Gravitational bonding energy of a galaxy

Question

If most mass from a proton comes from the nuclear force's bonding energy between quarks that make-up this proton, why wouldn't most mass from a galaxy come from the gravitational force's bonding energy between all traditional matter that make-up the galaxy?

Answer 1

Although it is often said that the mass of the proton comes mainly from the binding energy this is not really the case. Binding energy reduces the mass of the bound system so the bound system has a lower mass than its component parts. I go into this in my answer to the question Quarks in a hadron: where does the mass come from? The extra mass of the proton does come from strong force interactions, but to call it binding energy is at best an oversimplification. The binding energy is the change in energy when we bring in the component objects from infinity. For forces like gravity or the electrostatic this is well defined because the force goes to zero at infinity. The strong force does not go to zero at infinity, so the change in energy when we bring in the component objects from infinity cannot be defined.

So in principle the total mass of a gravitationally bound system, like the Earth and the Moon, is slightly less than the total you get if you add together the mass of the Earth and the mass of the Moon. However gravity is a weak force and leaving aside extreme systems like black holes the gravitational binding energy is a tiny fraction of the total mass.

To take a specific example, the gravitational binding energy of the Milky Way is around $10^{61}$ ergs and this is equivalent to about $10^6$ solar masses. But the mass of the Milky way is in the region of $10^{12}$ solar masses. So the gravitational binding energy of the Milky Way is about 0.0001% of its mass.