Order and chaos in a new 3D dynamical model describing motion in non axially symmetric galaxies
aa r X i v : . [ a s t r o - ph . GA ] F e b Nonlinear Dynamics manuscript No. (will be inserted by the editor)
Order and chaos in a new 3D dynamical model describing motion in nonaxially symmetric galaxies
Euaggelos E. Zotos · Nicolaos D. Caranicolas
Received: 25 June 2013 / Accepted: 1 August 2013 / Published online: 7 September 2013
Abstract
We present a new dynamical model describing 3Dmotion in non axially symmetric galaxies. The model coversa wide range of galaxies from a disk system to an ellipticalgalaxy by suitably choosing the dynamical parameters. Westudy the regular and chaotic character of orbits in the modeland try to connect the degree of chaos with the parameterdescribing the deviation of the system from axial symmetry.In order to obtain this, we use the Smaller ALingment In-dex (SALI) technique by numerically integrating the basicequations of motion, as well as the variational equations forextensive samples of orbits. Our results suggest, that the in-fluence of the deviation parameter on the portion of chaoticorbits strongly depends on the vertical distance z from thegalactic plane of the orbits. Using di ff erent sets of initialconditions, we show that the chaotic motion is dominant ingalaxy models with low values of z , while in the case of starswith large values of z the regular motion is more abundant,both in elliptical and disk galaxy models. Keywords
Galaxies: kinematics and dynamics; Newmodels
To study the dynamical behavior of galaxies it is necessaryto build a model describing the properties of the system. In-formation on the construction of dynamical models of galax-ies are provided usually from observations, made after somenecessary simplifying assumptions. On the other hand, it is afact that galactic dynamical models are successful and real-
Department of Physics,Section of Astrophysics, Astronomy & Mechanics,Aristotle University of ThessalonikiGR-541 24, Thessaloniki, GreeceCorresponding author’s email: [email protected] istic, only if the modeled characteristics agree with the cor-responding observational data.In most cases, the galactic models are spherical or axi-ally symmetric, in an attempt to simplify the study of orbits.For instance, in a spherical system all three components ofthe angular momentum and of course the total angular mo-mentum is conserved. Thus, we have a plane motion, whichtakes place in the plane perpendicular to the vector of thetotal angular momentum. On the other hand, in an axiallysymmetric system, where the motion is described by a po-tential Φ ( R , z ) the L z component of the angular momentumis conserved and the motion takes place in the meridionalplane ( R , z ), which rotates non-uniformly around the axis ofsymmetry with angular velocity ˙ φ = L z / R .Spherical models for galaxies were studied by [12,22,33].Moreover, interesting axially symmetric galaxy models werepresented and studied by [11]. Recently, [34] used data de-rived from rotation curves of real galaxies, in order to con-struct a new axially symmetric model describing the motionin both elliptical and disk galaxy systems.Of particular interest, are the so-called composite galac-tic dynamical models. In those models the potential has sev-eral components each describing a part of the system. Sucha dynamical model composed of four components, that is adisk, a nucleus, a bulge and a dark halo was studied by [6].A new composite mass model describing motion in axiallysymmetric galaxies with dark matter was recently presentedand studied by [7]. Composite axially symmetric galaxy mod-els describing the orbital motion in the Galaxy were alsostudied by [3]. In these models the gravitational potential isgenerated by three superposed disks: one representing thegas layer, one the thin disk and one representing the thickdisk.Another interesting class of galaxy models is the self-consistent models. In order to built a self-consistent model,one must take into account Jeans Theorem (see e.g., [5]). E. E. Zotos & N. D. Caranicolas
According to this theorem, the distribution function f ofa steady state galaxy, depends on the integrals of motion,where only isolating integrals are taken into account. Theself-consistent problem is one of the most di ffi cult problemsin galactic dynamics. Nevertheless, there are several ways ofapproaching this problem. For instance, Binney & Tremaineused two di ff erent approaches in order to obtain the distribu-tion function. Using the first way, they start from f in orderto produce the mass density ρ , while in the second they startfrom ρ heading to f .An e ff ective technique to build self-consistent modelsfor galaxies is the Schwarzschild’s orbit superposition method.This method has been applied in order to study the dynam-ical behavior in both axially symmetric and triaxial galax-ies in a number of papers (see e.g., [17,25,28,30]). Self-consistent dynamical models for a disk galaxy with a triaxialhalo were constructed by [31], where the starting point wasa galaxy in equilibrium composed of an axially symmetricdisk-bulge and a halo. Then, applying an artificial acceler-ation he managed to obtain a system in equilibrium with atriaxial halo.Realistic triaxial models for galaxies with dark matterhaloes were provided in [24]. In their article, the authors ex-tended an earlier method used by [23] to three-dimensionalsystems by replacing the radial with an ellipsoidal symme-try in the mass density. Triaxial galaxy models were alsoconstructed by [2] and [19].The main goal of this article is to introduce a new com-posite mass model in order to use it for the dynamical inves-tigation of galaxies. The model consists of two parts. Themain galaxy body and a massive dense nucleus. The modelcan describe disk and elliptical galaxies as well. Further-more, the model is designed to describe not only galaxiesthat are close to axial symmetry but also systems with con-siderable deviation from axial symmetry. Our target is to in-vestigate the regular and chaotic character of orbits in thenew model and connect the degree of chaos with the param-eter describing the deviation of the model from axial sym-metry.The paper is organized as follows: In Section 2 we presentthe structure and the properties of our new galactic massmodel. In Section 3 we describe the computational methodswe used in order to explore the nature of orbits. In the fol-lowing Section, we investigate how the parameter describ-ing the deviation from axially symmetry influences the reg-ular or chaotic character of the 3D orbits. We conclude withSection 5, where the discussion and the conclusions of thisresearch are presented. Our new dynamical model consists of two components andthe total potential V is given by the equation V ( x , y , z ) = V G ( x , y , z ) + V n ( x , y , z ) , (1)where V G ( x , y , z ) = − GM G q b + x + λ y + (cid:16) α + √ h + z (cid:17) , (2)and V n ( x , y , z ) = − GM n p x + y + z + c . (3)The first part, Eq. (2), is a generalization of the [18] potential(see also [8,9]). In Eq. (2) G is the gravitation constant, M G is the mass of the galaxy, while α, b , h and λ are parametersconnected to the geometry of the galaxy. The parameter λ describes the deviation from the axial symmetry. In the casewhere b > α and α ≫ h the model describes a disk galaxywith a disk halo. Here, α is the disk’s scale-length, h is thedisk’s scale-height and b is the core radius of the disk halo.On the other hand, when b = h ≫ α the model de-scribes an elliptical galaxy with α and h being the horizon-tal and vertical scale-lengths respectively. The second part,Eq. (3), is the potential of a spherically symmetric nucleusin which M n and c n are the mass and the scale-length of thenucleus respectively. This potential has been used in the pastto model the central mass component of a galaxy (see, e.g.,[14,15,36]). Here we must point out, that potential (3) is notintended to represent a black hole nor any other compactobject, but a dense and massive nucleus therefore, we don’tinclude any relativistic e ff ects.We made this choice for two basic reasons. The first rea-son is that the majority of galaxies are not exactly axiallysymmetric. The axial symmetry is only a good approxima-tion, allowing us to perform the numerical calculations a loteasier. Therefore, the new model is useful because it is morerealistic. Furthermore, this model can be use to describea wide variety of galaxies. For instance, when λ is closeto unity it can describe a nearly axially symmetric galaxy,while for larger values of λ it describes systems that are farfrom axial symmetry. A second reason is that the regular orchaotic nature of orbits is drastically a ff ected by the valueof the parameter λ . Thus, using this new model we can drawuseful conclusions connecting the deviation from axial sym-metry with the character of orbits in non axially symmetricgalaxies.In this work, we use the well known system of galacticunits, where the unit of length is 1 kpc, the unit of mass is2 . × M ⊙ and the unit of time is 0 . × yr. The ve-locity units is 10 km / s, while G is equal to unity. The energyunit (per unit mass) is 100 km s − . In these units the valuesof the involved parameters are: M G = M n =
400 and rder and chaos in a new 3D dynamical model describing motion in non axially symmetric galaxies 3 Λ d m i n (a) Λ d m i n (b) Fig. 1
The evolution of the minimum distance d min at which negative value of density appears for the first time, as a function of the parameter λ for (a-left): the disk galaxy model (b-right): the elliptical galaxy model. c n = .
25. For the disk model we choose b = , α = h = .
15, while for the elliptical model we have set b = α = h =
10. The particular values of the system’sparameters were chosen having in mind a Milky Way-typegalaxy [1]. The parameter describing the deviation from axi-ally symmetry λ , on the other hand, is treated as a parameterand its value varies in the interval 0 . ≤ λ ≤ . d = p x + y + z described by themodel exceeds a minimum distance d min , which strongly de-pends on the parameter λ . Fig. 1(a-b) shows a plot of d min vs λ for the both disk and elliptical galaxy models. In allcases, we consider that the dimensions of our new model donot exceed d min . Therefore, the mass density is always posi-tive inside the galaxy described by the model, while is zeroelsewhere. Being more precise, our gravitational potential istruncated at d max =
20 kpc for both reasons: (i) otherwisethe total mass of the galaxy modeled by this potential wouldbe infinite, which is obviously not physical and (ii) to avoidthe existence of any negative density.The corresponding equations of motion are¨ x = − ∂ V ∂ x , ¨ y = − ∂ V ∂ y , ¨ z = − ∂ V ∂ z . (4)The evolution of a deviation vector δ v = ( δ x , δ y , δ z , δ ˙ x , δ ˙ y , δ ˙ z ),which joins the corresponding phase space points of two ini-tially nearby orbits, needed for the computation of the stan-dard indicators of chaos (the SALI in our case), can be mon-itored by the variational equations˙( δ x ) = δ ˙ x , ˙( δ y ) = δ ˙ y , ˙( δ z ) = δ ˙ z , ( ˙ δ ˙ x ) = − ∂ V ∂ x δ x − ∂ V ∂ x ∂ y δ y − ∂ V ∂ x ∂ z δ z , ( ˙ δ ˙ y ) = − ∂ V ∂ y ∂ x δ x − ∂ V ∂ y δ y − ∂ V ∂ y ∂ z δ z , ( ˙ δ ˙ z ) = − ∂ V ∂ z ∂ x δ x − ∂ V ∂ z ∂ y δ y − ∂ V ∂ z δ z . (5)The Hamiltonian which determines the motion of a testparticle (star) in our system is H = (cid:16) ˙ x + ˙ y + ˙ z (cid:17) + V ( x , y , z ) = E , (6)where where ˙ x , ˙ y and ˙ z are the momenta per unit mass con-jugate to x , y and z respectively, while E is the numericalvalue of the Hamiltonian, which is conserved. When studying the orbital structure of a dynamical system,knowing whether an orbit is regular or chaotic is an issueof significant importance. Over the years, several dynami-cal indicators have been developed in order to determine thenature of orbits. In our case, we chose to use the Smaller AL-ingment Index (SALI) method. The SALI [27] is undoubt-edly a very fast, reliable and e ff ective tool, which is definedasSALI(t) ≡ min(d − , d + ) , (7)where d − ≡ k w ( t ) − w ( t ) k and d + ≡ k w ( t ) + w ( t ) k are thealignments indices, while w ( t ) and w ( t ), are two devia-tions vectors which initially point in two random directions.For distinguishing between ordered and chaotic motion, allwe have to do is to compute the SALI for a relatively shorttime interval of numerical integration t max . More precisely,we track simultaneously the time-evolution of the main or-bit itself as well as the two deviation vectors w ( t ) and w ( t )in order to compute the SALI. The variational equations (5),as usual, are used for the evolution and computation of thedeviation vectors.The time-evolution of SALI strongly depends on the na-ture of the computed orbit since when the orbit is regularthe SALI exhibits small fluctuations around non zero val-ues, while on the other hand, in the case of chaotic orbits the E. E. Zotos & N. D. Caranicolas
Fig. 2
Evolution of the SALI of a regular orbit (green color - R), asticky orbit (orange color - S) and a chaotic orbit (red color - C) in ourmodel for a time period of 10 time units. The horizontal, blue, dashedline corresponds to the threshold value 10 − which separates regularfrom chaotic motion. The chaotic orbits needs only about 120 timeunits in order to cross the threshold value, while on the other hand, thesticky orbit requires a vast integration time of about 8000 time units soas to reveal its chaotic nature. SALI after a small transient period it tends exponentiallyto zero approaching the limit of the accuracy of the com-puter (10 − ). Therefore, the particular time-evolution of theSALI allow us to distinguish fast and safely between reg-ular and chaotic motion. Nevertheless, we have to define aspecific numerical threshold value for determining the tran-sition from regularity to chaos. After conducting extensivenumerical experiments, integrating many sets of orbits, weconclude that a safe threshold value for the SALI takinginto account the total integration time of 10 time units isthe value 10 − . In order to decide whether an orbit is regu-lar or chaotic, one may use the usual method according towhich we check after a certain and predefined time intervalof numerical integration, if the value of SALI has becomeless than the established threshold value. Therefore, if SALI ≤ − the orbit is chaotic, while if SALI > − the orbitis regular. The time evolution of a regular (R) and a chaotic(C) orbit for a time period of 10 time units is presentedin Fig. 2. The horizontal, blue, dashed line in Fig. 2 corre-sponds to that threshold value which separates regular fromchaotic motion. Therefore, the distinction between regularand chaotic motion is clear and beyond any doubt when us-ing the SALI method.In our study, each orbit was integrated numerically for atime interval of 10 time units (10 yr), which correspondsto a time span of the order of hundreds of orbital periods and about 100 Hubble times. The particular choice of the to-tal integration time is an element of great importance, espe-cially in the case of the so called “sticky orbits” (i.e., chaoticorbits that behave as regular ones during long periods oftime). A characteristic example of a sticky orbit (S) in ourgalactic system can be seen in Fig. 2, where we observe thatthe chaotic character of the particular sticky orbit is revealedafter a considerable long integration time of about 8000 timeunits. A sticky orbit could be easily misclassified as regularby any chaos indicator , if the total integration interval is toosmall, so that the orbit do not have enough time in order toreveal its true chaotic character. Thus, all the sets of orbitsof a given grid were integrated, as we already said, for 10 time units, thus avoiding sticky orbits with a stickiness atleast of the order of 100 Hubble times. All the sticky orbitswhich do not show any signs of chaoticity for 10 time unitsare counted as regular ones, since that vast sticky periods arecompletely out of scope of our research.For the study of our models, we need to define the sam-ple of orbits whose properties (chaos or regularity) we willidentify. The best method for this purpose, would have beento choose the sets of initial conditions of the orbits from adistribution function of the models. This, however, is notavailable so, we define, for each set of values of the pa-rameters of the potential, a grid of initial conditions ( x , ˙ x )regularly distributed in the area allowed by the value of theenergy. In each grid the step separation of the initial condi-tions along the x and ˙ x axis was controlled in such a way thatalways there are at least 25000 orbits. For each initial condi-tion, we integrated the equations of motion (4) as well as thevariational equations (5) using a double precision Bulirsch-Stoer FORTRAN algorithm [21] with a small time step oforder of 10 − , which is su ffi cient enough for the desired ac-curacy of our computations (i.e. our results practically donot change by halving the time step). In all cases, the en-ergy integral (Eq. (6)) was conserved better than one part in10 − , although for most orbits it was better than one part in10 − . In this Section, we shall present the results of our research.We start our presentation from the results obtained for thedisk galaxy model, which are presented in subsection 4.1.Moreover, subsection 4.2 is devoted to the results obtainedfor elliptical galaxy model. A simple qualitative way for dis-tinguishing between regular and chaotic motion in a Hamil-tonian system is by plotting the successive intersections of Generally, dynamical methods are broadly split into two types: (i)those based on the evolution of sets of deviation vectors in order tocharacterize an orbit and (ii) those based on the frequencies of the or-bits which extract information about the nature of motion only throughthe basic orbital elements without the use of deviation vectors.rder and chaos in a new 3D dynamical model describing motion in non axially symmetric galaxies 5 the orbits using a Poincar´e Surface of Section (PSS) [16].This method has been extensively applied to 2D models, asin these systems the PSS is a two-dimensional plane. In 3Dsystems, however, the PSS is four-dimensional and thus thebehavior of the orbits cannot be easily visualized.One way to overcome this issue is to project the PSS tophase spaces with lower dimensions, following the methodused in [35,37]. Let us start with initial conditions on a 4Dgrid of the PSS. In this way, we are able to identify again re-gions of order and chaos, which may be visualized, if we re-strict our investigation to a subspace of the whole 6D phasespace. We consider orbits with initial conditions ( x , z , ˙ x ), y = ˙ z =
0, while the initial value of ˙ y is always obtainedfrom the energy integral (6). In particular, we define a valueof z , which is kept constant and then we calculate the SALIof the 3D orbits with initial conditions ( x , ˙ x ), y = ˙ z = x , ˙ x ) plane but with an additional value of z , since we dealwith 3D motion. All the initial conditions of the 3D orbitslie inside the limiting curve defined by f ( x , ˙ x ; z ) =
12 ˙ x + V ( x , , z ) . (8)For the disk galaxy model we use the energy value − − x max ≃
15 kpc when z =
1, where x max is the maximum possible value of thecoordinate x on the ( x , ˙ x ) plane.4.1 Disk galaxy modelIn Fig. 3(a-d) we present the final SALI values obtainedfrom the selected grids of initial conditions for four di ff er-ent values of λ when z =
1. The values of all the otherparameters are as in Fig. 1a. Each point is coloured accord-ing to its log (SALI) value at the end of the integration. Inthese SALI plots, the reddish colors correspond to regularorbits, the blue / purple colors represent the chaotic regions,while all the intermediate colors between the two representthe so-called “sticky” orbits whose chaotic nature is revealedonly after long integration time. The outermost black solidline corresponds to the limiting curve defined by Eq. (8).Fig. 3a shows the SALI grid when λ = .
2. We observe avast chaotic sea, which implies that the majority of the or-bits are chaotic. However, we can identify several regions ofregular motion. The two large islands on either side of the x = ff erent in the grid depicted in Fig. 3bwhere λ = .
4. Here, the 1:1 resonance is still present, whilethe 2:1 resonance has disappeared completely. The four re-gions of stability observed inside the grid correspond to 3:2 resonant 3D orbits. Moreover, proceeding to Fig.3c and Fig.3d where λ = . λ = . λ , our galactic model ex-hibits several types of resonant orbits, which are stronglya ff ected by shifting the value of λ . Furthermore, we shouldpoint out that as we increase the value of λ approaching toaxial symmetry ( λ =
1) the amount of chaos decreases. Sim-ilar results can be obtained from the grids shown in Fig. 4(a-d) where λ = . , . , . , .
5. It is evident, that when λ > λ increases (this time movingaway from axial symmetry), the chaotic motion grows inpercentage at the expense of the 1:1 resonant orbits.We proceed now, investigating how the deviation pa-rameter λ influences 3D orbits with large values of z . Wechoose z =
10 as a fiducial value for our sets of initialconditions of orbits. Fig. 5(a-d) shows the SALI grids of ini-tial conditions when λ = . , . , . , .
8. In Fig. 5a where λ = . λ = . λ = .
6. Here, stickyorbits grow again in percentage and also two additional re-gions of regular motion emerge. In Fig. 5d where λ = . z =
10 the available areainside the ( x , ˙ x ) plane is considerably less than that when z =
1. What actually happens is that by shifting to higherlevels of z , both the maximum possible value of the x co-ordinate and the maximum possible value of the ˙ x velocityon the ( x , ˙ x ) plane are constantly reduced as it can be seenin Fig. 7a where a plot of x max (red color) and ˙( x max ) (greencolor) versus z is presented. Moreover, for large values of z we see that the outermost limiting curve tends to be circular,because the ratio ˙( x max ) / x max tends asymptotically to unitywith increasing z (see Fig. 7b). Similar behavior applies inthe case of the elliptical galaxy. Thus, we could argue, thatthis fact justifies in a way, the swarming percentages of reg-ular orbits (see e.g., Fig. 5d). Quite similar outcomes are ob-tained when λ > z =
10. Fig. 6a shows the grid when λ = .
01, that is an almost axially symmetric model. As ex-pected, the vast majority of the computed orbits are regular,while the small deviation from axially symmetry induces a
E. E. Zotos & N. D. Caranicolas (a) (b)(c) (d)
Fig. 3
SALI grids of initial conditions ( x , ˙ x ) for the disk galaxy model when z =
1. (a-upper left): λ = .
2, (b-upper right): λ = .
4, (c-lowerleft): λ = . λ = . (a) (b)(c) (d) Fig. 4
Similar to Fig. 3(a-d). (a-upper left): λ = .
01, (b-upper right): λ = .
1, (c-lower left): λ = . λ = . (a) (b)(c) (d) Fig. 5
SALI grids of initial conditions ( x , ˙ x ) for the disk galaxy model when z =
10. (a-upper left): λ = .
2, (b-upper right): λ = .
4, (c-lowerleft): λ = . λ = . (a) (b)(c) (d) Fig. 6
Similar to Fig. 5(a-d). (a-upper left): λ = .
01, (b-upper right): λ = .
1, (c-lower left): λ = . λ = .
5. E. E. Zotos & N. D. Caranicolas (a) (b)
Fig. 7 (a-left): Evolution of the maximum value of the x coordinate x max (red color) and the maximum value of the ˙ x velocity ˙( x max ) (green color)on the ( x , ˙ x ) plane versus z and (b-right): correlation between the ratio ˙( x max ) / x max and the z value in the λ = . à à à à à à à à à à à à à à à à à à à à à à à à à à àæ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ Λ P e r ce n t a g e H % L o f c h a o ti c o r b it s à z = æ z = Fig. 8
Evolution of the percentage of chaotic orbits in the disk galaxymodels as a function of the deviation parameter λ . Green color corre-sponds to z =
1, while red color to z = low percentage of chaos, which is confined at the left andright regions of the grid. Moreover, as the value of the de-viation parameter λ increases, there is a constant increase ofthe portion of the initial conditions of chaotic orbits, whichpenetrates deeper and deeper toward the central region ofthe grid (see, Figs. 6(b-d)).Thus, calculating from all the SALI grids the percent-age of chaotic orbits (including also the sticky orbits, whichif we integrate them using more time will eventually revealtheir chaotic nature), we are able to follow how this fractionvaries as a function of the deviation parameter λ . As it isclear from Fig. 8, the pattern of the evolution of the chaoticpercentage is quite similar in both studied cases ( z = z = . . λ <
1, while this tendencyis reversed when λ >
1. For λ > . z = z = λ = z =
10, the chaotic percent-age tends asymptotically to zero when λ =
1. On the con-trary, chaos remains at high levels around 25% when z = ff erent values of λ when z =
1. Thevalues of all the other parameters are as in Fig. 1b corre-sponding to the elliptical galaxy model. Fig. 9a depicts thestructure of the ( x , ˙ x ) plane when λ = .
2. We see that a largeunified chaotic sea exists which however, surrounds severalregions of regular motion. We identify three di ff erent typesof regular 3D orbits: (i) 1:1 resonant orbits producing theset of the two islands of stability located at the outer partsof the grid, (ii) 2:1 resonant orbits with initial conditions in-side the two big and distinct islands and (iii) 3:1 resonantorbits, which produce the two small islands near the center.Things are quite di ff erent though according to Fig. 9b where λ = .
4. Here, the islands of the 1:1 resonance at the outerparts of the grid have been increased in size, while the 2:1resonance occupy now the central region replacing, in a way,the 3:1 resonance. With a more closer look, one may alsoidentify a set of four small islands of stability above the 2:1resonance. These islands correspond to the 3:2 resonance. InFig. 9c where λ = . rder and chaos in a new 3D dynamical model describing motion in non axially symmetric galaxies 9 considerably. Therefore, we may conclude, that the increaseof the values of λ in the elliptical model causes the extinctionof secondary resonances, while at the same time reveals thepredominance of the 1:1 orbits. Indeed, when λ = . λ = . , . , . , . λ = .
01 however, the grid is coveredentirely by initial conditions corresponding to 1:1 regularorbits. Furthermore, when λ = . λ = . λ thus having ellipticalgalaxy models away from axial symmetry. Taking into ac-count the numerical results presented in Fig. 10(a-b) when λ > λ influences the regular or chaotic character of the3D orbits when z =
10. In Fig. 11(a-d) we provide fourSALI grids corresponding to λ = . , . , . , . λ = . λ = . λ = .
6. Here, the same orbits with initial condi-tions located at the central region of the grid have changedonce more their character this time from chaotic to regular.Moreover, all the chaotic initial conditions are confined atthe outer parts of the grid. Things are quite similar when λ = .
8. Thus, we may conclude that for small values of λ there is an interplay between regular and chaotic motion,while for larger values tending to axial symmetry, almost allthe orbits are 1:1 regular orbits. In Fig. 12(a-b) we presentthe structure of SALI grids when λ >
1. We see, that infact, the results are very similar to those discussed earlier inFig. 10(a-d). In particular, when λ = .
01 the entire grid isoccupied only by initial conditions corresponding to regularorbits. However, the amount of chaos is increasing slowlybut steadily as we proceed to models with larger values ofdeviation parameter (see Fig. 12(b-d)). à à à à à à à à à à à à à à à à à à à à à à à à à à àæææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ Λ P e r ce n t a g e H % L o f c h a o ti c o r b it s à z = æ z = Fig. 13
Evolution of the percentage of chaotic orbits in the ellipticalgalaxy models as a function of the deviation parameter λ . Green colorcorresponds to z =
1, while red color to z = Table 1
Type, model and initial conditions for the 3D orbits shown inFig. 15(a-f). In all cases, y = ˙ z =
0, while ˙ y is found from the energyintegral given by Eq. (6).Figure Type of orbit model λ x z ˙ x
15a 1:1:0 disk 0.8 10.6 1 0.0015b 1:0:1 elliptical 0.8 0.00 10 10.215c 0:1:1 disk 1.3 0.10 10 0.0015d 1:2:0 disk 0.2 9.85 1 0.0015e 2:3:0 disk 0.4 5.10 1 14.415f chaotic elliptical 0.6 1.50 10 0.00
Of particular interest is the evolution of the percentage ofchaotic orbits in the elliptical galaxy models as a function ofthe deviation parameter λ , which is presented in Fig. 13. Weobserve, that the basic pattern of the evolution is the same inboth cases ( z = z = ff erent regions at the curves: (i) for 0 . < λ . . λ > . λ = λ > z = z = ff erent families, byusing the technique of frequency analysis used by [10,20].Initially, [4] proposed a technique, dubbed spectral dynam-ics, for this particular purpose. Later on, this method has (a) (b)(c) (d) Fig. 9
SALI grids of initial conditions ( x , ˙ x ) for the elliptical galaxy model when z =
1. (a-upper left): λ = .
2, (b-upper right): λ = . λ = . λ = . (a) (b)(c) (d) Fig. 10
Similar to Fig. 9(a-d). (a-upper left): λ = .
01, (b-upper right): λ = .
1, (c-lower left): λ = . λ = . (a) (b)(c) (d) Fig. 11
SALI grids of initial conditions ( x , ˙ x ) for the elliptical galaxy model when z =
10. (a-upper left): λ = .
2, (b-upper right): λ = . λ = . λ = . (a) (b)(c) (d) Fig. 12
Similar to Fig. 11(a-d). (a-upper left): λ = .
01, (b-upper right): λ = .
1, (c-lower left): λ = . λ = . (a) (b) Fig. 14
Orbital structure of the ( x , ˙ x ) plane for (a-left): an elliptical galaxy model with λ = . z =
1; (b-right): a disk galaxy model with λ = . z = been extended and improved by [10], while the extractionof basic frequencies was obtained with the frequency modi-fied Fourier transform which was refined by [26]. In generalterms, this method computes the Fourier transform of thecoordinates of an orbit, identifies its peaks, extracts the cor-responding frequencies and search for the fundamental fre-quencies and their possible resonances. Thus, we can easilyidentify the various families of regular orbits and also rec-ognize the secondary resonances that bifurcate from them.In Fig. 14(a-b) we present two characteristic examples thusdemonstrating the reconstruction of the orbital structure ofthe ( x , ˙ x ) plane, which enable us now to distinguish not onlybetween regular and chaotic motion but also between di ff er-ent families of regular orbits. Fig. 14a depicts the case of anelliptical galaxy model with λ = . z =
1, while inFig. 14b we see a similar grid for a disk galaxy model with λ = . z =
10. Here we should point out, that thesetwo grids are in fact advanced versions of the regular / chaoticSALI grids given in Fig. 10c and Fig. 5b respectively.We shall close this section by presenting in Fig. 15(a-f)several characteristic examples of 3D orbits that are encoun-tered in our galaxy model. All orbits were computed until t =
200 time units. The exact type (resonance), the modelused to produce each orbits and the initial conditions aregiven in Table 1. It is worth noticing that the 1:1 resonanceis usually the hallmark of loop orbits, both coordinates os-cillating with the same frequency in their main motion. Pre-viously, we have seen that in general terms, the dominanttype of regular orbits is the 1:1 resonant orbits. Our exten-sive numerical calculations indicate, that the 1:1 resonancein our model appears in three di ff erent forms, according towhich axes the oscillations take place. In particular, we see in Figs. 15(a-c) three di ff erent types of 1:1 resonant orbits.In Fig. 15a we have the 1:1:0 resonance since the oscilla-tions take place to the x and y axes. On the other hand, inFig. 15b we see a 1:0:1 resonant orbit oscillating at x and z axes, while in Fig. 15c the oscillations take place at the y and z axis and therefore we have a 0:1:1 resonant 3D or-bit. The first type, that is the 1:1:0, is very common to bothdisk and elliptical galaxy models with low values of z . Ingalaxy models with large values of z (i.e., z =
10) thedominant resonance transforms to the other two types (1:0:1and 0:1:1). So, one may conclude, that orbits possessing lowvalues of z should circulate horizontally (parallel with thegalactic plane) around the nucleus, while for 3D orbits hav-ing large values of z we expect them to perform loop orbitsperpendicularly to the galactic plane. Astronomers built and use dynamical models in order to rep-resent and therefore study the structure and the evolutionof galaxies. The data needed for the construction of thesemodels, consist mainly of images and spectra obtained us-ing ground-based observations as well as the Hubble SpaceTelescope (HST).Axially symmetric models for the central parts of galax-ies, containing a central black hole (BH) were constructedby [13]. In their paper, the authors combined ground baseddata from the Michigan-Dartmouth-MIT (MDM) observa-tory with similar input from HST. In particular, the tech-nique of numerical orbit superposition was applied, in or-der to built galactic models with distribution functions withthree isolating integrals of motion. Then, the mass of the rder and chaos in a new 3D dynamical model describing motion in non axially symmetric galaxies 13 (a) (b) (c)(d) (e) (f )
Fig. 15
Characteristic examples of 3D orbits encountered in our disk / elliptical galaxy models. More details are given in the text. central BH, the mass to light ratio and also the orbital struc-ture of the system could be obtained from those models.Moreover, axially symmetric models, for oblate ellipti-cal galaxies, with a distribution function depending on twointegrals of motion were constructed by [29]. In their work,the authors used high quality data from the HST. Using themodels, they managed to obtain the dynamical mass to lightratio M / L and the corresponding rotation rate of each galaxy.They also found that the brightest galaxies rotate too slow toaccount for their flattening.From all the above, it becomes clear that nearly axiallysymmetric, triaxial and asymmetric galaxies cannot be rep-resented by axially symmetric models. Therefore, it seemsnecessary to construct a new dynamical model in order tobe able to describe the properties of motion in non axiallysymmetric galaxies.In this paper, we have presented a new dynamical massmodel for non axially symmetric galaxies. The model con-sists of two parts. The first part represents the main body ofthe galaxy, while the second part represents a massive anddense central nucleus. We made this choice for a number ofreasons. A first reason is that in most galaxies, the axial sym-metry is just an approximation in order to make the math-ematical study of galaxies more convenient. On the otherhand, there is no doubt that there are galaxies that are closeto axial symmetry, as well as galaxies that are not axiallysymmetric. Since our model covers a large variety of types of galaxies, the model could be considered more realistic.For a second reason, we can argue the following: As thereis evidence that most galaxies host massive objects [32], intheir centres, we constructed a model with a spherical mas-sive nucleus. We believe that with this additional massivenucleus, we describe more precisely a real galaxy. A thirdreason is that the dense and massive nucleus plays a vitalrole on the nature of motion, that is the regular or chaoticcharacter of orbits (see [6,36] and references therein).An additional advantage of the new dynamical model isthat it can describe motion in disk as well as in ellipticalgalaxies. This is obtained by suitably choosing the values ofthe parameters ( α, b , h ), while the deviation from axial sym-metry is regulated by the quantity λ . Here we must makeclear, that we consider that the dimensions of the new galaxydynamical model are taken such as the mass density is al-ways positive inside the galaxy and zero elsewhere (see Fig.1a-b).We would also like to remind to the reader, that we haveconstructed this model in order to investigate the regular orchaotic nature of orbits and to try to connect it with the pa-rameter λ . In order to obtain this, we used the SALI method.Using the same technique, we obtained interesting resultson describing the di ff erent families of regular orbits that arepresent in the model. Our results referring to the dynamicalproperties of the new galactic model can be summarized asfollows: (1). Taking into account that our dynamical model isthree-dimensional (3D), we had to find a way to define thesample of orbits whose properties (order or chaos) would beexamined. A very convenient technique, which is describedin Sec. 4, was used and thus, we were able to study 3D or-bits with initial condition in ( x , ˙ x ) plane with an additionalvalue of z . We then studied how the particular value of z controls the amount of chaos in the system by choosing twodi ff erent values of z , leading to the conclusion that z is akey element regarding the nature of orbits. (2). We conducted a thorough investigation in severalcases, using di ff erent values of the parameter in the range0 . < λ < .
5. Our numerical results, indicate that the pa-rameter λ , which describes the deviation from axially sym-metry is indeed very influential both in the disk and the el-liptical galaxy models. (3). When λ < z = ff erent types of res-onances appear. On the other hand, in galaxy models withlarge values of z , such as z =
10, all the secondary reso-nances are suppressed and the 1:1 loop orbits is the domi-nant type. (4).
In the case where λ > z . However, the exact structure of these orbits dif-fer significantly according to the value of z . In fact, starsmoving in 3D regular orbits with low values of z circulateparallel to the galactic plane, while for large values of z the3D loop orbits are performed vertically to the galactic plane. (5). We found a very strong correlation between the valueof the deviation parameter λ and the fraction of chaotic or-bits both in disk and elliptical models. According to ournumerical experiments, chaos turns out to be dominant ingalaxy models with su ffi cient deviation from axial symme-try. Moreover, the observed amount of chaos in the diskgalaxy models is significantly larger than in elliptical mod-els. Acknowledgments
The authors would like to thank the two anonymous refereesfor the careful reading of the manuscript, their positive com-ments and for all the aptly suggestions which allowed us toimprove both the quality and the clarity of our work.
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