Numerical Procedures for Obtaining P(e, i)

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  • This article presents numerical procedures for efficiently obtaining P(e, i) based on the framework of Hill's equations.
  • The procedures involve reducing the degrees of freedom of particle motion and simplifying the computation process to avoid difficulties.
  • Specific simplifications include neglecting certain orbits, using a two-body formula for collision determination, and omitting computations in certain regions.
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この文章の和訳を教えてください。

4. Numerical procedures for obtaining <P(e, i)> Based on the results in Sect. 3, we develop numerical procedures for efficiently obtaining <P(e, i)>. By using the framework of Hill’s equations, one can reduce the degrees of freedom of particle motion, as described in Paper I. Nevertheless, they are still too numerous to compute orbits densely and uniformly over the five-dimensional phase space (e, i, b, τ_s, ω_s) of initial conditions. Furthermore, we frequently find orbits which comes very close to the protoplanet or which revolve complicatedly many times around the protoplanet. In these cases, an enormously long computation time is needed to calculate them. To avoid these difficulties, we introduce the following three simplifications: (i) We neglect n-recurrent orbits in evaluating <P(e, i)> when n≧3; that is, we stop an orbital calculation if the particle does not collide with the protoplanet even after two close encounters. (ii) When the planetesimal crosses the sphere surface of the two-body approximation, the two-body formula determines whether or not a collision occurs. (iii)We omit orbital computations in some regions of τ_s, which depend on a given set of e, i, and b; it is found empirically that there are no collision orbits in these regions. よろしくお願いします。

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  • ddeana
  • ベストアンサー率74% (2976/4019)
回答No.1

4.<P(e, i)>を得る為の数値計算手法 3章における結果に基づき、効率的に<P(e, i)>得るための数値計算手法を開発する。論文Iで述べているように、ヒル方程式の枠組みを使うことにより粒子運動の自由度を縮小することが可能である。にもかかわらず、それらは初期条件の5次元位相空間（e, i, b, τ_s, ω_s）上で軌道を緻密で一様に計算するには、いまだ数が多すぎるのである。加えて、原始惑星の非常に近くまで接近するか、原始惑星の周りを何度も複雑に周回する軌道が頻繁にみつかる。こうしたケースではそれらを計算するのにコンピューターを使った尋常ではない長さの算出時間が必要となるのである。 これらの問題を避ける為に、次なる3つの単純化を紹介する。 (1)nが3と同等かそれよりも大きい場合、<P(e, i)>の評価において、n回帰性軌道を無視する。つまり、2つの接近遭遇後であっても粒子が原始惑星と衝突しなければ軌道計算をやめるということである。 (2)微惑星が、二体近似の球体表面を横切る時、衝突が起こるか否かにかかわらず２体公式を決定する。 (3)特定のe, i, そしてbに依存するτ_sのいくつかの領域での軌道計算を省略する。なぜならこうした領域に衝突軌道がないことが実験的にわかっているからだ。

mamomo3
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お礼

どうもありがとうございました。 感謝いたします。

その他の回答 (1)

  • ddeana
  • ベストアンサー率74% (2976/4019)
回答No.2

ddeanaです。前回答の一部、simplificationの訳を「簡略化」に変更いたします。 理由：計算式における時間の短縮化などを念頭にしたものなので、「単純」よりも複雑な行程を減らす「簡略」の方が適していると考えた為。

mamomo3
質問者

お礼

了解いたしました。 補足いただきまして、ありがとうございます。

  1. カテゴリ
  2. 学問・教育
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  4. 英語

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    3.1. Case of e=0 and i=0 We have first calculated 6000 orbits in the parameter range of b from b_min=1.9 to b_max=2.5 at intervals of 0.0001. It is already known from previous studies (Nishida, and Petit and Hénon) that no collision orbits exist outside this region. 　　　The orbits vary in a complicated way with the value of parameter b (see Petit and Hénon, 1986). In spite of the complex behavior of the orbits, we can classify them in terms of the number of encounters with the two-body sphere, from the standpoint of finding collision orbits. The classes are: (a) non-encounter orbit, (b) n-recurrent non-collision orbit, and (c) n-recurrent collision orbit, where the term “n-recurrent orbit” means the particle encounters n-times with the two-body sphere. That is, n-recurrent non-collision (or collision) orbits are those which fly off to infinity (or collide with the protoplanet ) after n-times encounters with the two-body sphere, while non-recurrent orbits are those which fly off without penetrating the two-body sphere. Examples of orbits in the classes (a), (b), and (c) are illustrated in Figs. 2,3, and 4, respectively. The above classification of orbits will be utilized for developing numerical procedures for obtaining <P(e, i)>, as described in the next section. 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。

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  • 和訳してください。よろしくお願いします。

    In the present paper, we first describe initial conditions for orbital calculations in the framework of Hill’s equations (Sect.2). In Sect.3, orbital calculations are made for particular sets of e and i, i.e., (e,i)=(0,0) and (1,0.5). Based on these calculations and previous works: Papers I and II, Nishida (1983), Hénon and Petit (1986), and Petit and Hénon (1986), we develop an efficient numerical procedure obtaining <P(e, i)> (Sect. 4).According to this procedure, we systemically integrate the orbits of relative motion, to find collision orbits. Furthermore, compiling results of these orbital calculations, we find <P(e, i)> for various sets of e and i and compare them with those in the two-body approximation <P(e, i)>_2B (Sects. 5 to 7). The results will be compared with those of Nishida (1983) and Wetherill and Cox (1985) who also studied the collisional rate taking account of the effect of solar gravity (but their studies were restricted, as mentioned later).

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    　 Before a detailed description of our numerical procedures, we comment on above simplifications. As seen in an example of e=1.0 and i=0.5, n-recurrent (n≧3) collision orbits contribute by only 1% or less to the collision rate. On the one hand, from calculations with other e and i, the degree of contribution by n-recurrent (n≧3) collision orbits is found to be largest in the case e≒1. Hence, the error in <P(e, i)> introduced by simplification (i) is of the order of 1% or less. As for the applicability of the two-body approximation, we have confirmed in PaperII that the orbit are well described by the two-body formula inside the two-body sphere, whose radius is given by Eq. (13). No appreciable error in <P(e, i)> comes from simplification (ii). Simplification (iii) follows the discussion in the last section. Using above simplifications, we have developed numerical procedures for obtaining <P(e, i)> efficiently; their flow chart is illustrated in Fig. 10. Choosing initial values of orbital elements (e, i, b, τ_s, ω_s), we start to compute numerically Hill’s equations (6) by an ordinary fourth-order Runge-Kutta method from a starting point given by Eqs. (21) and (22). The distance r is checked at every time step of the numerical integration. If the particle flies off to a sufficient distance from the protoplanet after approaching it, i.e., if |y|>y_0+2e, 　　　　　　　　　　　　　　　　　　　 (26) then, the orbital computation is stopped. If a particle approaches the protoplanet and crosses the two-body sphere surface, i.e., if r≦r_cr, the two-body formula is employed to predict whether or not a collision occurs. When no collision occurs at the first encounter, the numerical integration of Hill’s equations is continued. Since a particle which enters the two-body sphere inevitably escapes from the sphere (see Paper II), the particle follows alternatives: one is that it departs to such a distance that Eq. (26) is satisfied, and the other is that it crosses the two-body sphere surface again. In the former case, we stop the computation,　considering that the orbit is non-collisional. In the latter case, the occurrence of collision is checked in the same way as earlier by means of the two-body formula, and the orbital calculation is terminated. Using the numerical procedures developed in this way, we obtain <P(e, i)> in many sets of (e, i); the results are presented in the preceeding sections. Fig. 10. Flow chart of orbital calculation for finding collision orbits. Fig. 10. 拡大画像↓ http://www.fastpic.jp/images.php?file=2113240192.jpg 長文になりますが、よろしくお願いします。

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