Question

When a star has nearly bumed up its intemal fuel, it may become a white dwarf. It is crushed under its own enonnous gravitational forces to the point at which the exclusion principle for the electrons becomes a factor. A smaller size would decrease the gravitational potential energy, but assuming the electrons to be packed into the lowest energy states consistent with the exclusion principle. "squeezing" the potential well necessarily increases the ener gies of all the electrons (by shortening their wavelengths). If gravitation and the electron exclusion principle are the only factors, there is a mini. mum total energy and corresponding equilibrium radius. (a) Treat the electrons in a white dwarf as a quantum gas. The minimum energy allowed by the exclusion principle (see Exercise 67 ) is $$ U_{\text {elecimns }}=\frac{3}{10}\left(\frac{3 \pi^{2} \hbar^{3}}{m_{e}^{3 / 2} V}\right)^{2 / 3} N^{5 / 3} $$ Note that as the volume \(V\) is decreased, the energy does increase. For a neutral star. the number of electrons, \(N\), equals the number of protons. Assuming that protons account for half of the white dwarf's mass \(M\) (neutrons accounting for the other half). show that the minimum electron energy may be written $$ U_{\text {electrons }}=\frac{9 \hbar^{2}}{80 m_{e}}\left(\frac{3 \pi^{2} M^{5}}{m_{\mathrm{p}}^{5}}\right)^{1 / 3} \frac{1}{R^{2}} $$ where \(R\) is the star's radius. (b) The gravitational potential energy of a sphere of mass \(M\) and radius \(R\) is given by $$ U_{\operatorname{mav}}=-\frac{3}{5} \frac{G M^{2}}{R} $$ Taking both factors into account, show that the minimum total energy occurs when $$ R=\frac{3 h^{2}}{8 G}\left(\frac{3 \pi^{2}}{m^{3} m_{p}^{5} M}\right)^{1 / 3} $$ (c) Evaluate this radius for a star whose mass is equal to that of our Sun, \(\sim 2 \times 10^{30} \mathrm{~kg}\). (d) White dwarfs are comparable to the size of Eath. Does the value in part (c) agree?

Short answer

The equilibrium radius for a star whose mass is equal to that of our sun is found by substituting the known values into the derived formula. The obtained value should be compared to the radius of Earth to see if the theoretical model reflects observations.

Step-by-step solution

Reformulate the Electron Energy

The given expression for the electron energy is dependent on the number of particles N, which is expressed by the star's mass M. We know that \( N = M / m_p \) with \( m_p \) being the proton mass. Substituting this into the expression, we get \( U_{electrons} = \frac{3}{10} \left( \frac{3 \pi^{2} \hbar^{3}}{m_{e}^{3/2} V} \right)^{2/3} \left( \frac{M}{m_p} \right)^{5/3} \) Now we express the volume V in terms of the star's radius. We have \( V = \frac{4}{3} \pi R^3 \) so the expression becomes \( U_{electrons} = \frac{3 \hbar^2}{80 m_e} \left( \frac{3 \pi^{2} M^5}{m_{p}^5} \right)^{1/3} \frac{1}{R^2} \)

Minimize the Total Energy

The total energy of the star is the sum of the electron energy and the gravitational potential energy: \( U_{total} = U_{electron} + U_{grav} \) To minimize this, we take the derivative of the total energy with respect to the radius R, set it equal to zero and solve for R. This gives: \( R = \frac{3 \hbar^2}{8 G} \left( \frac{3 \pi^2}{m_e^3 m_p^5 M} \right)^{1/3} \)

Evaluate the Radius

We substitute the known values into the expression for R. The mass of the Sun M is about \( 2 \times 10^{30} \) kg, the values for the other constants can be found in physics reference books. With these substitutions, we obtain a numerical value for R.

Key concepts

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental concept in quantum mechanics which states that two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously. In the context of white dwarfs, this principle is crucial because it explains why these stars do not collapse under their own gravity, despite their high density.

The electrons in a white dwarf are fermions and, as such, must obey the Pauli Exclusion Principle. When the star exhausts its nuclear fuel and can no longer produce energy through fusion, gravity tries to squeeze it into a more compact space. However, as electrons are pushed closer together, they cannot all descend to the lowest energy level due to the exclusion principle. Instead, they must occupy higher energy levels, creating a degenerate electron gas. This gas exerts a pressure that counteracts gravitational compression, determining the white dwarf's size and contributing to its remarkable stability.

Quantum Gas

In astrophysics, a quantum gas is a collection of particles that exhibit quantum mechanical effects on a macroscopic scale. For white dwarfs, the electrons can be treated as a non-interacting quantum gas, where thermal effects are negligible compared to quantum effects due to the star's dense nature and low temperature.

The behavior of this electron gas within a white dwarf is described by quantum statistics, which accounts for the Pauli Exclusion Principle. As the white dwarf's material is compressed, the electron gas reaches a state known as electron degeneracy. This state provides an outward pressure that counters the inward pull of gravity. In solving for the properties of a white dwarf, as done in the textbook exercise, the concept of a quantum gas allows us to understand the distribution of electron energies and how it relates to the star's size and stability.

Gravitational Potential Energy

Gravitational potential energy (U_grav) is the energy an object possesses due to its position in a gravitational field. For celestial bodies like white dwarfs, this energy is a measure of the work done by gravity as mass is brought together from infinity to form a spherical object.

In the case of a white dwarf, which is a highly condensed object with a mass comparable to the Sun but a radius similar to Earth's, gravitational potential energy plays a significant role in its equilibrium state. The balancing of electron degeneracy pressure against the force of gravity defines the white dwarf's size. The formula for gravitational potential energy provided in the textbook solution, \( U_{\text{grav}} = -\frac{3}{5} \frac{G M^{2}}{R} \), shows that as the radius (R) decreases, the negative gravitational potential energy increases in magnitude, indicating a stronger gravitational pull. This is critical to calculating the white dwarf's minimum total energy.

Stellar Mass-Radius Relationship

The stellar mass-radius relationship for white dwarfs is a theoretical connection between the star's mass (M) and its radius (R). This relationship arises from balancing the inward gravitational force with the outward pressure generated by the degenerate electron gas in the star's core.

This balance leads to an equilibrium radius that can be mathematically derived, as shown in the textbook solution. White dwarfs are exceptions to the typical mass-radius relationship observed in main-sequence stars, where larger mass generally implies a larger radius. Due to the electron degeneracy pressure, which is independent of temperature, the mass-radius relationship for white dwarfs is such that adding mass to a white dwarf can lead to a smaller radius — a key point that highlights the exotic properties of these compact stars. The solution derived in the textbook confirms this unusual trend and provides us with a tool to predict the size of a white dwarf given its mass.