Mercury

Figures 38 and 39 show ecliptic longitude and latitude data, respectively, for Mercury, covering the years 1995-1996. As usual, we need seven pieces of information to fit our model to the data. Our two ascending node data-points are $t_1=19.739$ and $t_2=107.718$. Our three superior conjunction data-points are $t_3=103.530$, $t_4=208.091$, $t_5=326.228$ and $\lambda_3=24.107^\circ$, $\lambda_5=124.566^\circ$, $\lambda_5=240.404^\circ$. Our maximum elongation data-point is $t_6=130.0$ and $\lambda_6=71.244^\circ$. Finally, our maximal ecliptic latitude data-point is $t_7=36.$ and $\beta_7=3.665^\circ$. These data-points are indicated in Figs. 38 and 39.

Figure 38: The ecliptic longitude of Mercury versus time. The vertical green lines indicate times of the first three superior conjunctions. The vertical yellow line indicates the time of the maximum elongation from the guide-point.

Figure 39: The ecliptic latitude of Mercury versus time. The vertical cyan lines indicates the times of the first two ascending nodes. 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 Mercury listed in Tab. 1. The true elements are given in Tab. 2. It can be seen that our model determines the orbital elements of Mercury to reasonable accuracy.

Figure 40: The ecliptic latitude of Mercury versus its ecliptic longitude. The blue and red curves indicate the prediction of the updated Almagest model, and the original data, respectively.

Figure 41: The residual in the ecliptic longitude of Mercury versus time.

Figure 42: The residual in the ecliptic latitude of Mercury versus time.

Figure 40 shows the ecliptic latitude versus the ecliptic longitude of Mercury for the years period 1995-1996, 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. 41 and 42 give the residuals in the ecliptic longitude and latitude of Mercury, respectively, versus time. It can be seen that the maximum error in longitude is about $70'$, whereas the maximum error in latitude is about $20'$. Thus, our updated version of Ptolemy's model does a fair job of accounting for Mercury's apparent motion.