What would happen to the moon's orbit if we reduce (instantaneously) its mass? [closed]

Question

In my opinion the moon's orbit shouldn't change, because an orbiting body is a free falling object and the trajectory of a free falling object is not affected by its mass (because the inertial mass and the gravitational mass are the same. Of course the mass affects the initial energy needed to put the object in orbit).

But...

What about the shift of the center mass? How can it move without the trajectory of the orbiting body being affected? And what happens if we change the mass of the center body? If the sun disappears (mass = 0) planets will stop orbiting it and they will escape tangentially. But this means that orbits are affected by the mass of the center object, which seems to contradict what I said in the first paragraph. Why?

Answer 1

Mass is a conserved quantity. Having it simply disappear violates the known laws of physics. Once you have violated the laws of physics you can no longer ask what the laws of physics predict, since you have already assumed them to be false.

So, instead of having the mass simply disappear, we can change the question to something where, instead of disappearing, the mass of the moon splits into two halves. Now, if there is no net force between the two halves (gravitational force balances pressure force between the halves) then indeed the orbit will not change.

If we specify the system so that the boundaries of the system change so that before the split the system was the earth and the full moon and after the split the system is the earth and one part of the moon, then indeed the center of mass changes. This change in the center of mass is accounted for by the flux of mass through the boundary of the system.

Answer 2

I will assume moon's orbit around the earth is modelled as a Kepler problem. The moon's mass $m_m$ is about 80 times smaller than earth's mass $M_e$. But if you don't neglect it completely, reducing it will have some consequences.

The moon's orbit will change. Your argument would hold only if you approximated $m_m/M_e \approx 0$. On one hand reducing the mass would shift the system's momentum and hence the center of mass would suddenly change its velocity. On the other hand it would also change the effective potential and the total conserved energy and angular momentum (these depend non trivially on both masses and reduced mass, so I don't think anything cancels out).

The latter will very likely change the radius and eccentricity of the orbit (this follows from the usual analysis of the equilibrium point of the effective potential and it's energy compared to the system's energy). Orbital period depends on both masses as well so it should also change.

You can't reduce the moon's mass so that it escapes though. Usually, we say there are two types of trajectories in Kepler's problem, depending on whether the energy of the system is positive or negative. In this case the negative potential energy will predominate and you will always get bound trajectories. If you wanted to make the energy positive and let the moon escape, you should rather increase $m_m$ or decrease $M_e$.