Collisional Rate of Keplerian Particles and Its Expression

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  • In this article, we explore the collisional rate of Keplerian particles and derive its expression.
  • We focus on cases where the impact parameter is nonzero and assume a uniform distribution of particles in the phase space of other orbital elements.
  • The velocity of the gyrocenter of Keplerian particles can be expressed as -(3/2)bi~hv_K, where bi~ is the impact parameter and v_K is the Keplerian velocity of the circular orbit.
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以前回答をいただいたものです。

　　But the total collisional cross section has no physical meaning for collisional process of Keplerian particles. Thus, we are no longer concerned with the total collisional cross section. Now we will drive the expression of the collisional rate of Keplerian particles. Let n (bi~, v)dv be the number of particles per unit area with an impact parameter, bi~, and with the velocity of the gyrocenter (or the relative velocity between the particles and the planet) between v and v+dv. 　　　Here we are concerned only with the cases ei~=0 and 4, and we assume that particles are distributed uniformly and independently of bi~ in the phase space of the other orbital elements. Noticing that the velocity of the gyrocenter of Keplerian particles can be expressed as－(3/2)bi~hv_K(v_K being the Keplerian velocity of the circular orbit), we can write down n(bi~,v) as n(bi~,v)=n_sδ(v－3/2|bi~|hv_K), (4・3) where δ is the ordinary δ-function and n_s is the surface number density of particles. 長い文章ですが、教えていただけると助かります。

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回答No.1

　But the total collisional cross section has no physical meaning for collisional process of Keplerian particles. Thus, we are no longer concerned with the total collisional cross section. Now we will drive the expression of the collisional rate of Keplerian particles. Let n (bi~, v)dv be the number of particles per unit area with an impact parameter, bi~, and with the velocity of the gyrocenter (or the relative velocity between the particles and the planet) between v and v+dv. 　　　Here we are concerned only with the cases ei~=0 and 4, and we assume that particles are distributed uniformly and independently of bi~ in the phase space of the other orbital elements. Noticing that the velocity of the gyrocenter of Keplerian particles can be expressed as－(3/2)bi~hv_K(v_K being the Keplerian velocity of the circular orbit), we can write down n(bi~,v) as n(bi~,v)=n_sδ(v－3/2|bi~|hv_K), (4・3) where δ is the ordinary δ-function and n_s is the surface number density of particles. 　　　　　　　　　　　　　　　　　　↓ しかし、総衝突断面はケプラー粒子の衝突過程のための物理的な意味を持っていません。したがって、私たちはもはや総衝突断面積に関係ありません。 今、私たちは、ケプラーの粒子の衝突速度の発現を駆動します。 n個の（ビ～、v）のDVは影響パラメータ、双方向～、ととv及びv+ dvの間にgyrocenterの速度（または粒子と地球との間の相対速度）と単位面積当たりの粒子数とする。ここで我々は唯一のケースを懸念しているEI～=0～4、我々は粒子が他の軌道要素の位相空間中のBi～の一様かつ独立に分布していることを前提としています。 Biは～hv_K（v_Kは円軌道のケプラー速度である）、私たちはnとして（BI～、V）を書き留めることができますように - （3/2）ケプラー粒子のgyrocenterの速度を表現することができることに気づい N（BI～、V）=n_sδ（V-3/2| BI～| hv_K）、（4.3） δがどこにあるかδ関数通常かつN_Sは、粒子の表面数密度である。 以上です！

  1. カテゴリ
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関連するQ&A

  • これの和訳を教えてください。

    But the total collisional cross section has no physical meaning for collisional process of Keplerian particles. Thus, we are no longer concerned with the total collisional cross section. 　　　Now we will drive the expression of the collisional rate of Keplerian particles. Let n (bi~, v)dv be the number of particles per unit area with an impact parameter, bi~, and with the velocity of the gyrocenter (or the relative velocity between the particles and the planet) between v and v+dv. Here we are concerned only with the cases ei~=0 and 4, and we assume that particles are distributed uniformly and independently of bi~ in the phase space of the other orbital elements. Noticing that the velocity of the gyrocenter of Keplerian particles can be expressed as－(3/2)bi~hv_K(v_K being the Keplerian velocity of the circular orbit), we can write down n(bi~,v) as 　　　　　　　　　　　　　　　　n(bi~,v)=n_sδ(v－3/2|bi~|hv_K), (4・3) where δ is the ordinary δ-function and n_s is the surface number density of particles. 長い文章ですが、教えていただけると助かります。

  • 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いただけませんか。

    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。

  • 和訳を教えてください。

    　　　　　　　　　　　　　　　　　　　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.

  • この文章の和訳をお願いします。

    　　　In this and subsequent papers we will study extensively the collisional rate and the scattering rate of the two-body encounters between Keplerian particles in the framework of the plane circular Restricted Three-Body problem ( the plane circular RTB problem). よろしくお願いします。

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  • 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。

  • この英文の和訳をお願いします。

    In Fig.13, we compare our results with those of Nishiida (1983) and Wetherill and Cox (1985). Nishida studied the collision probability in the two-dimensional problem for the two cases: e=0 and 4. For the case of e=0, his result (renormalized so as to coincide with our present definition) agrees accurately with ours. But for e=4, his collisional rate is about 1.5 times as large as ours; it seems that the discrepancy comes from the fact that he did not try to compute a sufficient number of orbits for e=4, thus introducing a relatively large statistical error. The results of Wetherill and Cox are summarized in terms of v/v_e where v is the relative velocity at infinity and v_e the escape velocity from the protoplanet, while our results are in terms of e and i. Therefore we cannot compare our results exactly with theirs. If we adopt Eq. (2) as the relative velocity, we have (of course, i=0 in this case) (e^2+i^2)^(1/2)≒34(ρ/3gcm^-3)^(1/6)(a_0*/1AU)^(1/2)(v/v_e). (34) According to Eq. (34), their results are rediscribed in Fig.13. From this figure it follows that their results almost coincide with ours within a statistical uncertainty of their evaluation. 7. The collisional rate for the three-dimensional case Now, we take up a general case where i≠0. In this case, we selected 67 sets of (e,i), covering regions of 0.01≦i≦4 and 0≦e≦4 in the e-i diagram, and calculated a number of orbits with various b, τ,and ω for each set of (e,i). We evaluated R(e,i) for r_p=0.001 and 0.005 (for r_p=0.0002 we have not obtained a sufficient number of collision orbits), and found again its weak dependence on r_p (except for singular points, e.g., (e,i)=(0,3.0)) for such values of r_p. Hence almost all results of calculations will be presented for r_p=0.005 (i.e., at the Earth orbit) here. Fig.13. Comparison of the two-dimensional enhancement factor R(e,0) with those of Nishida (1983) and those of Wetherill and Cox (1985).Their results are renormalized so as to coincide with our definition of R(e,0). 長文ですが、よろしくお願いします。

  • これの和訳を教えてください。

    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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