Orbital Calculations and Starting Points
- Learn about the orbital calculations and starting points.
- Understand the equations used in the calculations.
- Table 1 provides the values of b_min* evaluated from an equation.
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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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我々の軌道計算では、出発点は方式で割り当てられている（方程式(7)を参照）。 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), ここでは t_0は,τ_sとω_sとは無関係の時間の原点（※1）であり、 y_0はプラズマの（Y方向における）始まる距離である。 表1：b*の最小値の数値は方程式（18）から評価した。数値的に見つけたbの最小値も示されている。 ※1：物体の座標変化などを求める時関係します。
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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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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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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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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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Thus, it is convenient to express a solution to Eq. (6) in terms of the instantaneous Keplerian orbital elements of the relative motion (e, i, b, τ, ω, φ): the orbit is given by x=b－ecos(t－τ), y=－3b(t－φ)/2+2esin(t－τ), z=isin(t－ω), ・・・・・・・(7) where b, e, I, τ, and ω are the semimajor axis, eccentricity, inclination, the longitude of perihelion, and that of ascending node of the relative motion, respectively.
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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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For the phases, we assume that τ_s and ω_s are distributed uniformly in the range between 0 and 2π, because we have found numerically that τ_s (or ω_s) shifts from τ (or ω) only by a certain small angle if r_s is sufficiently large, and further because both τ and ω are supposed to be distributed uniformly. We can therefore replace τ and ω in Eq. (14) by τ_s and ω_s, respectively. Thus, <P(e,i)> is evaluated, instead of Eq. (14), by <P(e, i)>=∫[b_max→b_min]db(3b/2π^2)∫[0→2π]dτ_s∫[0→π]dω_s・p_col(e, i, b_s(b), τ_s, ω_s). ・・・・・・・・・・・(20) どうかよろしくお願いします。
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Fig. 8. Contours of minimum separation distance r_min in the second encounter for the case of (e, i, b)=(1.0, 0.5, 2.8). Dots have the same meanings as those in Fig. 7. Fig. 9. The “Differential” collisional rate <p(e, i, b)> (defined by Eq. (24)) is plotted as a function of b in the case of (e, i)=(1.0, 0.5). Fig. 8.↓ http://www.fastpic.jp/images.php?file=4403322728.jpg Fig. 9.↓ http://www.fastpic.jp/images.php?file=9905093670.jpg 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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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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