What is the significance of the center of force in orbital mechanics?

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The center of force in orbital mechanics refers to the point from which gravitational forces are calculated, typically the center of mass of a system. When analyzing orbital problems, the concept of reduced mass simplifies the two-body problem into a one-body problem, allowing for easier calculations. In elliptical orbits, the center of mass is located at one of the foci of the ellipse. For spherically symmetric objects, the gravitational force can be treated as if all mass is concentrated at the center of mass, with the distance measured from this point. Understanding the center of force is crucial for accurately modeling orbital dynamics.
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What exactly is the "center of force"?

For instance, you use reduced mass to make an orbital problem one body. Is the center of force the r "seperation between the bodies"? Or is the center of force at the center of mass? In an ellipse, the center of mass is always at one of the foci, right?
 
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When dealing with gravity, a spherically symmetric object can be treated as if the entire mass was at the center of mass, so r in the force of gravity equation is measured from the center of mass.

Your statement about the ellipse is correct.
 
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