Perifocal coordinate system

The perifocal coordinate (PQW) system is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors ${\displaystyle \mathbf {\hat {p}} }$ and ${\displaystyle \mathbf {\hat {q}} }$ lie in the plane of the orbit. ${\displaystyle \mathbf {\hat {p}} }$ is directed towards the periapsis of the orbit and ${\displaystyle \mathbf {\hat {q}} }$ has a true anomaly (${\displaystyle \theta }$) of 90 degrees past the periapsis. The third unit vector ${\displaystyle \mathbf {\hat {w}} }$ is the angular momentum vector and is directed orthogonal to the orbital plane such that:^[1][2] $${\displaystyle \mathbf {\hat {w}} =\mathbf {\hat {p}} \times \mathbf {\hat {q}} }$$

And, since ${\displaystyle \mathbf {\hat {w}} }$ is the unit vector in the direction of the angular momentum vector, it may also be expressed as: $${\displaystyle \mathbf {\hat {w}} ={\frac {\mathbf {h} }{\|\mathbf {h} \|}}}$$ where h is the specific relative angular momentum.

The position and velocity vectors can be determined for any location of the orbit. The position vector, r, can be expressed as: $${\displaystyle \mathbf {r} =r\cos \theta \mathbf {\hat {p}} +r\sin \theta \mathbf {\hat {q}} }$$ where ${\displaystyle \theta }$ is the true anomaly and the radius ${\displaystyle r=\|\mathbf {r} \|}$ may be calculated from the orbit equation.

The velocity vector, v, is found by taking the time derivative of the position vector: $${\displaystyle \mathbf {v} =\mathbf {\dot {r}} =({\dot {r}}\cos \theta -r{\dot {\theta }}\sin \theta )\mathbf {\hat {p}} +({\dot {r}}\sin \theta +r{\dot {\theta }}\cos \theta )\mathbf {\hat {q}} }$$

A derivation from the orbit equation can be made to show that: $${\displaystyle {\dot {r}}={\frac {\mu }{h}}e\sin \theta }$$ where ${\displaystyle \mu }$ is the gravitational parameter of the focus, h is the specific relative angular momentum of the orbital body, e is the eccentricity of the orbit, and ${\displaystyle \theta }$ is the true anomaly. ${\displaystyle {\dot {r}}}$ is the radial component of the velocity vector (pointing inward toward the focus) and ${\displaystyle r{\dot {\theta }}}$ is the tangential component of the velocity vector. By substituting the equations for ${\displaystyle {\dot {r}}}$ and ${\displaystyle r{\dot {\theta }}}$ into the velocity vector equation and simplifying, the final form of the velocity vector equation is obtained as:^[3] $${\displaystyle \mathbf {v} ={\frac {\mu }{h}}\left[-\sin \theta \mathbf {\hat {p}} +(e+\cos \theta )\mathbf {\hat {q}} \right]}$$