Perifocal Frame

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 $x_{ω} y_{ω}$ plane of the perifocal frame lies in the same plane as the orbit. This means that the $z_{ω}$ direction is pointing in the same direction as the angular momentum.

The unit vectors that define this coordinate system are:

$\hat{p}$ pointing along the $x_{ω}$ axis
$\hat{q}$ pointing along the $y_{ω}$ axis
$\hat{w}$ pointing along the $z_{ω}$ axis

../_images/definition-of-perifocal-frame.svg — Fig. 42 The definition of the perifocal frame. The $\hat{w}$ direction is pointing direction towards the viewer.#

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

(164)#

\hat{w} = \frac{h}{h}

Position Vector#

In the perifocal frame, the position vector $r$ may be written in terms of the $x_{ω}$ and $y_{ω}$ Cartesian coordinates. Since the orbit is planar, the $\hat{w}$ component is zero.

(165)#

r = x_{ω} \hat{p} + y_{ω} \hat{q}

$x_{ω}$ and $y_{ω}$ can be transformed into the radial-true anomaly polar coordinate system by:

(166)#

\begin{aligned} x_{ω} & = r \cos ν & y_{ω} & = r \sin ν \end{aligned}

By plugging in the orbit equation, Eq. (113), $r$ can be written as:

(167)#

r = \frac{h^{2}}{μ} \frac{1}{1 + e \cos ν} (\cos ν \hat{p} + \sin ν \hat{q})

Velocity Vector#

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

(168)#

v = \dot{r} = \dot{x_{ω}} \hat{p} + \dot{y_{ω}} \hat{q}

Then we need to apply the product rule, because both $r$ and $ν$ are functions of time:

(169)#

\begin{array}{r} \begin{aligned} {\dot{x}}_{ω} & = \dot{r} \cos ν - r \dot{ν} \sin ν \\ {\dot{y}}_{ω} & = \dot{r} \sin ν + r \dot{ν} \cos ν \end{aligned} \end{array}

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

(170)#

v = \frac{μ}{h} [- \sin ν \hat{p} + (e + \cos ν) \hat{q}]

Perifocal Frame

Contents

Perifocal Frame#

Position Vector#

Velocity Vector#