The core of a star collapses during a supernova, forming a neutron star. Angular momentum of the core is conserved, so the neutron star spins rapidly. If the initial core radius is \(5.0 \times 10^{5} \mathrm{km}\) and it collapses to \(10.0 \mathrm{km}\), find the neutron star's angular velocity in revolutions per second, given the core's angular velocity was originally 1 revolution per 30.0 days.

A neutron star is the collapsed core remnants of a massive star after it has gone through a supernova explosion. Neutron stars are incredibly dense, typically having about 1.4 times the mass of the Sun but compressed into a sphere only around 10 kilometers in diameter. This density means that the material inside a neutron star is mostly neutrons, with a soup of quarks potentially existing at the very heart.

Because of their density, neutron stars also have immense gravitational fields and exhibit interesting behaviors, such as the emission of intense electromagnetic radiation that can be observed on Earth as pulsars. Moreover, a neutron star's small size and mass have profound effects on the rotational speed, especially when conservation of angular momentum is considered.

Angular velocity, often denoted as \(\omega\), is a measure of how fast an object spins or rotates around a certain axis. It's essentially the rotational equivalent of linear velocity but refers to angle covered per unit time. The SI unit for angular velocity is radians per second (rad/s), but it can also be expressed in terms of revolutions per second or per minute.

In the case of a neutron star, the rapid spin is due to the conservation of angular momentum. As the star's core collapses into a much smaller radius, the angular velocity increases dramatically to conserve the total angular momentum. This is mathematically expressed by \(\omega \propto 1/r^2\), implying that if the radius (\(r\)) decreases, the angular velocity must increase to maintain the consistency of angular momentum.

The moment of inertia is a measure of an object's resistance to changes in its rotation. It is the rotational equivalent of mass in linear motion. A key aspect of the moment of inertia is its dependency on the object's mass distribution relative to the axis of rotation: the farther the mass is from the axis, the higher the moment of inertia. It is mathematically expressed as \(I = k \cdot r^2\), where \(k\) is a constant that depends on the object's mass distribution, and \(r\) is the radius from the axis of rotation.

When a star's core collapses, the reduction in radius leads to a significant decrease in the moment of inertia. As seen from the conservation of angular momentum equation \(I_\text{initial}\omega_\text{initial} = I_\text{final}\omega_\text{final}\), a decrease in \(I\) necessitates an increase in \(\omega\), leading to the astounding rotational speeds of neutron stars post-supernova.