Proof: Kepler's 1st Law ( Planetary Law of Ellipses: Sun  Centered Model )
"Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God."  Johannes Kepler ( 1571  1630 )
Since we are trying to determine a planet's orbital path as an elliptical conic section with the sun at one foci with eccentricity, it's incumbent to express a planet's orbital radius as a function of angle and eccentricity as follows:
Therefore
However from
we also know that
Hence, a new relationship emerges
Let's now simplify the above 2nd  order differential equation into an even more simple 1st order differential equation as follows:
Now, as before in the derivation of "Mercury's Orbital Precession Advance of its Perihelion", we set
Therefore,
Some Further Eccentricity Relationships
§ References:
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