\u53c2\u8003\uff1a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c<\/a><\/li>\n<\/ul>\n<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n
\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306f\u76f8\u5bfe\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{r}\\) \u3068\u5168\u8cea\u91cf \\(M\\) \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\u7d50\u5c401\u4f53\u554f\u984c\u306b\u5e30\u7740\u3057\u3066<\/p>\n
$$\\frac{d^2 \\boldsymbol{r}}{dt^2} = – \\frac{GM}{r^3} \\boldsymbol{r}$$<\/p>\n
\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09<\/h3>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u4fdd\u5b58\u91cf\u306f2\u3064\u3002\uff08\u5358\u4f4d\u8cea\u91cf\u5f53\u305f\u308a\u306e\uff09\u89d2\u904b\u52d5\u91cf \\(\\boldsymbol{\\ell}\\)<\/p>\n
$$\\boldsymbol{\\ell} \\equiv \\boldsymbol{r} \\times \\dot{\\boldsymbol{r}} = \\mbox{const.}$$<\/p>\n
\u3068\uff08\u5358\u4f4d\u8cea\u91cf\u5f53\u305f\u308a\u306e\uff09\u5168\u30a8\u30cd\u30eb\u30ae\u30fc \\(\\varepsilon\\)<\/p>\n
$$\\varepsilon \\equiv \\frac{1}{2} \\dot{\\boldsymbol{r}} \\cdot\\dot{\\boldsymbol{r}} \u00a0 \u00a0 – \u00a0 \u00a0 \\frac{GM}{r}$$<\/p>\n
\u4e00\u5b9a\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{\\ell}\\) \u306e\u5411\u304d\u3092 \\(z\\) \u8ef8\u65b9\u5411\u306b\u3068\u308b\u3068\uff0c\u904b\u52d5\u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u5236\u9650\u3055\u308c\u3066<\/p>\n
$$\\boldsymbol{r} = (x, y, 0)$$\u3068\u304a\u3051\u308b\u3002<\/p>\n
\u521d\u671f\u6761\u4ef6\u3067\u8868\u3055\u308c\u308b\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09\u3068\u6955\u5186\u8ecc\u9053\u8981\u7d20<\/h3>\n
\u6570\u5024\u8a08\u7b97\u3092\u3059\u308b\u969b\uff0c\\(t = 0\\) \u3067 \\(x = x_0, \\ y = 0\\) \u3068\u3044\u3046 \\(x\\) \u8ef8\u4e0a\u304b\u3089\uff0c\\(v_x = 0\\) , \\( v_y = v_{y0}\\) \u3068\u3044\u3046\\(y\\) \u8ef8\u65b9\u5411\u3078\u306e\u521d\u901f\u5ea6\u3067\u306f\u3058\u3081\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n
\u521d\u901f\u5ea6 \\(v_{y0}\\) \u304c\u4ee5\u4e0b\u306e\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b \\(v_0\\) \u306e\u3068\u304d\uff0c\u8ecc\u9053\u306f\u5186\u306b\u306a\u308b\u3002<\/p>\n
$$\\frac{m v_0^2}{x_0} = \\frac{GM m}{x_0^2}, \\quad \\therefore\\ \\ v_0 = \\sqrt{\\frac{GM}{x_0}}$$<\/p>\n
\u306a\u306e\u3067\uff0c\u521d\u901f\u5ea6 \\(v_{y0}\\) \u3092 \\(v_0\\) \u306e\u5b9a\u6570\u500d\u3068\u3044\u3046\u3075\u3046\u306b\u304a\u304f\u3002<\/p>\n
$$v_{y0} = k v_0 = k \\sqrt{\\frac{GM}{x_0}}$$<\/p>\n
\u904b\u52d5\u306e\u5b9a\u6570\u3067\u3042\u308b \\(\\varepsilon\\) \u3084 \\(\\ell = |\\boldsymbol{\\ell}|\\) \u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u521d\u671f\u6761\u4ef6\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\varepsilon &=& \\frac{1}{2} \\left(v_{y0}\\right)^2 – \\frac{GM}{x_0} \\\\
\n&=& – \\frac{GM (2 – k^2)}{2 x_0}\u00a0 \\\\
\n\\ell &=& x_0 v_{y0} = k x_0 v_0 \\\\
\n&=& k x_0 \\sqrt{\\frac{GM}{x_0}}
\n\\end{eqnarray}<\/p>\n
\u6955\u5186\u8ecc\u9053\u306e\u9577\u534a\u5f84 \\(a\\) \u304a\u3088\u3073\u96e2\u5fc3\u7387 \\(e\\) \u3082\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u521d\u671f\u6761\u4ef6\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\na &=& \\frac{GM}{2 |\\varepsilon|} = \\frac{x_0}{2 – k^2} \\\\
\ne &=& \\sqrt{1 – \\frac{2 |\\varepsilon| \\ell^2}{(GM)^2} } \\\\
\n&=& \\sqrt{1 – k^2 (2-k^2)} =\u00a0 |k^2 – 1|
\n\\end{eqnarray}<\/p>\n
\u675f\u7e1b\u8ecc\u9053 \\(\\varepsilon < 0\\) \u3068\u3059\u308b\u306e\u3067\uff0c<\/p>\n
$$ 0 < k < \\sqrt{2}$$<\/p>\n
\u7121\u6b21\u5143\u5316<\/h3>\n
\u3053\u306e\u7cfb\u306b\u7279\u5fb4\u7684\u306a\u9577\u3055 \\(x_0\\) \u304a\u3088\u3073\u6642\u9593\u30b9\u30b1\u30fc\u30eb \\(\\displaystyle \\tau \\equiv \\frac{x_0}{v_0} = \\sqrt{\\frac{x_0^3}{GM}}\\) \u3067\u3053\u306e\u7cfb\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u898f\u683c\u5316\u3059\u308b\u3002<\/p>\n
\\begin{eqnarray}
\nX &\\equiv& \\frac{x}{x_0} \\\\
\nY &\\equiv& \\frac{y}{x_0} \\\\
\nT &\\equiv& \\frac{t}{\\tau} = \\frac{v_0 t}{x_0}
\n\\end{eqnarray}<\/p>\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
\\begin{eqnarray}
\n\\frac{d^2 X}{dT^2} = – \\frac{X}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}} \\\\
\n\\frac{d^2 Y}{dT^2} = – \\frac{Y}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}}
\n\\end{eqnarray}<\/p>\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u521d\u671f\u6761\u4ef6\u3068\u6955\u5186\u8ecc\u9053\u8981\u7d20<\/h3>\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u5909\u6570\u306b\u5bfe\u3059\u308b\u521d\u671f\u6761\u4ef6\u306f \\( T = 0\\) \u3067
\n$$X = 1, \\quad Y = 0, \\quad V_x = \\frac{dX}{dT} = 0, \\quad V_y = \\frac{dY}{dT} = \\frac{v_{y0}}{v_0} = k \\ \\ ( 0 < k < \\sqrt{2})$$<\/p>\n
\u3053\u306e\u3068\u304d\uff0c\u6570\u5024\u8a08\u7b97\u3059\u308b\u524d\u306b\u3082\u3046\uff0c\u8ecc\u9053\u306f\u898f\u683c\u5316\u3055\u308c\u305f\u9577\u534a\u5f84\u304c<\/p>\n
$$ A \\equiv \\frac{a}{x_0} = \\frac{1}{2 – k^2} $$<\/p>\n
\u96e2\u5fc3\u7387\u304c<\/p>\n
$$e = |k^2 – 1|$$<\/p>\n
\u306e\u6955\u5186\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3057\u307e\u3046\u3002\u307e\u305f\uff0c\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c3\u6cd5\u5247\u304b\u3089\uff0c\u5468\u671f\u3092 \\(p\\) \u3068\u3059\u308b\u3068<\/p>\n
$$ \\frac{a^3}{p^2} = \\frac{GM}{4\\pi^2}$$<\/p>\n
\u3088\u308a\uff0c\u898f\u683c\u5316\u3055\u308c\u305f\u5468\u671f\u304c<\/p>\n
$$P \\equiv \\frac{p}{\\tau} = 2 \\pi A^{\\frac{3}{2}} = \\frac{2 \\pi}{\\left(2 – k^2\\right)^{\\frac{3}{2}}}$$<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u3082\u308f\u304b\u3063\u3066\u3057\u307e\u3063\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n
$k$ \u306e\u7bc4\u56f2<\/h4>\n
\u306a\u304a\uff0c\u521d\u901f\u5ea6\u306e\u5927\u304d\u3055\u3092\u8868\u3059 $k$ \u306f\u675f\u7e1b\u8ecc\u9053\u3067\u3042\u308b\u3068\u3044\u3046\u6761\u4ef6\u304b\u3089 $ 0 < k < \\sqrt{2}$\u00a0 \u3067\u3042\u308b\u304c\uff0c\u7279\u306b\u65ad\u3089\u306a\u3044\u9650\u308a
\n$$ 1 \\le k < \\sqrt{2}$$<\/p>\n
\u3068\u3059\u308b\u3002\u3053\u306e\u7bc4\u56f2\u306b\u3059\u308b\u3053\u3068\u3067\u521d\u671f\u4f4d\u7f6e\u304c\u8fd1\u70b9\u306b\u306a\u308b\u3002\u306a\u305c\u8fd1\u70b9\u306b\u306a\u308b\u304b\u3068\u3044\u3046\u3068\uff0c$k=1$ \u3067\u5186\u8ecc\u9053\u3067\u3042\u308a\uff0c$k > 1$\u00a0 \u3068\u3059\u308c\u3070\u5186\u8ecc\u9053\u3068\u306a\u308b\u521d\u901f\u5ea6\u3088\u308a\u3082\u5927\u304d\u3044\u521d\u901f\u5ea6\u3067 $x$ \u8ef8\u304b\u3089\u6253\u3061\u51fa\u3055\u308c\u308b\u306e\u3067\uff0c\u53cd\u5bfe\u5074\u3067\u3088\u308a\u81a8\u308c\u308b\u4f4d\u7f6e\u3067 $x$ \u8ef8\u3092\u6a2a\u5207\u308b\u3053\u3068\u304c\u4e88\u60f3\u3055\u308c\u308b\u304b\u3089\u3002<\/p>\n
\u9577\u534a\u5f84 $a$ \u3092\u4e00\u5b9a\u306b\u4fdd\u3061\u306a\u304c\u3089\u96e2\u5fc3\u7387 $e$ \u3092\u5909\u5316\u3055\u305b\u308b\u521d\u671f\u6761\u4ef6<\/h3>\n
\u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0c\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u5909\u6570\u306b\u5bfe\u3059\u308b\u521d\u671f\u6761\u4ef6\u3092 \\( T = 0\\) \u3067
\n$$X = 1, \\quad Y = 0, \\quad V_x = \\frac{dX}{dT} = 0, \\quad V_y = \\frac{dY}{dT} =\u00a0 k$$<\/p>\n
\u3068\u3059\u308b\u3068\uff0c\u521d\u901f\u5ea6 $k$ \u306e\u53d6\u308a\u65b9\u3057\u3060\u3044\u3067\u6955\u5186\u8ecc\u9053\u306b\u306a\u308b\u306e\u3067\u3042\u308b\u304c\uff0c\u305d\u306e\u969b\uff0c\u96e2\u5fc3\u7387\u3060\u3051\u3067\u306a\u304f\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3082 $k$ \u306b\u4f9d\u5b58\u3057\u3066\u5909\u5316\u3057\uff0c\u3057\u305f\u304c\u3063\u3066\u5468\u671f\u3082 $k$ \u306b\u4f9d\u5b58\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
\u305d\u308c\u306f\u305d\u308c\u3067\u304a\u3082\u3057\u308d\u3044\u306e\u3067\u3042\u308b\u304c\uff0c\u3082\u3046\u5c11\u3057\u5de5\u592b\u3059\u308b\u3068\uff0c\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3092\u4e00\u5b9a\u306b\u4fdd\u3061\u306a\u304c\u3089\u96e2\u5fc3\u7387 $e$ \u304c\u5909\u5316\u3059\u308b\u3088\u3046\u306a\u521d\u671f\u6761\u4ef6\u3092\u8a2d\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002$a$ \u304c\u4e00\u5b9a\u306e\u307e\u307e\u3067\u3042\u308b\u304b\u3089\u5468\u671f\u3082\u5909\u5316\u3057\u306a\u3044\u3002\uff08\u898f\u683c\u5316\u3055\u308c\u305f\u5468\u671f $P$ \u306f $P = 2 \\pi$ \u306e\u307e\u307e\u3002\uff09<\/p>\n
\u521d\u671f\u6761\u4ef6\u3092<\/p>\n
$$ x_i = f x_0, \\quad y_i = 0, \\quad\u00a0 v_{x 0} = 0, \\quad v_{y0} = k v_0$$<\/p>\n
\u3068\u304a\u304f\u3002\u3053\u308c\u306f\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u5909\u6570\u306b\u5bfe\u3057\u3066\u306f<\/p>\n
$$X = f, \\quad Y = 0, \\quad V_x = \\frac{dX}{dT} = 0, \\quad V_y = \\frac{dY}{dT} =\u00a0 k$$<\/p>\n
\u3068\u3057\u305f\u3053\u3068\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u904b\u52d5\u306e\u5b9a\u6570 $\\varepsilon$ \u304c\u9577\u534a\u5f84 $a$ \u3067\u66f8\u3051\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\n\\varepsilon\u00a0 \\equiv – \\frac{GM}{2 a} &=& \\frac{1}{2}\u00a0 v_{y0}^2 – \\frac{GM}{x_i} \\\\
\n&=& \\frac{1}{2} (k v_0)^2 – \\frac{GM}{f x_0} \\\\
\n&=& \\frac{1}{2} k^2 \\frac{GM}{x_0} – \\frac{GM}{f x_0} \\\\
\n&=& \\frac{GM}{x_0} \\left( \\frac{k^2}{2} – \\frac{1}{f}\\right)
\n\\end{eqnarray}<\/p>\n
\u5186\u8ecc\u9053\u306e\u521d\u671f\u6761\u4ef6 $k = 1, \\ f = 1$ \u306e\u3068\u304d\u306f $\\displaystyle \\frac{a}{x_0} = 1$ \u3068\u306a\u308b\u3002 $k > 1$ \u306e\u3068\u304d\u3082 $\\displaystyle \\frac{a}{x_0} = 1$ \u3068\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u306b\u306f\uff0c<\/p>\n
\\begin{eqnarray}
\n– \\frac{1}{2} &=& \\frac{k^2}{2} – \\frac{1}{f} \\\\
\n\\therefore\\ \\ f &=& \\frac{2}{k^2 + 1}
\n\\end{eqnarray}<\/p>\n
\u3068\u3059\u308c\u3070\u3088\u3044\u3002\u3053\u306e\u3068\u304d\u306e\u96e2\u5fc3\u7387\u3092\u6c42\u3081\u308b\u306b\u306f\u307e\u305a\uff0c\u904b\u52d5\u306e\u5b9a\u6570 $\\ell$ \u3092\u521d\u671f\u6761\u4ef6\u304b\u3089\u6c42\u3081\u3066\uff0c<\/p>\n
\\begin{eqnarray}
\n\\ell &\\equiv& x \\frac{dy}{dt} – y \\frac{dx}{dt} \\\\
\n&=&x_i \\times v_{y0} \\\\
\n&=& f x_0 \\times k v_0 \\\\
\n&=& \\frac{2 k}{k^2 + 1} x_0 \\sqrt{\\frac{GM}{x_0}}
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066<\/p>\n
\\begin{eqnarray}
\ne &=& \\sqrt{1 – \\frac{2 |\\varepsilon| \\ell^2}{(GM)^2}} \\\\
\n&=& \\sqrt{1 – \\frac{2}{(GM)^2} \\frac{GM}{2a} \\left(\\frac{2k}{k^2 + 1}\\right)^2 x_0^2 \\frac{GM}{x_0} } \\\\
\n&=& \\sqrt{1 – \\left(\\frac{2k}{k^2 + 1}\\right)^2} \\\\
\n&=& \\frac{|k^2 – 1|}{k^2 + 1}
\n\\end{eqnarray}<\/p>\n
<\/p>\n
$k$ \u306e\u7bc4\u56f2<\/h4>\n
\u3053\u306e\u5834\u5408\u3082\uff0c\u521d\u671f\u4f4d\u7f6e\u304c\u8fd1\u70b9\u3068\u306a\u308b\u3088\u3046\u306b $ k \\ge 1$ \u3068\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\n
$$\\lim_{k \\rightarrow \\infty} e = 1$$<\/p>\n
\u306a\u306e\u3067\uff0c$k$ \u306e\u4e0a\u9650\u306f\u5b9f\u8cea\u4e0a\u306a\u3044\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n
$e$ \u306e\u5f0f\u3092 $k$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\uff0c
\n\\begin{eqnarray}
\ne &=&\\frac{k^2 – 1}{k^2 + 1} \\\\
\n\\therefore\\ \\ k &=& \\sqrt{\\frac{1+e}{1-e}}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"
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