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# Argument of periapsis

In an orbit, the argument of periapsis (ω) is the angle between the ascending node (the point where the orbiting body passes from the southern to the northern hemisphere) and the periapsis (the point of closest approach to the central body), measured in the body's orbital plane and in its direction of motion. It is undefined for equatorial orbits, where there is no defined ascending node, and for circular orbits, where there is no defined periapsis.

## Calculation

In astrodynamics the argument of periapsis $\omega\,$ can be calculated as follows:

$\omega = arccos { {\mathbf{n} \cdot \mathbf{e}} \over { \mathbf{\left |n \right |} \mathbf{\left |e \right |} }}$
(if $e_z < 0\,$ then $\omega = 2 \pi - \omega\,$)

where:

• $\mathbf{n}$ is the vector pointing towards the ascending node (i.e. the z-component of $\mathbf{n}$ is zero),
• $\mathbf{e }$ is the eccentricity vector (the vector pointing towards the periapsis).

In the case of equatorial orbits, though the argument is strictly undefined, it is often assumed that:

$\omega = arccos { {e_x} \over { \mathbf{\left |e \right |} }}$

where:

• $e_x\,$ is x-component of the eccentricity vector $\mathbf{e }\,$.

In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore $\omega=0\,$.