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3.2. Case of e=1.0 and i=0.5
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三次元の場合では、軌道は3のパラメーターb、τ_s、およびω_s(このセクションでは、私たちは、軌道計算の出発点について記述する添字「s」を省略するでしょう)が特徴です。そのb_maxのリコール、同等物からの<3.7(16)およびb_min>同等物からの1.3(18)現在の場合では、私たちは、最初に1.3と3.7の間の範囲の中でbを備えた軌道を検査します:位相空間(b、τ、ω)は、約22000のメッシュ(つまりb(1.3~3.7)の中の24、r(π~π)のうちの60および15ω(0~π))に分割されます。これらの軌道計算は、衝突軌道がないことを示します、どこで、b<1.5およびb>3.2:\b_maxとb_minは、3.7と1.3ではなく、3.2と1.5でそれぞれセットされます。前のサブセクション中の議論と平行に、私たちは、原始惑星と微惑星体の間の最小の分離r_minを考慮します。Fig.7では、最初の遭遇中のr_minの輪郭は、b=2.3(図7a)、2.8(b)および3.1(c)の3つのケース用のτ-ω図形中で説明されます。数値はそれぞれ5000の軌道の軌道計算から集計されます、つまり、τ-ω航空機は100(τの中で)×50(ωの中の)に分割されます。私たちは、図7aに最初に専念します。粗末に点のある地域で、どこで、r_min>1、粒子は、原始惑星の丘球体に入ることができません。そのような地方は私たちの関心を超えています。粒子が丘球体に入ることができる他の地方では、r_minは複雑なやり方でτとωに応じて変わります。の中で、特別、ポイント(τ、ω)=(0.24π、0.42π)の近くで、また(0.26π、0.06π)、r_minは、τ-ω図形中の小さなエリアにおいて徹底的に異なります。これらは無秩序なゾーンかもしれません。しかし、ほとんどすべての地方で、r_minは、τとωに応じて連続的に変わります。また、この意味で、軌道は規則的です。 素晴らしく点のある地方は、r_minが0.03未満(2つの身体の半径)になるものを示します。そのような軌道は無秩序なゾーンでの近接遭遇軌道と呼ばれるでしょう、通常のゾーンでのそれと比較されて、非常に小さい。 これは以前に到達するのと同じ結論です。
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Now, we shall concentrate on the collision orbits. Figure 5 illustrates the minimum separation distance r_min in the first encounter (solid curves), identical to that obtained by Petit and Hénon (1986). One sees immediately, that there are two different zones: the “regular” zones, in which r_min varies smoothly with a change of parameter b and the irregular (or “chaotic”) zones, where r_min changes greatly with tiny differences in the choice of b. The chaotic zones lie near b=1.93, 2.30 and 2.48, with very narrow ranges of b. In the regular zone, we find two broad bands of collision orbits around b=2.09 and 2.39. These collision bands were first found by Giuli (1968). The sum of width of the collision bands ⊿b is found to be about 0.098, if the planetary radius is 0.005. よろしくお願いします。
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In the chaotic zone, there are, of course, a great number of discrete collision orbits. Minimum separation distance in the chaotic zone near b=1.93 is enlarged in Fig.6, which is obtained from the calculation of 3000 orbits with b between 1.926 and 1.932. Even in this enlarged figure, r_min varies violently with b. Although the chaotic zones are not sufficiently resolved in our present study, the phase space occupied by collision orbits in the chaotic zones is much smaller than that in the regular collision bands. Even if all orbits in the chaotic zone are collisional, their contribution to the collision rate is less than 4% of the total: the width in b=2.30 and 2.48, we also found that the total width is much smaller than 0.001. This implies that in the evaluation of <P(e, i)>, we can neglect the contribution of collision orbits in the chaotic zones. These are n-recurrent collision orbits in the regular zones. Of these, 2-recurrent collision orbits are most important. The collisional band composed of them is found near b=2.34. Its width ⊿b is about 0.011, and the contribution to the collision is as large as 15%. No.3- and more –recurrent collision orbits were observed in regular zones. They were found only in the chaotic zones and, hence, can be neglected. 長いですが、よろしくお願いします。
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Fig. 3a and b. An example of recurrent non-collision orbits. The orbit with b=2.4784, e=0, and i=0 is illustrated. To see the orbital behavior near the protoplanet, the central region is enlarged in b; the circle shows the sphere of the two-body approximation and the small one the protoplanet. Fig. 4a and b. Same as Fig. 3 but b=2.341, e=0, and i=0. This is an example of recurrent collision orbits. Fig.5. Minimum separation distance r_min between the protoplanet and a planetesimal in the case of (e, i)=(0, 0). By solid curves, r_min in the first encounter is illustrated as a function of b. The level of protoplanetary radius is shown by a thin dashed line. The collision band in the second encounter orbits around b=2.34 is also shown by dashed curves. Fig. 6. Minimum distance r_min in the chaotic zone near b=1.93 for the case of (e, i)=(0, 0); it changes violently with b. Fig. 7a-c. Contours of minimum separation distance r_min in the first encounter for the case of (e, i)=(1.0, 0.5); b=2.3(a), 2.8(b), and 3.1 (c). Contours are drawn in terms of log_10 (r_min) and the contour interval is 0.5. Regions where r_min>1 are marked by coarse dots. Particles in these regions cannot enter the Hill sphere of the protoplanet. Fine dots denotes regions where r_min is smaller than the radius r_cr of the sphere of the two-body approximation (r_cr=0.03; log_10(r_cr)=-1.52). Fig. 3a and b. および Fig. 4a and b. ↓ http://www.fastpic.jp/images.php?file=3994206860.jpg Fig.5. および Fig. 7a-c. ↓ http://www.fastpic.jp/images.php?file=2041732569.jpg Fig. 6. ↓ http://www.fastpic.jp/images.php?file=2217998690.jpg お手数ですが、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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Figure 8 shows r_min in the second encounter for b=2.8. In this case, there are four zones of close-encounter orbit in the τ-ω diagram. Comparing Fig.8 with Fig. 7b, the total area occupied by the recurrent close-encounter orbits (the dotted regions in Fig. 8) is smaller than that in the first encounter but not small enough to be neglected. Collision orbits belong necessarily to close-encounter orbits. Consequently, to find collision orbits, we subdivided the τ-ω phase space of close-encounter orbits (i.e., the finely dotted regions in Fig.7) more densely (mesh width being as small as 0.002π in τ) and pursued orbits for each set of τ and ω. Furthermore, as the phase volume of τ and ω occupied by collision orbits, we evaluated a “differential” collisional rate <p(e, i, b)> given by <p(e, i, b)>=(1/(2π)^2)∫p_col (e, i, b, τ, ω)dτdω. (24) Here, we calculated <p(e, i, b)> separately for 1-, 2-, and more recurrent orbits. The results are shown in Fig. 9, from which we can see that 2-recurrent collision orbits exist for relatively large b, and n-current (n≧3) ones exist only for b≒b_max. That is, the recurrent collision orbits appear only in cases of relatively low energy. From Eq. (10), we have <P(e, i)>=∫【-∞→∞】(3/2)|b|<p(e, i, b)>db. (25) Using evaluated values of <p(e, i, b)> for various b, we finally obtain <P(e, i)>=0.114 for (e, i)=(1.0, 0.5); the contribution of 2-recurrent orbits is 5%, and that of 3- and more-recurrent orbits is less than 1%. For this case (e=1.0 and i=0.5), we observed 874 collision orbits. The statistical error in evaluating <P(e, i)> is therefore presumed to be of the order of 4%. Since the contribution of 3- and more-recurrent orbits is within the statistical fluctuation, it can be neglected. よろしくお願いします。
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2. Initial conditions for orbital integration To obtain <P(e, i)>, we must numerically compute a number of orbits of planetesimals with various values of b, τ, and ω for each set of (e, i) and then examine whether they collide with the protoplanet or not. In this section, we consider the ranges of b, τ, and ω to be assigned in orbital calculations, and give initial conditions for orbital integration. One can see in Eq. (6) that Hill’s equations are invariant under the transformation of z→-z and that of x→-x and y→-y; on the other hand, a solution to Hill’s equations is described by Eq. (7). From the above two characteristics, it follows that it is sufficient to examine only cases where 0≦ω≦π and b≧0. Furthermore, we are not interested in orbits with a very small b or a very large b; an orbit with a very small b bends greatly and returns backward like a horse shoe ( Petit and Hénon, 1986; Nishida, 1983 ), while that with a very large b passes by without any appreciable change in its orbital element. どうか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 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。
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Furthermore, ρ is the density of the planet and is taken to be 4.45g/cm^3 according to Lewis. The values of r_p/h are tabulated in Table I for various regions from the Sun. It is to be noticed that the planetary radius decreases as the increase in the distance from the Sun in the system of units adopted here. Orbital calculations are performed by means of the 4th order Runge-Kutter-Gill method with an accuracy of double precision. よろしくお願いします。
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Now, the orbital elements (e, i, b, τ, ω) appearing in Eq. (14) are those at infinity. However, we cannot carry out an orbital calculation from infinity. In practice, we start to compute orbits from a sufficiently large but finite distance. Hence, we have to find the relation between orbital elements at infinity and those at a starting point. For b, it is readily done by Eq. (17); denoting quantities at a starting point of orbital calculations by subscripts “s”, we obtain b_s=(b^2-8/r_s)^(1/2), ・・・・・・・・(19) where we assumed e_s^2=e^2 and i_s^2=i^2 in the same manner as earlier. Hénon and Petit (1986) obtained a more accurate and complicated expression of b_s in the two-dimensional case. However, since we are now interested only in the averaged collisional rate but not detailed behavior of orbital motion, it is sufficient to use the simple relation (19). よろしくお願いします。
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