A Study of Discontinuous Spikes in the Impact Parameter Range
- This study examines the presence of discontinuous spikes in the range of the impact parameter between 1.8 and 2.6.
- The change in the impact parameter and eccentricity for particles within the range of 1.8 < b_i~ < 2.6 is analyzed.
- The results show that there are multiple discontinuous spikes in the specified range, which is referred to as the discontinuous band.
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Now, Figs. 3(a) and (b) indicate that there seems to be several discontinuous spikes in the range of |b_i~| between 1.8 and 2.6, which herefter is called the discontinuous band. Fig.3. (a) The change of the impact parameter, Δb~ and (b) that of the eccentricity, Δe~ for particles with 1.8＜b_i~＜2.6 and with e_i~＝0. Marks on the upper abscissa indicate that with these initial b_i~ particles collide with the planet. お手数ですがよろしくお願いいたします。
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さて、図3(a)と(b)は、b_i~の絶対値、1.8から2.6の範囲内にいくつもの不連続な数値的突起部分があるらしいことを示しており、今後不連続バンドという名で呼ばれることとなる。 図3(a)はb_i~が1.8～2.6の間でe_i~が0の粒子における衝突パラメーターΔb~の変化を、(b)は離心率Δe~の変化を表したものである。上部横座標上の太線（※1）は初めの b_i~粒子が、惑星と衝突することを示している。 ※1：図の一番上の座標軸にところどころ横に太く線を引いた部分がありますので、そこのことかと考えます。
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Now, Figs. 3(a) and (b) indicate that there seems to be several discontinuous spikes in the range of |bi| between 1.8 and 2.6, which hereafter is called the discontinuous band. In order to see the detailed features of the scattering in this region, we have calculated the particle orbits in a fine division of bi. The results, which are illustrated in Figs. 3(a) and (b), show that in almost all of the region ⊿b continues smoothly with respect to bi but there remain the fine discontinuous bands near bi=1.92 and bi=2.38. Again we have tried to magnify the range of bi between 1.915 and 1.925 and found that there are still finer discontinuities near bi=1.919(see Figs. 4(a) and (b)).
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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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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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Fig.4. Magnified figures of one of the discontinuous bands in 1.915<b_i~<1.925 in Figs. 3(a) and (b). There are many fine discontinuous bands in itself. Fig.5. Examples of particle orbits which belong to the discontinuous bands in Fig.4. (a) and (b) are of particles which escape from the Hill sphere through the gate of L_1 and through that of L_2, respectively. The values of b_i~ are 1.91894 (a) and 1.91898 (b). Fig.6. Examples of purely Keplerian orbits with e_i~=4 for b_i~=2 and b_i~=10. The former curls (|b_i~|<4(e_i~/3) and the latter waves (|b_i~|>4(e_i~/3)). The values of δ* of both orbits are zero. ↓Fig.4. http://www.fastpic.jp/images.php?file=0045805708.jpg ↓Fig.5. http://www.fastpic.jp/images.php?file=9845810934.jpg ↓Fig.6. http://www.fastpic.jp/images.php?file=0162323295.jpg どうかよろしくお願いします。
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For regions where |b_i~|≦5.7, we have the same pattern in ⊿b~ and ⊿e~ as those for the case e_i~=0; i.e., they jump discontinuously by a bit change of δ* (see Fig.8). In the same manner as described for e_i~=0 we have tried to decompose up these discontinuous bands. Unfortunately we cannot succeed either in the complete decomposition, because there seems to exist an infinite series of the fine structure. But contributions of the fine structure to the average values, such as <⊿b~> and <⊿e~> are found to be negligible too. In Figs.9 (a) and (b) the average values of <⊿b~> and <⊿e~> are illustrated as a function of b_i~, where <⊿b~> and <⊿e~> are defined as <⊿b~>=(1/2π)∫⊿b~dδ* (3.8) and <⊿e~>=(1/2π)∫⊿e~dδ*, (3.9) respectively. Fig.8. Some discontinuous bands of the change of the impact parameter for continuous variation of δ* with b_i~=3.0 and with e_i~4. Fig.9. (a) The change of the impact parameter, <⊿b~> and (b) the change of the eccentricity, <⊿e~> averaged over the parameter δ* for various values of b_i~ between －10 and 10 with e_i~=4. http://www.fastpic.jp/images.php?file=2049108297.jpg ↑Fig.8. http://www.fastpic.jp/images.php?file=5493230572.jpg7 ↑Fig.9. (a) and (b) 長文ですが、よろしくお願いします。
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Fig.10. The collision probability P_c(b_i~) for e_i~=4 at the region near the present Earth’s orbit. ↓Fig.10. http://www.fastpic.jp/images.php?file=3466353586.jpg Fig.11. The coefficient C_K obtained from our orbital calculations for both cases e_i~=0 and 4. Each mark indicates the value of C_K at the position of each present planet. Solidlines show C_K∝R^－(1/2). ↓Fig.11. http://www.fastpic.jp/images.php?file=4992926857.jpg Table I. The ratio, γ, of the collisional rate of Keplerian particles to that of free space particles for the various regions from the Sun. The coefficient C_K, defined by Eq. (4・6), and the planetary radius, γ_p, in units of h are also tabulated. ↓Table I. http://www.fastpic.jp/images.php?file=9315075324.jpg For e_i~=0, the orbital element, δ_i, loses its meaning as mentioned in §3.1 and only impact parameter b_i~ determines whether a particle collides with the planet or not; i.e., the collision probability P_c(b_i~) is unity or zero. 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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In order to see the detailed features of the scattering in this region, we have calculated the particle orbits in a fine division of b_i~. The results, which are illustrated in Figs. 3(a) and (b), show that in almost all of the region ⊿b~ continues smoothly with respect to bi but there remain the fine discontinuous bands near b_i~=1.92 and b_i~=2.38. Again we have tried to magnify the range of b_i~ between 1.915 and 1.925 and found that there are still finer discontinuities near b_i~=1.919(see Figs. 4(a) and (b)). 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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Accordingly, in our study, it is sufficient to try to find collision orbits with a limited range of b, i.e., b_min<b<b_max. Thus, the collisional rate <P(e, i)> is written practically as (see Eq. (10)) <P(e, i)>=∫[b_max→b_max](3/2)db(4b/(2π)^2)∫[0→2π]dτ∫[0→π]dωp_col(e, i, b, τ,ω). ・・・・・・(14) Unfortunately, we cannot predict b_min and b_max definitely. As to b_max, we only know its upper limit from the Jacobi integral E of Hill’s equations, given by (see Hayashi et al., 1977 and Paper I), E=(1/2){e(r)^2+i(r)^2}-(3/8)b(r)^2-(3/r)+(9/2). ・・・・・・(15) Particles with E<0 cannot enter the Hill sphere and never collide with the protoplanet. The condition E>0 yields an upper limit of b_max, in terms of orbital elements at infinity (r→∞), b_max<{(4(e^2+i^2)/3)+12}^(1/2). ・・・・・・(16) By the following consideration, a lower limit of b_min can also be estimated. First, we will find a turn-off point of the horseshoe orbit of the guiding center (see Fig.1). In the two-dimensional case, Hénon and Petit (1986) have found that the variation of e is much smaller than that of b at a large distance. This suggests that, as a first order approximation, we can neglect the variations of e and I compared to the b variation also in the three-dimensional case. Thus, assuming that both e and I are invariant, we have from Eq. (15) b(r)^2+(8/r)=b^2, ・・・・・・(17) where b is the semimajor axis at infinity. Fig.1. Example of an orbit with small b. Dashed curve denotes the trajectory of the guiding center. Turn-off point of the guiding center y_t and Hill sphere (x^2+y^2=1) are also shown. 長文ですが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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