{"id":4376,"date":"2024-07-18T11:00:01","date_gmt":"2024-07-18T02:00:01","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4376"},"modified":"2024-12-17T10:19:38","modified_gmt":"2024-12-17T01:19:38","slug":"%e3%83%9a%e3%83%bc%e3%82%bb%e3%82%af%e3%81%ae%e5%ae%9a%e7%be%a9","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4376\/","title":{"rendered":"\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9\u3068\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u30fb\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62"},"content":{"rendered":"

\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9\u306b $1$ \u79d2\u89d2\u306e $\\tan$ \uff08\u3042\u308b\u3044\u306f $\\tan$ \u306e\u9006\u6570\u3067\u3042\u308b $\\cot$\uff09\u304c\u3044\u3089\u306a\u304f\u306a\u3063\u3066\uff0c$\\displaystyle 1 \\mbox{pc} = \\frac{\\mbox{au}}{\\tan 1^{\\prime\\prime}} = \\mbox{au}\\ \\cot 1^{\\prime\\prime}$ \u3067\u306f\u306a\u304f\uff0c\u65b0\u5b9a\u7fa9 (IAU 2015 Resolution B2<\/strong><\/span>) \u3067\u306f
\n$$1 \\mbox{pc} \\equiv \\frac{\\mbox{au}}{1^{\\prime\\prime}} = \\frac{648000}{\\pi} \\mbox{au}$$<\/p>\n

\u3068\u306a\u3063\u305f\u3068\u3044\u3046\u8a71\u3002<\/p>\n

<\/p>\n

Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h3>\n

\uff08\u3044\u305a\u308c\u4fee\u6b63\u3055\u308c\u308b\u3068\u601d\u3046\u306e\u3067\uff0c\u3042\u304f\u307e\u3067\u672c\u65e5\u95b2\u89a7\u6642\u70b9\u3067\u306e\uff09 Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af<\/a>\u306e\u5b9a\u7fa9\u306f<\/p>\n

$$1 \\ \\mbox{pc} = \\cot (\\pi\/648000) \\ \\mbox{au}$$<\/p>\n

\u3068\u3042\u308b\u3002\u304b\u3044\u3064\u307e\u3093\u3067\u8aac\u660e\u3059\u308b\u3068\uff0c<\/p>\n

\"Parsec\"<\/p>\n

1\u79d2\u89d2\u3092\u30e9\u30b8\u30a2\u30f3\u3067\u8868\u3057\u305f\u89d2\u5ea6 $\\theta$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\theta &=&1^{\\prime\\prime} = \\frac{\\pi}{180\\times 60 \\times 60} \\ \\mbox{rad}= \\frac{\\pi}{648000} \\ \\mbox{rad}\\\\
\n\\frac{\\mbox{au}}{\\mbox{pc}} &=& \\tan\\theta \\\\
\n\\therefore\\ \\ 1\\,\\mbox{pc} &\\equiv& \\frac{\\mbox{au}}{\\tan\\theta} \\\\
\n&=& \\frac{\\mbox{au}}{\\tan\\frac{\\pi}{648000}} \\\\
\n&=& \\mbox{au}\\,\\times\\, \\cot\\left(\\frac{\\pi}{648000} \\right)
\n\\end{eqnarray}<\/p>\n

$1 \\ \\mbox{pc}$ \u3068\u306f\u56f3\u306e $PS$ \u306e\u9577\u3055\u3067\u3042\u308a\uff0c\u76f4\u89d2\u4e09\u89d2\u5f62 $PST$ \u304b\u3089 $\\displaystyle \\tan 1^{\\prime\\prime} = \\frac{ST}{PS}$ \u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u3057\u304b\u3057\uff0c\u56f3\u3092\u307f\u3066\u3044\u308b\u3068\uff0c$1 \\ \\mbox{pc}$ \u3063\u3066\u5730\u7403\u307e\u3067\u306e\u8ddd\u96e2 $PT$ \u3058\u3083\u3042\u306a\u3044\uff1f\u3068\u304b\uff0c\u305d\u3082\u305d\u3082\u5929\u4f53 $P$ \u306f\u5341\u5206\u9060\u65b9\u306b\u3042\u308b\u306e\u3060\u304b\u3089 $PS$ \u3068 $PT$ \u306f\u7b49\u3057\u3044\u3068\u3057\u3066\u3044\u3044\u3088\u306d\uff1f\u3068\u304b\uff0c\u3044\u308d\u3044\u308d\u8003\u3048\u304c\u51fa\u3066\u304f\u308b\u3060\u308d\u3046\u3002<\/p>\n

\u305d\u3053\u3067…<\/p>\n

Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h3>\n

Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u9805<\/a>\u3067\u306f\uff0c\u63a1\u629e\u3055\u308c\u305f IAU 2015 Resolution B2<\/strong><\/span> \u306b\u5f93\u3044\uff0c\u53b3\u5bc6\u306b<\/p>\n

$$1 \\mbox{pc} \\equiv \\frac{\\mbox{au}}{\\theta} = \\frac{648000}{\\pi} \\mbox{au}$$<\/p>\n

\u3068\u3057\u3066\u3044\u308b\u3002\u3064\u307e\u308a\u306f $|\\theta| \\ll 1$ \u306e\u3068\u304d $\\tan\\theta \\simeq \\theta$ \u306a\u306e\u3067\uff0c\u3044\u3063\u305d\u306e\u3053\u3068\uff0c$\\tan\\theta$ \u306e\u9805\u3092 $\\theta$ \u3068\u3057\u3066\u3057\u307e\u3048\uff0c\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n

\u300c1 \u30d1\u30fc\u30bb\u30af (pc) \u3068\u306f\uff0c1 au \u306e\u5f27\u306e\u9577\u3055\u3092\u898b\u8fbc\u3080\u4e2d\u5fc3\u89d2\u304c1\u79d2\u89d2\u3068\u306a\u308b\u3068\u304d\u306e\u534a\u5f84\u306e\u9577\u3055\u3067\u3042\u308b<\/strong><\/span>\u300d\u3068\u3044\u3046\u5177\u5408\u306b\u89e3\u91c8\u3067\u304d\u308b\u3002<\/p>\n

\u56f3\u3067\u3044\u3046\u3068\uff0c$PS = PT$ \u3067\u3042\u308a\uff0c$ST$ \u306f\u5186\u5f27\u306e\u9577\u3055\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u672c\u7a3f\u57f7\u7b46\u6642\u70b9\u306eWikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u9805<\/strong><\/span><\/a>\u3067\u306f17\u6841\u306e\u6570\u5024\u3067 1 \u30d1\u30fc\u30bb\u30af\u306e\u5024\u3092\u8868\u3057\u3066\u3044\u308b\uff1a<\/p>\n

$$1~\\mbox{pc} = \\frac{96\\, 939\\, 420\\, 213\\, 600\\, 000}{\\pi}~\\mbox{m}
\n= 30\\, 856\\, 775\\, 814\\, 913\\, 673~\\mbox{m}$$<\/p>\n

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Maxima \u3067\u30d1\u30fc\u30bb\u30af\u3092\u8a08\u7b97\u3059\u308b<\/h3>\n

Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h4>\n

\uff08\u3044\u305a\u308c\u4fee\u6b63\u3055\u308c\u308b\u3068\u601d\u3046\u306e\u3067\uff0c\u3042\u304f\u307e\u3067\u672c\u65e5\u95b2\u89a7\u6642\u70b9\u3067\u306e\uff09 Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af<\/a>\u306e\u65e7\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n

1\u79d2\u89d2\u3092\u30e9\u30b8\u30a2\u30f3\u3067\u8868\u3057\u305f\u89d2\u5ea6 $\\theta$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\theta &=& 1^{\\prime\\prime} = \\frac{\\pi}{180\\times 60 \\times 60}\\ \\mbox{rad} = \\frac{\\pi}{648000}\\ \\mbox{rad} \\\\
\n\\frac{\\mbox{au}}{\\mbox{pc}} &=& \\tan\\theta \\\\
\n\\therefore\\ \\ 1\\,\\mbox{pc} &\\equiv& \\frac{\\mbox{au}}{\\tan\\theta}= \\mbox{au}\\times\\cot\\frac{\\pi}{648000}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h4>\n

Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u9805<\/a>\u3067\u306f\uff0c\u63a1\u629e\u3055\u308c\u305f IAU 2015 Resolution B2<\/strong> \u306b\u5f93\u3063\u305f\u65b0\u5b9a\u7fa9\u3067\u306f\u53b3\u5bc6\u306b<\/p>\n

$$1 \\mbox{pc} \\equiv \\frac{\\mbox{au}}{\\theta} = \\frac{648000}{\\pi} \\mbox{au}$$<\/p>\n

\u3068\u3057\u3066\u3044\u308b\u3002<\/p>\n

Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u9805<\/a>\u3067\u306f17\u6841\u306e\u6570\u5024\u3067 1 \u30d1\u30fc\u30bb\u30af\u306e\u5024\u3092\u8868\u3057\u3066\u3044\u308b\uff1a<\/p>\n

$$1~\\mbox{pc} = \\frac{96\\, 939\\, 420\\, 213\\, 600\\, 000}{\\pi}~\\mbox{m}
\n= 30\\, 856\\, 775\\, 814\\, 913\\, 673~\\mbox{m}$$<\/p>\n

\u304c\uff0cMaxima \u306e float()<\/code> \u3067\u306f 16 \u6841\u307e\u3067\u306a\u306e\u3067\uff0c\u4ee5\u4e0b\u3067\u306f bfloat()<\/code> \u3092\u4f7f\u3063\u3066 20 \u6841\uff08fpprec: 20$<\/code>\uff09\u306e\u7cbe\u5ea6\u3067\u8a08\u7b97\u3057\uff0cprintf()<\/code> \u306e bigfloat<\/code> \u306e\u66f8\u5f0f\u6307\u5b9a ~17,0h<\/code> \u3067 17 \u6841\u307e\u3067\u8868\u793a\u3055\u305b\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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fpprec<\/span>:<\/span> 20$\r\nfpprintprec<\/span>:<\/span> 0$\r\n\r\n\/* \u5929\u6587\u5358\u4f4d\u306f\u5b9a\u7fa9\u5024 m *\/<\/span>\r\nau<\/span>:<\/span> 149597870700$\r\npc1<\/span>:<\/span> bfloat<\/span>(<\/span>648000*<\/span>au<\/span>\/<\/span>%pi<\/span>)<\/span>$\r\n\r\n\/* bfloat \u306e\u66f8\u5f0f\u6307\u5b9a\u306f h *\/<\/span>\r\nprintf<\/span>(<\/span>true<\/span>, \"1 pc = ~17,0h m~%\"<\/span>, pc1<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1 pc = 30856775814913673. m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65e7\u5b9a\u7fa9 $\\tan\\theta$ \u306e\u307e\u307e\u8a08\u7b97\u3059\u308b\u3068\uff0c12\u6841\u76ee\u304b\u3089\u8aa4\u5dee\u304c\u51fa\u3066\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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pc2<\/span>:<\/span> bfloat<\/span>(<\/span>au<\/span>\/<\/span>tan<\/span>(<\/span>%pi<\/span>\/<\/span>648000<\/span>))<\/span>$\r\nprintf<\/span>(<\/span>true<\/span>, \"\u65b0\u5b9a\u7fa9 1 pc = ~17,0h m~%\"<\/span>, pc1<\/span>)<\/span>$\r\nprintf<\/span>(<\/span>true<\/span>, \"\u65e7\u5b9a\u7fa9 1 pc = ~17,0h m~%\"<\/span>, pc2<\/span>)<\/span>$\r\n\r\nprintf<\/span>(<\/span>true<\/span>, \"\u3000\u3000\u5dee      =  ~17,0h m~%\"<\/span>, (<\/span>pc1<\/span>-<\/span>pc2<\/span>))<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b0\u5b9a\u7fa9 1 pc = 30856775814913673. m\r\n\u65e7\u5b9a\u7fa9 1 pc = 30856775814671916. m\r\n\u3000\u3000\u5dee      =            241757. m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u30fb\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62<\/h4>\n
\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u306f\u6b74\u53f2\u7684\u7d4c\u7def\u3082\u3042\u308a\uff0c\u5358\u4f4d\u304c $\\mbox{km\/s\/Mpc}$ \u3068\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
$$H_0 = 100 h \\,\\frac{\\mbox{km}}{\\mbox{s} \\cdot \\mbox{Mpc}}
\n= 100 h \\, \\frac{\\mbox{km}}{\\mbox{Mpc}} \\cdot \\mbox{s}^{-1}$$<\/p>\n
\u3064\u307e\u308a\uff0c$1 \\,\\mbox{Mpc}$ \u3092 $\\mbox{m}$ \u3067\u8868\u3057\u305f\u5024\u3092 Mpc<\/code> \u3068\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\/* H0 \u306e\u7cbe\u5ea6\u306f\uff0c\u305b\u3044\u305c\u3044\u3053\u306e\u304f\u3089\u3044 *\/<\/span>\r\n\/* \u306a\u306e\u3067\u30d1\u30fc\u30bb\u30af\u306e\u65b0\u5b9a\u7fa9\u30fb\u65e7\u5b9a\u7fa9\u306e\u5dee\u306a\u3069\u7121\u95a2\u4fc2\u3067\u3059 *\/<\/span>\r\nfpprintprec<\/span>:<\/span> 4$\r\n\/* h \u3092\u521d\u671f\u5316\uff08\u5ff5\u306e\u305f\u3081\uff09*\/<\/span>\r\nkill<\/span>(<\/span>h<\/span>)<\/span>$\r\n\r\nau<\/span>:<\/span> 149597870700<\/span> *<\/span> m<\/span>$\r\n\/* Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002*\/<\/span>\r\nMpc<\/span>:<\/span> 648000\/<\/span>float<\/span>(<\/span>%pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>$\r\nkm<\/span>:<\/span> 1E3<\/span> *<\/span> m<\/span>$\r\n\r\nH0<\/span>:<\/span> 100<\/span> *<\/span> h<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}\\frac{3.241 \\times 10^{-18}\\,h}{s}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u306e\u5024\u306e\u9006\u6570\u304c\u79d2\u3067\u8868\u3057\u305f\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u3068\u306a\u308b\u3002\u300c\u5104\u5e74\u300d\u5358\u4f4d\u306b\u5909\u63db\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\/* 1\u5104\u5e74\u3092\u79d2\u3067\u8868\u3059 *\/<\/span>\r\nokunen<\/span>:<\/span> 60*<\/span>60*<\/span>24*<\/span>365.<\/span>25*<\/span>1E8<\/span> *<\/span> s<\/span>$\r\n\r\nprint<\/span>(<\/span>1\/<\/span>H0<\/span>\/<\/span>okunen<\/span>, \"\u5104\u5e74\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\(\\frac{97.78}{h}\\) \u5104\u5e74<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\/* Planck 2018 \u306e\u5024\u3092\u4f7f\u3046\u3068... *\/<\/span>\r\nh<\/span>:<\/span> 0.<\/span>674$\r\n\r\nprint<\/span>(<\/span>\"H0 = \"<\/span>, 100*<\/span>h<\/span>, \"\u306e\u3068\u304d\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f\"<\/span>, 1\/<\/span>ev<\/span>(<\/span>H0<\/span>)<\/span>\/<\/span>okunen<\/span>, \"\u5104\u5e74\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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H0 = \\(67.4\\) \u306e\u3068\u304d\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f \\(145.1\\) \u5104\u5e74<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\/* Hubble (1929) \u306e\u5024\u3092\u4f7f\u3046\u3068... *\/<\/span>\r\nh<\/span>:<\/span> 4.<\/span>239$\r\n\r\nprint<\/span>(<\/span>\"H0 = \"<\/span>, 100*<\/span>h<\/span>, \"\u306e\u3068\u304d\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f\"<\/span>, 1\/<\/span>ev<\/span>(<\/span>H0<\/span>)<\/span>\/<\/span>okunen<\/span>, \"\u5104\u5e74\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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H0 = \\(423.9\\) \u306e\u3068\u304d\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f \\(23.07\\) \u5104\u5e74<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Python \u3067\u30d1\u30fc\u30bb\u30af\u3092\u8a08\u7b97\u3059\u308b<\/h3>\n
\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h4>\n
17 \u6841\u76ee\u307e\u3067\u8a08\u7b97\u3059\u308b\u306e\u306b\uff0cSymPy \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Wikipedia \u82f1\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h4>\n
\u65b0\u5b9a\u7fa9\uff08IAU 2015 Resolution B2<\/strong>\uff09\u306b\u3088\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5024\u306f<\/p>\n
\\begin{eqnarray}
\n\\theta &=& 1^{\\prime\\prime} = \\frac{\\pi}{180\\times 60 \\times 60}\\ \\mbox{rad} = \\frac{\\pi}{648000}\\ \\mbox{rad} \\\\
\n1\\ \\mbox{pc} &\\equiv& \\frac{\\mbox{au}}{\\theta} = \\frac{648000}{\\pi}\\ \\mbox{au}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u5929\u6587\u5358\u4f4d\u306f\u5b9a\u7fa9\u5024<\/span>\r\nau<\/span> =<\/span> 149597870700<\/span> # \u5358\u4f4d\u306f m<\/span>\r\n\r\n# \u65b0\u5b9a\u7fa9\u306e\u30d1\u30fc\u30bb\u30af<\/span>\r\npc1<\/span> =<\/span> N<\/span>(<\/span>648000<\/span>\/<\/span>pi<\/span>*<\/span>au<\/span>,<\/span> 17<\/span>)<\/span>\r\nprint<\/span>(<\/span>'\u65b0\u5b9a\u7fa9 1 pc ='<\/span>,<\/span> pc1<\/span>,<\/span> 'm'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b0\u5b9a\u7fa9 1 pc = 30856775814913673. m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9<\/h4>\n
\uff08\u3044\u305a\u308c\u4fee\u6b63\u3055\u308c\u308b\u3068\u601d\u3046\u306e\u3067\uff0c\u3042\u304f\u307e\u3067\u672c\u65e5\u95b2\u89a7\u6642\u70b9\u3067\u306e\uff09 Wikipedia \u65e5\u672c\u8a9e\u7248\u306b\u304a\u3051\u308b\u30d1\u30fc\u30bb\u30af<\/a>\u306e\u65e7\u5b9a\u7fa9\u306b\u3088\u308b\u30d1\u30fc\u30bb\u30af\u306e\u5024\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n
1\u79d2\u89d2\u3092\u30e9\u30b8\u30a2\u30f3\u3067\u8868\u3057\u305f\u89d2\u5ea6 $\\theta$ \u306f<\/p>\n
\\begin{eqnarray}
\n\\theta &=& 1^{\\prime\\prime} = \\frac{\\pi}{180\\times 60 \\times 60}\\ \\mbox{rad} = \\frac{\\pi}{648000}\\ \\mbox{rad} \\\\
\n1\\ \\mbox{pc} &\\equiv& \\frac{\\mbox{au}}{\\tan\\theta}= \\mbox{au}\\times\\cot\\frac{\\pi}{648000}
\n\\end{eqnarray}<\/p>\n
\u4ee5\u4e0b\u3067\u308f\u304b\u308b\u3088\u3046\u306b\uff0c$\\tan\\theta$ \u3092\u4f7f\u3063\u305f\u65e7\u5b9a\u7fa9\u3067\u306f 12 \u6841\u76ee\u304b\u3089\u9055\u3044\u304c\u51fa\u3066\u304f\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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# \u65e7\u5b9a\u7fa9\u306e\u30d1\u30fc\u30bb\u30af<\/span>\r\npc2<\/span> =<\/span> N<\/span>(<\/span>au<\/span>\/<\/span>tan<\/span>(<\/span>pi<\/span>\/<\/span>648000<\/span>),<\/span> 17<\/span>)<\/span>\r\n\r\nprint<\/span>(<\/span>'\u65b0\u5b9a\u7fa9 1 pc ='<\/span>,<\/span> pc1<\/span>,<\/span> 'm'<\/span>)<\/span>\r\nprint<\/span>(<\/span>'\u65e7\u5b9a\u7fa9 1 pc ='<\/span>,<\/span> pc2<\/span>,<\/span> 'm'<\/span>)<\/span>\r\n# \u8868\u793a\u306e\u6841\u3092\u5408\u308f\u305b\u308b\u306e\u306b\u4e00\u82e6\u52b4<\/span>\r\nprint<\/span>(<\/span>'         \u5dee = '<\/span>+<\/span>' '<\/span>*<\/span>11<\/span>+<\/span>str<\/span>(<\/span>N<\/span>((<\/span>pc1<\/span>-<\/span>pc2<\/span>),<\/span> n<\/span>=<\/span>6<\/span>))<\/span>+<\/span>' m'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b0\u5b9a\u7fa9 1 pc = 30856775814913673. m\r\n\u65e7\u5b9a\u7fa9 1 pc = 30856775814671916. m\r\n         \u5dee =            241757. m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3061\u306a\u307f\u306b\uff0c%17d<\/code> \u3068\u304b %17.f<\/code> \u3068\u304b\u306e\u66f8\u5f0f\u6307\u5b9a\u3067\u8868\u793a\u5408\u308f\u305b\u3092\u3057\u3088\u3046\u3068\u3059\u308b\u3068\uff0c\u6700\u5f8c\u306e\u6841\u306e\u6570\u5024\u304c\u5909\u308f\u3063\u3066\u3057\u307e\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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print<\/span>(<\/span>'\u65b0\u5b9a\u7fa9 1 pc = <\/span>%17d<\/span> m'<\/span> %<\/span> pc1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b0\u5b9a\u7fa9 1 pc = 30856775814913672 m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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print<\/span>(<\/span>'\u65b0\u5b9a\u7fa9 1 pc = %17.f m'<\/span> %<\/span> pc1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b0\u5b9a\u7fa9 1 pc = 30856775814913672 m\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u30fb\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62<\/h4>\n
\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u306f\u6b74\u53f2\u7684\u7d4c\u7def\u3082\u3042\u308a\uff0c\u5358\u4f4d\u304c $\\mbox{km\/s\/Mpc}$ \u3068\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
$$H_0 = 100 \\,h \\,\\frac{\\mbox{km}}{\\mbox{s} \\cdot \\mbox{Mpc}}
\n= 100 \\,h \\, \\frac{\\mbox{km}}{\\mbox{Mpc}} \\cdot \\mbox{s}^{-1}$$<\/p>\n
\u3064\u307e\u308a\uff0c$1 \\,\\mbox{Mpc}$ \u3092 $\\mbox{m}$ \u3067\u8868\u3057\u305f\u5024\u3092 Mpc<\/code> \u3068\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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au<\/span> =<\/span> 149597870700<\/span> *<\/span> m<\/span>\r\n# Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002<\/span>\r\nMpc<\/span> =<\/span> 648000<\/span>\/<\/span>float<\/span>(<\/span>pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>\r\nkm<\/span> =<\/span> 1E3<\/span> *<\/span> m<\/span>\r\n\r\nH0<\/span> =<\/span> 100<\/span> *<\/span> h<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>\r\n\r\n# H0 \u306e\u7cbe\u5ea6\u306f\u305b\u3044\u305c\u3044 4 \u6841<\/span>\r\nN<\/span>(<\/span>H0<\/span>,<\/span> 4<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\frac{3.241 \\cdot 10^{-18} h}{s}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u306e\u5024\u306e\u9006\u6570\u304c\u79d2\u3067\u8868\u3057\u305f\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u3068\u306a\u308b\u3002\u300c\u5104\u5e74\u300d\u5358\u4f4d\u306b\u5909\u63db\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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# 1\u5104\u5e74\u3092\u79d2\u3067\u8868\u3059 <\/span>\r\nokunen<\/span> =<\/span> 60<\/span>*<\/span>60<\/span>*<\/span>24<\/span>*<\/span>365.25<\/span>*<\/span>1E8<\/span> *<\/span> s<\/span>\r\n\r\nprint<\/span>(<\/span>N<\/span>(<\/span>1<\/span>\/<\/span>H0<\/span>\/<\/span>okunen<\/span>,<\/span> 4<\/span>),<\/span>'\u5104\u5e74'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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97.78\/h \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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# Planck 2018 \u306e\u5024 h = 0.674 \u3092\u4f7f\u3046\u3068... <\/span>\r\nh1<\/span> =<\/span> 0.674<\/span>\r\nprint<\/span>(<\/span>\"H0 =\"<\/span>,<\/span> 100<\/span>*<\/span>h1<\/span>,<\/span> \"\u306e\u3068\u304d\uff0c\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f\"<\/span>,<\/span> \r\n      N<\/span>(<\/span>1<\/span>\/<\/span>H0<\/span>.<\/span>subs<\/span>(<\/span>h<\/span>,<\/span> h1<\/span>)<\/span>\/<\/span>okunen<\/span>,<\/span> 4<\/span>),<\/span>'\u5104\u5e74'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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H0 = 67.4 \u306e\u3068\u304d\uff0c\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f 145.1 \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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# Hubble (1929) \u306e\u5024 h = 4.239 \u3092\u4f7f\u3046\u3068... <\/span>\r\nh2<\/span> =<\/span> 4.239<\/span>\r\n\r\nprint<\/span>(<\/span>\"H0 =\"<\/span>,<\/span> 100<\/span>*<\/span>h2<\/span>,<\/span> \"\u306e\u3068\u304d\uff0c\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f\"<\/span>,<\/span> \r\n      N<\/span>(<\/span>1<\/span>\/<\/span>H0<\/span>.<\/span>subs<\/span>(<\/span>h<\/span>,<\/span> h2<\/span>)<\/span>\/<\/span>okunen<\/span>,<\/span> 4<\/span>),<\/span>'\u5104\u5e74'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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H0 = 423.9 \u306e\u3068\u304d\uff0c\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f 23.07 \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9\u306b $1$ \u79d2\u89d2\u306e $\\tan$ \uff08\u3042\u308b\u3044\u306f $\\tan$ \u306e\u9006\u6570\u3067\u3042\u308b $\\cot$\uff09\u304c\u3044\u3089\u306a\u304f\u306a\u3063\u3066\uff0c$\\displaystyle 1 \\mbox{pc} = \\frac{\\mbox{au}}{\\tan 1^{\\prime\\prime}} = \\mbox{au}\\ \\cot 1^{\\prime\\prime}$ \u3067\u306f\u306a\u304f\uff0c\u65b0\u5b9a\u7fa9 (IAU 2015 Resolution B2) \u3067\u306f $$1 \\mbox{pc} \\equiv \\frac{\\mbox{au}}{1^{\\prime\\prime}} = \\frac{648000}{\\pi} \\mbox{au}$$<\/p>\n
\u3068\u306a\u3063\u305f\u3068\u3044\u3046\u8a71\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\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":[14,12,20],"tags":[],"class_list":["post-4376","post","type-post","status-publish","format-standard","hentry","category-maxima","category-sympy","category-rel-cosmo","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4376","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=4376"}],"version-history":[{"count":34,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4376\/revisions"}],"predecessor-version":[{"id":9942,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4376\/revisions\/9942"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=4376"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=4376"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=4376"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}