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下記の文章どう訳したら良いでしょうか？添削お願いします
The value rs derived from 式(19) must be compared with the other value rs’ derived from 式(20). we only need to minimize the difference (｜rs-rs’｜) in order to reach an optimal determination of the five orbital parameters involved in such relations (a,ε,i,ω,Λ). 我々はそのような関係（ε,i、ω、Λ）に関係する5つの軌道のパラメータの最適決定に達するために違い（｜rs-rs'｜）を最小にする必要があります。
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式(19)で算出された値ｒｓは、式(20)で算出されたもうひとつの値ｒｓ’と、比較されなければなりません。 そのような関係に関わる5つの軌道パラメータ（ε,i、ω、Λ）を最適に決定するためには、その（ｒｓとｒｓ’の）差（｜rs－rs'｜）を最小にしさえすればよいのです。
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式(19)に由来する値rsは、rsのものが式(20)から演繹（えんえき）された他の値と比較されなければなりません。 我々は、そのような関係（ε,i、ω、Λ）に関係する5つの軌道のパラメータの最適決定に達するために違い（｜rs-rs'｜）を最小にする必要があるだけです。 この文章は数学に関係あると思われますが、数学の中でもどの分野でしょうか？ 調べてみると“difference”の和訳が、微分法などで「差分」、数列などで「階差」となるようです。 参考サイト→http://honyaku.yahoo.co.jp/transtext
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- 下記の文章どう訳したら良いでしょうか？添削お願いします
The algorithm based on rs and rs’ considers only the parameters that define the geometry of the orbit (a,ε,i,ω,Λ),but it does not consider the also necessary adjustment of the other parameters that defines the satellite position within the orbit, that is, the time variable t or the directly related orbital element called Mean Anomaly at the reference time, scene-center time). rsとrsに基づいたアルゴリズムは、軌道(a、ε、i、ω、Λ)の幾何学を定義するパラメーターだけを考慮します。しかし、それは、軌道(すなわち時間変数t、言及時、場面中心時間で中間変則と呼ばれる、直接関連する軌道要素)内の衛星位置を定義する、他のパラメーターのさらなる必要な調節を考慮しません。
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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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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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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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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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In order to see qualitatively the collision frequency, we will define the collision probability, P_c(b_i~) as follows. For a fixed value of b_i~, orbital calculations are made with N's different value of δ*_i, which are chosen so as to devide 2π(from –π to π) by equal intervals. Let n_c be the number of collision orbits, then P(b_i~, N)=n_c/N gives the probability of collision in the limit N→∞. In practice, however, we can evaluate the value of P_c(b_i~) approximately, but in sufficient accuracy in use, by the following procedures; i.e., at first for N=N_1(e.g., N_1=1000), P(b_i~,N_1) is found and, next, for N=2N_1, P(b_i~,N) is reevaluated. お手数ですが、どうか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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Furthermore, ρ is the density of the planet and is taken to be 4.45g/cm^3 according to Lewis. The values of r_p/h are tabulated in Table I for various regions from the Sun. It is to be noticed that the planetary radius decreases as the increase in the distance from the Sun in the system of units adopted here. Orbital calculations are performed by means of the 4th order Runge-Kutter-Gill method with an accuracy of double precision. 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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According to Paper I, the total collisional rate <Γ(e_1,i_1)> of planetesimals upon the protoplanet with the heliocentric orbital elements e_1 and i_1 is given by <Γ(e_1,i_1)>=2π^2∫<n_2>e_i<P(e,i)>dedi, ・・・・・・・(9) where <n_2> is the distribution function of planetesimals averaged by the phase angles τ_1 and ω_1 of the protoplanet, and e and i are the orbital elements of relative motion between the protoplanet and the planetesimal at infinity. よろしくお願いします。
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