# 偏近点角

## 计算

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

• ${\displaystyle \mathbf {r} \,\!}$是轨道上天体的位置向量。(线段sp),
• ${\displaystyle a\,\!}$是轨道的半长轴（线段cz），和
• ${\displaystyle e\,\!}$ 是轨道的离心率

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

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

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

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

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

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

## 参考资料

• 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.)