\u5b87\u5b99\u8ad6\u3068\u306f\uff0c\u9280\u6cb3\u3092\u6700\u5c0f\u69cb\u6210\u5358\u4f4d\u3068\u3059\u308b\u3088\u3046\u306a\u30b9\u30b1\u30fc\u30eb\u3067\uff0c\u7269\u8cea\u3084\u9280\u6cb3\u306e\u5165\u308c\u7269\u3068\u3057\u3066\u306e\u5b87\u5b99\u306e\u73fe\u5728\u30fb\u904e\u53bb\u30fb\u672a\u6765\u3092\u63a2\u308b\u5b66\u554f\u5206\u91ce\u3067\u3042\u308b\u3002<\/p>\n
<\/p>\n
\u3053\u3053\u3067\u306f\u307e\u305a\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247<\/strong><\/span>\u304b\u3089\uff0c\uff08\u4e00\u822c\u76f8\u5bfe\u8ad6\u3092\u4f7f\u308f\u305a\u306b\uff09\u5b87\u5b99\u306e\u81a8\u5f35\u3092\u8a18\u8ff0\u3059\u308b\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3092\u5c0e\u304f\u3002\u5c0e\u304b\u308c\u305f\u65b9\u7a0b\u5f0f\u306f\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u3082\uff0c\u305d\u306e\u307e\u307e\u306e\u5f62\u3067\u6210\u308a\u7acb\u3064\u3002\u306a\u306e\u3067\uff0c\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u3068\u306f\u4f55\u305e\u3084\u3068\u3044\u3046\u554f\u984c\u306b\u6df1\u304f\u95a2\u308f\u3089\u305a\u306b\u81a8\u5f35\u5b87\u5b99\u306e\u59ff\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u3092\u671f\u5f85\u3057\u3066\u3044\u308b\u3002<\/p>\n \u3053\u3053\u3067\u306f4\u5143\u30d9\u30af\u30c8\u30eb\u306e\u51fa\u756a\u306f\u306a\u304f\uff0c\u3059\u3079\u30663\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u3002\u306a\u306e\u3067\uff0c\u592a\u5b57\u3067\u30d9\u30af\u30c8\u30eb\u3092\u8868\u3057\uff0c \u5b87\u5b99\u7a7a\u9593\u306e\u3069\u3053\u304b\u306b\u539f\u70b9\u3092\u8a2d\u5b9a\u3057\uff0c\u305d\u3053\u304b\u3089\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{r}\\) \u306e\u5730\u70b9\u306b\u3042\u308b\u8cea\u91cf \\(m\\) \u306e\u30c6\u30b9\u30c8\u7c92\u5b50\u306b\u5bfe\u3059\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f \u3053\u3053\u3067 \\(\\boldsymbol{g}\\) \u306f\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\u4ee5\u4e0b\u306e\u65b9\u7a0b\u5f0f\u304b\u3089\u6c7a\u307e\u308b\u3002<\/p>\n \\begin{eqnarray} \u30cf\u30c3\u30d6\u30eb=\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u3068\u306f\uff0c\u9060\u65b9\u306e\uff08\u304a\u304a\u3080\u306d 10\u30e1\u30ac\u30d1\u30fc\u30bb\u30af\u4ee5\u4e0a\u96e2\u308c\u305f\uff09\u9280\u6cb3\u306e\u5f8c\u9000\u901f\u5ea6\u304c\u8ddd\u96e2\u306b\u6bd4\u4f8b\u3059\u308b<\/strong><\/span>\uff0c\u3068\u3044\u3046\u89b3\u6e2c\u5024\u306e\u9593\u306e\u95a2\u4fc2\u5f0f\u3067\u3042\u308b\u3002<\/p>\n \u9280\u6cb3\u307e\u3067\u306e\u8ddd\u96e2\u3092 \\(R\\) \u3068\u3059\u308b\u3068\uff0c\u3053\u306e\u6cd5\u5247\u306f \u3053\u306e\u6cd5\u5247\u306b\uff0c\u300c\u5927\u304d\u306a\u30b9\u30b1\u30fc\u30eb\u3067\u307f\u308c\u3070\uff0c\u5b87\u5b99\u306f\u4e00\u69d8\uff08\u5834\u6240\u306b\u3088\u3089\u306a\u3044\uff09\u304b\u3064\u7b49\u65b9\uff08\u65b9\u5411\u306b\u3088\u3089\u306a\u3044\uff09\u3067\u3042\u308b<\/strong><\/span>\u300d\u3068\u3044\u3046\u300c\u5b87\u5b99\u539f\u7406<\/strong><\/span>\u300d\u306e\u4eee\u5b9a\u3092\u52a0\u3048\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n \u307e\u305a\uff0c\u30cf\u30c3\u30d6\u30eb=\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u304c\u7b49\u65b9<\/strong><\/span>\u3067\u3042\u308b\u3068\u3059\u308b\u3068\uff0c\u7a7a\u95933\u65b9\u5411\u306b\u3064\u3044\u3066\uff0c\uff083\u65b9\u5411\u5225\u3005\u3067\u306f\u306a\u304f\uff09\u5171\u901a\u306e\u6bd4\u4f8b\u9805\uff08\u6bd4\u4f8b\u300c\u5b9a\u6570\u300d\u3067\u306f\u306a\u3044\uff09\\(H\\) \u3092\u4f7f\u3063\u3066 \u3068\u66f8\u3051\u308b\u3060\u308d\u3046\u3002\u3055\u3089\u306b\u3053\u306e\u6cd5\u5247\u306f\u4e00\u69d8<\/strong><\/span>\u3067\u3042\u308b\u3068\u3059\u308b\u3068\uff0c\\( H(t, \\boldsymbol{r})\\) \u306f\u5834\u6240\u306b\u3088\u3089\u306a\u3044\u3068\u3059\u3079\u304d\u3067\u3042\u308a\uff0c \u3042\u3089\u305f\u3081\u3066 \\(\\displaystyle H(t) \\equiv \\frac{\\dot{a}}{a}\\) \u3068\u304a\u304f\u3068\u30cf\u30c3\u30d6\u30eb=\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u306f $$\\boldsymbol{r} = a(t) \\boldsymbol{x}, \\quad \\frac{d\\boldsymbol{x}}{dt} = \\boldsymbol{0}$$<\/p>\n \u307e\u3068\u3081\u308b\u3068\uff0c\u30cf\u30c3\u30d6\u30eb=\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u3068\u3044\u3046\u89b3\u6e2c\u91cf\u306e\u9593\u306e\u95a2\u4fc2\u5f0f\u306b\uff0c\uff08\u4f5c\u696d\u4eee\u8aac\u3068\u3057\u3066\u306e\uff09\u300c\u5b87\u5b99\u539f\u7406<\/strong><\/span>\u300d\u3092\u4eee\u5b9a\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff0c\u5b87\u5b99\u7a7a\u9593\u306e\u9280\u6cb3\u306e\u4f4d\u7f6e\u3092\u8868\u3059\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{r}\\) \u306f\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50<\/strong><\/span> \\(a(t)\\) \u3068\u6642\u9593\u7684\u306b\u5909\u5316\u3057\u306a\u3044\u5ea7\u6a19 \\(\\boldsymbol{x}\\) \u3092\u4f7f\u3063\u3066\uff0c \u534a\u5f84 \\(r\\) \u5185\u306e\u5168\u8cea\u91cf \\(M_r\\) \u306f\u6642\u9593\u7684\u306b\u5909\u5316\u3057\u306a\u3044\uff0c\u3068\u3044\u3046\u306e\u304c\u8cea\u91cf\u4fdd\u5b58\u5247\u3067\u3042\u308b\u3002\u4e00\u69d8\u6027<\/strong><\/span>\u306e\u4eee\u5b9a\u3092\u3059\u308b\u3068\uff0c\u7269\u8cea\u5bc6\u5ea6\u5206\u5e03 \\(\\rho\\) \u306f\u7a7a\u9593\u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u3053\u3068\u306b\u306a\u308a\uff0c \u4e00\u69d8\u306a\u8cea\u91cf\u5bc6\u5ea6\u5206\u5e03\u306e\u5834\u5408\u306f\uff08\u7403\u5bfe\u79f0\u5206\u5e03\u306e\u5834\u5408\u306e\u5f0f\u304c\u305d\u306e\u307e\u307e\u4f7f\u3048\u3066\uff09\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n $$m\\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 = \\boldsymbol{F}$$<\/p>\n \u5de6\u8fba\u306f<\/p>\n $$m\\frac{d^2\\boldsymbol{r}}{dt^2} = m \\boldsymbol{x} \\frac{d^2 a}{dt^2}$$<\/p>\n \u53f3\u8fba\u306f\uff08\u9023\u7d9a\u7684\u306a\u8cea\u91cf\u5206\u5e03\u306e\u5834\u5408\u306e \\(\\boldsymbol{F}\\) \u306b\u3064\u3044\u3066\u306f\u88dc\u8db3<\/a>\u3092\u53c2\u7167\uff09<\/p>\n \\begin{eqnarray} \u6700\u7d42\u7684\u306b \u3055\u3089\u306b \\((2)\\) \u5f0f\u306e\u4e21\u8fba\u306b \\(2 \\dot{a}\\) \u3092\u304b\u3051\u308b\u3068\uff0c \u3053\u3053\u3067 \\( \\displaystyle \\frac{d}{dt} (\\rho a^3) = 0\\) \u3092\u4f7f\u3063\u305f\u3002 \u4e21\u8fba\u3092 \\(a^2\\) \u3067\u5272\u3063\u3066\uff0c\u9069\u5b9c\u79fb\u9805\u3057\u3066\u3084\u308b\u3068 \u3053\u308c\u304c\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50<\/strong><\/span> \\(a(t)\\)\uff0c\u3072\u3044\u3066\u306f\u81a8\u5f35\u5b87\u5b99\u306e\u73fe\u5728\u30fb\u904e\u53bb\u30fb\u672a\u6765\u3092\u6c7a\u3081\u308b\u65b9\u7a0b\u5f0f\u3067\u3042\u308a\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u306f\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u3044\u3046\u540d\u524d\u304c\u3064\u3051\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n \u30cb\u30e5\u30fc\u30c8\u30f3\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u5f97\u3089\u308c\u305f\u5f0f\u3092\u5f0f\u756a\u53f7\u540c\u3058\u306e\u307e\u307e\uff0c\u3053\u3053\u306b\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n $$\\dot{\\rho} + 3 \\frac{\\dot{a}}{a} \\rho = 0 \\tag{1}$$ \u3053\u3053\u3067\uff0c\\(\\rho(t)\\) \u306f\u7269\u8cea\u5bc6\u5ea6\u5206\u5e03\u3067\u3042\u308a\uff0c\u300c\u5b87\u5b99\u539f\u7406\u300d\u306e\u4eee\u5b9a\u306b\u3088\u308a\u4e00\u69d8\u3067\u7a7a\u9593\u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u3002 \\(a(t)\\) \u306f\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u3067\u3042\u308a\uff0c\u6642\u9593\u7684\u306b\u4e00\u5b9a\u306a\u5171\u52d5\u5ea7\u6a19 \\(\\boldsymbol{x}\\) \u306b\u56e0\u5b50\u3068\u3057\u3066\u304b\u3051\u3089\u308c\uff0c\u7a7a\u9593\u5ea7\u6a19 \\(\\boldsymbol{r}\\) \u3092\u8868\u3059\uff1a\\(\\boldsymbol{r} = a(t) \\boldsymbol{x}\\)\u3002<\/p>\n \u3053\u308c\u3089\u306e\u5f0f\u306f\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u3082\uff0c\u30c0\u30b9\u30c8\u7269\u8cea\u306e\u5834\u5408\u306f\u305d\u306e\u307e\u307e\u306e\u5f62\u3067\u6210\u308a\u7acb\u3064\u3002<\/strong><\/span>\uff08\u305f\u3060\u3057\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u306f\u5358\u306a\u308b\u7a4d\u5206\u5b9a\u6570\u3060\u3063\u305f \\(k\\) \u304c\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u306f3\u6b21\u5143\u7a7a\u9593\u306e\u66f2\u7387\u5b9a\u6570\u3068\u3044\u3046\u89e3\u91c8\u3068\u306a\u308b\u306a\u3069\uff0c\u89e3\u91c8\u306b\u5fae\u5999\u306a\u9055\u3044\u304c\u3067\u3066\u304f\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\u3002\uff09<\/p>\n \u3053\u3053\u307e\u3067\u304f\u308b\u3068\uff0c\u89e3\u304f\u3079\u304d\u65b9\u7a0b\u5f0f\u306f\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3051\u308b\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u3068\u5168\u304f\u540c\u3058\u3067\u3042\u308b\u304b\u3089\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u5b87\u5b99\u8ad6\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u306f\u3053\u308c\u304f\u3089\u3044\u306b\u3057\u3066\uff0c\u3042\u3068\u306f\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306e\u307b\u3046\u3067\u8a71\u3092\u7d9a\u3051\u3088\u3046\u3002<\/p>\n
\n$$\\boldsymbol{v} = (v_x, v_y, v_z)$$\u306a\u3069\u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247\u3068\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n
\n$$m \\frac{d^2\\boldsymbol{r}}{dt^2} = \\boldsymbol{F} = m \\boldsymbol{g}$$<\/p>\n
\n\\nabla\\cdot\\boldsymbol{g} &=& -4 \\pi G \\rho \\\\
\n\\boldsymbol{g} &\\equiv& – \\nabla\\phi \\\\
\n\\therefore\\ \\ \\nabla^2 \\phi &=& 4\\pi G \\rho
\n\\end{eqnarray}
\n\u3053\u3053\u3067 \\(\\rho\\) \u306f\u8cea\u91cf\u5bc6\u5ea6\u5206\u5e03\u3067\u3042\u308b\u3002\uff08\u88dc\u8db3<\/a>\u3092\u53c2\u7167\u3002\uff09<\/p>\n\u30cf\u30c3\u30d6\u30eb=\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247\u3068\u300c\u5b87\u5b99\u539f\u7406\u300d\u306e\u4eee\u5b9a<\/h3>\n
\n
\n$$\\frac{dR}{dt} \\propto R$$\u3068\u66f8\u3051\u308b\u3002<\/p>\n
\n$$\\frac{d\\boldsymbol{r}}{dt} = H(t, \\boldsymbol{r}) \\boldsymbol{r}$$<\/p>\n
\n$$H(t, \\boldsymbol{r}) \\Rightarrow H(t)$$\u3068\u306a\u308b\u3002\u305b\u3044\u305c\u3044\u4f9d\u5b58\u3057\u3066\u3082\u3088\u3044\u306e\u306f \\(t\\) \u3060\u3051\u3067\u3042\u308a\uff0c\u7a7a\u9593\u4f9d\u5b58\u6027\u306f\u306a\u3044\uff0c\u3068\u3059\u308b\u306e\u304c\u4e00\u69d8\u6027<\/strong><\/span>\u306e\u4eee\u5b9a\u3067\u3042\u308b\u3002<\/p>\n
\n$$ \\dot{\\boldsymbol{r}} = \\frac{\\dot{a}}{a} \\boldsymbol{r}$$\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\n$$\\boldsymbol{r} = a(t) \\boldsymbol{x}$$
\n\u3068\u66f8\u3051\u308b\u3053\u3068\u306b\u306a\u308b\u3002\uff08\u6d41\u4f53\u529b\u5b66\u7684\u306b\u8a00\u3048\u3070 \\(\\boldsymbol{r}\\) \u304c\u30aa\u30a4\u30e9\u30fc\u5ea7\u6a19\uff0c\\(\\boldsymbol{x}\\) \u304c\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u5ea7\u6a19\u3002\u307e\u305f\uff0c\u5b87\u5b99\u8ad6\u3067\u306f \\(\\boldsymbol{x}\\) \u3092\u7279\u306b\u5171\u52d5\u5ea7\u6a19<\/strong><\/span>\u3068\u3044\u3046\u3002\uff09<\/p>\n\u8cea\u91cf\u4fdd\u5b58\u5247\u3068\u300c\u5b87\u5b99\u539f\u7406\u300d\u306e\u4eee\u5b9a<\/h3>\n
\n\\begin{eqnarray}
\n\\frac{d M_r}{dt} &=& \\frac{d}{dt} \\iiint_V \\rho(t) dV \\\\
\n&=& \\frac{d}{dt} \\left(\\frac{4\\pi}{3} r^3 \\rho(t) \\right) \\\\
\n&=& \\frac{d}{dt} \\left(\\frac{4\\pi}{3} a^3 |\\boldsymbol{x}|^3\u00a0 \\rho(t) \\right) = 0
\n\\end{eqnarray}
\n$$\\therefore \\ \\ \\frac{d}{dt} (\\rho a^3) = 0, \\quad\\mbox{or}\\quad
\n\\dot{\\rho} + 3 \\frac{\\dot{a}}{a} \\rho = 0 \\tag{1}$$<\/p>\n\u300c\u5b87\u5b99\u539f\u7406\u300d\u306b\u3088\u308b\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247\u3068\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n
\n\\boldsymbol{F} &=& – \\frac{GM_r m }{r^3} \\boldsymbol{r}\\\\
\n&=& – m \\frac{4\\pi G \\rho a^3 |\\boldsymbol{x}|^3}{3} \\frac{a \\boldsymbol{x}}{a^3 |\\boldsymbol{x}|^3} \\\\
\n&=& -m \\boldsymbol{x}\\frac{4\\pi G \\rho a}{3}
\n\\end{eqnarray}<\/p>\n
\n$$\\ddot{a} = – \\frac{4\\pi G}{3} \\rho a, \\quad \\mbox{or}\\quad \\frac{\\ddot{a}}{a} = – \\frac{4\\pi G}{3} \\rho \\tag{2}$$<\/p>\n
\n\\begin{eqnarray}
\n2 \\dot{a} \\ddot{a} = \\frac{d}{dt}\\left(\\dot{a}^2\\right) &=& – \\frac{8\\pi G}{3} \\rho a \\dot{a} \\\\
\n&=& – \\frac{8\\pi G}{3} (\\rho a^3) \\frac{\\dot{a}}{a^2}\\\\
\n&=& \\frac{8\\pi G}{3} (\\rho a^3) \\frac{d}{dt}\\left(\\frac{1}{a}\\right)\\\\
\n&=& \\frac{d}{dt}\\left(\\frac{8\\pi G}{3} (\\rho a^3) \\frac{1}{a}\\right) \\\\
\n&=& \\frac{d}{dt}\\left(\\frac{8\\pi G}{3} \\rho a^2\\right)
\n\\end{eqnarray}<\/p>\n
\n\\begin{eqnarray}
\n\\therefore\\ \\ \\frac{d}{dt} \\left( \\dot{a}^2 – \\frac{8\\pi G}{3} \\rho a^2\\right) &=& 0 \\\\
\n\\therefore\\ \\ \\dot{a}^2 – \\frac{8\\pi G}{3} \\rho a^2 &=& \\mbox{const.} \\equiv -k
\n\\end{eqnarray}<\/p>\n
\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho \\tag{3}$$<\/p>\n\u3053\u3053\u307e\u3067\u306e\u307e\u3068\u3081<\/h3>\n
\n$$\\quad \\frac{\\ddot{a}}{a} = – \\frac{4\\pi G}{3} \\rho \\tag{2}$$
\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho \\tag{3}$$<\/p>\n\n