Abstract
It is shown that orbital period ratios of successive planets in the Solar System, of satellites in giant planet systems and of exoplanets in exoplanetary systems are preferentially closer to irreducible fractions formed with Fibonacci numbers between 1 and 8 than to other fractions, in a ratio of approximately 60% vs 40%. Furthermore, if sets of minor planets are chosen with gradually smaller inclinations and eccentricities, the proximity to Fibonacci fractions of their period ratios with Jupiter or Mars’ period tends to increase. Finally, a simple model explains why the resonance of the form \(\frac{P_{1}}{P_{2}} = \frac{p}{p+q}\), with \(P_{1}\) and \(P_{2}\) orbital periods of planets or satellites and \(p\) and \(q\) small integers, are stronger and more commonly observed for \(p\) and \(( p+q )\) being both small Fibonacci numbers than for other cases.
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References
Aschwanden, M.J.: Order out of chaos: self-organization processes in astrophysics (2017). arXiv:1708.03394v1. Accessed 10 Aug. 2017
Aschwanden, M.J.: Self-organizing systems in planetary physics: harmonic resonances of planet and moon orbits. New Astron. 58C, 107–123 (2018)
Aschwanden, M.J., Scholkmann, F.: Exoplanet predictions based on harmonic orbit resonances. Galaxies 5(4), 56 (2017). arXiv:1705.07138v1. Accessed 10 Aug. 2017
Aschwanden, M.J., Scholkmann, F., Bethune, W., Schmutz, W., Abramenko, V., Cheung, M.C.M., Mueller, D., Benz, A.O., Kurths, J., Chernov, G., Kritsuk, A.G., Scargle, J.D., Melatos, A., Wagoner, R.V., Trimble, V., Green, W.: Order out of randomness: self-organization processes in astrophysics. Space Sci. Rev. 214, 55 (2018). https://doi.org/10.1007/s11214-018-0489-2
Bailey, N.T.J.: Statistical Method in Biology, pp. 146–160. Hodder and Stoughton, London (1983)
Batygin, K., Deck, K.M., Holman, M.J.: Dynamical evolution of multi-resonant systems: the case of GJ876 (2015). arXiv:1504.00051v1. Accessed 10 Aug. 2017
Bazsó, A., Eybl, V., Dvorak, R., Pilat-Lohinger, E., Lhotka, C.: A survey of near-mean-motion resonances between Venus and Earth. Celest. Mech. Dyn. Astron. 107(1), 63–76 (2010). arXiv:0911.2357. Accessed 10 Aug. 2017
Chandra, P., Weisstein, E.W.: Fibonacci Number. MathWorld—A Wolfram Web Resource (online). Available at http://mathworld.wolfram.com/FibonacciNumber.html. Accessed 27 Oct. 2017
Dermott, S.F.: On the origin of commensurabilities in the Solar System—II the orbital period relation. Mon. Not. R. Astron. Soc. 141, 363–376 (1968). https://doi.org/10.1093/mnras/141.3.363
Dubrulle, B., Graner, F.: Titius-Bode laws in the solar system. 2: Build your own law from disk models. Astron. Astrophys. 282(1), 269–276 (1994)
Exoplanet TEAM: The Extrasolar Planets Encyclopaedia (online). Available at http://exoplanet.eu/ (2017). Accessed 10 Aug. 2017
Feltz, B., Crommelinck, M., Goujon, P. (eds.): Self-Organization and Emergence in Life Sciences, Synthese Library. Springer, Berlin (2006)
Gillon, M., Triaud, A.H.M.J., Demory, B.-O., Jehin, E., Agol, E., Deck, K.M., Lederer, S.M., De Wit, J., Burdanov, A., Ingalls, J.G., Bolmont, E., Leconte, J., Raymond, S.N., Selsis, F., Turbet, M., Barkaoui, K., Burgasser, A., Burleigh, M.R., Carey, S.J., Chaushev, A., Copperwheat, C.M., Delrez, L., Fernandes, C.S., Holdsworth, D.L., Kotze, E.J., Van Grootel, V., Almleaky, Y., Benkhaldoun, Z., Magain, P., Queloz, D.: Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1. Nature 542(7642), 456 (2017). https://doi.org/10.1038/nature21360
Gine, J.: On the origin of the gravitational quantization: the Titius-Bode law. Chaos Solitons Fractals 32(2), 362–369 (2007)
Goldreich, P.: An explanation of the frequent occurrence of commensurable mean motions in the Solar System. Mon. Not. R. Astron. Soc. 130(3), 159–181 (1965)
Goldreich, P., Porco, C.: Shepherding of the Uranian Rings. II. Dynamics. Astron. J. 93, 730 (1987). https://doi.org/10.1086/114355
Gor’kavyi, N.N., Fridman, A.M.: Self-organization in planetary rings. Priroda 1, 56–68 (1991). ISSN 0032-874X. In Russian. Astronomicheskii Sovet, Moscow, USSR
Graner, F., Dubrulle, B.: Titius-Bode laws in the solar system. 1: Scale invariance explains everything. Astron. Astrophys. 282(1), 262–268 (1994)
Haken, H.: Information and Self-Organization: A Macroscopic Approach to Complex Systems. Springer Series in Synergetics, 3rd edn. Springer, Berlin (2006)
Hu, Z.W., Chen, Z.X.: Distance law and formation of satellite systems. Astron. Nachr. 308(6), 359–362 (1987)
Kernbach, S.: Structural Self-Organization in Multi-Agents and Multi-Robotic Systems. Logos Verlag, Berlin (2008)
Kirkwood, D.: The Asteroids, or Minor Planets Between Mars and Jupiter J. B. Lippencott, Philadelphia (1888)
Koshy, T.: Fibonacci and Lucas Numbers with Applications. John Wiley & Sons, Inc., Hoboken (2001). ISBN 978-04-713-9969-8. https://doi.org/10.1002/9781118033067
Krinsky, V.I. (ed.): Self-Organization: Autowaves and Structures Far from Equilibrium. Springer Series in Synergetics. Springer, Berlin (1984)
Lemaitre, A.: Resonances: models and captures. In: Souchay, J., Dvorak, R. (eds.) Dynamics of Small Solar System Bodies and Exoplanets. Lecture Notes in Physics, pp. 1–62. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-04458-8_1.
Livadiotis, G., McComas, D.J.: Evidence of large-scale quantization in space plasmas. Entropy 15, 1118–1134 (2013). https://doi.org/10.3390/e15031118
Luger, R., Sestovic, M., Kruse, E., Grimm, S.L., Demory, B.O., Agol, E., Bolmont, E., Fabrycky, D., Fernandes, C.S., Van Grootel, V., Burgasser, A., Gillon, M., Ingalls, J.G., Jehin, E., Raymond, S.N., Selsis, F., Triaud, A.H.M.J., Barclay, T., Barentsen, G., Howell, S.B., Delrez, L., deWit, J., Foreman-Mackey, D., Holdsworth, D.L., Leconte, J., Lederer, S., Turbet, M., Almleaky, Y., Benkhaldoun, Z., Magain, P., Morris, B., Heng, K., Queloz, D.: A seven-planet resonant chain in TRAPPIST-1. arXiv:1703.04166v2. Accessed 10 Aug. 2017
Malhotra, R.: The origin of Pluto’s orbit—implications for the solar system beyond Neptune. Astron. J. 110, 420–429 (1995)
Marov, M.Y., Kolesnichenko, A.V.: Turbulence and Self-Organization: Modeling Astrophysical Objects. Astrophysics and Space Science Library, vol. 389. Springer, Berlin (2013). https://doi.org/10.1007/978-1-4614-5155-6
Minor Planet Center: (2017). Available at https://www.minorplanetcenter.net/iau/MPCORB/Distant.txt. Accessed 10 Aug. 2017
Minton, D.A., Malhotra, R.: A record of planet migration in the main asteroid belt. Nat. Lett. 457, 1109–1111 (2009). https://doi.org/10.1038/nature07778
Mishra, R.K., Maaß, D., Zwierlein, E. (eds.): On Self-Organization: An Interdisciplinary Search for a Unifying Principle. Springer Series in Synergetics, vol. 61. Springer, Berlin (1994)
Murray, C.D., Thompson, R.P.: Orbits of shepherd satellites deduced from structure of the rings of Uranus. Nature 348, 499–502 (1990)
Murray, C.D., Dermott, S.F. (eds.): Solar System Dynamics. Cambridge University Press, Cambridge (1999). ISBN 0-521-57295-9
Nowotny, E.: On the formation of the planetary system—the Titius-Bode law. Moon Planets 21, 257–274 (1979)
Ovenden, M.W.: Physical sciences: Bode’s law and the missing planet. Nature 239, 508–509 (1972)
Patterson, C.W.: Resonance, capture and evolution of the planets. Icarus 70, 319–333 (1987)
Patton, J.M.: On the dynamical derivation of the Titius-Bode law. Celest. Mech. 44, 365–391 (1988)
Peale, S.J.: Orbital resonances in the Solar System. Annu. Rev. Astron. Astrophys. 14, 215–246 (1976). https://doi.org/10.1146/annurev.aa.14.090176.001243
Pletser, V.: Exponential distance laws for satellite systems. Earth Moon Planets 36, 193–210 (1986)
Pletser, V.: On exponential distance relations in planetary and satellite systems, observations and origin. Ph.D. Thesis, UCLouvain, Louvain-la-Neuve (1990)
Pletser, V., Basano, L.: Exponential distance relation and near resonances in the Trappist-1 planetary system (2017). arXiv:1703.04545. Accessed 10 Aug. 2017
Pletser, V.: Compilation of period ratio of minor planets and Jupiter and Mars, and of TNOs and Neptune. ResearchGate (2017a). https://doi.org/10.13140/RG.2.2.22423.68009. Accessed 10 Aug. 2017
Pletser, V.: Compilation of period ratio of exoplanets. ResearchGate (2017b). https://doi.org/10.13140/RG.2.2.24940.26243. Accessed 10 Aug. 2017
Pletser, V.: Fibonacci numbers and the golden ratio in biology, physics, astrophysics, chemistry and technology: a non-exhaustive review. arXiv:1712.2117064. Accessed 10 Aug. 2017
Pletser, V.: Orbital period ratios and Fibonacci numbers in solar planetary and satellite systems and in exoplanetary systems (2018). Accessed 20 June 2019. arXiv:1803.02828
Pontes, J.: Determinism, chaos, self-organization and entropy. An. Acad. Bras. Ciênc. (Ann. Braz. Acad. Sci.) 88(2), 1151–1164 (2016). https://doi.org/10.1590/0001-3765201620140396. Accessed 10 Aug. 2017
Read, B.A.: Fibonacci series in the Solar System. Fibonacci Q. 8(4), 428–438 (1970)
Rica, S.: Pattern formation through gravitational instability. C. R. Acad. Sci., Sér. 2, Fasc. b, Tome 320-9, 489–496 (1995)
Robutel, P., Souchay, J.: An introduction to the dynamics of Trojan asteroids. In: Souchay, J., Dvorak, R. (eds.) Dynamics of Small Solar System Bodies and Exoplanets. Lecture Notes in Physics, pp. 195–228. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-04458-8_4
Roy, A.E., Ovenden, M.W.: On the occurrence of commensurable mean motions in the solar system. Mon. Not. R. Astron. Soc. 114, 232–241 (1954)
Ruediger, G., Tschaepe, R.: Gerlands Beitr. Geophys. 97(2), 97 (1988)
Tamayo, D., Rein, H., Petrovich, C., Murray, N.: Convergent Migration Renders TRAPPIST-1 Long-Lived (2017). arXiv:1704.02957v2. Accessed 10 Aug. 2017
Torbett, M., Greenberg, R., Smoluchowski, R.: Orbital resonances and planetary formation sites. Icarus 49, 313–326 (1982)
Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F.: Origin of the orbital architecture of the giant planets of the Solar System. Nat. Lett. 435, 459–461 (2005). https://doi.org/10.1038/nature03539
Veras, D., Ford, E.B.: Identifying non-resonant Kepler planetary systems. Mon. Not. R. Astron. Soc. 420, L23–L27 (2012). https://doi.org/10.1111/j.1745-3933.2011.01185.x
Wells, D.R.: Titius-Bode and the helicity connection: a quantized field theory of protostar formation. IEEE Trans. Plasma Sci. 14(6), 865–873 (1989a)
Wells, D.R.: Quantization effects in the plasma universe. IEEE Trans. Plasma Sci. 17(2), 270–281 (1989b)
Wells, D.R.: Was the Titius-Bode series dictated by the minimum energy states of the generic solar plasma? IEEE Trans. Plasma Sci. 19(1), 73–76 (1990)
Wells, D.R.: Unification of gravitational, electrical, and strong forces by a virtual plasma theory. IEEE Trans. Plasma Sci. 20(6), 939–943 (1992)
Winn, J.N., Fabrycky, D.C.: The occurrence and architecture of exoplanetary systems. Annu. Rev. Astron. Astrophys. 53, 409–447 (2015)
Wisdom, J.: The origin of the Kirkwood gaps: a mapping technique for asteroidal motion near the 3/1 commensurability. Astron. J. 87, 577–593 (1982)
Wisdom, J.: Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74 (1983)
Wisdom, J.: A perturbative treatment of motion near the 3/1 commensurability. Icarus 63, 272–289 (1985)
Acknowledgements
This research has made use of data and/or services provided by the International Astronomical Union’s Minor Planet Center. Prof. D. Huylebrouck and Prof. L. Basano kindly provided comments on early versions of the manuscript. Stimulating discussions with C. Ducrest are also acknowledged.
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Pletser, V. Prevalence of Fibonacci numbers in orbital period ratios in solar planetary and satellite systems and in exoplanetary systems. Astrophys Space Sci 364, 158 (2019). https://doi.org/10.1007/s10509-019-3649-2
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DOI: https://doi.org/10.1007/s10509-019-3649-2