Eccentricity vector

The eccentricity vector ${\displaystyle \mathbf {e} \,}$  is: ^[1]

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

which follows immediately from the vector identity:

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

where:

• ${\displaystyle \mathbf {r} \,\!}$  is position vector
• ${\displaystyle \mathbf {v} \,\!}$  is velocity vector
• ${\displaystyle \mathbf {h} \,\!}$  is specific angular momentum vector (equal to ${\displaystyle \mathbf {r} \times \mathbf {v} }$ )

• ${\displaystyle \mu \,\!}$  is standard gravitational parameter

• Kepler orbit
• Orbit
• Eccentricity
• Laplace–Runge–Lenz vector