Binary Star Explosion: Calculating Total Energy and Period

[bon]

Homework Statement

Two individual stars in a binary system (m1=mo, m2=2mo) are in circular orbit about their common centre of mass and are separated by a distance ro. At some stage, the more massive star explodes - resulting in the two stars having equal mass after the explosion

(a)Calculate the total energy, angular momentum and period of the binary star as viewed in the ZMF before the collision.

(b)Calculate the total energy and period of the equivalent single particle system orbit, before the explosion.

(c) Using the equivalent single particle system or otherwise, show that the binary star will remain bound after the explosion.

Homework Equations

The Attempt at a Solution

(a)

So is the total energy just the sum of the energy for each star?

I defined PE to equal 0 at one star and found total energy = -Gmo^2/ro

How do you do it for angular momentum? Just find the two individually, note they are in opposite direction, and add?

I worked out T = 2/3 pi ro / (1/3 Gmo/ro)^1/2 Is this right?

(b) I am using F int = ur'' where u is reduced mass, r relative position vector..

Using this method i get E total to be -1/3 Gmo^2/ro..which isn't equal to the above, why?

Also I get T = 2/3pi ro / (Gmo/ro)^1/2 which again isn't equal to the above.. :S

(c) Not sure how to do this...

Thanks!

 

[ehild]

Start with the two individual bodies first. They interact, and there is no external force. Choose a system of reference with the centre of mass as origin.

Both planets orbit around the CM along their individual circles, but with the same angular velocity. What are the radii? What is that angular velocity?

What is the KE of the whole system?

As for the potential energy of a system of material points, take into mind, that the negative gradient of the potential energy is the force, and the gradient is a function of the coordinates of both bodies. It can be shown for this binary system that PE =-(G m1 m2 )/r0. It is zero when the stars are at infinite distance apart, not at one star, where r0=0.

The angular momentum is m r^2 w at a circular orbit, but it is not opposite for the individual stars. Would the CM stay in rest if they move in opposite direction?

As for the equivalent single particle system, you have a "reduced" or "effective" mass u= 2/3 m0, orbiting around a mass of 3 m0. Calculate angular velocity, energy and angular momentum according to this.

ehild

 

[bon]

Ah ok this is really helpful thanks. For now, just stuck on one bit of what you say about potential energy.

So i know that for the start with mass 2mo, F = Gmo(2mo)/ro^2

so the PE is the -grad of this.but how do i take -grad of F? I haven't done grad in polar coords yet.. :S

Thanks

 

[zachzach]

Ah ok this is really helpful thanks. For now, just stuck on one bit of what you say about potential energy.

So i know that for the start with mass 2mo, F = Gmo(2mo)/ro^2

so the PE is the -grad of this.but how do i take -grad of F? I haven't done grad in polar coords yet.. :S

Thanks

No, the FORCE can be obtained by taking the gradient of the potential energy. Taking the gradient makes a vector from a scalar and force is a vector and potential energy is a scalar.