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Suppose we have a central mass $M$ and a smaller mass $m$ orbiting around the central mass in an ellipse: enter image description here The other point is the other focus. We know that elliptical orbits have the central mass in one of the two foci of the ellipse, I put the other one to emphasize this fact.

Also I know that in the case of the circle the formula is easy to derive. $a_c = \frac{v^2}{r}$, in other words the velocity needs to be such that its magnitude squared over the radius is equal to the centripetal acceleration, and its direction perpendicular to the acceleration. If there is any other component to the velocity, it would stop being in a uniform or circular orbit. I know from the law of universal gravitation that the centripetal acceleration is $a_c = \frac{\frac{GMm}{r^2}}{m} = \frac{GM}{r^2}$ If we put that into $a_c = \frac{v^2}{r}$, we get

$$ \frac{GM}{r^2} = \frac{v^2}{r} $$ $$ \therefore v = \sqrt{\frac{GM}{r}}$$

However for the ellipse, how can we conclude that the velocity needs to be $\sqrt{\frac{GM}{r}}$ at any point? We can't make the assumption that $a_c = \frac{v^2}{r}$ because we are no longer considering uniform circular motion. What can be done?

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    $\begingroup$ You can find the velocity in an elliptical orbit using the vis-viva equation, $$v^2 = GM\left(\frac2r - \frac1a \right)$$ I have some info about it in this recent answer: physics.stackexchange.com/a/675868/123208 Also, physics.stackexchange.com/a/676872/123208 $\endgroup$
    – PM 2Ring
    Commented Nov 30, 2021 at 10:51
  • $\begingroup$ Another thing that I want to confirm. If there are two elliptical orbits which intersect at particular points, and two masses are orbiting via those ellipses, they don't necessarily need to have the same velocity at those intersection points, right? i.e. as 1/a is different for the two orbits, right? $\endgroup$ Commented Nov 30, 2021 at 11:02
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    $\begingroup$ Correct. They have the same $r$, but different $a$, so they also have different $v$. $\endgroup$
    – PM 2Ring
    Commented Nov 30, 2021 at 11:05
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    $\begingroup$ Similarly, if a spacecraft is coasting on an elliptical orbit with a certain $v$, but then fires its engines to give it a new $v$ you can easily calculate the $a$ of the new ellipse. $\endgroup$
    – PM 2Ring
    Commented Nov 30, 2021 at 11:08

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