Eccentricity vector

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

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

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

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.

Alternatively it can also be computed from orbital angular momentum vector $\mathbf{h}$ :

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

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.

Since $\mathbf{h}$ is defined as $\mathbf{r}\times\mathbf{v}$ , it can be shown by the vector triple product identity that this formula is equivalent to the one above.

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Categories: Astrodynamics | Celestial mechanics

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