Argument of perigee, ω, illustrated with respect to the ascending node and satellite orbit. k̂, ĵ, and î represent the unit vectors. [Source]

The argument of perigee, ω [°], represents the orientation of an orbit within its orbital plane. Specifically, it the angle between the ascending node of the orbit (where it crosses the Earth’s equatorial plane), represented by the ascending node vector $\vec{n}$, and the perigee of the orbit (when the satellite is nearest to the Earth), represented by the eccentricity vector $\vec{e}$ .

$$ \omega=\cos^{-1}\left( \frac{\vec{n} \cdot \vec{e}}{|\vec{n}||\vec{e}|} \right) $$

Where:

• ω [°]: Argument of Perigee
• $\vec{n}$: ascending node vector
• $\vec{e}$: eccentricity vector.
  • Magnitude: equal to the eccentricity of the orbit
  • Direction: apogee to perigee

Assumptions:

• N/A

• Derivation

References

Chapter 3 – The Classical Orbital Elements (COEs)