{"id":1298,"date":"2022-01-20T17:26:27","date_gmt":"2022-01-20T08:26:27","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1298"},"modified":"2022-02-13T12:04:01","modified_gmt":"2022-02-13T03:04:01","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e6%a5%95%e5%86%86%e8%bb%8c%e9%81%93%ef%bc%8c%e5%8f%8c%e6%9b%b2%e7%b7%9a%e8%bb%8c%e9%81%93%ef%bc%8c%e6%94%be%e7%89%a9%e7%b7%9a%e8%bb%8c%e9%81%93","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\/%e4%b8%87%e6%9c%89%e5%bc%95%e5%8a%9b%e3%81%ae2%e4%bd%93%e5%95%8f%e9%a1%8c\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e6%a5%95%e5%86%86%e8%bb%8c%e9%81%93%ef%bc%8c%e5%8f%8c%e6%9b%b2%e7%b7%9a%e8%bb%8c%e9%81%93%ef%bc%8c%e6%94%be%e7%89%a9%e7%b7%9a%e8%bb%8c%e9%81%93\/","title":{"rendered":"\u88dc\u8db3\uff1a\u6955\u5186\u8ecc\u9053\uff0c\u53cc\u66f2\u7dda\u8ecc\u9053\uff0c\u653e\u7269\u7dda\u8ecc\u9053"},"content":{"rendered":"

\\(\\displaystyle s \\equiv \\frac{1}{r}\\) \u3068\u3057\u305f\u3068\u304d\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u306f<\/p>\n

$$\\left(\\frac{ds}{d\\phi}\\right)^2 = \\frac{1}{B^2} – \\left(s – \\frac{GM}{\\ell^2}\\right)^2 \\tag{1}$$<\/p>\n

\u3053\u3053\u3067
\n$$\\frac{1}{B} \\equiv \\sqrt{\\frac{2\\epsilon}{\\ell^2} + \\left(\\frac{GM}{\\ell^2}\\right)^2} \\tag{2}$$<\/p>\n

\u3067\u3042\u3063\u305f\u3002\uff08\u5c0f\u6587\u5b57 \\(b\\) \u306f\u5f8c\u3067\u4f7f\u3046\u77ed\u534a\u5f84\u3068\u30d0\u30c3\u30c6\u30a3\u30f3\u30b0\u3059\u308b\u306e\u3067\uff0c\u5927\u6587\u5b57 \\(B\\) \u3092\u4f7f\u3044\u307e\u3059\u3002\uff09<\/p>\n

\\(\\phi = 0\\) \u3067 \\(s\\) \u3057\u305f\u304c\u3063\u3066 \\(r\\) \u304c\u6975\u5024\u3092\u3082\u3064\u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3092\u8ab2\u3059\u3068\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

$$s = \\frac{1}{r} =\\frac{GM}{\\ell^2}+ \\frac{\\cos\\phi}{B} \\tag{3}$$<\/p>\n

\\(\\epsilon < 0\\) \u306e\u5834\u5408\uff1a\u6955\u5186\u8ecc\u9053<\/h3>\n

\u6955\u5186\u8ecc\u9053\u3067\u3042\u308b\u3053\u3068\u304c\u3072\u3068\u76ee\u3067\u308f\u304b\u308b\u3088\u3046\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u6570\u3092\u5b9a\u7fa9\u3057\u3066\u3084\u308b\u3002
\n$$ \\frac{GM}{\\ell^2} = \\frac{1}{a(1-e^2)}, \\quad\u00a0 \\frac{1}{B} = \\sqrt{ \\frac{2\\epsilon}{\\ell^2} + \\left(\\frac{GM}{\\ell^2}\\right)^2} = \\frac{e}{a(1-e^2)}$$ \u3059\u308b\u3068\uff0c<\/p>\n

$$\\frac{1}{r} = \\frac{1 + e\\cos\\phi}{a(1-e^2)}, \\quad \\therefore \\ r = \\frac{a(1-e^2)}{1 + e\\cos\\phi} \\tag{4}$$ \u3068\u306a\u308a\uff0c\u3053\u306e\u8ecc\u9053\u306f\uff0c\u9577\u534a\u5f84<\/strong><\/span> \\(a\\)\uff0c\u96e2\u5fc3\u7387<\/strong><\/span> \\(e\\) \u306e\u6955\u5186<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u4e00\u76ee\u77ad\u7136\u3068\u306a\u308b\u3002<\/p>\n

\"\"<\/p>\n

\\(a\\)\uff0c\\(e\\) \u3092\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09\u3092\u4f7f\u3063\u3066\u66f8\u304d\u76f4\u3057\u3066\u3084\u308b\u3068\uff0c\u675f\u7e1b\u8ecc\u9053\u306e\u5834\u5408\u306b \\( \\epsilon = -|\\epsilon|\\) \u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068
\n$$a = \\frac{GM}{2|\\epsilon|}, \\quad e = \\sqrt{1 – \\frac{2 |\\epsilon| \\ell^2}{(GM)^2}}$$<\/p>\n

\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u306e\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u306e\u6a19\u6e96\u5f62<\/h4>\n

$$\\ r = \\frac{a(1-e^2)}{1 + e\\cos\\phi}$$\u304c\u78ba\u304b\u306b\u6955\u5186\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u306b\u5909\u63db\u3059\u308b\u3068\uff0c<\/p>\n

$$ x = a e + r \\cos\\phi, \\quad y = r \\sin\\phi$$<\/p>\n

\\begin{eqnarray}
\n\\frac{x^2}{a^2} + \\frac{y^2}{a^2 (1-e^2)} &=&
\n\\left(\\frac{e + \\cos\\phi}{1 + e\\cos\\phi} \\right)^2
\n+ \\frac{(1-e^2) (1-\\cos^2\\phi)}{(1 + e\\cos\\phi)^2}\\\\
\n&=& 1\\\\
\n\\therefore\\ \\ \\frac{x^2}{a^2} + \\frac{y^2}{b^2} &=& 1, \\quad b \\equiv a \\sqrt{1-e^2}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u78ba\u304b\u306b\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u9577\u534a\u5f84<\/strong><\/span> \\(a\\)\uff0c\u77ed\u534a\u5f84<\/strong><\/span> \\(b\\) \u306e\u6955\u5186<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u307b\u3063\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u306f\u4e09\u89d2\u95a2\u6570\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3067\u304d\u307e\u3057\u305f\u3088\u306d\u3002<\/p>\n

$$x = a \\cos\\theta, \\quad y = b \\sin\\theta$$<\/p>\n

\\(\\epsilon > 0\\) \u306e\u5834\u5408\uff1a\u53cc\u66f2\u7dda\u8ecc\u9053<\/h3>\n

\\(\\epsilon > 0\\) \u306e\u5834\u5408\u306f\u6955\u5186\u8ecc\u9053\u306e\u89e3 \\((4)\\) \u3092\u305d\u306e\u307e\u307e\u4f7f\u3044\u56de\u3057\u3066<\/p>\n

$$a = – \\frac{GM}{2\\epsilon} \\equiv – \\alpha, \\ \\ \\alpha > 0,
\n\\quad e =\u00a0 \\sqrt{1 + \\frac{2 \\epsilon \\ell^2} {(GM)^2}} > 1$$
\n\u3068\u3059\u308c\u3070
\n$$r = \\frac{\\alpha (e^2 – 1)}{1 + e\\cos\\phi}$$<\/p>\n

\"\"<\/p>\n

\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u306e\u53cc\u66f2\u7dda\u306e\u65b9\u7a0b\u5f0f\u306e\u6a19\u6e96\u5f62<\/h4>\n

\u3053\u308c\u304c\u78ba\u304b\u306b\u53cc\u66f2\u7dda\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u306b\u5909\u63db\u3059\u308b\u3068\uff0c<\/p>\n

$$ x = -\\alpha e + r \\cos\\phi, \\quad y = r \\sin\\phi$$<\/p>\n

\\begin{eqnarray}
\n\\frac{x^2}{\\alpha^2} – \\frac{y^2}{\\alpha^2 (e^2-1)} =\\frac{x^2}{\\alpha^2} – \\frac{y^2}{\\beta^2}
\n&=& 1, \\quad \\beta \\equiv \\alpha \\sqrt{e^2 – 1}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u3063\u3066\uff0c\u78ba\u304b\u306b\u53cc\u66f2\u7dda<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3053\u306e\u5f0f\u304c\u53cc\u66f2\u7dda\u95a2\u6570<\/strong><\/span>\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3067\u304d\u308b\u3053\u3068\u3082\uff0c\u3042\u3041\u53cc\u66f2\u7dda\u3060\u306a\u3041<\/strong><\/span>\u3068\u611f\u3058\u3055\u305b\u307e\u3059\u3002<\/p>\n

$$x = \\alpha \\cosh t, \\quad y = \\beta \\sinh t$$<\/p>\n

\\(\\epsilon = 0\\) \u306e\u5834\u5408\uff1a\u653e\u7269\u7dda\u8ecc\u9053<\/h3>\n

\\(\\epsilon = 0\\) \u306e\u5834\u5408\u306f\u6700\u521d\u306e\u5f0f \\((2)\\) \u304b\u3089<\/p>\n

$$\\frac{1}{B} \\equiv \\sqrt{\\frac{2\\epsilon}{\\ell^2} + \\left(\\frac{GM}{\\ell^2}\\right)^2} \\Rightarrow \\frac{GM}{\\ell^2}$$<\/p>\n

\u3068\u306a\u308b\u306e\u3067\uff0c\u5f0f \\((3)\\) \u306f<\/p>\n

$$s = \\frac{1}{r} =\\frac{GM}{\\ell^2}+ \\frac{\\cos\\phi}{B} \\Rightarrow \\frac{1 + \\cos\\phi}{B}$$<\/p>\n

\u3064\u307e\u308a$$ r = \\frac{B}{1 + \\cos\\phi}$$<\/p>\n

\"\"<\/p>\n

\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u306e\u653e\u7269\u7dda\u306e\u65b9\u7a0b\u5f0f\u306e\u6a19\u6e96\u5f62<\/h4>\n

\u3053\u308c\u304c\u78ba\u304b\u306b\u653e\u7269\u7dda<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u306b\u5909\u63db\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\nx &=& r \\cos\\phi – \\frac{B}{2} \\\\
\n&=& – \\frac{B}{2} \\frac{1 – \\cos\\phi}{1 + \\cos\\phi}\\\\
\ny &=& r\\sin\\phi \\\\
\n&=& B \\frac{\\sin\\phi}{1 + \\cos\\phi}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\ny^2 &=& B^2 \\frac{1 – \\cos^2\\phi}{(1 + \\cos\\phi)^2}\\\\
\n&=& B^2 \\frac{1 – \\cos\\phi}{1 + \\cos\\phi}\\\\
\n&=& -2 B x\\\\
\n\\therefore\\ \\ x &=& -\\frac{1}{2B} y^2
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u3063\u3066\uff0c\u78ba\u304b\u306b\u653e\u7269\u7dda<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u306e\u5f0f\u306f $\\displaystyle y = \\frac{1}{2B} x^2$\u3092\u539f\u70b9\u306e\u307e\u308f\u308a\u306b \\(90\\)\u5ea6\uff0c\u53cd\u6642\u8a08\u56de\u308a\u306b\u56de\u8ee2\u3055\u305b\u3066\u6a2a\u5012\u3057\u306b\u3057\u305f\u653e\u7269\u7dda<\/strong><\/span>\u306e\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u3042\u3041\uff0c\u3059\u3063\u304d\u308a\u3057\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":1258,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1298","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