# Density of a Neutron Star

Quote of the Day

Some people get rich studying artificial intelligence. Me, I make money studying natural stupidity.

— Carl Icahn

## Introduction

Figure 1: Example of Neutron Star System. (Source)

Television science programs frequently talk about black holes and neutron stars. A common quote during these programs is something like "a teaspoon of neutron star stuff weighs a billion tons" or some other similar statement. I always find numbers like these interesting to look at in detail.

Let's dig in  ...

## Background

We are going to estimate the average density of a neutron stars, which means that we need to get some mass and volume information.

• According to the Chandrasekhar limit, a neutron star must have a mass at least 1.44 times that of the Sun (mass of the Sun = 1 solar mass).
• According to the Tolman-Oppenheimer-Volkoff limit, the upper limit on a neutron star's mass is between 2 and 3 solar masses. Collapsing stars more massive than the TOV limit form black holes instead of neutron stars.
• The radius of a neutron star is between 10 and 15 km

My calculation is for the average density of neutron star stuff. Note that a neutron star does have structure and it is composed of material whose properties vary with distance from the core.

During my research, it became very clear that the physics of matter in neutron stars and black holes is still being worked out. As in every field, to really know a subject you must understand the corner cases and astronomers are still working this area out. Personally, I think its great because this means that there is more interesting science to come.

## Analysis

Figure 2 shows the details of my analysis.

Figure 2: Estimate of Neutron Star Density.

So my estimate is ~2 billion tons for a teaspoon of neutron star stuff. Since this is a Fermi-type problem, my estimate agrees with the statement of one billion tons per teaspoon.

How does the average density of a neutron star compare to the density of neutron? Figure 3 shows my estimate for the density of a neutron.

Figure 3: Estimate of a Neutron's Density.

So the average density of a neutron and a neutron star are pretty much the same.

## Conclusion

A neutron star does have a density of about a billion tons per teaspoon. Pretty incredible number when you think of it.

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