ELLIPSES ARE EPICYCLES

 

INTRODUCTION

 

For those interested in the classical debates involving planetary motion, it may be interesting to know that ellipses are actually a subset of epicycles.  That is to say, every ellipse can be constructed from an epicycle.   But while some epicycles are ellipses, others are not.

 

It is thus an entirely false criticism of early efforts to decipher planetary motion that there was some inherent religious bias that stymied intellectual inquiry.   Rather the Catholic Church went to great lengths to sponsor astronomical studies and actually invented the scientific method.   At the time of Kepler, the best models of the day contained tens of epicycles tweaked to fit ever more accurate observations.  Kepler’s insight, which had escaped both Copernicus and Galileo whose models were grossly inferior to even that of Ptolemy more than 1400 earlier, was that only two epicycles, but the right two, were sufficient.

 

ONE SPECIAL EPICYCLE

 

Consider the epicycle described below.  A large circle of radius r1 rotates counterclockwise.   The smaller circle of radius r2 has its center attached to the periphery of the larger circle.   The smaller circle then rotates at the same rate as the larger but in a clockwise direction.

 

 

What is interesting is that any point on the smaller circle will trace out an ellipse.   But there is one point shown above on the smaller circle, whose ellipse will be centered on the x-axis.  Another point on the opposite side of the smaller circle, that is rotated by 180 °, will  trace out the same topological ellipse but rotated by exactly 90°, that is along the y-axis.

 

EQUATIONS OF MOTION

 

As previously mentioned, both circles rotate at the same rate making one complete revolution with a period of “T” seconds.   The angle with respect to the x-axis in radians as a function of time “t” is given by

 

Angle   =  w t  =  (2 π /T)  t

 

We can then calculate the general (x,y) coordinates of the planet shown above using the following diagram:

 

x = r1 cos(wt) + r2 cos(wt) = (r1 + r2) cos(wt)

y = r1 sin(wt)  - r2 sin(wt)   = (r1 - r2) sin(wt)

 

or rearranging

 

 

 

We can square each quantity and add the two equations to get

 

 

or if we note that the minor “y” and major “x” axes of the ellipse are

 

            a = (r1 - r2)

            b = (r1 + r2)

 

then finally

 

 

which is the well-known equation of an ellipse.

EQUATION OF AN ELLIPSE

The geometric definition of an ellipse is those points which have a constant sum of distances to two foci, which are at coordinates (c, 0) and (-c, 0), as shown below:

From the definition of an ellipse we have for any point (x,y)

                       

We can solve for the constant term by considering the point on the major axis, x, at (b,0) where

And if we consider the point (0, a) on the minor axis, y, we have from two identical right triangles

For the general point (x,y) we have

which we can rearrange and then square both sides as follows

And again rearranging terms, dividing by four, and then squaring we get

We see that the minor “y” axis length of “a” appears on both sides of the above equation, so substituting and normalizing we get

which again from different considerations is the same well known equation of an ellipse.

HISTORICAL COMMENTARY

Epicycles (literally “on the circle” in Greek) were first used to describe the motions of the moon, sun, and planets by Apollonius of Perga at the end of the 3rd century BC.  These were necessary to account for the “retrograde” behavior of planets moving across the night sky.  

Somewhat later epicycles were improved upon by Hipparchus of Rhodes and were formalized by Ptolemy in his 2nd-century AD astronomical treatise the Almagest.  It is from this work that such orbits are commonly called Ptolemaic though the concepts were actually advanced almost five centuries earlier.  Of note is the fact that the Earth was not the center of motion but rather that center was “deferred”  or displaced some distance away and was different for each heavenly body.

Epicycles approximated the motions of the five planets known at the time within the rough limits of naked eye observation.  They also explained the slight but noticeable change in the apparent distance of the moon as seen from Earth.

In 1542, Copernicus, who had taken minor Holy Orders in the Catholic Church, to include a vow of chastity observed more in the breach, noted that his calculations to determine the date of Easter were simpler if one assumed a sun centered rather than an earth centered arrangement.  Copernicus was reluctant to publish because his circular orbits, with only a minor reduction in the number of original epicycles, were notably less accurate than the more complicated earth-centered system of Ptolemy which been refined over many centuries.   But considering its lifelong sponsorship of his work, the insistence of his superiors in the Catholic clergy on publication, finally overcame his reluctance.  [1]

This revived a well known earlier speculation, put forward for similar reasons by the early Greek astronomer Aristarchus of Samos (c. 310 – c. 230 BC) who had also postulated a sun-centered solar system.

Galileo was an early admirer of the Copernican system as was the reigning Pope Urban VIII who was a mathematician and had earlier defended Galileo’s ideas and his right to publish.  Unfortunately Galileo came to believe his fame as an early, even revolutionary, astronomer gave him license to ridicule the Pope in print for his own aggrandizement and to boast that his speculations had solid observational and theoretical foundation.

Galileo had no such observations or calculations and his challenge to Church authority was punished on nitpicking grounds as an example to those considering like-minded challenges to established order.   The difficulty was that despite careful observations and measurement, neither the required solar parallax nor any centrifugal force appeared to exist.   The suggestion that perhaps the stars were too far away, even though correct, seemed too much an ad hoc excuse.  Nor was there any known framework to calculate the miniscule kinetic forces operating at the surface of a spinning earth.

Interestingly, assorted musings and speculations aside, the first solid evidence for a sun centered solar system was not experimental but theoretical.  In 1609 and 1619 by analyzing the meticulous observations of his mentor, Tycho Brahe, Kepler finally made the definitive discoveries that the orbits of planets are best described as ellipses, i.e. special cases of epicycles.  It was however necessary to put the sun “off-center” at one of the foci of the ellipse, a subtlety which had eluded Copernicus and ruined his predictions.  

This conceptual breakthrough allowed extremely accurate predictions of planetary motion even when compared with the best measurements made with modern telescopes and lead to its adoption by all Western scholars to include the Catholic Church, which as previously noted, invented and has always sponsored modern science.

Despite widespread acceptance, experimental evidence for the stars being at great distance and the forces due a spinning earth being miniscule were a long time in coming.  It finally arrived with a report by Giuseppe Calandrelli of Italy who observed stellar parallax in the star α-Lyrae and in 1838 by Friedrich Bessel of Germany with the first quantitative measurements on the star 61-Cygni.  The first physical demonstration of the kinetic forces for a spinning earth was the Foucault pendulum introduced in Paris in 1851.   Unfortunately these validations of earlier insightful speculation only came centuries after Aristarchus of Samos, Copernicus, Galileo, and others had long passed.

Despite widespread modern acceptance, a stationary sun and rotating earth apparently remains a hard pill to swallow because it is so contrary to common sense appearance.

REFERENCES

1.      Repcheck, Jack, “Copernicus’ Secret”, Simon and Schuster, (2007), pp xiv, 156.

Despite what many simply assume to have been true, a detailed examination of Copernicus’s letters and the history of Christian Churches who held a passionate regard for the advancement of human knowledge, the Catholic Church rather than objecting, actually encouraged Copernicus to publish.

“Copernicus … made no move to finish it or submit it to a publisher, despite strenuous urgings from friends and colleagues in high places.  He was not afraid of being declared a heretic, so many assume; rather, he was worried that parts of the theory were simply wrong, of if not wrong, incomplete.”

2.      xxx