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# この文章の和訳を教えてください。

It should be noticed that Λ_f is independent of both ei~ and R. In order to compare Λ_K with Λ_f clearly, we consider the ratio of these, that is, γ=Λ_K/Λ_f=3.0C_K(R/1AU)^1/2, (4・9) which is independent of the mass of the planet. The values of γare listed in Table I as well as those of C_K. As the parameter C_K is approximately proportional to R^-1/2 (see Fig.11), γ is almost independent of R and is 2.3 for ei~=0 and 1.4 for ei~=4. This indicates that, though the result is obtained in the limited framework of the two-dimensional particle motion, the collisional rate of Keplerian particles is enhanced by a factor of about 2.3 or 1.4 compared to that estimated in a free space formula, as long as we are concerned with the two cases of eccentricity. Furthermore, as seen from Table I, γ for the case ei~=4 is significantly smaller than that of the case ei~=0. This seems to confirm the conjecture that γ tends to unity in the high energy limit (i.e., ei~→∞), or in other words, the free space formula is right only in the high energy limit. お手数ではございますが、どうかよろしくお願いいたします。

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Λ_f がei~ と R双方と無関係であることに気づくべきである。 Λ_KとΛ_f を明確に比較する為に、我々はこれらの比率を以下のように見なす。 γ=Λ_K/Λ_f=3.0C_K(R/1AU)^1/2, (4・9) この公式は惑星の質量とは無関係である。 γの値に関してはC_Kの値同様、表1に記載されている。パラメーター C_K はほぼ R^-1/2に比例するので（図11参照のこと）γはRにほぼ依存しておらず、その値はei~=0の時には2.3で、ei~=4の時は1.4である。これは、二次元粒子運動の限られた枠組における結果であったとしても、ケプラー粒子の衝突速度は自由空間数式において予測されるものと比較して2.3もしくは1.4倍強められることを示している。更に表１で示した通り、ei~=4の場合のγはei~=0の場合よりも著しく小さい。これはγが高エネルギー限界(i.e., ei~→∞)で1になる傾向があるという推測を確かにするものであり、言い換えれば自由空間数式は高エネルギー限界領域においてのみ正解ということである。 「ケプラーの法則」は専門外なので不適当な訳があった場合はご容赦ください。

### 関連するQ&A

• これの和訳を教えてください。

The evaluated values of C_K are shown in Fig. 11 and listed in Table I for the cases ei~=0 and 4, respectively. As is seen from Fig.11, C_K is approximately proportional to R~-1/2 and v_K also has the same dependence on R. Therefore, Λ_K is almost independent of R. On the other hand, the collisional rate for a free space, Λ_f is given by Λ_f=n_s2R_p(1+2θ)^(1/2)v(ei~), (4・7) where v(ei~) is the mean random velocity of an ensemble of particles with an eccentricity, ei~, and is given by √(5/8)ei~hv_K and θ is the Safronov number, which is defined as GM/v^2(ei~)R_p. For the cases ei~≦4, θ is much larger than unity, and then Eq.(4・7) is approximately rewritten as 　　　　　　　　　　　　Λ_f=(8GMR_p)^(1/2)n_s. (4・8)

• これの和訳を教えてください。

The evaluated values of C_K are shown in Fig. 11 and listed in Table I for the cases ei~=0 and 4, respectively. As is seen from Fig.11, C_K is approximately proportional to R~-1/2 and v_K also has the same dependence on R. Therefore, Λ_K is almost independent of R. On the other hand, the collisional rate for a free space, Λ_f is given by Λ_f=n_s2R_p(1+2θ)^(1/2)v(ei~), (4・7) where v(ei~) is the mean random velocity of an ensemble of particles with an eccentricity, ei~, and is given by √(5/8)ei~hv_K and θ is the Safronov number, which is defined as GM/v^2(ei~)R_p. For the cases ei~≦4, θ is much larger than unity, and then Eq.(4・7) is approximately rewritten as 　　　　　　　　　　　　Λ_f=(8GMR_p)^(1/2)n_s. (4・8) よろしくお願いします。

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Conclusive remarks As mentioned above, the collisional rate of Keplerian particles is about 2.3 (ei~=0) or 1.4 (ei~=4) times larger than that of free space particles, in the limited framework of the two-dimensional particle motion. We are now carrying out orbital calculations for other values of eccentricity, ei~, of which results appear that for larger values of ei~ the collisional rate of Keplerian particles is not so much different from that of free space particles. These show that the time scale of protoplanery growth is shortened by a little, compared with that deduced from the free space formula and that the free space approximation is almost right within an accuracy of a factor of 2. In order to obtain more definitely the time scale of protoplanetary growth, however, we have to compare the collisional rate with the scattering rate. Moreover, we need to study the three-dimensional collisional process of Keplerian particles, taking into account inclinations of particle orbits.

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In this and subsequent papers we will study worken Keplerian particles in the framework of the plane circular Restricted Three-Body problem ( the plane circular RTB problem). The aim of the present paper is to represent, for example, the collisional rate by means of the numerical solutions to the plane circular RTB problem for special cases and to compare the result with that deduced from the formula in a free space.

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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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The corresponding velocity components are given by x_s'=esin(t_0-τ_s) y_s'=‐(3/2)b_s+2ecos(t_0-τ_s), (22) z_s'=icos(t_0-ω_s). In the above, we set t_0=-(2y_0/(3b)) without any loss of generality, for later convenience, which is equivalent to the choice of φ=0 in Eq. (7). In Eq. (21), y_0 is set to be max (40, 20e, 20i). The choice of y_0 is not essential to the evaluation of <P(e, i)> as long as y_0 is larger than this value since Eq. (19) is valid for such a large y_0. In Table 2, the evaluated values of <P(e,i)> are tabulated for various y_0 in the cases of (e, i) = (0,0) and (4,0): when y_0≧max(40, 20e, 20i), <P(e, i)> is almost independent of y_0 within an accuracy of 0.1% for the case of (e, i)=(0, 0) and 5% for (e, i)=(4, 0). よろしくお願いします。

• この文章の和訳をお願いします。

1. Introduction This is the third of a series of papers in which we have investigated the collisional probability between a protoplanet and a planetesimal, taking fully into account the effect of solar gravity. Until now, the collisional probability between Keplerian particles has not been well understood, despite of its importance, in the study of planetary formation and, as an expedient manner, the two-body (i.e., free space) approximation has been adopted. In the two-body approximation, the collisional rate is given by (e.g., Safronov,1969) σv=πr_p^2(1+(2Gm_p/r_pv^2))v, (1) where r_p and m_p are the sum of radii and the masses of the protoplanet and a colliding planetesimal, respectively. Furthermore, v is the relative velocity at infinity and usually taken to be equal to a mean random velocity of planetesimals, i.e., v=(<e_2*^2>+<i_2*^2>)^(1/2)v_K, (2) where <e_2*^2> and <i_2*^2> are the mean squares of heliocentric eccentricity and inclination of a swarm of planetesimals and v_K is the Keplerian velocity; in the planer problem (i.e., <i_2*^2>=0), the collisional rate is given, instead of Eq.(1), by (σ_2D)v=2r_p(1+(2Gm_p/r_pv^2))^(1/2)v. (3) Equations (1) and (3) will be referred to in later sections, to clarify the effect of solar gravity on the collisional rate. よろしくお願いします。

• この文章の和訳を教えてください。

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。

• この文章の和訳をよろしくお願いします。

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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Ｉｔ is only in the present dimension of time-that which lies between past and future,between what has already happened what is yet to comethat freedom and the priority of the political for the human world fully emerge in Arendt's thought.For her the political is by no means the be-all and end-all of human experience. It is distinct from “what we can do and create in the singular: in isolation like the artist,in solitude like the philosopher,in the inherently worldless relationship between human beings as it exists in love and sometimes in friendship.