A star the size of the sun has a mass 8 times as great. It rotates on its axis once every 12 days. It then collapses into a neutron star of radius 10km, losing three quarters of its mass in the process. How long does it take to rotate once after collapse? (it's spere though process, and the loss mass carries no angular momentum).
To find the answer, we need to use the principles of conservation of angular momentum. Since the initial star and the collapsed neutron star are the same object, their angular momentum should remain constant.
Let's break down the problem step by step:
1. Determine the initial angular momentum (L_initial) of the star before its collapse.
Angular momentum (L) is given by the equation: L = (moment of inertia) x (angular velocity)
The moment of inertia (I) of a sphere is given by the equation: I = (2/5) x (mass) x (radius^2)
The angular velocity (ω) is given by the equation: ω = 2π / (time for one rotation)
Given:
- Mass of the star = 8 times the mass of the sun = 8M (where M is the mass of the sun)
- Radius of the collapsed neutron star = 10 km = 10,000 meters
- Time for one rotation of the star before collapse = 12 days = 12 x 24 x 60 x 60 seconds
Let's calculate L_initial:
Moment of inertia (I_initial) = (2/5) x (mass_initial) x (radius_initial^2)
= (2/5) x (8M) x (radius_initial^2) [Since the mass of the initial star is 8 times the mass of the sun]
Angular velocity (ω_initial) = 2π / (time_initial)
= 2π / (12 x 24 x 60 x 60) [Converting days to seconds]
L_initial = I_initial x ω_initial
2. Determine the final angular momentum (L_final) of the collapsed neutron star.
After the collapse, the neutron star loses three-quarters of its mass. Therefore, the mass of the final neutron star is (1/4) times the mass of the initial star.
Let's calculate L_final:
Moment of inertia (I_final) = (2/5) x (mass_final) x (radius_final^2)
= (2/5) x ((1/4) x 8M) x (radius_final^2)
Angular velocity (ω_final) is the unknown we need to find.
L_final = I_final x ω_final
3. Equate L_initial and L_final to find ω_final.
Since angular momentum is conserved, we can set L_initial = L_final and solve for ω_final.
L_initial = L_final
⇒ I_initial x ω_initial = I_final x ω_final
Solve for ω_final:
ω_final = (I_initial x ω_initial) / I_final
4. Calculate the time for one rotation after collapse (time_final).
Using the angular velocity (ω_final) obtained above, we can calculate the time for one rotation after the collapse (time_final).
time_final = 2π / ω_final
Following these steps will allow us to determine how long it takes for the collapsed neutron star to rotate once.