この文章の和訳をお願いします。 -The obtained collisi-

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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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Progress of Theoretical Physics. Vol. 70, No. 1, July 1983 Collisional Processes of Planetesimals with a Protoplanet under the Gravity of the Proto-Sun Shuzo NISHIDA Department of Industrial and Systems Engineering Setsunan University, Neyagawa, Osaka 572 (Received March 4, 1983) Abstruct We investigate collisional processes of planetesimals with a protoplanet, assuming that the mass of the protoplanet is much larger than that of a planetesimal and the motion of the planetesimal is limited in the two-dimensional ecliptic plane. Then, we can describe the orbit by a solution to the plane circular Restricted Three-Body problem. Integrating numerically the equations of motion of the plane circular RTB problem for numerous sets of initial osculating orbital elements, we obtain the overall features of the encounters between the Keplerian particles. In this paper we will represent only the cases e=0 and 4h, where e is the eccentricity of the planetesimal far from the protoplanet and h is the normalized Hill radius of the protoplanet. We find that the collisional rate of Keplerian particles is enhanced by a factor of about 2.3 (e=0) or 1.4 (e=4h) compared with that of particles in a free space, as long as we are concerned with the two-dimensional motion of particles. よろしくお願いします。

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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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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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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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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. よろしくお願いします。

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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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　　　　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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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。

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It is to be noted here that e, i, and b are also scaled by h: e=e*/h, i=i*/h, and b=(a*－a_0*)/ha_0*, ・・・ (8) where e*, i*, and a* denote, respectively, the instantaneous eccentricity, inclination, and semimajor axis of usual use. Throughout the present paper, the normalized orbital elements given by Eq. (8) will be used.

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As mentioned above, there are two peaks in R(e,i) in the e-i diagram: one is at e≒1 (i<1) and the other at i≒3 (e<0.1). The former corresponds directly to the peak in R(e,0) at e≒1 found in the two-dimensional case. The latter is due to the peculiar nature to the three-dimensional case, as understood in the following way. Let us introduce r_min (i, b,ω) in the case of e=0, which is the minimum distance during encounter between the protoplanet and a planetesimal with orbital elements i, b, and ω. In Fig. 16, r_min(i,b)=min_ω{r_min(i,b,ω)} is plotted as a function of b for various i, where r_min<r_p (=0.005) means “collision”; there are two main collision bands at b≒2.1 and 2.4 for i=0. For i≦2, these bands still exist, shifting slightly to small b. This shift is because a planetesimal feels less gravitational attracting force of the protoplanet as i increases. As i increases further, the bands approach each other, and finally coalesce into one large collision band at i≒3.0; this large band vanish when i≧4. In this way, the peculiar orbital behavior of three-body problem makes the peak at i≒3 (e<0.1). Though there are the peaks in R(e,i), the peak values are not so large: at most it is as large as 5. This shows that the collisional rate is well described by that of the two-body approximation <P(e,i)>_2B except for in the vicinity of v, i→0 if we neglect a difference of a factor of 5. Now we propose a modified form of <P(e,i)>_2B which well approximates the calculated collisional rate even in the limit of v, i→0. We find in Fig.14 that <P(e,i)> is almost independent of i, i.e., it behaves two-dimensionally for i≦{0.1 (when e≦0.2), {0.02/e (when e≧0.2). (35) This transition from three-dimensional behavior to two-dimensional behavior comes from the fact that the isotropy of direction of incident particles breaks down for the case of very small i (the expression <P(e,i)>_2B given by Eq. (29) assumes the isotropy). In other words, as an order of magnitude, the scale height of planetesimals becomes smaller than the gravitational radius r_G=σ_2D/2 (σ_2D given by Eq. (3)) and the number density of planetesimals cannot be uniform within a slab with a thickness σ_2D for small i. Table 4. Numbers of three-dimensional collision events found by orbital calculations for the representative sets of e and i. In the table r_p is the radius of the protoplanet. Table 5. The three-dimensional collision rate <P(e,i)> for the case of r_p=0.005 (r_p being the protoplanetary radius), together with two-dimensional <P(e,i=0)> Fig. 14. Contours of the evaluated <P(e,i)>, drawn in terms of log_10<P(e,i)> Fig. 15. Contours of the enhancement factor R(e,i) Table 4.↓ http://www.fastpic.jp/images.php?file=1484661557.jpg Table 5.↓ http://www.fastpic.jp/images.php?file=6760884829.jpg Fig. 14. ＆Fig. 15.↓ http://www.fastpic.jp/images.php?file=8798441290.jpg 長文になりますが、よろしくお願いします。

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According to Paper I, the total collisional rate <Γ(e_1,i_1)> of planetesimals upon the protoplanet with the heliocentric orbital elements e_1 and i_1 is given by <Γ(e_1,i_1)>=2π^2∫<n_2>e_i<P(e,i)>dedi, ・・・・・・・(9) where <n_2> is the distribution function of planetesimals averaged by the phase angles τ_1 and ω_1 of the protoplanet, and e and i are the orbital elements of relative motion between the protoplanet and the planetesimal at infinity. よろしくお願いします。

『この文章の和訳をお願いします』という質問の補足

以前、ddeana様に和訳して頂いた英文の質問の補足です。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。

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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。

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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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When |b_i~| is relatively large (e.g., |b_i~|≧5), a particle passes through the region far from the Hill sphere of the planet without a significant influence of the gravity of the planet. As seen from Figs. 1(a) and (b), both the impact parameter, b_f~, and the eccentricity, e_f~, at the final stage are not so much different from those at the initial, i.e., a particle is hardly scattered in this case. When 3≦|b_i~|≦5, as seen from Fig. 1(a) and (b), a particle is scattered a little by the gravity of the planet and both |⊿b~| and |⊿e~| increase gradually with the decrease in |b_i~|, where ⊿b~=b_f~－b_i~ (3・1) and ⊿e~=e_f~－e_i~. 　　　 (3・2) 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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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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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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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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