Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T18:10:50.169Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular and chaotic dynamics of non-spherical bodies. Zeldovich’s pancakes and emission of very long gravitational waves

Part of: Macroscopic Randomness

Published online by Cambridge University Press:  13 July 2015

G. S. Bisnovatyi-Kogan  and
O. Yu. Tsupko
G. S. Bisnovatyi-Kogan*
Affiliation:
Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia
O. Yu. Tsupko
Affiliation:
Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia
*
†Email addresses for correspondence: gkogan@iki.rssi.ru, tsupko@iki.rssi.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we review a recently developed approximate method for investigation of dynamics of compressible ellipsoidal figures. Collapse and subsequent behaviour are described by a system of ordinary differential equations for time evolution of semi-axes of a uniformly rotating, three-axis, uniform-density ellipsoid. First, we apply this approach to investigate dynamic stability of non-spherical bodies. We solve the equations that describe, in a simplified way, the Newtonian dynamics of a self-gravitating non-rotating spheroidal body. We find that, after loss of stability, a contraction to a singularity occurs only in a pure spherical collapse, and deviations from spherical symmetry prevent the contraction to the singularity through a stabilizing action of nonlinear non-spherical oscillations. The development of instability leads to the formation of a regularly or chaotically oscillating body, in which dynamical motion prevents the formation of the singularity. We find regions of chaotic and regular pulsations by constructing a Poincaré diagram. A real collapse occurs after damping of the oscillations because of energy losses, shock wave formation or viscosity. We use our approach to investigate approximately the first stages of collapse during the large scale structure formation. The theory of this process started from ideas of Ya. B. Zeldovich, concerning the formation of strongly non-spherical structures during nonlinear stages of the development of gravitational instability, known as ‘Zeldovich’s pancakes’. In this paper the collapse of non-collisional dark matter and the formation of pancake structures are investigated approximately. Violent relaxation, mass and angular momentum losses are taken into account phenomenologically. We estimate an emission of very long gravitational waves during the collapse, and discuss the possibility of gravitational lensing and polarization of the cosmic microwave background by these waves.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 5 , October 2015 , 395810501
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonov, V. A. 1973 The Dynamics of Galaxies and Stellar Clusters. Nauka, (in Russian).Google Scholar
Ardeljan, N. V., Bisnovatyi-Kogan, G. S. & Moiseenko, S. G. 2005 Magnetorotational supernovae. Mon. Not. R. Astron. Soc. 359, 333–344.Google Scholar
Bisnovatyi-Kogan, G. S. 1989 Physical Problems in the Theory of Stellar Evolution. Nauka, (in Russian); English translation: Stellar Physics, Vols. 1 and 2. Springer, 2001.Google Scholar
Bisnovatyi-Kogan, G. S. 2004 A simplified model of the formation of structures in dark matter and a background of very long gravitational waves. Mon. Not. R. Astron. Soc. 347, 163–172.Google Scholar
Bisnovatyi-Kogan, G. S. & Tsupko, O. Yu. 2005 Approximate dynamics of dark matter ellipsoids. Mon. Not. R. Astron. Soc. 364, 833–842.Google Scholar
Bisnovatyi-Kogan, G. S. & Tsupko, O. Yu. 2008a Dynamic stabilization of non-spherical bodies against unlimited collapse. Mon. Not. R. Astron. Soc. 386, 1398–1403.Google Scholar
Bisnovatyi-Kogan, G. S. & Tsupko, O. Yu. 2008b Gravitational lensing by gravitational waves. Gravit. Cosmol. 14 (3), 226–229.Google Scholar
Boily, C. M., Athanassoula, E. & Kroupa, P. 2002 Scaling up tides in numerical models of galaxy and halo formation. Mon. Not. R. Astron. Soc. 332, 971–984.Google Scholar
Boily, C. M., Clarke, C. J. & Murray, S. D. 1999 Collapse and evolution of flattened star clusters. Mon. Not. R. Astron. Soc. 302, 399–412.Google Scholar
Braginsky, V. B., Kardashev, N. S., Polnarev, A. G. & Novikov, I. D. 1990 Propagation of electromagnetic radiation in a random field of gravitational waves and space radio interferometry. Nuovo Cimento B 105, 1141–1158.Google Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.Google Scholar
Damour, T. & Esposito-Farèse, G. 1998 Light deflection by gravitational waves from localized sources. Phys. Rev. D 58, 042003.Google Scholar
Doroshkevich, A. G., Kotok, E. V., Polyudov, A. N., Shandarin, S. F., Sigov, Yu. S. & Novikov, I. D. 1980 Two-dimensional simulation of the gravitational system dynamics and formation of the large-scale structure of the universe. Mon. Not. R. Astron. Soc. 192, 321–327.Google Scholar
Doroshkevich, A. G., Müller, V., Retzlaff, J. & Turchaninov, V. 1999 Superlarge-scale structure in N-body simulations. Mon. Not. R. Astron. Soc. 306, 575–591.CrossRefGoogle Scholar
Faraoni, V. 1992 Nonstationary gravitational lenses and the Fermat principle. Astrophys. J. 398, 425–428.Google Scholar
Fridman, A. M. & Polyachenko, V. L. 1985 Physics of Gravitating Systems. Springer.Google Scholar
Gorbunov, D. S. & Rubakov, V. A. 2011 Introduction to the Theory of the Early Universe. Cosmological Perturbations and Inflationary Theory. World Scientific.Google Scholar
Klypin, A. A. & Shandarin, S. F. 1983 Three-dimensional numerical model of the formation of large-scale structure in the universe. Mon. Not. R. Astron. Soc. 204, 891–907.Google Scholar
Lai, D., Rasio, F. A. & Shapiro, S. L. 1993 Ellipsoidal figures of equilibrium: compressible models. Astrophys. J. Suppl. 88, 205–252.Google Scholar
Lai, D., Rasio, F. A. & Shapiro, S. L. 1994a Equilibrium, stability, and orbital evolution of close binary systems. Astrophys. J. 423, 344–370.Google Scholar
Lai, D., Rasio, F. A. & Shapiro, S. L. 1994b Hydrodynamics of rotating stars and close binary interactions: compressible ellipsoid models. Astrophys. J. 437, 742–769.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1993 The Classical Theory of Fields. Pergamon.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.Google Scholar
Lin, C. C., Mestel, L. & Shu, F. H. 1965 The gravitational collapse of a uniform spheroid. Astrophys. J. 142, 1431–1446.CrossRefGoogle Scholar
Lynden-Bell, D. 1964 On large-scale instabilities during gravitational collapse and the evolution of shrinking Maclaurin spheroids. Astrophys. J. 139, 1195–1216.Google Scholar
Lynden-Bell, D. 1965 On the evolution of frictionless ellipsoids. Astrophys. J. 142, 1648–1649.Google Scholar
Lynden-Bell, D. 1967 Statistical mechanics of violent relaxation in stellar system. Mon. Not. R. Astron. Soc. 136, 101–121.Google Scholar
Lynden-Bell, D. 1996 Consequences of one spring researching with Chandrasekhar. Curr. Sci. 70, 789–799.Google Scholar
Mazets, E. P., Golenetskii, S. V., Il’inskii, V. N., Panov, V. N., Aptekar, R. L., Gur’yan, Y. A., Proskura, M. P., Sokolov, I. A., Sokolova, Z. Ya. & Kharitonova, T. V. 1981 Catalog of cosmic gamma-ray bursts from the KONUS experiment data. Astrophys. Space Sci. 80, 3–83.Google Scholar
Moiseenko, S. G., Bisnovatyi-Kogan, G. S. & Ardeljan, N. V. 2006 A magnetorotational core-collapse model with jets. Mon. Not. R. Astron. Soc. 370, 501–512.Google Scholar
Novikov, I. D. 1975 Gravitational radiation from a star that is contracting into a disk. Astron. Zh. 52, 657–659 (in Russian); English translation: Novikov, I. D. 1976 Gravitational radiation from a star collapsing into a disk. Sov. Astron. 19, 398–399.Google Scholar
Ostriker, J. P. & Peebles, P. J. E. 1973 A numerical study of the stability of flattened galaxies: or, can cold galaxies survive? Astrophys. J. 186, 467–480.Google Scholar
Polnarev, A. G. 1985 Polarization and anisotropy induced in the microwave background by cosmological gravitational waves. Sov. Astron. 29, 607–613.Google Scholar
Rodrigues, H. 2014 On determining the kinetic content of ellipsoidal configurations. Mon. Not. R. Astron. Soc. 440, 1519–1526.CrossRefGoogle Scholar
Rosensteel, G. & Tran, H. Q. 1991 Hamiltonian dynamics of self-gravitating ellipsoids. Astrophys. J. 366, 30–37.Google Scholar
Seljak, U. & Zaldarriaga, M. 1997 Signature of gravity waves in the polarization of the microwave background. Phys. Rev. Lett. 78, 2054–2057.Google Scholar
Shandarin, S. F. & Zeldovich, Ya. B. 1989 The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys. 61, 185–220.Google Scholar
Shapiro, S. L. 2004 The secular bar-mode instability in rapidly rotating stars revisited. Astrophys. J. 613, 1213–1220.CrossRefGoogle Scholar
Sharif, M. & Zaeem Ul Haq Bhatti, M. 2013 Stability analysis of restricted non-static axial symmetry. J. Cosmol. Astropart. Phys. (11), 014.Google Scholar
Sharif, M. & Zaeem Ul Haq Bhatti, M. 2014 On the stability of a class of radiating viscous self-gravitating stars with axial symmetry. Astropart. Phys. 56, 35–41.Google Scholar
Sheth, R. K., Mo, H. J. & Tormen, G. 2001 Ellipsoidal collapse and an improved model for the number and spatial distribution of dark matter haloes. Mon. Not. R. Astron. Soc. 323, 1–12.Google Scholar
Thuan, T. X. & Ostriker, J. P. 1974 Gravitational radiation from stellar collapse. Astrophys. J. 191, L105–L107.Google Scholar
Vandervoort, P. O. 2011 On chaos in the oscillations of galaxies. Mon. Not. R. Astron. Soc. 411, 37–53.CrossRefGoogle Scholar
Vandervoort, P. O. 2014 On chaos in the pulsations of stars. Mon. Not. R. Astron. Soc. 443, 504–521.CrossRefGoogle Scholar
Zeldovich, Ya. B. 1964 Newtonian and Einsteinian motion of homogeneous matter. Astron. Zh. 41, 873–883; English translation: Zeldovich Ya. B. 1965 Newtonian and Einsteinian motion of homogeneous matter. Sov. Astron. 8, 700–707.Google Scholar
Zeldovich, Ya. B. 1970a Separation of uniform matter into parts under the action of gravitation. Astrofizika 6, 319–335; English translation: Zeldovich Ya. B. 1970 Fragmentation of a homogeneous medium under the action of gravitation. Astrophysics 6, 164–174.Google Scholar
Zeldovich, Ya. B. 1970b Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89.Google Scholar
Zeldovich, Ya. B. & Novikov, I. D. 1967 Relativistic Astrophysics. Nauka, (in Russian).Google Scholar