- 締切済み
phase volumeとは何ですか？
These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. この文章の中で『phase volume』という単語が出てきますが、これは一体何のことでしょうか？ 検索して見ましたが、よくわかりませんでした。 ご存知の方がいらっしゃいましたら、教えていただけないでしょうか？
- mamomo3
- お礼率68% (87/127)
- 英語
- 回答数3
- ありがとう数0
- みんなの回答 （3）
- 専門家の回答
みんなの回答
- ddeana
- ベストアンサー率74% (2976/4019)
phase volume:位相空間内の体積 これを理解するにリウヴィルの定理（物理学）における配位空間と位相空間の定義を知る必要があります。 配置空間：一般的座標によって構成される空間のこと ※一般的座標系とは http://ja.wikipedia.org/wiki/%E4%B8%80%E8%88%AC%E5%8C%96%E5%BA%A7%E6%A8%99%E7%B3%BB 位相空間：一般化座標とそれと共役する一般化運動量によって構成される空間 ※共役とは http://kotobank.jp/word/%E5%85%B1%E5%BD%B9 位相空間のどこかの点（座標）において何か運動がある場合（この文章では衝突軌道による運動）、当然その運動に伴う一般化運動量があり、それは時間と共にある一定の体積を作り出します。 これが位相空間内の体積です。リウヴィルの定理では時間の経過によって位相空間の領域が変化していく場合があっても領域の体積が不変であることが分かるとされています。
- d-y
- ベストアンサー率46% (1528/3312)
「位相空間の体積」 私には何のことだか全く理解できませんが、ググったらそんな感じの言葉に辿り着きました。 http://encyclopedia2.thefreedictionary.com/_/dict.aspx?rd=1&word=Phase+Volume http://ir.nul.nagoya-u.ac.jp/jspui/bitstream/2237/16106/3/14-位相空間.pdf http://orangepepper.hatenablog.com/entry/2012/09/14/012507 https://ja.wikipedia.org/w/index.php?oldid=46923617
- hirotan1879
- ベストアンサー率20% (512/2470)
星雲レベルでの惑星同士の軌道上での衝突の話でしょうか？ 複数の）軌道を統合して簡略化すれば軌道計算の時間短縮が可能に成る、そして 軌道の初期段階では広い範囲で複数の軌道同士の衝突が支配的だということが分かる。 （初期段階）では多くの（量の）惑星が軌道上で衝突を重ねている という事かと
関連するQ&A
- この文章の和訳を教えてください。
Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳を教えてください。
Collisional probability of planetesimals revolving in the solar gravitational field.III Summary. We have calculated the collisional rate of planetesimals upon the protoplanet, taking fully into account the effect of solar gravity. Our numerical scheme is based on Hill’s equations describing approximately the three –body problem. By the adoption of Hill’s equations , we can reduce the degrees of freedom of orbital motion. Furthermore, we made some simplifications: First, an orbital motion is determined by the formula of the two-body approximation when the distance between the protoplanet and a planetesimal is smaller than a certain critical length. Second, collision orbits in the chaotic zones are neglected in evaluating the collisional rate because of their very small measure. These simplifications enable us to save considerable computation time of orbital integration and, hence, to find numerically the phase volume occupied by collision orbits over wide ranges of orbital initial conditions. よろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳をお願いします。
2. Initial conditions for orbital integration To obtain <P(e, i)>, we must numerically compute a number of orbits of planetesimals with various values of b, τ, and ω for each set of (e, i) and then examine whether they collide with the protoplanet or not. In this section, we consider the ranges of b, τ, and ω to be assigned in orbital calculations, and give initial conditions for orbital integration. One can see in Eq. (6) that Hill’s equations are invariant under the transformation of z→-z and that of x→-x and y→-y; on the other hand, a solution to Hill’s equations is described by Eq. (7). From the above two characteristics, it follows that it is sufficient to examine only cases where 0≦ω≦π and b≧0. Furthermore, we are not interested in orbits with a very small b or a very large b; an orbit with a very small b bends greatly and returns backward like a horse shoe ( Petit and Hénon, 1986; Nishida, 1983 ), while that with a very large b passes by without any appreciable change in its orbital element. どうかよろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳をお願いします。
In the chaotic zone, there are, of course, a great number of discrete collision orbits. Minimum separation distance in the chaotic zone near b=1.93 is enlarged in Fig.6, which is obtained from the calculation of 3000 orbits with b between 1.926 and 1.932. Even in this enlarged figure, r_min varies violently with b. Although the chaotic zones are not sufficiently resolved in our present study, the phase space occupied by collision orbits in the chaotic zones is much smaller than that in the regular collision bands. Even if all orbits in the chaotic zone are collisional, their contribution to the collision rate is less than 4% of the total: the width in b=2.30 and 2.48, we also found that the total width is much smaller than 0.001. This implies that in the evaluation of <P(e, i)>, we can neglect the contribution of collision orbits in the chaotic zones. These are n-recurrent collision orbits in the regular zones. Of these, 2-recurrent collision orbits are most important. The collisional band composed of them is found near b=2.34. Its width ⊿b is about 0.011, and the contribution to the collision is as large as 15%. No.3- and more –recurrent collision orbits were observed in regular zones. They were found only in the chaotic zones and, hence, can be neglected. 長いですが、よろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳をよろしくお願いします。
Figure 8 shows r_min in the second encounter for b=2.8. In this case, there are four zones of close-encounter orbit in the τ-ω diagram. Comparing Fig.8 with Fig. 7b, the total area occupied by the recurrent close-encounter orbits (the dotted regions in Fig. 8) is smaller than that in the first encounter but not small enough to be neglected. Collision orbits belong necessarily to close-encounter orbits. Consequently, to find collision orbits, we subdivided the τ-ω phase space of close-encounter orbits (i.e., the finely dotted regions in Fig.7) more densely (mesh width being as small as 0.002π in τ) and pursued orbits for each set of τ and ω. Furthermore, as the phase volume of τ and ω occupied by collision orbits, we evaluated a “differential” collisional rate <p(e, i, b)> given by <p(e, i, b)>=(1/(2π)^2)∫p_col (e, i, b, τ, ω)dτdω. (24) Here, we calculated <p(e, i, b)> separately for 1-, 2-, and more recurrent orbits. The results are shown in Fig. 9, from which we can see that 2-recurrent collision orbits exist for relatively large b, and n-current (n≧3) ones exist only for b≒b_max. That is, the recurrent collision orbits appear only in cases of relatively low energy. From Eq. (10), we have <P(e, i)>=∫【-∞→∞】(3/2)|b|<p(e, i, b)>db. (25) Using evaluated values of <p(e, i, b)> for various b, we finally obtain <P(e, i)>=0.114 for (e, i)=(1.0, 0.5); the contribution of 2-recurrent orbits is 5%, and that of 3- and more-recurrent orbits is less than 1%. For this case (e=1.0 and i=0.5), we observed 874 collision orbits. The statistical error in evaluating <P(e, i)> is therefore presumed to be of the order of 4%. Since the contribution of 3- and more-recurrent orbits is within the statistical fluctuation, it can be neglected. よろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳を教えてください。
4. Numerical procedures for obtaining <P(e, i)> Based on the results in Sect. 3, we develop numerical procedures for efficiently obtaining <P(e, i)>. By using the framework of Hill’s equations, one can reduce the degrees of freedom of particle motion, as described in Paper I. Nevertheless, they are still too numerous to compute orbits densely and uniformly over the five-dimensional phase space (e, i, b, τ_s, ω_s) of initial conditions. Furthermore, we frequently find orbits which comes very close to the protoplanet or which revolve complicatedly many times around the protoplanet. In these cases, an enormously long computation time is needed to calculate them. To avoid these difficulties, we introduce the following three simplifications: (i) We neglect n-recurrent orbits in evaluating <P(e, i)> when n≧3; that is, we stop an orbital calculation if the particle does not collide with the protoplanet even after two close encounters. (ii) When the planetesimal crosses the sphere surface of the two-body approximation, the two-body formula determines whether or not a collision occurs. (iii)We omit orbital computations in some regions of τ_s, which depend on a given set of e, i, and b; it is found empirically that there are no collision orbits in these regions. よろしくお願いします。
- ベストアンサー
- 英語
- この英文の和訳をお願いします。
Before a detailed description of our numerical procedures, we comment on above simplifications. As seen in an example of e=1.0 and i=0.5, n-recurrent (n≧3) collision orbits contribute by only 1% or less to the collision rate. On the one hand, from calculations with other e and i, the degree of contribution by n-recurrent (n≧3) collision orbits is found to be largest in the case e≒1. Hence, the error in <P(e, i)> introduced by simplification (i) is of the order of 1% or less. As for the applicability of the two-body approximation, we have confirmed in PaperII that the orbit are well described by the two-body formula inside the two-body sphere, whose radius is given by Eq. (13). No appreciable error in <P(e, i)> comes from simplification (ii). Simplification (iii) follows the discussion in the last section. Using above simplifications, we have developed numerical procedures for obtaining <P(e, i)> efficiently; their flow chart is illustrated in Fig. 10. Choosing initial values of orbital elements (e, i, b, τ_s, ω_s), we start to compute numerically Hill’s equations (6) by an ordinary fourth-order Runge-Kutta method from a starting point given by Eqs. (21) and (22). The distance r is checked at every time step of the numerical integration. If the particle flies off to a sufficient distance from the protoplanet after approaching it, i.e., if |y|>y_0+2e, (26) then, the orbital computation is stopped. If a particle approaches the protoplanet and crosses the two-body sphere surface, i.e., if r≦r_cr, the two-body formula is employed to predict whether or not a collision occurs. When no collision occurs at the first encounter, the numerical integration of Hill’s equations is continued. Since a particle which enters the two-body sphere inevitably escapes from the sphere (see Paper II), the particle follows alternatives: one is that it departs to such a distance that Eq. (26) is satisfied, and the other is that it crosses the two-body sphere surface again. In the former case, we stop the computation, considering that the orbit is non-collisional. In the latter case, the occurrence of collision is checked in the same way as earlier by means of the two-body formula, and the orbital calculation is terminated. Using the numerical procedures developed in this way, we obtain <P(e, i)> in many sets of (e, i); the results are presented in the preceeding sections. Fig. 10. Flow chart of orbital calculation for finding collision orbits. Fig. 10. 拡大画像↓ http://www.fastpic.jp/images.php?file=2113240192.jpg 長文になりますが、よろしくお願いします。
- ベストアンサー
- 英語
- 和訳をお願いします。
In order to obtain the total collisional rate <Γ(e_1, i_1)>, we have to find <P(e,i)> numerically by computing orbits of relative motion between the protoplanet and the planetesimal. Since <P(e,i)> should be provided for wide ranges of e and i with a sufficient accuracy, we are obliged to compute a very large number of orbits. In practice, it is an important problem to find an efficient method for numerical computation. In the second paper (Nakazawa et al., 1989b, referred to as Paper II), we have studied the validity of the two-body approximation and found that within the sphere of the two-body approximation (hereafter referred to as the two-body sphere), the relative motion can be well described by a solution to the two-body problem: the sphere radius has been found to be r_cr=0.03(a_0*/1AU)^(-1/4)(ε/0.01)^(1/2). ・・・・・(13) Within the sphere, the nearest distance can be predicted with an accuracy εby the well-known formula of the two-body encounter. We can expect the above result to be useful to reduce computation time for obtaining <P(e,i)> numerically. よろしくお願いします。
- ベストアンサー
- 英語
- この文章の和訳を教えてください。
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。
- 締切済み
- 英語
- この文章の和訳をお願いします。
Now, we shall concentrate on the collision orbits. Figure 5 illustrates the minimum separation distance r_min in the first encounter (solid curves), identical to that obtained by Petit and Hénon (1986). One sees immediately, that there are two different zones: the “regular” zones, in which r_min varies smoothly with a change of parameter b and the irregular (or “chaotic”) zones, where r_min changes greatly with tiny differences in the choice of b. The chaotic zones lie near b=1.93, 2.30 and 2.48, with very narrow ranges of b. In the regular zone, we find two broad bands of collision orbits around b=2.09 and 2.39. These collision bands were first found by Giuli (1968). The sum of width of the collision bands ⊿b is found to be about 0.098, if the planetary radius is 0.005. よろしくお願いします。
- ベストアンサー
- 英語
