ABSTRACT
Neutron stars are one of the most elusive and mysterious celestial objects to study due to their unique properties and composition. White dwarfs are similar but are formed from low mass stars Here we study and derive the speed of sound inside of a neutron star via Fermi-statistical techniques. We modeled a star as a system of identical particles producing the pressure in the star: neutrons for neutron stars and electrons for white dwarfs. Using the geometrical framework called phase-space, we were able to calculate the momentum and energy of the Fermi gases, followed by the degeneracy pressure and Bulk modulus. Even though astronomers know little about how matter behaves at such high densities, we were able to calculate the theoretical speed of sound in neutron stars to be 1/3 of the speed of light.
1. INTRODUCTION
The goal of this paper is to review and summarize previous works on the subject of compression waves (sound) in highly dense Fermi gas systems, such as neutron stars, white dwarfs, and atomic nuclei, for the purpose of guiding and teaching science students. Fermi-gas systems present states of matter at extreme conditions [1], such as high pressure and high density, and are very interesting from theoretical and experimental points of view. Within these systems, we aimed to calculate the speed of sound.
Neutron stars, for example, are one of the most dense macroscopic objects in the universe after black holes [2]. Weighing roughly twice as much as our Sun with a radius of about 10km, they are unique in the macroscopic, gravitational, and electromagnetic interactions, as well as in microscopic, weak, and nuclear interactions. This makes them one of the few points of intersection between classical and quantum physics.
Because these astronomical phenomena are still not completely understood today, and such conditions cannot be obtained under laboratory conditions, [3] the study of such interactions can provide complementary information to that obtained from particle accelerators such as the Large Hadron Collider, help with the construction of more correct models of neutron stars and with adequate interpretation of their observations. It may also lead to important findings on the theory of stellar evolution in general, and about the properties of fundamental physics at such extremes.
In this work, to calculate the speed of compression waves in neutron stars, white dwarfs, and nuclear medium we will treat them as Fermi-gas systems [4]. A Fermi-gas system is an ideal gas made of indistinguishable fermions; and fermions are particles of half-integer spin [1]. Some examples of Fermions are electrons, neutrons, and protons. Due to the Pauli Exclusion Principle, those particles can not occupy the same quantum energy levels [1]. This principle also determines the compression of the gas, and as a final consequence, the speed of sound in these systems [5] [6]. To derive the energy of Fermi-gas at low temperatures we will use the quantization rule. And then find the pressure and the bulk modulus of the gas, which will allow us to calculate the speed of sound in these systems.
2. THEORY OUTLINE
The speed of compression waves (or speed of sound) in a gas can be calculated as followings
here B is the Bulk modulus (which represents the compressibility of the medium) and ρ is the density of the medium. The Bulk modulus is defined as
where V is the volume, P is the pressure, and E is the total energy of the gas. Thus to find the speed of sound in a star medium (such as neutron stars and/or white dwarfs) we need to derive the pressure P and/or energy E as a function of volume. We will use a Fermi-gas model, which is the most simple and successful microscopic model when describing the property of stars. This model allows the description of the interior of stars and gives us the energy and the pressure of the star.
2.1 Fermi gas model
We are going to model a star as a system of identical particles producing the pressure in the star: neutrons for Neutron Stars and electrons for White Dwarfs. These particles are called Fermions because they have half-integer spin and obey Fermi-Dirac statistics. A gas of fermions is called Fermi gas and is considered a quantum gas due to the quantum nature of the particle-particle interaction. The Fermi-Dirac statistics means that no two identical fermions can occupy one quantum mechanical state, it is also called Pauli’s exclusion principle. In order to find the bulk modulus and the pressure inside stars, we will need to find the energy of the Fermi-gas system as a function of volume. We will have to introduce the 6-D phase space and quantize this phase-space. The details of these calculations will be provided in the following sections.
2.2 Phase-space and Quantization
To calculate the various properties of the Fermi gas we will use a geometrical framework called phase space. For 1-D the phase-space is position vs momentum plane: (x, p = mv) (Fig.1). Within this momentum-space square, particles occupy smaller “cells”.
Figure 1: One-dimensional Phase Space
In 3-dimensions case, this phase-space is six-dimensional: (x, y, z, Px, Py, Pz). In this six-dimensional space composed of six positions and six momentum coordinates (Fig.2), each point corresponds to a particular position and momentum, which is a unique state of the particle. The state of a system of particles corresponds to a certain distribution in the phase space.
Figure 2: Six-dimensional space of positions (left) and momentum coordinates (right)
In Quantum Mechanics the phase-space is quantized, meaning that a particle cannot be localized to one point and occupies a minimum volume on the phase-space diagram. This minimum volume is related to the Heisenberg’s uncertainty principle and equals to
The Pauli Exclusion Principle states that no two neutrons can occupy identical states. We calculate the number of states (and consequently the number of particles) by
This would mean
where the numerator is the total volume, R is the radius of the sphere, and pF is the Fermi momentum. Rearranging we would have the Fermi momentum
This equation will be important later on.
2.3 Fermi-gas energy and pressure
We used the kinetic energy of one particle and integrated it over all the particles to calculate the total kinetic energy of the Fermi-gas. We began by using the number of particles, the volume in the real space, and the volume in the momentum space.
So,
Following the Bernoulli’s Equation, we use the relationship that energy is equivalent to pressure distributed within a volume ∆E =−P∆V, and we were able to calculate the degeneracy pressure as follows
2.4 Speed of Sound
We were then finally able to calculate the speed of sound in the stars as
Inputting some numbers and we were able to get the number of particles in a neutron star
In the same way, the equation for a White Dwarf gives us
2.5 Results
Through our calculations, we were able to find the speed of sound in a neutron star
Comparing with the speed of light, we were able to find out that the speed of sound in a white dwarf is one third the speed of light. This result is 42% different from previous studies [6].
2.6 Conclusion
Neutron stars are amazing puzzles of nature, whose solution is sought by many unrelated branches of science, such as particle physics and gravitational physics [3]. This makes neutron star the perfect cosmic laboratories for the testing of basic physical concepts. Despite the extensive studies in multiple research centers over the last years, both theoretical and observational studies of such stars still have much to uncover.
2.7 Acknowledgments
We thank professor Roman Senkov (LaGuardia Community College) for providing us the knowledge to be able to understand and write about this very interesting topic. We thank the Honors Program of the LaGuardia Community College for offering the opportunity of learning and researching while undergraduate students. And we thank Heriberto Vasquez Carrasco, for reading this paper, for correcting our grammar, and for providing emotional support.
REFERENCES
[1] C. Kittel, Thermal physics. San Francisco: W. H. Freeman, 2d ed. ed., 1980.
[2] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of
Compact Objects. Wiley, 2008.
[3] A. Y. Potekhin, “The physics of neutron stars,” Physics-Uspekhi, vol. 53, pp. 1235–1256, dec 2010.
[4] R. Pathria, Statistical Mechanics. International series of monographs in natural philosophy, Elsevier
Science & Technology Books, 1972.
[5] W. Rapando, M. Amuyunzu, Y. Ayodo, and W. Isoe, “Internal pressure and speed of sound in
neutron stars,” 10 2018.
[6] P. Bedaque and A. Steiner, “Sound velocity bound and neutron stars,” Physical review letters,
vol. 114, 08 2014.
APPENDIX
The equation for the radius of a White Dwarf can be written as
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