{"id":6621,"date":"2023-07-06T12:07:36","date_gmt":"2023-07-06T03:07:36","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6621"},"modified":"2025-02-13T12:38:34","modified_gmt":"2025-02-13T03:38:34","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e3%82%b1%e3%83%97%e3%83%a9%e3%83%bc%e3%81%ae%e6%b3%95%e5%89%87","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\/%e5%8f%82%e8%80%83%ef%bc%9a%e3%82%b1%e3%83%97%e3%83%a9%e3%83%bc%e3%81%ae%e6%b3%95%e5%89%87\/","title":{"rendered":"\u53c2\u8003\uff1a\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247"},"content":{"rendered":"<p><!--more--><\/p>\n<h3>\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247<\/h3>\n<p>\u53c2\u8003\u6587\u732e\uff1a<\/p>\n<ul>\n<li><a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%82%B1%E3%83%97%E3%83%A9%E3%83%BC%E3%81%AE%E6%B3%95%E5%89%87\">\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247 &#8211; Wikipedia<\/a><\/li>\n<li><a href=\"https:\/\/www.utp.or.jp\/book\/b301872.html\">\u5929\u4f53\u3068\u8ecc\u9053\u306e\u529b\u5b66\uff08\u6728\u4e0b\u5b99\uff0c\u6771\u4eac\u5927\u5b66\u51fa\u7248\u4f1a\uff0c1998\uff09<\/a>\u3002\u672c\u7a3f\u57f7\u7b46\u6642\u70b9\u3067\u306f\u300c\u54c1\u5207\u308c\u30fb\u91cd\u7248\u672a\u5b9a\u300d\u3068\u306a\u3063\u3066\u304a\u308a\uff0c<a href=\"https:\/\/www.amazon.co.jp\/dp\/4130607219?SubscriptionId=AKIAIBX3OSRN6HXD25SQ&amp;tag=ut00-22&amp;linkCode=xm2&amp;camp=2025&amp;creative=165953&amp;creativeASIN=4130607219\">Amazon \u3067\u306f\u4e2d\u53e4\u54c1\u306b\u5b9a\u4fa1\u306e3\u500d\u4ee5\u4e0a\u306e\u5024\u6bb5\u304c\u3064\u3044\u3066\u3044\u308b<\/a>\u3002<\/li>\n<\/ul>\n<p>\u30b1\u30d7\u30e9\u30fc\u306f\uff0c\u30c6\u30a3\u30b3\u30fb\u30d6\u30e9\u30fc\u30a8\u306e\u89b3\u6e2c\u30c7\u30fc\u30bf\u304b\u3089\uff0c\u592a\u967d\u7cfb\u306e\u60d1\u661f\u306e\u904b\u52d5\u3092\u6b21\u306e3\u3064\u306e\u6cd5\u5247\u306e\u5f62\u306b\u307e\u3068\u3081\u305f\u3002\u592a\u967d\u7cfb\u306e\u60d1\u661f\u306e\u904b\u52d5\u306b\u95a2\u3059\u308b\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u3068\u306f\uff0c<\/p>\n<h4>\u7b2c1\u6cd5\u5247<\/h4>\n<p><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u60d1\u661f\u306f\u592a\u967d\u3092\u7126\u70b9\u306e\u3072\u3068\u3064\u3068\u3059\u308b\u6955\u5186\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3059\u308b<\/strong><\/span>\u3002<\/p>\n<p>\uff08\u60d1\u661f\u306e\u904b\u52d5\u306f\u592a\u967d\u3092\u901a\u308b\u5e73\u9762\u4e0a\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u304c\u4eee\u5b9a\u3055\u308c\u3066\u3044\u308b\u3002\uff09\u592a\u967d\u306e\u4f4d\u7f6e\u3092\u539f\u70b9\u3068\u3059\u308b\u6975\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3059\u3068\uff083\u6b21\u5143\u6975\u5ea7\u6a19\u3092\u4f7f\u3046\u3068 $(x, y, z) = (r \\sin\\theta\\cos\\phi, r \\sin\\theta\\cos\\phi, r \\cos\\theta)$ \u3060\u304c\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u904b\u52d5\u3092 $\\theta = \\pi\/2$ \u306e\u8d64\u9053\u9762\u4e0a\u306b\u9650\u308b\u3068\u3057\u3066\uff09\uff0c\u60d1\u661f\u306e\u4f4d\u7f6e $(x, y, z) = (r \\cos\\phi, r \\sin\\phi, 0)$ \u306f\u4ee5\u4e0b\u306e\u5f0f\uff1a<\/p>\n<p>$$ r = \\frac{a(1-e^2)}{1 + e \\cos\\phi}$$<\/p>\n<p>\u3053\u3053\u3067\uff0c$a$ \u306f\u6955\u5186\u306e\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3002<\/p>\n<h4>\u7b2c2\u6cd5\u5247<\/h4>\n<p>\uff08\u4e00\u3064\u306e\u60d1\u661f\u306b\u7740\u76ee\u3059\u308b\u3068\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9762\u7a4d\u901f\u5ea6<\/strong><\/span>\uff08\u60d1\u661f\u3068\u592a\u967d\u3092\u7d50\u3076\u7dda\u5206\u304c\u5358\u4f4d\u6642\u9593\u306b\u6383\u304f\u9762\u7a4d\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u306f\u4e00\u5b9a\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n<p>\u60d1\u661f\u304c\u6955\u5186\u8ecc\u9053\u4e0a\u3092\u904b\u52d5\u3059\u308b\u3053\u3068\u304b\u3089\uff0c\u60d1\u661f\u30fb\u592a\u967d\u9593\u306e\u8ddd\u96e2\u306f\u4e00\u5b9a\u3067\u306f\u306a\u304f\uff0c\u4e00\u822c\u306b\u5909\u5316\u3059\u308b\u3002\u7b2c2\u6cd5\u5247\u306f\uff0c\u60d1\u661f\u304c\u592a\u967d\u306b\u8fd1\u3044\u6642\u306f\u3059\u3070\u3084\u304f\u904b\u52d5\u3057\uff0c\u592a\u967d\u304b\u3089\u96e2\u308c\u3066\u3044\u308b\u3068\u304d\u306f\u3086\u3063\u304f\u308a\u3068\u904b\u52d5\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n<p>\u6955\u5186\u306e\u9762\u7a4d\u3092 $S$ \u3068\u3059\u308b\u3068\uff0c\u5fae\u5c0f\u6642\u9593\u9593\u9694 $dt$ \u306e\u9593\u306b\u60d1\u661f\u3068\u592a\u967d\u3092\u7d50\u3076\u7dda\u5206\u304c\u6383\u304f\u5fae\u5c0f\u9762\u7a4d\u306f $dS = \\dfrac{1}{2} r^2 d\\phi$ \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n<p>$$\\frac{dS}{dt} = \\frac{1}{2} r^2 \\frac{d\\phi}{dt} = \\mbox{const.}\u00a0 \\left(= \\frac{\\pi a b}{T} = \\frac{\\pi a^2 \\sqrt{1-e^2}}{T}\\right)$$<\/p>\n<p>\u3053\u3053\u3067\uff0c$T$ \u306f\u516c\u8ee2\u5468\u671f\uff08\u6955\u5186\u8ecc\u9053\u4e0a\u30921\u5468\u3059\u308b\u306e\u306b\u304b\u304b\u308b\u6642\u9593\uff09\u3002\u6955\u5186\u306e\u77ed\u534a\u5f84 $b = a \\sqrt{1-e^2}$ \u3092\u7528\u3044\u305f\u3002<\/p>\n<p>\u3042\u308b\u3044\u306f\uff0c\u4ee5\u4e0b\u3067\u8aac\u660e\u3059\u308b\u3088\u3046\u306b\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\u3092\u5ff5\u982d\u306b\u304a\u3044\u3066\uff08\u4e21\u8fba\u30922\u500d\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u3082\u3088\u3044\u3060\u308d\u3046\u3002<\/p>\n<p>$$r^2 \\frac{d\\phi}{dt} = \\mbox{const.} \\equiv \\ell$$<\/p>\n<h4>\u7b2c3\u6cd5\u5247<\/h4>\n<p>\uff08\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3084\u516c\u8ee2\u5468\u671f $T$ \u306f\u60d1\u661f\u3054\u3068\u306b\u7570\u306a\u308b\u304c\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u516c\u8ee2\u5468\u671f\u306e\u4e8c\u4e57 $T^2$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\u306e\u4e09\u4e57 $a^3$ \u306b\u6bd4\u4f8b\u3059\u308b<\/strong><\/span>\u3002<\/p>\n<p>\u8a00\u3044\u63db\u3048\u308c\u3070\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u516c\u8ee2\u5468\u671f\u306e\u4e8c\u4e57 $T^2$ \u3068\u8ecc\u9053\u9577\u534a\u5f84\u306e\u4e09\u4e57 $a^3$ \u306e\u6bd4\u306f\u60d1\u661f\u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\u3042\u308b<\/strong><\/span>\u3002<\/p>\n<h3>\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u306f\u4e07\u6709\u5f15\u529b\u306e1\u4f53\u554f\u984c\u304b\u3089\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b<\/h3>\n<p>\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u306f\uff0c\u3042\u304f\u307e\u3067\u5929\u4f53\u89b3\u6e2c\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u305f\u89b3\u6e2c\u91cf\u9593\u306e\u95a2\u4fc2\u5f0f\u3067\u3042\u308b\u3002\u306a\u305c\u305d\u306e\u3088\u3046\u306a\u95a2\u4fc2\u306b\u306a\u3063\u3066\u3044\u308b\u304b\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e1\u4f53\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3067\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<p>\u8cea\u91cf $M$ \u306e\u4e2d\u5fc3\u5929\u4f53\uff08\u300c\u592a\u967d\u300d\u306b\u7279\u5316\u305b\u305a\u306b\u4e00\u822c\u306b\u4e2d\u5fc3\u5929\u4f53\u3068\u3066\u3082\u3057\u3066\u304a\u3053\u3046\uff09\u304b\u3089\u306e\u4e07\u6709\u5f15\u529b\u3092\u53d7\u3051\u3066\u4e2d\u5fc3\u5929\u4f53\u306e\u307e\u308f\u308a\u3092\u904b\u52d5\u3059\u308b\u8cea\u91cf $m$ \u306e\u5929\u4f53\uff08\u300c\u60d1\u661f\u300d\u306b\u76f8\u5f53\uff09\u306f\u4ee5\u4e0b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306b\u5f93\u3046\uff1a<\/p>\n<p>$$ m \\frac{d^2 \\boldsymbol{r}}{dt^2} = -\\frac{G M m}{r^3} \\boldsymbol{r}$$<\/p>\n<p>\u3053\u3053\u3067\u306f $ m \\ll M$ \u3068\u3057\u3066\uff0c\u8cea\u91cf $M$ \u306e\u4e2d\u5fc3\u5929\u4f53\u306f\u539f\u70b9\u306b\u56fa\u5b9a\uff0c\u4e2d\u5fc3\u5929\u4f53\u306e\u307e\u308f\u308a\u3092\u904b\u52d5\u3059\u308b $m$ \u306e\u5929\u4f53\u304c\u4e2d\u5fc3\u5929\u4f53\u306b\u53ca\u307c\u3059\u91cd\u529b\u306f\u7121\u8996\u3067\u304d\u308b\u3068\u3059\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u8a2d\u5b9a\u3092\u300c1\u4f53\u554f\u984c\u300d\u3068\u3044\u3046\u30021\u4f53\u554f\u984c\u306e\u89e3\u6cd5\u306b\u3064\u3044\u3066\u306f\u5225\u9014\u8aac\u660e\u306f\u3057\u306a\u3044\u3002\u306a\u305c\u306a\u3089\uff0c\u4ee5\u4e0b\u306b\u793a\u3059\u3088\u3046\u306b\uff0c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f1\u4f53\u554f\u984c\u306e\u305d\u308c\u306b\u5e30\u7740\u3059\u308b\u304b\u3089\u3002<\/p>\n<h3>\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u304b\u3089\u5f97\u3089\u308c\u305f\u7d50\u679c\u3068\u306e\u6bd4\u8f03<\/h3>\n<p>\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u306f\uff0c\u3042\u304f\u307e\u3067\u5929\u4f53\u89b3\u6e2c\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u305f\u89b3\u6e2c\u91cf\u9593\u306e\u95a2\u4fc2\u5f0f\u3067\u3042\u308b\u3002\u306a\u305c\u305d\u306e\u3088\u3046\u306a\u95a2\u4fc2\u306b\u306a\u3063\u3066\u3044\u308b\u304b\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3067\u3082\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3060\u304c\uff0c\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u304c\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247\u3092\u5c0e\u304f\u969b\u306e\u6307\u5c0e\u539f\u7406\u3060\u3063\u305f\u3068\u601d\u308f\u308c\u308b\u306b\u3082\u95a2\u308f\u3089\u305a\uff0c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u3092\u89e3\u3044\u3066\u5f97\u3089\u308c\u305f\u7d50\u679c\u3092\u3088\u304f\u898b\u308b\u3068\uff0c\u5fc5\u305a\u3057\u3082\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u3092\u305d\u306e\u307e\u307e\u306e\u5f62\u3067\u5b8c\u5168\u518d\u73fe\u3059\u308b\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\u307e\u305a\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u3067\u306f\u4f5c\u7528\u30fb\u53cd\u4f5c\u7528\u306e\u6cd5\u5247\u304c\u3042\u308b\u305f\u3081\uff0c\u592a\u967d\u304c\u60d1\u661f\u306b\u4e07\u6709\u5f15\u529b\u3092\u53ca\u307c\u305b\u3070\uff0c\u592a\u967d\u3082\u307e\u305f\u60d1\u661f\u304b\u3089\u306e\u4e07\u6709\u5f15\u529b\u3092\u53d7\u3051\u308b\u305f\u3081\uff0c\u53b3\u5bc6\u306b\u306f\u591a\u4f53\u554f\u984c\u3068\u306a\u308b\u30023\u4f53\u554f\u984c\u4ee5\u4e0a\u306f\u4e00\u822c\u306b\u306f\u89e3\u6790\u7684\u306b\u89e3\u3051\u305a\uff0c2\u4f53\u554f\u984c\u306e\u307f\uff081\u4f53\u554f\u984c\u306b\u5e30\u7740\u3057\u3066\uff09\u89e3\u3051\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3067\u307e\u3068\u3081\u3066\u3044\u308b\u7d50\u679c\u306f\u592a\u967d\u30681\u500b\u306e\u60d1\u661f\u306e\u307f\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u306b\u53b3\u5bc6\u306b\u6210\u308a\u7acb\u3064\u3002\u592a\u967d\u7cfb\u306e\u3088\u3046\u306b\u8907\u6570\u500b\u306e\u60d1\u661f\u304c\u3042\u308b\u5834\u5408\u306b\u306f\u53b3\u5bc6\u306a\u610f\u5473\u3067\u306f\u6210\u308a\u7acb\u305f\u306a\u3044\u3002<\/p>\n<ul>\n<li><a href=\"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\/\">\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<h4>\u7b2c1\u6cd5\u5247<\/h4>\n<p>\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u3092\u89e3\u304f\u3068\uff0c\u307e\u305a\uff0c\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u30d9\u30af\u30c8\u30eb $\\boldsymbol{\\ell}$ \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u305d\u306e\u5411\u304d\u3092 $z$ \u306b\u53d6\u308b\u3068\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u904b\u52d5\u3092 $xy$ \u5e73\u9762\u4e0a\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\uff0c\u78ba\u304b\u306b\u76f8\u5bfe\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306b\u3064\u3044\u3066\u6955\u5186\u306e\u5f0f<\/p>\n<p>$$r = \\frac{a(1-e^2)}{1 + e\\cos\\phi}$$<\/p>\n<p>\u304c\u5c0e\u304b\u308c\u308b\u304c\uff0c\uff08\u4e0d\u52d5\u306e\uff09\u592a\u967d\u306e\u307e\u308f\u308a\u3092\u60d1\u661f\u300c\u3060\u3051\u300d\u304c\u52d5\u304f\uff0c\u3068\u3044\u3046\u3088\u308a\u306f\uff0c\u592a\u967d\u300c\u3082\u300d\u60d1\u661f\u300c\u3082\u300d\u4e92\u3044\u306e\u8cea\u91cf\u4e2d\u5fc3\u306e\u307e\u308f\u308a\u3092\u6955\u5186\u8ecc\u9053\u3092\u63cf\u3044\u3066\u904b\u52d5\u3059\u308b\uff0c\u3068\u3057\u305f\u307b\u3046\u304c\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<h4>\u7b2c2\u6cd5\u5247<\/h4>\n<p>2\u4f53\u554f\u984c\u3092\u89e3\u304f\u30682\u3064\u306e\u4fdd\u5b58\u5247\uff08\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\uff0c\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\uff09\u304c\u5f97\u3089\u308c\u308b\u304c\uff0c\u9762\u7a4d\u901f\u5ea6\u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u306e\u534a\u5206\u306b\u3042\u305f\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u30d9\u30af\u30c8\u30eb\u306f<\/p>\n<p>$$\\boldsymbol{\\ell} \\equiv \\boldsymbol{r} \\times \\dot{\\boldsymbol{r} }, \\qquad \\frac{d\\boldsymbol{\\ell}}{dt} = \\boldsymbol{0}$$<\/p>\n<p>\u3067\u3042\u308a\uff0c\uff08\u904b\u52d5\u306f\u8d64\u9053\u9762\u4e0a\u306b\u9650\u3089\u308c\u308b\u3068\u3057\u305f\u304b\u3089\uff09<\/p>\n<p>$$\\boldsymbol{r} = (x, y, 0) = (r \\cos\\phi, r \\sin\\phi, 0)$$<\/p>\n<p>\u3068\u3059\u308c\u3070\uff0c\u89d2\u904b\u52d5\u91cf\u306e $z$ \u6210\u5206\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\ell_z &amp;=&amp; x \\dot{y} &#8211; y \\dot{x} \\\\<br \/>\n&amp;=&amp; r \\cos\\phi \\left( \\dot{r} \\sin\\phi + r \\cos\\phi \\, \\dot{\\phi}\\right)<br \/>\n&#8211; r \\sin\\phi \\left( \\dot{r} \\cos\\phi &#8211; r \\sin\\phi\\, \\dot{\\phi}\\right) \\\\<br \/>\n&amp;=&amp; r^2 \\frac{d\\phi}{dt}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u4e00\u65b9\uff0c\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u5247\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{dS}{dt} &amp;=&amp; \\frac{1}{2} r^2 \\frac{d\\phi}{dt}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{dS}{dt} &amp;=&amp; \\frac{1}{2} \\ell_z = \\frac{1}{2} \\ell \\qquad (\\ell = \\sqrt{\\boldsymbol{\\ell}\\cdot\\boldsymbol{\\ell}})\\\\<br \/>\n\\because\\ \\ \\boldsymbol{\\ell} &amp;=&amp; (0, 0, \\ell_z)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3064\u307e\u308a\uff0c\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u3068\u306f\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\u306e\u3053\u3068\u3067\u3042\u3063\u305f\u3002<\/p>\n<p>\u307e\u305f\uff0c\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306b $\\ell = \\sqrt{GMa (1-e^2)}$ \u3068\u89e3\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\uff0c\u7b2c2\u6cd5\u5247\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u3002<\/p>\n<p>$$r^2 \\frac{d\\phi}{dt} = \\ell = \\sqrt{GMa (1-e^2)}$$<\/p>\n<h4>\u7b2c3\u6cd5\u5247<\/h4>\n<p>\u3082\u3046\u4e00\u3064\u306e\u4fdd\u5b58\u91cf\u3067\u3042\u308b\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc $\\varepsilon$ \u306f\uff0c\u6975\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\varepsilon<br \/>\n&amp;=&amp; \\frac{1}{2} \\left( \\dot{x}^2 + \\dot{y}^2\\right)- \\frac{GM}{r} \\\\<br \/>\n&amp;=&amp; \\frac{\\ell^2}{2} \\left\\{\\left(\\frac{1}{r^2}\\frac{dr}{d\\phi}\\right)^2 + \\frac{1}{r^2}\\right\\}- \\frac{GM}{r}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3067\u3042\u3063\u305f\u3002\u6955\u5186\u8ecc\u9053\u306e\u5834\u5408\u306f\uff0c$r_{\\rm min} = a (1-e)$ \u304a\u3088\u3073 $r_{\\rm max} = a (1+e)$ \u3067 $r$ \u304c\u6975\u5024\u3092\u3068\u308b\uff0c\u3059\u306a\u308f\u3061 $\\displaystyle \\frac{dr}{d\\phi} = 0$ \u306a\u306e\u3067\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\varepsilon<br \/>\n&amp;=&amp; \\frac{\\ell^2}{2 r_{\\rm min}^2} &#8211; \\frac{GM}{r_{\\rm min}} \\tag{1}\\\\<br \/>\n\\varepsilon<br \/>\n&amp;=&amp; \\frac{\\ell^2}{2 r_{\\rm max}^2} &#8211; \\frac{GM}{r_{\\rm max}} \\tag{2} \\\\<br \/>\n\\end{eqnarray}<\/p>\n<p>$(1)$ \u5f0f\u3068 $(2)$ \u5f0f\u3092\u9023\u7acb\u65b9\u7a0b\u5f0f\u3068\u3057\u3066 $\\varepsilon$ \u304a\u3088\u3073 $\\ell$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\varepsilon &amp;=&amp; &#8211; \\frac{GM}{r_{\\rm min} + r_{\\rm max}} = &#8211; \\frac{GM}{2a} \\\\<br \/>\n\\ell^2 &amp;=&amp; 2 GM \\frac{r_{\\rm min}\u00a0 r_{\\rm max}}{r_{\\rm min} +\u00a0 r_{\\rm max}} = GMa (1-e^2) \\\\<br \/>\n\\therefore \\ \\ \\ell &amp;=&amp; \\sqrt{GMa (1-e^2)}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u3068\u7b2c2\u6cd5\u5247\u3092\u3042\u308f\u305b\u308b\u3068<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{dS}{dt} = \\frac{\\pi a^2 \\sqrt{1-e^2}}{T} &amp;=&amp; \\frac{1}{2} \\ell = \\frac{1}{2} \\sqrt{GMa (1-e^2)} \\\\<br \/>\n\\therefore\\ \\ \\frac{\\pi a^2}{T} &amp;=&amp; \\frac{1}{2} \\sqrt{GMa}\u00a0 \\\\<br \/>\n\\therefore\\ \\ \\frac{a^3}{T^2} &amp;=&amp; \\frac{GM}{4\\pi^2}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u3044\u3046\u7b2c3\u6cd5\u5247\u7684\u306a\u95a2\u4fc2\u5f0f\u304c\u51fa\u3066\u304f\u308b\u3002<\/p>\n<p>\u5f37\u8abf\u3057\u3066\u304a\u304f\u3079\u304d\u306f\uff0c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306b\u304a\u3044\u3066\u306f $M = m_1 + m_2$ \u3067\u3042\u308a\uff0c$m_1$ \u3092\u592a\u967d\u8cea\u91cf\uff0c$m_2$ \u3092\u60d1\u661f\u8cea\u91cf\u3068\u3059\u308b\u3068\uff0c\u6bd4 $\\displaystyle \\frac{a^3}{T^2} $ \u306f<span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong>\u60d1\u661f\u306b\u3088\u3089\u305a\u4e00\u5b9a\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u304f\uff0c$M$ \u306e\u4e2d\u306e\u60d1\u661f\u8cea\u91cf $m_2$ \u306b\u4f9d\u5b58\u3059\u308b\uff01<\/strong><\/span>\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":1258,"menu_order":40,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6621","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6621","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6621"}],"version-history":[{"count":18,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6621\/revisions"}],"predecessor-version":[{"id":10203,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6621\/revisions\/10203"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1258"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6621"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}