３D衝突イベントの数と３D衝突率<P(e,i)>の結果| OKWAVE

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　　　　The numbers of collision orbits found in the present calculations are shown in Table 4 for the representative sets of (e,i). From these numbers we can expect the magnitude of statistical error in the evaluation of <P(e,i)> to be a few percent for small e, i and within 10% for large e, i for r_p=0.005 are shown in Table 5, together with those of the two-dimensional case. Interpolating these values, we have obtained the contour of <P(e,i)> and R(e,i) on the e-I plane. They are shown in Figs. 14 and 15. From Fig. 15 we can read out the general properties of the collisional rate in the three-dimensional case: (i) <P(e,i)> is enhanced over <P(e,i)>_2B except for small e and i, (ii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, and (iii) there are two peaks in R(e,i) near regions where e≒1 (i<1) and where i≒3 (e<0.1): the peak value is at most as large as 5. 　　　　 In the vicinity of small v(=(e^2+i^2)^(1/2)) and i, R(e,i) rapidly reduces to zero. This is due to a singularity of <P(e,i)>_2B at v=0 and i=0 in the ordinary expression given by Eq. (29) and hence unphysical; the behavior of collisional rate in the vicinity of small v and i will be discussed in detail later. Thus, we are able to assert, more strongly, the property (i) mentioned in the last paragraph: that is, solar gravity always enhances the collisional rate over that of the two-body approximation. 　　　　 One of the remarkable features of R(e,i) found in Fig. 15 is the property (ii). That is, the collisional rate between Keplerian particles is well described by the two-body approximation, for (e^2+i^2)^(1/2)≧4. This is corresponding to the two-dimensional result that R(e,0)≒1 for e≧4. よろしくお願いします。

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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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In Fig.13, we compare our results with those of Nishiida (1983) and Wetherill and Cox (1985). Nishida studied the collision probability in the two-dimensional problem for the two cases: e=0 and 4. For the case of e=0, his result (renormalized so as to coincide with our present definition) agrees accurately with ours. But for e=4, his collisional rate is about 1.5 times as large as ours; it seems that the discrepancy comes from the fact that he did not try to compute a sufficient number of orbits for e=4, thus introducing a relatively large statistical error. The results of Wetherill and Cox are summarized in terms of v/v_e where v is the relative velocity at infinity and v_e the escape velocity from the protoplanet, while our results are in terms of e and i. Therefore we cannot compare our results exactly with theirs. If we adopt Eq. (2) as the relative velocity, we have (of course, i=0 in this case) (e^2+i^2)^(1/2)≒34(ρ/3gcm^-3)^(1/6)(a_0*/1AU)^(1/2)(v/v_e). (34) According to Eq. (34), their results are rediscribed in Fig.13. From this figure it follows that their results almost coincide with ours within a statistical uncertainty of their evaluation. 7. The collisional rate for the three-dimensional case Now, we take up a general case where i≠0. In this case, we selected 67 sets of (e,i), covering regions of 0.01≦i≦4 and 0≦e≦4 in the e-i diagram, and calculated a number of orbits with various b, τ,and ω for each set of (e,i). We evaluated R(e,i) for r_p=0.001 and 0.005 (for r_p=0.0002 we have not obtained a sufficient number of collision orbits), and found again its weak dependence on r_p (except for singular points, e.g., (e,i)=(0,3.0)) for such values of r_p. Hence almost all results of calculations will be presented for r_p=0.005 (i.e., at the Earth orbit) here. Fig.13. Comparison of the two-dimensional enhancement factor R(e,0) with those of Nishida (1983) and those of Wetherill and Cox (1985).Their results are renormalized so as to coincide with our definition of R(e,0). 長文ですが、よろしくお願いします。

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　　　　We evaluated <P(e, 0)> for 12 cases of e between 0 and 6: e=0.0, 0.01, 0.1, 0.5, 0.75, 0.9, 1.0, 1.2, 1.5, 2.0, 4.0, and 6.0. As for r_p, we considered three cases: r_p=0.005, 0.001, and 0.0002. These are representative values of radii of protoplanets at the Earth, Jupiter, and Neptune orbits regions, respectively. The numbers of collision orbits found by our orbital calculation are shown in Table 3 for representative values of e. From Table 3 we can expect the statistical errors in the evaluated collisional rate to be within 5% for the cases of e≦1.5 and within 8% for e=4 and 6; they are smaller than that of the previous studies by Nishida (1983) and by Wetherill and Cox (1985). 　　　The calculated collisional rate is summarized in terms of the enhancement factor defined by Eq. (27) and shown in Fig.11, as a function of e and r_p. From Fig.11 one can see that the collisional rate is always enhanced by the effect of solar gravity, compared with that of the two-body approximation <P(e,0)>_2B. In particular, in regions where e≦1, R(e,0) is almost independent of e, having a value as large as 3. At e≦1, R(e,0) has a notable peak beyond which the enhancement factor decreases gradually with increasing e. For large values of e, i.e., e≧4, <P(e,0)> tends rapidly to <P(e,0)>_2B. As seen in the next section, we will find a similar dependence on e even in the three-dimensional case (i≠0) as long as we are concerned with cases where i≦2. お手数ですが、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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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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　 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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　　　　　　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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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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The second feature seen from Fig.11 is that the profile of R(e,0) does not depend significantly on r_p (for r_p=0.005 to 0.0002). Only an exception is found near e≒1, but this is, in some sense, a singular point in R(e,0), which appears in a narrow region around e≒1 ( in fact, for e=0.9 and 1.2, there is no appreciable difference between r_p=0.005 and 0.0002). Thus, neglecting such fine structures in R(e,0), we can conclude that R(e,0) does depend very weakly on r_p. In other words, the dependence on r_p of <P(e,0)> is well approximated by that of <P(e,0)>_2B given by Eq. (28). Now, we will phenomenalogically show what physical quantity is related to the peak at e≒1. We introduce the collisional flux F(e,E) for orbits with e and E, where E is the Jacobi energy given by (see Eq. (15)) E=e^2/2-(3b^2)/8+9/2. (31) The collisional flux F(e,E) is defined by F(e,E)=(2/π)∫【‐π→π】p_col(e,i=0, b(E), τ)dτ. (32) From Eqs. (11) and (31), we obtain <P(e,0)>=∫F(e,E)dE. (33) In Fig.12, F(e,E) is plotted as a function of E for the cases of e=0, 0.5, 1.0, and 2.0. We can see from this figure that in the case of e=1 a large fraction of low energy planetesimals contributes to the collisional rate compared to other cases (even to the cases with e<1). In general, in the case of high energy a solution for the three-body problem can be well described by the two-body approximation: in other words, in the case of low energy a large difference would exist between a solution for the three-body problem and that in the two-body approximation. As shown before, this difference appears as an enhancement of the collisional rate. Thereby an enhancement factor peak is formed at e≒1 where a large fraction of low-energy planetesimals contributes to the collisional rate. よろしくお願いいたします。

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The obtained collisional rate is summarized in terms of the normalized eccentricity e and inclination i of relative motion; the normalized eccentricity e and inclination i of relative motion; the normalization is based on Hill’s framework, i.e., e=e*/h and i=i*/h where e* and i* are ordinary orbital elements and h is the reduced Hill radius defined by (m_p/3M_? )^(1/3) (m_p being the protoplanet mass and M_? the solar mass). The properties of the obtained collisional rate <P(e,i)> are as follows: (i) <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 a two-dimensional region, <P(e,i)> is always enhanced over that in the two-body approximation <P(e,i)>_2B, (iii) <P(e,i)> reduces to <P(e,i)>_2B when (e^2+i^2)^(1/2)≧4, and (iv) there are two notable peaks in <P(e,i)>/ <P(e,i)>_2B at regions where e≒1 (i<1) and where i≒3 (e<0.1); the peak values are at most as large as 5. As an order of magnitude, the collisional rate between Keplerian particles can be described by that of the two-body approximation suitably modified in the two-dimensional region. However, the existence of the peaks in <P(e,i)>/ <P(e,i)>_2B are characteristic to the three-body problem and would give an important insight to the study of the planetary growth. お手数ですがよろしくお願いいたします。

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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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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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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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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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Accordingly, in our study, it is sufficient to try to find collision orbits with a limited range of b, i.e., b_min<b<b_max. Thus, the collisional rate <P(e, i)> is written practically as (see Eq. (10)) <P(e, i)>=∫[b_max→b_max](3/2)db(4b/(2π)^2)∫[0→2π]dτ∫[0→π]dωp_col(e, i, b, τ,ω). 　　　　　　　・・・・・・(14) Unfortunately, we cannot predict b_min and b_max definitely. As to b_max, we only know its upper limit from the Jacobi integral E of Hill’s equations, given by (see Hayashi et al., 1977 and Paper I), E=(1/2){e(r)^2+i(r)^2}-(3/8)b(r)^2-(3/r)+(9/2). ・・・・・・(15) Particles with E<0 cannot enter the Hill sphere and never collide with the protoplanet. The condition E>0 yields an upper limit of b_max, in terms of orbital elements at infinity (r→∞), b_max<{(4(e^2+i^2)/3)+12}^(1/2). 　　　・・・・・・(16) By the following consideration, a lower limit of b_min can also be estimated. First, we will find a turn-off point of the horseshoe orbit of the guiding center (see Fig.1). In the two-dimensional case, Hénon and Petit (1986) have found that the variation of e is much smaller than that of b at a large distance. This suggests that, as a first order approximation, we can neglect the variations of e and I compared to the b variation also in the three-dimensional case. Thus, assuming that both e and I are invariant, we have from Eq. (15) b(r)^2+(8/r)=b^2, 　　　　　　・・・・・・(17) where b is the semimajor axis at infinity. Fig.1. Example of an orbit with small b. Dashed curve denotes the trajectory of the guiding center. Turn-off point of the guiding center y_t and Hill sphere (x^2+y^2=1) are also shown. 長文ですがよろしくお願いします。

以前和訳していただいた英文の誤記について

以前、ddeana様に和訳して頂いた英文の補足です。文中のいくつかの式において、記号の誤植が見つかりましたので、お詫びして訂正させていただきます。該当質問URL↓ http://okwave.jp/qa/q8267587.html (1) (誤)<P(e,0)>_2B=(2/π)E(√(3/4))ρ_(2D)v, 　　　　　　　　　　　　　　　　　　　　(28) (正)<P(e,0)>_2B=(2/π)E(√(3/4))σ_(2D)v, 　　　　　　　　　　　　　　　　　　　　(28) ρ→σの誤りでした。同様に (誤)ρ_(2D)v (正)σ_(2D)v 以下は既に補足済みのものです。 (2) (誤) <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) (正) <P(e,i)>_2B=((2/π)^2)E(k)πr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2i), 　　　　　　(29) 以下は誤りを訂正した正しい英文です。 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))σ_(2D)v, 　　　(28) where E(k) is the second kind complete elliptic integral and σ_(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)^2)E(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). お寄せ頂いた和訳↓ 5．衝突速度の規格化まず、衝突速度<P(e, i)>と二体近似での衝突速度<P(e, i)>_2Bとの比として、促進係数を導入する。 R(e, i)=<P(e, i)>/<P(e, i)>_2B. (27) 係数R(e, i)は、太陽重力の影響による衝突速度増大の尺度を与えてくれる。二次元において<P(e, 0)>は方程式（11）の通りだが、<P(e, 0)>_2Bは次のように定義される。 <P(e, 0)>_2B=(2/π)E(√(3/4))σ_(2D)v, 　 (28) ここでのE(k)（※1）は第2種完全楕円積分（※2）であり、σ_(2D)vは方程式(3)を用い、<e(2/2)>をe^2に置き換えることで与えられる（単元が変更されることに留意されたし。例えばvは(e^2+i^2)^(1/2)となりGm_pは3となる)。数値係数2E(k)/π(=0.77)が導入され、衝突速度<P(e, 0)>_2Bは高エネルギー限界（vが限りなく無限大に近づくところ）において<P(e, 0)>と一致する（第一論文と1989年グリーンバーグとリシャールによる論文を参照のこと）。　3次元の場合、<P(e, i)>は方程式(10)によって求められるが、<P(e, i)>_2Bは方程式(1)の<e(2/2)>をe^2に、<i(2/2)>をi^2にそれぞれ置き換えることで求められる。なお、<P(e, i)>が表面数密度n_s単位あたりの次元を有することに留意されたい。その後nσv/n_sによって<P(e, i)>_2Bを定義する。尚、(n_s/n)は微惑星集団のスケールハイト（※3）（z方向での）の2倍に相当する。通常スケールハイトはi*a_0*とみなされている（すなわち、ここでの単位）。二次元の場合と同様に、<P(e, i)>_2Bは高エネルギー限界で必ず<P(e, i)>と一致することが要求される。その後数値係数(2/π)^2E(k) (=0.49～0.64)(第1論文参照）を導入することにより次のような式が求められる。 <P(e, i)>_2B=((2/π)^2)E(k)πr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2i), 　 (29) それと　 k^2=3e^2/4(e^2+i^2).　 (30) 6.二次元の場合の衝突速度この章では iが0である二次元の場合の衝突速度に焦点をあわせることとする。このケースでは相対運動の自由度における柔軟性により軌道運動の詳細な動きを研究することが可能となる。つまり方程式(11)に見られるように各eについて　b-τ二次元位相空間の中での衝突軌道のみを見つけることで十分なのである。 ※1：カッコ内のkは母数、modulusのことです。 ※2：楕円積分については下記をご参照ください。 http://ja.wikipedia.org/wiki/%E6%A5%95%E5%86%86%E7%A9%8D%E5%88%86 ※3：地表面の大気圧に対して気圧がe－1 になる高度のこと。また惑星の大気は高度とともに指数関数に従って減少しますが、どの高度でも同じ気圧を持つ仮想的な大気で惑星を脱出する大気の割合を考える場合は、この仮想大気の高度の上限のことをスケールハイトと呼びます。

この和訳をおしえて下さい。

In order to see qualitatively the collision frequency, we will define the collision probability, Pc(bi~) as follows. For a fixed value of bi~, orbital calculations are made with N's different value of δ, which are chosen so as to devide 2π(from -πto π) by equal intervals. Let n_c be the number of collision orbits, then P(bi~, N)=n_c/N gives the probability of collision in the limit N→∞. In practice, however, we can evaluate the value of Pc(bi~) approximately, but in sufficient accuracy in use, by the following procedures; i.e., at first for N=N1(e.g., N1=1000), P(bi~,N1) is found and, next, for N=2N1, P(bi~,N) is reevaluated. よくわからないのでよろしくお願いします。

和訳をお願いします。

In the two-dimensional case where the protoplanet and the planetesimal revolve in the sample plane around the protosun (i.e., i=0), the phase angle ω loses its meaning and Eq. (10) must be modified as <P(e, 0)>=∫(3/2) |b|(1/2π)p_col(e, i=0, b, τ)dτdb. ・・・・・(11) We consider that the planetesimal collides with the protoplanet if the separation distance becomes smaller than the sum of their radii; the protoplanet radius scaled by ha_0* is given by r_p=0.005(ρ/3gcm^-3)^-(1/3)(a_0*/1AU)^-1, ・・・・・(12) where ρ is the mean mass density of the protoplanet. よろしくお願いいたします。

３D衝突イベントの数と３D衝突率<P(e,i)>の結果

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お礼 2013/10/08 06:34

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