Initial Conditions for Orbital Integration| OKWAVE

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Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。

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Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。

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In the preceding papar (Nakazawa et al., 1989a,referred to as Papar I), we have proposed that a framework of Hill’s equations (Hill, 1878) is of great advantages to find precisely the collisional rate between Keplerian particles over wide ranges of initial conditions. First, in Hill’s equations, the relative motion separates from the barycenter motion, and the equation of the barycenter motion can be integrated analytically (see also Henon and petit, 1986). Second, the equation of motion can be scaled by h and Ω; h is the reduced Hill radius and Ω is the Keplerian angular velocity. They are given by h=(m_p/3M_?)^(1/3) (4) and Ω={G(M_?+m_p)/a_0*^3}^(1/2), (5) Where a_0* is the reference heliocentric distance (which is usually taken to be equal to the semimajor axis of the protoplanet) and M_? is the solar mass. The first characteristic of Hill’s equations permits us to reduce the degree of freedom of particle motion and, hence, to reduce greatly the number of orbits to be pursued numerically. The second permits us to apply the result of an orbital calculation with particular m_p and a_0* to orbital motion with other arbitrary mass and heliocentric distance. 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. 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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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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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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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 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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　　　Now, we shall concentrate on the collision orbits. Figure 5 illustrates the minimum separation distance r_min in the first encounter (solid curves), identical to that obtained by Petit and Hénon (1986). One sees immediately, that there are two different zones: the “regular” zones, in which r_min varies smoothly with a change of parameter b and the irregular (or “chaotic”) zones, where r_min changes greatly with tiny differences in the choice of b. The chaotic zones lie near b=1.93, 2.30 and 2.48, with very narrow ranges of b. In the regular zone, we find two broad bands of collision orbits around b=2.09 and 2.39. These collision bands were first found by Giuli (1968). The sum of width of the collision bands ⊿b is found to be about 0.098, if the planetary radius is 0.005. 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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In the present paper, we first describe initial conditions for orbital calculations in the framework of Hill’s equations (Sect.2). In Sect.3, orbital calculations are made for particular sets of e and i, i.e., (e,i)=(0,0) and (1,0.5). Based on these calculations and previous works: Papers I and II, Nishida (1983), Hénon and Petit (1986), and Petit and Hénon (1986), we develop an efficient numerical procedure obtaining <P(e, i)> (Sect. 4).According to this procedure, we systemically integrate the orbits of relative motion, to find collision orbits. Furthermore, compiling results of these orbital calculations, we find <P(e, i)> for various sets of e and i and compare them with those in the two-body approximation <P(e, i)>_2B (Sects. 5 to 7). The results will be compared with those of Nishida (1983) and Wetherill and Cox (1985) who also studied the collisional rate taking account of the effect of solar gravity (but their studies were restricted, as mentioned later).

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　　　In the chaotic zone, there are, of course, a great number of discrete collision orbits. Minimum separation distance in the chaotic zone near b=1.93 is enlarged in Fig.6, which is obtained from the calculation of 3000 orbits with b between 1.926 and 1.932. Even in this enlarged figure, r_min varies violently with b. Although the chaotic zones are not sufficiently resolved in our present study, the phase space occupied by collision orbits in the chaotic zones is much smaller than that in the regular collision bands. Even if all orbits in the chaotic zone are collisional, their contribution to the collision rate is less than 4% of the total: the width in b=2.30 and 2.48, we also found that the total width is much smaller than 0.001. This implies that in the evaluation of <P(e, i)>, we can neglect the contribution of collision orbits in the chaotic zones. 　　　These are n-recurrent collision orbits in the regular zones. Of these, 2-recurrent collision orbits are most important. The collisional band composed of them is found near b=2.34. Its width ⊿b is about 0.011, and the contribution to the collision is as large as 15%. No.3- and more –recurrent collision orbits were observed in regular zones. They were found only in the chaotic zones and, hence, can be neglected. 長いですが、よろしくお願いします。

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　　　　　　　　　　　　　　　　3. Results of the orbital calculations 　 In this paper, we describe our numerical results only for the special cases e_i~=0 and e_i~=4 in order to find the characteristic features of the two-body encounters of Keplerian particles. 3.1. For the case e_i~=0 　 In this case the orbital element, δ_i, loses its meaning because the particle orbits have no periheria and , hence, we can actually assign the initial condition by one parameter, b_i~. The orbital calculations are performed for about 3800 cases with various values of b_i~ from －10 to 10. よろしくお願いします。

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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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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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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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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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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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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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　　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。

Initial Conditions for Orbital Integration

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