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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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1.序論 これは太陽重力の影響を十分に考慮しながら、原始惑星と微惑星間の衝突確率を研究した論文シリーズの第三弾である。今までケプラー粒子間の衝突確率は、惑星形成の研究においてその重要性にもかかわらず、十分に理解されてこなかった。そして便宜的に二体（すなわち自由空間）近似が採用されてきた。二体近似における衝突速度（例えば 1969年のサフロノフ）は次の方程式によって与えられている。 σv=πr_p^2(1+(2Gm_p/r_pv^2))v, (1) ここでは r_p と m_pはそれぞれ半径の合計と原始惑星と衝突する微惑星の質量である。更に v は無限遠相対速度であり、通常は微惑星の平均ランダム速度に等しくなるよう選ばれる。すなわち v=(<e_2*^2>+<i_2*^2>)^(1/2)v_K, (2) ここでは <e_2*^2> と <i_2*^2>は太陽中心座標離心率の平均の2乗と微惑星群の軌道傾斜角であり、 v_Kはケプラー粒子の速度である。すなわち（例えば <i_2*^2>=0）のような平面問題において、衝突速度は方程式(1)のかわりに次のように与えられる。 (σ_2D)v=2r_p(1+(2Gm_p/r_pv^2))^(1/2)v. (3) 方程式(1)と(3)については、衝突速度における太陽重力の影響を明確にする為に、後の節にて言及する。
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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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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。
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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.
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In the two-dimensional case where the protoplanet and the planetesimal revolve in the sample plane around the protosun (i.e., i=0), the phase angle ω loses its meaning and Eq. (10) must be modified as <P(e, 0)>=∫(3/2) |b|(1/2π)p_col(e, i=0, b, τ)dτdb. ・・・・・(11) We consider that the planetesimal collides with the protoplanet if the separation distance becomes smaller than the sum of their radii; the protoplanet radius scaled by ha_0* is given by r_p=0.005(ρ/3gcm^-3)^-(1/3)(a_0*/1AU)^-1, ・・・・・(12) where ρ is the mean mass density of the protoplanet. よろしくお願いいたします。
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The obtained collisional rate is summarized in terms of the normalized eccentricity e and inclination i of relative motion; the normalized eccentricity e and inclination i of relative motion; the normalization is based on Hill’s framework, i.e., e=e*/h and i=i*/h where e* and i* are ordinary orbital elements and h is the reduced Hill radius defined by (m_p/3M_? )^(1/3) (m_p being the protoplanet mass and M_? the solar mass). The properties of the obtained collisional rate <P(e,i)> are as follows: (i) <P(e,i)> is like that in the two-dimensional case for i≦0.1 (when e≦0.2) and i≦0.02/e (when e≧0.2), (ii) except for such a two-dimensional region, <P(e,i)> is always enhanced over that in the two-body approximation <P(e,i)>_2B, (iii) <P(e,i)> reduces to <P(e,i)>_2B when (e^2+i^2)^(1/2)≧4, and (iv) there are two notable peaks in <P(e,i)>/ <P(e,i)>_2B at regions where e≒1 (i<1) and where i≒3 (e<0.1); the peak values are at most as large as 5. As an order of magnitude, the collisional rate between Keplerian particles can be described by that of the two-body approximation suitably modified in the two-dimensional region. However, the existence of the peaks in <P(e,i)>/ <P(e,i)>_2B are characteristic to the three-body problem and would give an important insight to the study of the planetary growth. お手数ですがよろしくお願いいたします。
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Fig. 3a and b. An example of recurrent non-collision orbits. The orbit with b=2.4784, e=0, and i=0 is illustrated. To see the orbital behavior near the protoplanet, the central region is enlarged in b; the circle shows the sphere of the two-body approximation and the small one the protoplanet. Fig. 4a and b. Same as Fig. 3 but b=2.341, e=0, and i=0. This is an example of recurrent collision orbits. Fig.5. Minimum separation distance r_min between the protoplanet and a planetesimal in the case of (e, i)=(0, 0). By solid curves, r_min in the first encounter is illustrated as a function of b. The level of protoplanetary radius is shown by a thin dashed line. The collision band in the second encounter orbits around b=2.34 is also shown by dashed curves. Fig. 6. Minimum distance r_min in the chaotic zone near b=1.93 for the case of (e, i)=(0, 0); it changes violently with b. Fig. 7a-c. Contours of minimum separation distance r_min in the first encounter for the case of (e, i)=(1.0, 0.5); b=2.3(a), 2.8(b), and 3.1 (c). Contours are drawn in terms of log_10 (r_min) and the contour interval is 0.5. Regions where r_min>1 are marked by coarse dots. Particles in these regions cannot enter the Hill sphere of the protoplanet. Fine dots denotes regions where r_min is smaller than the radius r_cr of the sphere of the two-body approximation (r_cr=0.03; log_10(r_cr)=-1.52). Fig. 3a and b. および Fig. 4a and b. ↓ http://www.fastpic.jp/images.php?file=3994206860.jpg Fig.5. および Fig. 7a-c. ↓ http://www.fastpic.jp/images.php?file=2041732569.jpg Fig. 6. ↓ http://www.fastpic.jp/images.php?file=2217998690.jpg お手数ですが、よろしくお願いします。
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The numbers of collision orbits found in the present calculations are shown in Table 4 for the representative sets of (e,i). From these numbers we can expect the magnitude of statistical error in the evaluation of <P(e,i)> to be a few percent for small e, i and within 10% for large e, i for r_p=0.005 are shown in Table 5, together with those of the two-dimensional case. Interpolating these values, we have obtained the contour of <P(e,i)> and R(e,i) on the e-I plane. They are shown in Figs. 14 and 15. From Fig. 15 we can read out the general properties of the collisional rate in the three-dimensional case: (i) <P(e,i)> is enhanced over <P(e,i)>_2B except for small e and i, (ii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, and (iii) there are two peaks in R(e,i) near regions where e≒1 (i<1) and where i≒3 (e<0.1): the peak value is at most as large as 5. In the vicinity of small v(=(e^2+i^2)^(1/2)) and i, R(e,i) rapidly reduces to zero. This is due to a singularity of <P(e,i)>_2B at v=0 and i=0 in the ordinary expression given by Eq. (29) and hence unphysical; the behavior of collisional rate in the vicinity of small v and i will be discussed in detail later. Thus, we are able to assert, more strongly, the property (i) mentioned in the last paragraph: that is, solar gravity always enhances the collisional rate over that of the two-body approximation. One of the remarkable features of R(e,i) found in Fig. 15 is the property (ii). That is, the collisional rate between Keplerian particles is well described by the two-body approximation, for (e^2+i^2)^(1/2)≧4. This is corresponding to the two-dimensional result that R(e,0)≒1 for e≧4. よろしくお願いします。
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As mentioned above, there are two peaks in R(e,i) in the e-i diagram: one is at e≒1 (i<1) and the other at i≒3 (e<0.1). The former corresponds directly to the peak in R(e,0) at e≒1 found in the two-dimensional case. The latter is due to the peculiar nature to the three-dimensional case, as understood in the following way. Let us introduce r_min (i, b,ω) in the case of e=0, which is the minimum distance during encounter between the protoplanet and a planetesimal with orbital elements i, b, and ω. In Fig. 16, r_min(i,b)=min_ω{r_min(i,b,ω)} is plotted as a function of b for various i, where r_min<r_p (=0.005) means “collision”; there are two main collision bands at b≒2.1 and 2.4 for i=0. For i≦2, these bands still exist, shifting slightly to small b. This shift is because a planetesimal feels less gravitational attracting force of the protoplanet as i increases. As i increases further, the bands approach each other, and finally coalesce into one large collision band at i≒3.0; this large band vanish when i≧4. In this way, the peculiar orbital behavior of three-body problem makes the peak at i≒3 (e<0.1). Though there are the peaks in R(e,i), the peak values are not so large: at most it is as large as 5. This shows that the collisional rate is well described by that of the two-body approximation <P(e,i)>_2B except for in the vicinity of v, i→0 if we neglect a difference of a factor of 5. Now we propose a modified form of <P(e,i)>_2B which well approximates the calculated collisional rate even in the limit of v, i→0. We find in Fig.14 that <P(e,i)> is almost independent of i, i.e., it behaves two-dimensionally for i≦{0.1 (when e≦0.2), {0.02/e (when e≧0.2). (35) This transition from three-dimensional behavior to two-dimensional behavior comes from the fact that the isotropy of direction of incident particles breaks down for the case of very small i (the expression <P(e,i)>_2B given by Eq. (29) assumes the isotropy). In other words, as an order of magnitude, the scale height of planetesimals becomes smaller than the gravitational radius r_G=σ_2D/2 (σ_2D given by Eq. (3)) and the number density of planetesimals cannot be uniform within a slab with a thickness σ_2D for small i. Table 4. Numbers of three-dimensional collision events found by orbital calculations for the representative sets of e and i. In the table r_p is the radius of the protoplanet. Table 5. The three-dimensional collision rate <P(e,i)> for the case of r_p=0.005 (r_p being the protoplanetary radius), together with two-dimensional <P(e,i=0)> Fig. 14. Contours of the evaluated <P(e,i)>, drawn in terms of log_10<P(e,i)> Fig. 15. Contours of the enhancement factor R(e,i) Table 4.↓ http://www.fastpic.jp/images.php?file=1484661557.jpg Table 5.↓ http://www.fastpic.jp/images.php?file=6760884829.jpg Fig. 14. ＆Fig. 15.↓ http://www.fastpic.jp/images.php?file=8798441290.jpg 長文になりますが、よろしくお願いします。
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The second feature seen from Fig.11 is that the profile of R(e,0) does not depend significantly on r_p (for r_p=0.005 to 0.0002). Only an exception is found near e≒1, but this is, in some sense, a singular point in R(e,0), which appears in a narrow region around e≒1 ( in fact, for e=0.9 and 1.2, there is no appreciable difference between r_p=0.005 and 0.0002). Thus, neglecting such fine structures in R(e,0), we can conclude that R(e,0) does depend very weakly on r_p. In other words, the dependence on r_p of <P(e,0)> is well approximated by that of <P(e,0)>_2B given by Eq. (28). Now, we will phenomenalogically show what physical quantity is related to the peak at e≒1. We introduce the collisional flux F(e,E) for orbits with e and E, where E is the Jacobi energy given by (see Eq. (15)) E=e^2/2-(3b^2)/8+9/2. (31) The collisional flux F(e,E) is defined by F(e,E)=(2/π)∫【‐π→π】p_col(e,i=0, b(E), τ)dτ. (32) From Eqs. (11) and (31), we obtain <P(e,0)>=∫F(e,E)dE. (33) In Fig.12, F(e,E) is plotted as a function of E for the cases of e=0, 0.5, 1.0, and 2.0. We can see from this figure that in the case of e=1 a large fraction of low energy planetesimals contributes to the collisional rate compared to other cases (even to the cases with e<1). In general, in the case of high energy a solution for the three-body problem can be well described by the two-body approximation: in other words, in the case of low energy a large difference would exist between a solution for the three-body problem and that in the two-body approximation. As shown before, this difference appears as an enhancement of the collisional rate. Thereby an enhancement factor peak is formed at e≒1 where a large fraction of low-energy planetesimals contributes to the collisional rate. よろしくお願いいたします。
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