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Well the formula for it is here:

enter image description 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?

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  • $\begingroup$ 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! $\endgroup$
    – uhoh
    Aug 16 '21 at 0:27
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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:

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

More info about them can be found on this lecture:

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    $\begingroup$ There are 3 images in that one imgur link, I didnt do 3 seperate ones since I had lots of chrome tabs open at the time (still do now) . I never knew you could import images directly into stack exchange, I will import them as soon as I get the time which should be in a few hours, thanks. $\endgroup$
    – Sam
    Aug 15 '21 at 23:24
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    $\begingroup$ @uhoh I finished! Unfourtanetly when drawing out the formulas in MS paint the size of the images are disproportionately large compared to other screenshots. $\endgroup$
    – Sam
    Aug 16 '21 at 0:04
  • $\begingroup$ Thanks a lot. Would it be appropiate to remove the embedded images since you have provided the MathJax to it? $\endgroup$
    – Sam
    Aug 16 '21 at 0:52
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – Sam
    Aug 16 '21 at 18:39

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