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When atoms bind together, their total energy is less than each individual's energy. When planets come together, their total energy is also less (i.e. nature of attractive force). The mass of each stays the same before and after the binding. However, their total energy decreases.

  1. Does this mean that when planets come together, their gravitational field will be reduced in comparison to when the two was still free? I know that it will increase since when two masses come together, then of course the field will increase.

  2. What I mean is, will it decrease compared to when each of them was free?

e.g. in atoms, their total energy decreases when the two atoms bind together

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  • $\begingroup$ Total energy? Or potential energy? $\endgroup$
    – DKNguyen
    Aug 24, 2020 at 13:24

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This is a sensible question and the answer is: yes it does.

Your scenario starts with two planets (or stars, or whatever) with masses $m_1,m_2$ some distance apart. At some position a long way away there will be a gravitational field.

Then for some reason the planets get closer. Gravitational potential energy is converted into some other form which is then lost (maybe gravity waves, maybe tidal forces producing heat which is radiated). They still have masses $m_1$ and $m_2$ (as defined by their inertia or their gravitational attraction) but the total system has a mass less than the sum of the components: $M=m_1+m_2-B/c^2$ where $B$ is the binding energy. Just like nucleons in the nucleus.

A long way away the gravitational field will be due (once the binding energy gravitational waves or IR photons or whatever have departed) to this total $M$, a little bit smaller than $m_1+m_2$.

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The mass of each stays the same before and after the binding.

Actually not. The best example for this is nuclear fusion. There is a measurable mass difference between the rest mass of a nucleus and the total rest mass of the protons and neutrons.

If I understand well, your question is about the equivalence of energy and mass.
I know it's a bit confusing, but in bound and free systems, the total mass is different. With gravity it's more complicated, as in classical physics the source of gravity is the mass its self. But also in the merging of two black holes there is a mass difference, what is radiated away with gravitational waves (mostly). (I'm not sure if they can measure the mass difference directly, though.) Although in most (types of) processes the mass difference is usually negligable/unmeasurable.

These articles explain this whole stuff in more details:

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  • $\begingroup$ Am I wrong in something? $\endgroup$
    – fanyul
    Aug 28, 2020 at 10:31
  • $\begingroup$ This is not relevant for gravitational problems at planetary scales. Masses stay essentially the same, there are no relativistic effects. The issue is thinking that binding energy is related to mass as it should be explained. $\endgroup$
    – ohneVal
    Aug 28, 2020 at 13:09

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