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                        Acknowledgements The author wishes to thank Professor C.Hayashi and Dr.K. Nakazawa for their useful advice and continuous encouragement. The numerical calculations were mainly performed by FACOM M-200 at Data Processing Center of Kyoto University. This work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture (Nos. 511409, 56110009 and 57103006). The publication of this paper is indebted to the aid of Central Research Laboratory of Osaka Institute of Technology. よろしくお願いします。

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                 謝辞 本論文の作成に当たり、C.Hayashi教授ならびにK. Nakazawa博士の適切な助言と絶え間ない励ましに感謝の意を表したい。数値計算は、主に京都大学大型計算機センター(※1)のFACOM M-200で行った。本研究は、文部科学省基盤研究費(Nos. 511409, 56110009及び 57103006)の助成を受けたものである(※2)。本論文の刊行にあたっては大阪工業大学中央研究所の多大な協力を頂戴している。 ※1:固有名詞なので下記から名称を引用いたしました。 https://ipsj.ixsq.nii.ac.jp/ej/index.php?active_action=repository_view_main_item_detail&item_id=35765&item_no=1&page_id=13&block_id=8 ※2:国費を使っての研究発表においてはその旨を謝辞で表示することが義務づけられておりますので下記表記ルールに従って訳しました。 http://www.mext.go.jp/a_menu/shinkou/hojyo/faq/1322968.htm

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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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    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 H&#233;non (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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    In the present paper, we first describe initial conditions for orbital calculations in the framework of Hill’s equations (Sect.2). In Sect.3, orbital calculations are made for particular sets of e and i, i.e., (e,i)=(0,0) and (1,0.5). Based on these calculations and previous works: Papers I and II, Nishida (1983), H&#233;non and Petit (1986), and Petit and H&#233;non (1986), we develop an efficient numerical procedure obtaining <P(e, i)> (Sect. 4).According to this procedure, we systemically integrate the orbits of relative motion, to find collision orbits. Furthermore, compiling results of these orbital calculations, we find <P(e, i)> for various sets of e and i and compare them with those in the two-body approximation <P(e, i)>_2B (Sects. 5 to 7). The results will be compared with those of Nishida (1983) and Wetherill and Cox (1985) who also studied the collisional rate taking account of the effect of solar gravity (but their studies were restricted, as mentioned later).

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謝辞 著者はC.林教授とK.中沢博士の有益なアドバイスと絶間ない励ましに対し感謝を述べたい。数値計算は、主に京都大学のデータ処理センターのFACOM M-200を使用した。この業績は、文部科学省の助成金の補助を受けている(511409、56110009と57103006)。またこの論文刊行に際しては、大阪工業大学の中央研究所の恩義に浴することとなった。

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    謝辞     著者は、有益なアドバイスと不断の激励をいただきました林教授並びに中沢博士に感謝いたします。     計算は、主として京都大学データ処理センターの FACOM M-200 で行いました。     本研究は文部科学省の学術研究助成金 (Nos. 511409, 56110009 and 57103006) の支持を受けました。本研究の公刊に際しましては大阪工業大学中央研究所の援助をいただいております。

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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 H&#233;non, 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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          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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      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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    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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    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。

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      We will add some comments on the choice of the initial conditions. It should be noticed that we need not survey every set of phase parameters, ε_i and δ_i, when the semi-major axis and the eccentricity of the particle orbit are once fixed. This is due to the fact that there are numberless same orbits with different sets of ε_i and δ_i. Therefore, noticing that δ_i has a 2π-modulous, we can represent, practically, all of the possible phase of the Keplerian orbits by ε_i=constant and &#65293;π≦δ≦π. Secondly, we will transform a_i and e_i to b_i~ and e_i~, respectively, according to                       a_i=1+b_i~h (2・9) and                       e_i=e_i~h. (2・10) Here h is the radius of the sphere within which the gravity of the planet overcomes the solar gravity, i.e., the radius of the Hill sphere, and is defined in the units adopted here as                 h=(μ/3)^(1/3)=2.15×10^(&#65293;3), (2・11) where the planetary mass M is chosen to be 5.977×10^25g, i.e., one-hundredth of the present terrestrial mass. As shown by Hayashi et al, there exists an approximate similarity law between solutions to the plane circular RTB problem as long as μ≦10^&#65293;4, which is well scaled by the above transformations and, hence, the results obtained for the special choice of μ (or M) can apply extensively to the problem with different value of μ. 長い文章ですが、ご教授いただけると助かります。

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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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    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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       According to Hayashi et al., We will adopt the system of units where the distance between the planet and the Sun, the sum of their masses and the angular velocity of the rotation of the planet are all unity. When the coordinate is chosen such that the (x,y) plane coincides with the rotational plane of the planet, i.e., the ecliptic plane, the Sun and the planet are at rest on the x-axis and the planet is at the origin, then the equations of motion are given by (Szebeheley^10)),            x’’&#65293;2y’=&#65293;∂U/∂x,          (2・1)             y’’+2x’=&#65293;∂U/∂y,         (2・2) where U is the effective potential, described as                U=&#65293;(μ/r_1)&#65293;((1&#65293;μ)/r_2)&#65293;((1/2)r^2)+U_0.         (2・3) 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。