Astronomy Principles and Practice Fourth Edition.pdf

Planetary orbits 441
Figure 24.28. The construction for determining the distance of Saturn from the Sun.
For the graphical representation of the orbits of the Earth and Saturn, it may be assumed that they
are coplanar. To a sufficient accuracy, it is only necessary to record the celestial longitude of Saturn. It
can be taken that the Earth moves through 360/365 ◦ per day. Its positions in an assumed circular orbit
can be plotted from reference to the angle between the Earth, the Sun and the first point of Aries, ,
for the dates when the position of Saturn is observed. For a starting point, on Sep 22nd (when the Sun
is at RA 12 h ), the Sun, Earth and are in line, the Sun being opposite to the direction of . Usinga
scale of 1 AU = 25 mm, plot the positions of the Earth in its orbit according to the number of elapsed
days from September 22nd. From those positions, draw the observed directions of Saturn in relation to
, these corresponding to values of λ as determined from equation (24.19).
Kepler’s second law states that the radius vector joining the Sun to a planet sweeps out equal areas
in equal times. If Saturn’s orbit is assumed to be circular, then, in this case, the planet will sweep out
equal arcs in equal times. There is, therefore, one particular circular orbit on which Saturn lies that
cuts the Earth–Saturn direction lines, giving equal arcs along its orbit.
By trial and error, draw this circle on the construction and, hence, determine the distance of Saturn
from the Sun.
Plots of the Earth orbit and the direction lines to Saturn are illustrated in figure 24.28. From the
construction, Saturn’s orbit might be placed at 8·2 AU from the Sun, an answer which is good to 10%.
(NB: The exercise is suitable only for the orbits of Jupiter and Saturn—extremely accurate observations
are required and allowance must be made for eccentricity if the orbit of Mars is to be constructed in
this way.)