Kepler's First Law Revisited 

Consider a binary star. The gravitational force from the primary star keeps its companion orbiting. But the primary must also feel the pull from the companion. So it must orbit the companion, right?

Actually, each star is on an orbit, but not around each other. They orbit the center of mass of the system.

Center of mass diagram

 

Defining the center of mass:

M1r1 + M2r2 = (M1+M2)rcom


(important note: r1, r2, and rcom are all vectors!)
 
and now let's set the position of the center of mass at the origin (rcom=0). Then we have
M1r1 + M2r2 = 0
or
M1/M2 = -r2/r1

In other words, the distances r1 and r2 can change (ie., r can change), but the ratio r2/r1 is fixed. Also, the center of mass always lies on the line connecting the two stars.

So here's that binary star orbit:


 


The gravity of the Sun makes the Earth (and planets) move in an orbit. But gravity is symmetric, so the Sun must feel the Earth's gravity. Does the Sun also then move on an orbit?
 
 

So Kepler's First Law, revised:

Each planet moves on an elliptical orbit, with the center of mass at one focus.