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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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3.軌道計算例 <P(e ,i)>を得るための効率的な数値計算手法をみつける為に、2つの典型的なケース、 (e, i)=(0, 0) と(1, 0.5)について詳細な軌道計算を行った。前者は単一パラメーターbを伴うもっとも単純なケースでジュリ(1968年）、西田(1983年）そしてプテイとエノン（1986年）による研究に相当する。後者では、軌道は3つのパラメーター b, τ, そして ωに伴い複雑な方法で変化する。この例により3次元での軌道の特性を見ることができる。 本例では、平均質量密度 3gcm^-3をもつ原始惑星が地球領域を周回するものと仮定する。さらに二体近似使用の制限的精度基準として1％を採用する。これらにより原始惑星および二体球の半径はそれぞれ、0.005 と 0.03となる（ 方程式(12) と (13)を参照のこと）。
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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 Hénon, 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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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。
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3.1. Case of e=0 and i=0 We have first calculated 6000 orbits in the parameter range of b from b_min=1.9 to b_max=2.5 at intervals of 0.0001. It is already known from previous studies (Nishida, and Petit and Hénon) that no collision orbits exist outside this region. The orbits vary in a complicated way with the value of parameter b (see Petit and Hénon, 1986). In spite of the complex behavior of the orbits, we can classify them in terms of the number of encounters with the two-body sphere, from the standpoint of finding collision orbits. The classes are: (a) non-encounter orbit, (b) n-recurrent non-collision orbit, and (c) n-recurrent collision orbit, where the term “n-recurrent orbit” means the particle encounters n-times with the two-body sphere. That is, n-recurrent non-collision (or collision) orbits are those which fly off to infinity (or collide with the protoplanet ) after n-times encounters with the two-body sphere, while non-recurrent orbits are those which fly off without penetrating the two-body sphere. Examples of orbits in the classes (a), (b), and (c) are illustrated in Figs. 2,3, and 4, respectively. The above classification of orbits will be utilized for developing numerical procedures for obtaining <P(e, i)>, as described in the next section. 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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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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Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
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Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
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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 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), Hénon and Petit (1986), and Petit and Hénon (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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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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