質問の要約
- Fig.10のe_i~=4における現在の地球の軌道付近の衝突確率P_c(b_i~)を調査しました。
- Fig.11のe_i~=0および4の両方のケースにおける軌道計算から得られた係数C_Kを調査しました。
- Table Iでは、太陽からの様々な領域におけるケプラー粒子の衝突率の比率γを示しています。また、式(4・6)によって定義された係数C_Kと惑星の半径γ_p（単位：h）も表にまとめています。
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Fig.10. The collision probability P_c(b_i~) for e_i~=4 at the region near the present Earth’s orbit. ↓Fig.10. http://www.fastpic.jp/images.php?file=3466353586.jpg Fig.11. The coefficient C_K obtained from our orbital calculations for both cases e_i~=0 and 4. Each mark indicates the value of C_K at the position of each present planet. Solidlines show C_K∝R^－(1/2). ↓Fig.11. http://www.fastpic.jp/images.php?file=4992926857.jpg Table I. The ratio, γ, of the collisional rate of Keplerian particles to that of free space particles for the various regions from the Sun. The coefficient C_K, defined by Eq. (4・6), and the planetary radius, γ_p, in units of h are also tabulated. ↓Table I. http://www.fastpic.jp/images.php?file=9315075324.jpg For e_i~=0, the orbital element, δ_i, loses its meaning as mentioned in §3.1 and only impact parameter b_i~ determines whether a particle collides with the planet or not; i.e., the collision probability P_c(b_i~) is unity or zero. 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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以前の回答同様、引用URL部分は割愛いたしました。 図10：現在の地球軌道近くの領域で、 e_i~が4の場合の衝突確率 P_c(b_i~) 図11：e_i~が0と4両方の場合での、我々の軌道計算から導き出した係数 C_K。各マークは、その時々の惑星位置でのC_K値を示している。実線はC_K∝R^－(1/2)を表す。 表1：太陽からのさまざまな領域における、自由空間粒子の衝突頻度とケプラー粒子の衝突頻度γの比率。hを単位として、方程式(4・6)によって定義付けされた係数C_Kと惑星の半径 γ_pもまた示されている。 3章の1で述べたとおり、e_i~が0の時、軌道要素δ_iはその意味を成さず、衝撃パラメーターb_i~だけが粒子が惑星に衝突するかどうかを決定づける。すなわち、衝突確率P_c(b_i~)は乗法単位元となる数の1かゼロである。
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Table 3. Numbers of two-dimensional collision orbits (i=0) found by our orbital calculations. Only four cases of e are tabulated as examples. The numbers of collision orbits decrease with the decrease in the planetary radius r_p. Fig.11. The two-dimensional enhancement factor R(e,0) as a function of e for r_p=0.005, 0.001, and 0.0002. The enhancement factor depends rather weakly on r_p. Fig.12. The collisional flux F(e,E) defined by Eq. (32) as a function of the Jacobi energy E for e=0, 0.5, 1.0, and 2.0. ↓Table 3. http://www.fastpic.jp/images.php?file=0416143752.jpg ↓Fig.11. http://www.fastpic.jp/images.php?file=9556242912.jpg ↓Fig.12. http://www.fastpic.jp/images.php?file=1594984078.jpg よろしくお願いします。
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Fig.4. Magnified figures of one of the discontinuous bands in 1.915<b_i~<1.925 in Figs. 3(a) and (b). There are many fine discontinuous bands in itself. Fig.5. Examples of particle orbits which belong to the discontinuous bands in Fig.4. (a) and (b) are of particles which escape from the Hill sphere through the gate of L_1 and through that of L_2, respectively. The values of b_i~ are 1.91894 (a) and 1.91898 (b). Fig.6. Examples of purely Keplerian orbits with e_i~=4 for b_i~=2 and b_i~=10. The former curls (|b_i~|<4(e_i~/3) and the latter waves (|b_i~|>4(e_i~/3)). The values of δ* of both orbits are zero. ↓Fig.4. http://www.fastpic.jp/images.php?file=0045805708.jpg ↓Fig.5. http://www.fastpic.jp/images.php?file=9845810934.jpg ↓Fig.6. http://www.fastpic.jp/images.php?file=0162323295.jpg どうか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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Fig. 8. Contours of minimum separation distance r_min in the second encounter for the case of (e, i, b)=(1.0, 0.5, 2.8). Dots have the same meanings as those in Fig. 7. Fig. 9. The “Differential” collisional rate <p(e, i, b)> (defined by Eq. (24)) is plotted as a function of b in the case of (e, i)=(1.0, 0.5). Fig. 8.↓ http://www.fastpic.jp/images.php?file=4403322728.jpg Fig. 9.↓ http://www.fastpic.jp/images.php?file=9905093670.jpg よろしくお願いします。
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In order to see qualitatively the collision frequency, we will define the collision probability, P_c(b_i~) as follows. For a fixed value of b_i~, orbital calculations are made with N's different value of δ*_i, which are chosen so as to devide 2π(from –π to π) by equal intervals. Let n_c be the number of collision orbits, then P(b_i~, N)=n_c/N gives the probability of collision in the limit N→∞. In practice, however, we can evaluate the value of P_c(b_i~) approximately, but in sufficient accuracy in use, by the following procedures; i.e., at first for N=N_1(e.g., N_1=1000), P(b_i~,N_1) is found and, next, for N=2N_1, P(b_i~,N) is reevaluated. お手数ですが、どうかよろしくお願いします。
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For regions where |b_i~|≦5.7, we have the same pattern in ⊿b~ and ⊿e~ as those for the case e_i~=0; i.e., they jump discontinuously by a bit change of δ* (see Fig.8). In the same manner as described for e_i~=0 we have tried to decompose up these discontinuous bands. Unfortunately we cannot succeed either in the complete decomposition, because there seems to exist an infinite series of the fine structure. But contributions of the fine structure to the average values, such as <⊿b~> and <⊿e~> are found to be negligible too. In Figs.9 (a) and (b) the average values of <⊿b~> and <⊿e~> are illustrated as a function of b_i~, where <⊿b~> and <⊿e~> are defined as <⊿b~>=(1/2π)∫⊿b~dδ* (3.8) and <⊿e~>=(1/2π)∫⊿e~dδ*, (3.9) respectively. Fig.8. Some discontinuous bands of the change of the impact parameter for continuous variation of δ* with b_i~=3.0 and with e_i~4. Fig.9. (a) The change of the impact parameter, <⊿b~> and (b) the change of the eccentricity, <⊿e~> averaged over the parameter δ* for various values of b_i~ between －10 and 10 with e_i~=4. http://www.fastpic.jp/images.php?file=2049108297.jpg ↑Fig.8. http://www.fastpic.jp/images.php?file=5493230572.jpg7 ↑Fig.9. (a) and (b) 長文ですが、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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If |P(b_i~,N_1)-P(b_i~,2N_1)|≦0.05×P(b_i~,N_1), then P(b_i~,N_1) (or P(b_i~,2N_1)) is regarded as the collision probability, P_c(b_i~). In Fig.10, P_c(b_i~) is illustrated as a function of b_i~ at the region near the orbit of the present Earth for the case e_i~=4. As seen from this figure, a particle whose impact parameter, b_i~, lies between 1.5 and 5.5, collides with the planet with certain probability. Especially, for |b_i~|≒2 collisions occur frequently. よろしくお願いいたします。
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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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Now, Figs. 3(a) and (b) indicate that there seems to be several discontinuous spikes in the range of |b_i~| between 1.8 and 2.6, which herefter is called the discontinuous band. Fig.3. (a) The change of the impact parameter, Δb~ and (b) that of the eccentricity, Δe~ for particles with 1.8＜b_i~＜2.6 and with e_i~＝0. Marks on the upper abscissa indicate that with these initial b_i~ particles collide with the planet. お手数ですがよろしくお願いいたします。
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