{"id":1041,"date":"2024-01-11T17:50:00","date_gmt":"2024-01-11T08:50:00","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1041"},"modified":"2024-12-03T11:46:00","modified_gmt":"2024-12-03T02:46:00","slug":"%e6%b0%b4%e6%98%9f%e3%81%ae%e8%bf%91%e6%97%a5%e7%82%b9%e7%a7%bb%e5%8b%95","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%e3%83%86%e3%82%b9%e3%83%88%e7%b2%92%e5%ad%90%e3%81%ae%e9%81%8b%e5%8b%95\/%e6%b0%b4%e6%98%9f%e3%81%ae%e8%bf%91%e6%97%a5%e7%82%b9%e7%a7%bb%e5%8b%95\/","title":{"rendered":"\u6c34\u661f\u306e\u8fd1\u65e5\u70b9\u79fb\u52d5"},"content":{"rendered":"

<\/p>\n

\u5f31\u91cd\u529b\u5834\u4e2d\u306e\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053<\/h3>\n

\\(r_g\\) \u306e1\u6b21\u307e\u3067\u306e\u7bc4\u56f2\u3067\uff08\u305f\u3060\u3057\uff0c\\(O(r_g e^2)\\) \u306e\u9805\u306f\u7121\u8996\u3057\u3066\uff09\u6c42\u3081\u305f\u30c6\u30b9\u30c8\u7c92\u5b50\u306e\u8ecc\u9053\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002<\/p>\n

$$r =\u00a0 \\frac{a(1-e^2)}{1 + e \\cos(\\gamma\\phi) }$$<\/p>\n

\u3053\u3053\u3067<\/p>\n

$$\\gamma = \\sqrt{ 1 \u2013 \\frac{3 r_g}{a(1-e^2)}} \\simeq 1 \u2013 \\frac{3 r_g}{2a(1-e^2)}
\n$$<\/p>\n

\u307e\u305f\uff0c$a$ \u304a\u3088\u3073 $e$ \u306f\u30cb\u30e5\u30fc\u30c8\u30f3\u7406\u8ad6\u3067\u306f\u6955\u5186\u306e\u300c\u8ecc\u9053\u9577\u534a\u5f84<\/strong><\/span>\u300d\u304a\u3088\u3073\u300c\u96e2\u5fc3\u7387<\/strong><\/span>\u300d\u306b\u5bfe\u5fdc\u3059\u308b\u304c\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u306a\u8fd1\u4f3c\u89e3\u3067\u306f\uff0c\u9589\u3058\u305f\u6955\u5186\u306b\u306a\u3089\u306a\u3044\u306e\u3067\uff0c\u53b3\u5bc6\u306b\u306f\u300c\u8ecc\u9053\u9577\u534a\u5f84<\/strong><\/span>\u300d\u3068\u304b\u300c\u96e2\u5fc3\u7387<\/strong><\/span>\u300d\u3068\u304b\u306a\u3069\u3068\u547c\u3076\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002<\/p>\n

\u3057\u304b\u3057\uff0c\u8ecc\u9053\u304c\u53b3\u5bc6\u306b\u306f\u89e3\u3051\u306a\u3044\u3068\u3057\u3066\u3082\uff0c\u904b\u52d5\u304c\u6709\u754c\u3067\u3042\u308c\u3070 $r$ \u304c\u7121\u9650\u5927\u306b\u306a\u3063\u305f\u308a\u30bc\u30ed\u306b\u306a\u3063\u305f\u308a\u3059\u308b\u3053\u3068\u306a\u304f\uff0c\u539f\u70b9\u306e\u307e\u308f\u308a\u3092\u6709\u9650\u306e\u7bc4\u56f2<\/p>\n

$$ r_g < r_{\\rm min} \\le r \\le r_{\\rm max}$$<\/p>\n

\u3067\uff0c\u6709\u754c\u306a\u675f\u7e1b\u904b\u52d5\u3092\u3059\u308b\u30cf\u30ba\u3067\u3042\u308b\u3002\u305d\u3053\u3067\uff0c<\/p>\n

\\begin{eqnarray}
\nr_{\\rm max} &\\equiv& a\\,(1+e) \\\\
\nr_{\\rm min} &\\equiv& a\\,(1-e)
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\u5909\u6570 $a$ \u304a\u3088\u3073 $e$ \u3092\u5b9a\u7fa9\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u8fd1\uff08\u65e5\uff09\u70b9\u79fb\u52d5\u89d2<\/h3>\n

\u307e\u305f\uff0c$0<\\gamma < 1$ \u3067\u3042\u308b\u305f\u3081\u306b\uff0c$\\phi = 0$ \u3067 $r$ \u304c\u6700\u5c0f\u5024 \\(r_{\\textrm{min}}\\) \u3092\u3068\u3063\u305f\u5f8c\uff0c\u6b21\u306e\u6700\u5c0f\u5024\u3068\u306a\u308b\u89d2\u5ea6 \\(\\phi\\) \u306f $2\\pi$ \u30e9\u30b8\u30a2\u30f3\u304b\u3089\u3055\u3089\u306b $\\varDelta$ \u3060\u3051\u5fc5\u8981\u3067\u3042\u308b\u3002\u3053\u306e\u73fe\u8c61\u3092\u8fd1\u70b9\u79fb\u52d5<\/strong><\/span>\uff08\u4e2d\u5fc3\u5929\u4f53\u304c\u592a\u967d\u306e\u5834\u5408\u306f\u8fd1\u65e5\u70b9\u79fb\u52d5<\/strong><\/span>\uff09\uff0c\u3053\u306e\u89d2\u5ea6 $\\varDelta$ \u3092\u8fd1\u70b9\u79fb\u52d5\u89d2<\/strong><\/span>\uff08\u4e2d\u5fc3\u5929\u4f53\u304c\u592a\u967d\u306e\u5834\u5408\u306f\u8fd1\u65e5\u70b9\u79fb\u52d5\u89d2<\/strong><\/span>\uff09\u3068\u547c\u3073\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u3089\u308c\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n(2\\pi + \\varDelta) \\gamma &=& 2\\pi\\\\
\n\\therefore\\ \\ \\varDelta &=& \\frac{2\\pi}{\\gamma} – 2 \\pi\\\\
\n&=&\\frac{3\\pi r_g}{a(1-e^2)} = \\frac{6\\pi GM}{c^2 a (1-e^2)}
\n\\end{eqnarray}<\/p>\n

Maxima-Jupyter \u3067\u306e\u8a08\u7b97\u4f8b<\/h3>\n
\n
\n
<\/div>\n
\n
\n

Maxima \u3067\u6c34\u661f\u306e\u8fd1\u65e5\u70b9\u79fb\u52d5<\/h4>\n

\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u52b9\u679c\u306b\u3088\u308a\uff0c\u8fd1\u65e5\u70b9\u304c1\u5468\u3042\u305f\u308a\u4ee5\u4e0b\u306e\u5024\u3060\u3051\u79fb\u52d5\u3002<\/p>\n

$$ \\Delta = \\frac{3\\pi r_g}{a(1-e^2)} = \\frac{6\\pi GM}{c^2 a (1-e^2)}$$<\/p>\n

\u3067\u306f\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u52b9\u679c\u306b\u3088\u308b\u6c34\u661f\u306e\u8fd1\u65e5\u70b9\u79fb\u52d5\u306f100\u5e74\u3042\u305f\u308a\u4f55\u79d2\u89d2\u3067\u3042\u308b\u304b\u3002<\/p>\n