Asymptotic Orbits in Barred Spiral Galaxies
Maria Harsoula, Constantinos Kalapotharakos, George Contopoulos
aa r X i v : . [ a s t r o - ph . I M ] O c t Mon. Not. R. Astron. Soc. , 1–17 (2007) Printed 29 November 2017 (MN LaTEX style file v2.2)
Asymptotic Orbits in Barred Spiral Galaxies
M. Harsoula C. Kalapotharakos and G. Contopoulos
Research Center for Astronomy, Academy of Athens, Soranou Efesiou 4, GR-115 27 Athens, Greecee-mail: [email protected], [email protected], [email protected]
Released 2007 April 12
ABSTRACT
We study the formation of the spiral structure of barred spiral galaxies, using an N -body model. The evolution of this N -body model in the adiabatic approximationmaintains a strong spiral pattern for more than 10 bar rotations. We find that thislongevity of the spiral arms is mainly due to the phenomenon of stickiness of chaoticorbits close to the unstable asymptotic manifolds originated from the main unstableperiodic orbits, both inside and outside corotation. The stickiness along the manifoldscorresponding to different energy levels supports parts of the spiral structure. Theloci of the disc velocity minima (where the particles spend most of their time, in theconfiguration space) reveal the density maxima and therefore the main morphologicalstructures of the system. We study the relation of these loci with those of the apoc-entres and pericentres at different energy levels. The diffusion of the sticky chaoticorbits outwards is slow and depends on the initial conditions and the correspondingJacobi constant. Key words: galaxies: structure, kinematics and dynamics, spiral.
Galactic dynamics together with celestial mechanics haveplayed a leading role in the study of orbits as well as inthe study of order and chaos. Many decades ago and for along time the main interest was about regular orbits sincethey were considered as the main building blocks of galaxies.The main idea was that regular orbits are structurally robustand therefore they are able to support various morphologi-cal features. The most extreme manifestation was ellipticalgalaxies for which no chaotic orbits were considered to ex-ist (de Zeeuw 1985; Binney & Tremaine 1987). In rotatinggalaxies the picture was more confused, especially after thestudy of the main periodic orbits in systems representingbarred galaxies. These studies showed the existence of manyunstable orbits (especially near corotation) which are gen-erators of chaos (Contopoulos 1981). Chaos was found inthe 80’s near the ends of the bar (Sparke & Sellwood 1987;Contopoulos & Grosbol 1989). However, chaos was not con-sidered as responsible for some of the main features of thegalaxies like the spiral arms beyond corotation.The importance of chaos was realized mainly in the90’s, with application both in elliptical and in barred spi-ral galaxies, mostly in steady state cases (Merritt 1996;Kaufmann & Contopoulos 1996).Only in the 2000’s people started to consider asymp-totic orbits emanating from the unstable periodic orbitsin explaining the main features of galaxies, like the spiralstructure in barred spiral galaxies. The asymptotic orbits have initial conditions on the asymptotic manifolds ofunstable periodic orbits. This study has been done eitherby using analytical potentials or by using N -body sim-ulations (Voglis et al. 2006a; Contopoulos & Patsis 2006;Romero-Gomez et al. 2006, 2007; Tsoutsis et al. 2008;Athanassoula et al. 2009a; Harsoula & Kalapotharakos2009; Athanassoula et al. 2009b). The former method is notself consistent, but it has the advantage of the analyticalform of the potential, allowing the free choice of all theparameters (like, for example, the strength of the bar andof the spiral perturbation, or the pattern speed of the bar).The latter method has the advantage of self consistency,but the time evolving potential and pattern speed implya secular evolution of the system that makes the studyharder.Photometric data have shown that spiral structures arerelated to the old stellar disk (Grosbol et al. 2004). Ana-lytical models and N -body dissipationless simulations haveshown that, the chaotic domains are small in normal spi-ral galaxies (non barred), because the spiral perturbationsare relatively small. Therefore, in many studies up to now itwas argued that orbits near some stable periodic orbits (reg-ular motions) can support the spiral pattern (see for exam-ple Lin & Shu 1964; Kalnajs 1971; Lynden-Bell & Kalnajs1972; Contopoulos 1975; Toomre 1977).In Kaufmann & Contopoulos (1996) the authorspointed out the role of the so-called ‘hot population’(Sparke & Sellwood 1987), in supporting the inner partsof the spiral arms extending beyond the bar. They found c (cid:13) M. Harsoula, C. Kalapotharakos and G. Contopoulos chaotic orbits that wander stochastically, partly inside andpartly outside corotation. In strong barred galaxies thechaotic domains of phase space have been proved very im-portant (see for example Shen & Sellwood 2004).The role of the chaotic orbits in the spiral structureof a galaxy was pursued by Voglis et al. (2006b), usingself-consistent N -body simulations of barred spiral galaxies,where they found long living spiral arms composed almostentirely of chaotic orbits.Most chaotic orbits are connected to the asymptoticmanifolds of the main unstable periodic orbits. Voglis et al.(2006a) and Tsoutsis et al. (2008) have considered the apoc-entric positions of the asymptotic orbits and the coalescenceof all the unstable manifolds in a certain range of energy lev-els. On the other hand Romero-Gomez et al. (2006, 2007)and Athanassoula et al. (2009a,b) put the emphasis on theasymptotic manifolds emanating only from the unstable pe-riodic orbits of families originating at the Lagrangian points L and L .In this paper we study a certain snapshot of a 3-D N -body simulation of a barred spiral galaxy, in order to givea more concrete and explicit description of the mechanismof construction of the spiral structure out of sticky chaoticorbits which are close to asymptotic orbits from various un-stable periodic orbits. Since the N -body simulation is a self-consistent process, it gives all the details of the secular evo-lution of the galactic system during a Hubble time, havinga potential and a pattern speed that vary with time. How-ever by using frozen potentials and constant pattern speedsthat correspond to certain snapshots of the evolution we areable to study the role of chaotic orbits in supporting a spiralstructure that survives for more than 10 bar rotations, i.e.for at least 1/3 of the Hubble time. The sequence of thesesnapshots can be considered as an adiabatic approximationof the real galaxy evolution.Chaotic orbits spend a long time close and along theunstable asymptotic manifolds of the unstable periodic or-bits in the phase space, due to phenomena of stickiness (seeContopoulos & Harsoula 2008). In our model, there are en-ergy levels where the phase space is not bounded outwardsand chaotic orbits can escape from the system. However, theescape rate is very small, due to stickiness. The orbits spendmost of their time close to velocity minima on the rotationplane of the galaxy.We show that the loci of velocity minima are connectedwith the maxima of the density distribution in the configu-ration space in every energy level and find the connection ofthese geometrical loci with the apocentres and the pericen-tres of the orbits.We find the characteristic diagrams of the most impor-tant 3-D periodic families and we reveal the spiral structureof the galaxy by integrating chaotic orbits that lie close andalong the main unstable periodic orbits. Finally, we con-struct the space configuration by superimposing orbits be-longing to different energy levels. The space distributionof the apocentres and pericentres of these orbits is verysimilar to the apocentric and pericentric manifolds of 2-Dasymptotic orbits having initial conditions along the unsta-ble asymptotic curves of the 2D periodic orbits.The paper is organized as follows: In section 2 we give adescription of the system providing its main properties. Wemake a frequency analysis separately for the chaotic and reg- Figure 1.
The density distribution (in the color scale indicatedin the top of the panel) of the projected particles of our model onthe rotation y − z plane made by the superposition of 20 snapshotscorresponding to the first 20 T hmct of the integration of the systemin the fixed effective potential. Black curves denote characteristicdensity contours revealing the spiral structure. Note that the colorbar shown in this figure applies for all the color scales used in thefollowing figures. ular populations. In section 3 we discuss the role of the apoc-entres and the pericentres of the orbits in comparison withthe loci of the minimum velocities on the rotation plane. Insection 4 we discuss the role of the sticky resonant orbitsin supporting the spiral structure of the system. We presentexamples of specific periodic orbits that are able to recon-struct the main features of the galaxy. Moreover, we discussthe role of the apocentric and pericentric intersections ofthe 2D asymptotic orbits. In section 5 we give a descriptionof the way the orbits escape from the system and finally insection 6 we present our conclusions. The initial conditions of our N -body model, were created byVoglis et al. (2006b) where four experiments with differentpattern speeds simulating barred spiral galaxies were pro-duced. In our paper we use the QR . × particles in our simulation. The timeunit is taken equal to the half mass crossing time of the sys-tem (hereafter T hmct ). A Hubble time corresponds to about300 T hmct . The length unit is taken equal to the half massradius of the system (hereafter r hm ). Finally, the plane ofrotation is the y − z plane (intermediate-long axes) and thesense of rotation is clockwise.The Jacobi constant E j of an orbit is given by the re- c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 2.
The time evolution of the amplitude of the spiral per-turbation δρ/ρ at r ≈ . r hm . We observe that the spiral struc-ture fades out gradually. lation E j = 12 ( ˙ x + ˙ y + ˙ z ) + V ( x, y, z ) −
12 Ω p R yz (1)where V ( x, y, z ) is the full 3D ‘frozen’ potential, given bya Smooth Field Code (SFC code) as an expansion of a bi-orthogonal basis set, ˙ x, ˙ y, ˙ z are the velocities in the rotatingframe of reference and Ω p is the angular velocity of the bar(or pattern speed) at the studied snapshot. The value R yz isthe distance from the rotation axis (cid:16)p y + z (cid:17) . The Jacobiconstant E j is called also “energy in the rotating frame” andcan be written as E j = 12 ( ˙ x + ˙ y + ˙ z ) + V eff ( x, y, z ) (2)where V eff ( x, y, z ) = V ( x, y, z ) − Ω p R yz is the effectivepotential.The energy of the orbit in the inertial frame is relatedto the Jacobi constant by the following relation E = E j + Ω p · L (3)where L is the angular momentum in the inertial frame ofreference and Ω p = − Ω p i , i being the unit vector along the x -axis. Therefore the inertial energy reads E = E j − Ω p ( yP z − zP y ) (4)where P y , P z are the momenta in the inertial frame of refer-ence. Since P y = ˙ y + Ω p z, P z = ˙ z − Ω p y (5)we have E = 12 ( ˙ x + ˙ y + ˙ z ) − Ω p ( ˙ zy − ˙ yz )+ V ( x, y, z )+ 12 Ω p R yz (6)Our study is made in a distinct snapshot of the N -bodysimulation that corresponds to 55 T hmct . At this snapshot thespiral structure of the galaxy is clearly visible and moreover Figure 3.
The distribution of the Jacobi constant values for allthe particles, for the particles moving in regular orbits and forthe particles moving in chaotic orbits. The maximum of the dis-tribution corresponds to chaotic particles around corotation. it survives for several rotations of the bar. In order to studythe mechanism that creates this spiral structure which lastsfor such a long time we fix the potential and the patternspeed and we study the role of chaotic orbits in this fixedpotential. The corresponding value of the pattern speed isΩ p ≈ π/ T hmct and is calculated in Voglis et al. (2006b)(see Fig. 4a of that paper). In real units this value corre-sponds to ≈ km sec − kpc − . This value is in the rangeof the values observed in late type barred spiral galaxies (seefor example Fathi et al. 2009).By integrating all the orbits in this fixed effective po-tential we reveal the longevity of the spiral structure for atleast 100 T hmct , while after much longer times this structuredisappears.In Fig. 1 we plot, on the rotation plane y − z , the super-position of 20 snapshots corresponding to the first 20 T hmct of the integration time of the system in the fixed effective po-tential. Such a superposition consists of more than 2 . × particles and allows us to reduce significantly the noise thatappears when the number of particles is small. Since thedensity is now clearly defined, we can safely quantify the spi-ral perturbation. In Fig. 2 we show the time evolution of theamplitude of the spiral perturbation in our model. More pre-cisely, we plot the density excess ( δρ/ρ ) caused by the spiralstructure in a specific radius that corresponds to ≈ . r hm as a function of time. After a time period of 100 T hmct , dur-ing which the bar has completed ≈
13 rotations, the spiralperturbation is still marginally detectable. This time corre-sponds to 1/3 of the Hubble time.Figure 3 gives the distribution of the Jacobi constants(Eq. 2) for all the particles and separately for chaotic andfor regular orbits. The distinction between the two popu-lations is done by the method introduced by Voglis et al.(2006b). The percentage of chaotic orbits is found to be ≈ E j ( L ) ≈ − E j ( L ) ≈ − L , and L , respectively.From Fig. 3 it is obvious that the regular motions are re-stricted inside corotation and therefore they support theshape of the bar, while chaotic orbits are responsible forthe spiral structure outside corotation.Figure 4 presents the frequency analysis of the regular c (cid:13)000
13 rotations, the spiralperturbation is still marginally detectable. This time corre-sponds to 1/3 of the Hubble time.Figure 3 gives the distribution of the Jacobi constants(Eq. 2) for all the particles and separately for chaotic andfor regular orbits. The distinction between the two popu-lations is done by the method introduced by Voglis et al.(2006b). The percentage of chaotic orbits is found to be ≈ E j ( L ) ≈ − E j ( L ) ≈ − L , and L , respectively.From Fig. 3 it is obvious that the regular motions are re-stricted inside corotation and therefore they support theshape of the bar, while chaotic orbits are responsible forthe spiral structure outside corotation.Figure 4 presents the frequency analysis of the regular c (cid:13)000 , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 4.
The frequency analysis of (a) the regular componentand (b) the chaotic component of the system. The frequencyratio q is given by Eq. (7). We see that the vast majority ofregular orbits lie inside corotation ( q >
0) (especially around2:1 resonance) while the chaotic orbits lie both inside ( q > q < (b) correspond to chaotic orbits sticking close to various resonances. component (Fig. 4a) and of the chaotic component (Fig. 4b)of the system. We are only interested in revealing the “disc”resonances on the rotation plane and therefore the frequencyanalysis is made in the 2D projection of the orbits on the y − z plane. However, there exist also vertical resonancesthat influence mainly the edge-on profiles of our systems(see Patsis et al. 2002).The “frequency ratio” q is a parameter given by therelation q = Ω − Ω p κ (7)where Ω is the angular velocity of a particle in the iner-tial frame of reference and κ is the epicyclic (radial) fre-quency (Binney & Tremaine 1987). Positive q values cor-respond to resonances inside corotation while negative q values correspond to resonances outside corotation, whichare related only to chaotic orbits. In Fig. 4a we see thatthe main resonance of the regular orbits (inside corotation)is the 2:1 resonance (or inner Lindblad resonance), whilethere is also a small number of particles around the 3:1resonance and an even smaller number around corotation( q = 0). The vast majority of these orbits support the bar ofthe system. Figure 4b shows that a rather small percentageof chaotic orbits inside corotation lie near important reso-nances (e.g. 2:1, 3:1 and 4:1). These represent sticky chaotic Figure 5.
The effective potential V eff along the y -axis (blacksolid line) and along the z -axis (black dotted line). The two curvesof kinetic energy E k (dotted and dashed curves) correspond totwo different energy levels E j = − < E j ( L ) (dotted line)and E j = − > E j ( L ) (dashed line). In the former casethe minimum velocity corresponds to apocentres for the motioninside corotation and to pericentres for the motion lying outsidecorotation. In the latter case the minimum velocity correspondsto the position of the maximum of V eff . orbits mainly supporting the outer layers of the bar (seeHarsoula & Kalapotharakos 2009) and the innermost partsof the spiral structure. The rest of the chaotic orbits movearound corotation and outside it and some of them stickaround specific resonances ( q =0, -0.5, -1, -1.5, -2, -2.5, -3).The q = 0 population corresponds to chaotic orbits locatedaround corotation and close to the P L , P L , P L and P L families (nomenclature after Voglis et al. 2006a). Below wewill show that the chaotic orbits sticking along the asymp-totic manifolds originating from these unstable periodic or-bits are responsible for the spiral structure of the system. In this section we discuss the role of apocentres or pericen-tres in revealing the morphological features of the systemand we relate their geometrical loci with the locus of thevelocity minima on the rotation plane.The distribution of the radial velocities ˙ r of the N -bodyorbits has a preferable concentration around the value zero,indicating that the apocentres and pericentres should playan important role for the system.The geometrical loci of the minima of the velocities v yz on the rotation plane should correspond to the maxima ofthe density distribution on the configuration space, becauseparticles spend most of their time there. It is of interest toinvestigate how these loci are related to the apocentres orthe pericentres of the orbits.In Fig. 5 we plot the effective potential of a 2Dapproximation of the system ( x = 0) along the z -axis( V eff (0 , , z ) = V (0 , , z ) − Ω p z , black dotted curve) and c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 6.
The distribution of the velocities of the apocentres, of the pericentres and of the velocity minima for (a) E j < E j ( L ) outsidecorotation (b) E j < E j ( L ) inside corotation and (c) for E j > E j ( L ) where the areas inside and outside corotation can communicate. Figure 7. (a)
The density distribution of the particles of Fig. 1 when integrated until their (cylindrical) radial velocities become zero( ˙ R = 0) for the first time, i.e. at the apocentres or the pericentres of their orbits. The density contours corresponding to the initialconditions of Fig. 1 are superimposed (black curves). (b) The density distribution of only the apocentres. The comparison with thedensity contours shows that the apocentres support the bar but not the extensive spiral arms. (c)
The density distribution of only thepericentres. The comparison with the density contours shows that the pericentres support the spiral structure but not the bar. along the y-axis ( V eff (0 , y,
0) = V (0 , y, − Ω p y , blacksolid curve). The maximum of the black dotted curve cor-responds to the Jacobi constant E j ( L ) of the Lagrangianpoints L , , while the maximum of the black solid curve cor-responds to the Jacobi constant E j ( L ) of the Lagrangianpoints L , . For any Jacobi constant value below E j ( L ),for example for E j = − E j values above E j ( L ), for example for E j = − v min (minimum of thedashed curve) is found at the position of the maximum of theeffective potential (i.e. the minimum difference of E j − V eff ).In this case, the geometrical locus of the velocity minima onthe configuration space is not directly related to the apoc-entres or the pericentres of the orbits.Figure 6 presents the distribution of the velocities ofthe pericentres of the apocentres and of the velocity minimafor three cases: (a) for orbits trapped outside corotationwith E j < E j ( L ), (Fig. 6a), (b) for orbits inside corotationwith E j < E j ( L ), (Fig. 6b) and (c) for orbits that canexplore the whole permissible phase space inside and outsidecorotation with E j > E j ( L ), (Fig. 6c). c (cid:13) , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 8.
The density distribution of the particles of Fig. 1 (with E j < E j ( L )) at the positions corresponding to their local planevelocity minima ( v yz = v min ). We observe that this distributionfits well the morphology of the galaxy (black contour lines). Inthis figure we have omitted the particles with E j > E j ( L ) whichare gathered along the geometrical locus of the maximum of theeffective potential (gray elliptical curve). In Fig. 6a we observe that the pericentres of the orbitsoutside corotation have smaller velocities than the apocen-tres. Moreover the distribution of the velocity minima givesgood agreement with the distribution of the velocities of thepericentres. At the same time, the distribution of the veloc-ities of the apocentres almost coincide with the distributionof the velocity maxima (not plotted in the panel). Therefore,the particles outside corotation spend most of their timenear the pericentres of their orbits which is consistent withthe description of Fig. 5. On the other hand the opposite istrue for particles inside corotation (Fig. 6b). The particlesinside corotation have smaller velocities at their apocentresthan at their pericentres. Moreover, the apocentric veloc-ity distribution almost coincides with the distribution of thevelocity minima. Here again the distribution of the velocitymaxima (not plotted in the panel) almost coincides with thedistribution of the velocities of the pericentres.For the orbits having Jacobi constant above E j ( L )(Fig. 6c) even though there is not a clear breakoff of thedistributions, the pericentres have statistically smaller ve-locities than the apocentres. Both pericentres and apocen-tres present a peak around the same value of velocity, whichalso coincides with the peak of the velocity minima distri-bution. The distribution of the apocentres, however, has asecond important peak around a somewhat greater value ofvelocity.In Fig. 7a we plot, in color scale, the density distributionof the particles of Fig. 1 taking each of them at a positioncorresponding to ˙ r = 0, i.e. at the apocentre or the pericen-tre of its orbit. The black curves correspond to the densitycontours of the space distribution shown in Fig. 1. Figure 7b presents, the density distribution of only the apocentres ofthe orbits. The corresponding density maxima inside coro-tation support the bar (since they correspond to loci of min-imum velocities, see Fig. 6b) as well as a small extension ofthe bar to the inner parts of the spiral structure. However,the density maxima outside corotation, do not correspondto the density contours of the real spiral arms (black curves)but they extend far beyond. Figure 7c presents the densitydistribution of only the pericentres of the orbits. We see thatthese density maxima support a spiral structure very closeto the real one, but a little more tight, and they do notsupport the bar.Thus, we conclude that, in general, the apocentres ofthe orbits support the shape of the bar and the origins ofthe spirals near corotation, while the pericentres of the or-bits better support the spiral structure near and outsidecorotation.Figure 8 presents, in gray scale, the density distribu-tion of the particles of Fig. 1 at positions correspondingto local minimum values of their rotation plane velocities( v yz = v min ). The comparison with the density contours(black curves) of the initial conditions shows that the loci ofthe velocity minima reveal satisfactorily the features of thegalaxy, i.e. the bar and the spiral arms. In this figure, how-ever we have removed the overdensity of the particles alongthe theoretical locus of v min (gray elliptical curve) that rep-resents the maximum of the effective potential ( V eff ) forJacobi constants above E j ( L ), because it does not corre-spond to the distribution of the real particles.Here we note that a density maximum in some area canbe produced either by frequent (but not too short-lasting)transits or by long-lasting (but not too rare) transits ofthe particles through this area. For Jacobi constant valueshigher than E j ( L ) the locus of the local velocity minima(red curve in Fig. 8) corresponds to velocity values that canbe comparable to the mean velocity of the whole orbits. InFig. 9 we have plotted the density distribution of the par-ticles with Jacobi constant values E j > − > E j ( L )corresponding to low and high values of the ratio λ =
The density distribution of the particles having E j > − > E j ( L ) and ratio λ =
25 together withthe locus corresponding to the effective potential maxima V ( eff ) max (red curve). The density maxima in this case are located near thecurve of V ( eff ) max but along particular directions. (b) The same as in (a) but for particles having λ =
6. The densitymaxima are located inside the curve of V ( eff ) max (see text for details). Figure 10. (a)
The characteristics of the main 3D periodic orbits of the system. (b)
A focusing in the inserted frame of (a) . while the periodic orbits P L and P L exist above the value E j ( L ) = − E j (close to the po-tential well) while the bottom right panel corresponds to the highest values of E j ’s. The corresponding values of theJacobi constant are marked on the top of each panel. Notethat outside corotation (in energy levels with E j > E j ( L ))there exist practically only chaotic orbits (see Fig. 3). A gen-eral trend is that the space distribution of particles havinglower values of Jacobi constants supports mostly the outerparts of the spiral arms as well as the shape of the bar, whileparticles having greater values of Jacobi constants supportthe inner parts of the spiral arms. This is obvious in Figs.11 b-h, where we see a gradual shift of the maximum of thedensity distribution of particles outside corotation from the c (cid:13) , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 11.
The density distribution on the y − z plane of the particles belonging to nine different Jacobi constant bins. The correspondingvalues are marked on the top of each panel. The top left panel corresponds to the smallest values of Jacobi constants E j (close to thepotential well) while the bottom right panel corresponds to the highest values of E j ’s. In this figure we reveal which structures aresupported by the particles corresponding to the different Jacobi constant values. outer parts of the spiral arms to the inner parts and theinnermost extensions of the bar (Figs. 11f,g). On the otherhand there are levels of the Jacobi constant that do not sup-port at all the spiral structure. This happens for the lowestvalues of the Jacobi constant near the potential well (Fig.11a), where the distribution of particles outside corotation isquite uniform, and for the highest values of the Jacobi con-stant (Fig. 11i), where only chaotic orbits exist, which donot support the spiral structure but rather the outer partsof the bar.Below we give some examples of the effect of stickiness (Contopoulos & Harsoula 2008) of chaotic orbits around theunstable asymptotic manifolds of the unstable periodic or-bits, in the phase space. We show that this stickiness sup-ports the formation of the spiral structure, while the (slow)diffusion along the paths of the manifolds is responsible forthe fadeout of this structure.In the next subsections we give four examples of 3Dchaotic orbits lying near and along some main unstable pe-riodic orbits, having initial conditions inside corotation, nearcorotation and outside corotation. c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 12. (a)
The density distribution on the y − z plane of 20000 3D orbits integrated for several T hmct having initial conditions closeto and along the unstable periodic orbit − E j = − (b) the density distribution of the pericentres superimposed with the high density contours of the correspondingenergy level. (c) the same as in (b) but for the apocentres. We observe that the pericentres (apocentres) fit (do not fit) the densitymaxima of the real particles. Figure 12 gives an example of the density distribution of or-bits around an unstable periodic orbit of the − E j = − x, y, z, v x , v y , v z ). Then we integrated all these orbitsfor a time corresponding to many rotations of the bar. InFig. 12a we plot the density distribution of these orbits onthe rotation plane. In this figure we have considered a timeinterval within 5-6 bar rotations. In the same figure we havesuperimposed the high density contours of the correspond-ing Jacobi constant (black curves) and we see that the orbitsclose to the − E j value. On the one hand the pericentric density maximaseem to follow a part of the spiral structure but at a slightlysmaller radius. On the other hand the apocentric distribu-tion does not fit well the real density maxima. Finally weremark that the loci of the velocity minima practically co-incide with the ones of the pericentres. Figure 13 gives a similar information as Fig. 12, but for anunstable periodic orbit of the 3:1 family, which is located in-side corotation for E j = − E j ( L ), and the areas inside and outside corotationcan communicate. We considered again 20000 initial condi-tions all along this 3D (3 : 1) unstable periodic orbit (and its symmetric with respect to the origin), with small deviationsfrom it, in the 6-dimensional space ( x, y, z, v x , v y , v z ). Theseorbits follow, in the phase space, the unstable directions ofthe asymptotic manifolds originating from the periodic or-bit. For a short time (the shortness depends on how large isthe initial divergence from the periodic orbit) these orbitsstay very close to the periodic orbit, but later they startdeviating considerably from it. In Fig. 13a we see the den-sity distribution (in color scale) of these orbits after withina time interval of 3-4 bar rotations. The black curves indi-cate the high density contours of the real particles for thesame Jacobi constant value. It is evident that these two dis-tributions fit one another very well. A similar behavior hasbeen found by Patsis (2006) for chaotic orbits near the 4:1resonance. This type of orbits was considered responsiblefor the inner parts of the spiral arms in an analytical modelrepresenting the spiral galaxy NGC 4314.In Fig. 13b we have plotted the positions of all theorbits for the time interval of Fig. 13a. The color of eachpoint shows the corresponding velocity value on the rota-tion plane. The color scale is the same as for the densities(see Fig. 1) which means that blue color corresponds to theminima and red color to the maxima of the velocity values.We observe that the density maxima are well correlated withthe areas of small velocity values (dark to light blue colorsin Fig. 13b). Figures 13c,d are similar to Figs. 12b,c. Weobserve that the pericentres of the orbits (Fig. 13c) form aslightly more closed spiral structure than that of the realparticles (black density contours). On the other hand theapocentres of the orbits (Fig. 13d) fit well only the densitycontours of the bar and the innermost parts of the spiralstructure that originate from the end of the bar. In Fig. 13ewe plot the density distribution of the loci of the local veloc-ity minima. We notice that this distribution matches betterthe distribution of the apocentres in what concerns the bar c (cid:13) , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos and the distribution of the pericentres in what concerns thespiral structure. x ) family The 2:1 (or x after the nomenclature ofContopoulos & Papayannopoulos 1980) family is gen-erally considered responsible for the formation and therobustness of the bar. This is due to its morphology,together with the fact that this family is usually stable forlow values of the Jacobi constant. According to this point ofview the bar should end not beyond the point where the x family turns from stable to unstable. Here we point out theusefulness of the unstable part of this family which supportsthe spiral structure. Figure 14a,b is similar to Fig. 13a,bbut for the x family and for the same Jacobi constantvalue. We observe that the density of the orbits startingclose to the x periodic orbit is similar to the density ofthe orbits with initial conditions near the 3:1 periodicorbit. Their density maxima support the morphologicalstructures (bar and spiral) of the real particles for the sameJacobi constant value (Fig. 14a). These maxima lie nearthe areas corresponding to low values of plane velocities v yz (Fig. 14b). P L , families Figure 15 is similar to Fig. 14 but for the unstable periodicorbits
P L , which are located near corotation and for thesame Jacobi constant value. In this case we establish thesame behavior (as for the cases 3:1 and x ) for the orbitsinitiating close to the symmetric periodic orbits P L , . Theorbits starting near the periodic orbits 3:1, 2:1, P L , followthe unstable directions of the asymptotic manifolds originat-ing from these periodic orbits. For the same Jacobi constantvalue these manifolds cannot intersect each other, and thismeans that these manifolds are parallel in the phase space.Thus, the orbits along the unstable directions of the variousmanifolds follow parallel paths and present a similar behav-ior in the configuration space. The major difference betweenthe different sets of orbits starting near the different unsta-ble periodic orbits (3:1, 2:1, P L , ) is the diffusion rate (seesection 5), which increases towards higher resonance orders(2 : 1 → → ∞ : 1 ≡ P L , ). All the other families of periodic orbits have similar behav-ior to those studied in Figs.12-15. Thus, the behavior of theorbits with − . E j < E L lying outside corotation(e.g. − − E j > E L (e.g. 4:1, P L , ) be-have similarly to the families 2:1, 3:1, P L , (Figs. 13-15)and support the bar and the inner parts of the spiral armsconnected to the bar. Note that there are chaotic orbits as-sociated with the 2:1, 3:1 and 4:1 families, located insidecorotation for E j < E L that support the envelope of thebar (Harsoula & Kalapotharakos 2009). Some of the char-acteristics of these families are plotted in Fig. 10. Moreover, in Fig. 4b we see many real particles that stick around var-ious resonances both inside and outside corotation (see thevarious spikes corresponding to different resonances). The chaotic orbits close to the unstable periodic orbits corre-sponding to different Jacobi constant values may form dif-ferent density distributions. However, the orbits at everyJacobi constant support the density distribution of the realparticles at the same value of the Jacobi constant. Here wedemonstrate what is the contribution of a family, all alongits characteristic, to the global morphology of the galaxy.This is done in Fig. 16 for the 3:1 family. In Fig. 16a we plot100 periodic orbits (black points) of the 3:1 family equallysampled along its whole characteristic together with the den-sity contours of the galaxy (black lines). These orbits arelocated inside corotation. In Fig. 16b we plot the densitydistribution of 20000 orbits having initial conditions closeand along the periodic orbits of Fig. 16a integrated for 3-4bar rotations. Although the initial conditions are all insidecorotation supporting the bar shape, after a short transientperiod of stickiness along the periodic orbits they are dif-fused outside corotation and form a spiral structure whichis slightly more closed than that of the galaxy (Fig. 16b).Figure 17 is similar to Fig. 16 but for the 2:1 (or x − q < The main results of the above study, for the 3D orbits alongspecific important resonances of the system can be also ac-quired if we study the space distribution of the pericentres orapocentres of the asymptotic orbits in a 2D approximationof the system ( x = 0 , ˙ x = 0 in Eq. 2). Namely, if we consider2D unstable periodic orbits (the most important for everyenergy level) and take initial conditions along their unsta-ble asymptotic curves, the integration of these asymptoticorbits can also reveal the morphological features of everyenergy level. In Fig. 20a, 20b we give an example of theprojection on the configuration space of the apocentric and c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 13. (a)
The density distribution on the y − z plane of 20000 3D orbits having initial conditions close to and along the unstableperiodic orbit 3:1 (and its symmetric with respect to the origin) at E j = − (b) The positions of all the orbits at the snapshot shown in (a) . The color of each point indicates the corresponding velocityvalue on the rotation plane. The color scale is the same as for the densities (see Fig. 1). (c)
The density distribution of the pericentressuperimposed with the density contours of the corresponding Jacobi constant value. (d)
The same as in (c) but for the apocentres. (e)
The same as in (c) but for the local velocity minima v min . Figure 14.
Similar to Figs. 13a,b but for the unstable periodic orbit 2:1 ( x ) and for the same Jacobi constant value.c (cid:13) , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 15.
Similar to Figs. 13a,b but for the unstable periodic orbits
P L , and for the same Jacobi constant value. Figure 16. (a)
The projection on the y − z plane of 100 periodic orbits (black points) of the 3:1 family equally sampled along its wholecharacteristic together with the density contours of the galaxy (black lines). (b) the density distribution on the y − z plane of 20000orbits with initial conditions close and along the periodic orbits shown in (a) after integration for 3-4 bar rotations. Figure 17.
Similar to Fig. 16 but for the unstable part of the 2:1 family.c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 18.
Similar to Fig. 16 but for the − Figure 19.
The superposition of all the distributions shown inFigs. 16b,17b,18b superimposed with the density contours of thegalaxy (black curves). All the main morphological features of thegalaxy are reconstructed by the superposition of these orbits. pericentric intersections (black points), respectively, of the2D manifold of the − E j = − < E j ( L ) where the areas in-side and outside corotation cannot communicate. Note thatthe phase space in the area outside corotation, in the 2D ap-proximation of the system, presents no tori or cantori withsmall holes, but only a chaotic sea with escapes to infinity.In each panel of Fig. 20 we have superimposed the densitydistribution (in color scale) of the real particles at the sameJacobi constant value. We observe that the apocentres donot coincide with the maxima of the density while the peri-centres trace well the areas of the density maxima, but forslightly smaller radii. This is compatible to the information provided by Fig. 12 according to which for Jacobi constantvalues below E j ( L ) and outside corotation the pericentresreveal the density maxima since they lie near low values ofthe plane velocity.Figure 21 is similar to Fig. 20 but for a different Ja-cobi constant E j =-1120000, which is just above E j ( L ), andtherefore the area inside corotation has just communicatedwith the area outside corotation. In this energy level and inthe 2D approximation there exist unstable periodic orbitsof the P L , P L and − P L , P L families) trace well thedensity maxima of the density distribution. This is com-patible to the information provided by Fig. 13 according towhich the pericentres and the apocentres for Jacobi constantvalues above E j ( L ) reveal the density maxima when theylie near low values of the plane velocity. The stickiness effect along the asymptotic manifolds of theunstable periodic orbits can last for a number of dynam-ical times reinforcing the spiral structure, as it has beendescribed in the previous sections, but then the chaotic or-bits can escape to very large distances and finally they mayescape to infinity. This diffusion is relatively slow and itstime scale can be compared to the age of the Universe. Inthis section we study the way these chaotic orbits escape toinfinity.Figure 22 presents the unstable asymptotic curve of the
P L orbit on a Poincar´ e surface of section z, ˙ z for y = 0and ˙ y > E j ( L ). We have taken a small initialsegment of 50000 points and of length 10 − from the un-stable periodic orbit and along the direction of the unstable c (cid:13)000
P L orbit on a Poincar´ e surface of section z, ˙ z for y = 0and ˙ y > E j ( L ). We have taken a small initialsegment of 50000 points and of length 10 − from the un-stable periodic orbit and along the direction of the unstable c (cid:13)000 , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 20. (a), (b)
The projection on the configuration space of the apocentric and pericentric intersections (black points), respectively,of the 2D manifold of the − E j = − < E j ( L ) where the areas inside and outsidecorotation cannot communicate superimposed on the density distribution of the real particles of this Jacobi constant value. Figure 21.
Similar to Fig. 20 but for the manifolds of the
P L , and − E j = − E j ( L ). asymptotic curve, and we have integrated all these pointsfor 10 iterations (Fig. 22a) 15 iterations (Fig. 22b) and 20iterations (Fig. 22c). We observe the formation of character-istic rays along the asymptotic curves when integrated for along time.In Fig. 23 we see the picture of the same Poincar´ e sur-face of section after the integration of 100 initial conditionsof test particles for 200 iterations in the 2D approximationof the system (black points). These points are concentratedclose to the unstable asymptotic curves of the families P L , P L . In Fig. 23 we also plot the curves corresponding tozero inertial energy E = 0 (thick black curves). The ana-lytic relation ˙ z = f ( z ) (for y = 0) describing these curves isfound as follows: By subtracting Eqs. 2 and 6 for E = 0 and˙ x = x = y = 0 we findΩ p ˙ yz + Ω p z = − E j (8) where ˙ y = ± q E j + Ω p z − V (0 , , z ) − ˙ z (9)Hence ˙ z = − V (0 , , z ) − E j Ω p z (10)This last equation gives two symmetric curves, one with˙ z < z >
0, shown by thick black curvesin Fig. 23. Orbits starting below the lower thick black curveescape to infinity without intersecting the E = 0 curve. Anexample is given by the blue dots, numbered 1,2,3,..., whichcorrespond to the successive iterations of an initial conditionnear the point 1, having E >
0. These iterations are locatedalong rays. The corresponding orbit escapes from the galaxyfollowing a spiral path, in the rotating configuration space(see inserted orbit in Fig. 23). On the other hand, if aninitial condition has
E <
0, then the successive iterations c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies Figure 22.
The asymptotic curves of the unstable periodic orbit
P L in the phase space ( z, ˙ z ) (for y = 0 and ˙ y >
0) for the 2Dapproximation of our model and for E j = − E j ( L ) after (a)
10 iterations (b)
15 iterations and (c)
20 iterations of asmall initial segment along the unstable directions of the manifold. (red dots) lie on rays again, but between the two E = 0curves, and at the same time they form “bell-type” curves(Contopoulos and Patsis 2006). Different bell-type curvescorrespond to different values of the energy E . The valueof E along an orbit remains roughly constant far from thegalaxy (left part of Fig. 23) forming a bell-type curve, butit changes abruptly whenever the orbit comes close to themain body of the galaxy (small absolute values of z ) andthen it follows a new bell-type curve. The changes of E areirregular, due to the chaotic character of the orbit. Afterseveral changes of its value, E becomes positive, and thenthe orbit escapes to infinity following a spiral-like path. Notethat this path corresponds to an open hyperbolic curve inan inertial frame of reference.Finally, Fig. 24 presents the fraction of the particlesstarting close to the 3:1, 2:1, P L , periodic orbits (seeFig. 13-15) that stay located inside a radius of 2 r hm asa function of time in T hmct . This figure testifies the phe-nomenon of stickiness along the manifolds originating fromthe unstable periodic orbits and gives a measure of thechaotic diffusion of the system. We observe two differentrates of diffusion: (a) the first one lasts for 50 to 100 T hmct where about the 20%, 25% and 45% of the orbits around the2:1, 3:1 and P L , periodic orbits respectively have been dif-fused outwards and (b) a quite slower diffusion starting after ≈ T hmct . In general, orbits with initial conditions closerto lower order n of resonance n : 1 present a slower diffusionrate. During the first diffusion period, the spiral structure isviable, because the flow of chaotic material coming from in-side corotation stay confined close and along the asymptoticmanifolds of the unstable periodic orbits and support thespiral structure of the galaxy. This time period correspondsto about 10 rotations of the bar. After that and during thesecond period of slow diffusion where chaotic orbits moveoutwards, spiral structure is not clearly observed in the sys-tem. Figure 23.
The phase space z , ˙ z (for y = 0 and ˙ y >
0) ofa 2-D approximation of our model for a Jacobi constant E j = − > E j ( L ), where the area inside corotation can com-municate with the area outside corotation. The thick black curvescorrespond to zero inertial energy. An initial condition near thepoint 1 and inertial energy E >
E <
In this paper we study the main factors that affect the for-mation and the longevity of the stellar spiral arms of barredspiral galaxies. We examine the role of the apocentres, ofthe pericentres and of the velocity minima as regards the c (cid:13) , 1–17 M. Harsoula, C. Kalapotharakos and G. Contopoulos
Figure 24.
The percentage of the orbits starting close to the 3:1,2:1,
P L , unstable periodic orbits (see Figs. 13-15) that staylocated inside R = 2 r hm as a function of time in T hmct . morphological features of a galaxy. We emphasize the roleof the asymptotic orbits, emanating from the unstable peri-odic orbits, in forming different segments of the spiral armscorresponding to different values of the Jacobi constant in-side and outside corotation. We show that there is stickinessalong these asymptotic orbits which affects the longevity ofthe spiral arms, which finally fade away, but after more than10 bar rotations. Below we present and discuss our main re-sults. (a) An objective way to find the maxima of density is byfinding the loci of the velocity minima on the rotation plane,since the particles spend most of their time there. These lociare:(i) The apocentres for the particles with Jacobi con-stants below E L that are allowed to move only insidecorotation. These apocentres, of regular or chaotic orbits,support the shape of the bar. The regular orbits form themain body of the bar, while the chaotic orbits form anenvelope of the bar.(ii) The pericentres for the particles with Jacobi con-stants below E L that are allowed to move only outsidecorotation. These pericentres support the outer parts ofthe spiral structure of the galaxy.(iii) An elliptical locus corresponding to the local maxi-mum of the effective potential V eff for the particles withJacobi constant E j > E j ( L ) that are allowed to moveboth inside and outside corotation. This locus is onlypartly realized in the distribution of the real N -body par-ticles, because an important percentage of particles with E j > E j ( L ) have minimum velocities v min that do notdiffer significantly from the mean velocity < v yz > allalong their orbits. Therefore, this locus is not reflected inthe global configuration of the particles although it is re-flected in the sub-population corresponding to low v min values compared to < v yz > . For Jacobi constants E j ( L ) > E j > E j ( L ) some of the lociof pericentres and apocentres that are close to low values ofthe velocity are correlated to the inner parts of the spiralstructure. (b) For each value of the Jacobi constant there is a set ofstable and unstable periodic orbits. It is well known that thedomain of phase space associated with the stable periodicorbits supports features similar to the morphologies of theseorbits.On the other hand in the present study we emphasize therole of the unstable periodic orbits and the manifolds em-anating from them. Although the unstable periodic orbitsthemselves do not support the spiral structure, the asymp-totic orbits that start near the unstable periodic orbits sup-port the spiral structure. As long as there is a particle popu-lation near the unstable periodic orbits of the main familiesof orbits (e.g. -1:1, -2:1,
P L , P L , P L , P L , 4:1, 3:1 and2:1) the spiral structure remains clearly visible. We showthat the superposition of orbits with initial conditions closeto the periodic orbits of only three main families (-1:1, 2:1,3:1) all along their characteristics are able to reconstruct allthe main morphological features of the galaxy. (c) For Jacobi constants above E j ( L ) chaotic diffusionleads to escapes. The diffusion is fast (for all the families)for a time interval lasting for about 10 rotations. After-wards, the diffusion rate slows down but the spiral structureis marginally discernible. For the families inside and close tocorotation (e.g. 2:1, 3:1, 4:1, P L , P L ) the diffusion rate issmaller for orbits closer to the center. On the other hand,chaotic orbits having initial conditions outside corotationpresent stickiness at the resonances − − N -body system. Such a study is capableof providing the main trends about the dynamical mech-anisms of the formation and the longevity of the variousstructures. However, the dynamical behavior of the real sys-tem is definitely more complicated since it shows an evolu-tion (decrease) of the pattern speed (although this decreaseis rather small). Another issue has to do with the evolu-tion of the density and consequently of the potential. In our“frozen” model the amplitude of the spiral perturbation inthe density decreases with time. The change of these two c (cid:13) , 1–17 symptotic Orbits in Barred Spiral Galaxies parameters affects the phase space structure. This implies adifferent network of unstable periodic orbits and also differ-ent asymptotic orbits with different stickiness around them.An extended study of the real evolving system requires theuse of dynamical methods that take into account the evolu-tion of both the pattern speed and the potential. We haveproceeded in this direction and we will present our resultsin a forthcoming paper. ACKNOWLEDGMENTS
This work has been partially supported by the ResearchCommittee of the Academy of Athens through the project200/739.
REFERENCES
Athanassoula, E.M., Romero-Gomez, M., Masdemont,J.J.,2009a, MNRAS, 394, 67.Athanassoula E., Romero-Gomez M., Bosma A., Masde-mont J.J., 2009b, MNRAS, 400, 1706.Binney J., Tremaine S., 1987, ”Galactic Dynamics”,Princeton University Press.Contopoulos, G., 1975, ApJ, 201, 566.Contopoulos G., Papayannopoulos T., 1980, A&A, 92, 33.Contopoulos, G., 1981, A&A, 102, 265.Contopoulos, G., Grosbol, P., 1989, A&A, 1, 261.Contopoulos, G., Patsis, P.A., 2006, MNRAS, 369, 1039.Contopoulos, G. Harsoula, M., 2008, Int. J. Bif. Chaos 18,2929.de Zeeuw, T., 1985, MNRAS, 216, 273.Fathi, K., Beckman, J. E., Pinol-Ferrer, N., Hernandez,O., Martinez-Valpuesta, I., Carignan, C., 2009, ApJ, 704,1657.Grosbol,P., Patsis, P.A., Pompei, E., 2004, A&A, 423, 849.Harsoula, M. Kalapotharakos, C., 2009, MNRAS, 394,1605.Kalnajs A.J., 1971, ApJ, 166, 275.Kaufmann D.E., Contopoulos G., 1996, A&A, 309, 381.Lin, C. C. and Shu, F. H., 1964, Astrophys. J. 140, 646.Lynden-Bell D., Kalnajs A.J., 1972, MNRAS, 157, 1.Merritt, D., 1996, Science, 271, 337.Patsis P.A., Skokos Ch., Athanassoula E., 2002, MNRAS,337, 578.Patsis, P.A., 2006, MNRAS, 369, L56.Romero-Gomez, M.,Masdemont, J.J., Athanassoula, E.M.,Garcia-Gomez, C. 2006, A&A, 453, 39.Romero-Gomez, M., Athanassoula, E.M.,Masdemont, J.J.,Garcia-Gomez, C., 2007, A&A, 472,63.Shen, J., Sellwood, J.A., 2004, 604, 614.Sparke L., Sellwood J.A., 1987, MNRAS, 225, 653.Toomre, A., 1977, Ann. Rev. Astron. Astrophys., 15, 437.Tsoutsis P., Efthymiopoulos C., Voglis N., 2008, MNRAS,387, 1264.Voglis N., Tsoutsis P., Efthymiopoulos C., 2006a, MNRAS,373, 280.Voglis N., Stavropoulos I., Kalapotharakos C., 2006b, MN-RAS, 372, 901. c (cid:13)000