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- セミメジャーアクシスと軌道離心率が固定された場合、さまざまなε_iとδ_iを持つ同じ軌道が存在することに注意してください。
- Kepler軌道のすべての可能な相を実質的にε_i=constantと－π≦δ≦πで表現できます。
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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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初期条件のチョイスについていくつか所感を付け加えることとする。粒子軌道における軌道長半径と軌道離心率を一度固定したら、位相パラメーターε_i and δ_i,のすべての組み合わせを調べる必要はないということに留意すべきである。これは、違った ε_i と δ_iの組み合わせを有する無数の同じ軌道が存在するという事実による為である。したがって、δ_i が2パイ係数をもつことに留意し、 ε_iが一定であり、－π≦δ≦πであることを用いて、実質的にケプラー軌道のおこりえる位相のすべてを表すことができる。次に、a_i と e_i から b_i~ までと e_i~をそれぞれ次の公式に準じて変換していく。 a_i=1+b_i~h (2・9) および e_i=e_i~h. (2・10) ここでは、hは例えばヒル球のように惑星の重力が太陽の重力に勝る球体の半径であり、この実験で用いられた単位系で次のように定義される。 h=(μ/3)^(1/3)=2.15×10^(－3), (2・11) 上記式では、例えば惑星質量Mは現在の地球質量の100分の1である5.977×10の25乗グラムになるよう選ばれている。林およびその他によって示された通り、上記変換によって上手に見積もられたμ≦10^－4である限り、平面3体問題とその解答の間には似通うところの多い、類似の法則が存在し、従って μ (もしくは M)の特殊なチョイスにより得られた結果は、異なるμの値をもつ問題に広く適応することができる。
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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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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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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。
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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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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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In the cases 1.5≦|b_i~|≦3, a particle comes near the planet and, in many cases, is scattered greatly by the complicated manner. Especially, as seen from Fig. 2, all particles with the impact parameter ｜b_i~| in the range between 1.8 and 2.5, of which interval is comparable, as an order of magnitude, to the Hill radius, enter the Hill sphere of the planet. Such a particle, entering the Hill sphere, revolves around the planet along the complicated orbit. After one or several revolutions around the planet it escapes out of the Hill sphere in most cases, but it sometimes happens to collide with the planet as described later. Fig.2. Examples of particle orbits with various values of b_i~ and with e_i~＝0. The dotted circle represents the Hill sphere. All the particles with 1.75＜b_i~＜2.50 enter the sphere. 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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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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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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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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