If you were to add up the mass of all the atoms in the Earth, you’d get a number 3 trillion tons more than the Earth’s actual mass. We didn’t have to count all the atoms down to the Earth’s core to figure this out. It’s a consequence of the laws of physics.

Let’s start with a related puzzle. You might have heard that nuclear reactions work by converting mass to energy, according to Einstein’s famous equation E=mc². Most nuclear reactors work by hitting atoms of Uranium-235 with neutrons, breaking the atoms apart and producing more neutrons in a chain reaction. But these nuclear reactions don’t destroy any protons or neutrons. So what mass are they converting into energy?

Here’s an example nuclear fission reaction you might find in a nuclear power plant.¹ At the start, we have one neutron hitting an atom of Uranium-235, which has 92 protons and 143 neutrons. So in total, we have 92 protons and 144 neutrons.

After the fission, we have Caesium-137 with 55 protons and 82 neutrons, Rubidium-96 with 37 protons and 59 neutrons, and 3 extra neutrons ready to continue the chain reaction. So in total, we have 92 protons and 144 neutrons – exactly the same as before.

Now, let’s add up the mass. The standard unit of atomic mass is “daltons”, defined as one twelfth of the mass of a Carbon-12 atom, or about 1.66×10^-27 kg. So how many daltons do we have before and after our reaction?

Somehow, we’ve lost 0.1853 daltons.² How, exactly?

The secret goes back to E=mc². The equation doesn’t just say that mass can become energy, or that energy can become mass. It’s that mass is energy, and energy is mass.

If you wanted to pull a Uranium-235 atom apart into separate protons, neutrons, and electrons, you’d need to put in a bunch of energy to overcome the strong nuclear force that’s holding the nucleus together in the first place.³  The amount of energy you’d need is called the “binding energy” of the U-235 nucleus.

Adding the binding energy to the system literally adds mass. A collection of 92 loose protons, 92 loose electrons, and 143 loose neutrons has a mass of 236.9590 daltons, which is 1.9151 daltons more than an atom of U-235. And using E=mc², we can tell that difference equals 2.858×10^-10 Joules – exactly the binding energy of U-235!

So because an assembled atom is a lower energy system than a bunch of loose particles, it also has a lower mass. And because the fission products are an even lower energy system than U-235 plus a neutron, they have an even lower mass. And again, this is literal mass – the atom has less inertia and less gravity all because it has less energy.

Speaking of gravity, the Earth has a binding energy too. Earth is held together by gravity, not the strong nuclear force, but the principle applies the same way. The gravitational binding energy of Earth is 2.49×10³² joules, which corresponds to a bit less than 3 trillion metric tons of missing mass. So the Earth has about 3 trillion tons less mass than the material that makes it up.

On the scale of the Earth, the amount of missing mass is negligible – less than one billionth the planet’s total mass. But neutron stars, which are many trillions of times denser than Earth, are missing about 15% of their total mass because their gravity is so strong.⁴ With such extreme objects in the universe, crazy physics is bound to happen.