Role of galactic bars in the formation of spiral arms: A study through orbital and escape dynamics -- I
aa r X i v : . [ a s t r o - ph . GA ] F e b MNRAS , 1–13 (2020) Preprint 26 February 2021 Compiled using MNRAS L A TEX style file v3.0
Role of galactic bars in the formation of spiral arms: A study throughorbital and escape dynamics - I
Debasish Mondal ⋆ and Tanuka Chattopadhyay † Department of Applied Mathematics, University of Calcutta, A. P. C. Road, Kolkata , India
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
In the present work we have developed a three-dimensional gravitational model of barred galaxies, in order to studyorbital and escape dynamics of the stars inside their central barred region. Our gravitational model is composed of fourcomponents, central nucleus, bar, disc and dark matter halo. Furthermore we have analysed the model for two differenttypes of bar potentials. The study has been carried out for a Hamiltonian system and thorough numerical studieshave been done in order to categorize regular and chaotic motions of stars. We have seen that escape mechanism hasonly seen near saddle points ( L , L and L ′ , L ′ ) of the Hamiltonian system. Orbital structures in x - y plane indicatethat this escaping motion corresponds to the two ends of the bar. Classifications of orbits are found by calculatingmaximal Lyapunov exponent of the stellar trajectories corresponding to a specific initial condition vector. Poincar´esurface section maps are studied in both x - y and x - p x ( p x is the momentum along x - direction) plane to get acomplete view of the escape properties of the system in the phase space. Also we studied in detail how the chaoticdynamics varies with mass, length and nature of the bar. We found that under suitable physical conditions the chaosplays a pivotal role behind the formation of grand design or poor spiral pattern for stronger bars and ring structuresfor weaker bars. Key words:
Galaxy: kinematics and dynamics – galaxies: structure – galaxies: bar – chaos
In Hubble’s classification of galaxies central stellar bar struc-ture is mainly observed in lenticular (example: NGC 1460,NGC 1533, NGC 2787) and spiral galaxies (example: M58,M91, M95, M109, NGC 1300, NGC 1365, NGC 1512, NGC2217, NGC 2903, NGC 3953, NGC 4314, NGC 4921, NGC7541, UGC 12158). Also some irregular galaxies like LargeMagellanic Cloud (LMC) and Small Magellanic Cloud (SMC)have off-centred stellar bar (Zhao & Evans 2000; Bekki 2009;Piatti 2017; Monteagudo et al. 2018; Strantzalis et al. 2019),though not all lenticular and spiral galaxies have stellar bars.All lenticulars and spirals are disc supported systems andthese stellar discs may or may not support stellar bars. Onlyone third of the local disc galaxies have determinable typeof bars (or strong bars) and another one third have indeter-minable type of bars (or weak bars) (Eskridge et al. 2000;Cheung et al. 2013; Yoon et al. 2019). Fraction of barredgalaxies among lenticulars and spirals is strongly depen-dent on red-shift, stellar mass, colour and bulge prominence(Abraham et al. 1999; Sheth et al. 2008; Nair & Abraham2010; Simmons et al. 2014; Zhou et al. 2020).Galactic bars are one of the robust substructure of thebarred galaxies. They are solid, dense stellar body rotat-ing around the central core. Pattern speed of the bar is ⋆ E-mail: dmappmath [email protected] † E-mail: [email protected] different than that of the disc. There are many theoriesbehind origin of the galactic bar (Miwa & Noguchi 1998;Bournaud & Combes 2002; Seo et al. 2019; Petersen et al.2019; Polyachenko & Shukhman 2020). Most evident theoryis galactic bars are rotational instabilities, which arises dueto density waves radiating outwards from the galactic core.These instabilities influence stellar orbits by redistributingtheir trajectories. As time goes these reshaped orbits fol-low an outward motion, which further creates a self stabi-lizing stellar structure, in the form of bar (Raha et al. 1991;Sellwood 2016; Bovy et al. 2019; Lokas 2019; Sanders et al.2019). Barred galaxies mostly have single bar structure em-bedded inside the bulge, though there are many exam-ples of double barred galaxies also (example: Milky Way,NGC 1291, NGC 1326, NGC 1543). In such cases thesmaller secondary bar is wrapped inside the larger primarybar (Erwin 2004; Debattista & Shen 2006; Du et al. 2016;De Lorenzo-C´aceres et al. 2019).Galactic bars have many shapes and sizes accord-ing to the functional form of potential energy. Thereare many three-dimensional bar potential models likespherical, homeoidal, ellipsoidal etc. (Ferrers 1877;De Vaucouleurs & Freeman 1972; Caranicolas 2002;Jung & Zotos 2015; Williams & Evans 2017) have ex-tensively studied till date. Ferrers’ triaxial potential(Ferrers 1877) is the most used realistic bar potentialmodel, though its functional form is very much complex © Mondal & Chattopadhyay and also it is computationally very much challenging.There are some other potentials like homeoidal poten-tial (De Vaucouleurs & Freeman 1972), ad hoc potential(Barbanis & Woltjer 1967; Dehnen 2000) etc. have simplerfunctional forms than Ferrers’ but still they are rigorous tohandle numerically. Also there are some simple realistic barpotential models used by Caranicolas (2002), Jung & Zotos(2015) etc.Due to influence of Galactic bar some of the stellar or-bits remain trapped inside the potential boundary and whileothers are escaped from that boundary, during their timeevolution. This problem can be studied from viewpoint ofthe problem of escape in an open Hamiltonian dynamicalsystem (Contopoulos & Efstathiou 2004; Ernst et al. 2008;Ernst & Peters 2014; Jung & Zotos 2015, 2016). An openHamiltonian system is a system where for energies above anescape threshold, the energy shell is non compact and as aresult a part of the stellar orbits explores (here from poten-tial holes to saddles) an infinite part of the position space.Also the Hamiltonian dynamics is a time reversal invariant(Jung & Zotos 2016). Now for a conservative dynamical sys-tem the Hamiltonian (or the total energy) is a constant ofmotion. Hence all the stellar orbits are confined inside thefive-dimensional energy hyper-surface of the six-dimensionalphase space of the Hamiltonian system. These stellar orbitsare either regular or chaotic in nature according to their ini-tial condition (or initial energy value). Orbits having initialenergy below the escape energy are remain trapped insidethe potential boundary and exhibit bounded motion (regu-lar or chaotic). While orbits having initial energy above theescape energy exhibit unbounded motion. They escape fromthe interior along the open zero velocity curves (exist alongthe escape channels of the potential). For bounded motionthere are orbits which initially look like regular orbits butrevealed their true chaotic nature in long time period. Theseorbits are known as trapped chaotic orbits. Existence of suchorbits make the stellar dynamics more geometrically com-plicated in the phase space. For unbounded motion orbitsare generally chaotic in nature. There are many dynamicalindicators which can classify these orbits according to theirregular or chaotic behaviour. Lyapunov exponent is one sucheffective dynamical indicator and its working mechanism isquite simple (Sandri 1996). It calculates the rate of separa-tion of two neighbouring trajectories during the entire timeperiod of evolution. Value of the Maximal Lyapunov Expo-nent (hereafter MLE) gives us more complete view about dy-namical nature of these orbits (regular or chaotic) in the vastsea of initial conditions of the phase space. MLE basically isthe highest separation between two neighbouring trajectoriesstarting from same initial condition in a designated time in-terval. If the value of MLE is positive then it indicates thatorbits are chaotic in nature, while MLE = 0 indicates that or-bits are periodic in nature (Strogatz 1994). To analyse escapeproperties of these orbits, one need to visualise Poincar´e sur-face section maps (Birkhoff 1927) in different two-dimensionalphase planes. Under suitable physical conditions these es-caped orbits through the potential saddles further fuel theformation of spiral arms, which means there is some kind ofbar driven spiral arm formation mechanism in case of barredspiral galaxies. Effect of dynamical chaos of the stellar orbitsbehind formation of the spiral arms have extensively stud- ied in the recent past (Lindblad 1947; Lynden-Bell & Kalnajs1972; Pfenniger 1984; Patsis 2012; Mestre et al. 2020).Most of the earlier studies focused on detecting the chaoticinvariant set of stellar orbits, which governs the general dy-namics in the central region. Also there are studies about howthese chaotic invariant set fuels the formation of spiral arms.Due to chaotic dynamics in the central region, the escapingstars produce tidal trails at the two ends of the bar. Thesetidal trails have tendency to create different morphologieslike ring or spiral arms due to non-axisymmetric perturba-tions. These studies relates the fate of escaping stars withsubsequent morphological structures (Di Matteo et al. 2005;Minchev et al. 2010; Quillen et al. 2011; Grand et al. 2012;D’Onghia et al. 2013; Ernst & Peters 2014; Jung & Zotos2016).In the present work we showed the same analogy i.e. thefate of escaping stars behind formation of spiral arms butfrom the viewpoint of the amount of chaos produced therein.Therefore we primarily focus on detection of the chaotic dy-namics in the central region of barred galaxies, under suitablerealistic bar potential models. We then figure out a compar-ison between these bar potential models in order to showhow these models affect the chaotic dynamics with respect tomass and length of the bar. We relate these measurements ofchaos with subsequent structure formations like spiral armsor rings. Also we have shown which bar model is more fea-sible for certain type of structure formations under suitableastrophysical circumstances.Here we used a four component three-dimensional gravi-tational model for the barred galaxies. The model consistsof a spherical bulge, a bar embedded in bulge, a flat discand a logarithmic dark matter halo. Modelling is done in twoparts corresponding to bar potential models, (i) an anhar-monic mass-model bar potential (Caranicolas 2002) and (ii)the Zotos bar potential (Jung & Zotos 2015). These two arethe most non arduous potential forms studied till date. Foreach of these potential models, first we have studied severalorbital structures corresponding to different initial conditionvectors. Also we calculate values of MLE for each of theseinitial condition vector, which gives us complete dynamicalnature of these orbits (regular or chaotic). Then we drawPoincar´e surface section maps in both x - y and x - p x sub-section of the phase space. These two surface section maps areimportant in order to visualise the motion along the galacticplane which contains the disc. Also these two surface sectionmaps are plotted for different energy values (energies higherthan the escape energy of the saddles of our gravitationalpotential models), in order to get idea about escape mecha-nism of the system. Finally we described how the nature oforbits varies with the bar parameters e.g. mass and length bycalculating the MLE for each of the two bar potential models.Our work is divided into four sections. Section 2 describesthe mathematical part of the barred galaxy model. Section 2has divided into two subsections. In subsection 2.1 we havediscussed the model for an anharmonic bar potential and insubsection 2.2 we have discussed the same model for the Zotosbar potential. Section 3 consists of numerical analysis part.Section 3 has divided into three subsections. In subsection 3.1several orbital structures are plotted in x - y plane. In sub-section 3.2 several Poincar´e surface section maps are plottedin both x - y and x - p x planes while in subsection 3.3 wehave discussed how the chaotic dynamics evolved with mass MNRAS , 1–13 (2020) ole of galactic bars in the formation of spiral arms and length of the bar under the influence of the two bar po-tential models. Finally discussion and conclusions are givenin Section 4. We consider a three-dimensional gravitational model ofbarred galaxies and investigate orbital motions of stars insidetheir central region. Our gravitational model has four com-ponents – (i) spherical bulge, (ii) bar embedded in the bulge,(iii) flat disc and (iv) logarithmic dark matter halo. Here allthe modelling and calculations are done in the Cartesian co-ordinate system. Let Φ t ( x, y, z ) be the total potential of thegalaxy. This Φ t ( x, y, z ) consists of four parts and they are (i)bulge potential – Φ B ( x, y, z ), (ii) bar potential – Φ b ( x, y, z ),(iii) disc potential – Φ d ( x, y, z ) and (iv) dark matter halopotential – Φ h ( x, y, z ). Therefore,Φ t ( x, y, z ) = Φ B ( x, y, z )+Φ b ( x, y, z )+Φ d ( x, y, z )+Φ h ( x, y, z ) . Density distribution ρ t ( x, y, z ) corresponds to Φ t ( x, y, z ) isgiven through Poisson equation, ∇ Φ t ( x, y, z ) = 4 πGρ t ( x, y, z ) , where G is the gravitational constant. Now let ~ Ω b ≡ (0 , , Ω b )be the constant angular velocity of the bar, which follows aclockwise rotation along z - axis. In this rotating frame theeffective potential is,Φ eff ( x, y, z ) = Φ t ( x, y, z ) −
12 Ω ( x + y ) . (1)The potential functions for different sub-structures are asfollows, • Bulge: Central bulge is the excess of luminosity fromthe surrounding galactic disc. The distribution of stars inthe galactic bulges are not exponential rather sphericallysymmetric and dominated by old red stars. That’s why wechoose a potential of the Plummer type (Plummer 1911)to describe the distribution of matter inside the bulge ofbarred galaxies (Binney & Tremaine 1987; Sofue & Rubin2001; Halle & Combes 2013; Salak et al. 2016). Now densitydistribution of matter in the bulge for Plummer potential is, ρ B ( x, y, z ) = 3 M B c π x + y + z + c ) , and associated form of the potential is,Φ B ( x, y, z ) = − GM B p x + y + z + c , where M B is the mass of the bulge and c B is the radial scalelength. Here we consider a massive dense bulge rather thana central supermassive black hole, so that we can exclude allrelativistic effects from our model. • Bar: Bar is an extended linear non-axisymmetric stellarstructure in the central region of a galaxy. For model 1, wechoose a strong bar potential, whose density in the centralregion is very high (i.e. a cuspy type, see Fig. 3). For this weconsider an anharmonic mass-model potential (Caranicolas 2002), which has a non arduous potential form and its densitydistribution of matter is, ρ b ( x, y, z ) = M b π [(2 + α ) c − (1 − α )( x − α y + z )]( x + α y + z + c ) , and associated form of the potential is,Φ b ( x, y, z ) = − GM b p x + ( αy ) + z + c , where M b is the mass of the bar, α is the bar flatteningparameter and c b is the radial scale length. • Disc: Disc is the most luminous component of a galaxy.Distribution of matter inside the disc is axisymmetric, flat-tened and exponentially falls off with galactocentric ra-dius. Structure of the disc can be thought as a flat-tened spheroid. For this flattened spheroid disc structure(Binney & Tremaine 1987; Flynn et al. 1996; Shin & Evans2007; Smet et al. 2015; An & Evans 2019), we use the gravi-tational potential model developed by Miyamoto and Nagai(Miyamoto & Nagai 1975), often termed as Miyamoto andNagai potential. This potential model has a simple analyticalform for the corresponding density distribution of matter, ρ d ( x, y, z ) = M d h π × k ( x + y ) + ( k + 3 √ h + z )( k + √ h + z ) [( x + y ) + ( k + √ h + z ) ] ( h + z ) . Also associated form of the potential is,Φ d ( x, y, z ) = − GM d q x + y + ( k + √ h + z ) , where M d is the mass of the disc and k , h are the horizontaland vertical scale lengths respectively. • Dark matter halo: Dark matter halo is the extended dis-tribution of non-luminous (non-baryonic) matter of a galaxy(Ostriker et al. 1974). Structure of the dark matter haloesin barred galaxies are flattened axisymmetric, as observedrotation curve becomes almost flat at larger galactocentricdistances. For this flattened axisymmetric dark matter halostructure (Binney & Tremaine 1987; Ernst & Peters 2014),we use a variant of logarithmic potential (Zotos 2012). Thecorresponding density distribution of matter is, ρ h ( x, y, z ) = v πG β x + (2 β − β ) y + β z + ( β + 2) c ( x + β y + z + c ) and associated form of the potential is,Φ h ( x, y, z ) = v x + β y + z + c ) , where v is the circular velocity of halo, β is the halo flatten-ing parameter and c h is the radial scale length.In this model we take the physical units as – • Unit of length: 1 kpc • Unit of mass: 2 . × M ⊙ • Unit of time: 0 . × yr • Unit of velocity: 10 km s − • Unit of angular momentum: 10 km kpc s − • Unit of energy per unit mass: 100 km s − . MNRAS , 1–13 (2020)
Mondal & Chattopadhyay
Without loss of any generality, we consider G = 1. The valuesof other physical parameters are taken from Zotos (2012) andJung & Zotos (2016) and are given in Table 1.Parameter Value M B c B M b c b α M d k h v β c h b H ) of the given system is defined as, H = 12 ( p x + p y + p z ) + Φ t ( x, y, z ) − Ω b L z = E, (2)where ~r ≡ ( x, y, z ) is the position vector of test particle attime t , ~p ≡ ( p x , p y , p z ) is the corresponding linear momentumvector, E is the total energy and L z = xp y − yp x is the z -component of the angular momentum vector, ~L = ~r × ~p . HenceHamilton’s equations of motion are,˙ x = p x + Ω b y, ˙ y = p y − Ω b x, ˙ z = p z , ˙ p x = − ∂ Φ t ∂x + Ω b p y , ˙ p y = − ∂ Φ t ∂y − Ω b p x , ˙ p z = − ∂ Φ t ∂z , (3)where ‘ · ’ ≡ ddt . This autonomous Hamiltonian system hasfive Lagrangian (or equilibrium) points namely L , L , L , L and L , which are solutions of the following equations: ∂ Φ eff ∂x = 0 , ∂ Φ eff ∂y = 0 , ∂ Φ eff ∂z = 0 . (4)Locations of these Lagrangian points and their stability na-ture are given in Fig. 1 and Table 2 respectively. Figure 1: Model 1 – The isoline contours of Φ eff ( x, y, z ) in x - y plane at z = 0. Location of the Lagrangian points aremarked in red.Name Location Local Stability Type L (0 , ,
0) Stable Centre L (20 . , ,
0) Unstable Saddle Point L (0 , . ,
0) Asymptotically Stable Node L ( − . , ,
0) Unstable Saddle Point L (0 , − . ,
0) Asymptotically Stable NodeTable 2: Model 1 – Stability nature of Lagrangian points.The values of E (or Jacobi value of integral of motion)at L and L are identical and that value is E L = − . E L . Similarly the values of E at L and L are identical and that value is E L = 20 . E L .Also the value of E at L is E L = − . • E L ≤ E < E L : In this energy range motion of orbitsare bounded. • E ≥ E L : In this energy range motion of orbits areunbounded and two symmetrical escape channels exist nearboth L and L . In this model we consider the three-dimensional gravitationalmodel as discussed in subsection 2.1, but change only the po-tential form of the galactic bar. Here we choose a compara-tively weak bar potential, whose density at the central regionis moderate (Fig. 3). For this model we used the bar potentialmodel developed by Jung & Zotos (2015), which has also anon arduous potential form. This model is often termed as
MNRAS , 1–13 (2020) ole of galactic bars in the formation of spiral arms Zotos bar potential. For this potential model density distri-bution of matter is, ρ b ( x, y, z ) = M b c πa [ f ( x + a, y, z ) − f ( x − a, y, z )] , where f ( x, y, z ) = x (2 x + 3 y + 3 z + 3 c )( y + z + c ) ( x + y + z + c ) and associated form of the potential is,Φ b ( x, y, z ) = GM b a ln( x − a + q ( x − a ) + y + z + c x + a + q ( x + a ) + y + z + c ) , where M b is the mass of the bar, a is the length of semi-majoraxis of the bar and c b is the radial scale length. For this barpotential we use a = 10 and the values of other all modelparameters due to bulge, disc and dark matter halo remainthe same as given in Table 1.Locations of the Lagrangian points, namely L ′ , L ′ , L ′ , L ′ & L ′ and their stability natures are given in Fig. 2 andTable 3 respectively.Figure 2: Model 2 – The isoline contours of Φ eff ( x, y, z ) in x - y plane at z = 0. Location of the Lagrangian points aremarked in red.Name Location Local Stability Type L ′ (0 , ,
0) Stable Centre L ′ (20 . , ,
0) Unstable Saddle Point L ′ (0 , . ,
0) Asymptotically Stable Node L ′ ( − . , ,
0) Unstable Saddle Point L ′ (0 , − . ,
0) Asymptotically Stable NodeTable 3: Model 2 – Stability nature of Lagrangian points.The values of E at L ′ and L ′ are identical and that value is E L ′ = − . E L ′ . Similarly the values of E at L ′ and L ′ are identical and that value is E L ′ = − . E L ′ . Also the value of E at L ′ is E L ′ = − . r ρ b Model 1Model 2
Figure 3: Radial ( r = p x + y ) distribution of the densitystructure of the bar ( ρ b ) for both models 1 and 2 at z = 0. To study the orbital and escape dynamics of stars along thegalactic plane (which contains the bar), we put z = 0 = p z inboth the gravitational models 1 and 2 respectively. Dependingupon the initial energy value, stellar orbits are either trappedor escaped from the system. We have seen that two symmet-rical escape channels exist only near L , L and L ′ , L ′ (Figs.1 and 2). Hence escape mechanism is only relevant near thoseLagrangian points. Studying nature of orbits near either of L or L or of L ′ , L ′ gives us complete information aboutthe escape dynamics of the system. In order to do that wehave to investigate in the following energy range: E ≥ E L or E ≥ E L ′ . Now for simpler analysis we replace E with thedimensionless energy parameter C (Ernst et al. 2008), whichis defined asfor model 1: C = E L − EE L = E L − EE L ( ∵ E L = E L )for model 2: C = E L ′ − EE L ′ = E L ′ − EE L ′ ( ∵ E L ′ = E L ′ ) . For
C > L . To study escape motionaround L we have to choose energy levels higher than theescape threshold. We choose our tested energy levels as C =0 .
01 and C = 0 .
1. Energy value at L for different valuesof C are given in Table 4. In the phase space for large seaof initial conditions we consider only those initial conditions MNRAS , 1–13 (2020)
Mondal & Chattopadhyay which correspond to the central barred region, i.e. if ( x , y )is an initial condition in x - y plane then, x + y ≤ r L ,where r L is the radial length of L . The same formalism iscarried out for the point L ′ also. C E E (Model 1) (Model 2)0 . − . − . . − . − . δx ( t ) be the initial sepa-ration vector of two neighbouring trajectories, where t is theinitial time. Also let δx ( t ) be the separation vector at time t . Then MLE for that designated initial condition vector isdefined as,MLE = lim t →∞ lim | δx ( t ) |→ t ln | δx ( t ) || δx ( t ) | . (5)To follow the evolution of orbits in long time we chooseour integration time as 10 units, which is equivalent to 10 years (typical age of barred galaxies). In this vast integrationtime, orbits starting from an initial condition will reveal theirtrue nature (regular or chaotic). We use a set of MATLAB pro-grammes in order to integrate the system of Eqs. (3). In orderto do that we use the ode45 MATLAB package with small timestep 10 − . Here all the calculated values are corrected up-toeight decimal places. All the presented graphics are producedby using MATLAB ( version - ) software. For studying the orbital dynamics in x - y plane we choosetwo initial conditions. One initial condition is ( x , y , p x ) ≡ (5 , , L and L ′ respectively.Also in the vicinity of this initial condition we can figure outthe properties of escape dynamics due to L and L ′ at thenearest end of the bar. Similarly in order to figure out how L and L ′ affects the orbital dynamics for an initial conditionstarting with a point away from it we choose another initialcondition as ( x , y , p x ) ≡ ( − , , p y is calculated from Eq. (2). • Model 1: In Figs. 4 and 5, stellar orbits in x - y planehave been plotted for values C = 0 .
01 and 0 . x = 5, y = 0 and p x = 15 . In bothcases we get escaping chaotic orbits. Similarly for Figs. 6 and7 initial condition is x = − y = 0 and p x = 15. Herein both cases we get non-escaping retrograde quasi-periodicrosette orbits. Value of p y for each figure is evaluated fromEq. (2) and the corresponding values of MLE are given inTable 5. Initial Condition C MLE( x , y , p x ) 0 .
01 0 . ≡ (5 , ,
15) 0 . . x , y , p x ) 0 .
01 0 . ≡ ( − , ,
15) 0 . . C .Figure 4: Model 1 – escaping chaotic orbit for C = 0 .
01 with( x , y , p x ) ≡ (5 , , C = 0 . x , y , p x ) ≡ (5 , , MNRAS , 1–13 (2020) ole of galactic bars in the formation of spiral arms Figure 6: Model 1 – non-escaping retrograde quasi-periodicrosette orbit for C = 0 .
01 with ( x , y , p x ) ≡ ( − , , C = 0 . x , y , p x ) ≡ ( − , , • Model 2: Similar as model 1, here also in Figs. 8 and9 stellar orbits in x - y plane have been plotted for values C = 0 .
01 and 0 . x = 5, y = 0 and p x = 15. In both cases we get non-escapingchaotic orbits. Similarly for Figs. 10 and 11 initial condi-tion is x = − y = 0 and p x = 15. Here in both caseswe get non-escaping retrograde quasi-periodic rosette orbits.Also value of p y for each figure is evaluated from Eq. (2) andcorresponding values of MLE are given in Table 6. Initial Condition C MLE( x , y , p x ) 0 .
01 0 . ≡ (5 , ,
15) 0 . . x , y , p x ) 0 .
01 0 . ≡ ( − , ,
15) 0 . . C .Figure 8: Model 2 – non-escaping chaotic orbit for C = 0 . x , y , p x ) ≡ (5 , , C = 0 . x , y , p x ) ≡ (5 , , MNRAS , 1–13 (2020)
Mondal & Chattopadhyay
Figure 10: Model 2 – non-escaping retrograde quasi-periodicrosette orbit for C = 0 .
01 with ( x , y , p x ) ≡ ( − , , C = 0 . x , y , p x ) ≡ ( − , , Poincar´e maps are two-dimensional cuts of the six-dimensional hyper-surface in case of the given gravitationalsystem. For model 1 Poincar´e surface section maps in x - y plane are plotted in Figs. 12 and 13 for different energy val-ues. For this we consider a 43 ×
43 grid of initial conditionswith step sizes ∆ x = 1 kpc and ∆ y = 1 kpc. All the ini-tial conditions in x - y plane are considered from the centralbarred region i.e. x + y ≤ r L . Initial conditions for p x and p y are p x = 0 and p y ( > p y is evaluated from Eq. (2). Also for Poincar´e maps in x - y plane surface crosssections are p x = 0 and p y ≤ x - p x plane areplotted in Figs. 14 and 15 for different energy values. For thiswe also consider a 43 ×
31 grid of initial conditions with stepsize ∆ x = 1 kpc and ∆ p x = 10 km s − . Here also all theinitial conditions are taken from the central barred region.Initial conditions for y and p y are y = 0 and p y ( > p y is evaluated from Eq. (2). Also for Poincar´e mapsin x - p x plane surface cross sections are y = 0 and p y ≤ x - y plane are plotted inFigs. 16 and 17 and Poincar´e surface section maps in x - p x plane are plotted in Figs. 18 and 19 respectively for model 2. • Model 1: In Fig. 12, we observed that a primary stabilityisland does exist near (5 ,
0) in x - y plane for energy value C =0 .
01, which has formed due to quasi-periodic motions. Againwhen the energy value is increased to C = 0 . C . Similarlya primary stability island has been observed in Figs. 14 and15 and it exists near (5 ,
0) in x - p x plane. In x - p x planewe also observed chaotic and escaping motions and amountof escaping orbits has been increased with increment of C .Figure 12: Model 1 – Poincar´e surface sections of p x = 0 and p y ≤ C = 0 . MNRAS , 1–13 (2020) ole of galactic bars in the formation of spiral arms Figure 13: Model 1 – Poincar´e surface sections of p x = 0 and p y ≤ C = 0 . y = 0 and p y ≤ C = 0 .
01. Figure 15: Model 1 – Poincar´e surface sections of y = 0 and p y ≤ C = 0 . • Model 2: Same as model 1, here also in Figs. 16 and 17,we observe that a primary stability island has formed due toquasi-periodic motions and it exists near (6 ,
0) in x - y plane.As the energy value increased from C = 0 .
01 to C = 0 .
1, thestability island gradually fades and motion becomes chaoticin that region. Also number of cross sectional points in x - y plane outside the central barred region (escaping points)are also increased with increment of C . Similarly in Figs. 18and 19 a primary stability island has formed and it is existnear (6 ,
0) in x - p x plane. Here also motion is chaotic andescaping cross sectional points in x - p x plane are increasedwith increment of C .Figure 16: Model 2 – Poincar´e surface sections of p x = 0 and p y ≤ C = 0 . MNRAS , 1–13 (2020) Mondal & Chattopadhyay
Figure 17: Model 2 – Poincar´e surface sections of p x = 0 and p y ≤ C = 0 . y = 0 and p y ≤ C = 0 .
01. Figure 19: Model 2 – Poincar´e surface sections of y = 0 and p y ≤ C = 0 . Here we have discussed how the chaotic dynamics evolvedover the vast integration time with respect to mass and lengthof the bar. In order to do that we have calculated MLE for dif-ferent values of mass and length of the bar for each of the twobar potential model. Our main focus is to study the effect ofchaotic dynamics in the vicinity of the Lagrangian points L and L ′ respectively. That is why we restrict the study in thissubsection only for the orbits starting with initial condition( x , y , p x ) ≡ (5 , , • Model 1: In Table 7 we have showed how the chaoticdynamics vary with the bar flattening parameter ( α ) and theenergy parameter C for model 1. Similarly in Table 8 we haveshowed how the chaotic dynamics vary with the bar mass M b and the energy parameter C . C α
MLE
C α
MLE0 .
01 1 0 . . . . . . . . . . . . . . . . . . . . . α and C with( x , y , p x ) ≡ (5 , , MNRAS , 1–13 (2020) ole of galactic bars in the formation of spiral arms C M b MLE
C M b MLE0 .
01 3100 0 . . . . . . . . . . . . . . . . . . . . . M b and C with ( x , y , p x ) ≡ (5 , , • Model 2: In Table 9 we have showed how the chaoticdynamics varies with the length of semi-major axis of thebar ( a ) and the energy parameter C for model 2. Similarlyin Table 10 we have showed how the chaotic dynamics varieswith the bar mass M b and the energy parameter C . C a
MLE
C a
MLE0 .
01 1 0 . . . . . . . . . . . . . . . . . . . . . a and C with( x , y , p x ) ≡ (5 , , C M b MLE
C M b MLE0 .
01 3100 0 . . . . . . . . . . . . . . . . . . . . . M b and C with ( x , y , p x ) ≡ (5 , , The present work describes the nature of orbits of the stars inbarred spiral galaxies and the influence of bars along with thedevelopment of spiral arms as a result of escape mechanism.We have considered the stellar orbits in barred spiral galax-ies in the presence of four components e.g. bulge, bar, disc anddark matter halo. We considered two types of bars namely (i)anharmonic bar, and (ii) Zotos bar. It is clear from Figs. 1and 2 that the bar area of the latter one is more and it is moreelongated in x - direction than the first one. Also the densitydistribution Fig. (3) for bar in model 1 is very high close tothe centre i.e. the nature is one of cuspy type compared tomodel 2. So it may be associated with strong bar. On theother hand in model 2 the density distribution is rather flatand slightly rising in the central region. This kind of distri-bution may be associated with weak bar.Regarding the nature of orbits, it depends upon the initialconditions as well as bar potential. These orbits may be es-caping and chaotic with high MLE for model 1, whereas theyare non escaping for model 2 for a particular initial point(e.g. x = 5, y = 0, p x = 15). On the other hand they areretrograde quasi-periodic with low MLE at x = − y = 0, p x = 15. For the first situation it may result in spiral armsfor escaping orbits and for the second case a ring may result.This is evident in S0 or ring galaxies where spiral arms aremore or less absent (Abadi et al. 1999; Van den Bergh 2009;Querejeta et al. 2015; Sil’chenko et al. 2018).The second aspect is to study the nature of chaotic orbitsof stars and gas in barred spiral galaxies. The understandingof the dynamics of these galaxies closely relates with chaoticmotions of stars and gas in the central region. The presence ofchaos is a manifestation of unstable orbits in these galaxies.The chaos propagates further over long period of time andinfluence the evolution of several structural components e.g.bar, disc and dark matter halo. Also integration of chaotictheory in the orbital and escaping stellar motions helps us tofigure out true shape of these components. In case of barredspiral galaxies these growing instability relates with forma-tion and strength of spiral arms.Galactic bars are density waves in the disc. Spiral armsare thought to be the result of these outward density wavepatterns. Observational studies in blue and near IR bandof barred spiral galaxies confirm that spiral arms are con-tinuations of the bar, which means spiral arms are out-comes of the bar driven mechanisms. While theoretical stud-ies also suggested that there is a correlation between patternspeed of bar and that of spiral arms (Elmegreen & Elmegreen1985; Buta et al. 2009). These density waves and stellar or-bits usually have different rotational speeds but there existssome sort of corotation region inside the disc. Spiral armsemerge from the two ends of the bar through that corota-tion region of bar and disc. In this corotation region chaoticdynamics dominates. Theoretical studies confirm that the-ses chaotic orbits in the corotation region are the build-ing blocks of the spiral arms (Contopoulos & Grosbøl 1989;Kaufmann & Contopoulos 1996; Skokos et al. 2002). Also astudy by Patsis et al. (1997) have shown that these chaoticorbits are the reason behind the characteristic outer boxyisophotes of the nearly face-on bar of the barred spiral galaxyNGC 4314. Under suitable physical circumstances these spiralarms survived the chaotic dynamics in the corotation region MNRAS , 1–13 (2020) Mondal & Chattopadhyay of galactic disc. Stronger bar leads to the formation of granddesign spirals. Nearly 70% of the barred spirals in field havetightly wounded two-armed spiral structure, while only 30%of the unbarred spirals in field have such kind of grand spiralpattern (Elmegreen & Elmegreen 1982).Thus it is clear from the above discussion that the spiralarms might be the continuation of the escaping orbits of starsemerging from the end of bars driven by the chaos. In our casethere are chaotic orbits when the energy exceeds the energyof second Lagrangian point (i.e.
E > E L or E > E L ′ ). Wehave computed MLE for various initial conditions as well asbar parameters. The following observations have been found.(i) The orbital structures (Figs. 4 - 7) for model 1, starting atinitial conditions ( x , y , p x ) ≡ (5 , ,
15) and ( x , y , p x ) ≡ ( − , , L and L ). Also the amountof chaos increases with the energy. Similarly from Poincar´esurface section maps (Figs. 12 - 15) the similar results havebeen found.(ii) For model 1, MLE increases with the flattening parameter( α ) up to a threshold length and again slowly decreases (viz.Table 7).(iii) The threshold length decreases with increasing valueof C . This implies when the escape energy of L is high( C ∼ . C = 0 .
01, the threshold value of α = 8whereas for C = 0 . α = 6 (viz. Table 7).(iv) From Table 8 it is clear that MLE more or less de-creases with bar mass. So, heavier (or strong) bars gen-erally oppose escape mechanism. Thus formation of spiralarms for smaller values of escape energy are not favourablebut as the energy value increases (may be due to cen-tral explosions, shocks etc.) i.e. if C ∼ . x , y , p x ) ≡ (5 , ,
15) and( x , y , p x ) ≡ ( − , , L ′ and L ′ ) rather the chaotic behaviour remained encapsulatedinside the central region. Also the amount of chaos increaseswith the energy value. Poincar´e surface section maps (Figs.16 - 19) also confirm the same phenomena.(vi) The MLE increases with semi-major axis of the bar ( a )(viz. Table 9) up to a threshold value ( a = 6) and does notvary much with further flattening and it is independent of energy levels. So, for this type of bar an optimal bar lengthencourages escaping orbits and formation of spiral arms.(vii) Weaker bars helps escape mechanism but the increaseof chaos with bar mass is very slow (viz. Table 10). Thismight be the reasons why spiral arms are not very promi-nent in many barred galaxies and as a result the formation ofS0 or ring galaxies (Regan & Teuben 2003; Byrd et al. 2006;Athanassoula et al. 2009; Baba et al. 2013; Proshina et al.2019). Observational evidence of such type of ring structureshad been found in NGC 1326 by Buta (1995). Similarly stud-ies by Sakamoto et al. (1999, 2000) had also identified suchring type of structures in NGC 5005. Hence bar potential usedin model 2 favours the formation of ring type of structuresor less prominent spiral arms which emerge from the ends ofthe bar.Thus from the above discussions we have come to the fol-lowing conclusions. • Barred galaxies with massive and stronger bar potential(viz. model 1) may lead to the formation of grand designspirals only when there are some kind of violent activitiesgoing inside their central region. SMBHs may be one of thereasons for that kind of violent activities. Again galaxies withstronger or massive stronger bars but without central SMBHsmay lead to the formation of less prominent spiral arms. • On the contrary barred galaxies with weaker bar poten-tial (viz. model 2) may lead to the formation of ring type ofstructures or less prominent spiral arms around their centralregion.In a subsequent work we will discuss how different typesdark matter halo profiles affect the orbital and escapedynamics in the central region and conceptualise whetherthey have any role behind further substructure formationsor not.
ACKNOWLEDGEMENTS
The author DM thanks the University Grants Commis-sion of India for providing Junior Research Fellowship (ID- 1263/(CSIRNETJUNE2019)), under which the work hasbeen done.
DATA AVAILABILITY
Both the authors confirm that the analysed data supportingthe findings of this study are available within the article.
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