Argument of periapsis

In astrodynamics the argument of periapsis ω can be calculated as follows:

${\displaystyle \omega =\arccos {{\mathbf {n} \cdot \mathbf {e} } \over {\mathbf {\left|n\right|} \mathbf {\left|e\right|} }}}$

If e_z < 0 then ω → 2π − ω.

where:

• n is a vector pointing towards the ascending node (i.e. the z-component of n is zero),
• e is the eccentricity vector (a vector pointing towards the periapsis).

In the case of equatorial orbits (which have no ascending node), the argument is strictly undefined. However, if the convention of setting the longitude of the ascending node Ω to 0 is followed, then the value of ω follows from the two-dimensional case:

${\displaystyle \omega =\arctan 2\left(e_{y},e_{x}\right)}$

If the orbit is clockwise (i.e. (r × v)_z < 0) then ω → 2π − ω.

where:

• e_x and e_y are the x- and y-components of the eccentricity vector e.

In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore ω = 0. However, in the professional exoplanet community, ω = 90° is more often assumed for circular orbits, which has the advantage that the time of a planet's inferior conjunction (which would be the time the planet would transit if the geometry were favorable) is equal to the time of its periastron.[1][2][3]

• Kepler orbit
• Orbital mechanics
• Orbital node

• Argument Of Perihelion in Swinburne University Astronomy Website