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Stability of Trojan planets in multi-planetary systems

Stability of Trojan planets in different dynamical systems

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Abstract

Today there are more than 340 extra-solar planets in about 270 extra-solar systems confirmed. Besides the observed planets there exists also the possibility of a Trojan planet moving in the same orbit as the Jupiter-like planet. In our investigation we take also into account the habitability of a Trojan planet and whether such a terrestrial planet stays in the habitable zone. Its stability was investigated for multi-planetary systems, where one of the detected giant planets moves partly or completely in the habitable zone. By using numerical computations, we studied the orbital behaviour up to 107 years and determined the size of the stable regions around the Lagrangian equilibrium points for different dynamical models for fictitious Trojans. We also examined the interaction of the Trojan planets with a second or third giant planet, by varying its semimajor axis and eccentricity. We have found two systems (HD 155358 and HD 69830) that can host habitable Trojan planets. Another aim of this work was to determine the size of the stable region around the Lagrangian equilibrium points in the restricted three body problem for small mass ratios μ of the primaries μ ≤ 0.001 (e.g. Neptune mass of the secondary and smaller masses). We established a simple relation for the size depending on μ and the eccentricity.

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References

  • Beaugé C., Sándor Zs., Érdi B., Süli Á.: Co-orbital terrestrial planets in exoplanetary systems: a formation scenario. Astron. Astrophys. 463, 359–367 (2007)

    Article  ADS  Google Scholar 

  • Bennett A.: Characteristic exponents of the five equilibrium solutions in the elliptically restricted problem. Icarus 4, 177–190 (1965)

    Article  ADS  Google Scholar 

  • Bowell E., Holt H.E., Levy D.H., Innanen K.A., Mikkola S., Shoemaker E.M.: 1990 MB: the first Mars Trojan. BAAS 22(4), 1357–1369 (1990)

    ADS  Google Scholar 

  • Cresswell P., Nelson R.P.: On the evolution of multiple protoplanets embedded in a protostellar disc. Astron. Astrophys. 450, 833–945 (2006)

    Article  ADS  Google Scholar 

  • Cresswell P., Nelson R.P.: On the growth and stability of Trojan planets. Astron. Astrophys. 493, 1141–1157 (2009)

    Article  MATH  ADS  Google Scholar 

  • Cochran W.D., Endl M., Wittenmyer R.A., Bean J.L.: A planetary system around HD 155358: the lowest metallicity planet host star. Astroph. J. 665, 1407–1423 (2007)

    Article  ADS  Google Scholar 

  • Danby J.M.A.: Stability of the triangular points in the elliptic restricted problem of three bodies. Astrophys. J. 69, 165–179 (1964)

    ADS  MathSciNet  Google Scholar 

  • Dvorak R., Pilat-Lohinger E., Schwarz R., Freistetter F.: Extrasolar Trojan planets close to habitable zones. Astron. Astrophys. 426, L37–L51 (2004)

    Article  ADS  Google Scholar 

  • Dvorak R., Schwarz R., Süli Á., Kotoulas T.: On the stability of the Neptune Trojans. MNRAS 382, 1324–1343 (2007)

    ADS  Google Scholar 

  • Dvorak R., Lhotka Ch., Schwarz R.: Dynamics of inclined Neptune Trojans. Cel. Mech. Dynam. Astron. 102, 97–110 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Érdi B.: The motion of the perihelion of Trojan asteroids. Cel. Mech. Dynam. Astron. 20, 59–71 (1979)

    MATH  Google Scholar 

  • Érdi B., Sándor Z.: Stability of co-orbital motion in exoplanetary systems. Cel. Mech. Dynam. Astron. 92, 113–123 (2005)

    Article  MATH  ADS  Google Scholar 

  • Érdi B., Nagy I., Sándor Zs., Süli Á., Fröhlich G.: Secondary resonances of co-orbital motions in exoplanetary systems. MNRAS 381, 33–45 (2007)

    Article  ADS  Google Scholar 

  • Ford E.B., Gaudi B.S.: Observational constraints on Trojans of transiting extrasolar planets. Astroph. J. 652, 137–144 (2006)

    Article  ADS  Google Scholar 

  • Ford E.B., Holman M.J.: Using transit timing observations to search for Trojans of transiting extrasolar planets. Astroph. J. 664, 51–67 (2007)

    Article  ADS  Google Scholar 

  • Freistetter F.: The size of the stability regions of Jupiter Trojans. Astron. Astrophys. 453, 353–370 (2006)

    Article  ADS  Google Scholar 

  • Goździewski K., Mikaszewski C., Musieliński A.: Stability constraints in modeling of multi-planet extrasolar systems. IAUS 249, 447–463 (2008)

    ADS  Google Scholar 

  • Hanslmeier A., Dvorak R.: Numerical integration with Lie series. Astron. Astrophys. 132, 203–222 (1984)

    MATH  ADS  MathSciNet  Google Scholar 

  • Ji J., Kinoshita H., Liu L., Li G.: The secular evolution and architecture of the Neptunian triplet planetary system HD69830. Astroph. J. 657, 1092–1109 (2007)

    Article  ADS  Google Scholar 

  • Kasting J.F., Whitmire D.P., Reynolds R.T.: Habitable zones around main sequence stars. Icarus 101, 108–141 (1993)

    Article  ADS  Google Scholar 

  • Laughlin G., Chambers J.E.: Extrasolar Trojans: the viability and detectability of planets in the 1:1 resonance. Astrophys. J. 124, 592–606 (2002)

    ADS  Google Scholar 

  • Lichtenegger H.: The dynamics of bodies with variable masses. Cel. Mech. Dynam. Astron. 34, 357–376 (1984)

    MATH  MathSciNet  Google Scholar 

  • Lohinger, E.: Stabilitätsbereiche um L4 und L5 des eingeschränkten Dreikörperproblems für verschiedene Massenverhältnisse. Diploma thesis, University of Vienna, (1991)

  • Lohinger E., Dvorak R.: Stability regions around L4 in the elliptic restricted problem. Astron. Astrophys. 280, 683–696 (1993)

    ADS  Google Scholar 

  • Marchal C.: The Three-Body Problem, vol. 192. Elsevier, New York (1991)

    Google Scholar 

  • Markellos, V.V., Papadakis, K.E., Perdios, E.A.: Non-linear stability zones around the triangular Lagrangian points. In: Roy, A.E, Steves, B.A From Newton to Chaos, pp. 371–377. Plenum Press, New York (1995)

  • Marzari F., Scholl H.: The growth of Jupiter and Saturn and the capture of Trojans. Astron. Astrophys. 339, 278–285 (1998)

    ADS  Google Scholar 

  • Morbidelli A., Levison H.F., Tsiganis K., Gomes R.: Chaotic capture of Jupiter’s Trojan asteroids in the early Solar System. Nature 435, 462–485 (2005)

    Article  ADS  Google Scholar 

  • Nauenberg M.: Stability and eccentricity for two planets in a 1:1 resonance, and their possible occurence in extrasolar planetary systems. Astrophys. J. 124, 2332–2351 (2002)

    ADS  Google Scholar 

  • Pittichova, J.A., Meech, K.J., Wasserman, L.H., Trilling, D.E., Millis, R.L., Buie, M.W., Kern, S.D., Clancy, K.B., Hutchison, L.E., Chiang, E., Marsden, B.G.: 2001 QR322. MPEC, 2003-A55 (2003)

  • Robutel P., Gabern F., Jorba A.: The observed Trojans and the global dynamics around the Lagrangian points of the Sun Jupiter System. Cel. Mech. Dynam. Astron. 92, 53–89 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Schwarz R.: Global stability of L4 and L5 Trojans. PHD thesis, University of Vienna. Online database at: http://media.obvsg.at/dissd (2005)

  • Schwarz R., Pilat-Lohinger E., Dvorak R., Érdi B., Sándor Zs.: Trojans in habitable zones. Astrobiol. J. 5, 579 (2005)

    Article  ADS  Google Scholar 

  • Schwarz R., Dvorak R., Pilat Lohinger E., Süli Á., Érdi B.: Trojan planets in HD108874?. Astron. Astrophys. 462, 1165–1171 (2007a)

    Article  ADS  Google Scholar 

  • Schwarz R., Dvorak R., Süli Á., Érdi B.: Survey of the stability region of hypothetical habitable Trojan planets. Astron. Astrophys. 474, 1023–1046 (2007b)

    Article  ADS  Google Scholar 

  • Schwarz R., Dvorak R., Süli Á., Érdi B.: Stability of fictitious Trojan planets in extrasolar systems. Astron. Notes 328, 785–799 (2007c)

    Article  ADS  Google Scholar 

  • Thommes E.W.: A safety net for fast migrators: interactions between gap-opening and sub-gap-opening bodies in a protoplanetary disk. Astroph. J. 626, 1033–1056 (2005)

    Article  ADS  Google Scholar 

  • Valenti J., Fischer D.: Stellar metallicity and planet formation. Astron. Soc. Pacific Conf. Series 384, 292–304 (2008)

    ADS  Google Scholar 

  • Wolf M.: Wiederauffindung des Planeten (588) [1906 TG]. Astron. Notes 174, 47–65 (1907)

    Article  ADS  Google Scholar 

  • Wasserman, L.H., Chiang, E., Jordan, A.B., Ryan, E.L., Buie, M.W., Millis, R.L., Kern, S.D., Elliot, J.L., Washburn, K.E., Marsden, B.G.: 2001 QQ322. MPEC. 2001-V11 (2001)

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Authors and Affiliations

  1. Department of Astronomy, Eötvös University, Pázmány Péter sétány 1/A, 1117, Budapest, Hungary

    R. Schwarz & Á. Süli

  2. Institute for Astronomy, University of Vienna, Türkenschanzstr. 17, 1180, Wien, Austria

    R. Dvorak & E. Pilat-Lohinger

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  1. R. Schwarz
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  2. Á. Süli
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  3. R. Dvorak
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  4. E. Pilat-Lohinger
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Correspondence to R. Schwarz.

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Schwarz, R., Süli, Á., Dvorak, R. et al. Stability of Trojan planets in multi-planetary systems. Celest Mech Dyn Astr 104, 69–84 (2009). https://doi.org/10.1007/s10569-009-9210-9

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  • DOI: https://doi.org/10.1007/s10569-009-9210-9

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