For two bodies attracting to each other, we use the reduced mass and distance between them (r) to solve the equation of motion. The final result is an elliptical obit under certain conditions.
My understanding is that this obit is the obit of the reduced mass (so a virtual body) rotating about a point (focus) which is in a distance r from it. Is it correct?
In this case, if one of the bodies is much heavier than the other, we can say the heavier body is at the focus and the lighter one is orbiting around it. So the locus of the lighter mass is the same as the locus (orbit) of the virtual body with reduced mass. And this is the 1st Kepler's Law. Is my understanding correct?
If the masses of the two bodies are comparable, then we don't have straight forward information about the loci of the bodies, right? We only know how r (distance between the two bodies) changes but not their individual locus in the laboratory frame, right? And Kepler's 1st Law is not applicable any more (i.e. not rotating about the focus). Is it right? Are their orbits still elliptical?