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Stellar Evolution, and the Effects of Pauli’s Exclusion Principle on Electron Degeneracy in White Dwarfs

Abstract

In this scientific review paper, humanity’s interest in stars relating to the importance of an understanding of astrophysics is discussed. It is followed with an explanation of stellar evolution, including aspects such as hydrostatic equilibrium and the mechanism for nuclear fusion. Following this, the Pauli Exclusion Principle is expanded upon, and how it pertains to electron degeneracy in white dwarf evolution. Finally, it is explained why stars and the study of White Dwarves could play an important role in the development of astrophysics, just as previous understanding already has.

Introduction

Stars are the most fundamental celestial objects found throughout the universe. They heat planets, their enormous gravities make solar systems spin like clockwork, and other phenomena they exert have inspired human interest in heliophysics (the study of the Sun) and astrophysics for thousands of years.

Human innovation and curiosity have led us to an understanding of stars’ phenomena since ancient times. The ancient Sumerians and Babylonians developed highly advanced ‘star charts’ on holy tablets, such as in Figure 1a. [1] In the lands of ancient Egypt, people first developed astronomy as a science, such as when the Egyptian astronomers first developed the planetary model, stating that the inner planets Venus and Mercury revolve around the Sun, instead of the Earth. [2] Ancient Greeks observed patterns (constellations) in the night sky and created endless tales of their endeavours with their deities. These constellations are drawn out over a map of the northern hemisphere’s night sky in Figure 1b. [3]

Since the Enlightenment, human understanding of astrophysics and the universe has increased dramatically. Through the use of scientific instruments and advancement, more is known about space and its many phenomena than ever before. One such advancement would be of stars and their cycle of creation.

In the following article, stars, their journey of creation, and a study of why their differences are so fascinating are discussed. An introduction to white dwarfs and their properties is also talked about.

Figure 1: Ancient astrology and basic human understanding of astrophysics.

  1. A Sumerian star chart depicting the stars above Ancient Babylon and Sumeria, and precursors to later constellations.
  2. The constellations observed by the Ancient Greeks thousands of years ago in the night sky, connected.

Stars

1: Stellar Evolution

Stars always start their long lives off in nebulae and stellar nurseries. Stellar nurseries are often found inside of nebulae, where large collections of gas and dust, namely hydrogen and helium, can be used as the fundamental building blocks for stars. Gravity then causes the clusters of dust and gas in these stellar wombs to attract one another, creating the fuel and materials necessary for the formation of stars and planets. After thousands or millions of years of attraction, this protostar quickly begins its first reaction of nuclear fusion and starts its life journey of a star.

According to the factors seen in Section 2, particularly mass and elemental composition, stars can become either blue supergiants, Sun-like stars, red dwarfs, or brown dwarfs. Depending on these first stages of a star’s evolution, its next stage can be determined. For example, a blue supergiant can become either a black hole (more information is available in Appendix A) or a nebula, after ‘going supernova’.

A star which is in hydrostatic equilibrium (more information is available in Section 3), or is still performing nuclear fusion until its supply of fuel has run out, is known as a main sequence star. An example of a main sequence star would be the Sun. For most stars, fusion ends once its supply of hydrogen as a fuel has been exhausted. For more massive stars, however, the production of more exotic elements from helium up until iron can occur through further fusion.

Once all nuclear fuels in the core of a Sun-like star have been exhausted in fusion, hydrogen then on the outside shell of the star starts to fuse. This causes the star’s volume to increase, creating a red giant in most stars. Eventually, a star’s fusion becomes unstable and sporadic, throwing off layers into the cosmos. This ’shedding’ of stars eventually results in the creation of white dwarves (this topic is expanded on heavily in Section 5). The Sun will one day become a white dwarf.

Stars that have a mass greater than eight solar masses (m > 8M, where m is the mass of the star, and M is one Solar Mass (the mass of the Sun) will one day become supernovae. One of the mechanisms suggested for explaining what happens when a star ‘goes supernova’ is its iron core collapses under its pressure and goes from a very large size of roughly 5000 miles to just a mere few. This collapse happens in only a few seconds and a temperature spike of over 100 billion degrees occurs. The energy released in a supernova is unparalleled to almost every other release of energy in the entire universe. For weeks, the photonic energy (light) released from a supernova can outshine an entire galaxy. Every single naturally occurring heavy (past iron) element found in the periodic table is created in supernovae, including the heaviest naturally-occurring element, uranium. This is explained in Appendix B.

Overall, in at least some form of manner, stars almost always come back to form a nebula, thus creating the life cycle of stellar evolution. A simplified diagram of this is available in Figure 2. [4]

Figure 2: The stellar evolution and life cycle of a star, represented in a visual diagram.

2: Why are Stars Always Different?

Out of the millions of billions of stars that scientists have observed for thousands of years, no two stars have ever been alike. It is generally said in the scientific community that many different factors affect differences in stars. [5] These include:

  • Density
    • Mass
    • Volume
  • Nebula origin and elemental composition
  • Colour
  • Temperature

3: How do Stars Work?

3.1: Hydrostatic Equilibrium in Stars

A star needs to have equal forces acting upon it to stay stable. An example of a stable star would be the sun. In physics (particularly Newtonian physics), balanced interacting forces on an object are known as ‘in equilibrium’. This specific type of equilibrium is known as hydrostatic equilibrium. It is visible in Figure 3. One of the forces pushing towards the centre of the star is quite obvious: gravity.

Figure 3: A free-body diagram showing the forces exerted onto and by a stable star in hydrostatic equilibrium.

In simple terms, gravity works by attracting all particles with mass to all other particles with mass. This means that if there is a large cluster of masses (such as in a star), gravity acts more strongly between all particles. This, in turn, means that gravity exerts a ‘collapsing pressure’ on a star.

So this means that for a star to stay stable, an equal force must act pushing outwards on the star, to counteract gravity. This force is known as solar radiation pressure. It is created by nuclear fusion, releasing incredibly immense amounts of energy. [6]

3.2: Nuclear Fusion

Physicists have observed that stars shine due to nuclear fusion occurring in their cores. Nuclear fusion occurs when two atoms of the same type of element atomically fuse to create a new element. The Sun performs the nuclear fusion of 60 million metric tonnes of hydrogen each second. The most common type of nuclear fusion that occurs in stars is the fusion of hydrogen into helium. A way of describing this nuclear reaction would be this nuclear equation, inspired by a written chemical equation: [7]



<math xmlns="http://www.w3.org/1998/Math/MathML" data-lazy-srcset=H12&#xA0;+&#xA0;H13&#xA0;&#x2192;&#xA0;He14&#xA0;+&#xA0;n01

“>

Or:

Hydrogen + Hydrogen → Helium

Physicists have also observed that nuclear fusion releases immense amounts of energy. This energy powers entire cities, nuclear bombs, and of course: stars. This energy in stars is so powerful in fact, that it can counteract gravity in a star, exerting solar radiation pressure and emitting heat and light. [8]

4: Colour in Stars

Some of the energy released during nuclear fusion is converted into photonic (light) energy. This energy corresponds to a frequency on the electromagnetic spectrum. A physicist named Wilhelm Wien created the following equation to determine the colour of a star in 1893, known as Wien’s Displacement Law:



<math xmlns="http://www.w3.org/1998/Math/MathML" data-lazy-srcset=&#x3BB;max=CT

“>

where lambda () represents the wavelength of the colour associated with the starlight, in metres (m), Wien’s Constant (C) represents the value 2.898 x 10-3 mK and temperature (T) represents the heat of the star, in degrees Kelvin (K).

This equation can easily be put into practice, using real astronomical data that scientists have observed. Let us take the famous star Betelgeuse in the constellation of Orion, as an example.

The temperature that scientists have recorded of Betelgeuse would be roughly 3300 K. Dividing Wien’s constant by this gives us a value of 9.05757576 x 107 m, roughly 905.757576 nm. When compared with the wavelength diagram in Figure 4, this shows us that much of Betelgeuse’s emitted light is in the infrared part of the spectrum. Further information as to why this is is available in Appendix C. [9]

Figure 4: The known electromagnetic spectrum shown on a simple wavelength diagram.

White Dwarfs and the Effects of Pauli’s Exclusion principle on Electron Degeneracy

5: White Dwarfs

White dwarfs are formed by stars which lack the gravitational attraction to create the immense pressures neutrons experience in neutron stars, but enough to not completely collapse into a lower equilibrial state. The value that separates these types of stars is known as the Chandrasekhar limit, at roughly 1.4 solar masses. This is equivalent to roughly 2.784 x 1030 kg. [10]

White dwarfs are some of the densest objects known, with roughly the mass of the Sun and the volume of the Earth. Usually, gravity will cause the star to collapse catastrophically. However, because these stars exist at all, there must be another force to keep the star in equilibrium, as discussed in Section 3. But if this star does not produce solar radiation pressure from nuclear fusion anymore, then there must be a different opposing force. This is where electron degeneracy occurs in white dwarfs as a direct astrophysical example of Pauli’s exclusion principle. [11]

6: Pauli’s Exclusion Principle

Pauli’s exclusion principle states that no two electrons in the same atomic shell can have the same four quantum numbers, or be in the same quantum state. This quantum number, in our case, is specific to quantum spin. Let us use a stable Lithium-7 atom, seen in Figure 5, as an example of Pauli’s exclusion principle. [12]

Figure 5: An diagram of the atomic makeup of a Lithium-7 atom.

There is, of course, an exclusion in Pauli’s exclusion principle. In a degenerate gas, energy levels (shells) can become closer, to allow the ‘empty space’ in an atom to compress. This, in turn, means that denser objects can still hold the same mass. This is called electron degeneracy. [13]

7: The Effects of Pauli’s Exclusion Principle on White Dwarfs

White dwarfs are formed from degenerate gases. This means that the exclusion in Pauli’s exclusion principle applies. In Section 5 about hydrostatic equilibrium, the fact that a different opposing force to counteract gravity is needed to defy collapse was discussed. This force comes in the form of electron degeneracy pressure, touched upon in Section 6. When gravity inevitably compresses a white dwarf, it will increase the number of electrons in the given volume. If the Pauli exclusion principle is applied to this, it is observed that this will only increase the kinetic energy in the electrons, which in turn increases pressure. This pressure is dependent only on volume and not temperature. [14] [15]

.

Discussion

This knowledge of white dwarf stars holds great importance in astrophysics. Knowing this, phenomena such as neutron stars (and beyond) can be understood better. For example, neutron stars are linked very closely to and share many phenomena with black holes, an area of great interest to physicists. If work to understand the phenomena of white dwarfs and expand our revelations past the Chandrasekhar limit is conducted, a great deal about neutron stars and black holes could be learnt.

One property of white dwarfs, for example, is their ability to live longer than almost everything else in the universe, perhaps even until the universe’s very demise. Interestingly, black holes share almost the same property. Many cosmologists believe that by learning about what will happen in the future of the universe, what has happened in the past could also be understood similarly. Another intriguing fact about white dwarfs and neutron stars is the rotating type of neutron stars, known as pulsars (pulsating stars). These stellar emitters are used in astronomy for celestial navigation, such as the ‘celestial map’ inscribed onto the golden records attached to the Voyager spacecraft.

Conclusion

In conclusion, humans have come far since they first looked up and pondered about the stars. Extremely complex quantum mechanics and astrophysics are now common knowledge to physicists, and scientists know more about the life and death of stars than ever before. There is no doubt that in the future that this species will learn more about space and its phenomena than ever imagined.

Appendices

A: Black Holes

A black hole is the most extreme and exotic object known to man. It is (in its most basic form) an infinite gravitational well. It is also the strongest concentration of gravity found in the entire universe. The gravitational strength of a black hole is so strong that not even light can escape. The boundary beyond which light cannot escape is known as the event horizon. No object can escape a black hole, as it requires faster-than-light travel, which is impossible according to Albert Einstein’s theory of general relativity, as all physical objects have mass. The discovery of black holes by famous physicists as Stephen Hawking and David Finkelstein is, in fact, a direct offspring of Einstein’s work.

At the centre of a black hole is a point known as a singularity. A singularity is a point of infinite density and gravity which then, in turn, means it exerts extremely strange phenomena. Another example of a singularity is the theoretical big bang. It is now believed that black holes are created when the gravitational attraction of a star is so strong that it not only does withstand the pressures of electron and neutron degeneracy but also can absorb light. This is why black holes are sometimes referred to as ‘frozen stars’.

Extensive research has been carried out on these incredible phenomena of the universe, such as with the Laser Interferometer Gravitational-Wave Observatory (LIGO) experiments in 2016 to detect gravitational waves, and the Event Horizon Telescope (EHT) to take the first-ever image of the effects of a black hole in 2019. Certainly, black holes will become a defining factor of astrophysics in the future. [16] [17] [18]

B: Heavy Element production in Supernovae

Uranium is a vital resource in everyday life for powering nuclear fission in reactors that in turn supply 16% of all of Earth’s modern electricity supply. It is then important to understand where this precious resource came from. It is most commonly accepted in the scientific community that almost all of the Earth’s supply of uranium was created in supernovae. Supernovae are created by massive stars which once having achieved iron cores, have wrung all the energy they can out of nuclear fusion – fusion reactions that form elements heavier than iron consume energy rather than produce it. [4]

This then means that stars rapidly lose energy and begin to collapse. During this rapid gravitational collapse, heavy elements undergo extreme nuclear fusion in just a matter of seconds until they create uranium, the heaviest naturally-occurring element. The energy released in this insane amount of fusion is a component of the incredible amounts of energy released in a supernova. [19]

C: Betelgeuse

Physicists and astronomers believe that the reason why Betelgeuse emits so much infrared radiation is that it is quite a cold star. It is not rare for a star to be this cold in the universe, however, it is rare to be as hot as Betelgeuse’s neighbour – Rigel – at 12,100 K. It is important to note that the reason that Betelgeuse has its distinct red colour in the night sky is believed by scientists to be due to the observation that not all of Betelgeuse’s photonic frequencies are in the infrared section of the electromagnetic spectrum. Scientists believe that roughly 85% of Betelgeuse’s emitted radiation is infrared, with the rest being visible. This is fair to observe as if Wien’s law is recalled: technically only the maximum frequency a star could emit is stated, indicated by the subscript max. This means that Betelgeuse can still radiate photons with frequencies within the visible part of the electromagnetic spectrum, albeit still in the cooler end of the spectrum (shown in Figure 4), at red. [20]

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About the author

Benjamin Santiago Williams is an English-Spanish young scientist and physicist studying the fields of astronomy, physics, and engineering, born in Bristol, UK, in 2006. He attends The Castle School Thornbury, where he is active with the CastleSTEM project. His hobbies include sci-fi and all things nerdy!