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   2. ¤·¤«¤·¡¢±ßµ°Æ»¤ò²¾Äꤹ¤ë¤È¡¢²ÐÀ±¤Î±¿Æ°¤òÀâÌÀ¤Ç¤­¤Ê¤«¤Ã¤¿
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   $F$:

   ¾ÇÅÀ

   $a$:

   ĹȾ·Â

   $b$:

   ûȾ·Â

   $e\equiv$

   $\sqrt{1-\left(\frac{b}{a}\right)^2}$: Î¥¿´Î¨

   ÂʱߤÎÃæ¿´¤«¤é¾ÇÅÀ¤Þ¤Ç¤ÎŤµ: $ae$

   ÂʱߤÎÊýÄø¼°¡§ $${\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1}$$

   ±ß¤Î¤È¤­¡¢ $a=b=r  \Rightarrow x^2+y^2=r^2$

2. ÌÌÀÑ®ÅÙ°ìÄê¤Îˡ§
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   • ¥¨¥Í¥ë¥®¡¼Êݸ§
   • ±¿Æ°ÎÌÊݸ§

   ±¿Æ°¥¨¥Í¥ë¥®¡¼ $T=\frac{1}{2}mv^2$

   ±¿Æ°ÎÌ $\mbox{\boldmath$p$}=m\v$

   ³Ñ±¿Æ°ÎÌ ${\L }=m\r\times\v$

   $\times$: ³°ÀÑ $\leftrightarrow$ ÆâÀÑ($\cdot$)
   

   
   

   $$\mbox{\boldmath$a$}\cdot\b =a_x b_x + a_y b_y = ab\cos\theta$$
   
   
   
   
   $$\mbox{\boldmath$a$}\times\b =(a_x b_y-a_y b_x)\hat{\mbox{\boldmath$z$}}=ab\sin\theta\hat{\mbox{\boldmath$z$}}$$
   
   
   ( $\mbox{\boldmath$a$}, \b$¤¬$xy$Ê¿ÌÌÆâ¤Î¾ì¹ç)

   $\Longrightarrow$ $\mbox{\boldmath$a$}\parallel\b$¤Ê¤é¡¢ $\mbox{\boldmath$a$}\times\b =0$.

   
   

   $$\frac{\d\L }{\d t}=m\frac{\d\r }{\d t}\times\v +m\r\times\f... ...,3){$\cdot$} \put(6,4){\circle{14}} \end{picture}\v\times\v =0$$
   
   
   ¤¤¤Þ¡¢ $\r\parallel\mbox{\boldmath$F$}$¤Ê¤Î¤Ç( $\mbox{\boldmath$F$}=-\frac{GMm}{r^2}\hat{\r }$)¡¢
   
   $$\frac{\d\L }{\d t}=0  \Longrightarrow  \L ={\rm const.  (³Ñ±¿Æ°ÎÌÊݸ)}$$
   
   
   

   ¤³¤³¤ÇÌÌÀÑ®ÅÙ¤Ï $${\frac{\d S}{\d t}}$$¤Ç¤¢¤ë¡£¤¹¤ë¤È¡¢
   

   $$\Delta S \simeq \frac{1}{2}\left\vert\r\times\v\Delta t\right\vert$$
   
   
   
   
   $$\Longrightarrow \frac{\Delta S}{\Delta t}\simeq\frac{\d S}{... ...vert\r\times\v\right\vert=\frac{1}{2}\vert\L\vert={\rm const.}$$
   
   
   
   
   $$\setlength{\unitlength}{1pt} \thinlines \begin{picture}(1... ...ure}   \frac{\d S}{\d t}={\rm const.,  ¤Ä¤Þ¤ê¡¢ÌÌÀÑ®ÅÙ°ìÄê//}$$
   
   
   

3. µ°Æ»Ä¹È¾·Â¤Î3¾è¤¬¸øž¼þ´ü¤Î2¾è¤ËÈæÎã(ĴϤÎˡ§)¡Ä $a^3\propto T^2$
   

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   ÎϤΤĤꤢ¤¤(ËüÍ­°úÎÏ=±ó¿´ÎÏ)¤è¤ê¡¢
   

   $$\frac{GMm}{r^2}=\frac{mv^2}{r}=mr\dot{\theta}^2=mr\omega^2,  v=r\omega$$
   
   
   ¤¢¤ë¤¤¤Ï¡¢±¿Æ°ÊýÄø¼°(¤¿¤À¤·$\ddot{r}=0$)¤è¤ê¡¢
   
   $$m(-r\omega^2)=-\frac{mv^2}{r}=-\frac{GMm}{r^2}$$
   
   
   ¤³¤ì¤è¤ê¡¢
   
   $$rv^2=GM.$$
   
   
   ¤¤¤Þ¡¢¼þ´ü$T$¤Ï¡¢
   
   $$T=\frac{2\pi r}{v}\left(=\frac{2\pi}{\omega}\right)$$
   
   
   ¤Ç¤¢¤ë¤«¤é¡¢
   
   $$v=\frac{2\pi r}{T}$$
   
   
   ¤³¤ì¤é¤è¤ê¡¢
   
   $$rv^2=\frac{4\pi^2 r^3}{T^2}=GM$$
   
   
   
   
   $$\Longrightarrow r^3\propto T^2.$$
   
   

   ¤¤¤Þ¡¢Ãϵå¤Ï1ǯ¤Ç1¼þ¤·¡¢µ°Æ»È¾·Â¤Ï1AU¤Ê¤Î¤Ç¡¢
   

   $$\left(\frac{r}{\rm AU}\right)^3=\left(\frac{T}{\rm yr}\right)^2$$
   
   
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$$m\frac{\d ^2 r}{\d t^2}=-\frac{GMm}{r^2}$$

¤è¤ê¡¢

$$m\frac{r}{T^2}\sim \frac{GMm}{r^2}$$

$$\Rightarrow~\frac{r^3}{T^2}\sim GM$$

$$\Rightarrow~r^3\propto T^2$$

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