Numerical Procedures for Obtaining <P(e, i)>

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  • We comment on the simplifications made before describing our numerical procedures.
  • We have developed efficient numerical procedures for obtaining <P(e, i)>.
  • The results of the numerical procedures are presented in the preceding sections.
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この英文の和訳をお願いします。

　 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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  • ddeana
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数値計算手法を詳細に説明する前に、上記簡略化について言及する。eが1.0でiが0.5の例でわかるように、n回帰性（nは3と同等かそれよりも大きい数字）衝突軌道は、衝突速度にわずか1％かそれ以下の影響しか与えない。その一方で、別なeとiを用いた計算ではn回帰性（nは3と同等かそれよりも大きい数字）衝突軌道の影響度合いは、eが1とほぼ等しい場合もっとも大きくなるとされている。つまり(1)の簡略化を用いることにより生じる <P(e, i)>の誤差が1%かそれ以下程度ということだ。二体近似の適応性に関しては、第二論文において、軌道は方程式(13）で示された半径をもつ二体球内での二体公式により十分に説明されている。 (2)の簡略化による特筆すべき<P(e, i)>の誤差はない。(3)の簡略化については最後の章の議論で分る。 上記簡略化を用いて効率的に<P(e, i)>得るための数値計算手法を開発した。そのフローチャートは図10に示した通りである。軌道要素（e, i, b, τ_s, ω_s）の初期値を選び、(21)と(22)の方程式で得た出発点から、通常の4段階ルンゲクッタ法によりヒルの方程式(6)を数値的に計算することから始める。数値積分のすべてのタイムステップ（※1）において距離rはチェックされる。もし粒子が原始惑星に近づいた後に惑星から十分な距離まで飛散、すなわち例えば yの絶対値がy_0+2e(26）よりも大きければ 軌道計算は停止される。もし粒子が原始惑星に近づき、二体球表面を横切ったならば（たとえば rがr_crと同等かそれよりも小さかった時など）、衝突が起こるか否かの予測に二体公式を用いる。最初の遭遇で衝突が起きない場合、ヒル方程式の数値積分を継続する。二体球に入る粒子は必然的に球から出て行くので（第二論文参照のこと）、粒子には複数の選択肢が可能である。ひとつは方程式(26)に当てはまる距離へと離れること、もうひとつは二体球表面を再度横切ることである。前者の場合軌道は非衝突性であると見なされ、計算は停止する。後者の場合は以前と同じように二体公式を用いて、衝突の発生をチェックし軌道計算を終了する。 このように開発した数値計算手法を用いて、多くの (e, i)の組み合わせについて<P(e, i)>を求めることとした。結果は前節で示してある。 図10：衝突軌道を発見する為の軌道計算のフローチャート ※1：時間で微分された式を数値積分する際の時間刻みのこと

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    5. Normalization of collisional rate First, we introduce an enhancement factor defined as the ratio of the collisional rate <P(e, i)> to that in the two-body approximation <P(e, i)>_2B: R(e, i)= <P(e, i)>/ <P(e, i)>_2B (27) The factor R(e, i) gives a measure of the collisional rate enhancement due to the effect of solar gravity. In the two-dimensional case, <P(e,0)> is given by Eq. (11) while <P(e, 0)>_2B is defined by <P(e,0)>_2B=(2/π)E(√(3/4))ρ_(2D)v, (28) where E(k) is the second kind complete elliptic integral and ρ_(2D)v is given by Eq. (3) with <e(2/2)> replaced by e^2 (note that the units are changed, i.e., v=(e^2+i^2)^(1/2) and Gm_p=3). The numerical coefficient 2E(k)/π(=0.77) is introduced so that the collisional rate <P(e,0)>_2B coincides with <P(e,0)> in the high energy limit, v→∞ (see Paper I and Greenzweig and Lissauer, 1989). In the three-dimensional case, <P(e,i)> is given by Eq. (10) while <P(e, i)>_2B by Eq. (1) with <e(2/2) > and <i(2/2)> replaced, respectively, by e^2 and i^2. It should be noticed that <P(e,i)> has the dimension per unit surface number density n_s. Then, we define <P(e,i)>_2B by nσv/n_s; (n_s/n) corresponds to twice the scale height (in the z-direction) of a swarm of planetesimals. Usually, the scale height is taken to be i*a_0* (i.e., i, in the units here). As in the two-dimensional case, we require that <P(e,i)>_2B must coincide with <P(e,i)> in the high energy limit. Then, by introducing the numerical coefficient (2/π)^2E(k) (=0.49~0.64) (see Paper I), we have <P(e,i)>_2B=(2/π)^2E(k)πr_p^2{1+(6/(r_p(e^2+i^2)) }(e^2+i^2)^(1/2)/(2i), (29) with k^2=3e^2/4(e^2+i^2). (30) 6. The collisional rate for the two-dimensional case In this section, we concentrate on the collisional rate for the two-dimensional case where i=0. In this case, the small degrees of freedom of relative motion allow us to investigate in detail behaviors of orbital motion: it is sufficient to find collision orbits only in the b-τ two-dimensional phase space for each e, as seen in Eq. (11). 長文ですが、よろしくお願いします。

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    As mentioned above, there are two peaks in R(e,i) in the e-i diagram: one is at e≒1 (i<1) and the other at i≒3 (e<0.1). The former corresponds directly to the peak in R(e,0) at e≒1 found in the two-dimensional case. The latter is due to the peculiar nature to the three-dimensional case, as understood in the following way. Let us introduce r_min (i, b,ω) in the case of e=0, which is the minimum distance during encounter between the protoplanet and a planetesimal with orbital elements i, b, and ω. In Fig. 16, r_min(i,b)=min_ω{r_min(i,b,ω)} is plotted as a function of b for various i, where r_min<r_p (=0.005) means “collision”; there are two main collision bands at b≒2.1 and 2.4 for i=0. For i≦2, these bands still exist, shifting slightly to small b. This shift is because a planetesimal feels less gravitational attracting force of the protoplanet as i increases. As i increases further, the bands approach each other, and finally coalesce into one large collision band at i≒3.0; this large band vanish when i≧4. In this way, the peculiar orbital behavior of three-body problem makes the peak at i≒3 (e<0.1). Though there are the peaks in R(e,i), the peak values are not so large: at most it is as large as 5. This shows that the collisional rate is well described by that of the two-body approximation <P(e,i)>_2B except for in the vicinity of v, i→0 if we neglect a difference of a factor of 5. Now we propose a modified form of <P(e,i)>_2B which well approximates the calculated collisional rate even in the limit of v, i→0. We find in Fig.14 that <P(e,i)> is almost independent of i, i.e., it behaves two-dimensionally for i≦{0.1 (when e≦0.2), {0.02/e (when e≧0.2). (35) This transition from three-dimensional behavior to two-dimensional behavior comes from the fact that the isotropy of direction of incident particles breaks down for the case of very small i (the expression <P(e,i)>_2B given by Eq. (29) assumes the isotropy). In other words, as an order of magnitude, the scale height of planetesimals becomes smaller than the gravitational radius r_G=σ_2D/2 (σ_2D given by Eq. (3)) and the number density of planetesimals cannot be uniform within a slab with a thickness σ_2D for small i. Table 4. Numbers of three-dimensional collision events found by orbital calculations for the representative sets of e and i. In the table r_p is the radius of the protoplanet. Table 5. The three-dimensional collision rate <P(e,i)> for the case of r_p=0.005 (r_p being the protoplanetary radius), together with two-dimensional <P(e,i=0)> Fig. 14. Contours of the evaluated <P(e,i)>, drawn in terms of log_10<P(e,i)> Fig. 15. Contours of the enhancement factor R(e,i) Table 4.↓ http://www.fastpic.jp/images.php?file=1484661557.jpg Table 5.↓ http://www.fastpic.jp/images.php?file=6760884829.jpg Fig. 14. ＆Fig. 15.↓ http://www.fastpic.jp/images.php?file=8798441290.jpg 長文になりますが、よろしくお願いします。

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