System GPE not at Zero.

If we examine the graphs of GPE with varies Orbs,
0.0000980665*63.71*(1-e^(-x/63.71)-0.0000162*17.37*e^(-a/17.37+x/17.37) where a is made to vary from 3400 to 4000 in increments of 100.

We notice that the descends of the graphs are parallel.  All intersections with the x-axis beyond Orbs is a fix distance from Orbs at about 3 times r_e.  Which is outside the Moon.  This is impossible, the GPE of a system is based on the position of the C.G.  The system is not at the lowest GPE possible.  The force F_ob acting through the C.G. of the Moon, the system does not fall to the lowest GPE.

If we consider gravity variation beyond Orbs, inside the Moon, as simply,

\(g={g}_{m}(1-\cfrac{x-Orbs}{{r}_{m}})\) where gravity due to earth is completely negligible.

Integrating this, we obtain work done by Moon's gravity which is negative as it is opposite to the direction of Earth's gravity.

\({GPE}_{m} = C -\cfrac{{g}_{m}{r}_{m}}{2}+\cfrac{{g}_{m}{r}_{m}}{2}(1-\cfrac{x-Orbs}{{r}_{m}} )^{2}\)

where \({GPE}_{m}(x = Orbs) = GPE(x = Orbs) ={g}_{e}{r}_{e}-{g}_{m}{r}_{m}\), for Orbs large

\({GPE}_{m} = {g}_{e}{r}_{e} -\cfrac{3{g}_{m}{r}_{m}}{2}+\cfrac{{g}_{m}{r}_{m}}{2}(1-\cfrac{x-Orbs}{{r}_{m}} )^{2}\)

And so we have a new expression for GPE along the line joining Earth and the Moon as

GPE(x) for x ≤ Orbs,  

\(GPE(x) = {g}_{e}{r}_{e}(1-e^{-\cfrac{x}{{r}_{e}}})+{g}_{m}{r}_{m}(e^{\cfrac{-Orbs}{{r}_{m}}}-e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})})\)

which is the same as g_e*r_e*(1-e^(-x/r_e)) + g_m*r_m*(e^(-Orbs/r_m)-e^(x/r_m-Orbs/r_m)).

and GPE_m(x) for Orbs + r_m > x > Orbs.

\({GPE}_{m}(x) = {g}_{e}{r}_{e} -\cfrac{3{g}_{m}{r}_{m}}{2}+\cfrac{{g}_{m}{r}_{m}}{2}(1-\cfrac{x-Orbs}{{r}_{m}} )^{2}\)

When x = r_m+Orbs, ie. at the C.G. of the Moon,

GPE_m=\({g}_{e}{r}_{e}-\cfrac{3{g}_{m}{r}_{m}}{2}+\cfrac{{g}_{m}{r}_{m}}{2}(1-\cfrac{x-Orbs}{{r}_{m}} )^{2}\)

= 0.00980665*6371-1.5*0.00162*(1737) =  58.253 km²s^-2

The total change in GPE is

\(\triangle GPE = {GPE}_{max} - {GPE}_{m}   \),  where \({GPE}_{max}= {g}_{e}{r}_{e}\)

\(\triangle GPE= {g}_{e}{r}_{e} - {GPE}_{m}(x = {r}_{m}+Orbs) =3\cfrac{{g}_{m}{r}_{m}}{2} \)

\(\triangle GPE \) = 1.5*0.00162*1737 = 62.478 - 58.253 = 4.225 km²s^-2.   This decrease in GPE is converted to Rotational KE, we have, for Moon speed

\(\frac{1}{2}.{v}^{2}_{o}=3\cfrac{{g}_{m}{r}_{m}}{2}\)= 4.225

\({v}_{o}=\sqrt{3{g}_{m}{r}_{m}}\)

\({v}_{o}=\sqrt{}\)8.44182 = 2.90 km^-1

Interestingly, GPE(x = Orbs) = 59.66  is the same for all values of Orbs, perigee and apogee.  The drop in GPE inside the Moon is also the same for valid Orbs.  This explains why Moon speed at these two extreme points are the same.  Both points have the same decrease in GPE at the C.G.  In fact the Moon has almost constant speed.

v=2.90 km^-1 a better approximation of 1.03 kms^-1 but it came from assuming that the system is not at its lowest GPE only slightly less.  The good part is the conclusion that Moon speed is almost a constant throughout its orbit, which is what is observed.