How Do You Apply Conservation of Angular Momentum to a Collapsing Star?

[rakshashredder]

Homework Statement

A massive star collapses under its own gravity once it’s no longer supported by the pressure produced through nuclear fusion and becomes a much smaller, rapidly rotating neutron star as shown in the sketch. The initial star has a mass of Mi = 20·10^30 kg, a radius of Ri = 106 km and rotates around its own axis with a period Ti = 10 yr. The (final) neutron star has a mass of Mf = 3 · 10^30 kg and rotates with an angular velocity of ωf = 700 rad/s around its own axis. To solve to following questions treat the stars as homogeneous, solid spheres and assume that the angular momentum is conserved.
(Note: 1 year = 31, 536, 000 seconds.)

a)Compute the radius Rf of the neutron star
b)Calculate the ratio Kf/Ki between the final and initial rotational
kinetic energies

Relevant Equations

Ioωo=Ifωf

For part a I know the conservation of angular momentum is used, but I am not sure how to formulate the equation from the information given in the problem. I know that after the equation has been set up you set it up to solve what Rf is. For part be isn't the finial and initial rotational kinetic energies the same?

 

[phyzguy]

The problem is poorly stated. You should ask for a clarification. Most of the mass of the initial star has been lost, and unless you know how much angular momentum was carried away by the lost mass, there is no way to solve the problem.

 

[rakshashredder]

The problem is poorly stated. You should ask for a clarification. Most of the mass of the initial star has been lost, and unless you know how much angular momentum was carried away by the lost mass, there is no way to solve the problem.

I can't ask for clarification at the moment unfortunately. I emailed my TA for clarification, but I haven't gotten a response and I don't think that I'll get a response in time.

 

[phyzguy]

I can't ask for clarification at the moment unfortunately. I emailed my TA for clarification, but I haven't gotten a response and I don't think that I'll get a response in time.

Then you'll need to make an assumption and state what you are assuming. I guess you could assume that the lost mass has zero angular momentum. To answer your question, often angular momentum is conserved but rotational kinetic energy is not because some energy is converted to heat. Can you write the initial and final angular momenta?

 

[haruspex]

The problem is poorly stated. You should ask for a clarification. Most of the mass of the initial star has been lost, and unless you know how much angular momentum was carried away by the lost mass, there is no way to solve the problem.

From what I see online, the sun loses several times as much mass via fusion as via solar wind. Besides, it does say to assume angular momentum is conserved. It is clear this means the angular momentum of the neutron star is the same as that of the original star.

@rakshashredder , you quoted Ioωo=Ifωf. Write the expressions for Io and If. You are given ωo and ωf.

 

[phyzguy]

From what I see online, the sun loses several times as much mass via fusion as via solar wind. Besides, it does say to assume angular momentum is conserved. It is clear this means the angular momentum of the neutron star is the same as that of the original star.

The sun loses only a small amount of mass to either fusion or the solar wind. In this case, 85% of the original mass is lost. Since the original star is spinning and has angular momentum, why is it reasonable to assume that it will shed 85% of its mass and that the 85% which is lost has zero angular momentum? I think this is highly unlikely.

 

[Vanadium 50]

Oh, for heaven's sake. This is an introductory physics question, and not a graduate class on stellar modeling. They don't expect one to consider the stellar wind. Yes, this is unrealistic, just like frictionless surfaces, stretchless ropes, ideal gasses...

I would say: L_final = L_initial + L_in the lost mass
I would then explicitly write L_in the lost mass = 0 and that this is an assumption because the problem did not specify (but is almost certainly what they mean) and solve.

 

[phyzguy]

Oh, for heaven's sake. This is an introductory physics question, and not a graduate class on stellar modeling. They don't expect one to consider the stellar wind. Yes, this is unrealistic, just like frictionless surfaces, stretchless ropes, ideal gasses...

I would say: L_final = L_initial + L_in the lost mass
I would then explicitly write L_in the lost mass = 0 and that this is an assumption because the problem did not specify (but is almost certainly what they mean) and solve.

Assuming the lost mass has zero angular momentum is what I recommended the OP do in Post #4. I was just pointing out that it is unrealistic. If a small amount of mass is lost, OK, but when 85% of the mass is lost, it really doesn't make sense. It's much more unrealistic than frictionless surfaces or ideal gases. The errors in these assumptions are typically small. In this problem, the real answer is that the lost mass probably carried away most of the angular momentum, not a small fraction of it.

Edit: If you want an analogy, it is about like telling a student to apply the ideal gas equation to liquid water. Look at the solar system. It was a collapsing cloud of matter with a certain amount of angular momentum. What percentage of the original angular momentum ended up in the central object (the sun)? Answer: 0.3%

 

[256bits]

To solve to following questions treat the stars as homogeneous, solid spheres and assume that the angular momentum is conserved.
(Note: 1 year = 31, 536, 000 seconds.)

Thus one can calculate the momentum of the lost mass from the original star, by treating it as as shell that s ejected, with the remaining central material forming the neutron star,

Edt: Calculate the amount of lost mass of the ejected material,

PS - is the star really only 106 km in radius,

 

[phyzguy]

PS - is the star really only 106 km in radius,

I suspect that is supposed to be 10^6 km in radius.

 

[haruspex]

Thus one can calculate the momentum of the lost mass from the original star, by treating it as as shell that s ejected, with the remaining central material forming the neutron star,

Edt: Calculate the amount of lost mass of the ejected material,

PS - is the star really only 106 km in radius,

I think we can all agree that is a reasonable way to find the new radius, but, as happens too often, the challenge for the student is to guess what the question setter intended. Note the word "assume". Why should we need to assume angular momentum is conserved unless we are knowingly ignoring a loss or gain somewhere?

Perhaps the best advice is to solve it both ways and present both answers, but that won’t work if it is assessed by software.

 

[256bits]

Yes, The assume seems to be in the wrong place.
Assume that the star(s) are solid and homogeneous would have been a more appropriate statement.