\(GPE=C-{g}_{e}{r}_{e}e^{-\cfrac{x}{{r}_{e}}}-{g}_{m}{r}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}\)
If GPE=0 when x=0,
\( C={ g }_{ e }{ r }_{ e }+{ g }_{ m }{ r }_{ m }e^{ -\cfrac { Orbs }{ { r }_{ m } } }\)
We can plot two component curves
\({g}_{e}{r}_{e}(1-e^{-\cfrac{x}{{r}_{e}}})\) which is totally independent of the satellite.
and
\({g}_{m}{r}_{m}(e^{\cfrac{-Orbs}{{r}_{m}}}-e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})})\) which is totally independent of the other body. The negative of this component is plotted to give a sum of zero for GPE. The actual curve plotted are,
0.0000980665*63.71*(1-e^(-x/63.71)) and 0.0000162*17.37e^(-3549.96/17.37+x/17.37).
This a graph of the two GPE components. The x-axis has been scaled to give 100 km. We can see two intercepts, one at x=0 and the other approximately at x=38250. The oribital radius as measured from the Earth's center is then Om= 382500+6371=388871 km.
The graph below is when Orbs is allowed to vary from 20000 to 384400. This is why solution for Orbs can not be obtained from considering GPE(x) or gravity, g(x), an intercept occurs for every values of Orbs.
