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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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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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    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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                    3. Results of the orbital calculations   In this paper, we describe our numerical results only for the special cases e_i~=0 and e_i~=4 in order to find the characteristic features of the two-body encounters of Keplerian particles. 3.1. For the case e_i~=0   In this case the orbital element, δ_i, loses its meaning because the particle orbits have no periheria and , hence, we can actually assign the initial condition by one parameter, b_i~. The orbital calculations are performed for about 3800 cases with various values of b_i~ from -10 to 10. よろしくお願いします。

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    In order to obtain the total collisional rate <Γ(e_1, i_1)>, we have to find <P(e,i)> numerically by computing orbits of relative motion between the protoplanet and the planetesimal. Since <P(e,i)> should be provided for wide ranges of e and i with a sufficient accuracy, we are obliged to compute a very large number of orbits. In practice, it is an important problem to find an efficient method for numerical computation. In the second paper (Nakazawa et al., 1989b, referred to as Paper II), we have studied the validity of the two-body approximation and found that within the sphere of the two-body approximation (hereafter referred to as the two-body sphere), the relative motion can be well described by a solution to the two-body problem: the sphere radius has been found to be r_cr=0.03(a_0*/1AU)^(-1/4)(ε/0.01)^(1/2). ・・・・・(13) Within the sphere, the nearest distance can be predicted with an accuracy εby the well-known formula of the two-body encounter. We can expect the above result to be useful to reduce computation time for obtaining <P(e,i)> numerically. よろしくお願いします。

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    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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    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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    3. Examples of orbital calculations To find efficient numerical procedures for obtaining <P(e ,i)>, we have made detailed orbital calculations for two typical cases, (e, i)=(0, 0) and (1, 0.5). The former is the simplest case with a single parameter b, and corresponds to that studied by Giuli (1968), Nishida (1983) and Petit and Hénon (1986). In the latter, the orbit changes with three parameters b, τ, and ω in a complicated manner. From this example, we can see the characteristic features of orbits in the three-dimensional case. In the present examples, it is supposed that a protoplanet with a mean mass density 3gcm^-3, orbits in the Earth’s region. Furthermore, we adopt 1% as the limiting accuracy criterion for use of the two-body approximation. These give 0.005 and 0.03 for the radius of the protoplanet and that of the two-body sphere, respectively (see Eqs. (12) and (13)). よろしくお願いします。

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    To avoid this difficulty, we consider the scale height to be (i+αr_G) rather than i, where α is a numerical factor; α must have a value of the order of 10 to be consistent with Eq. (35). For the requirements that in the limit of i=0, <P(e,i)>_2B has to naturally tend to <P(e,0)>_2B given by Eq. (28), we put the modified collisional rate in the two-body approximation to be <P(e,i)>_2B=Cπr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2(i+ατ_G))            (36) with C=((2/π)^2){E(k)(1-x)+2αE(√(3/4))x},                (37) where x is a variable which reduces to zero for i>>αr_G and to unity for i<<αr_G. The above equation reduces to Eq. (29) when i>>αr_G while it tends to the expression of the two-dimensional case (28) for i<<αr_G. Taking α to be 10 and x to be exp(-i/(αr_G)), <P(e,i)> scaled by Eq. (36) is shown in Fig. 17. Indeed, the modified <P(e,i)>_2B approximates <P(e,i)> within a factor of 5 in whole regions of the e-I plane, especially it is exact in the high energy limit (v→∞). However, two peaks remain at e≒1 and i≒3, which are closely related to the peculiar features of the three-body problem and hence cannot be reproduced by Eq. (36). Fig. 16a and b. Behaviors of r_min(i,b): a i=0, b i=2, 2.5, and 3.0. The level of the planetary radius (r_p=0.005) is denoted by a dashed line. Fig. 17. Contours of <P(e,i)> normalized by the modified <P(e,i)>_2B given by Eq. (36). Fig. 16a and b.↓ http://www.fastpic.jp/images.php?file=4940423993.jpg Fig. 17.↓ http://www.fastpic.jp/images.php?file=5825412982.jpg よろしくお願いします。

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    Fig.2. The orbit of relative motion with b=1.920, e=0 and i=0. The particle enters the Hill sphere (denoted by a circle in a fine line) of the protoplanet at origin. But it escapes away from the Hill sphere without close encounter.

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          Finally, we will add a comment on comparison of our result with those of Wetherill and Cox (1985). Wetherill and Cox examined three-dimensional calculation for a swarm of planetesimals with a special distribution, i.e., e_2 has one value and i_2 is distributed randomly between 0.3e_2 and 0.7e_2 (<i_2>=e_2/2) while e_1=i_1=0, which corresponds, in our notation of Eq. (9), to <n_2>={n_sδ(e_2-e)δ(i_2-i)/0.4π^2e(2/2)      for 0.3e_2<i_2<0.7e_2,    {0      otherwise.                   (38) Integrating <P(e,i)> with above <n_2> according to Eq. (9), we compare our results with theirs. Figure 18 shows that their results almost agree with ours (the slight quantitative difference may come from the difference in definition of the enhancement factor); but their results contain a large statistical uncertainty because they calculated only 10~35 collision orbits for each set of e and i while 100~6000 collision orbits were found in our calculation (see Table 4). Furthermore, our results are more general than theirs in the sense that their calculations are restricted to the special distribution of planetesimals as mentioned above, while the collisional rate for an arbitrary planetesimal distribution can be deduced from our results. 8. Concluding remarks Based on the efficient numerical procedures to find collision orbits developed in Sect. 2 to 4, we have evaluated numerically the collisional rate defined by Eq. (10). The results are summarized as follows: (i) the collisional rate <P(e,i)> is like that in the two-dimensional case for i≦0.1 (when e≦0.2) and i≦0.02/e (when e≧0.2), (ii) except for such two-dimensional region, <P(e,i)> is always enhanced by the solar gravity, (iii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, where <P(e,i)>_2B is the collisional rate in the two-body approximation, and (iv) there are two notable peaks in <P(e,i)>/<P(e,i)>_2B at e≒1 (i<1) and i≒3 (e<0.1); but the peak value is at most 4 to 5.          From the present numerical evaluation of <P(e,i)>, we have also found an approximate formula for <P(e,i)>, which can reproduce <P(e,i)> within a factor 5 but cannot express the peaks found at e≒1 (i<1) and i≒3 (e<0.1). These peaks are characteristic to the three-body problem. They are very important for the study of planetary growth, since they are closely related to the runaway growth of the protoplanet, as discussed by Wetherill and Cox (1985). This will be considered in the next paper (Ohtsuki and Ida, 1989), based on the results obtained in the present paper. Acknowledgements. Numerical calculations were made by HITAC M-680 of the Computer Center of the University of Tokyo. This work was supported by the Grant-in-Aid for Scientific Research on Priority Area (Nos. 62611006 and 63611006) of the Ministry of Education, Science and Culture of Japan. Fig. 18. Comparison of the enhancement factors with those of Wetherill and Cox (1985). The error bars in their results arise from a small number (10~35) of collision orbits which they found for each e. Our results are averaged by the distribution function which they used (see text). Fig. 18.↓ http://www.fastpic.jp/images.php?file=0990654048.jpg かなりの長文になりますが、どうかよろしくお願いします。

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    This result indicates that a limited reaction of the monopolist to changes in profits can stabilize the quantity produced. Onthe other hand turbulences in the market are generated by an overreaction. To shed some light on what really happens in themarket we employ a numerical analysis.