Pulsar Formation and Structure
How does degeneracy support a white dwarf from collapsing under its own mass? Degeneracy pressure is caused by the Pauli exclusion principle which says that two fermions cannot occupy the same quantum state, meaning they can’t have the same quantum number. Some examples of fermions include protons, neutrons and electrons which have antisymmetric wavefunctions. This property limits the number of particles you can stuff into a given volume because they cannot share an energy state, imposing a limit to the proximity of the particles. This means that white dwarfs and other objects in a degenerate state cannot be compacted down any farther.
This paper will describe the formation of another type of degenerate object, pulsars, which are a type of neutron star. As the name suggests neutron stars are neutron degenerate objects, so instead of being packed so tightly that the electrons can’t be packed any closer they are limited by the proximity of their neutrons. These stars can be formed by certain types of supernova.
The first way a supernova can form neutron stars is by having a white dwarf find a companion star that’s less dense than itself, such as a main sequence star. Due to its greater density, it can draw mass away from the less dense companion star. We can explain this ability to remove mass from another object by both Newtonian and Relativistic physics. Overall it boils down to the greater density more sharply curving spacetime around the white dwarf, or by fringe material on the companion star escaping its original gravitational pull.
However, despite giving us an acceptable reason Newtonian physics fails to give us the appropriate rate of mass loss and gain from the companion star to the white dwarf. This process is called mass accretion and allows a white dwarf to destabilize itself by exceeding the Chandrasekhar mass limit set by:
where ω03 is a constant derived from a from the lame-emden equation, µe is the average molecular weight of the star, mH is the mass of hydrogen, and mp is the planck mass. This formula gives us a maximum weight of approximately 1.4 times the mass of the sun, which is the upper stability limit for a white dwarf. Normally the white dwarf doesn’t have an ongoing process that increases its mass and can destabilize itself. However, if a companion star is present, this cannot occur, and the white dwarf will collapse. Electron degeneracy pressure will no longer be able to support it and so the star pretty much implodes as gravity wins out over its outward pressure. This happens so fast that the mass gets superheated and undergoes runaway nuclear fusion resulting in a supernova.
Another way a supernova could occur is if the star was simply extremely massive. High mass stars burn hotter and can progress through enough cycles of stellar evolution that it can fuse much heavier elements. This allows them to naturally exceed the Chandrasekhar limit at the end of their stellar lifetimes instead of reaching a proper white dwarf phase and undergo a supernova without the aid of a companion star.
Either way, a supernova has the potential to form a new celestial body. The leftover matter at the center of these explosions is extremely condensed, which can form a neutron star. It can reach this only if the mass lies between the Chandrasekhar mass and the Tolman - Oppenheimer - Volkoff mass limit, the upper bound of a white dwarf and the lower bound of a black hole.
Now we shall delve into the structure of these interesting stars, starting with the equation of state. These tell us the pressure, the density, and the temperature of the star as a function of radius. Currently, there aren’t consistent formulations for the equations of states. Some models include APR EOS, UU, EOS FPS, and L, all of which come up with different mass predictions. An example of one of the possible equations of state for pressure is
The integral over the mass is a volume integral adjusted by the central density rho to give the mass as a function of radius. The factor nabla shows that this is a general relativity equation as the factor 2Gm/c2 factor is known as the Schwarzschild radius, the radius of a black hole.
This term is commonly used in general relativity, and we can use the Schwarzschild metric here to show us some interesting properties that occur on the surface of a neutron star.The Schwarzschild metric is
By taking some average values for a neutron star, 1.5 solar masses for its mass, and 10 km for its radius, we get the Schwarzschild radius to be roughly 4.4 km. This makes the factor 1-rs/r roughly .66 which we can see will have a significant factor in warping the space around the neutron star. This means that at the surface of the neutron star, the actual distance is ~1.2 times the coordinate distance! This means that if you were able to walk on the surface of a neutron star, in order to walk 10 meters from an observer’s reference frame, such as a scientist floating in a spaceship, the walker would need to walk 12 meters in his own frame!
The exact way in which to incorporate this spacetime curvature into an equations of state for a neutron star is currently unknown and warrants further research. The main contention between the different models, is on the application of general relativistic corrections, so its clear that we can gain a greater understanding of quantum gravity from the creation of more accurate neutron star models through data gathering and analysis.
Another structural feature to look at is the neutron star’s temperature. In particular it is important to model the cooling process. Immediately after a supernova, a neutron star must be immensely hot, on the order of 10^11 kelvins. We know from observation that it can’t dissipate this heat simply through simple radiation of heat, so there needs to be explanations on methods to quicken this rate of heat loss. In addition, neutron stars often undergo periods of quiescence where it turns on/off its x-ray emission spectrum. The speed at which it turns off also requires explanation, and one of the most common explanations is the Direct URCA process. This is when the star emits neutrinos via the processes:
This cools the star because neutrinos don’t interact very strongly with matter, they can simply pass through and leave the core of the neutron star, carrying away with it large amounts of energy very quickly. This process occurs at a rate many orders of magnitude quicker than cooling by radiation. This means that a neutron star actually cools from the inside out rather than from outside in, which is a very interesting side effect of this process. Another cooling method is the modified URCA process in which 
For now the most accepted structure of a neutron star is as follows. The surface of the star is a solid lattice formed of degenerate electrons and heavy nuclei such as iron. Below this is an inner crust that contains a lattice of heavier nuclei such as krypton, and superfluid neutrons and electrons. It is around here that neutron drip occurs. Neutron drip is when a neutrons tunnel out of the nuclei to become free neutrons. Normally a free neutron would decay into a proton, however, since the star is neutron degenerate, there is no lower energy state to occupy, so instead of decaying into a proton and its antiparticle, it stays a neutron and tunnels out of the energy well, becoming a free floating neutron, then undergoes its decay. This forms a layer in the star of free neutrons and protons located below the inner crust, and below this is the unknown core.
That fermi sea though, causes many of the interesting properties associated with neutron stars, including the ability for it to become a pulsar. Not only are the particles in that area a superfluid they are also superconductors. Superfluidity is a state of matter where a liquid behaves as though it has no viscosity, and has infinite thermal conductivity. Due to the property of infinity thermal conductivity a superfluid is also isothermal. What this means is that since it has an infinite ability to distribute heat, its not possible to locally heat a superfluid, so it always has the same temperature throughout the entire liquid. Superfluidity also allows a liquid to maintain equilibrium in containers regardless of gravity, allowing for fluids to crawl up or down the outside surfaces of its container. Also turbulence that is self contained, like a vortex last indefinitely inside a superfluid since it has no friction.
The neutron star’s sea of protons, neutrons and electrons eventually behave like a superfluid because the degenerate neutrons can pair together to form a boson. Boson’s do not have to obey the pauli exclusion principle and can be in the same state at the same time, which allows them to pass freely through one another. This gives it even more freedom than a gas as it has a viscosity of 0, but is still considered a liquid due to the distance and bonds between the particles as a whole.
In addition a neutron star’s sea of particles is also superconductive which means that it has no electrical resistance, and can maintain currents indefinitely. Only lenz’s law can slow down the current in a superconductive material.
These properties heavily affect the structure of the neutron star. Since the sea of particles has some charge due to protons and electrons, once the movement of this sea becomes uniform, the magnetic field lines they form will become constant, making the surface which is also charged solid. The particles on the surface are degenerate electrons and heavy nuclei, which means it is purely ionic and its position is heavily fixed due to the magnetic field lines created by the rotating superfluid.
This effect seems to be akin to the effect of ambipolar diffusion in star formation. In that case, the dust of the Interstellar medium has ionic particles that couple to the magnetic field lines of a molecular cloud, which slow down the movement and helps in the accretion of mass around those field lines. This lends credence to the idea of a solid surface on a neutron star because the effect of the superfluid causes a magnetic field many times stronger than that of ambipolar diffusion so the increasing of density and slowing of movement should also increase.
When these fields are strong enough, they can force the electromagnetic radiation of the star to exit only at the poles of the star resulting in a pulsar. The only way for them to get this strong is if the superfluid has enough angular momentum through formation, but where does it come from?
It relates all the way back to the formation of the original star. In the giant molecular clouds that formed the original star. In order to shrink some of that gas down gravitationally, assuming there was some initial rotation, or turbulence in that cloud, angular momentum has to be conserved. This is why stars rotate, such as our sun, and why they have a magnetic field. The ambipolar diffusion previously mentioned is a process that explains how the rotation can be slowed by clumping of material and removal of momentum through magnetic fields. Though again ambipolar diffusion is its own topic with its own controversies.
However what holds true is that as you decrease the size of the object and pull in more particles together, the rate at which it spins has to go up. Once the rest of the star is blown away and only the compacted matter is left, the speed at which it rotates is enormous because it went from a huge cloud of dust to a few kilometers compounded with the moment of inertia calculated as r2. Neutron stars have been recorded to have a period of rotation on the order of a millisecond. Not only that it will maintain this rotation very accurately due to the superfluid nature underneath the crust, no friction to slow down the process.
However there are some strange events that can disrupt this regularity. Sometimes a pulsar will “ glitch” or “kick”. Some speculate that this is caused by asymmetric supernova, others think that there are starquakes in which the solid crust compacts, as deformities caused by landing meteorites or crust compacting as pressure towards the surface goes down, causing quakes on the surface of the star. That quake can then severely disrupt the period of rotation for that star as the motion will remain in the superfluid. Forces continue endlessly as there is no friction disrupting its motion, so even the smallest events could have major consequences.
In conclusion there are still a lot of unknowns pertaining to the pulsar. In particular we don’t know the composition of the core, or have a definitive explanation for anomalies or “kicks” in the pulsar, or a reliable equation of state for a neutron star, which makes it very difficult to model. However, further research into this area will help further our understanding of both general relativity and quantum mechanics and possibly how to reconcile the two.
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