A star the size of our Sun runs out of nuclear fuel and, without losing mass, collapses to a white dwarf star the size of our Earth. The radius of our Sun is 6.96×108m , the radius of Earth is 6.37×106m .
If the star initially rotates at the same rate as our Sun, which is once every 25 days, determine the rotation rate of the white dwarf.
To determine the rotation rate of the white dwarf, we need to understand the concept of conservation of angular momentum.
Angular momentum is a property of rotating objects and is given by the formula:
L = Iω,
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, we want to compare the angular momentum of the star (before it collapsed) to the angular momentum of the white dwarf (after it collapsed):
L_sun = L_wd
Since angular momentum is conserved, we can write:
I_sun * ω_sun = I_wd * ω_wd,
where I_sun and ω_sun are the moment of inertia and angular velocity of the Sun, and I_wd and ω_wd are the moment of inertia and angular velocity of the white dwarf.
The moment of inertia of an object depends on its mass and its distribution of mass about the axis of rotation. In this case, we assume that the mass of the star remains constant during the collapse, so the moment of inertia can be approximated as:
I_sun = 2/5 * M_sun * R_sun^2,
where M_sun is the mass of the Sun and R_sun is its radius. Similarly, the moment of inertia of the white dwarf can be approximated as:
I_wd = 2/5 * M_wd * R_wd^2,
where M_wd is the mass of the white dwarf (which is the same as the mass of the initial star) and R_wd is the radius of the white dwarf.
We are given the values for R_sun and R_wd, so we can plug them into the equations:
I_sun = 2/5 * M_sun * (6.96×10^8)^2,
I_wd = 2/5 * M_wd * (6.37×10^6)^2.
Since the mass of the star doesn't change, we can cancel out the mass terms, leaving us with:
(6.96×10^8)^2 * ω_sun = (6.37×10^6)^2 * ω_wd.
Finally, we can calculate the rotation rate of the white dwarf (ω_wd) by solving the equation for it:
ω_wd = (6.96×10^8)^2 * ω_sun / (6.37×10^6)^2.
Plug in the given value for ω_sun (once every 25 days) and calculate the answer.