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If we some point mass of mass $M$, in a spherical potential given by $$\Phi(R,z) = \frac{GM}{R},$$ then under what condition would we get a circular orbit, supposing our initial radius from a central body is $R_0$.

Personally, I can't see an elliptical, hyperbolic or parabolic orbit coming out of a spherical potential. How would the total energy and or the angular momentum of the body have to be changed for a circular orbit to arise?

This was left as a remark in one of my old lecture notes and I can't figure out why it would not be circular and what consequentially what conditions make it circular.

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A circular orbit occurs, if the test body has the "right" velocity: $$ v_\text{circ} = \sqrt\frac{GM}r $$ If the velocity is less, the body will enter a non circular elliptical orbit or fall onto the central body (if it has a non zero extension). If the velocity is higher, the body will enter a non circular elliptical orbit or escape in a hyperbolic orbit.

Moreover the direction of the movement, i.e. the velocity vector must be orthogonal to the direction towards the centre of the potential.

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  • $\begingroup$ Direction matters, too: the "right" velocity must be perpendicular to the line connecting the two bodies. $\endgroup$ – rob Jul 28 '16 at 16:43
  • $\begingroup$ @rob: you are right, I edit the answer. Also thanks for editing the formula :) $\endgroup$ – steffen Jul 29 '16 at 10:59
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Planets and comets have elliptical orbits. If the velocity at any point (eg initially) is not tangential, the orbit will not be circular. If the velocity is tangential but $\frac{mv^2}{r}$ is not equal to the gravitational force $G\frac{Mm}{r^2}$ then the orbit will not be circular either.

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