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All orbits are elliptical and special orbits are circular (like when both bodies have the same mass or when the second body's mass is negligibly small). In a circular orbit centrifugal force (which I know is just an apparent force, not a real force, but here represents the force felt due to inertia; you get the picture, and save your time admonishing me) is always equal to the gravitational force.

Assumption 1: if the "centrifugal force" equals the gravitational force, then a circular, not elliptical, orbit should be implied and follow.

An elliptical orbit

In this picture (of an elliptical orbit), assume the body is orbiting clockwise.

Assumption 2: On arc ABC, kinetic energy is decreasing while potential energy is increasing. So in this segment of orbit, "centrifugal force" is higher than gravitational force. And in arc CDA, K.E. is increasing while P.E. is decreasing, so gravitational force should be greater than "centrifugal force."

Assumption 3: If assumption 2 is correct, then at points A and C (when the force-tables turn) "centrifugal force" should equal gravitational force.

Discrepancy: If both aforementioned forces are equal at points A and C that would imply a circular orbit, which is not the case.

Either assumption 1 is wrong or I don't understand kepler orbits as well as I thought I did. What's the case

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There are several inaccuracies in your post.

First - you don't need equal mass (or one much smaller than the other) for a circular orbit: you need the least amount of energy for a given angular momentum. You can have unequal masses in circular orbit (about their center of gravity).

Second - while the force of gravity is pulling towards the focus of the ellipse, the direction of the orbit need not be perpendicular to the radial vector. When it is not, then part of the force will "slow down" or "speed up" the object in orbit, and part of it will "bend" the direction:

enter image description here

In this case, the red component is the one that is providing centripetal force, and the green component is accelerating the particle in orbit.

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  • $\begingroup$ Yes, when the vector of orbit is not perpendicular to the focus of the ellipse (the gravitational center), the object in orbit will be speeding up/slowing down, but when the forces ARE aligned, centripetal force equals force due to gravity. Wouldn't that set up a circular orbit? $\endgroup$ – Jean Valjean Feb 17 '16 at 22:54
  • $\begingroup$ @JeanValsean; Not if the satellite is traveling fast. In general, newtonian orbits are conic sections, like a hyperbola. In fact, you've heard of "gravity assist" to accelerate spacecraft? That's when a fast-moving object loops past a planet and picks up speed from it. In the planet's frame, the orbit is a hyperbola. $\endgroup$ – Mike Dunlavey Feb 17 '16 at 22:58
  • $\begingroup$ @JeanValsean - no. While the curvature at the point you describe is entirely due to gravity, the radius of curvature is not in general equal to the distance. This means the object either curves "more than a circular orbit" (perigee) or less (apogee). $\endgroup$ – Floris Feb 17 '16 at 23:02

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