Saturn

Figures 28 and 29 show ecliptic longitude and latitude data, respectively, for Saturn, covering the years 1995-2006. As usual, we need seven pieces of information to fit our model to the data. Unfortunately, our data-set is again not long enough to contain two ascending nodes of Saturn. As before, we shall assume that, if it were, then we could easily deduce Saturn's orbital period, $10759.1$ days, as the time difference between two successive ascending nodes. Our single ascending node data-point is $t_1=3659.08$. Our three opposition data-points are $t_3=256.639$, $t_4=634.800$, $t_5=1013.185$ and $\lambda_3=-8.662^\circ$, $\lambda_5=-3.966^\circ$, $\lambda_5=16.908^\circ$. Our maximum elongation data-point is $t_6=348.0$ and $\lambda_6=-11.528^\circ$. Finally, our maximal ecliptic latitude data-point is $t_7=1014.0$ and $\beta_7=-2.762^\circ$. These data-points are indicated in Figs. 28 and 29.

Figure 28: The ecliptic longitude of Saturn versus time. The vertical green lines indicate times of the first three oppositions. The vertical yellow line indicates the time of the maximum elongation from the guide-point.

Figure 29: The ecliptic latitude of Saturn versus time. The vertical cyan line indicates the time of the ascending node. The vertical green line indicates the time of the maximal latitude.

Making use of the above data, and the iterative method described in Sect. 2.6.4, we obtain the orbital elements for Saturn listed in Tab. 1. The true elements are given in Tab. 2. It can be seen that our model determines the orbital elements of Saturn to good accuracy.

Figure 30: The ecliptic latitude of Saturn versus its ecliptic longitude. The blue and red curves indicate the prediction of the updated Almagest model, and the original data, respectively.

Figure 31: The residual in the ecliptic longitude of Saturn versus time.

Figure 32: The residual in the ecliptic latitude of Saturn versus time.

Figure 30 shows the ecliptic latitude versus the ecliptic longitude of Saturn for the period 1995-2006, and compares the predictions of our updated Almagest model, made using the orbital elements given in Tab. 1, against the original data. As before, the two are essentially indistinguishable. Finally, Figs. 31 and 32 give the residuals in the ecliptic longitude and latitude of Saturn, respectively, versus time. It can be seen that the maximum error in longitude is about $15'$, whereas the maximum error in latitude is about $0.5'$. Thus, our updated version of Ptolemy's model does a good job of accounting for Saturn's apparent motion.