Question

A white dwarf is a star supported by the pressure of degenerate electrons. As a simplified model for such an object, consider a sphere of an ideal gas consisting of electrons and completely ionized $\mathrm{Si}^{28}$, and of constant density throughout the star. (Note that the assumption of a constant density is inconsistent with hydrostatic equilibrium, since the pressure is then also constant. The assumption that the gas is ideal is also not really tenable. These shortcomings of the model are, however, not crucial for the issues which we wish to consider.) Let $n_{\mathrm{i}}$ denote the density of the silicon ions, and let $n_e=14 n_{\mathrm{i}}$ denote the electron density. (The atomic number of silicon is 14). (a) Find the relation between the mean kinetic energy $\bar{E}_e$ of the electrons and the density $n_e$, assuming that the densities are such that the electrons are "extremely relativistic," i.e., such that the rest energy is negligible compared with the total energy. (b) Compute $\bar{E}_e$ (in MeV ) in the case that the (rest mass) density of the gas equals $\rho=10^{\circ} \mathrm{g} / \mathrm{cm}^3$. Also compute the mean kinetic energy $\bar{E}_{\mathrm{i}}$ of the silicon ions in the central region of the dwarf, assuming that the temperature is $10^8 \mathrm{~K}$ and assuming that the "ion gas" can be regarded as a Maxwell-Boltzmann gas, and hence convince yourself that $\bar{E}_{\mathrm{e}} \gg \bar{E}_1$. (c) If $M$ is the mass of the star, and if $R$ is its radius, then the gravitational potential energy is given by $$ U_{\mathrm{G}}=\frac{3 G M^2}{5 R} . $$ In the case in which the internal energy is dominated by extremely relativistic electrons (as in part (b) above), the virial theorem implies that the total internal energy is approximately equal to the gravitational potential energy. Assuming equality, and assuming that the electrons do not contribute significantly to the mass of the star, show that the stellar mass can be expressed in terms of fundamental physical constants alone. Evaluate your answer numerically and compare it with the mass of the sun, $2 \times 10^{30} \mathrm{~kg}$. (It can be shown that this is approximately the maximum possible mass of a white dwarf.)

Name: Problem 109, Chapter 2: Problems and Solutions on Thermodynamics and Statistical Mechanics
Uploaded: 2021-11-16T08:48:23-08:00
Duration: 6 min 50 s
Channel: Mahnoor Amin

   A white dwarf is a star supported by the pressure of degenerate electrons. As a simplified model for such an object, consider a sphere of an ideal gas consisting of electrons and completely ionized $\mathrm{Si}^{28}$, and of constant density throughout the star. (Note that the assumption of a constant density is inconsistent with hydrostatic equilibrium, since the pressure is then also constant. The assumption that the gas is ideal is also not really tenable. These shortcomings of the model are, however, not crucial for the issues which we wish to consider.) Let $n_{\mathrm{i}}$ denote the density of the silicon ions, and let $n_e=14 n_{\mathrm{i}}$ denote the electron density. (The atomic number of silicon is 14).
(a) Find the relation between the mean kinetic energy $\bar{E}_e$ of the electrons and the density $n_e$, assuming that the densities are such that the electrons are "extremely relativistic," i.e., such that the rest energy is negligible compared with the total energy.
(b) Compute $\bar{E}_e$ (in MeV ) in the case that the (rest mass) density of the gas equals $\rho=10^{\circ} \mathrm{g} / \mathrm{cm}^3$. Also compute the mean kinetic energy $\bar{E}_{\mathrm{i}}$ of the silicon ions in the central region of the dwarf, assuming that the temperature is $10^8 \mathrm{~K}$ and assuming that the "ion gas" can be regarded as a Maxwell-Boltzmann gas, and hence convince yourself that $\bar{E}_{\mathrm{e}} \gg \bar{E}_1$.
(c) If $M$ is the mass of the star, and if $R$ is its radius, then the gravitational potential energy is given by
$$
U_{\mathrm{G}}=\frac{3 G M^2}{5 R} .
$$

In the case in which the internal energy is dominated by extremely relativistic electrons (as in part (b) above), the virial theorem implies that the total internal energy is approximately equal to the gravitational potential energy. Assuming equality, and assuming that the electrons do not contribute significantly to the mass of the star, show that the stellar mass can be expressed in terms of fundamental physical constants alone. Evaluate your answer numerically and compare it with the mass of the sun, $2 \times 10^{30} \mathrm{~kg}$. (It can be shown that this is approximately the maximum possible mass of a white dwarf.)

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 109

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Solved by Expert Mahnoor Amin

11/16/2021

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The energy of a relativistic particle is given by: \[ E = \gamma m c^2 \] where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor. For extremely relativistic electrons, we can approximate $\gamma \approx \frac{E}{m c^2}$ and the mean Show more…

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A white dwarf is a star supported by the pressure of degenerate electrons. As a simplified model for such an object, consider a sphere of an ideal gas consisting of electrons and completely ionized $\mathrm{Si}^{28}$, and of constant density throughout the star. (Note that the assumption of a constant density is inconsistent with hydrostatic equilibrium, since the pressure is then also constant. The assumption that the gas is ideal is also not really tenable. These shortcomings of the model are, however, not crucial for the issues which we wish to consider.) Let $n_{\mathrm{i}}$ denote the density of the silicon ions, and let $n_e=14 n_{\mathrm{i}}$ denote the electron density. (The atomic number of silicon is 14). (a) Find the relation between the mean kinetic energy $\bar{E}_e$ of the electrons and the density $n_e$, assuming that the densities are such that the electrons are "extremely relativistic," i.e., such that the rest energy is negligible compared with the total energy. (b) Compute $\bar{E}_e$ (in MeV ) in the case that the (rest mass) density of the gas equals $\rho=10^{\circ} \mathrm{g} / \mathrm{cm}^3$. Also compute the mean kinetic energy $\bar{E}_{\mathrm{i}}$ of the silicon ions in the central region of the dwarf, assuming that the temperature is $10^8 \mathrm{~K}$ and assuming that the "ion gas" can be regarded as a Maxwell-Boltzmann gas, and hence convince yourself that $\bar{E}_{\mathrm{e}} \gg \bar{E}_1$. (c) If $M$ is the mass of the star, and if $R$ is its radius, then the gravitational potential energy is given by $$ U_{\mathrm{G}}=\frac{3 G M^2}{5 R} . $$ In the case in which the internal energy is dominated by extremely relativistic electrons (as in part (b) above), the virial theorem implies that the total internal energy is approximately equal to the gravitational potential energy. Assuming equality, and assuming that the electrons do not contribute significantly to the mass of the star, show that the stellar mass can be expressed in terms of fundamental physical constants alone. Evaluate your answer numerically and compare it with the mass of the sun, $2 \times 10^{30} \mathrm{~kg}$. (It can be shown that this is approximately the maximum possible mass of a white dwarf.)

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Key Concepts

Degenerate Electron Pressure

This concept refers to the pressure exerted by a gas of electrons in conditions where quantum mechanical effects become significant. In white dwarfs, electrons are packed so closely that the Pauli exclusion principle prevents them from all occupying the lowest energy state. This 'degeneracy pressure' is independent of temperature and provides the support against gravitational collapse in the absence of thermal pressure, making it a key player in the structure and stability of these stars.

Extremely Relativistic Electrons

In very dense stellar objects like white dwarfs, the electrons can be so energetic that their kinetic energy far exceeds their rest mass energy, making them 'extremely relativistic.' In this regime, the energy is approximately given by E ? pc instead of the non-relativistic E ? p²/(2m). This change in behavior alters the relationship between the electron kinetic energy and density, and it is central to understanding the pressure and energy balance in the star.

Maxwell-Boltzmann Statistics for Ions

While the electrons in a white dwarf form a degenerate Fermi gas, the heavier ions are typically less dense and can often be treated classically using Maxwell–Boltzmann statistics. This model assumes that the ions are in thermal equilibrium and that their distribution of speeds follows the Maxwell-Boltzmann distribution, leading to an average kinetic energy proportional to the temperature. This separation in treatment is important because it explains why, under stellar conditions, the electrons dominate the pressure while the ions contribute negligibly to the internal thermal energy.

Gravitational Potential Energy and Virial Theorem

The gravitational potential energy in a self-gravitating sphere, such as a white dwarf, is a measure of how bound the system is by gravity. The virial theorem connects this gravitational energy to the internal (kinetic) energy of the system. Specifically, for systems in equilibrium, a relation is established that balances the total gravitational potential energy with the total internal energy. This balance is crucial for understanding the overall stability and structure of the star and for deriving relationships between macroscopic stellar parameters like mass and radius.

Chandrasekhar Limit

The Chandrasekhar limit is the theoretical maximum mass for a stable white dwarf star, derived by equating the degeneracy pressure of electrons with the gravitational force attempting to compress the star. It emerges from fundamental constants, such as the gravitational constant, the speed of light, and the properties of electrons, and represents the threshold above which electron degeneracy pressure can no longer support the star against gravitational collapse. This limit explains why white dwarfs do not exist above a certain mass and plays a pivotal role in the evolution of stars in astrophysics.

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