Relativity Science Calculator - Proof: Kepler's 1st Law ( Planetary Law of Ellipses: Sun - Centered Model )

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 )

relativity approximations

Proof: Kepler's 2nd Law - Polar AnalysisProof: Kepler's 2nd Law

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:


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:

The Geometry of Elliptical Orbitsrocket escape velocity

Now, as before in the derivation of "Mercury's Orbital Precession Advance of its Perihelion", we set


angular momentum is a constant of integrationMercury's Orbital Precession Advance of its Perihelion

Some Further Eccentricity Relationships

constant angular momentumMercury's Orbital Precession in Advance of its PerihelionNewton's gravitational potential energy scalar

ยง References:

  1. "Mathematical Physics", 2nd edition, by Donald H. Menzel ( 1901 - 1976 ), Professor of Astrophysics, Harvard University, Dover Publications, Inc., first published in 1947 under the title "Theoretical Physics", Prentice-Hall, Inc.

  2. "Calculus and Analytic Geometry", 8th, 9th, 10th or 11th editions, by George Brinton Thomas, Jr. ( 1914 - 2006 ), M.I.T., Ross L. Finney, U.S. Naval Postgraduate School, and Maurice D. Weir, Professor Emeritus, U.S. Naval Postgraduate School, Addision-Wesley Publishing Company.