MNRAS 000, 1–13 (2020) Preprint 26 February 2021 Compiled using MNRAS LATEX style file v3.0 Role of galactic bars in the formation of spiral arms: A study through orbital and escape dynamics - I Debasish Mondal1⋆ and Tanuka Chattopadhyay1† 1 Department of Applied Mathematics, University of Calcutta, 92 A. P. C. Road, Kolkata 700009, 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 study orbital and escape dynamics of the stars inside their central barred region. Our gravitational model is composed of four components, central nucleus, bar, disc and dark matter halo. Furthermore we have analysed the model for two different types of bar potentials. The study has been carried out for a Hamiltonian system and thorough numerical studies have been done in order to categorize regular and chaotic motions of stars. We have seen that escape mechanism has ′ ′ only seen near saddle points (L2, L4 and L2, L4) of the Hamiltonian system. Orbital structures in x - y plane indicate that this escaping motion corresponds to the two ends of the bar. Classifications of orbits are found by calculating maximal Lyapunov exponent of the stellar trajectories corresponding to a specific initial condition vector. Poincar´e surface section maps are studied in both x - y and x - px (px is the momentum along x - direction) plane to get a complete view of the escape properties of the system in the phase space. Also we studied in detail how the chaotic dynamics varies with mass, length and nature of the bar. We found that under suitable physical conditions the chaos plays a pivotal role behind the formation of grand design or poor spiral pattern for stronger bars and ring structures for weaker bars. Key words: Galaxy: kinematics and dynamics – galaxies: structure – galaxies: bar – chaos 1 INTRODUCTION different than that of the disc. There are many theories behind origin of the galactic bar (Miwa & Noguchi 1998; In Hubble’s classification of galaxies central stellar bar struc- Bournaud & Combes 2002; Seo et al. 2019; Petersen et al. ture is mainly observed in lenticular (example: NGC 1460, 2019; Polyachenko & Shukhman 2020). Most evident theory NGC 1533, NGC 2787) and spiral galaxies (example: M58, is galactic bars are rotational instabilities, which arises due M91, M95, M109, NGC 1300, NGC 1365, NGC 1512, NGC to density waves radiating outwards from the galactic core. 2217, NGC 2903, NGC 3953, NGC 4314, NGC 4921, NGC These instabilities influence stellar orbits by redistributing 7541, UGC 12158). Also some irregular galaxies like Large their trajectories. As time goes these reshaped orbits fol- Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) low an outward motion, which further creates a self stabi- have off-centred stellar bar (Zhao & Evans 2000; Bekki 2009; lizing stellar structure, in the form of bar (Raha et al. 1991; Piatti 2017; Monteagudo et al. 2018; Strantzalis et al. 2019), Sellwood 2016; Bovy et al. 2019; Lokas 2019; Sanders et al. though not all lenticular and spiral galaxies have stellar bars. arXiv:2102.12889v1 [astro-ph.GA] 25 Feb 2021 2019). Barred galaxies mostly have single bar structure em- All lenticulars and spirals are disc supported systems and bedded inside the bulge, though there are many exam- these stellar discs may or may not support stellar bars. Only ples of double barred galaxies also (example: Milky Way, one third of the local disc galaxies have determinable type NGC 1291, NGC 1326, NGC 1543). In such cases the of bars (or strong bars) and another one third have indeter- smaller secondary bar is wrapped inside the larger primary minable type of bars (or weak bars) (Eskridge et al. 2000; bar (Erwin 2004; Debattista & Shen 2006; Du et al. 2016; Cheung et al. 2013; Yoon et al. 2019). Fraction of barred De Lorenzo-C´aceres et al. 2019). galaxies among lenticulars and spirals is strongly depen- dent on red-shift, stellar mass, colour and bulge prominence Galactic bars have many shapes and sizes accord- (Abraham et al. 1999; Sheth et al. 2008; Nair & Abraham ing to the functional form of potential energy. There 2010; Simmons et al. 2014; Zhou et al. 2020). are many three-dimensional bar potential models like Galactic bars are one of the robust substructure of the spherical, homeoidal, ellipsoidal etc. (Ferrers 1877; barred galaxies. They are solid, dense stellar body rotat- De Vaucouleurs & Freeman 1972; Caranicolas 2002; ing around the central core. Pattern speed of the bar is Jung & Zotos 2015; Williams & Evans 2017) have ex- tensively studied till date. Ferrers’ triaxial potential ⋆ E-mail: dmappmath [email protected] (Ferrers 1877) is the most used realistic bar potential † E-mail: [email protected] model, though its functional form is very much complex © 2020 The Authors 2 Mondal & Chattopadhyay and also it is computationally very much challenging. ied in the recent past (Lindblad 1947; Lynden-Bell & Kalnajs There are some other potentials like homeoidal poten- 1972; Pfenniger 1984; Patsis 2012; Mestre et al. 2020). tial (De Vaucouleurs & Freeman 1972), ad hoc potential Most of the earlier studies focused on detecting the chaotic (Barbanis & Woltjer 1967; Dehnen 2000) etc. have simpler invariant set of stellar orbits, which governs the general dy- functional forms than Ferrers’ but still they are rigorous to namics in the central region. Also there are studies about how handle numerically. Also there are some simple realistic bar these chaotic invariant set fuels the formation of spiral arms. potential models used by Caranicolas (2002), Jung & Zotos Due to chaotic dynamics in the central region, the escaping (2015) etc. stars produce tidal trails at the two ends of the bar. These tidal trails have tendency to create different morphologies Due to influence of Galactic bar some of the stellar or- like ring or spiral arms due to non-axisymmetric perturba- bits remain trapped inside the potential boundary and while tions. These studies relates the fate of escaping stars with others are escaped from that boundary, during their time subsequent morphological structures (Di Matteo et al. 2005; evolution. This problem can be studied from viewpoint of Minchev et al. 2010; Quillen et al. 2011; Grand et al. 2012; the problem of escape in an open Hamiltonian dynamical D’Onghia et al. 2013; Ernst & Peters 2014; Jung & Zotos system (Contopoulos & Efstathiou 2004; Ernst et al. 2008; 2016). Ernst & Peters 2014; Jung & Zotos 2015, 2016). An open In the present work we showed the same analogy i.e. the Hamiltonian system is a system where for energies above an fate of escaping stars behind formation of spiral arms but escape threshold, the energy shell is non compact and as a from the viewpoint of the amount of chaos produced therein. result a part of the stellar orbits explores (here from poten- Therefore we primarily focus on detection of the chaotic dy- tial holes to saddles) an infinite part of the position space. namics in the central region of barred galaxies, under suitable Also the Hamiltonian dynamics is a time reversal invariant realistic bar potential models. We then figure out a compar- (Jung & Zotos 2016). Now for a conservative dynamical sys- ison between these bar potential models in order to show tem the Hamiltonian (or the total energy) is a constant of how these models affect the chaotic dynamics with respect to motion. Hence all the stellar orbits are confined inside the mass and length of the bar. We relate these measurements of five-dimensional energy hyper-surface of the six-dimensional chaos with subsequent structure formations like spiral arms phase space of the Hamiltonian system. These stellar orbits or rings. Also we have shown which bar model is more fea- are either regular or chaotic in nature according to their ini- sible for certain type of structure formations under suitable tial condition (or initial energy value). Orbits having initial astrophysical circumstances. energy below the escape energy are remain trapped inside Here we used a four component three-dimensional gravi- the potential boundary and exhibit bounded motion (regu- tational model for the barred galaxies. The model consists lar or chaotic). While orbits having initial energy above the of a spherical bulge, a bar embedded in bulge, a flat disc escape energy exhibit unbounded motion. They escape from and a logarithmic dark matter halo. Modelling is done in two the interior along the open zero velocity curves (exist along parts corresponding to bar potential models, (i) an anhar- the escape channels of the potential). For bounded motion monic mass-model bar potential (Caranicolas 2002) and (ii) there are orbits which initially look like regular orbits but the Zotos bar potential (Jung & Zotos 2015). These two are revealed their true chaotic nature in long time period. These the most non arduous potential forms studied till date. For orbits are known as trapped chaotic orbits. Existence of such each of these potential models, first we have studied several orbits make the stellar dynamics more geometrically com- orbital structures corresponding to different initial condition plicated in the phase space. For unbounded motion orbits vectors. Also we calculate values of MLE for each of these are generally chaotic in nature. There are many dynamical initial condition vector, which gives us complete dynamical indicators which can classify these orbits according to their nature of these orbits (regular or chaotic). Then we draw regular or chaotic behaviour. Lyapunov exponent is one such Poincar´esurface section maps in both x - y and x - px sub- effective dynamical indicator and its working mechanism is section of the phase space. These two surface section maps are quite simple (Sandri 1996).