Orbital Dynamics in Realistic Galaxy Models: NGC 3726, NGC 3877 And

arXiv:1907.09573v1 [astro-ph.GA] 22 Jul 2019 by nti epc,priua teto eev h mod the deserve attention particular consi by respect, grown proposed has this galaxies In of types ably. different the of for field models itational analytical three-dimensional on literature ic h eia ae by paper seminal the Since Introduction 2019. 2 ria yaisi elsi aaymdl:NC32,NC3 NGC 3726, NGC models: galaxy realistic in dynamics Orbital rp eIvsiaio nRltvddyGravitación, Esc y Relatividad Investigación en de Grupo ∗ 1 orsodni:G .González, guillermo.gonzalez@sa A. G. Correspondencia: rp eIvsiaio aeds,Fcla eCeca Hu Ciencias de Facultad Investigación Cavendish, de Grupo Jaffe aarsclave: Palabras ca principalmente dinámica es pres la an estrellas opuestos, energ´ıa total, se para órbitas caóticas su casos encontrar acotadas Seguidamente, los bajos valores en 4010. o que NGC prueba part´ıcula de y la 3877 de Poincaré, tr encontran NGC de para secciones 3726, método (WSRT), de términos NGC del Telescope l´ Radio Major: imágenes de Synthesis Ursa de Westerbork conjunto el a un con espirales de del generalización tomaron galaxias se tres una observacionales de utilizando rotación observadas de trabajo, curvas presente el En Resumen 37 galáxias: NGC de realistas modelos en Dinámica orbital e words: those Key in dynamics stars the for orbits cases, bounded opposite chaotic the particl find test on to the while possibility of regular, momentum terms mainly angular in the is c analyzed of is Major surv values system larger Ursa imaging the for anal the synthesis of in dynamics cm-line the galaxies the 21 to Accordingly, particular a galaxies three from for spiral taken Telescope, specific been Miyamoto have three the data of of generalization curves a using rotation paper, present the In Abstract 18)and (1983) rd .Dubeibe L. Fredy tla yais aais ieaisaddnmc;Nonl dynamics; and kinematics Galaxies: dynamics; Stellar iaoo&Nagai & Miyamoto Hernquist i´ mc sea,Glxa:cnmaiaydinámica, Din cinemática y Galaxias: Dinámica estelar, 1 adaM Mart´ınez-Sicachá M. Sandra , 19) h derived who (1990), 17) the (1975), ead ´sc,UiesddIdsra eSnadr A.A. Santander, de Industrial F´ısica, Universidad de uela G 4010 NGC grav- aa el dccń nvria elsLao,Villavi Llanos, los de Educación, Universidad la de y manas e.i.d.o eiio eotbed 08 cpao 1 Aceptado: 2018; de octubre de 5 Recibido: ber.uis.edu.co, der- els ne nea aaisparticulares. galaxias esas en entes aa ikb naporaestigo h reparameters, free the Miyamoto- of setting the appropriate to an reduces while by that disk galaxies Nagai barred for potential respectivel dist galaxies, later, light elliptical years the and spherical approximate for closely bution which models analytical l ymaso aiyo est rfie ihdffrn cent different with profiles density of family a of means by els oqepr aoe rne e oet angular momento del grandes valores para que do adnaiae rniamnerglr mientras regular, principalmente dinámica es la atclrgalaxies. particular a eoiae iclrsaa´tcs o datos Los anal´ıticas. circulares velocidades las utr G 76 G 87adNC4010. NGC and 3877 NGC 3726, NGC luster: oeca eMymt-aa,s jsa las ajustan se Miyamoto-Nagai, de potencial rlwrvle t oa nrytedynamics the energy total its values lower or e o ia usr oeoar apsblddde posibilidad la abre modelo Nuestro ótica. nad 1cn´mto olıe I obtenidos l´ınea HI) cent´ımetros (o 21 de ınea n aa oeta eajse h observed the adjusted we potential Nagai and tclcrua eoiis h observational The velocities. circular ytical Dehnen ftePicrescin ehd nigthat finding method, Poincaré sections the of smil hoi.Ortymdloesthe opens model toy Our chaotic. mainly is sglxa atclrse lguod la de grupo el en particulares galaxias es yuigteWsebr ytei Radio Synthesis Westerbork the using ey 2 ulem .González A. Guillermo , 6 G 87yNC4010 NGC y 3877 NGC 26, og&Murali & Long 19)gnrlzdteJff n enus mod- Hernquist and Jaffe the generalized (1993) na yaisadchaos. and dynamics inear a ian ielycaos. y lineal ámica no lz adnaiadlsseaen sistema dinámica del la aliza 19)peetda analytical an presented (1992) 7,Bcrmna Colombia Bucaramanga, 678, 2 ∗ hsclSciences Physical eco Colombia cencio, 7 and 877 efbeode febrero de 4 .Afew A y. ral ri- slopes. More recently, Vogt & Letelier (2005) derived an an- radial and vertical perturbations. On the other hand, the dy- alytical expression for the gravitational field of galaxies, based namics of the orbits is studied through the Poincarésurfaces on the multipolar expansion up to the quadrupole term. Using of section, showing that the orbital motion exhibits a strong a different approach, González et al. (2010) obtained a family dependence on the angular momentum and energy of the test of finite thin-discs models for four galaxies in the Ursa major particles (stars). cluster in which the circular velocities were adjusted to fit the observed rotation curves. The paper is organized as follows: in the first section, we de- rive the generalized Miyamoto-Nagai model. Next, from the One advantage of an analytical galaxy model is the possibil- new potential the explicit expressions for the physical quanti- ity to study the dynamics (regular or chaotic) of orbits. This ties of interest are determined. In the second section we adjust can be considered one of the standing problems in galactic the observed rotation curves of three specific spiral galaxies dynamics because it could allow us to understand the for- (NGC 3726, NGC 3877 and NGC 4010) to the analytical circu- mation and evolution of galaxies (Contopoulos, 1979), as lar velocities derived with our model. Then, the mass-density shown by the pioneer simulations of Lindblad (1960). De- profiles are calculated, along with the vertical and epicyclic fre- spite the fact that early papers on this topic studied only regu- quencies, showing that our model not only is well-behaved but lar orbits in the meridional plane (Martinet & Mayer, 1975, also satisfy the stability conditions. A dynamical analysis in Manabe, 1979, Greiner, 1987, Lees & Schwarzschild, terms of the Poincarésurfaces of section is performed in the 1992), soon after, the existence of chaos on the orbital mo- third section. Finally, in the fourth section, we summarize our tion started to be considered by Caranicolas (1996) and main conclusions. Caranicolas & Papadopoulos (2003). In the majority of cases all these studies focused on the distinction between regular and chaotic orbits (Manos & Athanassoula, 2011, Generalized Miyamoto-Nagai model Bountis et. al., 2012, Manos et al., 2013) or the influence of the galaxy components (nucleus, bulge, disk, halo) on the char- Let us start considering the axially symmetric Laplace’s equa- acter of orbits, see e.g. (Zotos, 2012, Zotos & Caranicolas, tion in spherical coordinates Zotos 2013, , 2014). Notwithstanding the evidence that both 1 ∂ ∂Φ 1 ∂ ∂Φ chaotic and regular motions are possible in many axisymmet- ∇2Φ(r, θ)= r2 + sin θ = 0, (1) r2 ∂r ∂r r2 sin θ ∂θ ∂θ ric potentials, recent studies on generalized axisymmetric potentials suggest that a third integral of motion seems to exist whose general solution reads as ∞ for energy values closer to the escape energy (Dubeibe et al., l −(l+1) 2018, Zotos et al., 2018). Hence, such apparent ambiguity Φ(r, θ)= Al r − Bl r Pl(cos θ), (2) might only be solved by performing systematic studies of each Xl=0 particular model. where Al and Bl are constants to be determined, Pl are the 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 B P (cos θ) Φ(r, θ)= − l l . (3) adjusted very accurately to fit the observed rotation curve and rl+1 Xl=0 hence it is assumed that all (or most of) the components are 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 cos θ = z/r and r = R2 + z2, (4) to small vertical perturbations (see e.g. the cases of NGC 3877 p Satoh and NGC 4010), our models satisfy the stability conditions for and applying the additional transformation ( , 1980), ∗ z → z = a + z2 + b2, (5) p 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) p 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 obtained from the following expressions evaluated at z = 0 HaL HbL 1.0 (Binney & Tremaine, 2011) 150 0.8 2 ∂Φ L

v = R , (8) s 0.6

∂R

100 S km 2 H 0.4

∂ Φ 3 ∂Φ v k2 = + , (9) ∂R2 R ∂R 50 0.2 2 2 ∂ Φ ν = . (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, HcL k2 ≥ 0, and ν2 ≥ 0, where the last two inequalities are un- 0.10 derstood as stability conditions (Vogt & Letelier, 2005). 0.05 As is evident from the preceding paragraphs, the galactic mod- z 0.00 els 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 will discuss in detail in the next section. HdL HeL 1.0 1.0

0.8 0.8

Rotation curves fitting 0.6 0.6 2 2 k Ν 0.4 0.4 The observational data were taken from Verheijen & Sancisi (2001) for three specific galaxies in the Ursa Major cluster: 0.2 0.2 NGC 3726, NGC 3877 and NGC 4010. Following the procedure 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 outlined in González et al. (2010), we take the galaxy radius R R Rd as the given by the largest tabulated value of the data. ˜ Thus, introducing dimensionless variables R = R/Rd, z˜ = Figure 1. Model fitted to the rotation curve of NGC 3726 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) 2 ˜ 3 ˜ 4 B1/Rd, B2 = B2/Rd, and B3 = B3/Rd, the nonlinear least The solid curve indicates the rotation velocity calculated from ∗ It should be noted that setting B0 = GM,B1 = B2 = B3 = 0 in (6), we get the well-known Miyamoto-Nagai Potential (Miyamoto & Nagai, 1975). (8) while the error bars denote the velocity dispersions of the NGC 4010. The solid lines correspond to the analytical ex- observational data. (b) Normalized mass-density distribution pressions (8) fitted to the rotation curves. As can be seen, in Σ˜ at z = 0, calculated from (7). (c) Constant-density curves of each case the model fits the observed data with good accuracy. equation (7) in the meridional plane. (d) Epicyclic frequency Additionally, in panels (b) of Figures 1, 2, and 3, we plot the (9) evaluated on z = 0. (e) Vertical frequency (10) evaluated normalized mass-density distribution (7) at z = 0 for the three on z = 0. galaxies, as a function of the dimensionless radial coordinate R˜. Here, we obtain a well-behaved mass-density function, showing HaL HbL a maximum value at the center that decreases to zero at the 1.0 edge of the disk. On the other hand, in panels (c) of Figures 1, 150 0.8 2, and 3, we present four isodensity curves of the mass-density distribution (7) in the meridional plane (R,˜ z˜), showing that L s

0.6 100 each model corresponds to a very diﬀerent mass distribution. S km H 0.4

v Finally, from panels (d) and (e) of the same ﬁgures, it is note- 50 0.2 worthy that in the three 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 R R HaL HbL 1.0 HcL 120 0.10 0.8 100 L

0.05 s 80 0.6 S km

H 60 z 0.00 0.4 v 40 -0.05 0.2 20 -0.10 0 0.0 -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 R R R HdL HeL HcL 1.0 1.0 0.10

0.8 0.8 0.05 0.6 0.6 z 0.00 2 2 k Ν 0.4 0.4 -0.05

0.2 0.2 -0.10 -1.0 -0.5 0.0 0.5 1.0 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 R R R HdL HeL 1.0 1.0 Figure 2. Model ﬁtted to the rotation curve of NGC 3877 using the parameters given in the second column of Table 1. (a) 0.8 0.8 The solid curve indicates the rotation velocity calculated from 0.6 0.6 2 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, calculated from (7). (c) Constant-density curves of 0.2 0.2 equation (7) in the meridional plane. (d) Epicyclic frequency 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 (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 obser- using the parameters given in the third column of Table 1. (a) vational data (points) of the rotation curve with the cor- The solid curve indicates the rotation velocity calculated from responding velocity dispersions (error bars) as reported by (8) while the error bars denote the velocity dispersions of the Verheijen & Sancisi (2001) for NGC 3726, NGC 3877 and 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 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. and we scan the phase space with a large number of initial conditions for the radii R0, these three values allow us to determine 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 1 L = R˙ 2 +(Rφ˙)2 +z ˙2 − Φ(R,z), (11) 2 h i 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 where Φ(R,z) is given by Eq. (6) and its respective parameters Figure 4. Poincarésurfaces of section of NGC 3726 for differ- should be taken from Table 1. ent values of angular momentum Lz with h = −1. Figure 5. Poincarésurfaces of section of NGC 3726 for differ- Figure 7. Poincarésurfaces of section of NGC 3877 for different values of total energy h with Lz = 1. ent values of total energy h with Lz = 1.

Figure 6. Poincarésurfaces of section of NGC 3877 for differ- Figure 8. Poincarésurfaces of section of NGC 4010 for different values of angular momentum Lz with h = −1. ent values of angular momentum Lz with h = −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. particle. In particular, from the surfaces of section presented in Figs. 4, 6, and 8, we may infer that there exists an increase 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 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. 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. 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