Orbital Energy of a binary star system

[Aquafina20]

Homework Statement

Say you have a binary star sytem. Both stars have mass M and semimajor axis a. The orbits are extremely eccentric (e is approximately 1). How would you describe the energy of the system?

Homework Equations

SEE BELOW

The Attempt at a Solution

Basically I'm very unconfident about my answer and feel like I'm blanking on basic physics. I imagined the stars both at apocenter at some initial time. With such an eccentric orbit the velocities would be nearly zero here, so the total energy of each stars orbit would be quite nearly all potential energy.

-Gmm/r where r is distance to the barycenter or 2a so the potential energy of each star is -(Gm^2)/2a. So the sum of the PE's would be 2 times this amount and therefore the total energy of the orbit is -(Gm^2)/a

Is this valid?

 

[Astronuc]

Essentially zero velocity at apastron would be a very special case.

This might be of use - http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm

http://www.astro.cornell.edu/academics/courses/astro201/bin_orbits.htm
http://filer.case.edu/sjr16/advanced/stars_binvar.html

See this page - http://csep10.phys.utk.edu/astr162/lect/binaries/visual.html - which also has an applet.

Should the two stars have the same initial angular momentum?

I was trying to think of a way to generalize circular binary orbits - e.g.
http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#bo

 

[Aquafina20]

Essentially zero velocity at apastron would be a very special case.

This is understandable, but would it not be possible to have orbits so eccentric that apocenter velocity would be negligable in contribution to energy? Do not be concerned with practicality or realism. I imagine this being an approximate model for a collision scenerio with both objects starting at rest where the exaggerated eccentricity allows us to throw the objects as nearly directly towards one another as an orbit can allow.

 

[Astronuc]

If the kinetic energy was negligible, then at apastron, the total energy (neglecting the rotational energies of the stars) would simply be the gravitational potential energy at that distance/separation.

The java applets show that e=1 is extremely eccentric with little overlap of the orbits.

But I was wondering about a more general case. I think one is assuming not only the same mass, but same eccentricity and angular momentum, i.e. the same (or symmetric) orbital parameters for both stars.

 

[Aquafina20]

If the kinetic energy was negligible, then at apastron, the total energy (neglecting the rotational energies of the stars) would simply be the gravitational potential energy at that distance/separation.

Would -GM^2/2a be an acceptable description of total energy or am i misinterpreting center of mass? Could you not say the two masses are m/2 and 2m and turn it into a one body with one oject stationary at the barycenter and the other following the same orbit in the two body?

But I was wondering about a more general case. I think one is assuming not only the same mass, but same eccentricity and angular momentum, i.e. the same (or symmetric) orbital parameters for both stars.

Well, wouldn't that be implicit in having a two body problem where m1=m2=m?

 

[Astronuc]

Well, wouldn't that be implicit in having a two body problem where m1=m2=m?

No - why would it? In addition to m1 = m2, one would need m₁v₁r₁ = m₂v₂r₂, no? And the two mass would have to be in phase in their orbits.

Orbital mechanics in not my specialty, and it's been a long time since I've sat down and worked through such material.