Hence, under a separable potential, the kinematics of a pointaxial symmetric system is totally free from conditions of consistency in regard to a mixture of populations. The population’s mean velocities, the semiaxes of the velocity ellipsoids, and their orientations remain unconstrained.
4.2. Nonseparable potential
The nonseparable potential satisfies and Then, . According to Eqs. (16)and (17), the pointaxial symmetry assumption requires . Hence, Eqs. (11)and (12)provide the conditions (20)which separate Eq. (18)into two identically null equations. Hence, we can consider only one of them. Similarly, the same reasoning of the preceeding section (either in regard to the dependency on the angle or on the population) applied to Eqs. (13)–(15)yields the conditions Thus, we reach the same set of equations as for an axisymmetric model consistent with a flat velocity distribution (Paper I, Eqs. (13) and (15)), by providing the potential , although in the pointaxial model we still have to submit it to Eq. (20). Therefore, the resulting potential must adopt the separable form U = f_{1}(τ + ζ) + f_{2}(ζ), so that (23)where, by continuity conditions in the Galactic plane, an additional term proportional to is neglected.
5. Conditions of consistency
For a separable potential, there are no conditions of consistency, similarly to the axisymmetric model.
For a nonseparable potential, all the system dependency on θ is carried through K_{4}(θ). Therefore, according to Eq. (3), and bearing in mind that , in the Galactic plane the tensor elements A_{ϖθ} and A_{θz} are null, as in the axisymmetric model. Hence, the velocity ellipsoid has no vertex deviation in z = 0. In addition, according to Paper I, since K_{1} = k_{3} the ellipsoid has no tilt in a meridional Galactic plane (i.e., the intersection of the ellipsoid with a meridional Galactic plane has an axis pointing toward the Galactic centre), and the mean velocities Π_{0} and Z_{0} are the same as in the axisymmetric case. In the Galactic plane, the only moment depending on θ is μ_{zz}, whilst Θ_{0} and the other second moments are also axisymmetric.
In summary, in the Galactic plane the velocity distribution of such a stellar system is basically axisymmetric and does not provide the most important feature we expected a pointaxial system should provide, that is, the vertex deviation.
Similarly, as for the axisymmetric case, the potential of Eq. (23)constrains the mean velocity components Π_{0} and Z_{0} to satisfy . For a two population mixture we get and , unless, according to Paper I, the function k(t) is linearly independent among populations and the potential does not depend on . In that case, an apparent vertex deviation of the mixture distribution is possible. The potential allowing unconstrained population mean velocities must then satisfy the condition (24)obtained in Paper I, and the potential takes the quasistationary form (25)with B = const, which is a particular, spherical case in Paper I, Eq. (31).
6. Conclusions
The conditions of consistency studied in Paper I proved that a finite mixture of stellar populations was able to describe the main features of the local velocity distribution without having to change the axisymmetry hypothesis. However, in the Galactic plane, single populations had velocity ellipsoids without vertex deviation, so that the apparent vertex deviation of the disc velocity distribution was the result of different radial and rotation mean motions of the populations. Now, as a corollary, we have investigated the same problem under pointaxial symmetry, in order to see how it might improve the velocity distribution approximation.
For the pointaxial symmetry case, the local kinematic features are similarly derived from a mixture of stellar populations, each one according to a quadratic velocity distribution in the peculiar velocities satisfying the BCE, with a common potential allowing for the populations to be kinematically independent. This means that the populations should differ not only in rotation, but also in radial and vertical mean motions. Under the pointaxial hypothesis we should also expect single populations with velocity ellipsoids having nonnull vertex deviation and nonvanishing tilt, as well as a pointaxial mass distribution.
An important fact is that the potential must be axisymmetric in order to support a quadratic integral of motion for each population, which usually represents a stellar system in statistical equilibrium. That is, we assume that the stellar system has achieved relaxation and satisfies regularity conditions about the definition of the local standard of rest, continuity, and differentiability of its velocity, and that higherorder velocity moments exist. Although dissipative forces related to third and oddorder moments does not appear in the moment equations planned for a single population, they are indirectly connected with the assumption of the mixture model.
The first result we obtain is that the pointaxial symmetry is, in a natural way, consistent with the flat velocity distribution of a disc population, by providing potentials not depending on the population parameter K_{4}, which is responsible for nonisothermal velocity distributions. In axisymmetric systems, only a particular family of potentials is consistent with a flat velocity distribution, while in pointaxial systems any potential always is.
We find two possible solutions depending on the separability of the potential:

(a)
The pointaxial model admits a potential additively separablein cylindrical coordinates that is the harmonic potential. As in theaxisymmetric model, for a separable potential there is no need ofconditions of consistency in regard to a mixture distribution,since the potential only constrains the population parametersthrough the function A(t) (Paper I, Eq. (20)). For each population, the radial and vertical mean velocities can be different, and their velocity ellipsoids can have different orientations, including the both vertex deviation and tilt.

(b)
For a nonseparable potential, the condition given by Eq. (24)provides nearly nonconstrained population kinematics, by leading to a spherical and quasistationary potential. Then, the radial and the vertical mean velocities can differ among populations, although they are coupled, and they may produce an apparent vertex deviation of the whole velocity distribution. However, single population velocity ellipsoids have no vertex deviation in the Galactic plane and no tilt in their intersection with a meridional Galactic plane, similarly to the axisymmetric case.
In both of these cases, the potential for the pointaxial model becomes a particular function of the potential for the axisymmetric model. The nonseparable potential loses the dependency on the elevation angle, and the separable potential loses the nonharmonic term that they showed in Paper I (Eqs. (31) and (33), respectively).
We can check the foregoing cases according to the main local kinematic trends analysed in Paper I. Against option (a) is the evidence that there are halo stars near the Sun with no net rotation velocity for which the harmonic potential is not able to support their orbits. On the other hand, option (b) really provides a potential, Eq. (25), with a nonharmonic term, which may be associated with a repulsive force (if B> 0) produced by the outer dark matter halo, as discussed in Paper I. This allows stable orbits for stars with no net rotation. However, the potential forces the population velocity ellipsoids to point toward the Galactic centre, although, out of the Galactic plane, the ellipsoids may show some vertex deviation.
Then, according to the pointaxial model, how can we explain that the thick disc and the halo ellipsoids have no vanishing tilt, as CasettiDinescu (2011), Carollo et al. (2010), Fuchs et al. (2009), Smith et al. (2009a), and Siebert et al. (2008) suggest? By assuming that the harmonic potential is not realistic, under the pointaxial model we cannot explain it. Similarly, the pointaxial model is unable to explain the trend of the moment μ_{ϖz} for the thin disc population described by Pasetto et al. (2012b), which was only possible under a separable potential, as discussed in Paper I.
Conversely, if the thick disc and the halo ellipsoids actually have a nonvanishing tilt, the axisymmetric model is capable of making a reasonable approach to the local features of the local velocity distribution. Therefore, we must conclude that the velocity distribution in the solar neighbourhood reflects a basically axisymmetric Galaxy.
Nevertheless, we should not discard the fact that some of the stellar samples used to describe the thick disc and the halo could have stars that were not sufficiently mixed to produce well defined velocity ellipsoids, or were contaminated by disc stars, as Smith et al. (2009a) suggest. If newer and more accurate analyses yielded nontilted velocity ellipsoids for the thick disc and the halo, both models would be capable of describing the local velocity distribution from a nonseparable potential, which, in all cases, would provide an axisymmetric velocity distribution in the Galactic plane.
However, for the stellar density, pointaxial symmetry matters. We may assume a Schwarzschild velocity distribution without loss of generality to discuss the shape of mass distribution. In that case, the stellar density is (Eq. (40) in Appendix A.2 in Paper I) and depends on the angle through K_{4}(θ). Leaving aside the simple and unrealistic case of a separable and harmonic potential, for the nonseparable potential with k ≡ K_{1} = k_{3}, Eq. (25)is a particular case of Eq. (23)with . The function σ involved in the stellar density satisfies (26)For ζ = 0, σ does not depend on θ. However, for ζ = 0, we have (27)so that, in the Galactic plane, the stellar density N depends on θ. This dependency of the mass distribution on the angle is balanced out by the velocity distribution, which also depends on θ, while the potential maintains the axisymmetry. This is the only basic feature that the pointaxial model adds to the axisymmetric model. While, in the Galactic plane, for the velocity distribution, according to Eqs. (3) and (27), the tensor element A_{zz} and detA, which depend on θ, lead to moments μ_{ϖϖ},μ_{zz} also depending on the angle, although each population component is unable to provide a nonvanishing moment μ_{ϖθ}. Thus, similarly to the axisymmetric case, for ellipsoidal velocity distributions under a nonseparable potential, the apparent vertex deviation of the velocity distribution is a consequence of the coexistence of two or more populations with different radial and rotation mean velocities.
Then, a pointaxisymmetric stellar system would, in principle, be able to show a triaxial or barlike structure in any of the population’s mass distributions. It would be a matter of time that, for the specific population, the rotation curve could transform a barlike into a spirallike structure. In that situation, the pointaxial model would have the ability to describe a pointaxial mass distribution, while the axial model would not. However, we note that the degenerate case of a rigid rotating bar, with k_{2} = 0, which was the only remaining case that could admit, a priori, a noncylindrical potential as a solution of Eq. (9), cannot coexist with threedimensional ellipsoidal velocity distributions under a common differentiable pointaxial potential. Hence, any phase mixing process involving a rigid rotating bar with a noncilyndrical potential must be considered as a state previous to statistical equilibrium, which is not associated with stellar populations having ellipsoidal velocity distributions, even with pointaxial symmetry.
Appendix
The first three partial differential equations for the potential, obtained by taking the curl in Eq. (6), are (28)where a common factor proportional to was simplified; (29)where a common factor proportional to was simplified; and (30)Those equations which were proportional to become null at the Galactic plane.
The remaining three equations, which are obtained by taking the gradient in Eq. (7)and the time derivative in Eq. (6), are (31)(32)in the last one, a common factor was also simplified; and (33)These equations complete the first set of three equations given by SanzSubirana (1987). They may be simplified by assuming that the parameter β is constant, which was actually derived by this author in solving them for the separable potential case, and by JuanZornoza (1995) for the general case. This fact is not relevant for our purpose.
In Paper I, the asymptotic case K_{4} → 0 was called flat velocity distribution, which, according to Chandrasekhar (1962), applies to the velocity distribution of an ideal disc. Although a disc stellar population can be approximated by this model, the other populations have a velocity distribution that must depend on z. Therefore, in general we must assume that K_{4} is nonnull.
In this case, we first prove that, if , the equation , with the variables x = lnτ,y = ζ/τ, provides a solution proportional to an exponential function on the argument x + θ, which is not periodic and, hence, unacceptable. We then verify that the remaining terms of Eq. (8)do not vanish, otherwise, its solution, which takes the general form , has discontinuities either at ζ = 0 or at points satisfying .
Acknowledgments
The author wishes to thank the anonymous referee for his/her useful comments that have contributed to improving this paper.
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