About Kepler's 2nd Law

Question

Kepler's 2nd Law states that the area swept by the line joining the Sun and the Earth is constant per unit time. And here the Sun is at fixed position. (first statement)

How about the area swept by the line joining the earth and the center of mass (or by the line joining the sun and the center of mass)? (second statement)

I went over the derivation and it seems that, based on constant angular momentum of the reduced mass, one can only prove the "first statement". However, I also see that in binary stars, this law is applicable to individual star in the system about the center of the mass e.g.

http://www.astro.cornell.edu/academics/courses/astro201/kepler_binary.htm

So is the second statement also correct? How to prove it (or deduce from the first statement)?

Answer 1

Yes, the second statement is also correct; it's just a special case of the result in the binary star link you posted where $M_{sun} \gg M_{planets}$ and so the center of mass of the system can be approximated as the center of the sun. The following link gives a proof of Kepler's 2nd Law for each mass in a binary system from Newton's laws (there are pointers to a particular textbook given on that page for more details):

http://www.physicspages.com/2015/04/20/keplers-laws/