Keplar's First Law: Explaining Orbit and Eccentricity

[Luca 123]

I know that Keplar's First law is true, but it doesn't occur to me why do stars orbit the center of gravity. Is there a proof for that, or a way for me to visualise why?
Also, to measure the eccentricity of the orbit, e^2=1+(2E(L^2))/(GM(m^2)). How did they derive this equation?

 

[Simon Bridge]

The answer to both your questions is the same: derive from Newton's Law of Gravitation.
The derivations are available online. i.e. for your second question:
http://scienceworld.wolfram.com/physics/Eccentricity.html

 

[Luca 123]

The answer to both your questions is the same: derive from Newton's Law of Gravitation.

Then how exactly did we derive them? Is there some links available showing the derivation? Or if you are willing to, maybe you can show me how?

 

[Simon Bridge]

Then how exactly did we derive them? Is there some links available showing the derivation? Or if you are willing to, maybe you can show me how?

"We" didn't derive them - I did, but you need to ... it is a common and worthwhile exercise for students.
There are plenty of derivations and proofs online - it is also a wortwhile exercise to lear to use google to find this sort of thing.

The first should be sensible to you anyway - consider: if both objects have equal masses, then how else would they behave?

The second one I gave you in a link in post #2 (while you were writing #3 :) ) as an example.
I found it by using the seach term "orbit eccentricity from energy".

 

[Coffee_]

For a two mass system, each orbits the center of mass in an ellipse but from the frame of one of the masses the other one orbits in an ellipse (albeit a different one) as well.

 

[Vanadium 50]

Please. "Kepler's".