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

In summary, the problem at hand is poorly stated as it does not specify how much angular momentum was carried away by the lost mass, making it impossible to solve. The best course of action would be to ask for clarification, but assuming that the lost mass has zero angular momentum may lead to a solution. However, this assumption is highly unrealistic as the lost mass makes up 85% of the original mass and is likely to carry away most of the angular momentum.
  • #1
rakshashredder
2
0
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?
 
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  • #2
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.
 
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  • #3
phyzguy said:
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.
 
  • #4
rakshashredder said:
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?
 
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  • #5
phyzguy said:
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.
 
  • #6
haruspex said:
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.
 
  • #7
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.
 
  • #8
Vanadium 50 said:
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%
 
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  • #9
rakshashredder said:
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,
 
  • #10
256bits said:
PS - is the star really only 106 km in radius,
I suspect that is supposed to be 10^6 km in radius.
 
  • #11
256bits said:
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.
 
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  • #12
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.
 

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

1. What is a collapsing star?

A collapsing star is a massive celestial object that has used up all of its nuclear fuel and is no longer able to produce enough energy to counteract the force of gravity. This causes the star to collapse in on itself, resulting in a highly dense and compact object.

2. What causes a star to collapse?

The collapse of a star is caused by the depletion of its nuclear fuel. As a star burns through its fuel, it produces energy that counteracts the force of gravity and keeps the star stable. When the fuel runs out, there is no longer enough energy to counteract gravity, causing the star to collapse.

3. What happens during a star's collapse?

During a star's collapse, the core of the star becomes extremely dense and hot. This can trigger a supernova explosion, where the outer layers of the star are blown off into space. The remaining core may become a neutron star or a black hole, depending on its mass.

4. What is the significance of studying collapsing stars?

Studying collapsing stars can provide valuable insights into the life cycle of stars and the fundamental laws of physics. It can also help us better understand the formation of elements and the origins of the universe.

5. Can collapsing stars pose a threat to Earth?

While the collapse of a star can cause a supernova explosion, it is unlikely to pose a direct threat to Earth. However, the radiation from a nearby supernova could potentially harm Earth's atmosphere and affect life on our planet. Fortunately, the nearest known star that could potentially go supernova is over 100 light years away.

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