Heliocentric and Geocentric Equivalence

INTRODUCTION

Early attempts to describe the motions of heavenly bodies struggled between two competing ideas.  In the common sense ”Geocentric” model, the Earth was at the center of the universe and everything revolved around us.  The much less obvious, but more elegant, “Heliocentric” model postulated the Earth and planets revolved around a stationary sun.

The immediate difficulty in distinguishing between the two is that for the simplest case of perfectly circular orbits, the kinematics of both models are identical.   That is, the distance and direction of any planet as seen from Earth is calculated to be exactly the same for either model.   And this holds true over time as both execute a stately dance around the other.

In hindsight this is simply a consequence of the mathematics of vector addition.  To demonstrate this, please consider only the planets Earth and Mars.  We can diagram both systems as follows

In the Heliocentric system both planets orbit the Sun as illustrated by blue and red circles.  In the equivalent geocentric system, we simply displace the orbit of the Earth and fix its center to the orbit of Mars to create an “epicycle.”   Earth is displaced to the center where the Sun used to be and Mars is displaced to lie on the circumference of the epicycle that used to be Earth’s orbit.  The rate and direction of rotation of the blue and red circles remains the same in each system.

Note that while for simplicity we have considered only two planets, adding any number of others, to include the Sun and Moon, would still give identical results for either model.

MATHEMATICAL EQUIVALENCE

For a simple right triangle, the sides can be expressed as trigonometric functions as folows

In the Heliocentric system the planets rotate counter clockwise around the Sun at angular rates of wEarth and wMars.   These are given by

where TEarth and TMars are the lengths of the Earth and Marian year in Earth days.   For the Heliocentric model the (xH, yH) coordinates of the vector from the Earth to Mars is given by

where “t” is the time in days.   And for the Geocentric model this same vector (xG, yG) is

The constant of “pi” or 180 degrees is added to initially place the planets in the correct equivalent positions.

Note that these two equations are mathematically identical so (xH, yH) = (xG, yG) at all times.  We can plot the (x, y) coordinates representing the distance and direction of Mars as seen from Earth as a function of time as follows:

Note the “looping” or retrograde motion of Mars against the background stars which occurs once each opposition.   And by opposition we mean whenever the two planets are at the closest together in their orbits.

WHAT MAKES THEM DIFFERENT

So the question naturally arises, “Which one is correct?”  Since theoretically the simplest models predict EXACTLY the same planetary positions and distances, kinematical considerations are not initially sufficient to distinguish between the two.

Before the invention of the telescope in Holland in the early 1600’s, it seemed absurd to image an apparently stationary Earth spinning through space at incredible speeds.  Not just common sense and but careful scientific observation strongly argued against the Heliocentric model.  The simplest of considerations included the following

a)      Given that all of space was thought to be filled with air, a spinning earth should create a wind shear.  Houses, trees, and people would have to be moving at one thousand miles per hour at the equator.  The consequent winds should knock down everything in their path, but this was clearly not observed.

b)      The centrifugal forces due the Earth’s revolving around the Sun and spinning on its axis, which no one could calculate, were nevertheless feared to be immense and were also not observed.  Like on a merry-go-round amusement ride, we should all have long since been spun off into space.

c)      The fixed stars, whose relative positions had been carefully measured to within about 1/60th of a degree of arc with naked eye observation, did not show any parallax effects.  That is to say the constellations did not noticeably change their shape as the Earth moved around the Sun.   So unless stars were very much brighter than the planets and unimaginably further away, a moving Earth was clearly wrong.

Occam’s razor which prefers the simpler solution clearly was on the side of the Geocentric Model.

In hindsight, the definitive answer is that what makes them different is threefold.

a)      It is not mathematically possible to easily model small variations from perfectly circular motion in the Geocentric representation.   The exact description requires an infinite series expansion and an infinite number of deferents and epicycles.  In the Heliocentric model, one and only one epicycle (equivalent to an ellipse) and the same deferent for everything is sufficient.

b)      The two models predict the same locations and distances of planets as seen from Earth but very different relative positions among themselves.   In particular, the planet Venus goes thru phases like the moon in a crescent shape illuminated by the sun which agrees with the Heliocentric model but not the simplest Geocentric.  If one really wants a stationary Earth, the Geocentric model needs to be much more complicated.

c)      But the definitive answer is not the kinematics of their relative positions but rather the dynamical forces governing their motions

THEORETICAL MODELLING

For many millennia before the birth of Christ, all heavenly motion was well known to deviate slightly from being perfectly circular.  The only theoretical solution to account for such periodic variation is, of mathematical necessity, to add deferents and epicycles.   After millennia of refinement with the addition of an equant, Ptolemy (100-170 A.D.) successfully modeled the heavens to well within one degree of arc.  Despite subsequent musings and many false claims, this one model remained by far the most the most accurate representation for the next 1500 years.

In 1609, Johannes Kepler discovered that a Heliocentric model with planets moving on a single epicycle (equivalent to an ellipse) was sufficient.

PHYSICAL DYNAMICS

The first small breakthrough to modern thinking only occurred in the late 16th century fostered by the invention of the scientific method by the Catholic Church in Western Europe.  In a continuing attempt to improve the calendar, the Pope asked for scientific speculation.  Nicholas Copernicus responded with a model that was more complicated and less accurate than Ptolemy.  But his model was Heliocentric and more physically realizable.  The Catholic Church repeatedly begged him to publish his speculations, granted the publication official approval in an Imprimatur, and widely distributed the documents.  It was only after 80 years that the Church, under attack from the Protestant revolt, and Galileo’s writing that these ideas might lead to yet more heresy, that the Church put Copernicus on a restricted reading list requiring special approval until removed after another 180 years.

In the Geocentric model the planets were thought to be attached to giant, rigid, interlocking, crystalline (i.e. read invisible) spheres which rotated about and inside of each other.   This is diagramed below:

In the Heliocentric system the planets move through empty space carried on by inertia and only swayed from their course because of the central force of gravity directed between the sun and planet.  And this force of gravity must somehow work invisibly across great distance.

The breakthrough occurred only when it was demonstrated that the motion of objects on Earth, such as cannonballs, are roughly as predictable as the motion of planets using Newton’s formula of Universal Gravitation.  In more recent times, our astronauts have yet to discover or bump into any crystalline spheres.

This was a stunning intellectual achievement involving insights which effectively amounted to a curve fitting exercise, which is all that science can ever do.   Note that we now believe that Newton’s gravitational fields do not really exist and prefer to believe instead that space is warped in an invisible new dimension as described by Einstein’s General Theory of Gravitation.   Hopefully this will in turn be replaced by a Quantum Theory of Gravity which we know from other considerations and experiments must exist somehow because General Relativity gives wrong answers at very small distances.