## What is a Neutron Star?

A neutron star is a compact star composed mainly of neutrons. Its size is relatively small with a radius of around 10 to 20 km. However, the mass of a neutron star is about 1.4 to 2 times more than that of the Solar Mass (the mass of the Sun $M_o$) with the largest mass observed recorded at 2.01 $M_o$. This leads to high densities ranging from about $2.4$ to $9.1 \times 10^{17} kg/m^3$. The density distribution is lowest at the outer most part of the crust starting from about $1 \times 10^{9} kg/m^{3}$. It then increases with depth to more than $6 \times 10^{17} kg/m^3$ deep inside the core. The closest comparison to this density is the approximate density of an atomic nucleus of $3 \times 10^{17} kg/m^3$. Compared to the sun, a neutron star is of order $10^{24}$ times denser. The surface temperature is typically of around $6 \times 10^{5} K$. To put things into perspective, “a single teaspoon of neutron star material would weigh a billion tons — assuming you somehow managed to snag a sample without being captured by the body’s strong gravitational pull.” [1,2,3]

Figure 1: Structure of Neutron Star [11]

## The Formation of Neutron Stars

In ancient times, it was believed that the stars are ever-lasting. As time passed, this belief was proven false. A star is created, and goes through a long process. It spends its lifetime fusing matter together until its ‘fuel’ runs out. At this point fusion stops and the massive star collapses towards its center of mass due to the overwhelming influence of its own gravity. This process is called Gravitational collapse, and is the heart of structure formation in the universe. [1,4]

A new star is formed through the gradual gravitational collapse of a cloud of interstellar matter. The collapse causes compression which raises the temperature until nuclear fuel ignites in the center of the star. This produces a thermal pressure outwards, which balances the gravitational forces and brings about dynamic equilibrium to the star. When the fuel runs out again, the same process is repeated until a new equilibrium state is reached. However, this process can not go on indefinitely, and so at some point the star will reach its death. The type of death the star experiences depends mostly on its mass. [1,4]

Typically, small stars end their life as white dwarfs, glowing compact stars with mass up to 1.39 $M_o$, cooling over billions of years. The mass of a white dwarf is restricted by the Chandrasekhar limit which gives the maximum mass of a stable white dwarf star. This is the same ending that awaits our sun. For stars that exceed 10 $M_o$, the neutron degeneracy pressure is overcome. The force of gravity of the star is so large that it breaks down the particles. The resulting matter and everything else within its reach is compressed and a black hole is formed. The force of gravity of a black hole is so strong that anything that gets close enough is sucked inside including light. The smallest observed mass of a black hole is recorded at 3.8 $M_o$. [1,5]

In between the white dwarf and the black hole, compact star having mass ranging from 1.40 to 3 $M_o$ typically form neutron stars. To understand how neutron stars form, we must first understand how neutrons form. In 1920 Ernest Rutherford predicted the existence of neutrons and a few years later it was observed by James Chedwick. Soon after, astronomers Walter Baade and Fritz Zwichy predicted that a supernova could produce a neutron star and in 1967 a pulsating neutron star was first discovered. Apart from the neutrons found in the nucleus of atoms, these can also be produced through a process called electron capture. With enough force, an electron in an atom’s inner shell is drawn into the nucleus where it combines with a proton to form a neutron and a neutrino. Neutrinos, having mass considered to be negligible even when compared to subatomic particles, are extremely fast and elusive, so they escape from the atom while the neutrons stay behind. [1,6]

The process of electron capture can be related to how neutron stars form. The stars gravity is strong enough to combine the electrons and protons together to form neutrons and neutrinos. The neutrinos escape into space leaving behind a large sphere of neutrons. These neutrons are compressed together due to the large force of gravity. However, these stars are supported against further collapse by the quantum degeneracy pressure due to the Pauli Exclusion principle. This states that no two neutrons can occupy the same place and quantum state simultaneously. The result is a neutron star. [1,6]

Figure 6: Electron Capture Equation [13]

Between the neutron star and black holes, hypothetical intermediate-mass stars such as quark stars and electroweak stars have been proposed, however non have been shown to exist.

Figure 7: Life Cycle of a Star [12]

## Detecting Neutron Stars

Neutron stars are not composed entirely of neutrons. A number of protons and electrons are still present within the star. Some neutron stars rotate very rapidly (up to 716 times per second), and since the star contains charged particles a massive magnetic field is created. This magnetic field does not have to line up with the axis of rotation. Due to this magnetic field, beams of electromagnetic radiation are emitted as pulsars, like a stellar lighthouse. Neutron stars are sometimes called pulsars because of the pulsing signal. Another way of detecting neutron stars is through Gamma-rays. When rapidly rotating, high-mass stars collapse to form a neutron star, bursts of Gamma-rays are sometimes produced. [4,7]

In the Milky-way galaxy there are approximately an order of $10^{8}$neutron stars. However, they are only detectable in certain instances, as if they are a pulsar of a binary system. Non-rotating and non-accreting Neutron Stars are virtually undetectable, however the Hubble Space Telescope has observed one thermally radiating Neutron Star within the Corona Australis constellation. [4,7]

## The Equation of State

Neutron stars contain matter with the highest densities in the observable universe. Thus they make ideal testing grounds to unleash the full power of theoretical physics, and further the study of dense matter. However, constructing a model of a neutron stars crust requires atomic and plasma physics, as well as the theory of condensed matter, the physics of matter in strong magnetic fields, the theory of nuclear structure, nuclear reactions, the nuclear many-body problem, superfluidity, physical kinetics, hydrodynamics, the physics of liquid crystals, and the theory of elasticity. Furthermore, “theories must be applied under extreme physical conditions, very far from the domains where they were originally developed and tested.” [1]

For a model to be constructed the equation of state (EoS) is also required. This is determined by the interactions between the particles making up the neutron star. It is a relation between the pressure P, and the density $\rho$ which can be translated to a mass-radius relation. The EoS for the core of the neutron star is problematic as the structure of the matter is not known, therefore the EoS cannot be found. [1,8]

Figure 9: Particle Behaviour [16]

On the other hand, the structure of the crust should be less complicated to obtain. Since the density varies with depth, the density of the crust is much less than the average density of the star. At the outer part of the crust the density is of order $10^{9}$, which allows this part of the crust to be composed of neutrons, protons and electrons, just like the matter around us. The density here is “sub-nuclear”, and so, nuclear physics methods which have been developed and successfully applied on terrestrial planets, can be applied to this section of the neutron star crusts. However, the physical conditions are extreme and far from terrestrial ones. At lower levels in the crust, compression of matter increases by gravity. This causes the process of electron capturing and the protons and electrons in the crust neutralize. This creates an ‘excess’ of neutrons, which produces conditions alien to terrestrial planets. For densities approaching $10^{14} g/cm^{3}$, around 90 % of nucleons (subatomic particles) are neutrons. At densities over $10^{14} g/cm^{3}$ nuclei can no longer exist, they collapse into a uniform plasma of nearly-pure neutron matter, with a few percent admixture of protons and electrons. This is the bottom of the crust. [1,9]

The crust contains only a small percentage of a neutron’s star mass. This is crucial for many astrophysical phenomena involving neutron stars. Since the matter it contains is at sub-nuclear density, the interactions are known, and many-body theory techniques can be used. This led theoretical physicists to derive EoS. The two cases typically considered for the EoS of the crust are for cold catalyzed matter (ground state crust), and accreted crust matter. The ground state approximation at the outer crust is well established, so that the pressure at any given density is determined within a few percent of accuracy. However, only theoretical models can be formed for the inner crust since the nuclei are influenced by a gas of dripped neutrons. This leads the EoS to be much more uncertain for the inner crust. The accreted crust EoS was calculated by Haensel and Zdunik, using a compressible liquid drop model with a “single nucleus” scenario. This model related closely to cold catalyzed model for the lowest and highest values, but predicts different values in the midrange. [1,9]

## Conclusion

The extreme conditions inside the crusts of neutron stars are already far beyond anything that can be recreated in terrestrial laboratories. The dense core of the neutron star is even more extreme. The matter in neutron star crusts experiences high pressures, and also a huge magnetic field. Neutron star branches many different fields of physics together. However, they remain a fascinating test-bed for all extreme physics and studying the details of their interior is still an active area of research.

## References

[1] Chamel, N. & Haensel, P. Living Rev. Relative. 11, 10 (2008)
[2] http://www.space.com/22180-neutron-stars.html
[3] http://hypertextbook.com/facts/1998/AnthonyColgan.shtml
[4] http://kipac.stanford.edu/kipac/research/Neutronstarts_Pulsars
[5] http://www.pbs.org/wgbh/nova/blogs/physics/2012/01/