Eccentricity vector

In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. 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. The magnitude of the eccentricity vector represents the eccentricity of the orbit. Note that the velocity and position vectors need to be relative to the inertial frame of the central body.

Calculation

The eccentricity vector \( \mathbf{e} \, \) is: [1]

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

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