10.1.3.2. Neutron Stars

Table of Contents

Extreme compact stars

A neutron star is the collapsed core left behind after a massive star ends its life in a supernova. It is one of the most compact objects in the universe. A neutron star contains about the mass of the Sun compressed into a sphere only about $10$ to $15 \, \text{km}$ in radius. This means its density is enormous, much greater than the density of ordinary matter on Earth.

Neutron stars form when gravity crushes the core of a dying star so strongly that electrons and protons are forced together, producing neutrons. The object that remains is supported not by ordinary gas pressure, but mainly by quantum pressure associated with densely packed neutrons. At this stage, the star does not continue collapsing into a black hole unless its mass is too large.

A neutron star is an object with roughly stellar mass and city-sized radius, supported against further collapse mainly by neutron degeneracy pressure and nuclear forces.

Size, mass, and density

Neutron stars are tiny compared with normal stars, but they are extremely massive. A typical neutron star may have a mass around

$$
M \approx 1.4 M_\odot
$$

where $M_\odot$ is the mass of the Sun.

Its radius is roughly

$$
R \approx 10^4 \text{ to } 1.5 \times 10^4 \, \text{m}
$$

The average density can be estimated from

$$
\rho = \frac{M}{\frac{4}{3}\pi R^3}
$$

Using solar-scale mass and a radius of only a few kilometers gives densities of about

$$
\rho \sim 10^{17} \, \text{kg/m}^3
$$

This is comparable to the density of atomic nuclei.

Property	Typical neutron star value
Mass	about $1.4 M_\odot$
Radius	about $10$ to $15 \, \text{km}$
Average density	about $10^{17} \, \text{kg/m}^3$
Surface gravity	enormously larger than Earth's

Internal structure

A neutron star is not uniform all the way through. The outer part is a crust made of nuclei and electrons packed very closely together. Deeper inside, matter becomes more and more compressed, and neutrons dominate. The core may contain mostly neutrons, along with a smaller number of protons, electrons, and perhaps more exotic forms of matter. The exact composition of the deep core is still an active area of research.

The pressure inside a neutron star is so high that normal atomic structure cannot survive. Electrons are no longer orbiting nuclei in the usual way. Matter exists in forms not found naturally on Earth.

Rendering scientific sketch…

Simple structure of a neutron star

Gravity at the surface

The gravitational field at the surface of a neutron star is extremely strong. Using the Newtonian expression,

$$
g = \frac{GM}{R^2}
$$

we find that surface gravity is many billions of times stronger than gravity on Earth. A person could not stand on a neutron star in any ordinary sense. The gravity would crush matter severely.

Because gravity is so strong, light leaving the surface loses energy. This produces gravitational redshift. Time also passes more slowly near the surface than far away, according to general relativity.

For a neutron star, the small radius is just as important as the large mass. Since $g = \frac{GM}{R^2}$, a very small $R$ makes the surface gravity enormous.

Rotation

When the original stellar core collapses, it spins faster, just as an ice skater spins faster when pulling in their arms. This happens because angular momentum is conserved. As a result, neutron stars can rotate very rapidly. Some spin many times each second, and some spin hundreds of times each second.

These rapidly spinning neutron stars can produce regular pulses of radiation if their magnetic axis is not aligned with their rotation axis. Such objects are observed as pulsars, which are discussed elsewhere in more detail. Here it is enough to note that neutron stars are often remarkable for their fast rotation and stable timing.

Rendering scientific sketch…

Rotation and beam geometry of a neutron star

Magnetic fields

Neutron stars often have extremely strong magnetic fields. During collapse, the magnetic field of the original star can become greatly intensified because the star's size shrinks dramatically. Some neutron stars have magnetic fields far stronger than anything we can produce in laboratories on Earth.

A special class of neutron stars, called magnetars, has especially intense magnetic fields. These fields can affect the star's emission and can lead to violent bursts of radiation.

Escape speed and compactness

Because neutron stars are so compact, the escape speed from their surface is a large fraction of the speed of light. The Newtonian estimate is

$$
v_{\text{esc}} = \sqrt{\frac{2GM}{R}}
$$

For a neutron star, this value can be tens of percent of $c$. This shows how close neutron stars are to becoming black holes, though they still remain outside the black hole limit.

A neutron star is not a black hole. It has a real surface. Matter and radiation can still come from that surface, although the gravity is extremely strong.

How neutron stars are observed

Neutron stars are difficult to see directly because they are small and not very bright in visible light. They are usually detected through their effects. Some are seen as pulsars, producing regular radio, X-ray, or gamma-ray pulses. Others are found in binary systems, where gas from a companion star falls onto the neutron star and emits strong X-rays. Some are detected through gravitational effects on nearby objects.

Astronomers can learn about a neutron star's mass, spin, and magnetic activity from these observations.

Comparison with other stellar remnants

Neutron stars are more compact than white dwarfs, but less extreme than black holes. This makes them an important middle case in the study of collapsed stars.

Object	Typical mass	Typical size	Surface
White dwarf	about stellar remnant mass	Earth-sized	yes
Neutron star	about $1$ to $2$ solar masses	city-sized	yes
Black hole	can be similar or larger	event horizon scale	no ordinary surface

Mass limit

There is an upper limit to how massive a neutron star can be and still resist collapse. If too much mass is packed into the star, neutron degeneracy pressure and nuclear effects are no longer enough to support it. Then further collapse can produce a black hole. The exact maximum mass depends on the behavior of ultra-dense matter, which is still being studied.

Why neutron stars matter in physics

Neutron stars are natural laboratories for extreme physics. They allow scientists to study matter at nuclear density, very strong gravity, rapid rotation, and powerful magnetic fields, all in a single object. They connect ideas from stellar evolution, nuclear physics, relativity, and electromagnetism.

Key picture of a neutron star: solar-scale mass, radius of only about $10 \, \text{km}$, enormous density, intense gravity, rapid rotation, and often very strong magnetic fields.

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