How to calculate velocity vector in perifocal coordinates [closed]

Well the formula for it is here:

Though if r is the radius and f is the true anomaly (which I assume is radians), then what is r dot and f dot respectively?

• Thank you for digging in and posting an answer to your own question! Once you've finished adjusting the MathJax there you can come back here and add the MathJax to your question. Welcome to Stack Exchange!
– uhoh
Aug 16 '21 at 0:27

The formula for velocity vector in perifocal actually very simple, no rotation matrices required:

$$\mathbf{v_{perifocal}} = \begin{bmatrix} -\sqrt{\frac{\mu}{p}} \sin f \\ \sqrt{\frac{\mu}{p}} (e + \cos f) \\ 0 \\ \end{bmatrix}$$

the angular/orbital momentum is equal to the radius squared times by $$\dot{f}$$. As shown here (make sure to take the length of the orbital momentum vector):

$$h = r^2 \dot{f}$$

So we can re-arange the equation to find $$\dot{f}$$ :

$$\dot{f} = \frac{h}{r^2}$$

We get r dot by differentiating the polar equation. The formula for $$\dot{r}$$ is this :

$$\dot{r} = \sqrt{\frac{\mu}{p}} (e \sin f)$$

where $$p$$ is the semi-lactus rectum, $$\mu$$ the standard gravitational parameter and $$f$$ is the true anomaly.

If you dont want to use the orbital momentum vector in your calculation you can now simply re-arange the previous equation which will give your the formula for $$\dot{f}$$ f dot:

$$\dot{f} = \frac{\sqrt{\frac{\mu}{p}} (1 + e \cos f)}{r}$$