Assume that a neutron star is a highly degenerate...

Assume that a neutron star is a highly degenerate non-relativistic gas of neutrons in a spherically symmetric equilibrium configuration. It is held together by the gravitational pull of a heavy object with mass $M$ and radius $r_0$ at the center of the star. Neglect all interactions among the neutrons. Calculate the neutron density as a function of the distance from the center, $r$, for $r>r_0$.

Key Concepts

Fermi Gas and Degeneracy Pressure

In dense astrophysical objects, fermions such as neutrons obey the Pauli exclusion principle, which prevents them from all occupying the same quantum state. This creates a quantum mechanical pressure, known as degeneracy pressure, that exists even at zero temperature. This pressure is vital in balancing gravitational forces in compact objects like neutron stars, helping prevent their collapse.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance between the inward gravitational force and the outward pressure gradient within a star. In the context of stellar astrophysics, this equilibrium ensures that the star maintains a stable structure, and it is described by differential equations that relate the pressure gradient to the gravitational forces acting at each radius.

Gravitational Potential in Spherical Symmetry

For a spherically symmetric mass distribution, the gravitational potential depends only on the radial distance from the center of the mass. This concept is critical for determining how gravitational forces vary within a star and for setting up the equations that describe the balance of forces in astrophysical bodies.

Equation of State for a Non-relativistic Degenerate Gas

The equation of state for a non-relativistic degenerate Fermi gas links the pressure and density of the particles. In a neutron star where the particles are highly degenerate and interactions are neglected, this relationship governs how the pressure varies with density, which is crucial for solving the equilibrium configuration under a given gravitational potential.

Radial Density Profiles in Spherically Symmetric Systems

With spherical symmetry, the analysis of the star’s structure simplifies to understanding how density varies with the radial distance from the center. This radial density profile is derived by combining the equation of state with the conditions of hydrostatic equilibrium and the gravitational potential, resulting in a description of how the density decreases as one moves away from the central region.

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