Venus

Our treatment of the inferior planets differs from that of the superior planets in one respect. Inferior planets do not have oppositions, only conjunctions (i.e., points at which the ecliptic longitude the planet is the same as that of the Sun). There are, in fact, two types of conjunctions. Superior conjunctions occur when the Sun lies between the planet and the Earth. Conversely, inferior conjunctions occur when the planet lies between the Sun and the Earth. Conjuctions satisfy $P_x/P_y=S_x/S_y=(-S')/(-C')$, similarly to the oppositions of superior planets. Thus, our previous analysis remains valid for inferior planets, as long as we replace oppositions by (say) superior conjunctions.

Figures 33 and 34 show ecliptic longitude and latitude data, respectively, for Venus, covering the years 1995-2000. As usual, we need seven pieces of information to fit our model to the data. Our two ascending node data-points are $t_1=188.287$ and $t_2=412.972$. Our three superior conjunction data-points are $t_3=232.004$, $t_4=822.573$, $t_5=1398.285$ and $\lambda_3=147.491^\circ$, $\lambda_5=232.856^\circ$, $\lambda_5=1296.544^\circ$. Our maximum elongation data-point (which occurs approximately half-way between a superior and an inferior conjuction) is $t_6=380.0$ and $\lambda_6=330.989^\circ$. Finally, our maximal ecliptic latitude data-point is $t_7=1698.0$ and $\beta_7=-8.338^\circ$. These data-points are indicated in Figs. 33 and 34.

Figure 33: The ecliptic longitude of Venus 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 34: The ecliptic latitude of Venus 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 Venus listed in Tab. 1. The true elements are given in Tab. 2. It can be seen that our model determines the orbital elements of Venus to reasonable accuracy.

Figure 35: The ecliptic latitude of Venus versus its ecliptic longitude. The blue and red curves indicate the prediction of the updated Almagest model, and the original data, respectively.

Figure 36: The residual in the ecliptic longitude of Venus versus time.

Figure 37: The residual in the ecliptic latitude of Venus versus time.

Figure 35 shows the ecliptic latitude versus the ecliptic longitude of Venus for part of the period 1995-2000, and compares the predictions of our updated Almagest model, made using the orbital elements given in Tab. 1, against the original data. As usual, the two are essentially indistinguishable. Finally, Figs. 36 and 37 give the residuals in the ecliptic longitude and latitude of Venus, respectively, versus time. It can be seen that the maximum error in longitude is about $20'$, whereas the maximum error in latitude is about $3'$. Thus, our updated version of Ptolemy's model does a reasonably good job of accounting for Venus's apparent motion.