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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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初期条件のチョイスについていくつか所感を付け加えることとする。粒子軌道における軌道長半径と軌道離心率を一度固定したら、位相パラメーターε_i and δ_i,のすべての組み合わせを調べる必要はないということに留意すべきである。これは、違った ε_i と δ_iの組み合わせを有する無数の同じ軌道が存在するという事実による為である。したがって、δ_i が2パイ係数をもつことに留意し、 ε_iが一定であり、-π≦δ≦πであることを用いて、実質的にケプラー軌道のおこりえる位相のすべてを表すことができる。次に、a_i と e_i から b_i~ までと e_i~をそれぞれ次の公式に準じて変換していく。           a_i=1+b_i~h (2・9)              および           e_i=e_i~h. (2・10) ここでは、hは例えばヒル球のように惑星の重力が太陽の重力に勝る球体の半径であり、この実験で用いられた単位系で次のように定義される。          h=(μ/3)^(1/3)=2.15×10^(-3), (2・11) 上記式では、例えば惑星質量Mは現在の地球質量の100分の1である5.977×10の25乗グラムになるよう選ばれている。林およびその他によって示された通り、上記変換によって上手に見積もられたμ≦10^-4である限り、平面3体問題とその解答の間には似通うところの多い、類似の法則が存在し、従って μ (もしくは M)の特殊なチョイスにより得られた結果は、異なるμの値をもつ問題に広く適応することができる。

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