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This is the third of a series of papers in which we have investigated the collisional probability between a protoplanet and a planetesimal, taking fully into account the effect of solar gravity.
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＞This is the third of a series of papers in which we have investigated the collisional probability ＞between a protoplanet and a planetesimal, taking fully into account the effect of solar gravity. 「これは、我々が、太陽の重力の影響を十分に考慮して、原始惑星と微惑星との衝突の確率を研究してきた一連の論文の三番目のものです。」
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- この文章の和訳をお願いします。
1. Introduction This is the third of a series of papers in which we have investigated the collisional probability between a protoplanet and a planetesimal, taking fully into account the effect of solar gravity. Until now, the collisional probability between Keplerian particles has not been well understood, despite of its importance, in the study of planetary formation and, as an expedient manner, the two-body (i.e., free space) approximation has been adopted. In the two-body approximation, the collisional rate is given by (e.g., Safronov,1969) σv=πr_p^2(1+(2Gm_p/r_pv^2))v, (1) where r_p and m_p are the sum of radii and the masses of the protoplanet and a colliding planetesimal, respectively. Furthermore, v is the relative velocity at infinity and usually taken to be equal to a mean random velocity of planetesimals, i.e., v=(<e_2*^2>+<i_2*^2>)^(1/2)v_K, (2) where <e_2*^2> and <i_2*^2> are the mean squares of heliocentric eccentricity and inclination of a swarm of planetesimals and v_K is the Keplerian velocity; in the planer problem (i.e., <i_2*^2>=0), the collisional rate is given, instead of Eq.(1), by (σ_2D)v=2r_p(1+(2Gm_p/r_pv^2))^(1/2)v. (3) Equations (1) and (3) will be referred to in later sections, to clarify the effect of solar gravity on the collisional rate. よろしくお願いします。
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Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
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- この文章の和訳を教えてください。
Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
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Progress of Theoretical Physics. Vol. 70, No. 1, July 1983 Collisional Processes of Planetesimals with a Protoplanet under the Gravity of the Proto-Sun Shuzo NISHIDA Department of Industrial and Systems Engineering Setsunan University, Neyagawa, Osaka 572 (Received March 4, 1983) Abstruct We investigate collisional processes of planetesimals with a protoplanet, assuming that the mass of the protoplanet is much larger than that of a planetesimal and the motion of the planetesimal is limited in the two-dimensional ecliptic plane. Then, we can describe the orbit by a solution to the plane circular Restricted Three-Body problem. Integrating numerically the equations of motion of the plane circular RTB problem for numerous sets of initial osculating orbital elements, we obtain the overall features of the encounters between the Keplerian particles. In this paper we will represent only the cases e=0 and 4h, where e is the eccentricity of the planetesimal far from the protoplanet and h is the normalized Hill radius of the protoplanet. We find that the collisional rate of Keplerian particles is enhanced by a factor of about 2.3 (e=0) or 1.4 (e=4h) compared with that of particles in a free space, as long as we are concerned with the two-dimensional motion of particles. よろしくお願いします。
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In order to understand precisely the formation process of planets it is important to study extensively the two-body encounters between planetesimals taking fully into account the influence of the solar gravity. Some authors have tried to simulate numerically the growth process of N-planetesimals to the present planets, taking into account the influence of the Sun. But before executing the simulation of the N-bodies’ encountering process numerically, it is necessary to know more detailes of the fundamental features of the two-body encounters between Keplerian particles. Furthermore, according to Hayashi, the ratio of the collisional rate to the scattering rate as well as the collisional rate itself is one of the important parameters which determine essentially the time scale of the planetary growth.
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According to Paper I, the total collisional rate <Γ(e_1,i_1)> of planetesimals upon the protoplanet with the heliocentric orbital elements e_1 and i_1 is given by <Γ(e_1,i_1)>=2π^2∫<n_2>e_i<P(e,i)>dedi, ・・・・・・・(9) where <n_2> is the distribution function of planetesimals averaged by the phase angles τ_1 and ω_1 of the protoplanet, and e and i are the orbital elements of relative motion between the protoplanet and the planetesimal at infinity.
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According to Paper I, the total collisional rate <Γ(e_1,i_1)> of planetesimals upon the protoplanet with the heliocentric orbital elements e_1 and i_1 is given by <Γ(e_1,i_1)>=2π^2∫<n_2>e_i<P(e,i)>dedi, ・・・・・・・(9) where <n_2> is the distribution function of planetesimals averaged by the phase angles τ_1 and ω_1 of the protoplanet, and e and i are the orbital elements of relative motion between the protoplanet and the planetesimal at infinity. よろしくお願いします。
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< P(e,i) > is defined by < P(e,i) >=∫(3/2)|b|(1/(2π)^2)p_col(e, i, b, τ, ω)dτdωdb, ・・・・・・(10) where p_col(e, i, b, τ, ω) is collisional probability, that is, it is unity if the planetesimal collides with the protoplanet, and zero otherwise. よろしくお願いします。
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< P(e,i) > is defined by < P(e,i) >=∫(3/2)|b|(1/(2π)^2)p_col(e, i, b, τ, ω)dτdωdb, ・・・・・・(10) where p_col(e, i, b, τ, ω) is collisional probability, that is, it is unity if the planetesimal collides with the protoplanet, and zero otherwise. よろしくお願いします。
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In the two-dimensional case where the protoplanet and the planetesimal revolve in the sample plane around the protosun (i.e., i=0), the phase angle ω loses its meaning and Eq. (10) must be modified as <P(e, 0)>=∫(3/2) |b|(1/2π)p_col(e, i=0, b, τ)dτdb. ・・・・・(11) We consider that the planetesimal collides with the protoplanet if the separation distance becomes smaller than the sum of their radii; the protoplanet radius scaled by ha_0* is given by r_p=0.005(ρ/3gcm^-3)^-(1/3)(a_0*/1AU)^-1, ・・・・・(12) where ρ is the mean mass density of the protoplanet. よろしくお願いいたします。
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