瞬時ケプラー軌道要素に基づいた解の表現
- 瞬時ケプラー軌道要素に基づいて、式(6)の解を表現することは便利です。
- 相対運動の瞬時ケプラー軌道要素（e、i、b、τ、ω、φ）に関連する式(7)を使用すると、軌道が次のように与えられます。
- b、e、I、τ、およびωは、相対運動の軌道の軌道要素であり、それぞれの半長軸、離心率、軌道傾斜角、近日点の経度、昇交点の経度です。
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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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よって、相対運動（e, i, b, τ, ω, φ)の瞬間ケプラー軌道要素の観点から方程式(6)の解を説明したい。軌道は次のように与えられる。 x=b－ecos(t－τ), y=－3b(t－φ)/2+2esin(t－τ), z=isin(t－ω), ・・・・・・・(7) ここでは、b, e, I, τ, ωはそれぞれ、軌道長半径、離心率、軌道傾斜角、近日点経度、そして相対運動の昇交点のことである。
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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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The normalized Hill’s equations of the relative motion are described as (for definitions of coordinates, see PaperI) x’’=2y’+3x－3x/r^3, y’’=－2x’－3y/r^3, z’’=－z－3z/r^3, (6) where time is normalized by the Keplerian period Ω^-1 and length by the Hill radius ha_0*. 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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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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3.2. For the case e_i~=4 As seen from Fig.6, an orbit with a finite eccentricity curls (|b_i~|<4(e_i~/3)) or waves (|b_i~|>4(e_i~/3)) for the purely Keplerian motion which is realized when the planetary mass is very small or a particle is far from the planet. Therefore, δ, which is the relative phase differnce between the planet and the perigee point of the particle orbit near the planet, is expected to be one of the important parameters for determining the features of the two-body encounters. It is convenient to introduce, here, a phase parameter δ*, in place of δ, defined as δ*=δ-ε/(1-a^(3/2)). (3・7) どうかよろしくお願いします。
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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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In our orbital calculations, the starting points are assigned in the form (see Eq. (7)) x_s=b_s-ecos(t_0-τ_s), y_s=y_0+2esin(t_0-τ_s), ・・・・・・・・・・・(21) z_s=isin(t_0-ω_s), where t_0 is the origin of the time, independent of τ_s and ω_s, and y_0 is the starting distance (in the y-direction) of guiding center. Table 1. The values of b_min* evaluated from Eq. (18). The values of numerically found b_min are also tabulated. よろしくお願いいたします。
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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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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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