{"id":7027,"date":"2023-11-20T17:44:27","date_gmt":"2023-11-20T08:44:27","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=7027"},"modified":"2023-11-21T17:35:53","modified_gmt":"2023-11-21T08:35:53","slug":"%e4%b8%87%e6%9c%89%e5%bc%95%e5%8a%9b%e3%81%ae2%e4%bd%93%e5%95%8f%e9%a1%8c%e3%81%ae%e9%81%8b%e5%8b%95%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%82%92%e6%95%b0%e5%80%a4%e7%9a%84%e3%81%ab%e8%a7%a3%e3%81%8f%e5%89%8d","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/7027\/","title":{"rendered":"\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048 ver. 2"},"content":{"rendered":"<p>\u300c<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4221\/\">\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048<\/a>\u300d\u306e\u6539\u826f\u7248\u3002\u5909\u6570\u306e\u898f\u683c\u5316\u3092\u3082\u3046\u5c11\u3057\u30b7\u30f3\u30d7\u30eb\u306b\u3002<\/p>\n<p>\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>\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<p><!--more--><\/p>\n<h3>\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n<p>\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<p>$$\\frac{d^2 \\boldsymbol{r}}{dt^2} = &#8211; \\frac{GM}{r^3} \\boldsymbol{r}$$<\/p>\n<h3>\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09<\/h3>\n<p>\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<p>$$\\boldsymbol{\\ell} \\equiv \\boldsymbol{r} \\times \\dot{\\boldsymbol{r}} = \\mbox{const.}$$<\/p>\n<p>\u3068\uff08\u5358\u4f4d\u8cea\u91cf\u5f53\u305f\u308a\u306e\uff09\u5168\u30a8\u30cd\u30eb\u30ae\u30fc \\(\\varepsilon\\)<\/p>\n<p>$$\\varepsilon \\equiv \\frac{1}{2} \\dot{\\boldsymbol{r}} \\cdot\\dot{\\boldsymbol{r}} \u00a0 \u00a0 &#8211; \u00a0 \u00a0 \\frac{GM}{r}$$<\/p>\n<p>\u4e00\u5b9a\u306e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{\\ell}\\) \u306e\u5411\u304d\u3092 \\(z\\) \u8ef8\u65b9\u5411\u306b\u3068\u308b\u3068\uff0c<\/p>\n<p>$$ \\boldsymbol{\\ell} = (0, 0, \\ell)$$<\/p>\n<p>\u904b\u52d5\u306f \\(xy\\) \u5e73\u9762\u4e0a\u306b\u5236\u9650\u3055\u308c\u3066<\/p>\n<p>$$\\boldsymbol{r} = (x, y, 0)$$\u3068\u304a\u3051\u308b\u3002<\/p>\n<h3>\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09\u3068\u6955\u5186\u8ecc\u9053\u8981\u7d20<\/h3>\n<p>\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<p>$$x = r \\cos\\phi, \\quad y = r \\sin \\phi$$<\/p>\n<p>$$r = \\frac{a (1-e^2)}{1 + e \\cos\\phi}$$<\/p>\n<p>\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<p>\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<p>\\begin{eqnarray}<br \/>\n\\varepsilon &amp;=&amp;\u00a0 &#8211; \\frac{GM}{2 a} \\\\<br \/>\n\\ell &amp;=&amp; \\sqrt{GM a (1-e^2)}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<p>$$ \\frac{a^3}{P^2} = \\frac{GM}{4 \\pi^2}, \\quad \\therefore\\ \\ P = 2 \\pi \\frac{a \\sqrt{a}}{GM}$$<\/p>\n<h3>\u7121\u6b21\u5143\u5316<\/h3>\n<p>\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<p>\\begin{eqnarray}<br \/>\nX &amp;\\equiv&amp; \\frac{x}{a} \\\\<br \/>\nY &amp;\\equiv&amp; \\frac{y}{a} \\\\<br \/>\nT &amp;\\equiv&amp; \\frac{t}{P}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{d^2 X}{dT^2} = &#8211; 4\\pi^2 \\frac{X}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}}\u00a0 \\tag{1}\\\\<br \/>\n\\frac{d^2 Y}{dT^2} = &#8211; 4\\pi^2 \\frac{Y}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}} \\tag{2}<br \/>\n\\end{eqnarray}<\/p>\n<h3>\u521d\u671f\u6761\u4ef6<\/h3>\n<p>&nbsp;<\/p>\n<p>\u521d\u671f\u6761\u4ef6\u3092 \\( t = 0\\) \u3067<\/p>\n<p>\\begin{eqnarray}<br \/>\nx(0) &amp;=&amp; r_{\\rm min} = a(1-e) \\\\<br \/>\ny(0) &amp;=&amp; 0 \\\\<br \/>\nv_x(0) &amp;=&amp; 0 \\\\<br \/>\nv_y(0) &amp;=&amp; v_{y0}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<p>\\begin{eqnarray}<br \/>\n\\varepsilon = &#8211; \\frac{GM}{2a} &amp;=&amp; \\frac{1}{2} \\left(v_{y0}\\right)^2 &#8211; \\frac{GM}{a(1-e)} \\\\<br \/>\n\\therefore\\ \\ v_{y0} &amp;=&amp; \\sqrt{\\frac{GM (1+e)}{a(1-e)}}<br \/>\n\\end{eqnarray}<\/p>\n<h3>\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u521d\u671f\u6761\u4ef6<\/h3>\n<p>$T = t\/P = 0$ \u306e\u3068\u304d<\/p>\n<p>\\begin{eqnarray}<br \/>\nX(0) &amp;=&amp; \\frac{x(0)}{a} = 1-e \\\\<br \/>\nY(0) &amp;=&amp; \\frac{y(0)}{a} = 0 \\\\<br \/>\nV_x(0) &amp;=&amp; \\frac{P}{a} v_x(0) = 0 \\\\<br \/>\nV_y(0) &amp;=&amp; \\frac{P}{a} v_{y0} = 2 \\pi \\sqrt{\\frac{1+e}{1-e}}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<p>\u3058\u3083\u3042\uff0c\u305d\u3053\u307e\u3067\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u306a\u3089\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u306a\u305c\u308f\u3056\u308f\u3056\u6570\u5024\u8a08\u7b97\u3059\u308b\u306e\u304b<\/strong><\/span>\u3068\u3044\u3046\u3068\uff0c\u305d\u308c\u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u6955\u5186\u8ecc\u9053\u306e\u89e3\u304c\u6642\u9593 <\/strong><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":"<p>\u300c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048\u300d\u306e\u6539\u826f\u7248\u3002\u5909\u6570\u306e\u898f\u683c\u5316\u3092\u3082\u3046\u5c11\u3057\u30b7\u30f3\u30d7\u30eb\u306b\u3002<\/p>\n<p>\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><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/7027\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n<ul>\n<li>\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,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[22],"tags":[],"class_list":["post-7027","post","type-post","status-publish","format-standard","hentry","category-22","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7027","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"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=7027"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7027\/revisions"}],"predecessor-version":[{"id":7052,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7027\/revisions\/7052"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=7027"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=7027"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=7027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}