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# この文章の和訳を教えてください。

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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### 関連するQ&A

• 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。

• 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。

• この文章の和訳をよろしくお願いします。

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。

• この文章の和訳をよろしくお願いします。

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。

• 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－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。

4. Numerical procedures for obtaining <P(e, i)> Based on the results in Sect. 3, we develop numerical procedures for efficiently obtaining <P(e, i)>. By using the framework of Hill’s equations, one can reduce the degrees of freedom of particle motion, as described in Paper I. Nevertheless, they are still too numerous to compute orbits densely and uniformly over the five-dimensional phase space (e, i, b, τ_s, ω_s) of initial conditions. Furthermore, we frequently find orbits which comes very close to the protoplanet or which revolve complicatedly many times around the protoplanet. In these cases, an enormously long computation time is needed to calculate them. To avoid these difficulties, we introduce the following three simplifications: (i) We neglect n-recurrent orbits in evaluating <P(e, i)> when n≧3; that is, we stop an orbital calculation if the particle does not collide with the protoplanet even after two close encounters. (ii) When the planetesimal crosses the sphere surface of the two-body approximation, the two-body formula determines whether or not a collision occurs. (iii)We omit orbital computations in some regions of τ_s, which depend on a given set of e, i, and b; it is found empirically that there are no collision orbits in these regions. よろしくお願いします。

• この文章の和訳をお願いします。

2. Adopted assumptions and basic equations We consider two planetesimals revolving around the proto-Sun (being called the Sun). Here we assume that the mass of the one of these, which hereafter is called a protoplanet or simply a planet, is much larger than that of the other (being called a particle). We also assume that the planet moves circularly around the Sun without the influence of gravity of the particle. Furthermore, each orbit of the particle is limited in the ecliptic plane of the planetary orbit. Under these assumptions, the particle motion is simply given by a solution to the plane circular RTB problem. よろしくお願いします。

• この文章の和訳を教えていただけませんか。

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。

• この文章の和訳をお願いします。

Now, we shall concentrate on the collision orbits. Figure 5 illustrates the minimum separation distance r_min in the first encounter (solid curves), identical to that obtained by Petit and Hénon (1986). One sees immediately, that there are two different zones: the “regular” zones, in which r_min varies smoothly with a change of parameter b and the irregular (or “chaotic”) zones, where r_min changes greatly with tiny differences in the choice of b. The chaotic zones lie near b=1.93, 2.30 and 2.48, with very narrow ranges of b. In the regular zone, we find two broad bands of collision orbits around b=2.09 and 2.39. These collision bands were first found by Giuli (1968). The sum of width of the collision bands ⊿b is found to be about 0.098, if the planetary radius is 0.005. 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。