# Finding the moment of inertia of a neutron star

## Homework Statement

The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light years from the earth (the figure (Figure 1) ). It is the remnant of a star that underwent a supernova explosion, seen on earth in 1054 a.d. Energy is released by the Crab Nebula at a rate of about 5×10^31W, about 105 times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every 0.0331 s, and this period is increasing by 4.22×10^−13s for each second of time that elapses.

1)If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star.

2)Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers.

3)Assume that the neutron star is uniform and calculate its density.

## Homework Equations

as far as relevant equations go i would imagine you need τ=Iω, K=1/2Iω2for part 1)... as for 2) and 3) im not entirely sure. keep in mind I is moment of inertia

## The Attempt at a Solution

so far i've tried using the kinetic energy formula to find I....i.e. (assuming energy emitted by the nebula is equal to energy loss by the neutron star): 5x10^31=1/2Iω2. i found ω by 2π/T=2π/0.0331=189.8 rad/s.
then, 1x10^32=I*(189.8)^2. isolating for I, i got I=2.78x10^27...which is wrong apparently.

as for parts 2) and 3), i can't really being until i have part 1)....

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mfb
Mentor
so far i've tried using the kinetic energy formula to find I....i.e. (assuming energy emitted by the nebula is equal to energy loss by the neutron star): 5x10^31=1/2Iω2.
Be careful here. What is on the left side? In particular, what units do you have there? What is on the right side?
Can it make sense to set them equal?

Another hint that you did something wrong: you did not use the increasing period at all.

1x10^32=I*(189.8)^2
What is that?

gneill
Mentor
The formula ##KE_r = \frac{1}{2} I \omega^2## is for the total rotational kinetic energy of the rotating body. As such it will have units of Joules. Your radiated energy figure is a rate, energy per unit time, or units of Watts.

So, you'll have to do a little calculus to relate quantities. I'd start with the expression for the angular frequency given the period and head towards finding the angular acceleration. It will help if you identify the given information with their differential forms. For example, the rate of change of the rotation period with respect to time is just dT/dt...

EDIT: Oops. mfb beat me to it!