Two-Body Problem: Homework Statement, Equations & Attempted Solution

[unscientific]

Homework Statement

Homework Equations

The Attempt at a Solution

What puzzles me is part (c), I got V_min > E_tot which suggests that the orbit is no longer bounded..

 

[Simon Bridge]

So at some point the two new stars are inside an expanding shell of gas (etc) from the explosion so we only need to deal with the final fields of the two stars?

The second star (the one that did not blow up) may gain mass but does not change momentum from material that hits it?

What is wrong with the final system being unbound?

 

[unscientific]

So at some point the two new stars are inside an expanding shell of gas (etc) from the explosion so we only need to deal with the final fields of the two stars?

The second star (the one that did not blow up) may gain mass but does not change momentum from material that hits it?

What is wrong with the final system being unbound?

The question gives no information about whether the explosion affects angular momentum/energy or not...I'm assuming angular momentum and energy stays constant.

In part (c) of the question I'm asked to show that the binary star remains bound...

 

[ehild]

Think: What does it mean that the exploding star "suffers a spherically symmetric loss of mass"? Does its momentum change in the explosion?
Just after the explosion, the the stars are still at the same place they were- The momentum of the smaller one is unchanged, what about the momentum of the bigger one? Did that symmetric mass loss change its momentum?

Calculate the energy of the two starts after the explosion, taking into account the change of volume. ehild

 

[unscientific]

Think: What does it mean that the exploding star "suffers a spherically symmetric loss of mass"? Does its momentum change in the explosion?
Just after the explosion, the the stars are still at the same place they were- The momentum of the smaller one is unchanged, what about the momentum of the bigger one? Did that symmetric mass loss change its momentum?

Calculate the energy of the two starts after the explosion, taking into account the change of volume. ehild

i would assume that the mass just halved instantaneously, but still retains its original speed? Not sure what volume has to do here

 

[ehild]

i would assume that the mass just halved instantaneously, but still retains its original speed? Not sure what volume has to do here

What else can can you assume if the explosion is spherically symmetric?
Just after the explosion, you have the half star and a spherical shell around it. The CM of the shell coincides with the centre of the star, and travels with the initial velocity of the star, while the shell itself can expand with a high speed. You are left with two stars at distance r₀ from each other, having the same velocities v₁ and v₂ as before the explosion. The new CM of the two-star system travels in opposite direction as the shell, but that does not influence if the starts stay bounded to each other or not.

The energy of this binary star system is 1/2 m₀(v₁²+v₂²)-Gm₀²/r₀. Substitute v₁ and v₂ from the solution of the original system.

ehild

 

[unscientific]

What else can can you assume if the explosion is spherically symmetric?
Just after the explosion, you have the half star and a spherical shell around it. The CM of the shell coincides with the centre of the star, and travels with the initial velocity of the star, while the shell itself can expand with a high speed. You are left with two stars at distance r₀ from each other, having the same velocities v₁ and v₂ as before the explosion. The new CM of the two-star system travels in opposite direction as the shell, but that does not influence if the starts stay bounded to each other or not.

The energy of this binary star system is 1/2 m₀(v₁²+v₂²)-Gm₀²/r₀. Substitute v₁ and v₂ from the solution of the original system.

ehild

Yes, but the problem is the total energy comes out as less than the effective potential, implying a non-bound state..

 

[ehild]

I do not know what you call effective potential. If you think that the angular momentum of the new two-star system with respect to the original CM is the same as the initial angular momentum, you need to prove it.

I suggest to ask your teacher what is the meaning of "spherically symmetric loss of mass"

ehild

 

[ehild]

I see now that you added the angular term of KE to the potential energy, and called the sum "effective potential". Stay with the definition of KE as the sum of 1/2 mv² of both stars, using their velocities with respect to the new centre of mass.

ehild

 

[unscientific]

I see now that you added the angular term of KE to the potential energy, and called the sum "effective potential". Stay with the definition of KE as the sum of 1/2 mv² of both stars, using their velocities with respect to the new centre of mass.

ehild

I used the wrong angular momentum; i used the angular momentum before the collision and put it in the new energy equation..