Eccentricity vector

In celestial mechanics, the eccentricity vector of a Kepler orbit is the vector that points towards the periapsis and has 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 in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously. For the eccentricity and argument of periapsis parameters, eccentricity zero (circular orbit) corresponds to a singularity.

Calculation

The eccentricity vector $\mathbf{e} \,$ is:

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

which follows immediately from the vector identity:

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

where:

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

See also

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