Kepler ́ s Ellipse Observed from Newton ́ s Evolute ( 1687 ) , Horrebow ́ s Circle ( 1717 ) , Hamilton ́ s Pedal Curve ( 1847 ) , and Two Contrapedal Curves ( 28 . 10 . 2018 )

Johannes Kepler discovered the very elegant elliptical path of planets with the Sun in one focus of that ellipse in 1605. Kepler inspired generations of researchers to study properties hidden in those elliptical paths. The visible elliptical paths belong to the Aristotelian World. On the other side there are invisible mathematical objects in the Plato ́s Realm that might describe the mechanism behind those elliptical paths. One such curve belonging to the Plato ́s Realm discovered Isaac Newton in 1687 the locus of radii of curvature of that ellipse (the evolute of the ellipse). Are there more curves in the Plato ́s Realm that could reveal to us additional information about Kepler ́s ellipse? W.R. Hamilton in 1847 discovered the hodograph of the Kepler ́s ellipse using the pedal curve with pedal points in both foci (the auxiliary circle of that ellipse). This hodograph depicts the moment of the tangent momentum of orbiting planets. Inspired by the hodograph model we propose newly to use two contrapedal curves of the Kepler ́s ellipse with contrapedal points in both the Kepler ́s occupied and Ptolemy ́s empty foci. Observers travelling along those contrapedal curves might bring new valuable experimental data about the orbital angular velocity of planets and a new version of the Kepler ́s area law. Based on these contrapedal curves we have defined the moment of the normal momentum. The first derivation of the moment of the normal momentum reveals the torque of the ellipse. This torque of ellipse should contribute to the precession of the Kepler ́s ellipse. In the Library of forgotten works of Old Masters we have re-discovered the Horrebow ́s circle (1717) and the Colwell ́s anomaly H (1993) that might serve as an intermediate step in the solving of the Kepler ́s Equation (KE). Have we found the Arriadne ́s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?


Introduction
The famous quote of Heraclitus "Nature loves to hide" was described in details by Pierre Hadot in 2008.Hadot in his valuable book gives us many examples how Nature protects Her Secrets.In several situations the enormous research of many generations is strongly needed before the right "recipe" unlocking the true reality can be found.
The Old Masters of the Hellenistic and Alexandrian astronomy used the combinations of perfect circular motions for the description of the planet orbits.Johannes Kepler in 1605 made his great breakthrough when he discovered the elliptical path of Mars with the Sun in one focus of that ellipse.Generations of researchers were inspired by this Kepler´s ellipse and were searching for properties hidden in those elliptical paths.The great step made Isaac Newton in 1687 when he discovered the locus of radii of curvature of that ellipse (the evolute of that ellipse) and applied it for the calculation of the centripetal force.W.R. Hamilton in 1847 discovered a very elegant model of the hodograph using the pedal curve with pedal points located in both foci (the auxiliary circle).However, this classical model of the Kepler´s ellipse could not properly explain the precession of the planets and Albert Einstein in 1915 replaced this classical model with his concept of the elastic spacetime.
A possible chance for further classical development of the Kepler´s ellipse is to penetrate more deeply into the secrets of the Kepler´s ellipse and to reappear with some new hidden properties overlooked by earlier generations of

Moment of the tangent momentum and moment of the normal momentum of the Kepler´s Ellipse
Based on the formulae in Table II we can evaluate the moment of the tangent momentum L T and to introduce a new physical quantity -the moment of the normal momentum L N .
The moment of momentum L is defined as the product of the linear momentum with the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle.It is this definition: L = (length of moment arm) x (linear momentum).
The moment of the tangent momentum L T is given as: (7 where m is the mass of the planet, v T the tangent velocity of the planet and KK´ is the length of the moment arm (the distance between the Kepler´s occupied focus and the tangent).The moment of the tangent momentum L T is constant during the complete path of the Kepler´s ellipse.Therefore, there is no contribution to the torque from this moment of the tangent momentum.This is very well-known experimental fact documented in the existing literature.
The moment of the normal momentum L N is given as: (8 where m is the mass of the planet, v N the normal velocity of the planet and SK´ is the length of the moment arm (the distance between the Kepler´s occupied focus and the contrapedal curve).The moment of the normal momentum is not constant during the complete path of the Kepler´s ellipse.Therefore, we expect a contribution to the torque of the Kepler´s ellipse.(We did not study in details the properties of the curve aε 2 sin 2 E.)

Torque of the Kepler´s Ellipse (Moment of Force)
Torque is defined mathematically as the rate of the change of the moment of the momentum.As long as the moment of the tangent momentum is constant then there is no net torque applied.However, what about the moment of the normal momentum?
The derivation of the formula for the torque caused by the moment of the normal momentum would be as: e e e e e e  Once we know the Colwell´s anomaly H then with the help of the analytical geometry we will get the precise value of the eccentric anomaly E without any iteration process.From the known equation of the line KF and the auxiliary circle with the center C we will get the intersection point G (cosE, sinE).From the known eccentric anomaly E, we will get easily the desired true anomaly Θ.
Our hypothesis is based on an idea that the contrapedal curve with the contrapedal point in the Ptolemy´s empty focus condensed into the Ptolemy´s empty focus and we have got the mean anomaly M depicted by the second auxiliary circle with the center in the Ptolemy´s empty focus.This hypothesis has to be mathematically proved by the Readers of this Journal.

"Antikythera Mechanism" in the Solar System
We propose to use the very-well known Antikythera Mechanism as an analogy for the visible Kepler´s ellipse -a part of our Aristotelian World -connected deeply with invisible curves from the Plato´s Realm -Newton´s evolute (1687), Horrebow´s circle (1717) and Colwell´s anomaly (1993), Hamilton´s pedal curve (1847), two contrapedal curves (2018), there are two more curves describing the moment of the normal momentum and the torque of the Kepler´s ellipse (2018).
Are there some more hidden curves in the Plato´s Realm connected to the Kepler´s ellipse?

Conclusions
1) We have presented some quantitative properties of the Kepler´s ellipse in Table I and Figures 1 and 2. 2) We have discovered a new trigonometric formula for the radius of curvature in the Newton´s evolute of the Kepler´s ellipse.
3) In the pedal curve with the pedal points in both foci (the auxiliary circle) we have depicted the Hamilton´s hodograph.

4)
We have observed the Kepler´s ellipse from two contrapedal curves with contrapedal points in both foci and presented some relationships in Table II.

5)
We have derived formulae for the moment of the tangent momentum and the moment of the normal momentum.
6) We have derived the formula for the torque of the Kepler´s ellipse.

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Figure 5 s Ptolemy´s et orbiting in th S will get diffe en that observe J and S collecting experimental data along the contrapedal curves Contrapedal curve with the contrapedal point in the Ptolemy´s empty focus -constant orbital angular velocity Contrapedal curve with the contrapedal point in the Ptolemy´s empty focus -modified Kepler´s area law Contrapedal curve with the contrapedal point in the Kepler´s occupied focus -orbital angular velocity Contrapedal curve with the contrapedal point in the Kepler´s occupied focus -constant Kepler´s area law Figure

7)
In the Horrebow´ circle we have presented a hypothesis about the condensation of the contrapedal curve with the contrapedal point in the Ptolemy´s empty focus into the Ptolemy´s empty focus.8) Are there some more hidden curves in the Plato´s Realm connected to the Kepler´s ellipse?/de.wikipedia.org/wiki/Ellipse

Table I .
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