*Earth Planets Space,* Vol. 60 (No. 6), pp. 681-691, 2008

## Probability distribution of orbital crossing times in a protoplanetary system

Hiroyuki Emori^{1}, Kiyoshi Nakazawa^{2}, and Kazunori Iwasaki^{3}

^{1}Shumei University, Daigaku-Cho, Yachiyo-city, Chiba Prefecture, Japan

^{2}Tokyo Institute of Technology, 1-12-1 Oookayama, Meguro-ku, Tokyo, Japan

^{3}Laboratory for Atomospheric and Space Physics, Univ. Colorado, 392, UCB Boulder,
CO 80309-0392, USA

(Received June 4, 2007; Revised November 8, 2007; Accepted January 10, 2008; Online published July 4, 2008)

**Abstract: **
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 10^{0.2} times later than the average orbital crossing time.

**Key words: **
Protoplanet, orbital crossing, planetary formation.

Corresponding author E-mail: hemori@kuramae.ne.jp

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