When a massive star (>8 solar masses) dies, its degenerate iron core collapses and forms a neutron star, which rotates very rapidly. Suppose that the innermost 1.5 solar masses of the star collapses to form the neutron star. This region has an initial radius of 10,000 km and an initial rotational velocity of 75 km/sec, and when it collapses into a neutron star it has a final radius of 10 km. Using the conservation of angular momentum, what is the final rotational velocity of the neutron star in km/sec?
To solve this problem, we can use the conservation of angular momentum, which states that the initial angular momentum of the collapsing region of the star is equal to the final angular momentum of the neutron star.
Angular momentum (L) is given by the equation L = Iω, where:
L = angular momentum
I = moment of inertia
ω = angular velocity
The moment of inertia (I) of a sphere is given by the equation I = (2/5)mr^2, where:
m = mass of the object
r = radius of the object
Given:
Initial mass (m_initial) = 1.5 solar masses = 1.5 * (mass of the Sun) = 1.5 * 2 * 10^30 kg
Initial radius (r_initial) = 10,000 km = 10,000,000 m
Initial rotational velocity (ω_initial) = 75 km/sec = 75,000 m/sec
Final radius (r_final) = 10 km = 10,000 m
To find the initial angular momentum (L_initial), we first need to find the initial moment of inertia (I_initial):
I_initial = (2/5) * m_initial * r_initial^2
= (2/5) * 1.5 * 2 * 10^30 * (10,000,000)^2
Next, we can use the equation L_initial = I_initial * ω_initial to find the initial angular momentum.
To find the final angular velocity (ω_final), we can rearrange the equation L = Iω to solve for ω:
ω_final = L_initial / I_final
The final moment of inertia (I_final) can be calculated using the equation I_final = (2/5) * m_final * r_final^2, where:
m_final = 1.5 solar masses = 1.5 * (mass of the Sun) = 1.5 * 2 * 10^30 kg
Finally, we can substitute the values of L_initial and I_final to find ω_final.
Let's plug in the numbers to find the solution:
I_initial = (2/5) * 1.5 * 2 * 10^30 * (10,000,000)^2
= 6 * 10^27 * (10^8)^2
= 6 * 10^27 * 10^16
= 6 * 10^43 kg m^2
L_initial = I_initial * ω_initial
= 6 * 10^43 kg m^2 * 75,000 m/sec
= 4.5 * 10^20 kg m^2/sec
I_final = (2/5) * 1.5 * 2 * 10^30 * (10,000)^2
= 6 * 10^27 * (10^4)^2
= 6 * 10^27 * 10^8
= 6 * 10^35 kg m^2
ω_final = L_initial / I_final
= 4.5 * 10^20 kg m^2/sec / 6 * 10^35 kg m^2
= 7.5 * 10^-16 sec^-1
Therefore, the final rotational velocity of the neutron star is approximately 7.5 * 10^-16 km/sec.