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The corresponding velocity components are given by x_s'=esin(t_0-τ_s) y_s'=‐(3/2)b_s+2ecos(t_0-τ_s), (22) z_s'=icos(t_0-ω_s). In the above, we set t_0=-(2y_0/(3b)) without any loss of generality, for later convenience, which is equivalent to the choice of φ=0 in Eq. (7). In Eq. (21), y_0 is set to be max (40, 20e, 20i). The choice of y_0 is not essential to the evaluation of <P(e, i)> as long as y_0 is larger than this value since Eq. (19) is valid for such a large y_0. In Table 2, the evaluated values of <P(e,i)> are tabulated for various y_0 in the cases of (e, i) = (0,0) and (4,0): when y_0≧max(40, 20e, 20i), <P(e, i)> is almost independent of y_0 within an accuracy of 0.1% for the case of (e, i)=(0, 0) and 5% for (e, i)=(4, 0). よろしくお願いします。

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### 質問者からのお礼 2013/08/13 15:32

どうもありがとうございました！ 大変参考になりました。

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• この文章の和訳をお願いします。

Table 2. Change of evaluated <P(e,i)> due to different choices of y_0. Two cases are tabulated: (e, i)=(0, 0) and (4,0). The value of <P(e, i)> varies with y_0, but the difference becomes very small as long as y_0 is greater than 40 and 80 for the cases of e=0 and e=4, respectively. よろしくお願いします。

• この文章の和訳をお願いします。

In Figs. 7b and 7c (b=2.8 and 3.1) we find that the τ values of close-encounter orbits are confined in a region near τ=0 and, further, its width decreases with an increase in b. In particular, when b=3.1 (i.e., b very close to b_max), the τ values of the close-encounter orbits localize in a narrow region around τ=0. This comes from the choice of φ=0 in the initial conditions: When b is relatively large and the mutual gravity is weak, then a particle continues approximately its original Keplerian motion. When the guiding center of the particle comes across the x-axis (i.e., when t=0), its position is given by (see Eq. (7) with φ=0) x=b-cosτ, y=-2esinτ, (23) z=-isinω. The distance from the origin becomes minimum when τ<<π. Thus only particles with τ<<π can be disturbed drastically by the gravity of the protoplanet and have the possibility of encountering the two-body sphere. The fact that the width of τ in the finely dotted region (i.e., the τ of close-encounter orbits) decreases with an increase in b is also observed in other e and i, as long as b>e. This behavior is very useful to systematically find collision orbits. If we can once find an appropriate restricted region of close-encounter orbit, e.g., τ_1≦τ≦τ_2, for some b_1 (b_1>e), then for b>b_1 it is sufficient to search close-encounter orbits in the limited region between τ_1 and τ_2. 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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どうもありがとうございました。

### 関連するQ&A

• この文章の和訳を教えてください。

In Eq. (2・3) μis defined as 　　　　　　　　　μ=M/( M? +M), (2・4) where M is the mass of the planet, γ, γ_1 and γ_2 are the distances from the center of gravity, the planet (i.e., the origin) and the Sun, respectively, which are given by 　　　　　　　　　　　　r^2=(x+1&#65293;μ)^2+y^2, (2・5) 　　　　　　　　　　　　r_1^2=x^2+y^2 　　(2・6) and 　　　　　　　　　 　　　r_2^2=(x+1)^2+y^2. 　 (2・7) Furthermore, U_0 is a certain constant and, for convenience, is chosen such that U is zero at the Lagrangian point L_2. お手数ですがよろしくお願いいたします。

• この文章の和訳を教えてください。

It should be noticed that Λ_f is independent of both ei~ and R. In order to compare Λ_K with Λ_f clearly, we consider the ratio of these, that is, γ=Λ_K/Λ_f=3.0C_K(R/1AU)^1/2, (4・9) which is independent of the mass of the planet. The values of γare listed in Table I as well as those of C_K. As the parameter C_K is approximately proportional to R^-1/2 (see Fig.11), γ is almost independent of R and is 2.3 for ei~=0 and 1.4 for ei~=4. This indicates that, though the result is obtained in the limited framework of the two-dimensional particle motion, the collisional rate of Keplerian particles is enhanced by a factor of about 2.3 or 1.4 compared to that estimated in a free space formula, as long as we are concerned with the two cases of eccentricity. Furthermore, as seen from Table I, γ for the case ei~=4 is significantly smaller than that of the case ei~=0. This seems to confirm the conjecture that γ tends to unity in the high energy limit (i.e., ei~→∞), or in other words, the free space formula is right only in the high energy limit. お手数ではございますが、どうかよろしくお願いいたします。

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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, H&#233;non 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。

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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 &#8895;b is about 0.011, and the contribution to the collision is as large as 15%. No.3- and more &#8211;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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For regions where |b_i~|≦5.7, we have the same pattern in &#8895;b~ and &#8895;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 <&#8895;b~> and <&#8895;e~> are found to be negligible too. In Figs.9 (a) and (b) the average values of <&#8895;b~> and <&#8895;e~> are illustrated as a function of b_i~, where <&#8895;b~> and <&#8895;e~> are defined as <&#8895;b~>=(1/2π)∫&#8895;b~dδ* (3.8) and <&#8895;e~>=(1/2π)∫&#8895;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, <&#8895;b~> and (b) the change of the eccentricity, <&#8895;e~> averaged over the parameter δ* for various values of b_i~ between &#65293;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。

• この文章の和訳をよろしくお願いします。

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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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 &#8211;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。

• この文章の和訳をお願いします。

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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1. Introduction This is the third of a series of papers in which we have investigated the collisional probability between a protoplanet and a planetesimal, taking fully into account the effect of solar gravity. Until now, the collisional probability between Keplerian particles has not been well understood, despite of its importance, in the study of planetary formation and, as an expedient manner, the two-body (i.e., free space) approximation has been adopted. In the two-body approximation, the collisional rate is given by (e.g., Safronov,1969) σv=πr_p^2(1+(2Gm_p/r_pv^2))v, (1) where r_p and m_p are the sum of radii and the masses of the protoplanet and a colliding planetesimal, respectively. Furthermore, v is the relative velocity at infinity and usually taken to be equal to a mean random velocity of planetesimals, i.e., v=(<e_2*^2>+<i_2*^2>)^(1/2)v_K, (2) where <e_2*^2> and <i_2*^2> are the mean squares of heliocentric eccentricity and inclination of a swarm of planetesimals and v_K is the Keplerian velocity; in the planer problem (i.e., <i_2*^2>=0), the collisional rate is given, instead of Eq.(1), by (σ_2D)v=2r_p(1+(2Gm_p/r_pv^2))^(1/2)v. (3) Equations (1) and (3) will be referred to in later sections, to clarify the effect of solar gravity on the collisional rate. よろしくお願いします。

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For the phases, we assume that τ_s and ω_s are distributed uniformly in the range between 0 and 2π, because we have found numerically that τ_s (or ω_s) shifts from τ (or ω) only by a certain small angle if r_s is sufficiently large, and further because both τ and ω are supposed to be distributed uniformly. We can therefore replace τ and ω in Eq. (14) by τ_s and ω_s, respectively. Thus, <P(e,i)> is evaluated, instead of Eq. (14), by <P(e, i)>=∫[b_max→b_min]db(3b/2π^2)∫[0→2π]dτ_s∫[0→π]dω_s・p_col(e, i, b_s(b), τ_s, ω_s). 　　　　　　　　　　　　　　　　　　　　　　　　　　　 ・・・・・・・・・・・(20) どうかよろしくお願いします。