\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\uff0c\u30b1\u30d7\u30e9\u30fc\u904b\u52d5\u3059\u308b\uff08\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u306b\u5f93\u3063\u305f\u904b\u52d5\u3092\u3059\u308b\uff09\u5929\u4f53\u306b\u306f\u3069\u306e\u3088\u3046\u306a\u529b\u304c\u50cd\u3044\u3066\u3044\u308b\u306e\u304b\u3092\u8abf\u3079\uff0c\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247\u3092\u5c0e\u304f\u3002<\/p>\n
\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u3092\u5f0f\u3067\u3042\u3089\u308f\u3059\u3068\uff0c<\/p>\n
\u592a\u967d\u306e\u4f4d\u7f6e\u3092\u539f\u70b9\u3068\u3057\u305f\u3068\u304d\u306e\u60d1\u661f\u306e\uff08\u76f8\u5bfe\uff09\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306f<\/p>\n
\\begin{eqnarray}
\n\\boldsymbol{r} &=& (x, y, 0) \\\\
\n&=& (r \\cos\\phi, r\\sin\\phi, 0) \\\\
\nr &=& \\frac{a\\,(1-e^2)}{1 + e\\cos\\phi}
\n\\end{eqnarray}<\/p>\n
$$\\frac{1}{2} r^2 \\dot{\\phi} = \\mbox{const.} = \\frac{\\pi a^2 \\sqrt{1-e^2}}{T}$$<\/p>\n
$\\displaystyle \\frac{a^3}{T^2}$ \u306f\u60d1\u661f\u306b\u3088\u3089\u306a\u3044\u5b9a\u6570\u3002<\/p>\n
\u3055\u304d\u306b $\\dot{r}$ \u3092\u6c42\u3081\u3066\u304a\u304f\u3002<\/p>\n
\\begin{eqnarray}
\nr &=& \\frac{a\\,(1-e^2)}{1 + e \\cos\\phi} \\\\
\n\\dot{r} &=& – \\frac{a\\,(1-e^2)}{(1+e \\cos\\phi)^2} \\left( – e \\sin\\phi\\ \\dot{\\phi}\\right) \\\\
\n&=& \\frac{e \\sin\\phi}{a\\,(1-e^2)} \\, r^2 \\dot{\\phi}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c\u901f\u5ea6 $\\dot{\\boldsymbol{r}} = (\\dot{x}, \\dot{y}, 0)$ \u306f<\/p>\n
\\begin{eqnarray}
\n\\dot{x} &=& \\left( r \\cos\\phi \\right)^{\\dot{}}\\\\
\n&=& \\dot{r} \\cos\\phi – r \\sin\\phi\\ \\dot{\\phi} \\\\
\n&=& \\frac{e \\sin\\phi}{a\\,(1-e^2)} \\, r^2 \\dot{\\phi}\\, \\cos\\phi – \\frac{\\sin\\phi}{r} \\ r^2 \\dot{\\phi} \\\\
\n&=& – \\frac{\\sin\\phi}{a\\, (1-e^2)} \\ r^2 \\dot{\\phi}
\n\\end{eqnarray}<\/p>\n
\u540c\u69d8\u306b\u3057\u3066<\/p>\n
\\begin{eqnarray}
\n\\dot{y} &=& \\frac{e + \\cos\\phi}{a \\, (1-e^2)} \\ r^2 \\dot{\\phi}
\n\\end{eqnarray}<\/p>\n
$r^2 \\dot{\\phi}$ \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3063\u3066\u52a0\u901f\u5ea6 $\\ddot{\\boldsymbol{r}} = (\\ddot{x}, \\ddot{y}, 0)$ \u306f<\/p>\n
\\begin{eqnarray}
\n\\ddot{x} &=& – \\frac{\\cos\\phi\\ \\dot{\\phi}}{a \\, (1-e^2)} \\ r^2 \\dot{\\phi} \\\\
\n&=& – \\frac{r \\cos\\phi}{a \\, (1-e^2)} \\cdot \\left(r^2 \\dot{\\phi}\\right)^2 \\cdot \\frac{1}{r^3} \\\\
\n&=& – \\frac{4\\pi a^3}{T^2} \\frac{x}{r^3}\\\\
\n&\\equiv& – K \\frac{x}{r^3}
\n\\end{eqnarray}<\/p>\n
\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c3\u6cd5\u5247\u306b\u3088\u3063\u3066\uff0c$K$ \u306f\u60d1\u661f\u306b\u3088\u3089\u306a\u3044\u5b9a\u6570\u3002\u540c\u69d8\u306b\u3057\u3066<\/p>\n
\\begin{eqnarray}
\n\\ddot{y} &=& – K \\frac{y}{r^3}
\n\\end{eqnarray}<\/p>\n
\u30d9\u30af\u30c8\u30eb\u5f62\u3067\u66f8\u304f\u3068<\/p>\n
\\begin{eqnarray}
\n\\ddot{\\boldsymbol{r}} &=& – K \\frac{\\boldsymbol{r}}{r^3}
\n\\end{eqnarray}<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\uff0c\u8cea\u91cf $m$ \u306e\u60d1\u661f\u306b\u306f\u305f\u3089\u304f\u529b $\\boldsymbol{F}$ \u306f<\/p>\n
$$ \\boldsymbol{F} = m \\ddot{\\boldsymbol{r}} = – K m \\frac{\\boldsymbol{r}}{r^3}$$<\/p>\n
\u5b9a\u6570 $K$ \u306f\u592a\u967d\u8cea\u91cf $M$ \u306b\u6bd4\u4f8b\u3059\u308b\u3068\u3057\u3066\uff0c\u6bd4\u4f8b\u5b9a\u6570\u3092 $G$ \u3068\u66f8\u304f\u3068\uff0c\u3081\u3067\u305f\u304f\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247<\/p>\n
$$\\boldsymbol{F} =- \\frac{G M\u00a0 m \\ \\boldsymbol{r}}{r^3}$$<\/p>\n
\u304c\u5f97\u3089\u308c\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\uff0c\u30b1\u30d7\u30e9\u30fc\u904b\u52d5\u3059\u308b\uff08\u30b1\u30d7\u30e9\u30fc\u306e\u6cd5\u5247\u306b\u5f93\u3063\u305f\u904b\u52d5\u3092\u3059\u308b\uff09\u5929\u4f53\u306b\u306f\u3069\u306e\u3088\u3046\u306a\u529b\u304c\u50cd\u3044\u3066\u3044\u308b\u306e\u304b\u3092\u8abf\u3079\uff0c\u4e07\u6709\u5f15\u529b\u306e\u6cd5\u5247\u3092\u5c0e\u304f\u3002<\/p>