I know that we have the angular orbital velocity of a satellite in circle orbit as $w_0=\frac{v}{r}$, but I want another way to find out the angular orbital velocity of a satellite in the circle orbit.
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$\begingroup$ See if this helps. $\endgroup$– AJNCommented Oct 5, 2023 at 19:31
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$\begingroup$ This here might be of interest: m.youtube.com/… $\endgroup$– The Rocket fanCommented Oct 5, 2023 at 19:50
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$\begingroup$ This is a good question! In your example $v$ is a scalar velocity and it is the component of the vector velocity perpendicular to the radial direction. There is a good answer to this question as written and it should not be closed. voting to leave open so that that good answer can be posted and not prevented. $\endgroup$– uhohCommented Oct 6, 2023 at 0:35
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$\begingroup$ @uhoh since a circular orbit is specified, all of the velocity is in the tangential/horizontal direction anyway $\endgroup$– Erin AnneCommented Oct 6, 2023 at 0:58
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$\begingroup$ @ErinAnne OP asks for "another way" so the full equation, starting from a state vector would count. Yes it can be reduced to the given equation, but they're not the same thing. $\endgroup$– uhohCommented Oct 6, 2023 at 2:14
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1 Answer
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The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.
You've specified the orbit is circular, so the semi-major axis (typically $a$) is just the orbital radius $r$.
The derivation in the link above from Newton's law of gravitation to Kepler's Third Law is pretty straightforward and I won't reproduce it here. But as the Wikipedia page on Circular orbits that AJN's comment linked to notes, you can rearrange to
$\omega^2 r^3 = \mu$
where $\mu$ is the standard gravitational parameter of whatever the satellite is orbiting. Solve for
$\omega = \sqrt{\mu/r^3}$
Ensure that you pay attention to the units of $\mu$ to get sensible results.
I suspect the close votes would be withdrawn if you're clear about which information you have available to try to derive the angular rate from. Since you utilized the radius in the question this seemed like the obvious place to start.
