Gravity Influence on Particle Motion in the Hill Sphere
- 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.
- When 3≦|b_i~|≦5, 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~|.
- The impact parameter, b_f~, and the eccentricity, e_f~, at the final stage are not significantly different from those at the initial stage, indicating that a particle is hardly scattered in this case.
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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。
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b_i~の絶対値（※1）が相対的に大きい時（例えば付帯条件が |b_i~|≧5といった場合）、粒子は惑星の重力の多大な影響を受けずに、惑星のヒル球から遠く離れた領域を通過する。図(a）と(b)から分るように、最終段階における衝撃パラメーター b_f~と離新率 e_f~はどちらも初期段階とそれほど違いはない、つまりこの場合粒子はほとんど拡散しない。図(a）と(b)から分るように、b_i~の絶対値が3かそれよりも大きく、5かそれよりも小さい時、粒子は惑星の重力によってわずかに拡散し、⊿b~の絶対値と⊿e~の絶対値いずれもb_i~の絶対値の減少に伴い、徐々に増加する。その場合 ⊿b~=b_f~－b_i~ (3・1) であり、 ⊿e~=e_f~－e_i~. (3・2) である。 ※1： |b_i~| の | |は不等式における「絶対値」を意味します。
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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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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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For the cases where |b_i~| is smaller than about 5.7, a particle with δ*=0 can enter the Hill sphere. The same as for the case e_i~=0, a particle which enters the Hill sphere revolves around the planet and after the complicated motion it escapes from the Hill sphere. As a result of this, the particle has quite different osculating orbital elements at the final stage from the initial. At the same time, a particle sometimes happens to be able to collide with the planet during the revolution. As |b_i~| decreases, the region of δ*, with which a particle experiences a large angle scattering or a direct collision, propagate on both sides around δ*=0 as seen from Fig.7. We can also see from this that the region of |b_i~| where a particle is largely scattered or collides with the planet, is shifted outward (i.e., to the side of large values of |b_i~| in comparison with that for the case e_i~=0). This is due to the fact that a particle has a finite eccentricity in this case and it can come close to the Hill sphere near the perigee point even if the impact parameter |b_i~| is relatively large. Fig.7. The change of the eccentricity ⊿e~ versus the parameter δ*, for particle orbits with various values of b_i~ and with e_i~=4. As b_i~ decreases from 6.0 to 2.0, the regions where a large scattering occurs spread to both sides of δ*=0. For b_i~=1, particles return half way to an opposite side of b_i~=0, conserving the initial eccentricity. 長文になりますが、どうかよろしくお願いします。
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3.2. For the case e_i~=4 As seen from Fig.6, an orbit with a finite eccentricity curls (|b_i~|<4(e_i~/3)) or waves (|b_i~|>4(e_i~/3)) for the purely Keplerian motion which is realized when the planetary mass is very small or a particle is far from the planet. Therefore, δ, which is the relative phase differnce between the planet and the perigee point of the particle orbit near the planet, is expected to be one of the important parameters for determining the features of the two-body encounters. It is convenient to introduce, here, a phase parameter δ*, in place of δ, defined as δ*=δ-ε/(1-a^(3/2)). (3・7) どうか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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We have two kinds of the final stage of the particle orbit: One is the scattering case and another is the collisional case. For the scattering case, after the passage near the planet a particle orbit settles again to the Keplerian at the region far from the planet. The final orbital elements (b_f~, e_f~, ε_f, δ_f), of course, are different from the initial owing to the gravitational interaction with the planet. On the other hand, we regard the case as the collision of the particle with the planet when the distance between the particle and the center of the planet becomes smaller than the planetary radius, r_p, which is given in the units adopted here by r_p=R_p/R=(3M/4πρ)^(1/3)/R, =4.57×(10^－3)h/(R/1AU), (2・12) where R_p and R are the radius of the planet in ordinary units and the distance between the planet and the Sun, respectively. どうか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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We start our orbital calculation of a particle from a point far from the planet where the effect of the gravitational force of the planet can be neglected. At an initial point where a particle is governed almost completely by the solar gravity, the particle orbit can be approximated in a good accuracy by the Keplerian. Hence we will give the initial conditions for the orbital calculation in terms of the Keplerian orbital elements (a_i, e_i, ε_i, δ_i) in place of (x, x’, y, y’), where a, e and ε are the semi-major axis, the eccentricity and the mean longitude at t=0, respectively. Furthermore the parameter δ is defined as δ=－nt_0, (2・8) where n and t_0 are the mean angular velocity and the time of the perihelion passage, respectively. 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 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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