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In the present paper, using a generalization of the Miyamoto and Nagai potential we adjusted the observed rotation curves of three specific spiral galaxies to the analytical circular velocities. The observational data have been taken from a 21 cm-line synthesis imaging survey using the Westerbork Synthesis Radio Telescope, for three particular galaxies in the Ursa Major cluster: NGC 3726, NGC 3877 and NGC 4010. Accordingly, the dynamics of the system is analyzed in terms of the Poincaré sections method, finding that for larger values of the angular momentum of the test particle or lower values its total energy the dynamics is mainly regular, while on the opposite cases, the dynamics is mainly chaotic. Our toy model opens the possibility to find chaotic bounded orbits for stars in those particular galaxies.© 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.

En el presente trabajo, utilizando una generalización del potencial de Miyamoto-Nagai, se ajustan las curvas de rotación observadas de tres galaxias espirales a las velocidades circulares analíticas. Los datos observacionales se tomaron de un conjunto de imágenes de línea de 21 centímetros (o línea HI) obtenidos con el Westerbork Synthesis Radio Telescope (WSRT), para tres galaxias particulares en el grupo de la Ursa Major: NGC 3726, NGC 3877 y NGC 4010. Seguidamente, se analiza la dinámica del sistema en términos del método de secciones de Potincaré, encontrando que para valores grandes del momento angular de la partícula de prueba o valores bajos su energía total, la dinámica es principalmente regular, mientras que en los casos opuestos, la dinámica es principalmente caótica. Nuestro modelo abre la posibilidad de encontrar orbitas´ caóticas acotadas para estrellas presentes en esas galaxias particulares. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.

Since the seminal paper by

One advantage of an analytical galaxy model is the possibility to study the dynamics (regular or chaotic) of orbits. This can be considered one of the standing problems in galactic dynamics because it could allow us to understand the formation and evolution of galaxies (

In this paper, we are interested in meridional motions of free test particles (stars) in presence of analytical realistic galaxy models. Our models possess axial symmetry, which is a good approximation given the morphology of galaxies that are mainly approximate figures of revolution. Additionally, the galaxy components were not added one by one, instead of this, we derived a generalized Miyamoto-Nagai model that can be adjusted very accurately to fit the observed rotation curve and hence it is assumed that all (or most of) the components are taken into account. The determination of the specific values of the coeﬃcients of the series expansion let us calculate the corresponding surface densities and all the kinematic quantities characterizing the particular galaxy models. Unlike the models derived by

The paper is organized as follows: in the first section, we derive the generalized Miyamoto-Nagai model. Next, from the new potential the explicit expressions for the physical quantities of interest are determined. In the second section we adjust the observed rotation curves of three specific spiral galaxies (NGC 3726, NGC 3877 and NGC 4010) to the analytical circular velocities derived with our model. Then, the massdensity profiles are calculated, along with the vertical and epicyclic frequencies, showing that our model not only is well-behaved but also satisfy the stability conditions. A dynamical analysis in terms of the Poincaré surfaces of section is performed in the third section. Finally, in the fourth section, we summarize our main conclusions.

Let us start considering the axially symmetric Laplace’s equation in spherical coordinates

whose general solution reads as

where _{
l
} and _{
l
} are constants to be determined, P_{l} are the Legendre polynomials, and the notation (

Since Φ(r, θ) denotes the gravitational potential of an axisymmetric finite distribution of mass, the boundary condition lim_{r→∞} Φ(r, θ) = 0 must be satisfied, thus the solution (2) takes the form

Following

and applying the additional transformation (

with

Once the potential has been specified, the mass-density distribution Σ can be calculated directly from Poisson equation,

while the circular velocity v of particles in the galactic plane, the epicyclic frequency k, and the vertical frequency ν of small oscillations about the equilibrium circular orbit, can be obobtained from the following expressions evaluated at z = 0 (

From (8-10), it is important to emphasize that a feasible model must satisfy the constraints set by the conditions v^{2} ≥ 0, k^{2} ≥ 0, and ν^{2} ≥ 0, where the last two inequalities are understood as stability conditions (

As is evident from the preceding paragraphs, the galactic models and its associated physical quantities are uniquely determined by the set of constants a, b, B_{0}, B_{1}, B_{2}, and B_{3}, which (taking a pragmatic approach) can be estimated from the observational data of the corresponding rotation curves, as we will discuss in detail in the next section.

The observational data were taken from _{d} as the given by the largest tabulated value of the data.

Thus, introducing dimensionless variables

In panels (a) of

It is a well-known fact that using rough estimates of the dimensions of typical stars and galaxies, the collision interval between stars is about 10^{8} times longer than the average age for most galaxies (

The orbital motion of a test particle in an axisymmetric potential is governed by the Lagrangian

with (R, ø, z) the usual cylindrical coordinates. The generalized canonical momenta read as

and the Hamiltonian takes the form

With

Here, L_{z} = p_{φ} =constant, denotes the conserved component of angular momentum about the z-axis.

From (13), the resulting Hamilton’s equations of motion can be expressed as

where Φ(R, z) is given by

Since the Hamiltonian is autonomous, H is an integral of motion

with h the energy of an orbit.

The existence of an analytic integral of motion reduces the phase space dimensionality, and hence the Poincaré surface of section is an appropriate and well-established method to analyze the dynamics of the system. Taking into account the axial symmetry associated to the system, it is customary to choose the equatorial plane z = 0 as the Poincaré plane in order to represent the surface of sections in the (^{−14}. In all cases we set z_{0} = p_{R0} = 0 and we scan the phase space with a large number of initial conditions for the radii R_{0}, these three values allow us to determine the values of p_{z0} through the relation (19).

The transition from regularity to chaos (or viceversa) that takes place for the three considered galaxy models was inspected through the Poincaré sections in _{z} (_{z} and energy h of the test particle. In particular, from the surfaces of section presented in _{z} , i.e. if there exists a chaotic sea the increase of L_{z} will fill the phase with KAM islands, while the opposite eﬀect is observed for larger values of energy h (see

In the present paper, using the general solution to the Laplace equation, we have derived a generalized Miyamoto-Nagai potential. By means of the nonlinear least square fitting, the analytical velocity curves were adjusted to the observed ones of three specific spiral galaxies: NGC 3726, NGC 3877 and NGC 4010. The resulting analytical models were used to determine the mass-density distributions and the vertical and epicyclic frequencies, showing that unlike the results presented in

On the other hand, by using the Poincaré section method we have also studied the dynamics of the meridional orbits of stars in presence of the gravitational field of the galaxy models. From our results it may be inferred that there exists an increase in the regularity of the orbits for larger values of the angular momentum, while for larger values of energy the orbits tend to be more chaotic. Our toy models suggest that in the three galaxy models chaotic orbits are possible, however the chaotic behavior is very weak for the NGC 3877 model in comparison to NGC 3726 and NGC 4010. It should be noted that none of the studied models showed a fully chaotic phase space. Our results could have significant implications for the study of the dynamics and kinematics of these three specific galaxies, since the regular or chaotic behaviors could shed lights into the evolution and structure of these galaxies, i.e., in phase space, regular orbits are trapped in the vicinity of neighbor orbits, while chaotic orbits, by its own nature, will diverge exponentially in time from its neighbors by filling the phase space in an erratic manner.

We would like to thank the anonymous referees for their useful comments and remarks, which improved the clarity and quality of the manuscript. FLD, SMM and GAG gratefully acknowledges the financial support provided by COLCIENCIAS (Colombia) under Grants No. 8840 and 8863.

All authors make substantial contributions to conception, de-sign, analysis and interpretation of data. All authors participate in drafting the article and reviewed the final manuscript.

The authors declare that they have no conflict of interest.