{"id":7128,"date":"2023-12-15T16:00:50","date_gmt":"2023-12-15T07:00:50","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7128"},"modified":"2023-12-15T16:53:49","modified_gmt":"2023-12-15T07:53:49","slug":"%e6%a5%95%e5%86%86%e8%bb%8c%e9%81%93%e4%b8%8a%e3%81%ae%e6%99%82%e5%88%bb%e3%81%94%e3%81%a8%e3%81%ae%e4%bd%8d%e7%bd%ae%e3%82%92%e6%95%b0%e5%80%a4%e7%9a%84%e3%81%ab%e6%b1%82%e3%82%81%e3%82%8b%e6%ba%96","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/%e6%a5%95%e5%86%86%e8%bb%8c%e9%81%93%e4%b8%8a%e3%81%ae%e6%99%82%e5%88%bb%e3%81%94%e3%81%a8%e3%81%ae%e4%bd%8d%e7%bd%ae%e3%82%92%e6%95%b0%e5%80%a4%e7%9a%84%e3%81%ab%e6%b1%82%e3%82%81%e3%82%8b%e6%ba%96\/","title":{"rendered":"\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u6642\u523b\u3054\u3068\u306e\u4f4d\u7f6e\u3092\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u6e96\u5099"},"content":{"rendered":"

\u4e07\u6709\u5f15\u529b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u6955\u5186\u8ecc\u9053\u306b\u306a\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3060\u304c\uff0c\u6955\u5186\u8ecc\u9053\u306f $r(\\phi)$ \u3067\u8868\u3055\u308c\uff0c\u6642\u9593 $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u306f\u8868\u305b\u3066\u3044\u306a\u3044\u306e\u3067\uff0c$\\phi$ \u3092 $t$ \u306e\u95a2\u6570\u3068\u3057\u3066\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u305f\u3081\u306e\u4e0b\u3054\u3057\u3089\u3048\u3002<\/p>\n

\u300c\u4e07\u6709\u5f15\u529b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u6e96\u5099<\/a>\u300d\u3067\u306f\uff0c\u4e07\u6709\u5f15\u529b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048\u3092\u3057\u3066\u3044\u305f\u3002\u5b9f\u969b\u306b Python \u3067\u89e3\u304f\u306e\u306f\u300cPython \u3066\u3099\u4e07\u6709\u5f15\u529b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u6955\u5186\u8ecc\u9053\u306e\u30a2\u30cb\u30e1\u3092\u3064\u304f\u308b<\/a>\u300d\u306b\u66f8\u3044\u3066\u3044\u308b\u3002<\/p>\n

\u9023\u7acb2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u306e\uff08\u898f\u683c\u5316\u3055\u308c\u305f\uff09\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u306e\u3082\u6559\u80b2\u7684\u3067\u306f\u3042\u308b\u304c\uff0c\u3053\u306e\u554f\u984c\u306f\u6955\u5186\u8ecc\u9053\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3002\u305f\u3060\u3057\uff0c\u89e3\u304c<\/p>\n

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

\u3068\u66f8\u3051\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3060\u304c\uff0c$r$ \u3082 $\\phi$ \u3082\u6642\u9593 $t$ \u306e\u95a2\u6570\u3068\u3057\u3066\u306f\u3042\u304b\u3089\u3055\u307e\u306b\u4e0e\u3048\u3089\u308c\u3066\u3044\u306a\u3044\u305f\u3081\uff0c\u3042\u308b\u6642\u523b\u306b\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u3069\u3053\u306b\u3044\u308b\u304b\u306f\uff0c\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u307f\u306a\u3044\u3068\u308f\u304b\u3089\u306a\u3044\u3002<\/p>\n

\u3053\u3053\u3067\u306f\uff0c\u6955\u5186\u8ecc\u9053\u306f $r(\\phi)$ \u3067\u8868\u3055\u308c\u308b\u306e\u3067\uff0c$\\phi$ \u3092 $t$ \u306e\u95a2\u6570\u3068\u3057\u3066\u6570\u5024\u7684\u306b\u3082\u3068\u3081\uff0c$\\phi(t)$ \u304c\u6570\u5024\u7684\u306b\u6c42\u307e\u308c\u3070\uff0c<\/p>\n

$$x(t) = r(\\phi(t))\\,\\cos\\phi(t), \\quad\u00a0 y(t) = r(\\phi(t))\\, \\sin\\phi(t)$$<\/p>\n

\u306e\u3088\u3046\u306b\uff0c\u6642\u523b\u3092\u4e0e\u3048\u308c\u3070\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u4f4d\u7f6e\u304c\u308f\u304b\u308b\u3088\u3046\u306b\u3057\u3066\u307f\u308b\u3002\u4ee5\u4e0b\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\u304b\u3089\u8981\u70b9\u3092\u629c\u7c8b\u3057\u3066\u5229\u7528\u3059\u308b\u3002<\/p>\n

\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306f1\u4f53\u554f\u984c\u306b\u5e30\u7740\u3059\u308b<\/h3>\n

\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306f1\u4f53\u554f\u984c\u306b\u5e30\u7740\u3057\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$\\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 = \u2013 \\frac{GM \\boldsymbol{r}}{r^3}$$<\/p>\n

\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09<\/h3>\n

\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\uff0c2\u3064\u306e\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09\u304c\u5f97\u3089\u308c\u308b\u30021\u3064\u306f\uff0c(\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\uff09\u89d2\u904b\u52d5\u91cf\u306e $z$ \u6210\u5206\u306b\u76f8\u5f53\u3059\u308b $\\ell$\uff1a<\/p>\n

$$\\ell \\equiv r^2 \\frac{d\\phi}{dt}$$<\/p>\n

\u3082\u30461\u3064\u306f\uff08\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\uff09\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u76f8\u5f53\u3059\u308b $\\varepsilon$\uff1a<\/p>\n

$$\\varepsilon \\equiv \\frac{1}{2} \\left\\{\\left(\\frac{dr}{dt}\\right)^2 + \\frac{\\ell^2}{r^2}\\right\\} – \\frac{GM}{r}$$<\/p>\n

\u89e3\uff1a\u6955\u5186\u8ecc\u9053<\/h3>\n

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

\u3053\u3053\u3067\uff0c\u8ecc\u9053\u9577\u534a\u5f84 $a$\uff0c\u96e2\u5fc3\u7387 $e$ \u3068\u4fdd\u5b58\u91cf\u306e\u95a2\u4fc2\u306f<\/p>\n

$$\\ell = \\sqrt{GM a (1-e^2)}, \\quad \\varepsilon = – \\frac{GM}{2a}$$<\/p>\n

\u89e3\u304f\u3079\u304d\u65b9\u7a0b\u5f0f<\/h3>\n

\u57fa\u672c\u7684\u306b\u4ee5\u4e0b\u306e1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3060\u3051\u3067\u3088\u3044\u3002<\/p>\n

$$\\frac{d\\phi}{dt} = \\frac{\\ell}{r^2}, \\quad r = \\frac{a(1-e^2)}{1 + e \\cos\\phi}$$<\/p>\n

\u7121\u6b21\u5143\u5316<\/h3>\n

\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c3\u6cd5\u5247<\/p>\n

$$\\frac{a^3}{P^2} = \\frac{GM}{4\\pi^2}, \\quad \\therefore\\ \\ P = \\frac{2\\pi a^2}{\\sqrt{GM a}}$$<\/p>\n

\u8ecc\u9053\u5468\u671f $P$ \u3092\u4f7f\u3063\u3066\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u6642\u9593 $T$ \u3092<\/p>\n

$$T \\equiv \\frac{t}{P}$$<\/p>\n

\u3068\u3059\u308b\u3068\uff0c<\/p>\n

$$\\frac{d\\phi}{dT} = 2 \\pi \\frac{(1+e \\cos\\phi)^2}{(1-e^2)^{3\/2}} \\tag{1}$$<\/p>\n

\u3053\u308c\u3092\u6570\u5024\u7684\u306b\u89e3\u304d\uff0c$\\phi(T)$ \u304c\u308f\u304b\u3063\u305f\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\nR(T) &\\equiv& \\frac{r}{a} = \\frac{1-e^2}{1+e\\cos\\phi(T)} \\tag{2}\\\\
\nX(T) &\\equiv& R(T) \\cos\\phi(T) \\tag{3}\\\\
\nY(T) &\\equiv& R(T) \\sin\\phi(T) \\tag{4}
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\uff0c\u898f\u683c\u5316\u3055\u308c\u305f $X$ \u5ea7\u6a19\uff0c$Y$ \u5ea7\u6a19\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e07\u6709\u5f15\u529b\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u6955\u5186\u8ecc\u9053\u306b\u306a\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3060\u304c\uff0c\u6955\u5186\u8ecc\u9053\u306f $r(\\phi)$ \u3067\u8868\u3055\u308c\uff0c\u6642\u9593 $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u306f\u8868\u305b\u3066\u3044\u306a\u3044\u306e\u3067\uff0c$\\phi$ \u3092 $t$ \u306e\u95a2\u6570\u3068\u3057\u3066\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u305f\u3081\u306e\u4e0b\u3054\u3057\u3089\u3048\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2826,"menu_order":80,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7128","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