Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T02:56:49.189Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  06 April 2020

Seppo Mikkola
Affiliation:
University of Turku, Finland
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Gravitational Few-Body Dynamics
A Numerical Approach
, pp. 237 - 242
Publisher: Cambridge University Press
Print publication year: 2020

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

Aarseth, S. J. (2010). Gravitational N-Body Simulations. Cambridge, UK: Cambridge University Press.Google Scholar
Aarseth, S. J. and Zare, K. (1974). A regularization of the three-body problem. Celestial Mechanics, 10, 185205.Google Scholar
Blanchet, L. and Iyer, B. R. (2003). Third post-Newtonian dynamics of compact binaries: equations of motion in the centre-of-mass frame. Classical and Quantum Gravity, 20, 755776.CrossRefGoogle Scholar
Bulirsch, R. and Stoer, J. (1966). Numerical treatment of ordinary differential equations by extrapolation methods. Numerische Mathematik, 8, 113.CrossRefGoogle Scholar
Burkardt, T. M. and Danby, J. M. A. (1983). The solution of Kepler’s equation, II. Celestial Mechanics, 31, 317328.CrossRefGoogle Scholar
Burrau, C. (1913). Numerische Berechnung eines Spezialfalles des Dreikörperproblems. Astronomische Nachrichten, 195, 113116.Google Scholar
Chambers, J. (1999). A hybrid symplectic integrator that permits close encounters between massive bodies. Monthly Notices of the Royal Astronomical Society, 304, 793799.Google Scholar
Chambers, J. E. and Murison, M. A. (2000). Pseudo-high-order symplectic integrators. The Astronomical Journal, 119, 425433.Google Scholar
Choi, J. S. and Tapley, B. D. (1973). An extended canonical perturbation method. Celestial Mechanics, 7, 7790.Google Scholar
Conway, B. A. (1986). An improved algorithm due to Laguerre for the solution of Kepler’s equation. Celestial Mechanics, 39, 199211.Google Scholar
Cordeiro, R. R., Gomes, R. S. and Martins, R. V. (1997). A mapping for nonconservative systems. Celestial Mechanics and Dynamical Astronomy, 65, 407419.Google Scholar
Danby, J. M. A. (1992). Fundamentals of Celestial Mechanics, 2nd ed. Richmond, VA: Willmann-Bell.Google Scholar
Danby, J. M. A. and Burkardt, T. M. (1983). The solution of Kepler’s equation, I. Celestial Mechanics, 31, 95107.Google Scholar
Dey, L., Valtonen, M. J., Gopakumar, A., et al. (2018). Authenticating the presence of a relativistic massive black hole binary in OJ 287 using its general relativity centenary flare: improved orbital parameters. The Astrophysical Journal, 866, 11.Google Scholar
Duncan, M. J., Levison, H. F. and Lee, M. H. (1998). A multiple timestep symplectic algorithm for integrating close encounters. The Astronomical Journal, 116, 20672077.Google Scholar
Fukushima, T. (1996). A fast procedure solving Kepler’s equation for elliptic case. The Astronomical Journal, 112, 2858.Google Scholar
Fukushima, T. (1997a). A method solving Kepler’s equation without transcendental function evaluations. Celestial Mechanics and Dynamical Astronomy, 66, 309319.Google Scholar
Fukushima, T. (1997b). A procedure solving the extended Kepler’s equation for the hyperbolic case. The Astronomical Journal, 113, 1920.Google Scholar
Fukushima, T. (1998). A fast procedure solving Gauss’ form of Kepler’s equation. Proc. 30th Symp. on Celestial Mechanics, p. 217.Google Scholar
Fukushima, T. (1999). Fast procedure solving universal Kepler’s equation. Celestial Mechanics and Dynamical Astronomy, 75, 201226.Google Scholar
Fukushima, T. (2008). An orbital element formulation without solving Kepler’s equation. The Astronomical Journal, 135, 7282.Google Scholar
Funato, Y., Hut, P., McMillan, S. and Makino, J. (1996). Time-symmetrized Kustaanheimo–Stiefel regularization. The Astronomical Journal, 112, 1697.CrossRefGoogle Scholar
Goldstein, H., Poole, C. P., Jr and Safko, J. L. (2002). Classical Mechanics, 3rd ed. San Francisco, CA: Addison-Wesley.Google Scholar
Gragg, W. B. (1964). Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. PhD thesis, University of California, Los Angeles.Google Scholar
Gragg, W. B. (1965). On extrapolation algorithms for ordinary initial value problems. SIAM Journal on Numerical Analysis, 2, 384403.Google Scholar
Hagihara, Y. (1970). Dynamical Principles and Transformation Theory. Cambridge, MA: MIT Press.Google Scholar
Hand, L. N. and Finch, J. D. (2008). Analytical Mechanics. Cambridge, UK: Cambridge University Press.Google Scholar
Heggie, D. C. (1974). A global regularisation of the gravitational N-body problem. Celestial Mechanics, 10, 217241.Google Scholar
Hellström, C. and Mikkola, S. (2009). Universal formulation of quasi-Keplerian motion, and its applications. New Astronomy, 14, 607614.Google Scholar
Hellström, C. and Mikkola, S. (2010). Explicit algorithmic regularization in the few-body problem for velocity-dependent perturbations. Celestial Mechanics and Dynamical Astronomy, 106, 143156.Google Scholar
Herrick, S. (1972). Astrodynamics, Vol. II. London, UK: Van Nostrand Reinhold.Google Scholar
von Hoerner, S. (1960). Die numerische Integration des n-Körper-Problems für Sternhaufen. I. Zeitschrift für Astrophysik, 50, 184.Google Scholar
von Hoerner, S. (1963 ). Die numerische Integration des n-Körper-Problems für Sternhaufen. II. Zeitschrift für Astrophysik, 57, 47.Google Scholar
Hopf, H. (1931). Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen, 104, 637665. Reprinted: (1964). Selecta Heinz Hopf. Berlin, Germany: Springer, pp. 38–63.Google Scholar
Huang, W. and Leimkuhler, B. (1997). The adaptive Verlet method. SIAM Journal on Scientific Computing, 18, 239256.Google Scholar
Hut, P., Makino, J. and McMillan, S. (1995). Building a better leapfrog. The Astrophysical Journal, 443, L93L96.Google Scholar
Kamel, A. A. (1971). Lie transformations and the Hamiltonization of non-Hamiltonian systems. Celestial Mechanics, 4, 397405.Google Scholar
Kinoshita, H., Yoshida, H. and Nakai, H. (1991). Symplectic integrators and their application in dynamical astronomy. Celestial Mechanics and Dynamical Astronomy, 50, 5971.CrossRefGoogle Scholar
Kiseleva, L. G., Eggleton, P. P. and Mikkola, S. (1998). Tidal friction in triple stars. Monthly Notices of the Royal Astronomical Society, 300, 292302.Google Scholar
Koseleff, P.-V. (1993). Relations among Lie formal series and construction of symplectic integrators. In Cohen, G., Mora, T. and Moreno, O., eds, Applied Algebra, Algebraic Algorithms and Error Correcting Codes, AAECC-10. New York, NY: Springer, pp. 213230.Google Scholar
Kustaanheimo, P. (1964). Die Spinordarstellung der energetischen Identitäten der Keplerbewegung. In Stiefel, E., ed., Mathematische Methoden der Himmelsmechanik und Astronautik, Mathematisches Forschungsinstitut Oberwolfach, Berichte 1. Mannheim, Germany: Bibliographisches Institut Mannheim, pp. 333340.Google Scholar
Kustaanheimo, P. and Stiefel, E. (1965). Perturbation theory of Kepler motion based on spinor regularization. Journal für die Reine und Angewandte Mathematik, 218, 204219.Google Scholar
Laskar, J. and Robutel, P. (2001), High order symplectic integrators for perturbed Hamiltonian systems. Celestial Mechanics and Dynamical Astronomy, 80, 3962.Google Scholar
Levi-Civita, T. (1920). Sur la régularisation du problème des trois corps. Acta Mathematica, 42, 99144.Google Scholar
Makino, J. and Aarseth, S. J. (1992). On a Hermite integrator with Ahmad–Cohen scheme for gravitational many-body problems. Publications of the Astronomical Society of Japan, 44, 141151.Google Scholar
McLachlan, R. I. (1995). Composition methods in the presence of small parameters. BIT Numerical Mathematics, 35(2), 258268.Google Scholar
Mikkola, S. (1983). Encounters of binaries – I. Equal energies. Monthly Notices of the Royal Astronomical Society, 203, 11071121.Google Scholar
Mikkola, S. (1984a). Encounters of binaries. II – Unequal energies. Monthly Notices of the Royal Astronomical Society, 207, 115126.Google Scholar
Mikkola, S. (1984b). Encounters of binaries. III – Fly-bys. Monthly Notices of the Royal Astronomical Society, 208, 7582.Google Scholar
Mikkola, S. (1985). A practical and regular formulation of the N-body equations. Monthly Notices of the Royal Astronomical Society, 215, 171177.CrossRefGoogle Scholar
Mikkola, S. (1987). A cubic approximation for Kepler’s equation. Celestial Mechanics, 40, 329334.CrossRefGoogle Scholar
Mikkola, S. (1997). Practical symplectic methods with time transformation for the few-body problem. Celestial Mechanics and Dynamical Astronomy, 67, 145165.Google Scholar
Mikkola, S. (1998). Non-canonical perturbations in symplectic integration. Celestial Mechanics and Dynamical Astronomy, 68, 249255.CrossRefGoogle Scholar
Mikkola, S. and Aarseth, S. J. (1990). A chain regularization method for the few-body problem. Celestial Mechanics, 47, 375390.CrossRefGoogle Scholar
Mikkola, S. and Aarseth, S. J. (1993). An implementation of N-body chain regularization. Celestial Mechanics and Dynamical Astronomy, 57, 439.Google Scholar
Mikkola, S. and Aarseth, S. J. (1996). A slow-down treatment of close binaries. Celestial Mechanics and Dynamical Astronomy, 64, 197208.Google Scholar
Mikkola, S. and Aarseth, S. J. (1998). An efficient integration method for binaries in N-body simulations. New Astronomy, 3, 309320.Google Scholar
Mikkola, S. and Aarseth, S. J. (2002). A time-transformed leapfrog scheme. Celestial Mechanics and Dynamical Astronomy, 84, 343354.Google Scholar
Mikkola, S. and Innanen, K. (1999). Symplectic tangent map for planetary motions. Celestial Mechanics and Dynamical Astronomy, 74, 5967.Google Scholar
Mikkola, S. and Innanen, K. (2002). Individual accuracy checks for massive bodies and particles in symplectic integration. The Astronomical Journal, 124, 34453448.Google Scholar
Mikkola, S. and Merritt, D. (2006). Algorithmic regularization with velocitydependent forces. Monthly Notices of the Royal Astronomical Society, 372, 219223.Google Scholar
Mikkola, S. and Merritt, D. (2008). Implementing few-body algorithmic regularization with post-Newtonian terms. The Astronomical Journal, 135, 2398– 2405.Google Scholar
Mikkola, S. and Palmer, P. (2001). Simple derivation of symplectic integrators with first order correctors. Celestial Mechanics and Dynamical Astronomy, 77, 305317.Google Scholar
Mikkola, S. and Tanikawa, K. (1999a). Algorithmic regularization of the few-body problem. Monthly Notices of the Royal Astronomical Society, 310, 745749.CrossRefGoogle Scholar
Mikkola, S. and Tanikawa, K. (1999b). Explicit symplectic algorithms for time-transformed Hamiltonians. Celestial Mechanics and Dynamical Astronomy, 74, 287295.Google Scholar
Mikkola, S. and Tanikawa, K. (2007). Correlation of macroscopic instability and Lyapunov times in the general three-body problem. Monthly Notices of the Royal Astronomical Society, 379, L21L24.Google Scholar
Mikkola, S. and Tanikawa, K. (2013a). Implementation of an efficient logarithmic-Hamiltonian three-body code. New Astronomy, 20, 3841.Google Scholar
Mikkola, S. and Tanikawa, K. (2013b). Regularizing dynamical problems with the symplectic logarithmic Hamiltonian leapfrog. Monthly Notices of the Royal Astronomical Society, 430, 28222827.Google Scholar
Mikkola, S. and Wiegert, P. (2002). Regularizing time transformations in symplectic and composite integration. Celestial Mechanics and Dynamical Astronomy, 82, 375390.Google Scholar
Mikkola, S., Palmer, P. and Hashida, Y. (2002). An implementation of the logarithmic Hamiltonian method for artificial satellite orbit determination. Celestial Mechanics and Dynamical Astronomy, 82, 391411.Google Scholar
Mikkola, S., Brasser, R., Wiegert, P. and Innanen, K. (2004). Asteroid 2002 VE68, a quasi-satellite of Venus. Monthly Notices of the Royal Astronomical Society, 351, L63–L65.Google Scholar
Mora, T. and Will, C. M. (2004). Post-Newtonian diagnostic of quasiequilibrium binary configurations of compact objects. Physical Review D, 69, 104021.Google Scholar
Poincaré, H. (1895). Les Méthodes Nouvelles de la Mécanique Céleste. Paris, France: Gauthier-Villars. Republished in English: (1993). New Methods in Celestial Mechanics. New York, NY: AIP Press.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1986). Numerical Recipes. Cambridge, UK: Cambridge University Press.Google Scholar
Preto, M. and Tremaine, S. (1999). A class of symplectic integrators with adaptive time step for separable Hamiltonian systems. The Astronomical Journal, 118, 25322541.CrossRefGoogle Scholar
Rauch, K. P. and Holman, M. (1999). Dynamical chaos in the Wisdom–Holman integrator: origins and solution. The Astronomical Journal, 117, 10871102.Google Scholar
Saha, P. and Tremaine, S. D. (1992). Symplectic integrators for Solar System dynamics. The Astronomical Journal, 104, 16331640.Google Scholar
Saha, P. and Tremaine, S. D. (1994). Long-term planetary integrations with individual timesteps. The Astronomical Journal, 108, 19621969.Google Scholar
Sanz-Serna, J. M. (1992). Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica, 1, 243286.Google Scholar
Soffel, M. H. (1989). Relativity in Astrometry, Celestial Mechanics and Geodesy. Berlin, Germany: Springer, p. 141.Google Scholar
Steffensen, J. F. (1907). Über die Integration des Dreikörperproblems in der Ebene. Astronomische Nachrichten, 176, 221.Google Scholar
Stiefel, E. L. and Scheifele, G. (1971). Linear and Regular Celestial Mechanics. Berlin, Germany: Springer.Google Scholar
Stumpff, K. (1959). Himmelsmechanik, Band I. Berlin, Germany: VEB Deutscher Verlag der Wissenschaften.Google Scholar
Szebehely, V. and Peters, C. F. (1967). Complete solution of a general problem of three bodies. The Astronomical Journal, 72, 876.Google Scholar
Szebehely, V. and Zare, K. (1975). Time transformations in the extended phasespace. Celestial Mechanics, 11, 469.Google Scholar
Tanikawa, K., Saito, M. M. and Mikkola, S. (2019). A search for triple collision orbits inside the domain of the free-fall three-body problem. Celestial Mechanics and Dynamical Astronomy, 131, 24.Google Scholar
Tremaine, S. (2001). Canonical elements for collision orbits. Celestial Mechanics and Dynamical Astronomy, 79, 231233.Google Scholar
Valtonen, M. J., Mikkola, S., Merritt, D., et al. (2010). Measuring the spin of the primary black hole in OJ287. The Astrophysical Journal, 709, 725732.CrossRefGoogle Scholar
Waldvogel, J. (1997). Symplectic integrators for Hill’s lunar problem. In Dvorak, R. and Henrard, J., eds, The Dynamical Behaviour of Our Planetary System. Dordrecht, The Netherlands: Kluwer Academic, pp. 291305.Google Scholar
Waldvogel, J. (2008). Quaternions for regularizing celestial mechanics: the right way. Celestial Mechanics and Dynamical Astronomy, 102, 149162.Google Scholar
Wisdom, J. and Holman, M. (1991). Symplectic maps for the N-body problem. The Astronomical Journal, 102, 15201538.Google Scholar
Wisdom, J., Holman, M. and Touma, J. (1996). Symplectic correctors. In Marsden, J. E., Patrick, G. W. and Shadwick, W. F., eds, Integration Algorithms and Classical Mechanics (Proc. Integration Methods in Classical Mechanics Meeting, Waterloo, ON, 14–18 October 1993). Fields Institute Communications, Vol. 10. Providence, RI: American Mathematical Society, p. 217.Google Scholar
Yoshida, H. (1990). Construction of higher order symplectic integrators. Physics Letters A, 150, 262268.Google Scholar
Zare, K. (1974). A regularization of multiple encounters in gravitational N-body problems. Celestial Mechanics, 10, 207215.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Seppo Mikkola, University of Turku, Finland
  • Book: Gravitational Few-Body Dynamics
  • Online publication: 06 April 2020
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Seppo Mikkola, University of Turku, Finland
  • Book: Gravitational Few-Body Dynamics
  • Online publication: 06 April 2020
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Seppo Mikkola, University of Turku, Finland
  • Book: Gravitational Few-Body Dynamics
  • Online publication: 06 April 2020
Available formats
×