Gravitational binding energy as alternative to dark matter? Pondering this question: Casimir effect and negative mass
and, in particular, the response of John Rennie "as the mass of any bound system is slightly less than the mass of its parts" I thought: "This statement hold true even for gravitationally bounded system?"
If we have m1 and m2 so close that their gravitational interaction is not negligible, the system composed of m1 and m2 has a total weight less than m1 + m2 ?
If so, if we consider 2 galaxies, since each galaxy is composed by masses (M) and gravitational binding energy between those masses (E), it is possible that the gravitational attraction between those 2 galaxies is proportional to (M1 + E1/c^2) * (M2 + E2/c^2)  ? (with E1 and E2 < 0)
Googling "Gravitational binding energy as alternative to dark matter" i have found this that should be relevant for the discussion: An explanation for dark matter and dark energy consistent with the standard model of particle physics and General Relativity
 A: It isn't an explanation because as an effect it is many orders of magnitude too small and it is in the wrong direction. Dark matter increases the gravitational influence of galaxies.
Approximately speaking, you could assume that the binding energy (which is negative) is at most equal to the gravitational potential energy. Dividing that by $c^2$ then gives a correction to the gravitational mass. So the size of the effect is roughly
$$\frac{\Delta M}{M} \sim -\frac{GM}{Rc^2}$$
For a galaxy like the Milky Way, the mass, including dark matter is roughly $10^{12}M_\odot$ within a radius of 100 kpc. The fraction above is $-5\times 10^{-7}$.
A: That statement about "any bound" system is inaccurate. Some bound systems' mass can be greater than sum of masses of its parts, due to repulsive forces between the parts. For example, mass of nucleus of uranium 235 is greater than sum of masses of its decay products, because of the strong enough EM repulsive forces between protons.
Of course, for gravity, we don't expect such repulsive forces being involved. Gravity is known to be always attractive, so total energy of purely gravitationally bound system is lower than energy of parts in a disassembled state.

If we have m1 and m2 so close that their gravitational interaction is not negligible, the system composed of m1 and m2 has a total weight less than m1 + m2 ?

It's possible, although we don't have as direct experimental evidence for this as we have for nuclear and electromagnetic forces, and gravity has some very different properties from other interactions.
But the extrapolation does make some sense; it order for the system to decrease its gravitational potential energy, it has to get rid of it somehow. For example, via EM radiation. This should decrease its total mass.
Assuming this is true, this still does not help us explain rotation curves of galaxies better than dark matter, because assuming only Newtonian gravity, observations suggest more gravitational mass being present than sum of visible masses being observed, not less.
But hypothetically, if some weak long-range repulsive forces were present between stars/parts of galaxies, analogously to repulsive forces between protons in the uranium nucleus, bound state (galaxy) would have greater energy than expected with gravity forces alone.
Maybe this increase in energy and implied increase in gravitational mass and its gravity could be enough to explain the rotation curves, but I have not done any calculations on this.
