Two-Body Encounters in Orbit with Different Eccentricities
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- In the case when the eccentricity is not equal to 4, the orbit with a finite eccentricity either curls or waves for the purely Keplerian motion.
- The relative phase difference between the planet and the perigee point of the particle orbit near the planet, denoted as δ, is expected to be an important parameter for determining the features of the two-body encounters.
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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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3.2. e_i~が4の場合 図6からわかるように、有限の軌道離心率をもつ軌道は、惑星としての質量が非常に小さいか、粒子が惑星から離れている場合におこると認識されている、純粋なケプラー運動の為にらせん状に動くか（b_i~の絶対値が、3分のe_i~×4よりも小さい時）、うねる（b_i~の絶対値が3分のe_i~×4よりも大きい時）。それゆえ、惑星と惑星付近の粒子軌道の近日点との間の相対位相差δは、2体衝突の特徴を決定する上での1つの重要なパラメーターと予想される。ここでは便宜上、δの代わりに、下記のように定義した位相パラメーターδ*を適用する。 δ*=δ-ε/(1-a^(3/2)). (3・7)
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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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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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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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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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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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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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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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If |P(b_i~,N_1)-P(b_i~,2N_1)|≦0.05×P(b_i~,N_1), then P(b_i~,N_1) (or P(b_i~,2N_1)) is regarded as the collision probability, P_c(b_i~). In Fig.10, P_c(b_i~) is illustrated as a function of b_i~ at the region near the orbit of the present Earth for the case e_i~=4. As seen from this figure, a particle whose impact parameter, b_i~, lies between 1.5 and 5.5, collides with the planet with certain probability. Especially, for |b_i~|≒2 collisions occur frequently. よろしくお願いいたします。
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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。
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