Article | Open | Published:
Probability distribution of orbital crossing times in a protoplanetary system
Earth, Planets and Spacevolume 60, pages681–691 (2008)
Long term behavior of five protoplanets was studied under the same conditions as those used in Chambers et al. (1996). One major difference was the number of calculations carried out for one parameter set of initial orbital distance. We reconfirmed their result for the orbital crossing times among five protoplanets starting from circular and co-planar orbits with an equal distance of their semi major axes. For each distance, the distribution of orbital crossing times was calculated from 500 sets of azimuthal positions of protoplanets randomly chosen. The distribution of the times around the average value resembles each other for almost all orbital distance cases. Based on a statistical certification we conclude that the fluctuations in orbital crossing times take “the log normal function”. The dispersion of the log normal distribution function is equal to 0.2. This means that 70% of the events of orbital crossing occurs in the range between 10−0.2 times earlier and 100.2 times later than the average orbital crossing time.
Adachi, I., C. Hayashi, and K. Nakazawa, The gas drag effect on the elliptical motion of a solid body in the primordial solar nebula, Prog. Theor. Phys., 56, 1756–1771, 1976.
Chambers, J. E., G. W. Wetherill, and A. P. Boss, The Stability of Multi-Planet Systems, Icarus, 119, 261–268, 1996.
Hasegawa, M. and K. Nakazawa, Distant encounter between Keplerian particles, Astron. Astrophys., 227(2), 619–627, 1990.
Ida, S., Stirring and dynamical friction rates of planetesimals in the solar gravitational field, Icarus, 88, 129–145, 1990.
Ikoma, M., K. Nakazawa, and H. Emori, Formation of giant planets: Dependencies on core accretion rate and grain opacity, Astrophys. J., 537, 1013-1025.
Ito, T. and K. Tanikawa, Stability and Instability of the terrestrial Proto-planet System and Their Possible Roles in the Final Stage of Planet Formation, Icarus, 139, 336–349, 1999.
Iwasaki, K., H. Tanaka, K. Nakazawa, and H. Emori, The Gas-Drag Effect on the Orbital Instability of a Protoplanets System, Publ. Astron. Soc. Jpn., 53, 321–329, 2001.
Iwasaki, K., H. Emori, K. Nakazawa, and H. Tanaka, Orbital Stability of a Protoplanet System under a Drag Force Proportional to the Random Velocity, Publ. Astron. Soc. Jpn., 54(3), 471–479, 2002.
Kokubo, E. and S. Ida, On Runaway Growth of Planetesimals, Icarus, 123(1), 180–191, 1996.
Kokubo, E. and S. Ida, Oligarchic Growth of Protoplanets, Icarus, 131(1), 171–178, 1998.
Mizuno, H., K. Nakazawa, and C. Hayashi, Instability of a gaseous envelope surrounding a planetary core and formation of giant planets, Prog. Theor. Phys., 60, 699–710, 1978.
Nakazawa, K. and S. Ida, Hill’s approximation in the three-body problem, Prog. Theor. Phys. Suppl., 96, 167–174, 1988.
Petti, J.-M. and M. Henon, A numerical simulation of planetary rings. I— Binary encounters, Astoron. Astrophys., 173(2), 389–404, 1987.
Press, W. H., B. P. Flannery, S. A. Teukolosky, and W. T. Vetterling, Numerical Recipes, Cambridge Univ. Press, London/New York, 1986.
Safronov, V. S., Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Chap. 7, Nauka Press, Moscow, 1969.
Stewart, G. R. and G. W. Wetherill, Evolution of planetesimal velocities, Icarus, 74, 542–553, 1988.
Strom, S. E., S. Edwards, and M. F. Skrutskie, Evolutionary time scales for circumstellar material associated with solar-type pre-main-sequence stars: A possible comstraint on the timescale for planet building, in Protostars and Planets III, edited by E. H. Levy and J. I. Lunine, 837–866, University of Arizona Press., Tucson, 1993.
Strom, S. E., Initial frequency, lifetime and evolution of YSO disks, Rev. Mex. Astron. Astrofis, 1, 317–328, 1995.
Yoshinaga, K., E. Kokubo, and J. Makino, The Stability of Protoplanet Systems, Icarus, 139, 328–335, 1999.
Wetherill, G. W and G. R. Stewart, Accumulation of a swarm of small planetesimals, Icarus, 77, 330–357, 1989.