Sun's rotation rate if it collapsed to a 10 km neutron star

[Vitani11]

Homework Statement

The sun is a pretty typical star with a mass of 1.99x10³⁰kg and a radius of 6.69x10⁸ m. Since it isn't solid, it doesn't rotate uniformly, but has an average rotation rate of 1rev/25d. A star with a mass about about three times that of the Sun eventually explodes as a supernova and leaves behind its collapsed remnants - a neutron star or a black hole. Neutron stars masses are somewhat lower than the masses of the original stars, but have much smaller radii, less than 10 km. Newly-collapsed neutron stars emit beams of radio waves; since they rotate so quickly, the beams may intersect Earth at regular intervals. To produce the observed pulsar rates, the rotation rates must vary from 1rev/s to 1000rev/s.

What would be the approximate rotation rate of the Sun if it became a neutron star with a radius of 10km? Assume it is spherical, with a uniform mass distribution, and that its moment of inertia if 0.059MR² (Since it is not a solid you can not use I for a solid sphere).

Homework Equations

I₁ω_o=I₂ω_f conservation of angular momentum
ω_avg = 1rev/25d
ω₀ = 1rev/(25x2x6.69x10⁸m) = 2.99x10^-11 rev/s
I₁=0.059(1.99x10³⁰kg)(6.69x10⁸m)²
I₂ = 0.059(1.99x10³⁰)(10m)²
ω_f = ?

The Attempt at a Solution

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I used I₁ω_o=I₂ω_f and solved for ω_f. After I solved for this I got 133,800rev. I then divided this final answer by 25d, or 25x2x10km since the suns radius is now that of a neutron star instead of it's original 6.69x10⁸ m. This gave me a value of 267.7 rev/s, which according to the problem statement makes sense. I'm not exactly confident in my method though so I would like to know if this is correct.

 

[Vitani11]

I used I₁ω_o=I₂ω_f and solved for ω_f. After I solved for this I got 133,800rev. I then divided this final answer by 25d, or 25x2x10km since the suns radius is now that of a neutron star instead of it's original 6.69x10⁸ m. This gave me a value of 267.7 rev/s, which according to the problem statement makes sense. I'm not exactly confident in my method though so I would like to know if this is correct.

 

[haruspex]

1rev/(25x2x6.69x10⁸m)

Why are you dividing the rate by the radius?

I₂ = 0.059(1.99x10³⁰)(10m)²

Having collapsed to a neutron star, it is no longer a ball of gas.
Also, the new radius is 10km, not 10m.

 

[Vitani11]

Okay am I supposed to assume that a neutron star is to be regarded as a point mass? sphere? Also what is d? I saw it as a rotation rate as a function of radius which is why I did that

 

[gneill]

Okay am I supposed to assume that a neutron star is to be regarded as a point mass? sphere? Also what is d? I saw it as a rotation rate as a function of radius which is why I did that

A 10 km radius point would be a good trick Treat it as a solid sphere. The "d" in 25d is meant to be the units, "days". You could have googled "sun rotation" to pick up on that.

 

[Vitani11]

Okay. Now I'm getting a number that disturbs me. 30,565 revolutions per second?!

 

[gneill]

Okay. Now I'm getting a number that disturbs me. 30,565 revolutions per second?!

Looks a bit high (a couple of orders of magnitude). Check your calculations and see if you've picked up a factor of 100 somewhere (say via unit conversions or mucking some power of ten exponent work).

Failing that, present some details of your calculation steps.

 

[Vitani11]

Beautiful thank you. I got the correct answer which I'm confident in. It's reassuring (but also not) that the issue lies in the algebra and interpretation rather than the physics, lol