Algebraic function

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 $\mathbf{e}$ can be calculated

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

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

or equivalently:

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

See also

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

References

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