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I was thinking about orbital velocities, and came across this question (Velocity of satellites greater than required velocity). Does the answer to this question imply that for planets going round the sun, or satellites going round a planet, the eccentricity of the orbit depends on the initial velocity of the orbiting body when it is first captured by the larger body? That the closer the initial velocity being equal to the actual orbital velocity, the more circular the orbit?

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The concept of a large celestial body passively "capturing" a smaller one is a bit of a misconception. Imagine a body outside the solar system that by chance heads toward the sun, and begins to feel appreciable gravitational pull from it. It accelerates toward the sun to the point of closest approach, but at that point, it has enough speed to just coast out of the solar system again, even though it's being pulled back in by the sun. Unless the body actively slows down by aerobraking or firing thrusters, it will not remain in orbit around the sun if it didn't start there. In absence of non-conservative forces, orbits are reversible in time. If you could start outside the solar system and fall into a stable orbit around the sun without any active maneuvering, that would imply that you could do the reverse - begin in a stable orbit around the sun and then fling yourself out of the solar system without expending any effort. This spontaneous exit from orbit is clearly and intuitively impossible, but it's fundamentally the same as the reverse of spontaneously entering orbit.

So, we can see that planets in stable orbit around the sun didn't get there by wandering through the universe until they got close to the sun and then just falling into orbit. Instead, they coalesced slowly from a pre-solar system nebula. The idea of a fully-formed planet being "captured" by the sun and falling into orbit is not a good way to think about why planets are in the orbits they are in.

Comets, on the other hand, are good examples of bodies that somehow got kicked into orbit around the sun. Some comets come from the Oort cloud, a sphere of small objects in roughly circular orbits around the sun, far beyond Neptune. If a gravitational interaction with a nearby body significantly changes the velocity of another body, it might find itself in a significantly altered orbit, which may be an ellipse with the sun at one focus.

But in general, an orbit's eccentricity is related to the orbital velocity and position. For a particular orbital height, there is one velocity that will yield a circular orbit, and larger derivations from that will result in larger eccentricity.

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