In this section, we investigate how the apparent motion between Sun and Earth changes when we shift from a heliocentric, or sun-in-the-centre, frame of reference to a geocentric, or earth-in-the-centre, frame of reference. A similar discussion is provided in the later part of this section concerning the Sun, the Earth and the planets. The former is based on page 118 of “The Exact Sciences in Antiquity” by O. Neugebauer while the latter is based on page 1010 of James Evans’ article entitled “The division of the Martian eccentricity from Hipparchos to Kepler: A history of the approximations to Kepler motion”.
Consider a system with only Sun S and Earth E. We know that the Earth revolves around the Sun and completes its orbit in a year. See Figure 5(a).
By arresting the motion of the Earth, we would also observe the Sun to revolve around Earth on a circular path but in the opposite direction. This is shown in Figure 5(b). The apparent path of the Sun around the Earth is called the ecliptic.
Now, consider the motion of a superior planet, or one that is farther than the Earth with respect to the Sun, on a heliocentric theory.
With reference to Figure 6(a), the Earth E executes an orbit about the stationary Sun S. In the course of a year, the position vector rotates anticlockwise about S. The angular speed of varies slightly in the course of the year, and so does the length of the vector. Similarly, a superior planet Z revolves around the Sun in a larger orbit. Vector varies at its own slightly different rate. At any instant, the line of sight from the Earth to the planet coincides with which is also equal to Since these vectors may be added in either order, is also equivalent to The new form of the addition is shown in Figure 6(b).
Here we begin at the Earth E. A vector equal to is drawn with its tail at E; let the head of this vector be called K. Then rotates in Figure 6(b) at the same rate as rotates in Figure 6(a). At K, place the tail of a second vector, equal to with its head at planet Z. This series of steps brings about a transformation in a superior planet from a heliocentric model (Figure 6(a)) to a geocentric model (Figure 6(b)). Point K serves as the centre of a small circle, or an epicycle, upon which Z revolves while K itself moves on a large carrying circle, or deferent, about the Earth E. The equivalence of Figures 6(a) and 6(b) is a consequence of the commutative property of vector addition. Hence, in the case of a superior planet, the epicycle corresponds to the orbit of the Earth about the Sun, and the deferent, to the heliocentric orbit of the planet itself.
In the case of an inferior planet, or one that is between the Sun and the Earth in the planetary system, the transformation is similar to that as explained for superior planets, but with adjustments made to the vectors and vector addition.
With reference to figures 7(a) and 7(b), S, E and Z retain their original representations. Notice in Figure 7(a), however, that Z is on the inner orbit and vector is equal to We draw a vector equivalent to with its tail at E and its head at a new point called K on the deferent. At K, we add a vector equal to and call it Once again, Figures 7(a) and 7(b) are equivalent to each other because of the commutative property of vector addition. Thus, for an inferior planet, the epicycle corresponds to its heliocentric orbit whilst the deferent corresponds to the orbit of the Earth about the Sun.
As a remark, the above discussion has assumed circular orbits for Earth and planets about Sun and hence, results in circular deferents and epicycles. Yet if Keplerian motion (elaborated on later), in particular, Kepler’s First Law, is taken into account, then the same discussion would involve orbits that are elliptical, elliptical deferents and epi-ellipses.