# 離心率向量

## 計算

{\displaystyle {\begin{aligned}\mathbf {e} &={\mathbf {v} \times \mathbf {h} \over {\mu }}-{\mathbf {r} \over {\left|\mathbf {r} \right|}}\\&=\left({\mathbf {\left|v\right|} ^{2} \over {\mu }}-{1 \over {\left|\mathbf {r} \right|}}\right)\mathbf {r} -{\mathbf {r} \cdot \mathbf {v} \over {\mu }}\mathbf {v} \\\end{aligned}}}

${\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} }$
${\displaystyle \mathbf {v} \times \left(\mathbf {r} \times \mathbf {v} \right)=\left(\mathbf {v} \cdot \mathbf {v} \right)\mathbf {r} -\left(\mathbf {r} \cdot \mathbf {v} \right)\mathbf {v} }$

• ${\displaystyle \mathbf {r} \,\!}$位置向量 (position vector)
• ${\displaystyle \mathbf {v} \,\!}$速度向量 (velocity vector)
• ${\displaystyle \mathbf {h} \,\!}$比角動量向量 (specific angular momentum vector) (${\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} }$)
• ${\displaystyle \mu \,\!}$標準重力參數 (standard gravitational parameter)。

${\displaystyle \mathbf {e} ={\mathbf {v} \times \mathbf {L} \over {m\mu }}-{\mathbf {r} \over {\left|\mathbf {r} \right|}}}$

## 參閱

1. ^ Cordani, Bruno. The Kepler Problem. Birkhaeuser. 2003: 22. ISBN 3-7643-6902-7.