\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\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<\/p>\n
$$ \\boldsymbol{\\ell} = (0, 0, \\ell)$$<\/p>\n
\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
\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\u307e\u3067\u3082\u306a\u304f\uff0c\u3053\u306e\u89e3\u306f\u6955\u5186\u8ecc\u9053\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3002$xy$ \u5e73\u9762\u4e0a\u306e\u6975\u5ea7\u6a19 $r, \\phi$ \u3067\u3042\u3089\u308f\u3059\u3068\uff0c<\/p>\n
$$x = r \\cos\\phi, \\quad y = r \\sin \\phi$$<\/p>\n
$$r = \\frac{a (1-e^2)}{1 + e \\cos\\phi}$$<\/p>\n
\u3053\u3053\u3067\uff0c$a$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3002$\\phi = 0 $ \u3067 $r = r_{\\rm min} = a (1-e)$ \u306a\u308b\u8fd1\u70b9\u8ddd\u96e2\u306b\u306a\u308b\u3088\u3046\u306b\u7a4d\u5206\u5b9a\u6570\u3092\u9078\u3093\u3067\u3044\u308b\u3002<\/p>\n
\u904b\u52d5\u306e\u5b9a\u6570\u3067\u3042\u308b \\(\\varepsilon\\) \u3084 \\(\\ell\\) \u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6955\u5186\u8ecc\u9053\u8981\u7d20\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\varepsilon &=&\u00a0 – \\frac{GM}{2 a} \\\\
\n\\ell &=& \\sqrt{GM a (1-e^2)}
\n\\end{eqnarray}<\/p>\n
\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3068\u5468\u671f $P$ \u306e\u9593\u306b\u306f\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c3\u6cd5\u5247\u304c\u6210\u308a\u7acb\u3061\uff0c<\/p>\n
$$ \\frac{a^3}{P^2} = \\frac{GM}{4 \\pi^2}, \\quad \\therefore\\ \\ P = 2 \\pi \\frac{a \\sqrt{a}}{GM}$$<\/p>\n
\u7121\u6b21\u5143\u5316<\/h3>\n
\u3053\u306e\u7cfb\u306b\u7279\u5fb4\u7684\u306a\u9577\u3055\u3067\u3042\u308b\u8ecc\u9053\u9577\u534a\u5f84 \\(a\\) \u304a\u3088\u3073\u6642\u9593\u30b9\u30b1\u30fc\u30eb\u3067\u3042\u308b\u5468\u671f \\(P\\) \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}{a} \\\\
\nY &\\equiv& \\frac{y}{a} \\\\
\nT &\\equiv& \\frac{t}{P}
\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} = – 4\\pi^2 \\frac{X}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}}\u00a0 \\tag{1}\\\\
\n\\frac{d^2 Y}{dT^2} = – 4\\pi^2 \\frac{Y}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}} \\tag{2}
\n\\end{eqnarray}<\/p>\n
\u521d\u671f\u6761\u4ef6<\/h3>\n
<\/p>\n
\u521d\u671f\u6761\u4ef6\u3092 \\( t = 0\\) \u3067<\/p>\n
\\begin{eqnarray}
\nx(0) &=& r_{\\rm min} = a(1-e) \\\\
\ny(0) &=& 0 \\\\
\nv_x(0) &=& 0 \\\\
\nv_y(0) &=& v_{y0}
\n\\end{eqnarray}<\/p>\n
\u3068\u3059\u308b\u3002\u3053\u3053\u3067 $v_{y0}$ \u306f $t = 0$ \u3067\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5f0f\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\varepsilon = – \\frac{GM}{2a} &=& \\frac{1}{2} \\left(v_{y0}\\right)^2 – \\frac{GM}{a(1-e)} \\\\
\n\\therefore\\ \\ v_{y0} &=& \\sqrt{\\frac{GM (1+e)}{a(1-e)}}
\n\\end{eqnarray}<\/p>\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u521d\u671f\u6761\u4ef6<\/h3>\n
$T = t\/P = 0$ \u306e\u3068\u304d<\/p>\n
\\begin{eqnarray}
\nX(0) &=& \\frac{x(0)}{a} = 1-e \\\\
\nY(0) &=& \\frac{y(0)}{a} = 0 \\\\
\nV_x(0) &=& \\frac{P}{a} v_x(0) = 0 \\\\
\nV_y(0) &=& \\frac{P}{a} v_{y0} = 2 \\pi \\sqrt{\\frac{1+e}{1-e}}
\n\\end{eqnarray}<\/p>\n
\u3053\u306e\u521d\u671f\u6761\u4ef6\u3067\uff0c\u5f0f $(1)$, $(2)$ \u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u3068\uff0c\uff08\u898f\u683c\u5316\u3055\u308c\u305f\uff09\u9577\u534a\u5f84 $1$\uff0c\u96e2\u5fc3\u7387 $e$ \u306e\u6955\u5186\u306e\u8ecc\u9053\u304c\u5f97\u3089\u308c\u308b\u30cf\u30ba\u3002<\/p>\n
\u3058\u3083\u3042\uff0c\u305d\u3053\u307e\u3067\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u306a\u3089\uff0c\u306a\u305c\u308f\u3056\u308f\u3056\u6570\u5024\u8a08\u7b97\u3059\u308b\u306e\u304b<\/strong><\/span>\u3068\u3044\u3046\u3068\uff0c\u305d\u308c\u306f\u6955\u5186\u8ecc\u9053\u306e\u89e3\u304c\u6642\u9593 <\/strong> $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u304b\u3089<\/strong><\/span>\u3002\u6642\u523b $t$ \u306e\u3068\u304d\uff0c\u3069\u3053\u306b\u3044\u308b\u304b\u304c\u89e3\u6790\u7684\u306b\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u306e\u3067\uff0c\u305d\u308c\u3092\u77e5\u308a\u305f\u3044\u304b\u3089\u6570\u5024\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\uff08\u306f\u7d50\u5c401\u4f53\u554f\u984c\u306b\u5e30\u7740\u3059\u308b\u3093\u3060\u3051\u3069\uff09\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048\u3002\u95c7\u96f2\u306a\u521d\u671f\u6761\u4ef6\u304b\u3089\u306f\u3058\u3081\u308b\u306e\u3067\u306f\u306a\u304f\uff0c\u305d\u3082\u305d\u3082\u6570\u5024\u8a08\u7b97\u3057\u306a\u304f\u3066\u3082\u6955\u5186\u306b\u306a\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3060\u304b\u3089\uff0c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306a\u308a\u3092\u4f7f\u3063\u3066\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306b\uff0c\u305d\u308c\u306a\u308a\u306e\u4e0b\u3054\u3057\u3089\u3048\u3092\u3057\u3066\u304a\u3053\u3046\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n\n- \u53c2\u8003\uff1a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c<\/li>\n<\/ul>\n","protected":false},"author":33,"featured_media":0,"parent":2826,"menu_order":70,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7110","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t