Jupiter

The ecliptic longitude of Jupiter can be determined with the aid of Tables 50-52. Table 50 allows the mean longitude, $\bar{\lambda}$, and the mean anomaly, $M$, of Jupiter to be calculated as functions of time. Next, Table 51 permits the equation of center, $q$, and the radial anomaly, $\zeta$, to be determined as functions of the mean anomaly. Finally, Table 52 allows the quantities $\delta\theta_-$, $\bar{\theta}$, and $\delta\theta_+$ to be calculated as functions of the epicyclic anomaly, $\mu$. The procedure for using the tables is analogous to the previously described procedure for using the Mars tables. One example of this procedure is given below.

 
Example: May 5, 2005 CE, 00:00 UT:
 
From before, $t-t_0=1\,950.5$ JD, $\lambda_S = 44.602^\circ$, $M_S\simeq 120^\circ$, and $\zeta_S= -8.56\times 10^{-3}$. Making use of Table 50, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
+1000	$83.125$	$83.081$
+900	$74.813$	$74.773$
+50	$4.156$	$4.154$
+.5	$0.042$	$0.042$
Epoch	$34.365$	$19.348$
	$196.501$	$181.398$
Modulus	$196.501$	$181.398$

Given that $M\simeq 181^\circ$, Table 51 yields

$$q(181^\circ)= -0.091^\circ,\mbox{\hspace{0.5cm}}\zeta(181^\circ)=-4.838\times 10^{-2}.$$

Thus,

$$\mu=\lambda_S - \bar{\lambda}-q = 44.602-196.501 + 0.091= -151.808\simeq 208^\circ,$$

where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 52 that

$$\delta\theta_-(208^\circ) = -0.447^\circ,\mbox{\hspace{0.5cm... ...\mbox{\hspace{0.5cm}}\delta\theta_+(208^\circ) = -0.522^\circ.$$

Now,

$$z= (1-\zeta)/(1-\zeta_S) = (1+4.838\times 10^{-2})/(1+8.56\times 10^{-3}) = 1.0395.$$

However, from Table 44, $\bar{z}= 1.00109$ and $\delta z = 0.06512$, so

$$\xi = (\bar{z}-z)/\delta z = (1.00109-1.0395)/0.06512 \simeq -0.59.$$

According to Table 45,

$$\Theta_-(-0.59) = -0.469, \mbox{\hspace{0.5cm}}\Theta_+(-0.59) = -0.121,$$

so

$$\theta = \Theta_-\,\delta\theta_- + \bar{\theta}+\Theta_+\,\... ..._+ = 0.469\times 0.447-6.194+0.121\times 0.522 =- 5.921^\circ.$$

Finally,

$$\lambda=\bar{\lambda} + q + \theta= 196.501-0.091-5.921=190.489 \simeq 190^\circ 29'.$$

Thus, the ecliptic longitude of Jupiter at 00:00 UT on May 5, 2005 CE was 10LI29.

The conjunctions, oppositions, and stations of Jupiter can be investigated using analogous methods to those employed earlier to examine the conjunctions, oppositions, and stations of Mars. We find that the mean time period between successive oppositions or conjunctions of Jupiter is 1.09 yr. Furthermore, on average, the retrograde and direct stations of Jupiter occur when the epicyclic anomaly takes the values $\mu=125.6^\circ$ and $234.4^\circ$, respectively. Finally, the mean time period between a retrograde station and the following opposition, or between the opposition and the following direct station, is 60 JD. The conjunctions, oppositions, and stations of Jupiter during the years 2000-2010 CE are shown in Table 53.