Abstract
We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form \(Z(R)\propto [\varSigma (R)]^{-1/3}\), where R is the radial galactocentric coordinate, Z is the z-amplitude (vertical amplitude) of the orbit and \(\varSigma \) represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.
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Notes
Here, the term “small” means small enough to neglect variations in the projection of the orbit on the \(z=0\) plane.
References
Bienaymé, O., Robin, A.C., Famaey, B.: Quasi integral of motion for axisymmetric potentials. Astron. Astrophys. 581, A123 (2015). doi:10.1051/0004-6361/201526516
Bienaymé, O., Traven, G.: Approximate integrals of motion. Astron. Astrophys. 549, A89 (2013). doi:10.1051/0004-6361/201220008
Binney, J.: Distribution functions for the Milky Way. Mon. Not. R. Astron. Soc. 401, 2318–2330 (2010). doi:10.1111/j.1365-2966.2009.15845.x
Binney, J.: More dynamical models of our Galaxy. Mon. Not. R. Astron. Soc. 426, 1328–1337 (2012). doi:10.1111/j.1365-2966.2012.21692.x
Binney, J., McMillan, P.: Models of our Galaxy-II. Mon. Not. R. Astron. Soc. 413, 1889–1898 (2011). doi:10.1111/j.1365-2966.2011.18268.x
Binney, J., Sanders, J.L.: Dynamical models and Galaxy surveys. In: Feltzing, S., Zhao, Walton, G., Whitelock, P. (eds.) IAU Symposium, volume 298 of IAU Symposium, pp. 117–129. doi:10.1017/S1743921313006297 (2014)
Binney, J., Tremaine, S.: Galactic Dynamics, 2nd edn. Princeton University Press, Princeton (2008)
Contopoulos, G.: A third Integral of Motion in a Galaxy. Z. Astrophys. 49, 273 (1960)
Contopoulos, G.: On the existence of a third integral of motion. Astron. J. 68, 1 (1963). doi:10.1086/108903
Contopoulos, G.: The development of nonlinear dynamics in astronomy. Found. Phys. 31, 89–114 (2001). doi:10.1023/A:1004155905361
Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin (2002)
de Zeeuw, P.T.: Dynamical models for axisymmetric and triaxial galaxies. In: de Zeeuw, P.T. (eds.) Structure and Dynamics of Elliptical Galaxies, volume 127 of IAU Symposium, pp. 271–289 (1987). doi:10.1007/978-94-009-3971-4_23
Freeman, K.C.: On the Disks of Spiral and so Galaxies. Astron. J. 160, 811 (1970). doi:10.1086/150474
González, G.A., Plata-Plata, S.M., Ramos-Caro, J.: Finite thin disc models of four galaxies in the UrsaMajor cluster: NGC3877, NGC3917, NGC3949 and NGC4010. Mon. Not. R. Astron. Soc. 404, 468–474 (2010). doi:10.1111/j.1365-2966.2010.16303.x
González, G.A., Reina, J.I.: An infinite family of generalized Kalnajs discs. Mon. Not. R. Astron. Soc. 371, 1873–1876 (2006). doi:10.1111/j.1365-2966.2006.10819.x
Hietarinta, J.: Direct methods for the search of the second invariant. Phys. Rep. 147, 87–154 (1987). doi:10.1016/0370-1573(87)90089-5
Hunter, C.: The structure and stability of self-gravitating disks. Mon. Not. R. Astron. Soc. 126, 299 (1963). doi:10.1093/mnras/126.4.299
Hunter, C.: Disk-Crossing Orbits. In: Contopoulos, G., Voglis, N. (eds.) Galaxies and Chaos, volume 626 of Lecture Notes in Physics, pp. 137–153. Springer, Berlin (2003). doi:10.1007/978-3-540-45040-5_11
Hunter, C.: Chaos in orbits due to disk crossings. Ann. New York Acad. Sci. 1045, 120 (2005). doi:10.1196/annals.1350.011
Iorio, L.: Orbital perturbations due to massive rings. Earth Moon Planets 108, 189–217 (2012). doi:10.1007/s11038-012-9391-1
Kent, S.M.: Dark matter in spiral galaxies. I-Galaxies with optical rotation curves. Astron. J. 91, 1301–1327 (1986). doi:10.1086/114106
Kent, S.M.: Dark matter in spiral galaxies. II-Galaxies with H I rotation curves. Astron. J. 93, 816–832 (1987). doi:10.1086/114366
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Course of Theoretical Physics, 2nd edn. Elsevier, Oxford (2007)
Lemos, J.P.S., Letelier, P.S.: Exact general relativistic thin disks around black holes. Phys. Rev. D 49, 5135–5143 (1994). doi:10.1103/PhysRevD.49.5135
Letelier, P.S.: Stability of circular orbits of particles moving around black holes surrounded by axially symmetric structures. Phys. Rev. D 68, 104002 (2003). doi:10.1103/PhysRevD.68.104002
Letelier, P.S.: Simple potential-density pairs for flat rings. Mon. Not. R. Astron. Soc. 381, 1031–1034 (2007). doi:10.1111/j.1365-2966.2007.12128.x
Lora-Clavijo, F.D., Ospina-Henao, P.A., Pedraza, J.F.: Charged annular disks and Reissner-Nordström type black holes from extremal dust. Phys. Rev. D 82, 084005 (2010). doi:10.1103/PhysRevD.82.084005
Morgan, T., Morgan, L.: The gravitational field of a disk. Phys. Rev. 183, 1097–1101 (1969). doi:10.1103/PhysRev.183.1097
Pedraza, J.F., Ramos-Caro, J., González, G.A.: An infinite family of self-consistent models for axisymmetric flat galaxies. Mon. Not. R. Astron. Soc. 390, 1587–1597 (2008). doi:10.1111/j.1365-2966.2008.13846.x
Ramos-Caro, J., López-Suspes, F., González, G.A.: Chaotic and regular motion around generalized Kalnajs discs. Mon. Not. R. Astron. Soc. 386, 440–446 (2008). doi:10.1111/j.1365-2966.2008.13047.x
Ramos-Caro, J., Pedraza, J.F., Letelier, P.S.: Motion around a monopole+ ring system-I. Stability of equatorial circular orbits versus regularity of three-dimensional motion. Mon. Not. R. Astron. Soc. 414, 3105–3116 (2011). doi:10.1111/j.1365-2966.2011.18618.x
Saa, A., Venegeroles, R.: Chaos around the superposition of a black-hole and a thin disk. Phys. Lett. A 259, 201–206 (1999). doi:10.1016/S0375-9601(99)00447-8
Semerák, O., Suková, P.: Free motion around black holes with discs or rings: between integrability and chaos-I. Mon. Not. R. Astron. Soc. 404, 545–574 (2010). doi:10.1111/j.1365-2966.2009.16003.x
Sofue, Y., Honma, M., Omodaka, T.: Unified rotation curve of the Galaxy–decomposition into de Vaucouleurs bulge, disk, dark halo, and the 9-kpc rotation dip. Publ. Astron. Soc. Jpn. 61, 227 (2009). doi:10.1093/pasj/61.2.227
Varvoglis, H.: Non ergodic particle motion in a c0 potential. J. Phys. 46, 495–502 (1985)
Vieira, R.S.S., Ramos-Caro, J.: A simple formula for the third integral of motion of disk-crossing stars in the Galaxy. Astron. J. 786, 27 (2014). doi:10.1088/0004-637X/786/1/27
Vieira, R.S.S., Ramos-Caro, J.: On the stability of circular orbits in galactic dynamics: Newtonian thin disks. In: Rosquist, K., Jantzen, R.T., Ruffini, R. (eds.) The Thirteenth Marcel Grossmann Meeting, pp. 2346–2348. World Scientific, Singapore (2015). doi:10.1142/9789814623995_0438
Vogt, D., Letelier, P.S.: Analytical potential-density pairs for flat rings and toroidal structures. Mon. Not. R. Astron. Soc. 396, 1487–1498 (2009). doi:10.1111/j.1365-2966.2009.14803.x
Acknowledgments
The work of R.S.S.V. is supported by the “São Paulo Research Foundation” (FAPESP), grants 2010/00487-9 and 2015/10577-9. RSSV thanks Alberto Saa for fruitful discussions and Sylvio Ferraz Mello for helpful remarks on the accepted version of the manuscript. The authors thank the anonymous referees for insightful comments which helped us improve the final version of the manuscript. The authors also acknowledge the support from FAPESP grant 2013/09357-9.
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Vieira, R.S.S., Ramos-Caro, J. Integrability of motion around galactic razor-thin disks. Celest Mech Dyn Astr 126, 483–500 (2016). https://doi.org/10.1007/s10569-016-9705-0
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DOI: https://doi.org/10.1007/s10569-016-9705-0