Heliocentric and Geocentric Equivalence

 

INTRODUCTION

 

Our earliest attempts to describe the motions of heavenly bodies generally produced two competing models.   The obvious and common sense ”Geocentric” model placed a stationary earth at the center of the universe while the more elegant “Heliocentric” model had the Earth and planets revolve around a stationary sun.   One difficulty in distinguishing between the two is that for the simplest case of perfectly circular orbits, the kinematics of both models are surprisingly identical.   That is, the distance and direction of any planet as seen from Earth is 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.  If we consider only the planets of Earth and Mars, we can graphically diagram both systems as

 

In the Heliocentric system both planets orbit the Sun as illustrated by blue and red circles.  To create an 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 would give identical results for either system as viewed from Earth.

 

MATHEMATICAL EQUIVALENCE

 

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

 

 

 

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?”   The answer is that what makes them different is not the kinematics of their relative positions but rather the dynamical forces governing their motions.

 

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.

Before the invention of the telescope in Holland in the early 1600’s, it seemed absurd to imaging the apparently stationary Earth spinning through space at incredible speeds.  Occam’s razor which prefers the simpler solution clearly was on the side of the Geocentric Model.   In addition, several scientific observations militated against the Heliocentric model to include

 

a)      Given that all of space was thought to be filled with air, a spinning earth should create a wind shear which is not observed at the Earth’s surface.

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 not observed.

c)      The fixed stars, whose positions had been carefully measured to within about 10 arc seconds did not demonstrate any parallax effects.   So unless stars were much brighter than the planets and unimaginably further away, a moving Earth could not be right.

 

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.