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When a massive star goes through the final stages of its life cycle, it can collapse under its own gravity, becoming a neutron star. This process is akin to putting the mass of the Sun into a sphere merely 16 kilometers across. Neutron stars are one of the densest objects in the universe, second only to black holes. As a result, a neutron star's density is extraordinarily high, roughly \(10^{14}\) times that of ordinary matter.
This immense density is due to the combination of extreme gravitational compression during collapse and the properties of neutron-rich matter, primarily made up of neutrons packed tightly together. In our exercise, we studied a star that went from a radius similar to the Sun's to the compact size of a neutron star, radically changing its physical characteristics while conserving certain properties like angular momentum.

The concept of moment of inertia is fundamental in understanding how an object rotates. It depends on the object's mass distribution relative to the axis of rotation. For a solid sphere, the moment of inertia \(I\) is calculated as \(\frac{2}{5} m R^2\), where \(m\) is the mass and \(R\) is the radius.
During the transformation from a large star to a neutron star, the star's radius decreases significantly, thereby affecting its moment of inertia. As the radius decreases, the moment of inertia decreases, since it's proportionate to the square of the radius \(R^2\). This decrease is crucial in understanding how the neutron star's angular speed changes. Angular momentum need to be conserved, so when the moment of inertia drops, the angular speed has to increase to keep the product \(I \cdot \omega\) constant.

Angular speed \(\omega\) is an expression of how fast something spins around an axis, measured in radians per second. In our exercise, the star's angular speed drastically changes due to the conservation of angular momentum. Initially, the star rotates slowly, once every 30 days, which is a slow angular speed of \(\omega_i = \frac{2\pi}{T}\) with \( T \) being the period in seconds.
When the star collapses to form a neutron star with a much smaller radius, its angular speed must increase. This is described by the equation \( \omega_f = \omega_i \frac{R_i^2}{R_f^2} \). Here, the original angular speed \( \omega_i \) and the ratio of the squares of the initial and final radii determine the new angular speed \( \omega_f \). From this, we calculated that the neutron star ends up spinning about 5.6 radians per second—a huge increase due to the significant decrease in radius.