In general, the gravity profile of a two body system can be derive from assuming a space density profile
\({ d }_{ s }(x)=Ae^{(-\cfrac{x}{{r}_{e}})}+Be^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}+C\)
from which we derive gravity, with the appropriate boundary conditions
\(g=-{g}_{e}.e^{-\cfrac{x}{{r}_{e}}}+{g}_{m}.e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})})\)
The expression for gravity is then integrated to give Gravitational Potential Energy, GPE
GPE = \({g}_{e}{r}_{e}(1-e^{-\cfrac{x}{{r}_{e}}}) + {g}_{m}{r}_{m}(e^{\cfrac{-Orbs}{{r}_{m}}}-e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})})\)
It was assumed that at x = 0, GPE = 0.
We have also derived an expression for gravity inside the Moon,
\(gm(1-\cfrac { x-Orbs }{ { r }_{ m } } )\)
and the corresponding GPEm
\({GPE}_{m} = {g}_{e}{r}_{e} -\cfrac{3{g}_{m}{r}_{m}}{2}+\cfrac{{g}_{m}{r}_{m}}{2}(1-\cfrac{x-Orbs}{{r}_{m}} )^{2}\)
where the boundary condition is obtained from GPE when x = Orbs.
It is proposed that the amount of reduction in GPE from the maximum value attained, GPEmax, at the C.G., is converted to Rotational Kinetic Enegry, RKE.
\(RKE =\triangle GPE={GPE}_{max} - {GPE}_{m}( x ={r}_{m} + Orbs) =3\cfrac{{g}_{m}{r}_{m}}{2} \),
\(x ={r}_{m} + Orbs\) is at the C.G. of the Moon and \({GPE}_{max} = {g}_{e}{r}_{e}\)
RKE turns out to be a constant, because GPE(x = Orbs) is constant for all Orbs large, Orbs being the distance between the bodies, surface to surface and GPEm is independent of Orbs. It follows then orbital speed is also constant. If this decrease in GPE is converted to Rotational Potential Enegry, RPE
\(\cfrac{1}{2}{v}^{2}= RKE=3\cfrac{{g}_{m}{r}_{m}}{2}\) and so,
\( v =\sqrt{3{g}_{m}{r}_{m}} \)
Furthermore, when we differentiated GPE with respect to Orbs,
\(\cfrac{d GPE(Orbs)}{d Orbs}={F}_{ob}=-{ g }_{ m }e^{ -\cfrac { Orbs }{ { r }_{ m } } }+{g}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}\)
we obtain a force per unit mass opposing an increase in Orbs without energy input, despite a positive gravity outwards. This is the force that accounts for the centripetal force needed for orbital motion.
Moon dear, how I know thee....
