Orbital Dynamics in Realistic Galaxy Models

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 doi: http://dx.doi.org/10.18257/raccefyn.774

Original article Physical Sciences Physical Sciences

Orbital dynamicsOrbital in realistic dynamics galaxy in realistic models: galaxy NGC models: 3726, NGC Physical 3877 Sciences and Orbital dynamicsNGC in realistic 3726, NGC galaxyNGC 3877 models: 4010 and NGC NGC 4010 3726, NGC 3877 and ∗ FredyFredy L. Dubeibe L. Dubeibe1,1 Sandra, Sandra M. M.NGC Mart´ınez-Sicach´a Martínez-Sicachá 4010 2, 2 , Guillermo Guillermo A. A.González Gonz´alez2,* 2

1 Grupo1 de Investigación Cavendish, Facultad1 de Ciencias Humanas y de la Educación,2 Universidad de los Llanos, Villavi2 ∗ cencio, Colombia 2 Grupo de InvestigaciónFredy Cavendish, L. Dubeibe Facultad, de Sandra Ciencias M. Humanas Mart´ınez-Sicachá y de la Educación,, Universidad Guillermo de los A. Llanos, González Villavicencio, Colombia. Grupo de2Grupo Investigaciónen de Investigación Relatividad en Relatividad y Gravitación, y Gravitación,Physical Escuela Escuela Sciences de F´ısica, de Física, Universidad Universidad Industrial Industrial de de Santander, Santander, A.A.Bucaramanga,678, Bucaramanga, Colombia. Colombia 1Grupo de Investigación Cavendish, Facultad de Ciencias Humanas y de la Educación, Universidad de los Llanos, Villavicencio, Colombia 2Grupo de Investigaciónen Relatividad y Gravitación, Escuela de F´ısica, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia

Orbital dynamics in realistic galaxy models:Abstract NGC 3726, NGC 3877 and NGC 4010 InAbstract 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 1 2 2 ∗ Fredy L. Dubeibe , Sandra M. Mart´ınez-Sicachá , GuillermodataIn the have present been A. paper, Gonzáleztaken usingfrom a generalization 21 cm-line synthesis of the Miyamoto imaging survandey Nagai using potential the Westerbork we adjusted Synthesis the observed Radio 1 Telescope,rotation curves for three of particularthree specific galaxies spiral in galaxies the Ursa to Major the anal cluster:ytical NGCcircular 3726, velocities. NGC 3877 The and observational NGC 4010. Grupo de Investigación Cavendish, Facultad de Ciencias Humanas y de la Educación, Universidad de los Llanos, Villavicencio, Colombia 2 data have been taken from a 21 cm-line synthesis imaging survey using the Westerbork Synthesis Radio Grupo de Investigaciónen Relatividad y Gravitación, Escuela de F´ısica, Universidad IndustrialAccordingly, de Santander, the dynamics A.A. 678, of Bucaramanga, the system is Colombia analyzed in terms of the Poincarésections method, finding that forTelescope, larger values for three of the particular angular momentum galaxies inthe of the Ursa test Major particl cluster:e or lower NGC values 3726, its NGC total 3877 energy and the NGC dynamics 4010. isAccordingly, mainly regular, the dynamics while on ofthe the opposite system cases,is analyzed the dynamics in terms ofis the mainly Poincarésections chaotic. Our method, toy model finding opens that the possibilityfor larger to values find of chaotic the angular bounded momentum orbits for of stars the test in those particlparticulare or lower galaxies. values its© total2019. energy Acad. Colomb. the dynamics Cienc. Abstract Ex.is Fis. mainly Nat. 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. Key words: Stellar dynamics; Galaxies: kinematics and dynamics; Nonlinear dynamics and chaos. 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 circularDinámicaKey velocities.words: orbitalStellar The en dynamics; observational modelos Galaxies: realistas kinematics de galáxias: and dynamics; NGC 37 Nonl26, NGCinear dynamics 3877 y NGC and chaos. 4010 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: NGCResumenDinámica 3726, NGC orbital 3877 and en modelos NGC 4010. realistas de galáxias: NGC 3726, NGC 3877 y NGC 4010 Accordingly, the dynamics of the system is analyzed in terms of the Poincarésections method, finding that for larger values of the angular momentum of the test particle or lowerEn valuesResumen el presente its total trabajo,energy the utilizando dynamics una generalización del potencial de Miyamoto-Nagai, se ajustan las is mainly regular, while on the opposite cases, the dynamics is mainlycurvas chaotic. de rotaciónobservadas Our toy model opens de the tres galaxias espirales a las velocidades circulares anal´ıticas. Los datos En el presente trabajo, utilizando una generalización del potencial de Miyamoto-Nagai, se ajustan las possibility to find chaotic bounded orbits for stars in those particularobservacionales galaxies. se tomaron de un conjunto de imágenes de l´ınea de 21 cent´ımetros (o l´ınea HI) obtenidos curvas de rotaciónobservadas de tres galaxias espirales a las velocidades circulares anal´ıticas. Los datos con el Westerbork Synthesis Radio Telescope (WSRT), para tres galaxias particulares en el grupo de la observacionales se tomaron de un conjunto de imágenes de l´ınea de 21 cent´ımetros (o l´ınea HI) obtenidos Key words: Stellar dynamics; Galaxies: kinematics and dynamics; NonlUrsainear Major: dynamics NGC and 3726, chaos. NGC 3877 y NGC 4010. Seguidamente, se analiza la dinámica del sistema en con el Westerbork Synthesis Radio Telescope (WSRT), para tres galaxias particulares en el grupo de la términos del método de secciones de Poincaré,encontrando que para valores grandes del momento angular Dinámica orbital en modelos realistas de galáxias: NGC 3726,Ursa NGC Major: 3877NGC y NGC 3726, 4010 NGC 3877 y NGC 4010. Seguidamente, se analiza la dinámica del sistema en detérminos la part´ıcula del método de prueba de secciones o valores de bajos Poincaré,encontran su energ´ıa total,dola que dinámica para valores es principalmente grandes del momento regular, angular mientras que en los casos opuestos, la dinámica es principalmente caótica. Nuestro modelo abre la posibilidad de Resumen de la part´ıcula de prueba o valores bajos su energ´ıa total, la dinámica es principalmente regular, mientras encontrarque en los órbitas casos caóticas opuestos, acotadas la dinámica para es estrellas principalmente presentes ca enótica. esas galaxiasNuestro modeloparticulares. abre la posibilidad© 2019. Acad. de En el presente trabajo, utilizando una generalización del potencialColomb. deencontrar Miyamoto-Nagai, Cienc. órbitas Ex. caóticasFis. se Nat. ajustan acotadas las para estrellas presentes en esas galaxias particulares. curvas de rotaciónobservadas de tres galaxias espirales a las velocidadesPalabras circulares clave: anal´ıticas.Dinámica Los estelar, datos Galaxias: cinemáticaslopes. y dinámica, More recently, DinámicaVogt no lineal & Letelier y caos. (2005) derived an anof section, showing that the orbital motion exhibits a strong observacionales se tomaron de un conjunto de imágenes de l´ınea de 21Palabras cent´ımetros clave: (o l´ıneaDinámica HI) obtenidos estelar, Galaxias: cinemáticaalytical y dinámica, expression Dinámica for the no linealgravitational y caos. field of galaxies, based dependence on the angular momentum and energy of the test con el Westerbork Synthesis Radio Telescope (WSRT), para tres galaxias particulares en el grupo de la on the multipolar expansion up to the quadrupole term. Using particles (stars). Ursa Major: NGC 3726, NGC 3877 y NGC 4010. Seguidamente, se analiza la dinámica del sistema en Introduction aanalytical different approach, models whichGonzález closelyet approximate al. (2010) obtained the light a family distri- términos del método de secciones de Poincaré,encontrando que para valores grandes del momento angular The paper is organized as follows: in the first section, we de- Introduction ofanalyticalbution finite for thin-discs models spherical whichmodels and closelyelliptical for four approximate galaxiesgalaxies, in respectivel the Ursalighty. majodist Ari- fewr de la part´ıcula de prueba o valores bajos su energ´ıa total, la dinámica es principalmente regular, mientras rive the generalized Miyamoto-Nagai model. Next, from the que en los casos opuestos, la dinámica es principalmente caótica. Nuestro modelo abre la posibilidad de clusterbutionyears later, infor which sphericalLong the & andcircular Murali elliptical velocities(1992) galaxies, presented were respectivel adjusted an analytical toy. fit Athe few po- Since the seminal paper by Miyamoto & Nagai (1975), the new potential the explicit expressions for the physical quanti- encontrar órbitas caóticas acotadas para estrellas presentes en esas galaxias particulares. observedyears later, rotationLong &curves. Murali (1992) presented an analytical po- literatureSince the on seminal three-dimensional paper by Miyamoto analytical & models Nagai for(1975), the grav- the tential for barred galaxies that reduces to the Miyamoto-Nagai ties of interest are determined. In the second section we adjust tential for barred galaxies that reduces to the Miyamoto-Nagai itationalliterature field on of three-dimensional different types of analytical galaxies has models grown for consi the grav-der- Onedisk advantage by an appropriate of an analytical setting galaxy of the model free parameters, is the possibil- while the observed rotation curves of three specific spiral galaxies Palabras clave: Dinámica estelar, Galaxias: cinemática y dinámica, Dinámica no lineal y caos. diskDehnen by an appropriate setting of the free parameters, while ably.itational In this field respect, of different particular types of attention galaxiesdeserve has grown the consi modder-els ity to study(1993) the dynamics generalized (regular the Jaffe or chaotic) and Hernquist of orbits. models This (NGC 3726, NGC 3877 and NGC 4010) to the analytical circu- Dehnen (1993) generalized the Jaffe and Hernquist models proposedably. In by thisJaffe respect,(1983) particular and Hernquist attention(1990), deserve who the derived models canby means be considered of a family one of of the density standing profiles problems with indifferent galactic central dy- lar velocities derived with our model. Then, the mass-density by means of a family of density profiles with different central proposed by Jaffe (1983) and Hernquist (1990), who derived namics because it could allow us to understand the formation profiles are calculated, along with the vertical and epicyclic fre- Introduction analytical∗ models which closely approximate the light distri- Contopoulos Correspondencia: G. A. González, [email protected],and evolution Recibido: of 5 galaxies de octubre ( de 2018; Aceptado:, 1979), 1 as4 de shown febrero by de quencies, showing that our model not only is well-behaved but bution∗ for spherical and elliptical galaxies, respectively. A few 2019. Correspondencia: G. A. González, [email protected],the pioneer Recibido: simulations 5 de octubre of Lindblad de 2018; Aceptado:(1960). Despite 14 de febrero the fact de also satisfy the stability conditions. A dynamical analysis in Long & Murali Since the seminal paper by Miyamoto & Nagai (1975), the years2019. later, (1992) presented an analytical po- that early papers on this topic studied only regular orbits in terms of the Poincarésurfaces of section is performed in the literature on three-dimensional analytical models for the grav- tential for barred galaxies that reduces to the Miyamoto-Nagai the meridional plane (Martinet & Mayer, 1975, Manabe, third section. Finally, in the fourth section, we summarize our itational field of different types of galaxies has grown consider- disk by an appropriate setting of the free parameters, while 1979, Greiner, 1987, Lees & Schwarzschild, 1992), soon af- main conclusions. Dehnen ably. In this respect, particular attention deserve the models (1993) generalized the Jaffe and Hernquist models ter, the existence of chaos on the orbital motion started to be proposed by Jaffe (1983) and Hernquist (1990), who derived by means of a family of density profiles with different central considered by Caranicolas (1996) and Caranicolas & Pa- slopes. More recently, Vogt & Letelier (2005) derived an an- padopoulosof*Corresponding section, showing(2003). autor: that In the the majority orbital motion of cases exhibits all these a stron stud-g G. A. González; [email protected] Generalized Miyamoto-Nagai model ∗ alytical expression for the gravitational field of galaxies, based iesdependence focused on on the the distinction angular momentum between regular and energy and chaotic of the test or- Correspondencia: G. A. González, [email protected], Recibido: 5 de octubre de 2018; Aceptado: 14 de febrero de Received: October 5, 2018 on the multipolar expansion up to the quadrupole term. Using bitsparticles (Manos (stars). & Athanassoula, 2011, Bountis et. al., 2012, 2019. Accepted: February 14, 2019 Let us start considering the axially symmetric Laplace’s equa- a different approach, González et al. (2010) obtained a family ManosEditor: Románet al. Castañeda, 2013) Sepulveda or the influence of the galaxy compo- tion in spherical coordinates of finite thin-discs models for four galaxies in the Ursa major nentsThe paper (nucleus, is organized bulge, disk, as follows: halo) on in the characterfirst section, of orbitwe de-s, cluster in which the circular velocities were adjusted to fit the seerive e.g. the ( generalizedZotos, 2012, Miyamoto-NagaiZotos & Caranicolas model. Next,, 2013, fromZotos the, 1 ∂ ∂Φ 1 ∂ ∂Φ ∇2Φ(r, θ)= r2 + sin θ =0, (1) observed24 rotation curves. 2014).new potential Notwithstanding the explicit the expressions evidence for that the both physical chaotic qua andnti- r2 ∂r ∂r r2 sin θ ∂θ ∂θ regularties of interest motions are are determined. possible in manyIn the axisymmetric second section potentia we adjustls, whose general solution reads as One advantage of an analytical galaxy model is the possibil- recentthe observed studies rotation on generalized curves axisymmetric of three specific potentials spiral galaxi suggestes ity to study the dynamics (regular or chaotic) of orbits. This (NGC 3726, NGC 3877 and NGC 4010) to the analytical circu- ∞ that a third integral of motion seems to exist for energy values l −(l+1) can be considered one of the standing problems in galactic dy- closerlar velocities to the escape derived energy with ( ourDubeibe model.et Then, al., 2018, theZotos mass-densiet al.ty, Φ(r, θ)= Al r − Bl r Pl(cos θ), (2) namics because it could allow us to understand the formation 2018).profiles Hence, are calculated, such apparent along with ambiguity the vertical might and only epicycl be solvedic fre- l=0 and evolution of galaxies (Contopoulos, 1979), as shown by byquencies, performing showing systematic that our studies model of not each only particular is well-behaved model. but where Al and Bl are constants to be determined, Pl are the the pioneer simulations of Lindblad (1960). Despite the fact also satisfy the stability conditions. A dynamical analysis in Legendre polynomials, and the notation (r, θ, φ) means (radial, that early papers on this topic studied only regular orbits in Interms this of paper, the Poincarésurfaces we are interested of section in meridional is performed motions in th ofe polar, azimuthal) coordinates, respectively. the meridional plane (Martinet & Mayer, 1975, Manabe, freethird test section. particles Finally, (stars) in the in fourth presence section, of analytical we summarize realisticour 1979, Greiner, 1987, Lees & Schwarzschild, 1992), soon af- galaxymain conclusions. models. Our models possess axial symmetry, which is Since Φ(r, θ) denotes the gravitational potential of an ax- ter, the existence of chaos on the orbital motion started to be a good approximation given the morphology of galaxies that isymmetric finite distribution of mass, the boundary condition considered by Caranicolas (1996) and Caranicolas & Pa- are mainly approximate figures of revolution. Additionally, the limr→∞ Φ(r, θ) = 0 must be satisfied, thus the solution (2) padopoulos (2003). In the majority of cases all these stud- galaxyGeneralized components Miyamoto-Nagai were not added one by model one, instead of this, takes the form ies focused on the distinction between regular and chaotic or- we derived a generalized Miyamoto-Nagai model that can be ∞ BlPl(cos θ) bits (Manos & Athanassoula, 2011, Bountis et. al., 2012, adjusted very accurately to fit the observed rotation curve and Φ(r, θ)=− . (3) rl+1 Manos et al., 2013) or the influence of the galaxy compo- henceLet us it start is assumed considering that the all axially (or most symmetric of) the components Laplace’s eq areua- l=0 tion in spherical coordinates nents (nucleus, bulge, disk, halo) on the character of orbits, taken into account. The determination of the specific values Following Vogt & Letelier (2005), in order to obtain a gener- see e.g. (Zotos, 2012, Zotos & Caranicolas, 2013, Zotos, of the2 coefficients1 ∂ of the2 ∂Φ series expansion1 ∂ let us∂ calculateΦ the alized Miyamoto-Nagai model and for the sake of simplicity, we ∇ Φ(r, θ)= r + sin θ =0, (1) 2014). Notwithstanding the evidence that both chaotic and correspondingr surface2 ∂r densities∂r andr2 sin allθ the∂θ kinematic ∂θ quantities shall consider terms up to l = 3 in (3), therefore, transforming regular motions are possible in many axisymmetric potentials, characterizing the particular galaxy models. Unlike the models whose general solution reads as to cylindrical coordinates (R, z) by means of the relations recent studies on generalized axisymmetric potentials suggest derived by González et al. (2010), which exhibit instabilities to small vertical perturbations∞ (see e.g. the cases of NGC 3877 cos θ = z/r and r = R2 + z2, (4) that a third integral of motion seems to exist for energy values l −(l+1) closer to the escape energy (Dubeibe et al., 2018, Zotos et al., and NGCΦ( 4010),r, θ)= our modelsAl r satisfy− Bl r the stabilityPl(cos conditionsθ), f(2)or and applying the additional transformation (Satoh, 1980), 2018). Hence, such apparent ambiguity might only be solved radial and vertical perturbations.l=0 On the other hand, the dy- ∗ by performing systematic studies of each particular model. namicswhere A ofl and the orbitsBl are is constants studied through to be determined, the PoincarésurfacPl are thees z → z = a + z2 + b2, (5) Legendre polynomials, and the notation (r, θ, φ) means (radial, In this paper, we are interested in meridional motions of polar, azimuthal) coordinates, respectively. free test particles (stars) in presence of analytical realistic galaxy models. Our models possess axial symmetry, which is Since Φ(r, θ) denotes the gravitational potential of an ax- a good approximation given the morphology of galaxies that isymmetric finite distribution of mass, the boundary condition are mainly approximate figures of revolution. Additionally, the limr→∞ Φ(r, θ) = 0 must be satisfied, thus the solution (2) galaxy components were not added one by one, instead of this, takes the form we derived a generalized Miyamoto-Nagai model that can be ∞ BlPl(cos θ) adjusted very accurately to fit the observed rotation curve and Φ(r, θ)=− . (3) rl+1 hence it is assumed that all (or most of) the components are l=0 taken into account. The determination of the specific values Following Vogt & Letelier (2005), in order to obtain a gener- of the coefficients of the series expansion let us calculate the alized Miyamoto-Nagai model and for the sake of simplicity, we corresponding surface densities and all the kinematic quantities shall consider terms up to l = 3 in (3), therefore, transforming characterizing the particular galaxy models. Unlike the models to cylindrical coordinates (R, z) by means of the relations derived by González et al. (2010), which exhibit instabilities to small vertical perturbations (see e.g. the cases of NGC 3877 cos θ = z/r and r = R2 + z2, (4) and NGC 4010), our models satisfy the stability conditions for and applying the additional transformation (Satoh, 1980), radial and vertical perturbations. On the other hand, the dy- ∗ namics of the orbits is studied through the Poincarésurfaces z → z = a + z2 + b2, (5) slopes. More recently, Vogt & Letelier (2005) derived an anof section, showing that the orbital motion exhibits a strong alytical expression for the gravitational field of galaxies, based dependence on the angular momentum and energy of the test on the multipolar expansion up to the quadrupole term. Using particles (stars). a different approach, González et al. (2010) obtained a family of finite thin-discs models for four galaxies in the Ursa major The paper is organized as follows: in the first section, we de- cluster in which the circular velocities were adjusted to fit the rive the generalized Miyamoto-Nagai model. Next, from the observed rotation curves. new potential the explicit expressions for the physical quantities of interest are determined. In the second section we adjust One advantage of an analytical galaxy model is the possibil- the observed rotation curves of three specific spiral galaxies ity to study the dynamics (regular or chaotic) of orbits. This (NGC 3726, NGC 3877 and NGC 4010) to the analytical circu- canslopes. be considered More recently, one ofVogt the standing& Letelier problems(2005) in derived galactic an dy an-- larofsection, velocities showing derived that with the our orbital model. motion Then, theexhibits mass-densi a strontyg namicsalytical because expression it could for the allow gravitational us to understand field of galaxies the formation, based profilesdependence are calculated, on the angular along momentum with the vertical and energy and epicycl of theic fre-test andon the evolution multipolar of galaxies expansion (Contopoulos up to the quadrupole, 1979), as term. shown Using by quencies,particles (stars).showing that our model not only is well-behaved but thea different pioneer approach, simulationsGonzález of Lindbladet al.(1960).(2010) obtained Despite the a family fact also satisfy the stability conditions. A dynamical analysis in thatof finite early thin-discs papers on models this topic for four studied galaxies only in regular the Ursa orbits majo inr termsThe paper of the is Poincarésurfaces organized as follows: of section in the is first performed section, inwe th de-e thecluster meridional in which plane the circular (Martinet velocities & Mayer were adjusted, 1975, Manabe to fit the, thirdrive the section. generalized Finally, Miyamoto-Nagai in the fourth section, model. we Next,summarize fromour the 1979,observedGreiner rotation, 1987, curves.Lees & Schwarzschild, 1992), soon af- mainnew potential conclusions. the explicit expressions for the physical quanti- ter, the existence of chaos on the orbital motion started to be ties of interest are determined. In the second section we adjust consideredOne advantage by Caranicolas of an analytical(1996) galaxy and Caranicolas model is the possibil- & Pa- the observed rotation curves of three specific spiral galaxies padopoulosity to study the(2003). dynamics In the (regular majority or ofchaotic) casesall of orbits. these stud- This Generalized(NGC 3726, NGC Miyamoto-Nagai 3877 and NGC 4010) to model the analytical circu- iescan focused be considered on the onedistinction of the standing between problems regular and in galactic chaotic odyr-- lar velocities derived with our model. Then, the mass-density bitsnamics (Manos because & Athanassoulait could allow us, 2011, to understandBountis et. the al.formation, 2012, profiles are calculated, along with the vertical and epicyclic fre- Manosand evolutionet al., of 2013) galaxies or the (Contopoulos influence of, the 1979), galaxy as shown compo- by Letquencies, us start showing considering that our the model axially not symmetric only is well-behaved Laplace’s equa- but nentsthe pioneer (nucleus, simulations bulge, disk, of Lindblad halo) on(1960). the character Despite of the orbit facts, tionalso in satisfy spherical thestability coordinates conditions. A dynamical analysis in that early papers on this topic studied only regular orbits in terms of the Poincarésurfaces of section is performed in the see e.g. (Zotos, 2012, Zotos & Caranicolas, 2013, Zotos, 2 1 ∂ 2 ∂Φ 1 ∂ ∂Φ Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 ∇ Φ(r, θ)= r Orbital+ dynamics in sinrealisticθ galaxy=0 models, (1) 2014).the meridional Notwithstanding plane (Martinet the evidence & Mayer that both, 1975, chaoticManabe and, third section.r Finally,2 ∂r in∂r the fourthr2 sin section,θ ∂θ we summarize∂θ our doi:regular1979, http://dx.doi.org/10.18257/raccefyn.774Greiner motions,are 1987, possibleLees & in Schwarzschild many axisymmetric, 1992), potentia soonls, af- main conclusions. whose general solution reads as recentter, the studies existence on generalized of chaos on axisymmetric the orbital motion potentials started sug togest be considered by Caranicolas (1996) and Caranicolas & Pa- ∞ that a third integral of motion seems to exist for energy values l −(l+1) closerpadopoulos to the escape(2003). energy In the (Dubeibe majorityet of al. cases, 2018, allZotos theseet stud- al., GeneralizedΦ(r, θ)= Miyamoto-NagaiAl r − Bl r modelPl(cos θ), (2) 2018).ies focused Hence, on such the distinction apparent ambiguity between regular might andonly chaotic be solved or- l=0 et. al. bybits performing (Manos & systematic Athanassoula studies, 2011,of eachBountis particular model., 2012, where Al and Bl are constants to be determined, Pl are the Let us start considering the axially symmetric Laplace’s equa- Manos et al., 2013) or the influence of the galaxy compo- Legendre polynomials, and the notation (r, θ, φ) means (radial, tion in spherical coordinates Innents this (nucleus, paper, we bulge, are disk, interested halo) on in themeridional character motions of orbit ofs, polar, azimuthal) coordinates, respectively. freesee e.g. test ( particlesZotos, 2012, (stars)Zotos in presence & Caranicolas of analytical, 2013, realiZotosstic, 1 ∂ ∂Φ 1 ∂ ∂Φ ∇2Φ(r, θ)= r2 + sin θ =0, (1) galaxy2014). models. Notwithstanding Our models the possess evidence axial that symmetry, both chaotic which and is Since Φ(r, θ)r denotes2 ∂r the∂r gravitationalr2 sin θ ∂θ potential ∂θ of an ax- aregular good approximation motions are possible given in the many morphology axisymmetric of galaxies potentia thatls, isymmetric finite distribution of mass, the boundary condition whose general solution reads as arerecent mainly studies approximate on generalized figures axisymmetric of revolution. potentials Additionally sug,gest the limr→∞ Φ(r, θ) = 0 must be satisfied, thus the solution (2) galaxy components were not added one by one, instead of this, takes the form ∞ that a third integral of motion seems to exist for energy values l −(l+1) wecloser derived to the a escape generalized energy Miyamoto-Nagai (Dubeibe et al., model 2018, Zotos that canet al.be, Φ(r, θ)= Al r ∞− Bl r Pl(cos θ), (2) BlPl(cos θ) adjusted2018). Hence, very accurately such apparent to fit theambiguity observed might rotation only curve be solved and Φ(r,l=0 θ)=− . (3) rl+1 henceby performing it is assumed systematic that all studies (or most of each of) particular the components model. are where Al and Bl are constantsl=0 to be determined, Pl are the taken into account. The determination of the specific values FollowingLegendreVogt polynomials, & Letelier and the(2005), notation in order (r, θ, to φ) obtain means a (radial, gener- ofIn the this coefficients paper, we of are the interested series expansion in meridional let us calculate motions th ofe alizedpolar, Miyamoto-Nagai azimuthal) coordinates, model and respectively. for the sake of simplicity, we correspondingfree test particles surface (stars) densities in presenceand all the of kinematic analytical quan realititiesstic shall consider terms up to l = 3 in (3), therefore, transforming Since Φ(r, θ) denotes the gravitational potential of an ax- characterizinggalaxy models. the Our particular models galaxy possess models. axial symmetry, Unlike the which models is to cylindrical coordinates (R, z) by means of the relations deriveda good by approximationGonzález et given al. (2010), the morphology which exhibit of galaxies instabilities that isymmetric finite distribution of mass, the boundary condition toare small mainly vertical approximate perturbations figures (see of revolution.e.g. the cases Additionally of NGC 3877, the limr→∞ Φ(r, θcos) =θ = 0z/r mustand be satisfied,r = R thus2 + z the2, solution(4) (2) galaxy components were not added one by one, instead of this, takes the form and NGC 4010), our models satisfy the stability conditions for Satoh we derived a generalized Miyamoto-Nagai model that can be and applying the additional transformation∞ ( , 1980), radial and vertical perturbations. On the other hand, the dy- BlPl(cos θ) adjusted very accurately to fit the observed rotation curve and Φ(r, θ)=∗ − . (3) namics of the orbits is studied through the Poincarésurfaces z → z = a + z2rl++1b2, (5) hence it is assumed that all (or most of) the components are l=0 taken into account. The determination of the specific values Following Vogt & Letelier (2005), in order to obtain a gener- of the coefficients of the series expansion let us calculate the withalizeda and Miyamoto-Nagaib two arbitrary model parameters, and for the the sake generalized of simplicity, poten-we square curve fitting method allows us to calculate the numer- ∗ corresponding surface densities and all the kinematic quantities tialshall takes consider the form terms up to l = 3 in (3), therefore, transforming ical values of the parameters for each particular galaxy. The ∗ ˜ ˜ ˜ ˜ ˜ characterizing the particular galaxy models. Unlike the models to cylindrical coordinatesB (R, z) by meansB z of the relations resulting values ofa, ˜ b, B0, B1, B2, and B3, for the three galax- − 0 − 1 derived by González et al. (2010), which exhibit instabilities Φ(R, z)= 3/2 ies under consideration, are given in Table 1. R2 + z∗2 (R2 + z∗2) to small vertical perturbations (see e.g. the cases of NGC 3877 cos θ = z/r and r = R2 + z2, (4) 2 ∗2 2 ∗ ∗3 and NGC 4010), our models satisfy the stability conditions for B2 R − 2z B3 3R z − 2z NGC 3726 NGC 3877 NGC 4010 and applying+ the additional transformation+ (Satoh, 1980),. (6) radial and vertical perturbations. On the other hand, the dy- 2(R2 + z∗2)5/2 2(R2 + z∗2)7/2 a˜ 0.6773 0.8491 1.143 ∗ − − − namics of the orbits is studied through the Poincarésurfaces z → z = a + z2 + b2, (5) ˜b −1.045 × 10 6 −2.929 × 10 7 −9.568 × 10 6 4 4 4 slopes. More recently, Vogt & Letelier (2005) derived an anof section, showing that the orbital motion exhibits a strong Once the potential has been specified, the mass-density distri- B˜0 −7.183 × 10 −9.859 × 10 −3.146 × 10 5 5 4 alytical expression for the gravitational field of galaxies, based dependence on the angular momentum and energy of the test bution Σ can be calculated directly from Poisson equation, B˜1 1.342 × 10 1.820 × 10 4.735 × 10 on the multipolar expansion up to the quadrupole term. Using particles (stars). B˜ −8.337 × 104 −1.098 × 105 1.464 × 104 1 ∂2Φ 1 ∂Φ ∂2Φ 2 a different approach, González et al. (2010) obtained a family ˜ 4 4 3 Σ= 2 + + 2 , (7) B3 2.616 × 10 3.674 × 10 −7.815 × 10 of finite thin-discs models for four galaxies in the Ursa major The paper is organized as follows: in the first section, we de- 4πG ∂R R ∂R ∂z rive the generalized Miyamoto-Nagai model. Next, from the cluster in which the circular velocities were adjusted to fit the while the circular velocity v of particles in the galactic plane, Table 1. Parameters for each particular galaxy model. observed rotation curves. new potential the explicit expressions for the physical quanti- the epicyclic frequency k, and the vertical frequency ν of small ties of interest are determined. In the second section we adjust oscillations about the equilibrium circular orbit, can be ob- One advantage of an analytical galaxy model is the possibil- the observed rotation curves of three specific spiral galaxies a b tained from the following expressions evaluated at z =0 1.0 ity to study the dynamics (regular or chaotic) of orbits. This (NGC 3726, NGC 3877 and NGC 4010) to the analytical circu- (Binney & Tremaine, 2011) can be considered one of the standing problems in galactic dy- lar velocities derived with our model. Then, the mass-density 150 0.8 2 ∂Φ namics because it could allow us to understand the formation profiles are calculated, along with the vertical and epicyclic fre- v = R , (8) 0.6 100

∂R � and evolution of galaxies (Contopoulos, 1979), as shown by quencies, showing that our model not only is well-behaved but �

2 km s 0.4

Lindblad 2 ∂ Φ 3 ∂Φ v the pioneer simulations of (1960). Despite the fact also satisfy the stability conditions. A dynamical analysis in k = + , (9) that early papers on this topic studied only regular orbits in terms of the Poincarésurfaces of section is performed in the ∂R2 R ∂R 50 0.2 the meridional plane (Martinet & Mayer, 1975, Manabe, third section. Finally, in the fourth section, we summarize our ∂2Φ ν2 = . (10) 0 0.0 1979, Greiner, 1987, Lees & Schwarzschild, 1992), soon af- main conclusions. ∂z2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � � ter, the existence of chaos on the orbital motion started to be From (8-10), it is important to emphasize that a feasible model R R considered by Caranicolas (1996) and Caranicolas & Pa- must satisfy the constraints set by the conditions v2 ≥ 0, c 0.10 padopoulos (2003). In the majority of cases all these stud- Generalized Miyamoto-Nagai model k2 ≥ 0, and ν2 ≥ 0, where the last two inequalities are un- ies focused on the distinction between regular and chaotic or- derstood as stability conditions (Vogt & Letelier, 2005). 0.05 bits (Manos & Athanassoula, 2011, Bountis et. al., 2012, � Let us start considering the axially symmetric Laplace’s equa- z 0.00 Manos et al., 2013) or the influence of the galaxy compo- As is evident from the preceding paragraphs, the galactic mod- tion in spherical coordinates nents (nucleus, bulge, disk, halo) on the character of orbits, els and its associated physical quantities are uniquely deter- �0.05 see e.g. (Zotos, 2012, Zotos & Caranicolas, 2013, Zotos, 1 ∂ ∂Φ 1 ∂ ∂Φ mined by the set of constants a, b, B0,B1,B2, and B3, which ∇2Φ(r, θ)= r2 + sin θ =0, (1) �0.10 2014). Notwithstanding the evidence that both chaotic and r2 ∂r ∂r r2 sin θ ∂θ ∂θ (taking a pragmatic approach) can be estimated from the ob- �1.0 �0.5 0.0 0.5 1.0 � regular motions are possible in many axisymmetric potentials, servational data of the corresponding rotation curves, as we R whose general solution reads as recent studies on generalized axisymmetric potentials suggest will discuss in detail in the next section. d e ∞ 1.0 1.0 that a third integral of motion seems to exist for energy values l −(l+1) closer to the escape energy (Dubeibe et al., 2018, Zotos et al., Φ(r, θ)= Al r − Bl r Pl(cos θ), (2) 0.8 0.8 2018). Hence, such apparent ambiguity might only be solved l=0 Rotation curves fitting 0.6 0.6 2 by performing systematic studies of each particular model. where Al and Bl are constants to be determined, Pl are the 2 � � k Legendre polynomials, and the notation (r, θ, φ) means (radial, 25 0.4 Ν 0.4 In this paper, we are interested in meridional motions of The observational data were taken from Verheijen & San- polar, azimuthal) coordinates, respectively. 0.2 0.2 free test particles (stars) in presence of analytical realistic cisi (2001) for three specific galaxies in the Ursa Major cluster: NGC 3726, NGC 3877 and NGC 4010. Following the procedure 0.0 0.0 galaxy models. Our models possess axial symmetry, which is Since Φ(r, θ) denotes the gravitational potential of an ax- 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 isymmetric finite distribution of mass, the boundary condition outlined in González et al. (2010), we take the galaxy radius � � a good approximation given the morphology of galaxies that R R are mainly approximate figures of revolution. Additionally, the limr→∞ Φ(r, θ) = 0 must be satisfied, thus the solution (2) Rd as the given by the largest tabulated value of the data. ˜ galaxy components were not added one by one, instead of this, takes the form Thus, introducing dimensionless variables R = R/Rd, z˜ = Figure 1. Model fitted to the rotation curve of NGC 3726 ˜ ˜ ˜ we derived a generalized Miyamoto-Nagai model that can be ∞ z/Rd, a˜ = a/Rd, b = b/Rd and setting B0 = B0/Rd, B1 = using the parameters given in the first column of Table 1. (a) BlPl(cos θ) 2 3 4 adjusted very accurately to fit the observed rotation curve and Φ(r, θ)=− . (3) B1/R , B˜2 = B2/R , and B˜3 = B3/R , the nonlinear least The solid curve indicates the rotation velocity calculated from rl+1 d d d hence it is assumed that all (or most of) the components are l=0 ∗ B GM, B B B Miyamoto & taken into account. The determination of the specific values It should be noted that setting 0 = 1 = 2 = 3 = 0 in (6), we get the well-known Miyamoto-Nagai Potential ( Following Vogt & Letelier (2005), in order to obtain a gener- Nagai, 1975). of the coefficients of the series expansion let us calculate the alized Miyamoto-Nagai model and for the sake of simplicity, we corresponding surface densities and all the kinematic quantities shall consider terms up to l = 3 in (3), therefore, transforming characterizing the particular galaxy models. Unlike the models to cylindrical coordinates (R, z) by means of the relations derived by González et al. (2010), which exhibit instabilities to small vertical perturbations (see e.g. the cases of NGC 3877 cos θ = z/r and r = R2 + z2, (4) and NGC 4010), our models satisfy the stability conditions for and applying the additional transformation (Satoh, 1980), radial and vertical perturbations. On the other hand, the dy- ∗ namics of the orbits is studied through the Poincarésurfaces z → z = a + z2 + b2, (5) with a and b two arbitrary parameters, the generalized poten- square curve fitting method allows us to calculate the numer- tial takes the form∗ ical values of the parameters for each particular galaxy. The ∗ ˜ ˜ ˜ ˜ ˜ B B z resulting values ofa, ˜ b, B0, B1, B2, and B3, for the three galax- − 0 − 1 Φ(R, z)= 3/2 ies under consideration, are given in Table 1. R2 + z∗2 (R2 + z∗2) 2 ∗2 2 ∗ ∗3 B2 R − 2z B3 3R z − 2z + + . (6) NGC 3726 NGC 3877 NGC 4010 2(R2 + z∗2)5/2 2(R2 + z∗2)7/2 a˜ 0.6773 0.8491 1.143 − − − ˜b −1.045 × 10 6 −2.929 × 10 7 −9.568 × 10 6 4 4 4 Once the potential has been specified, the mass-density distri- B˜0 −7.183 × 10 −9.859 × 10 −3.146 × 10 5 5 4 bution Σ can be calculated directly from Poisson equation, B˜1 1.342 × 10 1.820 × 10 4.735 × 10 B˜ −8.337 × 104 −1.098 × 105 1.464 × 104 1 ∂2Φ 1 ∂Φ ∂2Φ 2 Σ= + + , (7) B˜ 2.616 × 104 3.674 × 104 −7.815 × 103 4πG ∂R2 R ∂R ∂z2 3 while the circular velocity v of particles in the galactic plane, Table 1. Parameters for each particular galaxy model. the epicyclic frequency k, and the vertical frequency ν of small oscillations about the equilibrium circular orbit, can be ob- a b tained from the following expressions evaluated at z =0 1.0 (Binney & Tremaine, 2011) 150 0.8 2 ∂Φ v = R , (8) 0.6 100

∂R � �

2 km s 0.4

∂ Φ 3 ∂Φ v k2 = + , (9) ∂R2 R ∂R 50 0.2 ∂2Φ ν2 = . (10) 0 0.0 ∂z2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � � From (8-10), it is important to emphasize that a feasible model R R must satisfy the constraints set by the conditions v2 ≥ 0, c 0.10 k2 ≥ 0, and ν2 ≥ 0, where the last two inequalities are un- derstood as stability conditions (Vogt & Letelier, 2005). 0.05 � z 0.00 As is evident from the preceding paragraphs, the galactic models and its associated physical quantities are uniquely deter- �0.05

mined by the set of constants a, b, B0,B1,B2, and B3, which �0.10 (taking a pragmatic approach) can be estimated from the ob- �1.0 �0.5 0.0 0.5 1.0 � servational data of the corresponding rotation curves, as we R Dubeibe FL, Martínez-Sicachá SM, González GA Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 will discuss in detail in the next section. d e 1.0 doi: http://dx.doi.org/10.18257/raccefyn.7741.0 0.8 0.8 (8) while the error bars denote the velocity dispersions of the solid lines correspond to the analytical expressions (8) fitted to Rotation curves fitting 0.6 0.6 2 observational2 data. (b) Normalized mass-density distribution the rotation curves. As can be seen, in each case the model fits � � k Ν Σ˜ at z =0.4 0, calculated from (7). (c)0.4 Constant-density curves of the observed data with good accuracy. Additionally, in panels The observational data were taken from Verheijen & San- equation (7) in the meridional plane. (d) Epicyclic frequency (b) of Figures 1, 2, and 3, we plot the normalized mass-density cisi (2001) for three specific galaxies in the Ursa Major cluster: 0.2 0.2 (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated distribution (7) at z = 0 for the three galaxies, as a function of NGC 3726, NGC 3877 and NGC 4010. Following the procedure 0.0 0.0 on z = 0. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 the dimensionless radial coordinate R˜. Here, we obtain a well- outlined in González et al. (2010), we take the galaxy radius � � R R behaved mass-density function, showing a maximum value at R as the given by the largest tabulated value of the data. d the center that decreases to zero at the edge of the disk. On the Thus, introducing dimensionless variables R˜ = R/R , z˜ = a b d Figure 1. Model fitted to the rotation1.0 curve of NGC 3726 other hand, in panels (c) of Figures 1, 2, and 3, we present four z/Rd, a˜ = a/Rd, ˜b = b/Rd and setting B˜0 = B0/Rd, B˜1 = using the parameters given in the first column of Table 1. (a) 150 isodensity curves of the mass-density distribution (7) in the 2 ˜ 3 ˜ 4 0.8 B1/Rd, B2 = B2/Rd, and B3 = B3/Rd, the nonlinear least The solid curve indicates the rotation velocity calculated from meridional plane (R,˜ z˜), showing that each model corresponds 0.6 with a and b two arbitrary parameters, the generalized poten- square∗ curve fitting method allows us to calculate the numer- 100

� to a very different mass distribution. Finally, from panels (d) ∗ withIta shouldand b betwo noted arbitrary that setting parameters,B0 = GM, the B generalized1 = B2 = B poten-3 = 0 in (6),square we get curve the well-known fitting method Miyamoto-Nagai allows� us to Potential calculate (Miyamoto the numer- & tial takes the form ical values of the parameters∗ for each particular galaxy. The km s 0.4 tialNagai takes, 1975). the form ical valuesv of the parameters for each particular galaxy. The and (e) of the same figures, it is noteworthy that in the three ∗ 50 resulting values ofa, ˜ ˜b, B˜0, B˜1, B˜2, and B˜3, for∗ the three galax- ˜ ˜ ˜ ˜ ˜ B0 B1 z B B z resulting values ofa, ˜ b, B0, B1, B2, and0.2 B3, for the three galax- cases the stability conditions are fully satisfied. Φ(R, z)=− − − 0 − 1 2 ∗2 2 ∗2 3/2 ies underΦ(R, consideration, z)= are given in Table 1. 3/2 ies under consideration, are given in Table 1. R + z (R + z ) 2 ∗2 (R2 + z∗2) 0 0.0 R + z 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 2 ∗2 2 ∗ ∗3 ∗ ∗ ∗ B2 R − 2z B3 3R z − 2z 2 − 2 2 − 3 � � a b + + . (6) NGC 3726B2 R NGC2z 3877B3 3R NGCz 40102z NGC 3726R NGC 3877 NGCR 4010 ∗ 5/2 ∗ 7/2 + + . (6) 1.0 2 2 2 2 ∗2 5/2 ∗2 7/2 2(R + z ) 2(R + z ) a˜ 0.67732(R2 + 0z.8491) 2(R 12.143+ z ) a˜ 0.6773 0.8491 c 1.143 120 ˜ −6 −7 −6 − − − 0.8 b −1.045 × 10 −2.929 × 10 −9.568 × 10 ˜b −1.0450.10 × 10 6 −2.929 × 10 7 −9.568 × 10 6 100 4 4 4 B˜ −7.183 × 10 −9.859 × 10 −3.146 × 10 4 4 4 80 0.6 Once the potential has been specified, the mass-density distri- Once the0 potential has been specified, the mass-density distri- B˜0 −7.1830.05 × 10 −9.859 × 10 −3.146 × 10 � ˜ 5 5 4 5 5 4 �

× × × km s 60 bution Σ can be calculated directly from Poisson equation, butionB1 Σ1 can.342 be calculated10 1. directly820 10 from Poisson4.735 equation,10 B˜1 1.342 × 10 1.820 × 10 4.735 × 10 0.4 v � 4 5 4 z 0.00 2 2 B˜2 −8.337 × 10 −1.098 × 10 1.464 × 10 ˜ 4 5 4 40 1 ∂ Φ 1 ∂Φ ∂ Φ 2 2 B2 −8.337 × 10 −1.098 × 10 1.464 × 10 0.2 ˜ 41 ∂ Φ 1 ∂Φ4 ∂ Φ 3 Σ= + + , (7) B3 2.616Σ=× 10 3.674+× 10 + −7.815, × 10 (7) ˜ �0.05 4 4 3 20 4πG ∂R2 R ∂R ∂z2 2 2 B3 2.616 × 10 3.674 × 10 −7.815 × 10 4πG ∂R R ∂R ∂z 0 0.0 �0.10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 while the circular velocity v of particles in the galactic plane, whileTable the 1. circularParameters velocity for eachv of particular particles in galaxy the galactic model. plane, Table 1. �1.0 �0.5 0.0 0.5 1.0 � � Parameters for each particular� galaxy model. R R the epicyclic frequency k, and the vertical frequency ν of small the epicyclic frequency k, and the vertical frequency ν of small R oscillations about the equilibrium circular orbit, can be ob- c oscillations abouta the equilibrium circular orbit,b can be ob- d e 0.10 1.0 a 1.0 b tained from the following expressions evaluated at z =0 tained from the following expressions1.0 evaluated at z =0 1.0 0.05 (Binney & Tremaine, 2011) Binney & Tremaine 0.8 0.8 ( , 2011) 0.8 150 150 0.8 � z 0.00 2 ∂Φ v = R , (8) 2 ∂Φ 0.6 0.6 0.6 2 v = R , (8) 2 0.6 � � k 100 Ν

∂R � � 100 �0.05

∂R � 0.4 � 0.4 2 km s 2 0.4 km s 2 ∂ Φ 3 ∂Φ v 0.4 2 ∂ Φ 3 ∂Φ v �0.10 k = 2 + , (9) 50 k = + , (9) 0.2 0.2 ∂R R ∂R ∂R2 0.2R ∂R 50 0.2 �1.0 �0.5 0.0 0.5 1.0 2 � ∂ Φ 2 0.0 0.0 R 2 0 2 ∂ Φ 0.0 0.0 ν = . (10) 0.0 0.2 0.4 ν0.6 0.8= 1.0 . 0.0 0.2 0.4 0.6 0.8 1.0 (10) 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ∂z2 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � ∂z � � � d e R R � R R� 1.0 1.0 From (8-10), it is important to emphasize that a feasible model From (8-10), it is important to emphasize that a feasible model R R 2 c Figure 2. Model fitted to the rotation curve of NGC 3877 us- 0.8 0.8 must satisfy the constraints set by the conditions v ≥ 0, must satisfy the constraints set by the conditions v2 ≥ 0, c 2 2 0.10 ing the parameters0.10 given in the second column of Table 1. (a) k ≥ 0, and ν ≥ 0, where the last two inequalities are un- k2 ≥ 0, and ν2 ≥ 0, where the last two inequalities are un- 0.6 0.6 2 0.05 The solid curve indicates the rotation velocity calculated from 2 � � k derstood as stability conditions (Vogt & Letelier, 2005). derstood as stability conditions (Vogt & Letelier, 2005). 0.05 Ν (8) while the error bars denote the velocity dispersions of the 0.4 0.4 � z 0.00 � z 0.00 As is evident from the preceding paragraphs, the galactic mod- As is evident from the preceding paragraphs, the galactic mod- observational data. (b) Normalized mass-density distribution 0.2 0.2 els and its associated physical quantities are uniquely deter- �0.05 Σ˜ at z = 0, calculated�0.05 from (7). (c) Constant-density curves of els and its associated physical quantities are uniquely deter- 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 mined by the set of constants a, b, B0,B1,B2, and B3, which mined by the�0.10 set of constants a, b, B0,B1,B2, and B3, which equation (7) in the meridional plane. (d) Epicyclic frequency �0.10 � � (taking a pragmatic approach) can be estimated from the ob- (taking a pragmatic�1.0 approach)�0.5 can0.0 be estimated0.5 from1.0 the ob- (9) evaluated on�1.0z = 0.� (e)0.5 Vertical0.0 frequency0.5 (10)1.0 evaluated R R � � servational data of the corresponding rotation curves, as we servational data of the correspondingR rotation curves, as we on z = 0. R will discuss in detail in the next section. d e Figure 3. Model fitted to the rotation curve of NGC 4010 will discuss in detail in the next section. d e 1.0 1.0 In panels1.0 (a) of Figures 1, 2, and 3,1.0 we show the observational using the parameters given in the third column of Table 1. (a) data (points) of the rotation curve with the corresponding ve- The solid curve indicates the rotation velocity calculated from 0.8 0.8 0.8 0.8 Rotation curves fitting locity dispersions (error bars) as reported by Verheijen & (8) while the error bars denote the velocity dispersions of the Rotation0.6 curves fitting 0.6 Sancisi 0.6 0.6 2 2 (2001) for NGC 3726, NGC 3877 and NGC 4010. The observational data. (b) Normalized mass-density distribution � 2 2 � k Ν � � k (8) while0.4 the error bars denote the0.4 velocity dispersions of the solid lines0.4 correspond to the analyticalΝ 0.4 expressions (8) fitted to The observational data were taken from Verheijen & San- Theobservational observational data. data (b) wereNormalized taken from mass-densityVerheijen distribu & San-tion the rotation curves. As can be seen, in each case the model fits cisi (2001) for three specific galaxies in the Ursa Major cluster: 0.2 0.2 0.2 0.2 cisiΣ˜ at(2001)z = 0, for calculated three specific from (7). galaxies (c) Constant-density in the Ursa Major curves cluster: of the observed data with good accuracy. Additionally, in panels NGC 3726, NGC 3877 and NGC 4010. Following the procedure NGC 3726,0.0 NGC 3877 and NGC 4010.0.0 Following the procedure 0.0 0.0 equation (7)0.0 in0.2 the0.4 meridional0.6 0.8 1.0 plane.0.0 (d)0.2 Epicyclic0.4 0.6 0.8 frequen1.0 cy (b) of Figures0.0 0.2 1, 2,0.4 and0.6 3,0.8 we1.0 plot the0.0 normalized0.2 0.4 0.6 mass-densit0.8 1.0 y outlined in González et al. (2010), we take the galaxy radius � � outlined(9) evaluated in González on z =R 0.et (e) al. (2010), Vertical we frequency take theR (10) galaxy evaluated radius distribution (7) at z� = 0 for the three galaxies,� as a function of R as the given by the largest tabulated value of the data. R R d Rond zas= the 0. given by the largest tabulated value of the data. the dimensionless radial coordinate R˜. Here, we obtain a well- Thus, introducing dimensionless variables R˜ = R/Rd, z˜ = Figure 1. ˜ Thus, introducingModel fitteddimensionless to the rotation variables curveR = ofR/R NGCd, 3726z˜ = Figurebehaved 1. mass-densityModel fitted function, to the showing rotation a curve maximum of NGC value 3726 at z/R , a˜ = a/R , ˜b = b/R and setting B˜ = B /R , B˜ = d d d 0 0 d 1 z/Rusingd, thea˜ = parametersa/Rd, ˜b = givenb/Rd inand the setting first columnB˜0 = ofB Table0/Rd, B 1.˜1 (a)= using the parameters given in the first column of Table 1. (a) 2 3 4 a b the center that decreases to zero at the edge of the disk. On the B1/R , B˜2 = B2/R , and B˜3 = B3/R , the nonlinear least The solid2 ˜ curve indicates3 the˜ rotation velocity4 calculated from d d d B1/Rd, B2 = B2/Rd, and B3 = B1.03/Rd, the nonlinear least Theother solid hand, curve in panels indicates (c) of the Figures rotation 1, 2, velocity and 3, calculated we presentfrom four ∗ ∗ 150 isodensity curves of the mass-density distribution (7) in the It should be noted that setting B0 = GM, B1 = B2 = B3 = 0 in (6), weIt get should the well-known be noted that Miyamoto-Nagai setting B 0.8= GM, Potential B = (MiyamotoB = B = 0 & in (6), we get the well-known Miyamoto-Nagai Potential (Miyamoto & 0 1 2 3 meridional plane (R,˜ z˜), showing that each model corresponds Nagai, 1975). Nagai, 1975). 0.6 100

� to a very diﬀerent mass distribution. Finally, from panels (d) �

km s 0.4

v and (e) of the same ﬁgures, it is noteworthy that in the three 50 26 0.2 cases the stability conditions are fully satisﬁed. 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � � a b R R 1.0 c 120 0.8 0.10 100 0.6 0.05 80 � �

km s 60 0.4 v � z 0.00 40 0.2 �0.05 20 0 0.0 �0.10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 �1.0 �0.5 0.0 0.5 1.0 � � � R R R c d e 0.10 1.0 1.0 0.05 0.8 0.8 � z 0.00 0.6 0.6 2 2 � � k Ν �0.05 0.4 0.4 �0.10 0.2 0.2 �1.0 �0.5 0.0 0.5 1.0 � 0.0 0.0 R 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � � d e R R 1.0 1.0

Figure 2. Model ﬁtted to the rotation curve of NGC 3877 us- 0.8 0.8 ing the parameters given in the second column of Table 1. (a) 0.6 0.6 2 The solid curve indicates the rotation velocity calculated from 2 � � k Ν (8) while the error bars denote the velocity dispersions of the 0.4 0.4 observational data. (b) Normalized mass-density distribution 0.2 0.2 Σ˜ at z = 0, calculated from (7). (c) Constant-density curves of 0.0 0.0 equation (7) in the meridional plane. (d) Epicyclic frequency 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 � � (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated R R on z = 0. Figure 3. Model ﬁtted to the rotation curve of NGC 4010 In panels (a) of Figures 1, 2, and 3, we show the observational using the parameters given in the third column of Table 1. (a) data (points) of the rotation curve with the corresponding ve- The solid curve indicates the rotation velocity calculated from locity dispersions (error bars) as reported by Verheijen & (8) while the error bars denote the velocity dispersions of the Sancisi (2001) for NGC 3726, NGC 3877 and NGC 4010. The observational data. (b) Normalized mass-density distribution Σ˜ at z = 0, calculated from (7). (c) Constant-density curves of Since the Hamiltonian is autonomous, H is an integral of mo- equation (7) in the meridional plane. (d) Epicyclic frequency tion (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated

on z = 0. H(R,z,pR,pz)=H(R0,z0,pR0 ,pz0 )=h, (19)

with h the energy of an orbit. Stellar Dynamics The existence of an analytic integral of motion reduces the phase space dimensionality, and hence the Poincarésurface of It is a well-known fact that using rough estimates of the dimen- section is an appropriate and well-established method to ana- sions of typical stars and galaxies, the collision interval between lyze the dynamics of the system. Taking into account the axial stars is about 108 times longer than the average age for most symmetry associated to the system, it is customary to choose (8) while the error bars denote the velocity dispersions of the solid lines correspond to the analytical expressions (8) fitted to galaxies (Binney & Tremaine, 2011). This implies that the the equatorial plane z = 0 as the Poincaréplane in order to observational data. (b) Normalized mass-density distribution the rotation curves. As can be seen, in each case the model fits star’s motion can be determined solely by the gravitational at- ˜ ˜ ˜˙ Σ at z = 0, calculated from (7). (c) Constant-density curves of the observed data with good accuracy. Additionally, in panels traction of the galaxy and that collisions between stars are so represent the surface of sections in the (R, R)-plane. The orbits equation (7) in the meridional plane. (d) Epicyclic frequency (b) of Figures 1, 2, and 3, we plot the normalized mass-density rare that are irrelevant (Maoz, 2016). Therefore, as a first ap- were numerically integrated forward in time for 1000 units of (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated distribution (7) at z = 0 for the three galaxies, as a function of proximation, the orbital dynamics of a star in a given galaxy time by using a Runge-Kutta-Fehlberg Method (RKF45), with ˜ on z = 0. the dimensionless radial coordinate R. Here, we obtain a well- can be studied following the usual Lagrangian and Hamiltonian this setting the numerical error related to the conservation of Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 Orbital dynamics in realistic galaxy models −14 behaved mass-density function, showing a maximum value at approaches for the motion of a test particle in the presence of the energy is at most 10 . In all cases we set z0 = pR0 =0 doi: http://dx.doi.org/10.18257/raccefyn.774 and we scan the phase space with a large number of initial con- a b the center that decreases to zero at the edge of the disk. On the an estimated gravitational potential. 1.0 other hand, in panels (c) of Figures 1, 2, and 3, we present four ditions for the radii R0, these three values allow us to determine 150 0.8 isodensity curves of the mass-density distribution (7) in the The orbital motion of a test particle in an axisymmetric poten- the values of pz0 through the relation (19). meridional plane (R,˜ z˜), showing that each model corresponds tial is governed by the Lagrangian 0.6 100

� to a very different mass distribution. Finally, from panels (d) � 1 2 2 2 km s 0.4 L = R˙ +(Rφ˙) +˙z − Φ(R, z), (11) v and (e) of the same figures, it is noteworthy that in the three 50 2 0.2 cases the stability conditions are fully satisfied. with (R, φ, z) the usual cylindrical coordinates. The general- 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ized canonical momenta read as � � a b R R 2 1.0 pR = R,˙ pφ = R φ,˙ pz =˙z, (12) c 120 0.8 0.10 100 and the Hamiltonian takes the form 0.6 0.05 80 1 2 2 � � H = pR + pz +Φeff (R, z), (13) km s 60 0.4 2 v � z 0.00 40 0.2 with �0.05 20 2 Lz 0 0.0 Φeff (R, z)= + Φ(R, z). (14) �0.10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 2R2 �1.0 �0.5 0.0 0.5 1.0 � � � R R Here, Lz = pφ =constant, denotes the conserved component of R c angular momentum about the z-axis. d e 0.10 1.0 1.0 0.05 From (13), the resulting Hamilton’s equations of motion can 0.8 0.8 be expressed as � z 0.00 0.6 0.6 ˙ 2 2 R = pR, (15) � � k Ν �0.05 0.4 0.4 z˙ = pz, (16) �0.10 0.2 0.2 2 �1.0 �0.5 0.0 0.5 1.0 Lz ∂Φ(R, z) � p˙R = − , (17) 0.0 0.0 R R3 ∂R 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ∂Φ(R, z) � � d e p˙ = − , (18) R R 1.0 1.0 z ∂z Figure 2. Model fitted to the rotation curve of NGC 3877 us- 0.8 0.8 where Φ(R, z) is given by Eq. (6) and its respective parameters Figure 4. Poincarésurfaces of section of NGC 3726 for differ- ing the parameters given in the second column of Table 1. (a) 0.6 0.6 should be taken from Table 1. ent values of angular momentum Lz with h = −1. 2 The solid curve indicates the rotation velocity calculated from 2 � � k Ν (8) while the error bars denote the velocity dispersions of the 0.4 0.4 observational data. (b) Normalized mass-density distribution Σ˜ at z =0.2 0, calculated from (7). (c)0.2 Constant-density curves of Since the Hamiltonian is autonomous, H is an integral of mo- Σ˜ at z = 0, calculated from (7). (c) Constant-density curves of equation0.0 (7) in the meridional plane.0.0 (d) Epicyclic frequency tion equation (7) in the meridional plane. (d) Epicyclic frequency (9) evaluated0.0 0.2 on z0.4=0.6 0. (e)0.8 Vertical1.0 0.0 frequency0.2 0.4 0.6 (10)0.8 evaluated1.0 � � (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated R R on z = 0. H(R,z,pR,pz)=H(R0,z0,pR0 ,pz0 )=h, (19) on z = 0. Figure 3. Model fitted to the rotation curve of NGC 4010 In panels (a) of Figures 1, 2, and 3, we show the observational using the parameters given in the third column of Table 1. (a) with h the energy of an orbit. data (points) of the rotation curve with the corresponding ve- StellarThe solid Dynamics curve indicates the rotation velocity calculated from locity dispersions (error bars) as reported by Verheijen & (8) while the error bars denote the velocity dispersions of the The existence of an analytic integral of motion reduces the phase space dimensionality, and hence the Poincarésurface of Sancisi (2001) for NGC 3726, NGC 3877 and NGC 4010. The Itobservational is a well-known data. fact (b) that Normalized using rough mass-density estimates of distribu the dimetionn- section is an appropriate and well-established method to ana- sions˜ of typical stars and galaxies, the collision interval between Σ at z = 0, calculated from (7). (c) Constant-density curves of lyzeSince the the dynamics Hamiltonian of the is system. autonomous, TakingH intois an account integral the of axia mo-l stars is about 108 times longer than the average age for most equation (7) in the meridional plane. (d) Epicyclic frequency symmetrytion associated to the system, it is customary to choose galaxies (Binney & Tremaine, 2011). This implies that the (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated the equatorial plane z = 0 as the Poincaréplane in order to star’s motion can be determined solely by the gravitational at- on z = 0. H(R,z,pR,pz)=H(R0,z0,pR˜0 ,p˜˙ z0 )=h, (19) traction of the galaxy and that collisions between stars are so represent the surface of sections in the (R, R)-plane. The orbits rare that are irrelevant (Maoz, 2016). Therefore, as a first ap- were numerically integrated forward in time for 1000 units of proximation, the orbital dynamics of a star in a given galaxy timewith byh the using energy a Runge-Kutta-Fehlberg of an orbit. Method (RKF45), with canStellar be studied Dynamics following the usual Lagrangian and Hamiltonian this setting the numerical error related to the conservation of The existence of an analytic−14 integral of motion reduces the approaches for the motion of a test particle in the presence of the energy is at most 10 . In all cases we set z0 = pR0 =0 andphase we space scan thedimensionality, phase space andwith hence a large the number Poincarésurfac of initial con-e of anIt is estimated a well-known gravitational fact that potential.using rough estimates of the dimen- ditionssection for is an the appropriate radii R , these and three well-established values allow method us to determine to ana- sions of typical stars and galaxies, the collision interval between 0 thelyze values the dynamics of p through of the system. the relation Taking (19). into account the axial Thestars orbital is about motion 108 times of a test longer particle than in the an average axisymmetric age for pote mostn- z0 symmetry associated to the system, it is customary to choose tialgalaxies is governed (Binney by & the Tremaine Lagrangian, 2011). This implies that the the equatorial plane z = 0 as the Poincaréplane in order to star’s motion can be1 determined solely by the gravitational at- L ˙ 2 ˙ 2 2 − ˜ ˜˙ traction of the= galaxyR and+( thatRφ) collisions+˙z Φ( betweenR, z), stars are(11)so represent the surface of sections in the (R, R)-plane. The orbits 2 rare that are irrelevant (Maoz, 2016). Therefore, as a first ap- were numerically integrated forward in time for 1000 units of with (R, φ, z) the usual cylindrical coordinates. The general- proximation, the orbital dynamics of a star in a given galaxy time by using a Runge-Kutta-Fehlberg Method (RKF45), with ized canonical momenta read as this setting the numerical error related to the conservation of can be studied following the usual Lagrangian and Hamiltonian −14 2 the energy is at most 10 . In all cases we set z0 = pR0 =0 approaches for thepR = motionR,˙ p ofφ = a testR φ,˙ particle pz =˙ inz, the presence(12) of an estimated gravitational potential. and we scan the phase space with a large number of initial con- and the Hamiltonian takes the form ditions for the radii R0, these three values allow us to determine The orbital motion of1 a test particle in an axisymmetric poten- the values of pz0 through the relation (19). H = p2 + p2 +Φ (R, z), (13) tial is governed by the2 LagrangianR z eff 27 1 with L ˙ 2 ˙ 2 2 − = R +(Rφ) 2+˙z Φ(R, z), (11) 2 Lz Φeff (R, z)= + Φ(R, z). (14) with (R, φ, z) the usual cylindrical2R2 coordinates. The general- Here,ized canonicalLz = pφ =constant, momenta read denotes as the conserved component of

angular momentum about the z-axis.2 pR = R,˙ pφ = R φ,˙ pz =˙z, (12) From (13), the resulting Hamilton’s equations of motion can and the Hamiltonian takes the form be expressed as 1 2 2 H =˙ pR + pz +Φeﬀ (R, z), (13) R 2 = pR, (15) with z˙ = pz, (16) L2 L2 ∂Φ(R, z) Φp˙ (R,= z)=z −z + Φ(R, z,). (17)(14) eﬀR R32R2 ∂R

Here, Lz = pφ =constant, denotes∂Φ(R, the z) conserved component of p˙z = − , (18) angular momentum about the z-axis.∂z where Φ(R, z) is given by Eq. (6) and its respective parameters Figure 4. Poincar´esurfaces of section of NGC 3726 for diﬀer- From (13), the resulting Hamilton’s equations of motion can should be taken from Table 1. ent values of angular momentum L with h = −1. be expressed as z

R˙ = pR, (15)

z˙ = pz, (16) L2 ∂Φ(R, z) p˙ = z − , (17) R R3 ∂R ∂Φ(R, z) p˙ = − , (18) z ∂z where Φ(R, z) is given by Eq. (6) and its respective parameters Figure 4. Poincar´esurfaces of section of NGC 3726 for diﬀer- should be taken from Table 1. ent values of angular momentum Lz with h = −1. Σ˜ at z = 0, calculated from (7). (c) Constant-density curves of Since the Hamiltonian is autonomous, H is an integral of mo- equation (7) in the meridional plane. (d) Epicyclic frequency tion (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated on z = 0. H(R,z,pR,pz)=H(R0,z0,pR0 ,pz0 )=h, (19)

with h the energy of an orbit. Stellar Dynamics The existence of an analytic integral of motion reduces the phase space dimensionality, and hence the Poincarésurface of It is a well-known fact that using rough estimates of the dimen- section is an appropriate and well-established method to ana- sions of typical stars and galaxies, the collision interval between lyze the dynamics of the system. Taking into account the axial stars is about 108 times longer than the average age for most symmetry associated to the system, it is customary to choose galaxies (Binney & Tremaine, 2011). This implies that the the equatorial plane z = 0 as the Poincaréplane in order to star’s motion can be determined solely by the gravitational at- ˜ ˜˙ traction of the galaxy and that collisions between stars are so represent the surface of sections in the (R, R)-plane. The orbits rare that are irrelevant (Maoz, 2016). Therefore, as a first ap- were numerically integrated forward in time for 1000 units of proximation, the orbital dynamics of a star in a given galaxy time by using a Runge-Kutta-Fehlberg Method (RKF45), with this setting the numerical error related to the conservation of can be studied following the usual Lagrangian and Hamiltonian −14 approaches for the motion of a test particle in the presence of the energy is at most 10 . In all cases we set z0 = pR0 =0 an estimated gravitational potential. andDubeibe we FL, scan Martínez-Sicachá the phase space SM, González with a large GA number of initial con- Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 ditions for the radii R0, these three values allow us to determine doi: http://dx.doi.org/10.18257/raccefyn.774 The orbital motion of a test particle in an axisymmetric poten- the values of pz0 through the relation (19). tial is governed by the Lagrangian Figure 5. Poincarésurfaces of section of NGC 3726 for differ- Figure 7. Poincarésurfaces of section of NGC 3877 for differ- 1 ent values of total energy h with Lz = 1. ent values of total energy h with Lz = 1. L = R˙ 2 +(Rφ˙)2 +˙z2 − Φ(R, z), (11) 2 with (R, φ, z) the usual cylindrical coordinates. The generalized canonical momenta read as

2 pR = R,˙ pφ = R φ,˙ pz =˙z, (12) and the Hamiltonian takes the form 1 H = p2 + p2 +Φ (R, z), (13) 2 R z eﬀ with L2 Φ (R, z)= z + Φ(R, z). (14) eﬀ 2R2

Here, Lz = pφ =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

R˙ = pR, (15) z˙ = pz, (16) L2 ∂Φ(R, z) p˙ = z − , (17) R R3 ∂R ∂Φ(R, z) p˙ = − , (18) z ∂z Figure 6. Poincarésurfaces of section of NGC 3877 for differ- Figure 8. Poincarésurfaces of section of NGC 4010 for differ- Figure 4. where Φ(R, z) is given by Eq. (6) and its respective parameters Poincarésurfaces of section of NGC 3726 for different values of angular momentum Lz with h = −1. ent values of angular momentum Lz with h = −10. should be taken from Table 1. ent values of angular momentum Lz with h = −1.

FigureFigure 5. 5.PoincarésurfacesPoincarésurfaces of of section section of of NGC NGC 3726 3726 for for differ- differ- FigureFigure 7. 7.PoincarésurfacesPoincarésurfaces of of section section of of NGC NGC 3877 3877 for for differ- differ- entent values values of of total total energy energyhhwithwithLLzz== 1. 1. entent values values of of total total energy energyhhwithwithLLzz== 1. 1.

FigureFigure 6. 6.PoincarésurfacesPoincarésurfaces of of section section of of NGC NGC 3877 3877 for for differ- differ- FigureFigure 8. 8.PoincarésurfacesPoincarésurfaces of of section section of of NGC NGC 4010 4010 for for differ- differ- entent values values of of angular angular momentum momentumLLzzwithwithhh==−−1.1. entent values values of of angular angular momentum momentumLLzzwithwithhh==−−10.10. epicyclic frequencies, showing that unlike the results presented in González et al. (2010) for NGC 3877 and NGC 4010, our models satisfy the stability conditions for radial and vertical perturbations. Even though the set of models presented here should be considered as a rough approximation, the circular velocities were shown to fit very accurately to the observed rotation curves and in the three cases the stability conditions were fully satisfied. Here, it is important to note that con- trary to the observed tendency in the Miyamoto-Nagay model, where the limit a → 0 reduces to the Plummer sphere, our models exhibit a tendency to an spherical mass distribution with increasing of the parameter a.

On the other hand, by using the Poincarésection 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 compari- son to NGC 3726 and NGC 4010. It should be noted that Figure 9. Poincarésurfaces of section of NGC 4010 for differ- none of the studied models showed a fully chaotic phase space. ent values of total energy h with Lz = 5. Our results could have significant implications for the study of the dynamics and kinematics of these three specific galaxies, The transition from regularity to chaos (or viceversa) that takes since the regular or chaotic behaviors could shed lights into the place for the three considered galaxy models was inspected evolution and structure of these galaxies, i.e., in phase space, through the Poincarésections in Figs. 4-9, by using different regular orbits are trapped in the vicinity of neighbor orbits, values of L (Figs. 4, 6 and 8) and h (Figs. 5, 7 and 9). It can z while chaotic orbits, by its own nature, will diverge exponen- be observed that the orbital motion exhibits a strong depen- tially in time from its neighbors by filling the phase space in dence on the angular momentum L and energy h of the test z an erratic manner. Figure 5. Poincarésurfaces of section of NGC 3726 for differ- Figure 7. Poincarésurfaces of section of NGC 3877 for differ- particle. In particular, from the surfaces of section presented Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 in Figs. 4, 6, and 8, we mayOrbital infer thatdynamics there in realistic exists galaxy an increase models ent values of total energy h with Lz = 1. ent values of total energy h with Lz = 1. doi: http://dx.doi.org/10.18257/raccefyn.774 in the regularity of the system for larger values of the angular momentum L , i.e. if there exists a chaotic sea the increase z Acknowledgments of Lz will fill the phase with KAM islands, while the opposite effect is observed for larger values of energy h (see Figs. 5, 7, and 9), where the KAM islands deform and shrink giving place We would like to thank the anonymous referees for their use- to larger regions of chaos. ful comments and remarks, which improved the clarity and quality of the manuscript. FLD, SMM and GAG gratefully ac- knowledges the financial support provided by COLCIENCIAS Concluding remarks (Colombia) under Grants No. 8840 and 8863.

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 Authors’ contributions epicyclicanalytical frequencies, velocity curves showing were that adjusted unlike tothe the results observed presented ones inofGonzález three specificet al. spiral(2010) galaxies: for NGC NGC 3877 3726, and NGC 4010, 3877 andour All authors make substantial contributions to conception, de- modelsepicyclicNGC 4010. satisfy frequencies, The the resulting stability showing analytical conditions that unlike models for the radial resultswere and used pre vertsented toical de- sign, analysis and interpretation of data. All authors partici- perturbations.intermineGonzález the mass-densityet Even al. (2010) though distributionsfor the NGC set 3877of models and and the NGC presented vertical 4010, here anourd pate in drafting the article and reviewed the final manuscript. modelsshouldepicyclic be satisfy frequencies, considered the stability showing as a rough conditions that approximation, unlike for the radial results and the pre circularvertsentedical perturbations.velocitiesin González wereet Evenshown al. (2010) though to fit for thevery NGC set accurately 3877of models and to NGC presented the 4010, observed here our modelsshouldrotation be satisfy curves considered the and stability in as the a three rough conditions cases approximation, the for stability radial and the conditi circularverticalons perturbations.velocitieswere fully were satisfied. Evenshown thoughHere, to fit it thevery is importantset accurately of models to to notepresented the that observed herecon- shouldrotationtrary to be the curves considered observed and in tendency as the a three rough in cases the approximation, Miyamoto-Nagay the stability the conditi circular model,ons → velocitieswerewhere fully the were satisfied.limit showna Here,0 to reduces fit it very is importantto accurately the Plummer to to note the sphere, that observed con- our rotationtrarymodels to exhibit the curves observed anda tendency in tendency the three to inan cases the spherical Miyamoto-Nagay the stability mass distribution conditi model,ons Figure 6. Poincarésurfaces of section of NGC 3877 for differ- Figure 8. Poincarésurfaces of section of NGC 4010 for differ- werewherewith increasing fully the satisfied.limit ofa the→ Here, parameter0 reduces it is importanttoa. the Plummer to note sphere, that con- our ent values of angular momentum Lz with h = −1. ent values of angular momentum Lz with h = −10. models exhibit a tendency to an spherical mass distribution traryOn the to other the observed hand, by tendency using the in thePoincarésection Miyamoto-Nagay method model, we with increasing of the→ parameter a. wherehave also the studied limit a the dynamics0 reduces of to the the meridional Plummer orbits sphere, of st ourars models exhibit a tendency to an spherical mass distribution Onin presence the other of the hand, gravitational by using the field Poincarésection of the galaxy models. method From we withepicyclic increasing frequencies, of the showing parameter thata. unlike the results presented haveour results also studied it may the be dynamics inferred that of the there meridional exists an orbits increase of stars in in González et al. (2010) for NGC 3877 and NGC 4010, our inthe presence regularity of the of the gravitational orbits for largerfield of values the galaxy of the models. angular Frommo- Onmodels the other satisfy hand, the stability by using conditions the Poincarésection for radial and method vertical we mentum,our results while it may for be larger inferred values that of there energy exists theorbits an increase tend toin haveperturbations. also studied Even the dynamics though the of theset meridionalof models presented orbits of st herears bethe more regularity chaotic. of the Our orbits toy for models larger suggest values of that the in angular the threemo- inshould presence be ofconsidered the gravitational as a rough field approximation, of the galaxy models. the circular From mentum,galaxy models while chaotic for larger orbits values are possible, of energy however the orbits the tend chaotic to ourvelocities results were it may shown be inferred to fit very that accurately there exists to an the increase observed in bebehavior more chaotic.is very weak Our for toy the models NGC suggest 3877 model that in in the compari- three therotation regularity curves of and the orbits in the for three larger cases values the stability of the angular conditimo-ons galaxyson to models NGC 3726 chaotic and orbits NGC are 4010. possible, It should however be thenoted chao thattic Figure 9. Poincarésurfaces of section of NGC 4010 for differ- mentum,were fully while satisfied. for larger Here, values it is ofimportant energy the to noteorbits that tend con- to behaviornone of the is studied very weak models for theshowed NGC a fully 3877 chaotic model phase in compari- space. ent values of total energy h with L = 5. betrary more to the chaotic. observed Our tendency toy models in the suggest Miyamoto-Nagay that in the model, three z sonOur to results NGC could 3726 have and significant NGC 4010. implications It should for be the noted stud thaty of galaxywhere modelsthe limit chaotica → orbits0 reduces are possible, to the Plummer however thesphere, chao ourtic Figure 9. Poincarésurfaces of section of NGC 4010 for differ- nonethe dynamics of the studied and kinematics models showed of these a fully three chaotic specific phase galaxies space,. The transition from regularity to chaos (or viceversa) that takes behaviormodels exhibit is very a weak tendency for the to NGC an spherical 3877 model mass in distribution compari- ent values of total energy h with Lz = 5. Oursince results the regular could or have chaotic significant behaviors implications could shed for lights the stud intoy the of place for the three considered galaxy models was inspected sonwith to increasing NGC 3726 of the and parameter NGC 4010.a. It should be noted that Figure 9. Poincarésurfaces of section of NGC 4010 for differ- theevolution dynamics and andstructure kinematics of these of galaxies, these three i.e., specific in phase galaxies space,, Thethrough transition the Poincarésections from regularity to in chaos Figs. (or 4-9, viceversa) by using that differetakesnt none of the studied models showed a fully chaotic phase space. ent values of total energy h with Lz = 5. sinceregular the orbits regular are or trapped chaotic behaviors in the vicinity could ofshed neighbor lights int orbito thes, placevalues for of L thez (Figs. three 4, considered 6 and 8) and galaxyh (Figs. models 5, 7was and 9).inspected It can OurOn theresults other could hand, have by significant using the implications Poincarésection for the method study weof thewhileevolutionhave dynamics also chaotic andstudied orbits, andstructure the kinematics by dynamics ofits these own of of nature, galaxies, these the meridional three will i.e., diverge specific in orbits phase expone galaxies of sp stace,arsn-, beThethrough observed transition the Poincarésectionsthat from the regularity orbital motion to in chaos Figs. exhibits (or 4-9, viceversa) by a using strong that differe depen-takesnt sincetiallyregularin presence the in orbits time regular of fromthe are or gravitational trapped chaotic its neighbors behaviors in the field by vicinity of filling could the galaxy ofshedthe neighbor phase lights models. space int orbito From the ins, denceplacevalues foron of L the thez (Figs. angular three 4, considered 6 momentum and 8) and galaxyLhz (Figs.and models energy 5, 7was andh of 9).inspected the It test can evolutionanwhileour erratic results chaotic and manner. it orbits, maystructure be by inferred ofits these own that nature, galaxies, there will existsi.e., diverge in an phase increase expone space,n- in beparticle.through observed the In particular, Poincarésectionsthat the orbital from themotion in surfaces Figs. exhibits 4-9, of by section a using strong prese differe depen-ntednt regulartiallythe regularity in orbits time from areof the trapped its orbits neighbors forin the larger by vicinity filling values ofthe of neighbor the phase angular space orbitmo- ins, denceinvalues Figs. on of 4,L the 6,(Figs. andangular 8, 4, we 6 momentum and may 8) infer and thatLhz (Figs.and there energy 5, exists 7 andh anof 9). increasethe It test can z whileanmentum, erratic chaotic whilemanner. orbits, for larger by its values own nature, of energy will the diverge orbits expone tendn- to beparticle.in the observed regularity In particular, that of the the orbital systemfrom themotion for surfaces larger exhibits values of section a ofstrong the prese angul depen-ntedar tiallybe more in time chaotic. fromOur its neighbors toy models by suggestfilling the that phase in the space three in denceinmomentum Figs. on 4, the 6,L andangularz, i.e. 8, we if momentum there may infer exists thatL aand chaotic there energy exists seah the anof increasethe test z Acknowledgmentsangalaxy erratic models manner. chaotic orbits are possible, however the chaotic particle.inof L thez will regularity In fill particular, the of phase the systemfrom with KAMthe for surfaces larger islands, values of while section of the the prese opposite angulntedar behavior is very weak for the NGC 3877 model in compari- effectinmomentum Figs. is 4,observed 6,L andz, i.e. for8, we if larger there may values infer exists that of a energy chaotic there existsh sea(see the an Figs. increase 5, 7, andof L 9),will where fill the the phase KAM withislands KAM deform islands, and shrink while the giving opposite place WeAcknowledgmentsson would to NGC like 3726 to thank and theNGC anonymous 4010. It refereesshould be for noted their use- that inFigure thez regularity 9. Poincarésurfaces of the system of for section larger of values NGC 4010of the for angul differ-ar effectto larger is observed regions of for chaos. larger values of energy h (see Figs. 5, 7, fulnone comments of the studied and remarks, models showed which a improved fully chaotic the phase clarity space and. momentument values ofLz total, i.e. energy if thereh with existsL a= chaotic 5. sea the increase z WeAcknowledgmentsqualityOur would results of the like could manuscript. to thank have significant the FLD, anonymous SMM implications and referees GAG for gratefullyfor the their stud use-y ac- of andof Lz 9),will where fill the the phase KAM withislands KAM deform islands, and shrink while the giving opposite place knowledgesthe dynamics the and financial kinematics support of provided these three by specificCOLCIENCIAS galaxies, effecttoThe larger transition is observed regions from of for chaos. regularity larger values to chaos of energy (or viceversa)h (see Figs. that takes 5, 7, ful comments and remarks, which improved the clarity and (Colombia)since the regular under or Grants chaotic No. behaviors 8840 and could 8863. shed lights into the Concludingandplace 9), for where the the three remarks KAM considered islands deform galaxy and models shrink was giving inspected place Wequality would of the like manuscript. to thank the FLD, anonymous SMM and referees GAG gratefullyfor their use- ac- evolution and structure of these galaxies, i.e., in phase space, tothrough larger regionsthe Poincarésections of chaos. in Figs. 4-9, by using different fulknowledges comments the and financial remarks, support which provided improved by COLCIENCIAS the clarity and regular orbits are trapped in the vicinity of neighbor orbits, InConcludingvalues the present of L (Figs. paper, remarks 4, using 6 and the 8) andgeneralh (Figs. solution 5, 7 to and the 9). Lapla It cance quality(Colombia) of the under manuscript. Grants No. FLD, 8840 SMM and and 8863. GAG gratefully ac- z while chaotic orbits, by its own nature, will diverge exponen- equation,be observed we that have the derived orbital a generalized motion exhibits Miyamoto-Nagai a strong depen- po- knowledges the financial support provided by COLCIENCIAS Authors’tially in time contributions from its neighbors by filling the phase space in Concludingtential.dence on By the means angular remarks of momentum the nonlinearLz leastand energy squareh fitting,of the thetest (Colombia) under Grants No. 8840 and 8863. In the present paper, using the general solution to the Laplace an erratic manner. equation,analyticalparticle. Inwe velocity particular, have derivedcurves from were a generalized the adjusted surfaces toMiyamoto-Nagai of the section observed prese onnted po-es Authors’ contributions Intential.ofin thethree Figs. present 4,Byspecific 6, means and paper, spiral 8, of we using the galaxies: may nonlinearthe infer general NGC that least solution there 3726, square exists NGC to the fitting, an 3877 Lapla increase and thece All authors make substantial contributions to conception, de- in the regularity of the system for larger values of the angular equation,analyticalNGC 4010. we velocity The have resulting derivedcurves were analytical a generalized adjusted models toMiyamoto-Nagai the were observed used to on po-de-es sign, analysis and interpretation of data. All authors partici- terminemomentum theL mass-densityz, i.e. if there distributions exists a chaotic and the sea vertical the increase and Authors’pate in drafting contributions the article and reviewed the final manuscript. tential.of three Byspecific means spiral of the galaxies: nonlinear NGC least 3726, square NGC fitting, 3877 and the AllAcknowledgments authors make substantial contributions to conception, de- analyticalNGCof Lz 4010.will velocity fill The the resulting phase curves with were analytical KAM adjusted islands, models to thewhile were observed the used opposite to on de-es sign, analysis and interpretation of data. All authors partici- effect is observed for larger values of energy h (see Figs. 5, 7, oftermine three the specific mass-density spiral galaxies: distributions NGC 3726, and the NGC vertical 3877 anandd Allpate authors in drafting make the substantial article and contributions reviewed the to final conception, manuscripde-t. We would like to thank the anonymous referees for their use- NGCand 9), 4010. where The the resulting KAM islands analytical deform models and shrink were usedgiving to place de- sign, analysis and interpretation of data. All authors partici-29 ful comments and remarks, which improved the clarity and termineto larger the regions mass-density of chaos. distributions and the vertical and pate in drafting the article and reviewed the final manuscript. quality of the manuscript. FLD, SMM and GAG gratefully ac- knowledges the financial support provided by COLCIENCIAS Concluding remarks (Colombia) under Grants No. 8840 and 8863.

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 Authors’ contributions analytical velocity curves were adjusted to the observed ones of three specific spiral galaxies: NGC 3726, NGC 3877 and All authors make substantial contributions to conception, de- NGC 4010. The resulting analytical models were used to de- sign, analysis and interpretation of data. All authors partici- termine the mass-density distributions and the vertical and pate in drafting the article and reviewed the final manuscript. Dubeibe FL, Martínez-Sicachá SM, González GA Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 43(166):24-30, enero-marzo de 2019 doi: http://dx.doi.org/10.18257/raccefyn.774

Conflict of interest Lindblad, P. O. (1960). Stockholm Obs. Ann. 21, No. 3-4. Long, K., Murali, C. (1992). Analytical potentials for barred galaxies. The Astrophysical Journal, 397, 44-48. The authors declare that they have no conflict of interest. Manabe, S. (1979). Applicability of approximate third integral of motion for stellar orbits in the galaxy. Publications of the Astro- References nomical Society of Japan, 31, 369-394. Manos, T. and Athanassoula, E. (2011). Regular and chaotic orbits in barred galaxies-I. Applying the SALI/GALI method to Binney, J. and Tremaine, S. (2011). Galactic dynamics. Prince- explore their distribution in several models. Monthly Notices of ton university press. the Royal Astronomical Society, 415(1), 629-642. Bountis, T., Manos, T. and Antonopoulos, C. (2012). Com- Manos, T., Bountis, T. and Skokos, C. (2013). Interplay be- plex statistics in Hamiltonian barred galaxy models. Celestial Me- tween chaotic and regular motion in a time-dependent barred chanics and Dynamical Astronomy, 113(1), 63-80. galaxy model. Journal of Physics A: Mathematical and Theoreti- Caranicolas, N. D. (1996). The structure of motion in a 4- cal, 46(25), 254017. component galaxy mass model. Astrophysics and Space Science, Maoz, D. (2016). Astrophysics in a Nutshell: Second Edition. 246(1), 15-28. Princeton university press. Caranicolas, N. D. and Papadopoulos, N. J (2003). Chaotic Martinet, L. and Mayer, F. (1975). Galactic orbits and integrals orbits in a galaxy model with a massive nucleus. Astronomy & of motion for stars of old galactic populations. III-Conclusions and Astrophysics, 399(3), 957-960. applications. Astronomy and Astrophysics, 44, 45-57. Contopoulos, G. (1979). Stochastic behavior in classical and quan- Miyamoto, M. and Nagai, R. (1975). Three-dimensional mod- tum Hamiltonian systems, G. Casati and J. Ford Eds., p. 1-17. els for the distribution of mass in galaxies. Publications of the Dehnen, W. (1993). A family of potential-density pairs for spherical Astronomical Society of Japan, 27, 533-543. galaxies and bulges. Monthly Notices of the Royal Astronomical Satoh, C. (1980). Dynamical models of axisymmetric galaxies and Society, 265(1), 250-256. their applications to the elliptical galaxy NGC4697. Publications Dubeibe, F. L. and Bermúdez-Almanza, L. D. (2014). Opti- of the Astronomical Society of Japan, 32, 41. mal conditions for the numerical calculation of the largest Lya- Verheijen, M. A. W. and Sancisi, R. (2001). The Ursa Ma- punov exponent for systems of ordinary differential equations. In- jor cluster of galaxies-IV. HI synthesis observations. Astronomy ternational Journal of Modern Physics C, 25(07), 1450024. & Astrophysics, 370(3), 765-867. Dubeibe, F. L., Riaño-Doncel, A., and Zotos, E. E (2018). Vogt, D. and Letelier, P. S. (2005). On multipolar analytical Dynamical analysis of bounded and unbounded orbits in a gener- potentials for galaxies. Publications of the Astronomical Society alized Hénon-Heiles system. Physics Letters A, 382(13), 904-910. of Japan, 57(6), 871-875. González, G. A., Plata-Plata, S. M. and Ramos-Caro J. Zotos, E. E. (2012). Exploring the nature of orbits in a galactic (2010). Finite thin disc models of four galaxies in the Ursa Ma- model with a massive nucleus. New Astronomy, 17(6), 576-588. jor cluster: NGC 3877, NGC 3917, NGC 3949 and NGC 4010. Zotos, E. E. and Caranicolas, N. D. (2013). Revealing the in- Monthly Notices of the Royal Astronomical Society, 404(1), 468- fluence of dark matter on the nature of motion and the families of 474. orbits in axisymmetric galaxy models. Astronomy & Astrophysics, Greiner, J. (1987). A new kind of stellar orbit in a galactic poten- 560, A110. tial. Cel. Mech. 40, 171. Zotos, E. E. (2014). Classifying orbits in galaxy models with a Hernquist, L. (1990). An analytical model for spherical galaxies prolate or an oblate dark matter halo component. Astronomy & and bulges. The Astrophysical Journal, 356, 359-364. Astrophysics, 563, A19. Jaffe, W. (1983). A simple model for the distribution of light in Zotos, E. E., Riaño-Doncel, A., and Dubeibe, F. L. (2018). spherical galaxies. Monthly Notices of the Royal Astronomical So- Basins of convergence of equilibrium points in the generalized ciety, 202(4), 995-999. Hénon-Heiles system. International Journal of Non-Linear Me- Lees, J. F. and Schwarzschild, M. (1992). The orbital structure chanics, 99, 218-228. of galactic halos. The Astrophysical Journal, 384, 491-501.