Conservation of angular momentum for a nonrigid body

Question

Question:

The sun is not a rigid body but a hot ball of gas. The period of rotation varies from 37 days at the pole to 26 days at the equator. The mean radius of the sun is $7\times 10^8\text{ m}$. Suppose the sun collapses into a neutron star of radius approximately $10^4\text{ m}$. Assume that the neutron star is a spherical ball. Estimate the final period of rotation of the neutron star. You may assume it is a rigid sphere for this approximate calculation.

So I understand how this is a conservation of angular momentum problem and the neutron star side is just $I\omega$ where $I$ is for a solid sphere, but I don't know what to do about the sun side. It seems like it would be an integral of all the disks of the sphere for $I\omega$, but it doesn't say anywhere about whether or not density is uniform throughout the sun and how $\omega$ varies (linearly, by curve of the sphere?). It seems like density should be uniform, so I could probably figure out the I using that. Then I thought I could use the average value of $\omega$ and just multiply it with the integral for $I$, but I can't do this since I don't know the function for $\omega$ so I wouldn't be able to integrate it along the interval and then divide by the interval. Can anyone help me with this? Thanks!

Answer 1

It just tells you to "estimate" the result and allows you to "assume the Sun is a rigid sphere" (and I am not sure whether they really mean a "sphere" or a "ball": the difference in the results won't be radical) so all your excuses not to calculate the result are explicitly refused by the very formulation of the problem! Just consider the Sun as another rigid sphere and use 31 days as the average periodicity! Obviously, they don't tell you the precise profile of the velocity as the function of the latitude so you can't do any better.