Determination of Conjunction and Greatest Elongation Dates

The geocentric orbit of an inferior planet is similar to that of the superior planet shown in Fig. 32, except for the fact that the sun is coincident with guide-point $G'$ in the former case. It follows that it is impossible for an inferior planet to have an opposition with the sun (i.e, for the earth to lie directly between the planet and the sun). However, inferior planets do have two different kinds of conjunctions with the sun. A superior conjuction takes place when the sun lies directly between the planet and the earth. Conversely, an inferior conjunction takes place when the planet lies directly between the sun and the earth. It is clear from Fig. 32 that a superior conjunction corresponds to $\mu=0^\circ$, and an inferior conjunction to $\mu=180^\circ$. Now, the equation of the epicycle, $\theta$, measures the angular separation between the planet and the sun (since the sun lies at the guide-point). It is evident from Figure 32 that $\theta$ attains a maximum and a minimum value each time the planet revolves around its epicycle. In other words, there is a limit to how large the angular separation between an inferior planet and the sun can become. The maximum value is termed the greatest eastern elongation of the planet, whereas the modulus of the minimum value is termed the greatest western elongation.

Tables 58-60 can be used to determine the dates of the conjunctions and greatest elongations of Venus. Consider the first superior conjunction after the epoch (January 1, 2000 CE). We can estimate the time at which this event occurs by approximating the epicyclic anomaly as the mean epicyclic anomaly:

$$\mu \simeq \bar{\mu} = \bar{\lambda}-\bar{\lambda}_S = \bar{... ...da}_{0\,S} + (n-n_S)\,(t-t_0) = 261.515 + 0.61652137\,(t-t_0).$$

Thus,

$$t\simeq t_0 + (360-261.515)/0.61652137\simeq t_0 + 160\, {\rm JD}.$$

A calculation of the epicyclic anomaly at this time, using Tables 58-60, yields $\mu = -1.267^\circ$. Now, the actual conjunction takes place when $\mu=0^\circ$. Hence, our final estimate is

$$t=t_0+160+1.267/0.61652137= t_0 + 162.1\,{\rm JD},$$

which corresponds to June 11, 2000 CE.

Consider the first inferior conjunction of Venus after the epoch. Our first estimate of the time at which this event takes place is

$$t\simeq t_0+(540-261.515)/0.61652137\simeq t_0 + 452\,{\rm JD}.$$

A calculation of the epicyclic anomaly at this time yields $\mu=178.900^\circ$. Now, the actual conjunction takes place when $\mu=180^\circ$. Hence, our final estimate is

$$t = t_0 +452+1.100/0.61652137 = t_0+453.8\,{\rm JD},$$

which corresponds to March 30, 2001 CE. Incidentally, it is clear from the above analysis that the mean time period between successive superior, or inferior, conjunctions of Venus is $360/0.61652137= 583.9$ JD, which is equivalent to $1.60$ years.

Consider the greatest elongations of Venus. We can approximate the equation of the epicycle as

$$\theta\simeq \bar{\theta} = \tan^{-1}\left(\frac{\sin \bar{\mu}}{\bar{a}^{-1}+\cos\bar{\mu}}\right),$$ (202)

where $\bar{\mu}$ is the mean epicyclic anomaly, and $\bar{a}=a/\bar{z}$. It follows that

$$\frac{d\bar{\theta}}{d\bar{\mu}} = \frac{\bar{a}^{-1}\,\cos\bar{\mu}+1} {1 + 2\,\bar{a}^{-1}\,\cos\bar{\mu} + \bar{a}^{-2}}.$$ (203)

Now, $\bar{\theta}$ attains its maximum or minimum value when $d\bar{\theta}/d\bar{\mu}=0$: i.e., when

$$\bar{\mu} = \cos^{-1}(-\bar{a}).$$ (204)

For the case of Venus, we obtain $\bar{\mu} = 136.3^\circ$ or $223.7^\circ$. The first solution corresponds to the greatest eastern elongation, and the second to the greatest western elongation. Substituting back into Eq. (202), we find that $\bar{\theta} = \pm 46.3^\circ$. Hence, the mean value of the greatest eastern or western elongation of Venus is $46.3^\circ$. The mean time period between a greatest eastern elongation and the following inferior conjunction, or between an inferior conjunction and the following greatest western elongation, is $(180-136.3)/0.61652137\simeq 71$ JD. Unfortunately, the only option for accurately determining the dates at which the greatest elongations occur is to calculate the equation of the epicycle of Venus over a range of days centered 71 days before and after an inferior conjunction.

Table 61 shows the conjunctions, and greatest elongations of Venus for the years 2000-2015 CE, calculated using the techniques described above.