# 開普勒定律

## 開普勒定律

### 開普勒第二定律

${\displaystyle S_{AB}=S_{CD}=S_{EK}}$

### 開普勒第三定律

${\displaystyle {\frac {\tau ^{2}}{a^{3}}}=K}$

## 數學推導：由牛頓萬有引力定律導出開普勒定律

### 開普勒第二定律推導

${\displaystyle {\boldsymbol {F}}=-G{\frac {mM}{r^{2}}}\ {\hat {\boldsymbol {r}}}}$

${\displaystyle {\boldsymbol {F}}=m{\ddot {\boldsymbol {r}}}}$

${\displaystyle {\ddot {\boldsymbol {r}}}=-G{\frac {M}{r^{2}}}\ {\hat {\boldsymbol {r}}}}$(1)

${\displaystyle {\dot {\boldsymbol {r}}}={\dot {r}}{\hat {\boldsymbol {r}}}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}}}$
${\displaystyle {\ddot {\boldsymbol {r}}}=\left({\ddot {r}}{\hat {\boldsymbol {r}}}+{\dot {r}}{\frac {\mathrm {d} {\hat {\boldsymbol {r}}}}{\mathrm {d} t}}\right)+\left({\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\dot {\theta }}{\frac {\mathrm {d} {\hat {\boldsymbol {\theta }}}}{\mathrm {d} t}}\right)=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\boldsymbol {r}}}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}}$(2)

${\displaystyle {\frac {\mathrm {d} {\hat {\boldsymbol {r}}}}{\mathrm {d} t}}={\dot {\theta }}{\hat {\boldsymbol {\theta }}}}$
${\displaystyle {\frac {\mathrm {d} {\hat {\boldsymbol {\theta }}}}{\mathrm {d} t}}=-{\dot {\theta }}{\hat {\boldsymbol {r}}}}$

${\displaystyle ({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\boldsymbol {r}}}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}=-{\frac {GM}{r^{2}}}{\hat {\boldsymbol {r}}}}$

${\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {GM}{r^{2}}}}$(3)
${\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}=0}$(4)

${\displaystyle {\dot {\ell }}=mr(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }})=0}$

${\displaystyle \Delta A=\int _{t_{1}}^{t_{2}}{\frac {1}{2}}\cdot r\cdot r{\dot {\theta }}\cdot \mathrm {d} t=\int _{t_{1}}^{t_{2}}{\frac {\ell }{2m}}\mathrm {d} t={\frac {\ell }{2m}}\cdot (t_{2}-t_{1})}$

### 開普勒第一定律推導

${\displaystyle {\dot {\theta }}={\frac {\ell }{mr^{2}}}={\frac {\ell u^{2}}{m}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}={\dot {\theta }}{\frac {\mathrm {d} }{\mathrm {d} \theta }}={\frac {\ell u^{2}}{m}}{\frac {\mathrm {d} }{\mathrm {d} \theta }}}$

${\displaystyle {\dot {r}}={\frac {\ell u^{2}}{m}}{\frac {\mathrm {d} }{\mathrm {d} \theta }}{\frac {1}{u}}=-{\frac {\ell u^{2}}{m}}{\frac {1}{u^{2}}}{\frac {\mathrm {d} u}{\mathrm {d} \theta }}=-{\frac {\ell }{m}}{\frac {\mathrm {d} u}{\mathrm {d} \theta }}}$

${\displaystyle {\ddot {r}}={\frac {\ell u^{2}}{m}}{\frac {\mathrm {d} {\dot {r}}}{\mathrm {d} \theta }}={\frac {\ell u^{2}}{m}}{\frac {\mathrm {d} }{\mathrm {d} \theta }}(-{\frac {\ell }{m}}{\frac {\mathrm {d} u}{\mathrm {d} \theta }})=-{\frac {\ell ^{2}u^{2}}{m^{2}}}{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}}$

${\displaystyle -{\frac {\ell ^{2}u^{2}}{m^{2}}}{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}-{\frac {\ell ^{2}u^{3}}{m^{2}}}=-GMu^{2}}$

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {GMm^{2}}{\ell ^{2}}}}$

${\displaystyle u={\frac {GMm^{2}}{\ell ^{2}}}}$

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u=0}$

${\displaystyle u=C\ \cos(\theta -\theta _{0})}$

${\displaystyle u={\frac {GMm^{2}}{\ell ^{2}}}+C\ \cos(\theta -\theta _{0})}$

${\displaystyle {\frac {1}{r}}={\frac {GMm^{2}}{\ell ^{2}}}(1+e\ \cos {\theta })}$

### 開普勒第三定律推導

${\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}={\frac {\ell }{2m}}}$

${\displaystyle \tau ={\frac {2m\pi ab}{\ell }}}$(5)

${\displaystyle a={\frac {r_{A}+r_{B}}{2}}}$(6)
${\displaystyle b={\sqrt {r_{A}\ r_{B}}}}$(7)

${\displaystyle E={\frac {1}{2}}m{\dot {r}}^{2}+{\frac {1}{2}}mr^{2}{\dot {\theta }}^{2}-G{\frac {mM}{r}}}$

${\displaystyle {\dot {r}}=0}$

${\displaystyle E={\frac {1}{2}}mr^{2}{\dot {\theta }}^{2}-G{\frac {mM}{r}}={\frac {\ell ^{2}}{2mr^{2}}}-G{\frac {mM}{r}}}$

${\displaystyle r^{2}+{\frac {GmM}{E}}r-{\frac {\ell ^{2}}{2mE}}=0}$

${\displaystyle r_{A}=-{\frac {{\frac {GmM}{E}}-{\sqrt {\left({\frac {GmM}{E}}\right)^{2}+{\frac {2\ell ^{2}}{mE}}}}}{2}}}$
${\displaystyle r_{B}=-{\frac {{\frac {GmM}{E}}+{\sqrt {\left({\frac {GmM}{E}}\right)^{2}+{\frac {2\ell ^{2}}{mE}}}}}{2}}}$

${\displaystyle a=-{\frac {GmM}{2E}}}$
${\displaystyle b={\frac {\ell }{\sqrt {-2mE}}}={\frac {\ell }{m}}{\frac {\sqrt {a}}{\sqrt {GM}}}}$

${\displaystyle \tau ={\frac {2\pi a^{3/2}}{\sqrt {GM}}}}$

## 參考資料

1. 克卜勒的行星運動三定律 (PDF). 高苑科技大學. [2015-07-26].