Abstract
In this 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 bulge, 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 investigations has been done in order to explore the regular and chaotic motions of stars. We have seen that escape mechanism has only seen near saddle points (\(L_1\), \(L_2\) and \(L_1^{'}\), \(L_2^{'}\)) of the Hamiltonian system. Orbital structures in the \(x-y\) 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. Poincaré surface section maps are studied in both the \(x-y\) and \(x-p_x\) (\(p_x\) is the momentum along the x-direction) planes to get a complete information about the escape properties of the system in the phase space. Also, we studied in detail how the chaotic dynamics varies with the 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 less prominent spiral patterns for stronger bars and ring structures for weaker bars.
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References
Abraham, R.G., Merrifield, M.R., Ellis, R.S., Tanvir, N.R., Brinchmann, J.: The evolution of barred spiral galaxies in the Hubble Deep Fields North and South. Mon. Not. R. Astron. Soc. 308, 569–576 (1999)
Al-Baidhany, I., Seigar, M., Treuthardt, P., Sierra, A., Davis, B., et al.: Study of the relation between the spiral arm pitch angle and the kinetic energy of random motions of the host spiral galaxies. A. J. Ark. Acad. Sci. 6(8), 25–36 (2014)
An, J., Evans, N.W.: Self-consistent potential-density pairs of thick discs and flattened galaxies. Mon. Not. R. Astron. Soc. 486, 3915–3926 (2019)
Athanassoula, E.: The existence and shapes of dust lanes in galactic bars. Mon. Not. R. Astron. Soc. 259, 345–364 (1992)
Barbanis, B., Woltjer, L.: Orbits in spiral galaxies and the velocity dispersion of population I stars. Astrophys. J. 150, 461–468 (1967)
Basu, B., Kanjilal, T.: Explosion-triggered star formation in the central region of the galaxy. Astrophys. Space Sci. 152, 203–214 (1989)
Bekki, K.: Formation of the off-centre bar in the large Magellanic cloud: a collision with a dark satellite? Mon. Not. R. Astron. Soc. 393, L60–L64 (2009)
Berrier, J.C., Davis, B.L., Kennefick, D., Kennefick, J.D., Seigar, M.S., et al.: Further evidence for a supermassive black hole mass-pitch angle relation. Astrophys. J. 769, 132 (2013)
Binney, J., Tremaine, S.: Galactic Dynamics. Princeton Univ. Press, New Jersey (1987)
Birkhoff, G.D.: Dynamical Systems. Amer. Math. Soc. Colloq. Publ, Rhode Island (1927)
Bournaud, F., Combes, F.: Gas accretion on spiral galaxies: bar formation and renewal. Astron. Astrophys. 392, 83–102 (2002)
Bovy, J., Leung, H.W., Hunt, J.A., Mackereth, J.T., García-Hernández, D.A., et al.: Life in the fast lane: a direct view of the dynamics, formation, and evolution of the Milky Way’s bar. Mon. Not. R. Astron. Soc. 490, 4740–4747 (2019)
Buta, R.: The catalog of southern ringed galaxies. Astrophys. J. Suppl. Ser. 96, 39–116 (1995)
Buta, R.J., Knapen, J.H., Elmegreen, B.G., Salo, H., Laurikainen, E., et al.: Do bars drive spiral density waves? Astron. J. 137, 4487–4516 (2009)
Byrd, G.G., Freeman, T., Buta, R.J.: The inner resonance ring of NGC 3081. II. Star formation, bar strength, disk surface mass density, and mass-to-light ratio. Astron. J. 131, 1377–1393 (2006)
Caranicolas, N.D.: Connecting global to local parameters in barred galaxy models. J. Astrophys. Astron. 23, 173–183 (2002)
Cedrés, B., Cepa, J., Bongiovanni, Á., Castañeda, H., Sánchez-Portal, M., et al.: Density waves and star formation in grand-design spirals. Astron. Astrophys. 560, A59 (2013)
Cheung, E., Athanassoula, E., Masters, K.L., Nichol, R.C., Bosma, A., et al.: Galaxy zoo: observing secular evolution through bars. Astrophys. J. 779, 162 (2013)
Contopoulos, G., Grosbøl, P.: Orbits in barred galaxies. Astron. Astrophys. Rev. 1, 261–289 (1989)
Contopoulos, G., Efstathiou, K.: Escapes and recurrence in a simple Hamiltonian system. Celest. Mech. Dyn. Astron. 88, 163–183 (2004)
Dauphas, N.: The U/Th production ratio and the age of the Milky Way from meteorites and Galactic halo stars. Nature 435, 1203–1205 (2005)
Davis, B.L., Graham, A.W., Seigar, M.S.: Updating the (supermassive black hole mass)-(spiral arm pitch angle) relation: a strong correlation for galaxies with pseudobulges. Mon. Not. R. Astron. Soc. 471, 2187–2203 (2017)
de Lorenzo-Cáceres, A., Méndez-Abreu, J., Thorne, B., Costantin, L.: Deconstructing double-barred galaxies in 2D and 3D-I. Classical nature of the dominant bulges. Mon. Not. R. Astron. Soc. 484, 665–686 (2019)
De Vaucouleurs, G., Freeman, K.C.: Structure and dynamics of barred spiral galaxies, in particular of the Magellanic type. Vistas Astron. 14, 163–294 (1972)
Debattista, V.P., Shen, J.: Long-lived double-barred galaxies from pseudobulges. Astrophys. J. 654, L127–L130 (2006)
Dehnen, W.: The effect of the outer lindblad resonance of the galactic bar on the local stellar velocity distribution. Astron. J. 119, 800–812 (2000)
Di Matteo, P., Dolcetta, R.C., Miocchi, P.: Clumpy substructures in globular cluster tidal tails. Celest. Mech. Dyn. Astron. 91, 59–73 (2005)
D’Onghia, E., Vogelsberger, M., Hernquist, L.: Self-perpetuating spiral arms in disk galaxies. Astrophys. J. 766, 34 (2013)
Efthymiopoulos, C., Kyziropoulos, P.E., Páez, R.I., Zouloumi, K., Gravvanis, G.A.: Manifold spirals, disc–halo interactions, and the secular evolution in N-body models of barred galaxies. Mon. Not. R. Astron. Soc. 484, 1487–1505
Elmegreen, D.M., Elmegreen, B.G.: Flocculent and grand design spiral structure in field, binary and group galaxies. Mon. Not. R. Astron. Soc. 201, 1021–1034 (1982)
Elmegreen, B.G., Elmegreen, D.M.: Properties of barred spiral galaxies. Astrophys. J. 288, 438–455 (1985)
Ernst, A., Just, A., Spurzem, R., Porth, O.: Escape from the vicinity of fractal basin boundaries of a star cluster. Mon. Not. R. Astron. Soc. 383, 897–906 (2008)
Ernst, A., Peters, T.: Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar. Mon. Not. R. Astron. Soc. 443, 2579–2589 (2014)
Erwin, P.: Double-barred galaxies-I. A catalog of barred galaxies with stellar secondary bars and inner disks. Astron. Astrophys. 415, 941–957 (2004)
Eskridge, P.B., Frogel, J.A., Pogge, R.W., Quillen, A.C., Davies, R.L., et al.: The frequency of barred spiral galaxies in the near-infrared. Astron. J. 119, 536–544 (2000)
Ferrers, N.M.: On the potentials of ellipsoids, ellipsoidal shells, elliptic laminae and elliptic rings of variable densities. Q. J. Pure Appl. Math. 14, 1–22 (1877)
Frick, P., Stepanov, R., Beck, R., Sokoloff, D., Shukurov, A., et al.: Magnetic and gaseous spiral arms in M83. Astron. Astrophys. 585, A21 (2016)
Gonidakis, I., Livanou, E., Kontizas, E., Klein, U., Kontizas, M., et al.: Structure of the SMC-Stellar component distribution from 2MASS data. Astron. Astrophys. 496, 375–380 (2009)
Grand, R.J.J., Kawata, D., Cropper, M.: The dynamics of stars around spiral arms. Mon. Not. R. Astron. Soc. 421, 1529–1538 (2012)
Halle, A., Combes, F.: Influence of baryonic physics in galaxy simulations: a semi-analytic treatment of the molecular component. Astron. Astrophys. 559, A55 (2013)
James, P.A., Percival, S.M.: Star formation suppression and bar ages in nearby barred galaxies. Mon. Not. R. Astron. Soc. 474, 3101–3109 (2017)
Jung, C., Zotos, E.E.: Introducing a new 3D dynamical model for barred galaxies. Publ. Astron. Soc. Aust. 32, e042 (2015)
Jung, C., Zotos, E.E.: Orbital and escape dynamics in barred galaxies-I. The 2D system. Mon. Not. R. Astron. Soc. 457, 2583–2603 (2016)
Kaufmann, D.E., Contopoulos, G.: Self-consistent models of barred spiral galaxies. Astron. Astrophys. 309, 381–402 (1996)
Kaviraj, S., Shabala, S.S., Deller, A.T., Middelberg, E.: Radio AGN in spiral galaxies. Mon. Not. R. Astron. Soc. 454, 1595–1604 (2015)
Kim, M., Choi, Y.Y., Kim, S.S.: Effect of bars on evolution of SDSS spiral galaxies. Mon. Not. R. Astron. Soc. 494, 5839–5850 (2020)
Laurikainen, E., Salo, H., Buta, R., Knapen, J.H.: Properties of bars and bulges in the Hubble sequence. Mon. Not. R. Astron. Soc. 381, 401–417 (2007)
Lindblad, B.: On the dynamics of the barred spiral nebulae. Publ. Astron. Soc. Pac. 59, 305–309 (1947)
Łokas, E.L.: Anatomy of a buckling galactic bar. Astron. Astrophys. 629, A52 (2019)
Long, K., Murali, C.: Analytical potentials for barred galaxies. Astrophys. J. 397, 44–48 (1992)
Lynden-Bell, D., Kalnajs, A.J.: On the generating mechanism of spiral structure. Mon. Not. R. Astron. Soc. 157, 1–30 (1972)
Maragoudaki, F., Kontizas, M., Morgan, D.H., Kontizas, E., Dapergolas, A., et al.: The recent structural evolution of the SMC. Astron. Astrophys. 379, 864–869 (2001)
Melia, F., Falcke, H.: The supermassive black hole at the Galactic Center. Annu. Rev. Astron. Astrophys. 39, 309–352 (2001)
Mestre, M., Llinares, C., Carpintero, D.D.: Effects of chaos on the detectability of stellar streams. Mon. Not. R. Astron. Soc. 492, 4398–4408 (2020)
Minchev, I., Boily, C., Siebert, A., Bienayme, O.: Low-velocity streams in the solar neighbourhood caused by the Galactic bar. Mon. Not. R. Astron. Soc. 407, 2122–2130 (2010)
Miyamoto, M., Nagai, R.: Three-dimensional models for the distribution of mass in galaxies. Publ. Astron. Soc. Jpn. 27, 533–543 (1975)
Mondal, D., Chattopadhyay, T.: Star formation under explosion mechanism in a magnetized medium. Bulg. Astron. J. 31, 16–29 (2019)
Monteagudo, L., Gallart, C., Monelli, M., Bernard, E.J., Stetson, P.B.: The origin of the LMC stellar bar: clues from the SFH of the bar and inner disc. Mon. Not. R. Astron. Soc. 473, L16–L20 (2018)
Nair, P.B., Abraham, R.G.: On the fraction of barred spiral galaxies. Astrophys. J. Lett. 714, L260–L264 (2010)
Nishiyama, S., Nagata, T.: IRSF/SIRIUS team: is the milky way a double-barred galaxy? J. Phys. Conf. Ser. 54, 62–66 (2006)
Ostriker, J.P., Peebles, P.J.E., Yahil, A.: The size and mass of galaxies, and the mass of the universe. Astrophys. J. 193, L1–L4 (1974)
Pastorini, G., Marconi, A., Capetti, A., Axon, D.J., Alonso-Herrero, A., et al.: Supermassive black holes in the Sbc spiral galaxies NGC 3310, NGC 4303 and NGC 4258. Astron. Astrophys. 469, 405–423 (2007)
Patsis, P.A., Athanassoula, E., Quillen, A.C.: Orbits in the Bar of NGC 4314. Astrophys. J. 483, 731–744 (1997)
Patsis, P.A.: Structures out of Chaos in barred-spiral galaxies. Int. J. Bifurcation Chaos 22, 1230029 (2012)
Petersen, M.S., Weinberg, M.D., Katz, N.: Using torque to understand barred galaxy models. Mon. Not. R. Astron. Soc. 490, 3616–3632 (2019)
Pfenniger, D.: The 3D dynamics of barred galaxies. Astron. Astrophys. 134, 373–386 (1984)
Piatti, A.E.: The real population of star clusters in the bar of the Large Magellanic Cloud. Astron. Astrophys. 606, A21 (2017)
Plummer, H.C.: On the problem of distribution in globular star clusters. Mon. Not. R. Astron. Soc. 71, 460–470 (1911)
Polyachenko, E.V., Shukhman, I.G.: On the Lynden-Bell bar formation mechanism in galactic disks. Astron. Lett. 46, 12–23 (2020)
Proshina, I.S., Kniazev, A.Y., Sil’chenko, O.K.: Star-forming rings in lenticular galaxies: origin of the gas. Astron. J. 158, 5 (2019)
Querejeta, M., Eliche-Moral, M.D.C., Tapia, T., Borlaff, A., van de Ven, G., et al.: Formation of S0 galaxies through mergers-explaining angular momentum and concentration change from spirals to S0s. Astron. Astrophys. 579, L2 (2015)
Quillen, A.C., Dougherty, J., Bagley, M.B., Minchev, I., Comparetta, J.: Structure in phase space associated with spiral and bar density waves in an N-body hybrid galactic disc. Mon. Not. R. Astron. Soc. 417, 762–784 (2011)
Raha, N., Sellwood, J.A., James, R.A., Kahn, F.D.: A dynamical instability of bars in disk galaxies. Nature 352, 411–412 (1991)
Regan, M.W., Teuben, P.: The formation of nuclear rings in barred spiral galaxies. Astrophys. J. 582, 723–742 (2003)
Romero-Gómez, M., Masdemont, J.J., Athanassoula, E., García-Gómez, C.: The origin of rR1 ring structures in barred galaxies. Astron. Astrophys. 453, 39–45 (2006)
Romero-Gómez, M., Athanassoula, E., Masdemont, J.J., García-Gómez, C.: The formation of spiral arms and rings in barred galaxies. Astron. Astrophys. 472, 63–75 (2007)
Sakamoto, K., Okumura, S.K., Ishizuki, S., Scoville, N.Z.: Bar-driven transport of molecular gas to galactic centers and its consequences. Astrophys. J. 525, 691–701 (1999)
Sakamoto, K., Baker, A.J., Scoville, N.Z.: Gasdynamics in the liner galaxy NGC 5005: episodic fueling of a nuclear disk. Astrophys. J. 533, 149–161 (2000)
Sanchez-Martin, P., Romero-Gomez, M., Masdemont, J.J.: Warp evidence in precessing galactic bar models. Astron. Astrophys. 588, A76 (2016)
Sanders, J.L., Smith, L., Evans, N.W.: The pattern speed of the Milky Way bar from transverse velocities. Mon. Not. R. Astron. Soc. 488, 4552–4564 (2019)
Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. 6, 78–84 (1996)
Seigar, M.S., Kennefick, D., Kennefick, J., Lacy, C.H.: Discovery of a relationship between spiral arm morphology and supermassive black hole mass in disk galaxies. Astrophys. J. 678, L93–L96 (2008)
Sellwood, J.A.: Bar instability in disk-halo systems. Astrophys. J. 819, 92 (2016)
Seo, W.Y., Kim, W.T., Kwak, S., Hsieh, P.Y., Han, C., et al.: Effects of gas on formation and evolution of stellar bars and nuclear rings in disk galaxies. Astrophys. J. 872, 5 (2019)
Sharma, S., Stello, D., Bland-Hawthorn, J., Hayden, M.R., Zinn, J.C., et al.: The K2-HERMES survey: age and metallicity of the thick disc. Mon. Not. R. Astron. Soc. 490, 5335–5352 (2019)
Sheth, K., Elmegreen, D.M., Elmegreen, B.G., Capak, P., Abraham, R.G., et al.: Evolution of the bar fraction in COSMOS: quantifying the assembly of the Hubble sequence. Astrophys. J. 675, 1141–1155 (2008)
Sil’chenko, O., Kostiuk, I., Burenkov, A., Parul, H.: Star formation in outer rings of S0 galaxies-I. NGC 6534 and MCG 11-22-015. Astron. Astrophys. 620, L7 (2018)
Simmons, B.D., Melvin, T., Lintott, C., Masters, K.L., Willett, K.W., et al.: Galaxy zoo: CANDELS barred discs and bar fractions. Mon. Not. R. Astron. Soc. 445, 3466–3474 (2014)
Smet, C.O., Posacki, S., Ciotti, L.: Miyamoto-Nagai discs embedded in the Binney logarithmic potential: analytical solution of the two-integrals Jeans equations. Mon. Not. R. Astron. Soc. 448, 3466–3474 (2015)
Sofue, Y., Rubin, V.: Rotation curves of spiral galaxies. Annu. Rev. Astron. Astrophys. 39, 137–174 (2001)
Strantzalis, A., Hatzidimitriou, D., Zezas, A., Antoniou, V., Lianou, S., et al.: Discrete star formation events in the central bar of the Small Magellanic Cloud. Mon. Not. R. Astron. Soc. 489, 5087–5097 (2019)
Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Addison–Wesley, Reading (1994)
Van den Bergh, S.: Lenticular galaxies and their environments. Astrophys. J. 702, 1502–1506 (2009)
Voglis, N., Tsoutsis, P., Efthymiopoulos, C.: Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies. Mon. Not. R. Astron. Soc. 373, 280–294 (2006)
Weżgowiec, M., Ehle, M., Beck, R.: Hot gas and magnetic arms of NGC 6946: indications for reconnection heating? Astron. Astrophys. 585, A3 (2016)
Williams, A.A., Evans, N.W.: Models of bars-I. Flattish profiles for early-type galaxies. Mon. Not. R. Astron. Soc. 469, 4414–4421 (2017)
Yoon, Y., Im, M., Lee, G.H., Lee, S.K., Lim, G.: Observational evidence for bar formation in disk galaxies via cluster-cluster interaction. Nat. Astron. 3, 844–850 (2019)
Zotos, E.E.: Order and chaos in a galactic model with a strong nuclear bar. Res. Astron. Astrophys. 12, 500–512 (2012)
Zotos, E.E.: Distinguishing between order and chaos in a simple barred galaxy model. Astron. Nachr. 338, 614–620 (2017)
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The author DM thanks the University Grants Commission of India for providing Junior Research Fellowship (ID - 1263/(CSIRNETJUNE2019)), under which the work has been done. The authors are highly grateful to the referee for his/her detailed comments and fruitful suggestions, which improved the quality of the work to a great extent.
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Mondal, D., Chattopadhyay, T. Role of galactic bars in the formation of spiral arms: a study through orbital and escape dynamics—I. Celest Mech Dyn Astr 133, 43 (2021). https://doi.org/10.1007/s10569-021-10037-5
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DOI: https://doi.org/10.1007/s10569-021-10037-5