## 惑星の運動方程式についての和訳と要約

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• この質問文章では、Hayashiらによる運動方程式についての和訳を求めています。
• 和訳された文章では、惑星と太陽の距離、質量の合計、惑星の回転の角速度が全て単位になっているシステムを採用しています。
• また、和訳された要約文を3つ作成し、運動方程式とその効果的なポテンシャルに関する情報をまとめています。

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# この文章の和訳をお願いします。

According to Hayashi et al., We will adopt the system of units where the distance between the planet and the Sun, the sum of their masses and the angular velocity of the rotation of the planet are all unity. When the coordinate is chosen such that the (x,y) plane coincides with the rotational plane of the planet, i.e., the ecliptic plane, the Sun and the planet are at rest on the x-axis and the planet is at the origin, then the equations of motion are given by (Szebeheley^10)), 　　　　　　　　　　　x’’－2y’=－∂U/∂x, 　　　　　　　　 (2・1) 　　　　　　　　　　　 y’’+2x’=－∂U/∂y, 　　　　　　　　(2・2) where U is the effective potential, described as 　　　　　　　　　　　　　　　U=－(μ/r_1)－((1－μ)/r_2)－((1/2)r^2)+U_0. 　　　　　　　　(2・3) よろしくお願いします。

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### 関連するQ&A

• この文章の和訳を教えてください。

In Eq. (2・3) μis defined as 　　　　　　　　　μ=M/( M? +M), (2・4) where M is the mass of the planet, γ, γ_1 and γ_2 are the distances from the center of gravity, the planet (i.e., the origin) and the Sun, respectively, which are given by 　　　　　　　　　　　　r^2=(x+1－μ)^2+y^2, (2・5) 　　　　　　　　　　　　r_1^2=x^2+y^2 　　(2・6) and 　　　　　　　　　 　　　r_2^2=(x+1)^2+y^2. 　 (2・7) Furthermore, U_0 is a certain constant and, for convenience, is chosen such that U is zero at the Lagrangian point L_2. お手数ですがよろしくお願いいたします。

• この文章の和訳を教えてください。

We start our orbital calculation of a particle from a point far from the planet where the effect of the gravitational force of the planet can be neglected. At an initial point where a particle is governed almost completely by the solar gravity, the particle orbit can be approximated in a good accuracy by the Keplerian. Hence we will give the initial conditions for the orbital calculation in terms of the Keplerian orbital elements (a_i, e_i, ε_i, δ_i) in place of (x, x’, y, y’), where a, e and ε are the semi-major axis, the eccentricity and the mean longitude at t=0, respectively. Furthermore the parameter δ is defined as 　　　　　　　　　　　　　　　　　　　δ=－nt_0,　　　　　　　　　　　　 (2・8) where n and t_0 are the mean angular velocity and the time of the perihelion passage, respectively. よろしくお願いします。

• この文章の和訳をお願いします。

We have two kinds of the final stage of the particle orbit: One is the scattering case and another is the collisional case. For the scattering case, after the passage near the planet a particle orbit settles again to the Keplerian at the region far from the planet. The final orbital elements (b_f~, e_f~, ε_f, δ_f), of course, are different from the initial owing to the gravitational interaction with the planet. On the other hand, we regard the case as the collision of the particle with the planet when the distance between the particle and the center of the planet becomes smaller than the planetary radius, r_p, which is given in the units adopted here by 　　　　　　r_p=R_p/R=(3M/4πρ)^(1/3)/R, 　　　　　　 =4.57×(10^－3)h/(R/1AU), 　　　　　　　　(2・12) where R_p and R are the radius of the planet in ordinary units and the distance between the planet and the Sun, respectively. どうかよろしくお願いします。

• この文章の和訳をお願いします。

2. Adopted assumptions and basic equations We consider two planetesimals revolving around the proto-Sun (being called the Sun). Here we assume that the mass of the one of these, which hereafter is called a protoplanet or simply a planet, is much larger than that of the other (being called a particle). We also assume that the planet moves circularly around the Sun without the influence of gravity of the particle. Furthermore, each orbit of the particle is limited in the ecliptic plane of the planetary orbit. Under these assumptions, the particle motion is simply given by a solution to the plane circular RTB problem. よろしくお願いします。

• この文章の和訳を教えてください。

In the preceding papar (Nakazawa et al., 1989a,referred to as Papar I), we have proposed that a framework of Hill’s equations (Hill, 1878) is of great advantages to find precisely the collisional rate between Keplerian particles over wide ranges of initial conditions. First, in Hill’s equations, the relative motion separates from the barycenter motion, and the equation of the barycenter motion can be integrated analytically (see also Henon and petit, 1986). Second, the equation of motion can be scaled by h and Ω; h is the reduced Hill radius and Ω is the Keplerian angular velocity. They are given by h=(m_p/3M_?)^(1/3) (4) and Ω={G(M_?+m_p)/a_0*^3}^(1/2), (5) Where a_0* is the reference heliocentric distance (which is usually taken to be equal to the semimajor axis of the protoplanet) and M_? is the solar mass. The first characteristic of Hill’s equations permits us to reduce the degree of freedom of particle motion and, hence, to reduce greatly the number of orbits to be pursued numerically. The second permits us to apply the result of an orbital calculation with particular m_p and a_0* to orbital motion with other arbitrary mass and heliocentric distance. 長文になりますが、どうかよろしくお願いします。

• この文章の和訳をお願いします。

When |b_i~| is relatively large (e.g., |b_i~|≧5), a particle passes through the region far from the Hill sphere of the planet without a significant influence of the gravity of the planet. As seen from Figs. 1(a) and (b), both the impact parameter, b_f~, and the eccentricity, e_f~, at the final stage are not so much different from those at the initial, i.e., a particle is hardly scattered in this case. When 3≦|b_i~|≦5, as seen from Fig. 1(a) and (b), a particle is scattered a little by the gravity of the planet and both |⊿b~| and |⊿e~| increase gradually with the decrease in |b_i~|, where ⊿b~=b_f~－b_i~ (3・1) and ⊿e~=e_f~－e_i~. 　　　 (3・2) よろしくお願いします。

• この文章の和訳を教えてください。

In our orbital calculations, the starting points are assigned in the form (see Eq. (7)) x_s=b_s-ecos(t_0-τ_s), y_s=y_0+2esin(t_0-τ_s), ・・・・・・・・・・・(21) z_s=isin(t_0-ω_s), where t_0 is the origin of the time, independent of τ_s and ω_s, and y_0 is the starting distance (in the y-direction) of guiding center. Table 1. The values of b_min* evaluated from Eq. (18). The values of numerically found b_min are also tabulated. よろしくお願いいたします。

• 和訳をお願いします。

The normalized Hill’s equations of the relative motion are described as (for definitions of coordinates, see PaperI) x’’=2y’+3x－3x/r^3, y’’=－2x’－3y/r^3, z’’=－z－3z/r^3, (6) where time is normalized by the Keplerian period Ω^-1 and length by the Hill radius ha_0*. よろしくお願いします。

• この文章の和訳をお願いします。

These come from the fact that for small values of b~ and μ, as considered here, Eqs.(2・1) and (2.2) are approximately symmetric with respect to the y-axis. Now, we will see features of collision of a particle with the planet. It should be noticed that, as mentioned in §2, the radius of the planet changes with the distance between the Sun and the planet is assigned. Here we will concentrate on the collision near the orbit of the present Earth. よろしくお願いします。

• この文章の和訳をしてくれませんか。

In the above, x, y, and z are the Hill coordinates and r is the distance between two particles. As shown in Paper I, a solution of Keplerian motion satisfies Eq. (6) if the mutual interaction term 3r/r^3 can be neglected. よろしくお願いします。