#### HIPPARCHUS’ MODEL OF THE MOTION OF THE SUN

Greek astronomers believe that all orbits of luminaries and planets treated in astronomy should be circles or components of circles. The simplest manner to represent the apparent motion of the Sun S as observed from the Earth E would be a circle in the plane of the ecliptic, centered on the Earth E. This may be seen in Figure 8.

Figure 8

The solstices and equinoxes are located 90˚ apart as seen from the Earth E, and if we take y as the number of days in a year, we would expect the interval between the seasons to be exactly of length days. However, the observed facts show otherwise: the seasons are not equal. In particular, Hipparchus found the Sun to move 90˚ in the ecliptic plane from Spring equinox to Summer solstice in 94˝ days and 90˚ from Summer solstice to Autumn equinox in 92˝ days. Modern values for the lengths of the seasons are as follows: Spring – 92 days, 18 hours, 20 minutes or 92.764 days; Summer – 93 days, 15 hours, 31 minutes or 93.647 days; Autumn – 89 days, 20 hours, 4 minutes or 89.836 days; and Winter – 88 days, 23 hours, 56 minutes or 88.997 days. To explain this phenomenon, ancient astronomers would rather regard the inequalities in the speed of the Sun as an optical illusion than rule that the Sun does not move at uniform angular speed. For more information on how the illusion works, please refer to pages 104 – 105 in Heilbron’s book. The essence of their solution was to displace the Earth from the centre of the Sun’s orbit.

Figure 9

Much of the information in the following paragraph concerning Hipparchus’ simple model of the orbit of the Sun is quoted from page 41 of “The Cambridge Illustrated of Astronomy” by Michael Hoskin.

In Figure 9, Hipparchus’ solar model is given. The Earth is stationary at E, while the Sun S moves around the circle at a uniform angular speed about the centre of the circle C. The Sun’s circle is thus said to be eccentric to the Earth. To generate the longer intervals between solstice and equinox, the Earth had to be removed from the centre in the opposite direction so that the corresponding arcs as seen from the Earth would each be more than  of the circle, and it would take longer than days to traverse them. Hipparchus’ calculations showed that the distance between the Earth and centre has to be 1/24 of the radius of the circle and that the line from the Earth to centre had to make an angle of 65˝˚ with the Spring equinox.

At this point, I would like to highlight some of the terminology used in Heilbron’s book as well as this paper.

Figure 10

With reference to Figure 10, suppose a is the radius of the circle centered at C and Z is a body moving on the circle. The eccentricity e is defined as the ratio of the distance between O and C to radius a, that is,

This is similar to the definition of eccentricity for an ellipse. In Hipparchus’ model for the Sun’s orbit mentioned earlier, the figure “1/24” is in fact the eccentricity. The distance of separation OC is thus equivalent to ae.

Perigee P occurs where Z is closest to O whilst apogee A occurs where Z is furthest from O.

The line joining the perigee and apogee is known as the line of apsides.