Along the line of space density conceptualization, where space is most compressed at the surface of the body, we would them expect the space density to be high, at first, on moving from earth towards the moon. It then decreases along a exponential profile, reaches a floor value, but then rises up again on the surface of the moon.
We have seen from previously, that gravity is the first derivative of the space density profile,
\(g=D.\cfrac{d({d}_{s}(x))}{dx}\)
where D is a constant of proportionality dependent on the way the space density profile is established.
We also know that space density around a body can be presented as,
\({ d }_{ s }(x)={ e }^{ -bx }({ d }_{ e }{ -d }_{ n })+{ d }_{ n }\) where \(b=\cfrac{1}{r}\)
where \(r\) is the radius of the body modeled as a sphere, and \({d}_{e}\), \({d}_{n}\) are boundary conditions. This is the singular body space density profile.
In a similar way, the space density profile between two bodies can be modeled as,
\({ d }_{ s }(x)=Ae^{(-\cfrac{x}{{r}_{a}})}+Be^{(-\cfrac{Orbs}{{r}_{b}}+\cfrac{x}{{r}_{b}})}+C\)
This equation is the sum of 2 singular body space density profile, with one reversed on the x-axis and translated to Orbs unit apart. Each of the singular profile is scaled by A, B respectively, and a floor value C is added.
We know that gravity on earth surface is -9.8 ms-2, and on the moon surface +1.62 ms-2 . Earth radius is about 6 times that of the moon, and the distance between Earth and the Moon is about 221 times the radius of the Moon. A and B are chosen in order that, the end points of the derivative of the profile (at x = 0 and x = 221 are at about -9.8 and 1.6. We will set the floor value to 10 arbitrarily. We model space density between the moon and earth in the first approximation to be,
\({ d }_{ s }(x)=68.52e^{(-x/6)}+1.62*e^{(-221+x)}+10\)
If gravity between earth and the moon, is just the derivative of this space density function ie. D = 1, we have, for illustration only,
Observe that the red curve, the derivative of the blue profile is at -9.8 x = 0 on the earth side. Its absolute value decreases reaching zero and then rises up again to 1.6 positive, on the Moon side x=221. On the earth side, gravity is pointed at earth's center (negative value) and on the Moon side gravity is pointed away from earth towards the Moon (positive). This space density profile models the change in gravity along the line between Earth and the Moon centers.
