Eccentricity vector

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.

Calculation

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):: $mathbf{e} = {mathbf{left |v
ight ^2 mathbf{r} over {mu - {(mathbf{r} cdot mathbf{v} ) mathbf{v} over{mu - {mathbf{r}over{left|mathbf{r}
ight|$ 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 $mathbf{h}$ :

: $mathbf{e} = {mathbf{v} imesmathbf{h}over{mu - {mathbf{r}over{left|mathbf{r}
ight|$ 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 $mathbf{r} imesmathbf{v}$ , it can be shown by the vector triple product identity that this formula is equivalent to the one above.

ee also

*Eccentricity
*Orbit
*Laplace-Runge-Lenz vector

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