粒子フラックスと衝突率の式
- この文章では、粒子フラックスと衝突率についての式が示されています。
- 粒子フラックスは粒子の数密度と速度の積分で表され、衝突率はフラックスと衝突面積の積分で表されます。
- これらの式は、ケプラー運動をする粒子に対して適用されます。
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※先ほど投稿した文章とは連続性がありませんのでご注意ください。 Then, the particle flux can be expressed as F=∫n(b_i~,v)vdv=(3/2)n_s|b_i~|hv_K, (4・4) and the collisional rate per unit time for Keplerian particles, Λ_K is given by Λ_K=∫Fdσ_c(b_i~)=(C_K)(h^2)n_sRv_K, (4・5) where C_K=(3/2)∫P_c(b_i~)|b_i~|db_i~. (4・6) よろしくお願いします。
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それから、粒子流束は下記のように表すことが出来る。 F=∫n(b_i~,v)vdv=(3/2)n_s|b_i~|hv_K, (4・4) また、ケプラー粒子Λ_K の時間単位あたりの衝突速度は次式で求められる。 Λ_K=∫Fdσ_c(b_i~)=(C_K)(h^2)n_sRv_K, (4・5) ここでは、 C_K=(3/2)∫P_c(b_i~)|b_i~|db_i~. (4・6) である。
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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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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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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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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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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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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。
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If |P(b_i~,N_1)-P(b_i~,2N_1)|≦0.05×P(b_i~,N_1), then P(b_i~,N_1) (or P(b_i~,2N_1)) is regarded as the collision probability, P_c(b_i~). In Fig.10, P_c(b_i~) is illustrated as a function of b_i~ at the region near the orbit of the present Earth for the case e_i~=4. As seen from this figure, a particle whose impact parameter, b_i~, lies between 1.5 and 5.5, collides with the planet with certain probability. Especially, for |b_i~|≒2 collisions occur frequently. よろしくお願いいたします。
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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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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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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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