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Pulsars, like the Crab pulsar, are fascinating cosmic objects. They rotate rapidly and send out beams of electromagnetic radiation. This rotation leads to each pulsar having a specific period, or the time it takes for one complete spin. On rare occasions, a phenomenon known as a 'glitch' can occur. This is when a pulsar unexpectedly speeds up, shortening its period. In the case of the Crab pulsar, a glitch caused the period to decrease by \(|\Delta P| \approx 10^{-8} P\). Such tiny changes might seem negligible but, in the context of neutron stars, they are significant. Glitches provide insights into the internal and dynamic processes of neutron stars. By understanding period changes, scientists can study the alterations in the rotational dynamics and internal structure of these stars.

The conservation of angular momentum is a pivotal principle in physics, applying extensively to rotating systems like pulsars. Angular momentum (\(L\) ) is the product of an object's moment of inertia (\(I\) ) and its angular velocity (\(\omega\) ). For a rotating celestial object, this remains unchanged in the absence of external forces. This means any change in the pulsar's radius or structure must lead to a compensatory change in rotational speed. \(L = I\omega\) illustrates this balance. During a glitch, a drop in the period indicates a rise in \(\omega\) . Given \(L = I_{1}\omega_{1} = I_{2}\omega_{2}\), when \(\omega\) increases, \(I\) must decrease. Such intricacies of angular momentum help explain why contraction results in faster rotations, akin to a figure skater pulling in their arms to spin faster.

The moment of inertia (\(I\) ) is crucial for understanding how mass distribution affects an object's rotational characteristics. In astrophysics, it's calculated differently than on Earth due to the unique settings. For a neutron star presumed to have a uniform density and spherical shape, the formula \(I = \frac{2}{5} m r^2\) is used. Here, \(m\) represents the neutron star's mass and \(r\) its radius. By using this formula, one can relate rotational changes to physical changes within the neutron star. As the radius contracts slightly during a glitch, \(I\) diminishes, leading to an increase in \(\omega\) to conserve \(L\). The interplay between \(I\), rotational speed, and period changes provides a window into the star's internal workings.

Pulsar contraction has direct implications on their rotational dynamics. When a neutron star like a pulsar contracts, its radius \(r\) decreases. This affects its entire rotational behavior under the conservation laws. For the Crab pulsar, this contraction was a major factor in the period change after a glitch. Knowing \(r_1\) and calculating \(r_2\) using \(r_2 = r_1 \sqrt{1 - 10^{-8}}\) shows how small contractions can result in notable period changes. Such calculations are vital for assessing how internal structural changes impact a pulsar's observable properties. These contractions are subtle but hold significant importance in terms of energy release, rotational kinetics, and neutron star stability. Observing these changes aids scientists in verifying theoretical models about neutron star dynamics and internal compositions.