The Physics behind the stability of Degenerate Stars

 

Introduction:

 

A star from the day it is formed undergoes multiple phases until no more fuel, that keeps it going, ends. A star is stabilized by the process of stellar nucleosynthesis, i.e., the process of creation of chemical elements within the core of a star by the process of Nuclear Fusion. A star is mostly made up of hydrogen atoms and when this concentration of hydrogen atoms is maximum or more than the percentage of composition of other elements, the star is said to be relatively younger. The process of nuclear fusion helps in fusing the hydrogen atoms to form helium atoms which again fuses to produce heavier mass atoms and along with this fusion process, there is an extreme amount of thermal energy released which plays a significant role in the stability of a star. The thermal energy so released, exerts a large amount of thermal radiation pressure on the star in an outward direction, and this balances out the gravitational pressure exerted by the star on itself and hence, the stability of a star is ensured.

         

However, this process is not going to go for a long time. As the core of the sun keeps on fusing lighter elements into heavier elements, the heavier elements are hard to fuse, and hence, after a certain element, mostly Iron(Fe), the core of the star is unable to fuse the iron atoms to produce a heavier mass nuclei because of the extraordinary stability of the iron nucleus, which then requires very high energy applied on it externally, to make the fusion possible and because of this situation, the amount of thermal energy released in the fusion of heavier elements is less and they start requiring external energy for their fusion to be possible. So, the day most of the core becomes iron, the thermal pressure fails to stabilize the star and the gravitational force dominates the situation, resulting in the implosion of the star, also known as the Gravitational Collapse. This implosion of a star produces an immense amount of energy resulting in an outward blast of the star which is very popularly known as a Supernova.

 

A star since the day it is born out of gases has its end already written with it as the death of a star is inevitable. However, not all seem to have ended with the end of a bright beautiful star. Deep within its core, something else is brewing but not as alive as the star was. Depending upon its mass and size, we can predict the end fate of a star. A star on its implosion can produce a particular astronomical object such as a white dwarf star, neutron star, black holes, quasars, pulsar, etc. Now, although we know about how stellar nucleosynthesis helps in stabilizing a star, it’s quite interesting to know how all these remains of a star, are so stable after such a violent death of their parent star?

 

In this post, we shall discuss the stability of a neutron star and a white dwarf star (mostly neutron stars). We will be discussing the quantum mechanical effects that take place in these stars and how they ensure their stability. Starting from degenerate matter, the Pauli exclusion principle, degenerate pressure, and eventually the stability of the stars will be discussed. 

Neutron Stars and White Dwarf Stars, what are they?

 

A neutron star as it sounds is made up of neutrons. It is the collapsed core of a massive supergiant star having a mass between 10 to 25 solar masses and even more if the star was metal-rich. Neutron stars are an exceptional outcome of the death of a supermassive star. Unlike black holes, neutron stars are the smallest but densest stellar object yet known. Neutron stars are so tiny that they have a radius of about 10KMs only but a mass of about 1.4 Solar mass (Note: 1 Solar mass represents the mass of our sun). The concentration of this much amount of mass in such a tiny volume result in a greater density of matter in the star. Neutron stars are the result of a massive star’s explosion along with the extreme gravitational collapse of the core of the star, which compresses the core of the star beyond the white dwarf star density to that of the atomic nuclei.

Neutron stars once formed, no longer actively generate heat and start cooling down over time. However, due to collision with another object or because of the presence of an accretion disk around it, the star may still evolve with time.

Neutron stars are partially supported against gravitational collapse by Neutron Degeneracy Pressure, similar to Electron Degeneracy Pressure supporting the white dwarf star against the gravitational collapse due to their gravity. However, Neutron stars are also supported by the repulsive nuclear forces acting between the nucleons due to them being extremely close to each other because of the gravity compressing the core of the star.

White Dwarf Stars are the stellar core remnant of a star that was not massive enough to become a neutron star or a black hole and is mostly made up of electron degenerate matter. It too is one of the densest objects in the universe. No fusion takes place in both neutron stars and a white dwarf. Due to some residual thermal energy present in the white dwarf, it still has some faint luminosity. White dwarfs have a mass as that of about 1 solar mass, all concentrated in a spherical volume of radius as close to that of Earth’s. This results in white dwarf stars being very dense in mass, similar to neutron stars. White Dwarfs as discussed above are supported by electron degeneracy pressure which acts outwards and opposes the gravitational collapse of the star.

 

Degenerate matter And Degeneracy Pressure

 

We have talked about degeneracy pressure. But what exactly it is? Degenerate matter is a highly dense state of fermionic matter i.e., the matter consisting of fermions, in which due to Pauli’s exclusion principle, an immense amount of pressure is generated. Fermions are quantum particles that have a half-integral spin and obey Pauli’s exclusion principle that specifically tells that a set of quantum numbers is unique to an individual fermion and no two fermions can have the same set of quantum numbers. In simple words, no two fermions can coexist in the same state.

The term degenerate matter is commonly used in astrophysics to refer to dense stellar objects where the gravitational pressure is so extremely strong that the quantum mechanical effects become highly significant. In a degenerate star such as a neutron star or white dwarf star, the gravitational force is so strong that it tries to pull down all the fermions to their ground state. However, as per Pauli Exclusion Principle, no 2 fermions can coexist in the same quantum state, and hence, this results in the generation of pressure that pushes out the fermions trying to occupy the same quantum state as that of another fermion that is already present in that state. This pressure which is a result of the Pauli exclusion principle, present in degenerate matter, is known as Degeneracy Pressure.

Degeneracy pressure ensures the stability of degenerate matter stars. Degenerate Gases are gases composed of fermions rather than molecules of ordinary matter. These are made up of a large collection of freely moving fermions and their examples can be taken from the electron cloud present in metals and the interiors of a white dwarf. 

Electron Degeneracy Pressure

 

In an ordinary fermion gas in which thermal effects dominate, most of the available electron energy levels are unfilled and the electrons are free to move to these states. As particle density is increased, electrons progressively fill the lower energy states and additional electrons are forced to occupy states of higher energy even at low temperatures. Degenerate gases strongly resist further compression because the electrons cannot move to already filled lower energy levels due to the Pauli exclusion principle. Since electrons cannot give up energy by moving to lower energy states, no thermal energy can be extracted. The momentum of the fermions in the fermion gas nevertheless generates pressure, termed "degeneracy pressure".

Under high densities, the matter becomes a degenerate gas when all electrons are stripped from their parent atoms. The core of a star, once hydrogen burning in nuclear fusion reactions stops, becomes a collection of positively charged ions, largely helium and carbon nuclei, floating in a sea of electrons, which have been stripped from the nuclei. White Dwarfs are luminous not because they are generating energy but rather because they have trapped a large amount of heat which is gradually radiated away. Normal gas exerts higher pressure when it is heated and expands, but the pressure in a degenerate gas does not depend on the temperature. When gas becomes super-compressed, particles position right up against each other to produce degenerate gas that behaves more like a solid. In degenerate gases, the kinetic energies of electrons are quite high and the rate of collision between electrons and other particles is quite low, therefore degenerate electrons can travel great distances at velocities that approach the speed of light. Instead of temperature, the pressure in a degenerate gas depends only on the speed of the degenerate particles; however, adding heat does not increase the speed of most of the electrons, because they are stuck in fully occupied quantum states. Pressure is increased only by the mass of the particles (due to them traveling at relativistic speeds), which increases the gravitational force pulling the particles closer together. Therefore, the phenomenon is the opposite of that normally found in the matter where if the mass of the matter is increased, the object becomes bigger. In degenerate gas, when the mass is increased, the particles become spaced closer together due to gravity (and the pressure is increased), so the object becomes smaller.

Neutron Degeneracy Pressure

 

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are partially supported by the pressure from a degenerate neutron gas. The collapse happens when the core of a white dwarf exceeds approximately 1.4 solar masses, which is the Chandrasekhar Limit, above which the collapse is not halted by the pressure of degenerate electrons. As the star collapses, due to such a high density of mass, the electrons are captured by the protons to produce neutrons (via inverse beta decay, also termed electron capture). The result is an extremely compact star composed of nuclear matter, which is predominantly a degenerate neutron gas, sometimes called neutronium, with a small admixture of degenerate proton and electron gases.

Neutrons in a degenerate neutron gas are spaced much more closely than electrons in an electron-degenerate gas because the more massive neutron has a much shorter wavelength at a given energy. In the case of neutron stars and white dwarfs, this phenomenon is compounded by the fact that the pressures within neutron stars are much higher than those in white dwarfs. The pressure increase is caused by the fact that, the compactness of a neutron star causes gravitational forces to be much higher than in a less compact body with similar mass and hence the degeneracy pressure becomes high as well. The result is a star with a diameter on the order of a thousandth that of a white dwarf.

The stability of degenerate stars

Quantum Mechanical Effects observed in a Degenerate star

Degeneracy Pressure approximation:

 

Since electron degeneracy pressure and neutron degeneracy pressure are similar to each other, we are going to calculate that neutron degeneracy pressure and electron degeneracy pressure can similarly be approximated.

When all the lowest energy states in a neutron star are completely filled by the neutrons, other neutrons are forced to exist in higher energy states due to the Pauli Exclusion Principle and that results in pressure.

This pressure can be calculated if we consider the neutron star to be a 3-Dimensional Potential well in which the neutrons are trapped and have restricted motion along all the 3 spatial directions. Now let us consider the degenerate matter to be a plasma that is cooled under increasing pressure, due to Pauli Exclusion Principle, after some point, we will eventually won't be able to compress the plasma further as compression would lead to more than 1 fermion trying to obtain the same Quantum state.

Now, in this highly compressed state of matter, there is no extra space for the particles to move properly. Therefore, the position of the particle is highly localized and more certain. Then according to Heisenberg’s uncertainty principle, the particle’s momentum becomes highly uncertain and this momentum uncertainty comes if the particle’s speed is really high. This means that, even though the degenerate matter is cold, its particles are moving very fast with very high kinetic energies. Hence, in order to compress an object into a very small space, a tremendous amount of force is required to control the constituent particle’s momentum.

This is Heisenberg’s uncertainty principle where (del x) represents the uncertainty in the position of a particle, (del p) represents the uncertainty in the momentum of the particle and 'h' represents Planck’s constant.

So, as per Heisenberg’s uncertainty principle, we have ‘p’ representing the fermi momentum of neutrons in this case. As the core collapses under high gravitational pressure, the neutrons rush to fill the higher energy levels resulting in an increase in the fermi momentum of the neutrons. Hence degeneracy pressure is released and the gravitational pressure is opposed.

Gravitational Pressure Approximation

In order to obtain the relation between the mass and radius of a degenerate star, we also need to approximate the amount of gravitational pressure the star is exerting on itself.

The gravitational Pressure acting on the star due to its own mass helps in the formation of the electron degeneracy pressure and neutron degeneracy pressure. 

Under the Equilibrium conditions, the two pressures are equal in magnitude but are directed opposite to each other, such that P_n+P_g=0. 

References:

 

 

1. Branson, Jim M. "Filling the Box with Fermions." Quantum Physics. USCD, 22 Apr. 2013. Web. 23 May 2013.

2. Latimer, J. M., & Prakash, M. (2004). The Physics of Neutron Stars (Master's thesis). 13MayRetrieved June 1, 2013, from arXiv (arXiv:astro-ph/0405262).

3. Martin, B. R. (2012). Nuclear and Particle Physics (Second Edition ed.). West Sussex, United Kingdom: John Wiley.

4. Pasachoff, Jay M., and Marc L. Kutner. University Astrophysics. First Edition ed. Philadelphia: W.B Saunders Company, 1978. Print

 

5. Wikipedia: Degenerate matter

 

6. Wikipedia: Electron Degeneracy Pressure

 

7. Wikipedia: Neutron Degeneracy pressure