An extremely dense neutron star with mass equal to that of the Sun has a radius of about 10 km −about the size of Manhattan Island. These stars are thought to rotate once about their axis every 0.03 to 4 s, depending on their size and mass. Suppose that the neutron star described in the first sentence rotates once every 0.040 s
If the volume of the neutron star then expanded to occupy a uniform sphere of radius 1.4×108m (most of the Sun's mass is in a sphere of this size) with no change in mass or rotational momentum, what time interval would be required for one rotation? By comparison, the Sun rotates once about its axis each month.
To determine the time interval required for one rotation of the neutron star after it has expanded to a sphere with a radius of 1.4x10^8 m, we can use the principle of conservation of angular momentum.
The formula for angular momentum is given by:
L = Iω
Where:
L = Angular momentum
I = Moment of inertia
ω = Angular velocity
Since there is no change in mass or rotational momentum, the angular momentum remains constant.
The moment of inertia (I) for a uniform sphere is given by:
I = (2/5) * m * r^2
Where:
m = Mass of the sphere
r = Radius of the sphere
Let's calculate the initial moment of inertia (I_initial) for the neutron star, assuming its mass is equal to that of the Sun (M_sun = 1.989 x 10^30 kg) and its radius is 10 km (r_initial = 10,000 m).
I_initial = (2/5) * M_sun * r_initial^2
= (2/5) * (1.989 x 10^30 kg) * (10,000 m)^2
Next, let's calculate the final moment of inertia (I_final) for the expanded sphere with a radius of 1.4x10^8 m.
I_final = (2/5) * M_sun * r_final^2
= (2/5) * (1.989 x 10^30 kg) * (1.4x10^8 m)^2
Since the angular momentum is conserved, we can equate the initial and final angular momentum.
I_initial * ω_initial = I_final * ω_final
Solving for ω_final, we get:
ω_final = (I_initial * ω_initial) / I_final
Now, substitute the given values:
ω_initial = 2π / 0.040 s (since the neutron star rotates once every 0.040 s)
I_initial = (2/5) * (1.989 x 10^30 kg) * (10,000 m)^2
I_final = (2/5) * (1.989 x 10^30 kg) * (1.4x10^8 m)^2
Calculate ω_final using the above formula,
ω_final = (I_initial * ω_initial) / I_final
And the time interval required for one rotation can be calculated as:
T_final = 2π / ω_final
Performing the calculations will give us the desired result.
To solve this problem, we need to find the time interval required for one rotation of the expanded neutron star. We are given that the original neutron star rotates once every 0.040 seconds.
We can use the principle of conservation of angular momentum to solve this problem. Angular momentum is given by the equation:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Since the mass and rotational momentum of the neutron star remain constant when it expands, its moment of inertia (I) will remain the same.
The moment of inertia of a uniform sphere can be given by the equation:
I = (2/5) * m * r^2
Where m is the mass of the sphere and r is its radius.
First, let's calculate the moment of inertia of the original neutron star using the given radius of 10 km:
I_original = (2/5) * m * (10,000 m)^2
Next, let's calculate the moment of inertia of the expanded neutron star using the radius of 1.4×10^8 m:
I_expanded = (2/5) * m * (1.4×10^8 m)^2
Since the mass of the neutron star does not change, we can equate the two equations:
I_original = I_expanded
Solving for m:
(2/5) * m_old * (10,000 m)^2 = (2/5) * m_new * (1.4×10^8 m)^2
Simplifying:
m_old = m_new * ((1.4×10^8 m)^2 / (10,000 m)^2)
m_old = m_new * 1.96×10^12
Now, we need to find the new time interval for one rotation. Let's denote it as T_new. We know that the angular momentum remains constant, so:
L_original = L_expanded
I_original * ω_original = I_expanded * ω_expanded
Since we are looking for the new time interval, we need to find ω_expanded in terms of T_new. ω_expanded can be calculated as:
ω_expanded = (2π / T_new)
Substituting the values into the equation:
I_original * ω_original = I_expanded * ω_expanded
I_original * ω_original = I_expanded * (2π / T_new)
Substituting the expressions for the moments of inertia:
(2/5) * m_old * (10,000 m)^2 * ω_original = (2/5) * m_new * (1.4×10^8 m)^2 * (2π / T_new)
Canceling the common terms:
(10,000 m)^2 * ω_original = (1.4×10^8 m)^2 * (2π / T_new)
Simplifying:
ω_original = (1.96×10^12) * ω_new
Substituting the value of ω_original:
(10,000 m)^2 * (1.96×10^12) * ω_new = (1.4×10^8 m)^2 * (2π / T_new)
Canceling the common terms:
(10,000 m)^2 * (1.96×10^12) * ω_new = (1.4×10^8 m)^2 * (2π / T_new)
Simplifying:
(1.96×10^16) * ω_new = (1.96×10^16) * (2π / T_new)
Cancelling (1.96×10^16) from both sides:
ω_new = 2π / T_new
Finally, substituting the value of ω_new and solving for T_new:
2π / T_new = (2π / 0.040 s)
Cancelling 2π from both sides:
1 / T_new = 1 / 0.040 s
Therefore, T_new = 0.040 s.
So, the time interval required for one rotation of the expanded neutron star is the same as the original neutron star, which is 0.040 seconds.