Eccentricity vector - Definition and Overview

In astrodynamics the eccentricity vector of a conic section orbit is the vector pointing towards the periapsis and with length equal to the orbit's scalar eccentricity.

Calculation

The eccentricity vector <math> \mathbf{e} \,<math> can be calculated from the orbital state vectors <math> \mathbf{v} \,<math> and <math> \mathbf{r} \,<math> at any time (the result is constant):

<math> \mathbf{e} = {1 \over {\mu}} \left [\left (v^2 - {\mu \over {\mathbf{\left |r \right |}}}\right)

\mathbf{r} - (\mathbf{r} \cdot \mathbf{v} ) \mathbf{v} \right ]<math> where:

• <math> \mathbf{v} \,<math> is velocity vector of the orbital state vectors,
• <math> \mathbf{r} \,<math> is position vector of the orbital state vectors,
• <math> \mu \,<math> is standard gravitational parameter.

Alternatively it can also be computed from orbital angular momentum vector h:

<math> \mathbf{e} = {\mathbf{v}\times\mathbf{h}\over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}}<math>

where:

• <math> \mathbf{v}\,\!<math> is orbital velocity vector,
• <math>\mathbf{h}\,\!<math> is orbital angular momentum vector,
• <math>\mathbf{r}\,\!<math> is orbital position vector,
• <math>\mu\,\!<math> is standard gravitational parameter.

See also

• Eccentricity
• Orbit
• Laplace-Runge-Lenz vector