TY - JOUR

T1 - Distribution of planetesimals around a protoplanet in the nebula gas I. The semi-analytic calculation of the gravitational scattering by a protoplanet

AU - Tanaka, Hidekazu

AU - Ida, Shigeru

N1 - Funding Information:
The authors are grateful to Dr. Jean-Marc Petit for many helpful suggestions. They thank Dr. Kiyoshi Nakazawa and Dr. Hiroyuki Emori for valuable comments. They also thank Ms. Kei Kudo for help in preparing the manuscript. This work was supported by the foundation for Research Fellowships of the Japan Society for the promotion of Science for Young Scientists. This is Contribution No. 24 of the Department of Earth and Planetary Sciences, Tokyo Institute of Technology.

PY - 1996/4

Y1 - 1996/4

N2 - We have developed a semi-analytic method of calculating the changes in heliocentric Keplerian orbital elements due to gravitational scattering by a protoplanet as a three-body problem. In encounters with high incident velocities, either the gravity of the protoplanet or the solar gravity can be regarded as perturbation force. In close encounters, by taking into account the solar gravity as a perturbation, we modified the two-body gravitational scattering. On the other hand, in slightly distant encounters, we apply the perturbing force of the protoplanet to the heliocentric Keplerian orbit of planetesimals. As a result, as for high-velocity encounters, the three-body problem is semi-analytically solvable. Our semi-analytic method can reproduce the numerical result of the orbital changes of individual planetesimals for the broad range of high-energy encounters with surprising high accuracy. We found that our method is valid under the conditions (i) b0 ≳ 2 and (ii) (e20 + i20 - 3/4 b20)1/2 ≳ 4, where e0 and i0 are eccentricity and inclination of relative motion normalized by the reduced Hill radius and b0 is the difference between semimajor axes normalized by the Hill radius. Though our method needs some numerical procedure, its cpu time is negligibly short compared with that of the direct orbital integration. In simulation of orbital evolution of planetesimals around a protoplanet in the gas, which we will perform in the subsequent paper, most encounters can be calculated by the semi-analytic method. This makes it possible to perform the long term (≃105 years) orbital calculation of ≃103-4 planetesimals.

AB - We have developed a semi-analytic method of calculating the changes in heliocentric Keplerian orbital elements due to gravitational scattering by a protoplanet as a three-body problem. In encounters with high incident velocities, either the gravity of the protoplanet or the solar gravity can be regarded as perturbation force. In close encounters, by taking into account the solar gravity as a perturbation, we modified the two-body gravitational scattering. On the other hand, in slightly distant encounters, we apply the perturbing force of the protoplanet to the heliocentric Keplerian orbit of planetesimals. As a result, as for high-velocity encounters, the three-body problem is semi-analytically solvable. Our semi-analytic method can reproduce the numerical result of the orbital changes of individual planetesimals for the broad range of high-energy encounters with surprising high accuracy. We found that our method is valid under the conditions (i) b0 ≳ 2 and (ii) (e20 + i20 - 3/4 b20)1/2 ≳ 4, where e0 and i0 are eccentricity and inclination of relative motion normalized by the reduced Hill radius and b0 is the difference between semimajor axes normalized by the Hill radius. Though our method needs some numerical procedure, its cpu time is negligibly short compared with that of the direct orbital integration. In simulation of orbital evolution of planetesimals around a protoplanet in the gas, which we will perform in the subsequent paper, most encounters can be calculated by the semi-analytic method. This makes it possible to perform the long term (≃105 years) orbital calculation of ≃103-4 planetesimals.

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U2 - 10.1006/icar.1996.0057

DO - 10.1006/icar.1996.0057

M3 - Article

AN - SCOPUS:0030120027

SN - 0019-1035

VL - 120

SP - 371

EP - 386

JO - Icarus

JF - Icarus

IS - 2

ER -