# What is the binding energy of a neutron star?

Neutrons which constitute a neutron star have a rest mass that is greater when separated from the star because they are bound with a certain potential energy. This potential energy causes the system to have less mass. So my question is how much less? Is it significant? Do we measure it?

• It's a bit confusing to call binding energy as potential energy. It's negative potential energy… a deficit that needs to be added to disassemble the system. – Blackbody Blacklight Jul 26 '15 at 4:55
• Because the potential energy causes the system to have more mass. When you lift a brick you do work on it. You add energy to it. You give it potential energy. You increase its mass. When you drop it, this potential energy is converted into kinetic energy which gets dissipated, then you've got a mass deficit. – John Duffield Jul 26 '15 at 9:40

The gravitational mass of a neutron star is quite a lot less than its baryonic rest mass (plus the mass associated with the kinetic energy of its contents), because a bound neutron star, by definition, must have a total energy (the sum of its internal energy and gravitational potential energy) that is less than zero.

In a “normal star” this is also true, bit the difference is that the gravitational potential energy of a neutron star can be comparable with its rest mass energy.

How significant is this? It depends on the baryonic mass of the neutron star and the equation of state of the dense matter. For a typical neutron star of 1.4 solar masses and 10km radius, the order of magnitude estimate for binding energy as a multiple of rest mass energy, $GM/Rc^2$, is about 0.2, suggesting a significant reduction in the gravitational mass compared with the baryonic mass.

• Rob, is there a typo above? It reads as if the total energy of the neutron star is less than zero. – John Duffield Jul 26 '15 at 9:35
• @JohnDuffield Precisely so, as I have defined the total energy. The sum of the internal (kinetic) energy and gravitional PE is less than zero for a bound star. – Rob Jeffries Jul 26 '15 at 9:43

This paper is interesting. It uses the method of calculating the number of nucleons in the neutron star, $N$, based on the radius, $r$, the number density as a function of radius, $n(r)$, and the metric function $\lambda$, which comes from the equations of general relativity: $$N=\int_0^R 4\pi r^2e^{\lambda/2}n(r)dr=\int_o^R4\pi r^2 n(r)\left[1-\frac{2m(r)}{r}\right]^{\frac{1}{2}}dr$$ The binding energy, $BE$, is then $$BE=Nm_b-M$$ where $m_b$ is the mass of a nucleon and $M \equiv m(R)$.

What is the binding energy of a neutron star?

What Rob said is about right. It's about a fifth of the original mass-energy. See Wikipedia: "Its mass fraction gravitational binding energy would then be 0.187".

Neutrons which constitute a neutron star have a rest mass that is greater when separated from the star because they are bound with a certain potential energy.

Yes. When you lift a neutron all the way out of a neutron star, you increase its mass. It's a bit like lifting a brick. You do work on it. You add energy to it. You give it potential energy. You increase its mass. When you drop it, some of the internal E=mc² kinetic energy is converted into external kinetic energy. This typically gets radiated away, and then you're left with a mass deficit. Of course, to be absolutely precise you consider the Earth too, but it's so much bigger than the brick that whilst momentum is equal and opposite, kinetic energy is not. It's so slight for the Earth that we disregard it.

This potential energy causes the system to have less mass.

Like Blackbody Blacklight said in the comment, potential energy causes the system to have more mass, not less. See what I said above about the brick. Then imagine you start with a zillion neutrons in space. You drop them and they all fall together. As this occurs some of their internal E=mc² kinetic energy is converted into external kinetic energy. This typically gets radiated away, and then you're left with a mass deficit. The neutron star mass is about 20% less than the mass of the original neutrons. That's significant.

Note though that things can get a bit confusing because the zero level for gravitational potential energy is set at infinity. See hyperphysics. So we say that the gravitational potential energy of the neutrons in the neutron star is negative. In similar vein we say the binding energy is negative. Note though that there isn't anything in there that's actually made of negative energy. All you've got in there is neutrons, and they're made of positive energy. There's just less of it, that's all. The mass of each and every neutron has decreased. You hear of things like invariant mass, but when it comes to general relativity, mass varies. See Wikipedia. In fact, it isn't limited to general relativity. The mass of a hydrogen atom is less than the mass of an electron plus the mass of a proton. And because the electron is less massive than the proton, most of the mass deficit can be attributed to the electron. It plays the part of the brick.