Neutron Stars
After a supernova occurs, one of the objects that can result from the explosion is a neutron star. There are black holes, which will be discussed later, and theoretical astronomical bodies like quark stars that could form but for now, let's keep it simple and restrict it to neutron stars. Neutron stars are similar to white dwarves because they are supported by repulsive nuclear forces along with neutron degeneracy pressure, like how white dwarves are supported by electron degeneracy pressure. Neutron stars that theoretically don't rotate have a mass limit defined by the Tolman-Oppenheimer-Volkoff Limit, which is 2.16 solar masses as of 2019. However, since many neutron stars rotate, they can support sufficiently greater masses.
Neutron stars are known to be extremely astronomically compact. They have masses between 1.4 and 3 solar masses, but their sizes are on the levels of small towns. This makes them incredibly dense and due to their density, their gravitational attractions are 1011 times Earth's surface gravity.
As the star's core collapses due to the burning of the shell, the rotational rate of the neutron star will increase. This is simply due to the conservation of angular momentum from mechanics. Recall that the angular momentum of an object can be expressed as shown by Equation 1 on the corresponding section of the Science Reference Astronomy Equation Sheet. Since no external torques act on the system of the neutron star(we can neglect other astronomical systems because the gravitational forces from them are negligible), the angular momentum of the neutron star must be conserved(Equation 2) as it collapses when it goes supernova. The radius of the star decreases so the moment of inertia, which can be simplified to Equation 3, decreases. If the moment of inertia decreases but the angular momentum is constant, then the angular velocity, the rate at which the star rotates, represented by the lower-case omega, must increase. This is why neutron stars rotate significantly quicker than their earlier stages. Neutron stars do have two main subtypes: magnetars and pulsars. There are also neutron stars that have properties of both magnetars and pulsars combined.
Pulsars
Pulsars are special types of neutron stars that spin extremely rapidly and have a strong magnetic field. These two properties cause its defining characteristic, which are pulses of electromagnetic radiation many times every few seconds. These pulses are caused by the rapidly moving magnetic field(caused by the fast rotation of the pulsar). The magnetic field accelerates charged particles, mainly electrons and protons, towards the poles, the places where the magnetic field is weakest. This causes the particles to jet out from the magnetic poles of the neutron star. These beams of particles emit intense electromagnetic radiation which is why they're known as pulses.
However, you may be wondering why exactly humans can observe the pulses from pulsars. Remember that light in the universe has to travel in very specific directions to come to us so why would we be able to observe so many pulsars? This is where the pulsar's rotation is key. The magnetic poles of the pulsar are usually not perfectly aligned with the axis of rotation. This means that the magnetic poles are constantly moving and rotating, like the lenses from a lighthouse. At certain portions during the rotations of the star, the pulses will be directed towards Earth for us to view. This is how we can view pulsars: eventually, they'll be directed(due to the misalignment of the magnetic poles) towards Earth.
Magnetars
A magnetar is a neutron star that has an extraordinarily strong magnetic field(on the order of 10^10 T). Due to the magnetic field of magnetars, we often see them emit strong beams of electromagnetic radiation, like X-Rays and Gamma Rays. Above is an artist's impression of a magnetar emitting electromagnetic radiation near its surrounding supernova Astronomers aren't entirely sure as to how magnetars form their immense magnetic fields, and there aren't many magnetars to help investigate.
Some theories believe that the formation of strong magnetic fields is due to the conservation of magnetic flux, given by Equation 6. The equation for magnetic flux itself is Equation 5 on the Science Reference Astronomy Equation Sheet. When a star collapses into a neutron star, its radius decreases, which means its surface area, denoted by A in the equation, decreases even more. This means that the magnetic field must significantly increase to keep the magnetic flux constant. The term with the angle, theta, is necessary for magnetic flux but the previous deductions assume the angles of the magnetic field lines wouldn't change much.
The other theory is that magnetars form their magnetic fields due to a dynamo mechanism caused by the extreme sizes and temperatures of the neutron star. A dynamo mechanism is a theoretical way for astronomical bodies to create their own magnetic fields. It simply asserts that at the core of the body, electrically conductive fluid is charged and moving such that it creates a magnetic field on the surface. Due to the extreme conditions of most magnetars, this could be possible, too. However, neither of these are substantial enough given that we lack samples of magnetars to test theoretical models from.
Citations/Attributions
File:Artist’s impression of a gamma-ray burst and supernova powered by a magnetar.jpg. Provided by: Wikimedia commons. Located at: https://commons.wikimedia.org/wiki/File:Artist%E2%80%99s_impression_of_a_gamma-ray_burst_and_supernova_powered_by_a_magnetar.jpg. License: CC BY 4.0