{"id":9652,"date":"2024-11-14T10:50:15","date_gmt":"2024-11-14T01:50:15","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9652"},"modified":"2024-11-19T15:36:28","modified_gmt":"2024-11-19T06:36:28","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e5%85%89%e3%81%ae%e6%9b%b2%e3%81%8c%e3%82%8a%e8%a7%92%e3%82%92%e3%83%8b%e3%83%a5%e3%83%bc%e3%83%88%e3%83%b3%e7%90%86%e8%ab%96%e3%81%a7%e8%a8%88%e7%ae%97%e3%81%99%e3%82%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%87%8d%e5%8a%9b%e5%a0%b4%e4%b8%ad%e3%81%ae%e5%85%89%e3%81%ae%e4%bc%9d%e6%92%ad\/%e5%8f%82%e8%80%83%ef%bc%9a%e5%85%89%e3%81%ae%e6%9b%b2%e3%81%8c%e3%82%8a%e8%a7%92%e3%82%92%e3%83%8b%e3%83%a5%e3%83%bc%e3%83%88%e3%83%b3%e7%90%86%e8%ab%96%e3%81%a7%e8%a8%88%e7%ae%97%e3%81%99%e3%82%8b\/","title":{"rendered":"\u53c2\u8003\uff1a\u5149\u306e\u66f2\u304c\u308a\u89d2\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u3067\u8a08\u7b97\u3059\u308b"},"content":{"rendered":"

\u91cd\u529b\u5834\u4e2d\u306e\u5149\u306e\u7d4c\u8def\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6<\/strong><\/span>\uff08\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f<\/strong><\/span>\uff0b\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247<\/strong><\/span>\uff09\u3067\u8a08\u7b97\u3057\uff0c\u5149\u306e\u66f2\u304c\u308a\u89d2\u304c\u4e00\u822c\u76f8\u5bfe\u8ad6\u306e\u4e88\u8a00\u306e\u534a\u5206\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

\u5149\u306e\u7c92\u5b50\u8aac<\/h3>\n

\u5149\u306e\u7c92\u5b50\u8aac\u306e\u7acb\u5834\u3067\u306f\uff0c\u5149\u306f\u300c\u5149\u5b50<\/strong><\/span>\u300d\uff08\u307f\u3064\u3053\u3067\u306f\u306a\u304f\u3053\u3046\u3057\uff09\u3068\u3044\u3046\u7c92\u5b50\u304b\u3089\u306a\u308b\u3068\u3059\u308b\u3002<\/p>\n

\u73fe\u5728\u307e\u3067\u306e\u3068\u3053\u308d\uff0c\u5149\u5b50\u306b\u6709\u9650\u306e\u8cea\u91cf\u304c\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u78ba\u8a8d\u3055\u308c\u3066\u3044\u306a\u3044\u304c\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u3067\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u306f\uff0c\u5149\u5b50\u306b\u306f\uff0c\u5b9f\u9a13\u7684\u306b\u78ba\u8a8d\u3055\u308c\u306a\u3044\u307b\u3069\u308f\u305a\u304b\u3067\u3042\u3063\u3066\u3082\uff0c\u305d\u308c\u3067\u3082\u30bc\u30ed\u3067\u306a\u3044\u6709\u9650\u306e\u8cea\u91cf $m$ \u304c\u3042\u308b\u3068\u4eee\u5b9a\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002\uff08\u4e07\u6709\u5f15\u529b\u306f\u8cea\u91cf\u3092\u3082\u3063\u305f\u7269\u4f53\u9593\u306b\u50cd\u304f\u5f15\u529b\u3067\u3042\u308b\u304b\u3089\u3002\uff09<\/p>\n

\u4e07\u6709\u5f15\u529b\u306e1\u4f53\u554f\u984c<\/h3>\n

\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u3092\u53d7\u3051\u308b\u7269\u4f53\u306e\u904b\u52d5\u306b\u3064\u3044\u3066\u306f\uff0c\u300c\u53c2\u8003\uff1a\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c<\/a>\u300d\u306b\u307e\u3068\u3081\u3066\u3044\u308b\u304c\uff0c\u3053\u3053\u3067\u306f1\u4f53\u554f\u984c\u3068\u3057\u3066\u3042\u3089\u305f\u3081\u3066\u304a\u3055\u3089\u3044\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u5149\u5b50\u306b\u5bfe\u3059\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n

\u5149\u5b50\u304c\u6709\u9650\u306e\u8cea\u91cf $m$ \u3092\u3082\u3064\u3068\u3059\u308b\u3068\uff0c\u539f\u70b9\u306b\u3042\u308b\u8cea\u91cf $M$ \u306e\u5929\u4f53\u304c\u3064\u304f\u308b\u91cd\u529b\u5834\u4e2d\u3092\u904b\u52d5\u3059\u308b\u8cea\u91cf $m$ \u306e\u8cea\u70b9\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n

\\begin{eqnarray}
\nm \\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 &=& \u2013\\frac{GM\u00a0 m}{r^3}\\boldsymbol{r} \\\\
\n\\therefore\\ \\ \\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 &=& \u2013\\frac{GM }{r^3}\\boldsymbol{r} \\tag{1}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\u306f\u5149\u5b50\u3092\u60f3\u5b9a\u3057\u3066\u3044\u308b\u304c\uff0c$m$ \u304c\u30ad\u30e3\u30f3\u30bb\u30eb\u3055\u308c\u305f\u4ee5\u4e0a\uff0c\u6728\u661f\u3068\u304b\u5730\u7403\u3068\u304b\u306e\u60d1\u661f\u3067\u3042\u3063\u3066\u3082\u3053\u306e\u65b9\u7a0b\u5f0f\u306b\u5f93\u3063\u3066\u904b\u52d5\u3059\u308b\u3053\u3068\u306b\u306a\u308a\uff0c\u91cd\u529b\u5834\u4e2d\u306e\u904b\u52d5\u306f\u8cea\u70b9\u306e\u8cea\u91cf\u306b\u3088\u3089\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3093\u3060\u3063\u305f\u306d\u3002\u3053\u308c\u30924\u5b57\u719f\u8a9e\u3067\u7b49\u4fa1\u539f\u7406<\/strong><\/span>\u3068\u3044\u3063\u305f\u308a\u3059\u308b\u3002<\/p>\n

\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\u5c0e\u304b\u308c\u308b\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09<\/h4>\n

\u4e07\u6709\u5f15\u529b<\/strong><\/span>\u306f\u300c\u4e2d\u5fc3\u529b<\/strong><\/span>\u300d\uff08\u91cd\u529b\u6e90\u3092\u539f\u70b9\u3068\u3057\u305f\u3068\u304d\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306b\u6bd4\u4f8b\u3059\u308b\u529b\uff09\u3067\u3042\u308b\u305f\u3081\uff0c\u89d2\u904b\u52d5\u91cf<\/strong><\/span>\u304c\u4fdd\u5b58\u3055\u308c\u308b\u3002\u307e\u305f\uff0c\u300c\u4fdd\u5b58\u529b<\/strong><\/span>\u300d\uff08\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u52fe\u914d grad \u3067\u66f8\u3051\u308b\u529b\uff09\u3067\u3082\u3042\u308b\u305f\u3081\uff0c\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc<\/strong><\/span>\uff08\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u3082\u3044\u3046\uff09\u3082\u4fdd\u5b58\u3055\u308c\u308b\u3002\u5148\u306b\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\u304b\u3089\u793a\u3057\u3066\u304a\u304f\u3068\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u8ecc\u9053\u3092 $xy$ \u5e73\u9762\u4e0a\u306b\u5236\u9650\u3067\u304d\u308b\u306e\u3067\u8a71\u304c\u7c21\u5358\u306b\u306a\u308a\uff0c\u5409\u3002<\/p>\n

\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58<\/h5>\n

(1) \u5f0f\u306b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u3092\u5de6\u304b\u3089\u5916\u7a4d\u3057\u3066\u3084\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{r}\\times \\frac{d^2\\boldsymbol{r}}{dt^2} &=& -\\frac{GM}{r^3} \\boldsymbol{r}\\times \\boldsymbol{r} = \\boldsymbol{0}\\\\
\n\\therefore\\\u00a0 \\boldsymbol{r}\\times \\frac{d^2\\boldsymbol{r}}{dt^2} &=&
\n\\frac{d}{dt}\\left(\\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} \\right) = \\boldsymbol{0} \\\\
\n\\therefore\\\u00a0 \\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} &=& \\mbox{const.} \\equiv \\boldsymbol{\\ell}
\n\\end{eqnarray}<\/p>\n

\u30d9\u30af\u30c8\u30eb $\\boldsymbol{\\ell}$ \u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u306b\u76f8\u5f53\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\u4e00\u5b9a\u3067\u3042\u308b\u305d\u306e\u5411\u304d\u3092\uff08\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\uff09$z$ \u8ef8\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3059\u308b\u3068\uff0c$\\boldsymbol{\\ell}$ \u306f $z$ \u6210\u5206\u306e\u307f\u3068\u306a\u308a\uff0c<\/p>\n

$$\\boldsymbol{\\ell} = (0, 0, \\ell)$$<\/p>\n

\u6b21\u306b\uff0c\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u304c\u3053\u306e $\\boldsymbol{\\ell}$ \u306b\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u793a\u3059\u3002\u30d9\u30af\u30c8\u30eb\u306e\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/a>\uff08\u30d9\u30af\u30c8\u30eb\u306a\u306e\u304b\u30b9\u30ab\u30e9\u30fc\u306a\u306e\u304b\u307e\u304e\u3089\u308f\u3057\u3044\uff09\u306e\u6027\u8cea\u3092\u4f7f\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{r}\\cdot\\boldsymbol{\\ell} &=&
\n\\boldsymbol{r}\\cdot\\left(\\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} \\right) \\\\
\n&=&
\n\\frac{d\\boldsymbol{r}}{dt}\\cdot(\\boldsymbol{r}\\times\\boldsymbol{r}) = 0, \\\\
\n\\therefore\\ \\boldsymbol{r} &\\perp& \\boldsymbol{\\ell}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u3053\u308c\u3082\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f $\\boldsymbol{r}$ \u3092 $xy$ \u5e73\u9762\uff08\u300c\u8d64\u9053\u9762\u300d\u3068\u3082\u8a00\u3063\u305f\u308a\u3059\u308b\uff09\u5185\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u3066\uff0c<\/p>\n

$$\\boldsymbol{r} = (x, y, 0) = (r\\cos\\phi, r\\sin\\phi, 0)$$<\/p>\n

\u5927\u304d\u3055\uff08\u3042\u308b\u3044\u306f $z$ \u6210\u5206\uff09$\\ell$ \u3092\u6975\u5ea7\u6a19 $r, \\phi$ \u3067\u8868\u3059\u3068\uff08\u30c9\u30c3\u30c8 ( $\\dot{\\ }$ ) \u306f\u6642\u9593\u5fae\u5206\uff09\uff0c<\/p>\n

\\begin{eqnarray}
\n\\ell = |\\boldsymbol{\\ell}| &=& x \\dot{y} -y \\dot{x} \\\\
\n&=& r^2 \\dot{\\phi} \\\\
\n\\therefore\\ \\\u00a0 \\frac{d \\phi}{dt} &=& \\frac{\\ell}{r^2} \\tag{2}
\n\\end{eqnarray}<\/p>\n

\u5b66\u751f\u8af8\u541b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066 $x \\dot{y} -y \\dot{x}\u00a0 = r^2 \\dot{\\phi}$ \u3068\u306a\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u304a\u304f\u3053\u3068\u3002<\/p>\n

\\begin{eqnarray}
\nx &=& r \\cos\\phi, \\\\
\ny &=& r \\sin\\phi, \\\\
\n\\dot{x} &=& \\dot{r} \\cos \\phi + r \\left(\\cos \\phi\\right)\\dot{} \\\\
\n&=& \\dot{r} \\cos \\phi + r \\left( -\\sin\\phi\\cdot \\dot{\\phi}\\right)\\\\
\n&=& \\dot{r} \\cos \\phi -r \\sin\\phi\\cdot \\dot{\\phi}\\\\
\n\\dot{y} &=& \\cdots \\\\
\n\\therefore\\ \\ x \\dot{y} -y \\dot{x} &=& \\cdots
\n\\end{eqnarray}<\/p>\n

\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58<\/h5>\n

(1) \u5f0f\u306b\u901f\u5ea6\u30d9\u30af\u30c8\u30eb $\\displaystyle \\boldsymbol{v} \\equiv \\frac{d \\boldsymbol{r}}{dt}$ \u3092\u5185\u7a4d\u3057\u3066\u3084\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 &=&
\n\u2013\\frac{GM}{r^3} \\boldsymbol{r}\\cdot\\frac{d\\boldsymbol{r}}{dt} \\\\
\n&=&\\frac{d\\boldsymbol{r}}{dt}\\cdot \\nabla \\left( \\frac{GM}{r}\\right) \\\\
\n&=& \\frac{d}{dt} \\left( \\frac{GM}{r}\\right) \\\\
\n\\frac{d}{dt}\\left(\\frac{1}{2}\u00a0 \\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d\\boldsymbol{r}}{dt}\\right)&=&
\n\\frac{d}{dt} \\left( \\frac{GM}{r}\\right)
\n\\end{eqnarray}<\/p>\n

$$\\therefore\\ \\ \\frac{d}{dt} \\left(\\frac{1}{2}\u00a0 \\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d\\boldsymbol{r}}{dt}-\\frac{GM}{r} \\right) = 0$$
\n$$\\therefore\\ \\\u00a0 \\frac{1}{2}\u00a0 \\boldsymbol{v}\\cdot\\boldsymbol{v}-\\frac{GM}{r}\u00a0 =\\mbox{const.} \\equiv \\varepsilon$$<\/p>\n

$\\varepsilon$ \u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u548c\uff09\u306b\u76f8\u5f53\u3059\u308b\u5b9a\u6570\u3067\u3042\u308a\uff0c\u6975\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n

\\begin{eqnarray}
\n\\varepsilon &=& \\frac{1}{2} v^2 -\\frac{GM}{r} \\tag{3}\\\\
\n&=& \\frac{1}{2} \\left\\{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2 \\right\\} -\\frac{GM}{r} \\\\
\n&=& \\frac{1}{2}\\left\\{\\left(\\frac{dr}{dt}\\right)^2 +r^2\\left(\\frac{d\\phi}{dt}\\right)^2 \\right\\} -\\frac{GM}{r}\\\\
\n&=& \\frac{1}{2}\\left\\{\\left(\\frac{dr}{dt}\\right)^2 +\\frac{\\ell^2}{r^2} \\right\\} -\\frac{GM}{r} \\tag{4}\\\\
\n\\therefore\\ \\ \\left(\\frac{dr}{dt}\\right)^2 &=& 2 \\varepsilon + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2} \\\\ \\ \\\\
\n\\mbox{\u3068\u3053\u308d\u3067}\\quad\\frac{dr}{dt} &=& \\frac{dr}{d\\phi} \\frac{d\\phi}{dt} = \\frac{\\ell}{r^2} \\frac{dr}{d\\phi}\u00a0 \\quad\\mbox{\u3067\u3042\u308b\u304b\u3089}\\\\ \\ \\\\
\n\\therefore\\ \\ \\left(\\frac{\\ell}{r^2} \\frac{dr}{d\\phi}\\right)^2 &=& 2 \\varepsilon + \\frac{2 G M}{r}\u00a0 -\\frac{\\ell^2}{r^2} \\tag{5}
\n\\end{eqnarray}<\/p>\n

\u5b66\u751f\u8af8\u541b\u306f $\\dot{x}^2 + \\dot{y}^2 = \\dot{r}^2 + r^2 \\dot{\\phi}^2$ \u3068\u306a\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u304a\u304f\u3053\u3068\u3002<\/p>\n

(5) \u5f0f\u3092 $r$ \u306b\u3064\u3044\u3066\u89e3\u304d\u305f\u3044\u306e\u3067\u3042\u308b\u304c\uff0c$r$ \u304c\u3053\u3068\u3054\u3068\u304f\u5206\u6bcd\u306b\u304d\u3066\u3044\u308b\u3082\u306e\u3060\u304b\u3089\uff0c\u3044\u3063\u305d\u306e\u3053\u3068 $\\displaystyle s \\equiv \\frac{1}{r}$ \u3068\u3057\u3066\u5909\u6570\u5909\u63db\u3057\u3066\u3084\u308b\u3068<\/p>\n

\\begin{eqnarray}
\ns &\\equiv& \\frac{1}{r} \\\\
\n\\frac{d s}{d\\phi} &=& -\\frac{1}{r^2} \\frac{dr}{d\\phi}
\n\\end{eqnarray}<\/p>\n

(5) \u5f0f\u306e\u4e21\u8fba\u3092 $\\ell^2$ \u3067\u5272\u3063\u3066\uff0c$s$ \u3067\u66f8\u304d\u76f4\u3057\u3066\u3084\u308b\u3068<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{d s}{d\\phi} \\right)^2 &=& \\frac{2\\varepsilon}{\\ell^2} + 2\\frac{GM}{\\ell^2} s -s^2 \\\\
\n&=& \\frac{2\\varepsilon}{\\ell^2}\u00a0 + \\left(\\frac{GM}{\\ell^2}\\right)^2 -\\left(s -\\frac{GM}{\\ell^2} \\right)^2 \\tag{6}
\n\\end{eqnarray}<\/p>\n

\u5149\u5b50\u8ecc\u9053\u306b\u5bfe\u3059\u308b\u5883\u754c\u6761\u4ef6<\/h4>\n

\u5149\u306f $r \\rightarrow \\infty$ \u304b\u3089\u901f\u3055 $c$ \u3067\u3084\u3063\u3066\u304d\u3066\uff0c$\\displaystyle \\phi = \\frac{\\pi}{2}$ \u3067\u6700\u8fd1\u63a5\u8ddd\u96e2 $r = b$ \u3092\u3068\u308a\uff0c\u307e\u305f $r \\rightarrow \\infty$ \u3078\u53bb\u3063\u3066\u3044\u304f\u3068\u3059\u308b\u3002<\/p>\n

\u6700\u8fd1\u63a5\u8ddd\u96e2 $r = b$ \u3059\u306a\u308f\u3061 $\\displaystyle s = \\frac{1}{r}$ \u3067 $\\displaystyle \\frac{ds}{d\\phi} = 0$ \u3067\u3042\u308b\u304b\u3089 (6) \u5f0f\u3088\u308a<\/p>\n

\\begin{eqnarray}
\n0 &=& \\frac{2\\varepsilon}{\\ell^2}\u00a0 + \\left(\\frac{GM}{\\ell^2}\\right)^2 -\\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right)^2 \\\\
\n\\therefore\\ \\ \\frac{2\\varepsilon}{\\ell^2}\u00a0 + \\left(\\frac{GM}{\\ell^2}\\right)^2 &=& \\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right)^2
\n\\end{eqnarray}<\/p>\n

$b$ \u3092\u4f7f\u3063\u3066\uff0c\u3042\u305f\u3089\u3081\u3066 (6) \u5f0f\u3092\u66f8\u304d\u306a\u304a\u3059\u3068\uff0c<\/p>\n

$$\\left(\\frac{d s}{d\\phi} \\right)^2 =
\n\\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right)^2
\n-\\left(s -\\frac{GM}{\\ell^2} \\right)^2$$<\/p>\n

\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u4f55\u5ea6\u3082\u51fa\u3066\u3044\u308b\u5f62<\/a>\u3067\u3042\u308a\uff0c$\\phi=\\frac{\\pi}{2}$ \u3067 $s$ \u304c\u6975\u5024\u3092\u3068\u308b\u3053\u3068\u304b\u3089\u7a4d\u5206\u5b9a\u6570\u304c\u6c7a\u307e\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

$$ s =\\frac{1}{r} = \\frac{GM}{\\ell^2} + \\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right) \\sin \\phi \\tag{7}$$<\/p>\n

\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u8a73\u7d30<\/h5>\n

\u5ff5\u306e\u305f\u3081\uff0c\u89e3\u3092\u6c42\u3081\u308b\u8a73\u7d30\u3092\u304b\u3044\u3064\u307e\u3093\u3067\u307e\u3068\u3081\u308b\u3068\uff0c<\/p>\n

$$a \\equiv \\frac{1}{b} -\\frac{GM}{\\ell^2}, \\quad y \\equiv s -\\frac{GM}{\\ell^2}$$<\/p>\n

\u3068\u304a\u304f\u3068\uff0c\u89e3\u304f\u3079\u304d\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{dy}{d\\phi}\\right)^2 &=& a^2 -y^2 \\\\
\n\\frac{dy}{d\\phi} &=& \\pm \\sqrt{a^2 -y^2} \\\\
\n\\frac{dy}{\\sqrt{a^2 -y^2} } &=& \\pm d\\phi \\\\
\n\\int \\frac{dy}{\\sqrt{a^2 -y^2} }\u00a0 &=& \\pm \\int d\\phi \\\\
\n\\sin^{-1} \\frac{y}{a} &=& \\pm \\phi + C \\\\
\n\\therefore\\ \\ y &=& a \\sin(\\pm \\phi + C)
\n\\end{eqnarray}<\/p>\n

$\\displaystyle \\phi = \\frac{\\pi}{2}$ \u3067 $\\displaystyle \\frac{dy}{d\\phi} = 0, \\ y = a$ \u3068\u3044\u3046\u6761\u4ef6\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ny &=& a \\sin \\phi \\\\
\ns -\\frac{GM}{\\ell^2} &=& \\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right) \\sin\\phi \\\\
\n\\therefore \\ \\ s =\\frac{1}{r} &=& \\frac{GM}{\\ell^2} + \\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right) \\sin \\phi
\n\\end{eqnarray}<\/p>\n

\u6f38\u8fd1\u7684\u632f\u308b\u821e\u3044<\/h4>\n

\u89e3 (7) \u5f0f\u304b\u3089\uff0c$r \\rightarrow \\infty$ \u3068\u306a\u308b\u306e\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{GM}{\\ell^2} + \\left(\\frac{1}{b} -\\frac{GM}{\\ell^2} \\right) \\sin \\phi\u00a0 &=&0 \\\\
\n\\therefore\\ \\ \\sin\\phi &=& -\\frac{\\frac{GM}{\\ell^2}}{\\frac{1}{b} -\\frac{GM}{\\ell^2}} \\\\
\n&=& -\\frac{\\frac{GM b}{\\ell^2}}{1 -\\frac{GM b}{\\ell^2}} \\tag{8}
\n\\end{eqnarray}<\/p>\n

\u3092\u6e80\u305f\u3059\u89d2\u5ea6 $\\phi$ \u306e\u3068\u304d\u3067\u3042\u308b\u3002<\/p>\n

$r \\rightarrow \\infty$\u00a0 \u3067 $v \\rightarrow c$ \u3068\u3044\u3046\u6761\u4ef6\u304b\u3089 (3) \u5f0f\u3088\u308a<\/p>\n

$$\\varepsilon = \\frac{1}{2} c^2 $$<\/p>\n

\u307e\u305f\uff0c\u6700\u8fd1\u63a5\u8ddd\u96e2 $r = b$ \u3067\u306f\uff08$r$ \u304c\u6975\u5024\u3092\u3068\u308b\u3053\u3068\u304b\u3089\uff09$\\displaystyle \\frac{dr}{dt} = 0$ \u3068\u306a\u308a\uff0c(4) \u5f0f\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\varepsilon = \\frac{1}{2} c^2 &=& \\frac{1}{2} v^2 -\\frac{GM}{b}\\\\
\n&=& \\frac{1}{2} \\frac{\\ell^2}{b^2} -\\frac{GM}{b}
\n\\end{eqnarray}<\/p>\n

\u6700\u8fd1\u63a5\u70b9 $r=b$ \u306b\u304a\u3044\u3066\u3082\u5149\u306e\u901f\u3055\u306f\u307b\u307c\u5149\u901f\u3067\u3042\u308b\u3068\u3059\u308b\u3068 $v \\simeq c$ \uff0c\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u307b\u3046\u304c\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u3088\u308a\u3082\u5727\u5012\u7684\u306b\u5927\u304d\u3044\u3053\u3068\u306b\u306a\u308a\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{2} v^2 &\\gg& \\frac{GM}{b} \\\\
\n\\therefore\\ \\ \\frac{1}{2} c^2 &\\gg& \\frac{GM}{b} \\\\
\n\\therefore\\ \\ \\frac{2 GM}{b c^2}&\\ll&\u00a0 1\\\\
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u72b6\u6cc1\u3067\u306f<\/p>\n

\\begin{eqnarray}
\n\\varepsilon = \\frac{1}{2} c^2 &\\simeq& \\frac{1}{2} \\frac{\\ell^2}{b^2} \\\\
\n\\therefore\\ \\ \\ell &\\simeq& b c
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a<\/p>\n

\\begin{eqnarray}
\n\\frac{GM b}{\\ell^2} &\\simeq& \\frac{GM}{b c^2} \\ll 1
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c$r \\rightarrow \\infty$ \u3068\u306a\u308b\u89d2\u5ea6 $\\phi$ \u3092\u6c42\u3081\u308b (8) \u5f0f\u306f<\/p>\n

$$\\sin\\phi \\simeq -\\frac{GM}{b c^2}$$<\/p>\n

\u3053\u308c\u306f $|\\sin \\phi| \\ll 1$ \u306e\u3082\u3068\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\phi &\\simeq& -\\frac{GM}{b c^2}, \\quad \\pi + \\frac{GM}{b c^2}
\n\\end{eqnarray}<\/p>\n

\u5149\u306e\u66f2\u304c\u308a\u89d2<\/h4>\n

\u3064\u307e\u308a\uff0c\u3053\u306e\u5149\u306e\u7d4c\u8def\u306e\u6f38\u8fd1\u7684\u632f\u308b\u821e\u3044\u306f\uff0c<\/p>\n