Eccentricity vector

In astrodynamics, the eccentricity vector of a Kepler orbit is the vector pointing towards the periapsis having a magnitude equal to the orbit's scalar eccentricity. The magnitude is unitless. For Kepler orbits the eccentricity vector is a constant of motion. Its main use is for almost circular orbits as perturbing (non-Keplarian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously as opposed to the parameters eccentricity and argument of perigee for which eccentricity zero (circular orbit) corresponds to a singularity. See the article Kepler orbit

Calculation

The eccentricity vector $\mathbf{e} \,$ can be calculated

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

where:

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

or equivalently from the orbital state vectors by

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

where:

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

See also

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