[Frankwopsy (OP)]

Just wondering.

How much reduction of mined material from the Moon would it take to start making the Moons orbit change?

[chiralSPO]

Hi Frankwopsy. As a mod, I have taken the liberty of splitting this post into its own topic (it has nothing to do with measuring photon energy, so doesn't belong in that thread).

In the future, if you would like to start a new thread, navigate to the subforum that makes the most sense for the topic (physics/astronomy, chemistry, new theories etc.) and click the button (it's dark blue in my browser) that says "New Topic":
Screen Shot 2019-03-28 at 9.51.14 AM.png (163.05 kB . 2180x526 - viewed 2612 times)

And welcome to the forum!

[Janus]

Any amount of mass we could ever remove from the Moon by mining would have an insignificant effect on its orbit.

The Earth and Moon orbit around a common barycenter which is located some distance below the surface of the Earth. The location of that barycenter is determined by the relative masses of the two and their distance apart. 

Removing mass from the Moon would change the ratio of masses and shift the barycenter.

To give you an idea of the effect this would have, we'll consider how this would change the orbital period.
The orbital period for this situation is found by
T = 2 pi sqrt(d^3/G(Me+Mm)
where d is the distance between Earth and moon, G the universal gravitational constant, Me the mass of the Earth, and Mm the mass of the Moon
We'll use 384,000,000 m for d
6.673e-11 for G
6e24 kg for Me
7.35e22 kg for Mm

This gives 2348532.771 sec or 27.18209225 days for the the orbit.

Now lets say that we moved even 1% of the Moon's mass (way more than we'd ever removed by mining.)

This reduces Mm to 7.2765e22 kg and gives a new orbital period of 2348674.89 sec, or just 142 sec (2.37 min)longer

We are now extracting some 50 billion kg of material from the Earth a year (much of this includes fossil fuels which would not be on the Moon).

At this rate, it would take some 14.7 million years to remove 1% of the Moon's mass and increase its orbital period by that 2.37 min. 
Meanwhile, the Moon is already slowly moving away on it own accord by some 4 cm a year, increasing its orbital period all by itself.  At this rate, in 14.7 million years it will have increased its orbit by some 588 km and its orbital period by 5396 sec or ~1.5 hours. 

There is just no way that we could remove enough material from the Moon to have anywhere near the effect on its orbit as natural processes already have.

[PmbPhy]

You can't change the orbit of the moon by decreasing its mass by any amount. An orbit is determined by its initial location and its initial velocity. This is akin to the fact that a bullets trajectory being determined entirely by where its pointed and its speed pointed out of the gun (forgetting drag for the time being since there's no drag on the moon by drag. However there is tidal friction on the moon due to the Earth.

[Halc]

You can't change the orbit of the moon by decreasing its mass by any amount. An orbit is determined by its initial location and its initial velocity.

This is approximately true only for small orbiting masses, and the moon is not a trivial mass.  Janus calculated the change in orbit time for a 1% change in mass and holding radius constant (rather than orbital speed constant) and got a 2 minute increase in orbit time.  If the velocity were held constant, the radius would drop and there would be a similar decrease in orbital period.

To illustrate why mass matters, compute the oribital period an Earth/moon pair where the 'moon'  had the mass of Venus at the same distance.  The period would be much shorter.

This is akin to the fact that a bullets trajectory being determined entirely by where its pointed and its speed pointed out of the gun (forgetting drag for the time being since there's no drag on the moon by drag. However there is tidal friction on the moon due to the Earth.

Common misconception.  If I dropped a 1kg object and an object that massed the same as Venus from the leaning tower of Pisa, the latter would meet the ground in less time.

As for my contribution to the question asked in the OP: Much depends on how the mass is removed.  The likely method would be a horizontal rail-gun which, if aimed consistently in one direction with or against the orbital path, would add thrust (and spin if they do it at a high enough rate) to the moon with or against its motion, affecting the orbit accordingly.  The logical place to put the collection of such material is at one of the Lagrange points (probably L1), but it is unclear which direction would most efficiently reach L1 on a ballistic trajectory.

[chiralSPO]

Launching the mass from the moon might have some influence on the orbit of the moon, but this will depend on the trajectory of the launch...


oops, I see I crossed posts with Halc

[PmbPhy]

The question assumes the earth doesn't move when the moons mass is decrease etc. Otherwise the derivation is here;
http://www.newenglandphysics.org/physics_world/cm/two_accel.htm

You really need to state what approximations you're using when stating a conclusion. What you assumed is never an assumption used in general relativity or in classical mechanics. It also contradicts Kepler's laws if i recally correctly.