Kepler 3rd law for circular orbits

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SUMMARY

Kepler's Third Law applies to two bodies of different masses, M and m, orbiting their common center of mass in circular orbits. Despite having different masses, both bodies share the same orbital period, which leads to the conclusion that they must have the same radius of orbit around the center of mass. However, this interpretation is valid only when considering the system as a whole; analyzing each body individually does not conform to the traditional Kepler problem. Thus, the law holds true for the entire system rather than for individual bodies.

PREREQUISITES
  • Understanding of Kepler's Laws of Planetary Motion
  • Basic knowledge of circular motion and gravitational forces
  • Familiarity with the concept of center of mass
  • Fundamentals of orbital mechanics
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  • Study the implications of Kepler's Third Law in multi-body systems
  • Explore the mathematical derivation of Kepler's Laws
  • Learn about the dynamics of orbits in gravitational systems
  • Investigate the differences between individual and system-wide orbital analysis
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Astronomy students, physicists, and anyone interested in understanding orbital mechanics and the application of Kepler's Laws in celestial dynamics.

Silviu
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Hello! I am a bit confused about Kepler 3rd law for circular orbits. So assume you have 2 bodies of masses M>m orbiting in circles around their center of mass. As the center of mass passes through the line connecting them, they must have the same period. But according to Kepler's 3rd law, this means that, as they rotate around the same center, they must also have the same radius of the orbits. But this is not possible as they have different masses. So how does this work? Thank you!
 
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Kepler's third law holds for the system as a whole with the distance between the objects as the radius.

If you only look at each individual body, you do not have a Kepler problem.
 

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