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- e, i, and b are scaled by h in Eq. (8)
- Normalized orbital elements given by Eq. (8) are used
- The values e*, i*, and a* denote eccentricity, inclination, and semimajor axis
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It is to be noted here that e, i, and b are also scaled by h: e=e*/h, i=i*/h, and b=(a*－a_0*)/ha_0*, ・・・ (8) where e*, i*, and a* denote, respectively, the instantaneous eccentricity, inclination, and semimajor axis of usual use. Throughout the present paper, the normalized orbital elements given by Eq. (8) will be used.
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ここでは、e, i, b もまた h により拡大縮小されるということも注意すべきである。 e=e*/h, i=i*/h, and b=(a*－a_0*)/ha_0*, ・・・ (8) ここでのe*, i*, a* は、それぞれ瞬間離心率、軌道傾斜角、通常使用での軌道長半径を意味する。本論文中では、方程式(8)により与えられた正規化された軌道要素が使用される。
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- Nakay702
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ここで注目すべきは、e, i, および b はまた h によっても計測されることである。 e=e*/h, i=i*/h, and b=(a*－a_0*)/ha_0*, ・・・ (8) 上で、e*, i*, および a* はそれぞれ、通常使用からの離心、傾斜、軸ズレ（？）を示す。本稿全体を通して、(8) 式によって与えられる正常化軌道因子が用いられる。
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- 和訳してください。 よろしくお願いします。
Thus, it is convenient to express a solution to Eq. (6) in terms of the instantaneous Keplerian orbital elements of the relative motion (e, i, b, τ, ω, φ): the orbit is given by x=b－ecos(t－τ), y=－3b(t－φ)/2+2esin(t－τ), z=isin(t－ω), ・・・・・・・(7) where b, e, I, τ, and ω are the semimajor axis, eccentricity, inclination, the longitude of perihelion, and that of ascending node of the relative motion, respectively.
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We start our orbital calculation of a particle from a point far from the planet where the effect of the gravitational force of the planet can be neglected. At an initial point where a particle is governed almost completely by the solar gravity, the particle orbit can be approximated in a good accuracy by the Keplerian. Hence we will give the initial conditions for the orbital calculation in terms of the Keplerian orbital elements (a_i, e_i, ε_i, δ_i) in place of (x, x’, y, y’), where a, e and ε are the semi-major axis, the eccentricity and the mean longitude at t=0, respectively. Furthermore the parameter δ is defined as δ=－nt_0, (2・8) where n and t_0 are the mean angular velocity and the time of the perihelion passage, respectively. よろしくお願いします。
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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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We will add some comments on the choice of the initial conditions. It should be noticed that we need not survey every set of phase parameters, ε_i and δ_i, when the semi-major axis and the eccentricity of the particle orbit are once fixed. This is due to the fact that there are numberless same orbits with different sets of ε_i and δ_i. Therefore, noticing that δ_i has a 2π-modulous, we can represent, practically, all of the possible phase of the Keplerian orbits by ε_i=constant and －π≦δ≦π. Secondly, we will transform a_i and e_i to b_i~ and e_i~, respectively, according to a_i=1+b_i~h (2・9) and e_i=e_i~h. (2・10) Here h is the radius of the sphere within which the gravity of the planet overcomes the solar gravity, i.e., the radius of the Hill sphere, and is defined in the units adopted here as h=(μ/3)^(1/3)=2.15×10^(－3), (2・11) where the planetary mass M is chosen to be 5.977×10^25g, i.e., one-hundredth of the present terrestrial mass. As shown by Hayashi et al, there exists an approximate similarity law between solutions to the plane circular RTB problem as long as μ≦10^－4, which is well scaled by the above transformations and, hence, the results obtained for the special choice of μ (or M) can apply extensively to the problem with different value of μ. 長い文章ですが、ご教授いただけると助かります。
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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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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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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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To avoid this difficulty, we consider the scale height to be (i+αr_G) rather than i, where α is a numerical factor; α must have a value of the order of 10 to be consistent with Eq. (35). For the requirements that in the limit of i=0, <P(e,i)>_2B has to naturally tend to <P(e,0)>_2B given by Eq. (28), we put the modified collisional rate in the two-body approximation to be <P(e,i)>_2B=Cπr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2(i+ατ_G)) (36) with C=((2/π)^2){E(k)(1-x)+2αE(√(3/4))x}, (37) where x is a variable which reduces to zero for i>>αr_G and to unity for i<<αr_G. The above equation reduces to Eq. (29) when i>>αr_G while it tends to the expression of the two-dimensional case (28) for i<<αr_G. Taking α to be 10 and x to be exp(-i/(αr_G)), <P(e,i)> scaled by Eq. (36) is shown in Fig. 17. Indeed, the modified <P(e,i)>_2B approximates <P(e,i)> within a factor of 5 in whole regions of the e-I plane, especially it is exact in the high energy limit (v→∞). However, two peaks remain at e≒1 and i≒3, which are closely related to the peculiar features of the three-body problem and hence cannot be reproduced by Eq. (36). Fig. 16a and b. Behaviors of r_min(i,b): a i=0, b i=2, 2.5, and 3.0. The level of the planetary radius (r_p=0.005) is denoted by a dashed line. Fig. 17. Contours of <P(e,i)> normalized by the modified <P(e,i)>_2B given by Eq. (36). Fig. 16a and b.↓ http://www.fastpic.jp/images.php?file=4940423993.jpg Fig. 17.↓ http://www.fastpic.jp/images.php?file=5825412982.jpg よろしくお願いします。
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In the preceding papar (Nakazawa et al., 1989a,referred to as Papar I), we have proposed that a framework of Hill’s equations (Hill, 1878) is of great advantages to find precisely the collisional rate between Keplerian particles over wide ranges of initial conditions. First, in Hill’s equations, the relative motion separates from the barycenter motion, and the equation of the barycenter motion can be integrated analytically (see also Henon and petit, 1986). Second, the equation of motion can be scaled by h and Ω; h is the reduced Hill radius and Ω is the Keplerian angular velocity. They are given by h=(m_p/3M_?)^(1/3) (4) and Ω={G(M_?+m_p)/a_0*^3}^(1/2), (5) Where a_0* is the reference heliocentric distance (which is usually taken to be equal to the semimajor axis of the protoplanet) and M_? is the solar mass. The first characteristic of Hill’s equations permits us to reduce the degree of freedom of particle motion and, hence, to reduce greatly the number of orbits to be pursued numerically. The second permits us to apply the result of an orbital calculation with particular m_p and a_0* to orbital motion with other arbitrary mass and heliocentric distance. 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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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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