# 偏近點角

## 計算

${\displaystyle E=\arccos {{1-\left|\mathbf {r} \right|/a} \over e}}$

• ${\displaystyle \mathbf {r} \,\!}$ 是軌道上天體的位置向量。(線段sp),
• ${\displaystyle a\,\!}$ 是軌道的半長軸（線段cz），和
• ${\displaystyle e\,\!}$  是軌道的離心率

${\displaystyle M=E-e\cdot \sin {E}.\,\!}$

• ${\displaystyle E_{1}=M+e\,\sin M}$
• ${\displaystyle E_{2}=M+e\,\sin M+{\frac {1}{2}}e^{2}\sin 2M}$
• ${\displaystyle E_{3}=M+e\,\sin M+{\frac {1}{2}}e^{2}\sin 2M+{\frac {1}{8}}e^{3}(3\sin 3M-\sin M)}$ .

${\displaystyle \cos {T}={{\cos {E}-e} \over {1-e\cdot \cos {E}}}}$

${\displaystyle \tan {T \over 2}={\sqrt {{1+e} \over {1-e}}}\tan {E \over 2}.\,}$

${\displaystyle r=a\left(1-e\cdot \cos {E}\right)\,\!}$

${\displaystyle r=a{(1-e^{2}) \over (1+e\cdot \cos {T})}.\,\!}$

## 參考資料

• Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.
• Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)