[mbaker335 (OP)]

With the Moon increasing its orbit by what to me is a startling rapid rate of 38 mm a year will the barycenter of the Earth moon system move? If it does and seeing that it is already 4671km from the Earths centre then in a relatively short time could it leave the earths surface?

My thinking here is if that was to happen the Earth would become a binary planet with the Moon. So 9 planets become 7 (no Pluto and Neptune is not a planet as it also has not cleared its orbit) then back to 8 when the Moon becomes a planet. The new planetary definition is a mess.

[Kryptid]

The equation used to calculate the location of the barycenter is as follows:

r = a/(1 + (m₁/m₂)), where:

r = the distance from the center of the primary body to the barycenter
a = the distance between the centers of the two bodies
m₁ = the mass of the primary body
m₂ = the mass of the secondary body

Putting in the numbers for the current Earth-Moon system, we get:

r = (384,000 km)/(1 + (1 Earth mass/0.0123 Earth masses))
r = (384,000 km)/(1 + 81.3)
r= (384,000 km)/(82.3)
r = 4,665.86 km

The distance of the barycenter varies linearly with the distance between the two bodies. So in order to increase the existing barycenter from 4,665.86 kilometers to 6,371 kilometers (the Earth's average radius), you would need to increase the Moon's distance from the Earth by the factor 6,371/4,665.86 = 1.365 times. That is equivalent to increasing the Moon's average orbital distance from 384,000 kilometers to about 524,300 kilometers.

I recall from some past reading that the Moon is expected to continue to move away from the Earth until it appears to be only about 40% of its current size in the sky (sorry, I don't recall where I read this). That's an increase in distance of 2.5-fold (960,000 km). If this is true, then yes, the barycenter should definitely move outside of the Earth's surface in the future if the Earth-Moon system survives long enough (the Sun turning into a red giant might complicate this).

[evan_au]

in a relatively short time could the barycenter leave the earths surface?

Just because the Earth/Moon barycenter is outside the Earth's surface does not mean that the Moon will drift off to infinity - it is still gravitationally bound to both Earth and Sun.

The Sun/Jupiter barycenter is outside the Sun's surface, but Jupiter remains in orbit around the Sun.

the Earth would become a binary planet with the Moon

The Earth/Moon system is already close to being a binary planet.
The ratio of Earth:Moon mass is really big compared to other Planet:Moons in our Solar System.

However, under the new official Planet definition (which demoted Pluto to dwarf planet), the Moon would not become a planet, because it does not dominate its orbit - there is still this big thing called Earth in the same orbit around the Sun.

[Dave Lev]

The gravity force is based directly on the distance between the objects.
If we set the calculation, we can find that the current Sun - Moon gravitational force is stronger than the Earth - Moon gravitational force.
So, technically, based on that force ratio, it is expected that the Moon will prefer to orbit directly around the Sun.
We can assume that the answer for that is: As historically the moon orbits around the Earth, it still keeps its momentum.
I wonder if and when there will be a breaking point - as the Sun -Moon gravity force will be much more significantly stronger than the Earth -Moon force.

Don't forget that the Earth is also drifting from the Sun. So, in early days (when the solar system had been formed), the Sun - Earth Gravity force (and the Sun -Moon) gravity force was much stronger that the current gravity forces.
However, it is clear that at that time the Earth-Moon gravity force was significantly stronger than the Sun-moon gravity force.

[Kryptid]

If we set the calculation, we can find that the current Sun - Moon gravitational force is stronger than the Earth - Moon gravitational force.
So, technically, based on that force ratio, it is expected that the Moon will prefer to orbit directly around the Sun.

The Moon is inside of the Earth's Hill Sphere (the region of space where the Earth's gravity is dominant over other sources of gravity), so I'm very skeptical about this claim.

[Janus]

If we set the calculation, we can find that the current Sun - Moon gravitational force is stronger than the Earth - Moon gravitational force.
So, technically, based on that force ratio, it is expected that the Moon will prefer to orbit directly around the Sun.

The Moon is inside of the Earth's Hill Sphere (the region of space where the Earth's gravity is dominant over other sources of gravity), so I'm very skeptical about this claim.

It is true.  Gravitational acceleration of Moon due to the Earth is 0.0027 m/s^2.    Gravitational acceleration of Moon due to the Sun is ~0.006 m/s^2.    As an interesting result, the path of the Moon is always concave towards the Sun.
The reason the Moon still holds the Earth in orbit is the fact that the acceleration of the Earth due to the Sun is also ~0.006 m/s^2, so this ratio hasn't much to do with whether the Earth holds  the Moon in orbit.  What does have an effect is the differential of the Sun's pull, or tidal force across the diameter of the Moon's orbit.  It is this that determines the Hill sphere.
For example:  Move the Earth-Moon system in to 1/2 it present distance from the Sun while reducing the mass of the Sun by a a factor of 4 and you end up with exactly the same acceleration values we got above, but the Hill sphere reduces to ~ 72% of its present radius.