# Perifocal Frame#

In this section, we define the **perifocal reference frame**, a convenient reference frame for describing the planar orbits that we’ve been discussing. The perifocal frame is a Cartesian reference centered at the occupied focus of the orbit, the location where \(m_1\) exists. Since all the orbits we’ve looked at so far are planar, the \(\pf{x}\pf{y}\) plane of the perifocal frame lies in the same plane as the orbit. This means that the \(\pf{z}\) direction is pointing in the same direction as the angular momentum.

The unit vectors that define this coordinate system are:

\(\uvec{p}\) pointing along the \(\pf{x}\) axis

\(\uvec{q}\) pointing along the \(\pf{y}\) axis

\(\uvec{w}\) pointing along the \(\pf{z}\) axis

As shown in Fig. 42, \(\uvec{p}\) points along the apse line to the right of the focus, towards periapsis. \(\uvec{q}\) points 90° true anomaly from \(\uvec{p}\). Finally, \(\uvec{w}\) points in the same direction as the angular momentum, and can be defined by:

## Position Vector#

In the perifocal frame, the position vector \(\vector{r}\) may be written in terms of the \(\pf{x}\) and \(\pf{y}\) Cartesian coordinates. Since the orbit is planar, the \(\uvec{w}\) component is zero.

\(\pf{x}\) and \(\pf{y}\) can be transformed into the radial-true anomaly polar coordinate system by:

By plugging in the orbit equation, Eq. (113), \(\vector{r}\) can be written as:

## Velocity Vector#

The velocity is found by taking the time derivative of the position:

Then we need to apply the product rule, because both \(r\) and \(\nu\) are functions of time:

Substituting some of the relationships from previously, we can simplify the velocity vector as: