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Equation (17) shows that the guiding center intersects the y-axis at y_1=8/b^2 (where b(r)=0) and turns back to infinity (see Fig.1). On the other hand, the radius of an elliptic oscillation around the guiding center is equal to 2e in the y-direction. Thus, it follows that if (y_1-2e) is greater than a certain distance (say, 2.5 times as large as the Hill radius) no particle enters the Hill sphere and collide with the protoplanet. Inversely, for the occurrence of collision, (y_1-2e)<2.5 must be statisfied; explicitly we have b_min^2>b_min*^2≒8/(2.5+2e). ・・・・・・・・・・(18) Equation (18) is confirmed by Table 1, where b_min* and numerically found b_min are shown for representative e. よろしくお願いします。
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方程式(17)は、プラズマが、y_1=8/b^2（b(r)は0）で、Y軸と交差し無限遠に折り返すことを示している（図1を参照）。一方プラズマの周りの楕円振動（※1）の半径は、Y方向の2eに等しい。よって、(y_1-2e)がある一定の距離（ヒル半径の2.5倍の長さとしよう）よりも大きい場合、ヒル球に入り原子惑星と衝突する粒子はまったくないということになる。逆に、衝突が生じる為には、 (y_1-2e)が2.5よりも小さいという条件が満たされていなければならない。つまり明確に次のようになる。 b_min^2>b_min*^2≒8/(2.5+2e). ・・・・・・・・・・(18) 方程式(18)はeを表現する為の bの最小値*と数値的に見つけたbの最小値が示された表1から確認される。 ※1：プラズマからの距離に比例する力を受けて運動する物体は、プラズマを中心として楕円軌道を描くように振動するということ。
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- ddeana
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ddeanaです。ひとつ訂正を。 ×bの最小値*→〇b* の最小値 です。失礼いたしました。
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了解いたしました。
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- この文章の和訳をお願いします。
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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A particle, which escapes again from the Hill sphere, is scattered largely whose orbital elements at the final stage are much different from those of the initial； i.e., in some cases Δb~ and Δe~ are about ten times as large as the Hill radius (see Figs. 1(a) and (b)). Nevertheless, we have found that ｜b_i~｜≦｜b_f~｜≦7； the former is easily deduced. Noticing that the Jacobi integral, which is given by, E_J＝((e~^2)/2)－((3b~^2)/8)＋9/2)h^2＋O(h^3), (3・3) is conserved during the particle motion, we have b_f~^2＝((4e_f~^2)/3)＋b_i~^2＞b_i~^2, (3・4) for the case e_i~＝0. Fig. 1. (a) The relations between b_i~ and b_f~ and (b) between b_i~ and e_f~ for the case e_i~=0. よろしくお願いします。
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3.2. Case of e=1.0 and i=0.5 In the three-dimensional case, the orbit is characterized by three parameters b, τ_s, and ω_s (in this section, we will omit the subscript “s” describing a starting point of orbital calculations). Recalling that b_max<3.7 from Eq. (16) and b_min>1.3 from Eq. (18) in the present case, we first examine orbits with b in the range between 1.3 and 3.7: Phase space (b, τ, ω) is divided into about 22000 meshes, i.e., 24 in b (1.3~3.7), 60 in τ (-π~π), and 15 ω (0~π). These orbital calculations show that there is no collision orbit where b<1.5 and b>3.2: b_max and b_min are set at 3.2 and 1.5, respectively, rather than 3.7 and 1.3. In parallel with the argument in the previous subsection, we consider the minimum separation r_min between the protoplanet and the planetesimal. In Fig.7, contours of r_min in the first encounter are illustrated in the τ-ω diagram for the three cases of b=2.3 (Fig. 7a), 2.8 (b), and 3.1 (c). Each figure is compiled from the orbital calculations of 5000 orbits, i.e., the τ-ω plane is divided into 100 (in τ)×50 (in ω). We concentrate first on Fig. 7a. In the coarsely dotted region where r_min>1, particles cannot enter the Hill sphere of the protoplanet. Such regions are beyond our interest. In the other regions where particles can enter the Hill sphere, r_min varies with τ and ω in a complicated manner. In particular, near the points (τ, ω)=(-0.24π,0.42π) and (-0.26π, 0.06π), r_min varies drastically in a small area in the τ-ω diagram. These may be chaotic zones. But in almost all regions, r_min varies continuously with τ and ω, and in this sense the orbits are regular. The finely dotted regions show those in which r_min becomes smaller than 0.03 (the radius of the two-body).Such orbits will be called close-encounter orbits in the chaotic zones is very small compared with that in the regular zones. This is the same conclusion as reached earlier. 長文ですが、どうかよろしくお願いします。
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Fig. 3a and b. An example of recurrent non-collision orbits. The orbit with b=2.4784, e=0, and i=0 is illustrated. To see the orbital behavior near the protoplanet, the central region is enlarged in b; the circle shows the sphere of the two-body approximation and the small one the protoplanet. Fig. 4a and b. Same as Fig. 3 but b=2.341, e=0, and i=0. This is an example of recurrent collision orbits. Fig.5. Minimum separation distance r_min between the protoplanet and a planetesimal in the case of (e, i)=(0, 0). By solid curves, r_min in the first encounter is illustrated as a function of b. The level of protoplanetary radius is shown by a thin dashed line. The collision band in the second encounter orbits around b=2.34 is also shown by dashed curves. Fig. 6. Minimum distance r_min in the chaotic zone near b=1.93 for the case of (e, i)=(0, 0); it changes violently with b. Fig. 7a-c. Contours of minimum separation distance r_min in the first encounter for the case of (e, i)=(1.0, 0.5); b=2.3(a), 2.8(b), and 3.1 (c). Contours are drawn in terms of log_10 (r_min) and the contour interval is 0.5. Regions where r_min>1 are marked by coarse dots. Particles in these regions cannot enter the Hill sphere of the protoplanet. Fine dots denotes regions where r_min is smaller than the radius r_cr of the sphere of the two-body approximation (r_cr=0.03; log_10(r_cr)=-1.52). Fig. 3a and b. および Fig. 4a and b. ↓ http://www.fastpic.jp/images.php?file=3994206860.jpg Fig.5. および Fig. 7a-c. ↓ http://www.fastpic.jp/images.php?file=2041732569.jpg Fig. 6. ↓ http://www.fastpic.jp/images.php?file=2217998690.jpg お手数ですが、よろしくお願いします。
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At first sight, from these figures, a symmetric pattern with respect to b_i~=0 holds approximately and a large scattering occurs in the region 1.5≦|b_i~|≦6. This is because a particle of which |b_i~| lies between 1.5 and 6 enters the Hill sphere and, hence, is scattered heavily. When |b_i~| is less than about 1.0, the feature of the scattering takes a new aspect. The gyrocenter of a particle comes slowly close to the planet whereas the particle rotates rapidly around the gyrocenter. When the distance between the planet and the gyrocenter is several times as short as the Hill radius, the gyrocenter of the particle gradually turns back to the opposite side of the y-axis, as if it were reflected by a mirror. The feature of the scattering is the same as that for the case e_i~=0. For the final impact parameter, b_f~, and the eccentricity, e_f~, we have again the relations b_f~=-b_i~ and e_f~=e_i~. (3・10) 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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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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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 －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^(－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^－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 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－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。
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It should be noticed that when δ*=0 the perigee of a purely Keplerian orbit lies on the x-axis. For the orbital calculations, b_i~ is taken in the range between －10 to 10 and δ* is varied from －π to π. When |b_i~| is about 10, the perigee distance from the planet is larger than several times the Hill radius. Thus, the particle is hardly scattered and keeps almost the same orbital elements as the initial during the encounter. When |b_i~| is as small as 6, however, a particle begins to interact with the planet appreciably only in the vicinity of δ*=0 and the orbital elements are slightly changed. よろしくお願いします。
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