Neutron stars form from supernova explosions, when the remaining core does not collapse into a black hole. They have a diameter of around 20km, which is the size of a city, but they also contain the equivalent of 1.4 mass of the Sun, making them extremely dense. You probably have heard the famous anecdote that a teaspoon of a neutron star weighs a few tonnes. The supernova explosion also gives them tremendous angular momentum, making them spin up to 43000 times per minute as with PSR J1311-3430.
As the name suggests, they are made mostly of neutrons. However, they also made of protons and neutrons, and their structure is very complex. For instance, the surface of a neutron star is composed of a lattice of regular atomic nuclei with electrons flowing between them. It is suggested that they are mostly iron, since it is the most stable nucleus with the highest binding energy per nucleon.
Neutron stars also exhibit some of the strongest magnetic fields of the universe, trillions of gauss stronger than the magnetic field of Earth, which protects us from the Sun’s powerful radiation. The stars with the strongest magnetic fields are known as magnetars, which have relatively short lifespans and rotate slowly every few seconds. The disturbances in the magnetic field known as Starquakes, result in very powerful gamma ray bursts. One of these bursts in 1979 helped detect magnetars, when two Soviet probes and Helios 2 were hit by a burst of energy for a few milliseconds. A bigger burst could prove deadly to life on Earth, again showing how fragile the existence of our species is.
Some of these highly magnetised stars can also emit X-Ray bursts, and they are known as pulsars. They are the sources of very powerful cosmic rays, which could have been the cause of a cosmic bit flip during the 2003 Belgian election, which is an absolutely crazy story. They are of particular interest to astronomers, as they allowed for the detection of the first exoplanets and they can be used as clocks more accurate than atomic clocks. Some scientists claim they can be used for interstellar navigation, although the massive bursts of radiation would easily destroy a ship from a few parsecs away.
Furthermore, stars rarely exist by themselves. Binary star systems can often emerge, where you have two stars orbiting a common center of mass. This creates a lot of interesting possibilities for calculation and applications of chaos theory. Take a look at this interesting video by Fabio Pacucci on the three body problem, where three objects exert gravitational forces on each other. For high school students wanting a challenge, take a look at problem 20 from the 2017 Oxford Physics Aptitude Test. Interesting things arise when these binary stars collide in a collision known as a kilonova. In 2017, a merger allowed for another detection of gravitational waves. Some scientist claim that these star mergers are the origin of some heavy elements, such as gold.
My personal favourite pulsar is The Black Widow Pulsar (PSR B1957+20). I just think it has an awesome name and also orbits a brown dwarf, which I think is unique. The upper limit for its estimated mass is 2.4 times the Sun’s mass, which makes it the heaviest neutron star. This puts it close to the Tolman–Oppenheimer–Volkoff limit, which states that a neutron star with a mass greater than 3 times the mass of the Sun, would become a black hole. It is fascinating that a star, an object we can imagine fairly well can become a black hole, something that bends the laws of physics just by adding a little bit more mass.
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How small are neutron stars? (2020, March 16). Astronomy.Com. https://astronomy.com/news/2020/03/how-big-are-neutron-stars
Tillman, N. T. (2018, February 24). Neutron Stars: Definition & Facts. Space.Com. https://www.space.com/22180-neutron-stars.html
Wall, M. (2012, October 25). Super-Dense Neutron Star Is Fastest Ever Seen. Space.Com. https://www.space.com/18218-fastest-orbiting-pulsar-neutron-star.html
What are neutron stars made out of? | NewCompStar. (n.d.). NewCompStar. https://compstar.uni-frankfurt.de/outreach/short-articles/what-are-neutron-stars-made-out-of/