The Superior Planets

Figure 25 compares and contrasts heliocentric and geocentric models of the motion of a superior planet (i.e., a planet which is further from the sun than the earth), $P$, as seen from the earth, $G$. The sun is at $S$. In the heliocentric model, we can write the earth-planet displacement vector, ${\bf P}$, as the sum of the earth-sun displacement vector, ${\bf S}$, and the sun-planet displacement vector, ${\bf P}'$. The geocentric model, which is entirely equivalent to the heliocentric model as far as the relative motion of the planet with respect to the earth is concerned, and is much more convenient, relies on the simple vector identity

$${\bf P} = {\bf S}+{\bf P}' \equiv{\bf P}'+{\bf S}.$$ (141)

In other words, we can get from the earth to the planet by one of two different routes. The first route corresponds to the heliocentric model, and the second to the geocentric model. In the latter model, ${\bf P}'$ gives the displacement of the so-called guide-point, $G'$, from the earth. Since ${\bf P}'$ is also the displacement of the planet, $P$, from the sun, $S$, it is clear that $G'$ executes a Keplerian orbit about the earth whose elements are the same as those of the orbit of the planet about the sun. The ellipse traced out by $G'$ is termed the deferent. The vector ${\bf S}$ gives the displacement of the planet from the guide-point. However, ${\bf S}$ is also the displacement of the sun from the earth. Hence, it is clear that the planet, $P$, executes a Keplerian orbit about the guide-point, $G'$, whose elements are the same as the sun's apparent orbit about the earth. The ellipse traced out by $P$ about $G'$ is termed the epicycle.

Figure 25: Heliocentric and geocentric models of the motion of a superior planet. Here, $S$ is the sun, $G$ the earth, and $P$ the planet. View is from the northern ecliptic pole.

Figure 26: Planetary longitude model. View is from northern ecliptic pole.

Figure 26 illustrates in more detail how the deferent-epicycle model is used to determine the ecliptic longitude of a superior planet. The planet $P$ orbits (counterclockwise) on a small Keplerian orbit $\Pi'PA'$ about guide-point $G'$, which, in turn, orbits the earth, $G$, (counterclockwise) on a large Keplerian orbit $\Pi G'A$. As has already been mentioned, the small orbit is termed the epicycle, and the large orbit the deferent. Both orbits are assumed to lie in the plane of the ecliptic. This approximation does not introduce a large error into our calculations because the orbital inclinations of the visible planets to the ecliptic plane are all fairly small. Let $C$, $A$, $\Pi$, $a$, $e$, $\varpi$, and $T$ denote the geometric center, apapsis (i.e., the point of furthest distance from the central object), periapsis (i.e., the point of closest approach to the central object), major radius, eccentricity, longitude of the periapsis, and true anomaly of the deferent, respectively. Let $C'$, $A'$, $\Pi'$, $a'$, $e'$, $\varpi'$, and $T'$ denote the corresponding quantities for the epicycle.

Figure 27: The triangle $GBP$.

Let the line $GG'$ be produced, and let the perpendicular $PB$ be dropped to it from $P$, as shown in Fig. 27. The angle $\mu\equiv PG'B$ is termed the epicyclic anomaly (see Fig. 28), and takes the form

$$\mu =T'+\varpi' - T - \varpi = \bar{\lambda}' + q' - \bar{\lambda}-q,$$ (142)

where $\bar{\lambda}$ and $q$ are the mean longitude and equation of center for the deferent, whereas $\bar{\lambda}'$ and $q'$ are the corresponding quantities for the epicycle--see Sect. 5. The epicyclic anomaly is generally written in the range $0^\circ$ to $360^\circ$. The angle $\theta\equiv PGG'$ is termed the equation of the epicycle, and is usually written in the range $-180^\circ$ to $+180^\circ$. It is clear from the figure that

$$\tan\theta = \frac{\sin\mu}{r/r'+ \cos\mu},$$ (143)

where $r\equiv GG'$ and $r'\equiv G'P$ are the radial polar coordinates for the deferent and epicycle, respectively. Moreover, according to Equation (22), $r/r' = (a/a')\,z$, where

$$z = \frac{1-\zeta}{1-\zeta'},$$ (144)

and

$$\zeta = e\,\cos\,M -e^{\,2}\,\sin^2\,M,$$ (145)

$$\zeta' = e'\cos\,M' -e'^{\,2}\sin^2\,M'$$ (146)

are termed radial anomalies. Finally, the ecliptic longitude of the planet is given by (see Fig. 28)

$$\lambda = \bar{\lambda} + q + \theta.$$ (147)

Now, the function

$$\theta(\mu,z) \equiv \tan^{-1}\left[ \frac{\sin\mu}{(a/a')\,z+ \cos\mu}\right],$$ (148)

has a strong dependence on $\mu$, but only a fairly weak dependence on $z$. In fact, it is easily seen that $z$ varies between $z_{\rm min} = \bar{z} - \delta z$ and $z_{\rm max} = \bar{z}+\delta z$, where

$$\bar{z} = \frac{1+e\,e'}{1-e'^{\,2}},$$ (149)

$$\delta z = \frac{e+e'}{1-e'^{\,2}}.$$ (150)

Let us define

$$\xi = \frac{\bar{z}-z}{\delta z}.$$ (151)

This variable takes the value $-1$ when $z=z_{\rm max}$, the value $0$ when $z=\bar{z}$, and the value $+1$ when $z=z_{\rm min}$. Thus, using quadratic interpolation, we can write

$$\theta(\mu,z)\simeq \Theta_-(\xi)\,\delta\theta_-(\mu) + \bar{\theta}(\mu) + \Theta_+(\xi)\,\delta\theta_+(\mu),$$ (152)

where

$$\bar{\theta}(\mu) = \theta(\mu,\bar{z}),$$ (153)

$$\delta\theta_-(\mu) = \theta(\mu,\bar{z}) - \theta(\mu,z_{\rm max}),$$ (154)

$$\delta\theta_+(\mu) = \theta(\mu,z_{\rm min}) - \theta(\mu,\bar{z}),$$ (155)

and

$$\Theta_-(\xi) = - (1/2)\,\xi\,(\xi-1),$$ (156)

$$\Theta_+(\xi) = + (1/2)\,\xi\,(\xi+1).$$ (157)

Our procedure for determining the ecliptic longitude of a superior planet is described below. It is assumed that the ecliptic longitude, $\lambda_S$, and the radial anomaly, $\zeta_S$, of the sun have already been calculated. The latter quantity is tabulated as a function of the solar mean anomaly in Table 33. In the following, $a$, $e$, $n$, $\tilde{n}$, $\bar{\lambda}_0$, and $M_0$ represent elements of the orbit of the planet in question about the sun, and $e_S$ represents the eccentricity of the sun's apparent orbit about the earth. (In general, the subscript $S$ denotes the sun.) In particular, $a$ is the major radius of the planetary orbit in units in which the major radius of the sun's apparent orbit about the earth is unity. The requisite elements for all of the superior planets at the J2000 epoch ( $t_0= 2\,451\,545.0$ JD) are listed in Table 30. The ecliptic longitude of a superior planet is specified by the following formulae:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (158)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (159)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (160)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (161)

$$\mu = \lambda_S - \bar{\lambda}-q,$$ (162)

$$\bar{\theta} = \theta(\mu,\bar{z})\equiv \tan^{-1} \left(\frac{\sin\mu}{a\,\bar{z}+\cos\mu}\right),$$ (163)

$$\delta\theta_- = \theta(\mu,\bar{z}) - \theta(\mu,z_{\rm max}),$$ (164)

$$\delta\theta_+ = \theta(\mu,z_{\rm min}) - \theta(\mu,\bar{z}),$$ (165)

$$z = \frac{1-\zeta}{1-\zeta_S},$$ (166)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (167)

$$\theta = \Theta_-(\xi)\,\delta\theta_-+ \bar{\theta} + \Theta_+(\xi)\,\delta\theta_+,$$ (168)

$$\lambda = \bar{\lambda} + q+ \theta.$$ (169)

Here, $\bar{z} = (1+e\,e_S)/(1-e_S^{\,2})$, $\delta z = (e+e_S)/(1-e_S^{\,2})$, $z_{\rm min} = \bar{z} - \delta z$, and $z_{\rm max} = \bar{z}+\delta z$. The constants $\bar{z}$, $\delta z$, $z_{\rm min}$, and $z_{\rm max}$ for each of the superior planets are listed in Table 44. Finally, the functions $\Theta_\pm$ are tabulated in Table 45.

For the case of Mars, the above formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of $3'$ and a maximum error of $14'$. For the case of Jupiter, the mean error is $1.6'$ and the maximum error $4'$. Finally, for the case of Saturn, the mean error is $0.5'$ and the maximum error $1'$.

The ecliptic longitude of Mars can be determined with the aid of Tables 46-48. Table 46 allows the mean longitude, $\bar{\lambda}$, and the mean anomaly, $M$, of Mars to be calculated as functions of time. Next, Table 47 permits the equation of center, $q$, and the radial anomaly, $\zeta$, to be determined as functions of the mean anomaly. Finally, Table 48 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 as follows:

1. Determine the fractional Julian day number, $t$, corresponding to the date and time at which the ecliptic longitude is to be calculated with the aid of Tables 1-3. Form $\Delta t = t-t_0$, where $t_0= 2\,451\,545.0$ is the epoch.
2. Calculate the ecliptic longitude, $\lambda_S$, and radial anomaly, $\zeta_S$, of the sun using the procedure set out in Sect. 5.
3. Enter Table 46 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta\bar{\lambda}$ and $\Delta M$. If $\Delta t$ is negative then the corresponding values are also negative. The value of the mean longitude, $\bar{\lambda}$, is the sum of all the $\Delta\bar{\lambda}$ values plus the value of $\bar{\lambda}$ at the epoch. Likewise, the value of the mean anomaly, $M$, is the sum of all the $\Delta M$ values plus the value of $M$ at the epoch. Add as many multiples of $360^\circ$ to $\bar{\lambda}$ and $M$ as is required to make them both fall in the range $0^\circ$ to $360^\circ$. Round $M$ to the nearest degree.
4. Enter Table 47 with the value of $M$ and take out the corresponding value of the equation of center, $q$, and the radial anomaly, $\zeta$. It is necessary to interpolate if $M$ is odd.
5. Form the epicyclic anomaly, $\mu = \lambda_S-\bar{\lambda}-q$. Add as many multiples of $360^\circ$ to $\mu$ as is required to make it fall in the range $0^\circ$ to $360^\circ$. Round $\mu$ to the nearest degree.
6. Enter Table 48 with the value of $\mu$ and take out the corresponding values of $\delta\theta_-$, $\bar{\theta}$, and $\delta\theta_+$. If $\mu > 180^\circ$ then it is necessary to make use of the identities $\delta\theta_\pm(360^\circ - \mu) =-\delta\theta_\pm(\mu)$ and $\bar{\theta}(360^\circ - \mu) =-\bar{\theta}(\mu)$.
7. Form $z = (1-\zeta)/(1-\zeta_S)$.
8. Obtain the values of $\bar{z}$ and $\delta z$ from Table 44. Form $\xi = (\bar{z}-z)/\delta z$.
9. Enter Table 45 with the value of $\xi$ and take out the corresponding values of $\Theta_-$ and $\Theta_+$. If $\xi<0$ then it is necessary to use the identities $\Theta_+(\xi)=-\Theta_-(-\xi)$ and $\Theta_-(\xi)=-\Theta_+(-\xi)$.
10. Form the equation of the epicycle, $\theta = \Theta_-\,\delta\theta_-+ \bar{\theta} + \Theta_+\,\delta\theta_+$.
11. The ecliptic longitude, $\lambda$, is the sum of the mean longitude, $\bar{\lambda}$, the equation of center, $q$, and the equation of the epicycle, $\theta$. If necessary convert $\lambda$ into an angle in the range $0^\circ$ to $360^\circ$. The decimal fraction can be converted into arc minutes using Table 31. Round to the nearest arc minute. The final result can be written in terms of the signs of the zodiac using the table in Sect. 4.6.

Two examples of this procedure are given below.

Example 1: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 5, $t-t_0=1\,950.5$ JD, $\lambda_S = 44.602^\circ$, $M_S\simeq 120^\circ$. Hence, it follows from Table 33 that $\zeta_S= -8.56\times 10^{-3}$. Making use of Table 46, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
+1000	$164.071$	$164.021$
+900	$111.664$	$111.619$
+50	$26.204$	$26.201$
+.5	$0.262$	$0.262$
Epoch	$355.460$	$19.388$
	$657.661$	$321.491$
Modulus	$297.661$	$321.491$

Given that $M\simeq 321^\circ$, Table 47 yields

$$q(321^\circ)= -7.345^\circ,\mbox{\hspace{0.5cm}}\zeta(321^\circ)=6.912\times 10^{-2}.$$

Thus,

$$\mu=\lambda_S - \bar{\lambda}-q = 44.602-297.661 + 7.345= 114.286\simeq 114^\circ,$$

where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 48 that

$$\delta\theta_-(114^\circ) = 3.853^\circ,\mbox{\hspace{0.5cm}... ... \mbox{\hspace{0.5cm}}\delta\theta_+(114^\circ) = 4.612^\circ.$$

Now,

$$z= (1-\zeta)/(1-\zeta_S) = (1-6.912\times 10^{-2})/(1+8.56\times 10^{-3}) = 0.9230.$$

However, from Table 44, $\bar{z}= 1.00184$ and $\delta z = 0.11014$, so

$$\xi = (\bar{z}-z)/\delta z = (1.00184-0.9230)/0.11014 \simeq 0.72.$$

According to Table 45,

$$\Theta_-(0.72) = 0.101, \mbox{\hspace{0.5cm}}\Theta_+(0.72) = 0.619,$$

so

$$\theta = \Theta_-\,\delta\theta_- + \bar{\theta}+\Theta_+\,\... ...+ = 0.101\times 3.853+39.209+0.619\times 4.612 = 42.453^\circ.$$

Finally,

$$\lambda=\bar{\lambda} + q + \theta= 297.661-7.345+42.453=332.769 \simeq 332^\circ 46'.$$

Thus, the ecliptic longitude of Mars at 00:00 GMT on May 5, 2005 CE was 2PI46.

Example 2: December 25, 1800 CE, 00:00 GMT:
 
From Sect. 5, $t-t_0=-72\,690.5$ JD, $\lambda_S = 273.055^\circ$, $M_S\simeq 354^\circ$. Hence, it follows from Table 33 that $\zeta_S= 1.662\times 10^{-2}$. Making use of Table 46, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
-70,000	$-324.983$	$-321.453$
-2,000	$-328.142$	$-328.042$
-600	$-314.443$	$-314.412$
-90	$-47.166$	$-47.162$
-.5	$-0.262$	$-0.262$
Epoch	$355.460$	$19.388$
	$-659.536$	$-991.943$
Modulus	$60.464$	$88.057$

Given that $M\simeq 88^\circ$, Table 47 yields

$$q(88^\circ)= 10.739^\circ, \mbox{\hspace{0.5cm}}\zeta(88^\circ)=-5.45\times 10^{-3},$$

so

$$\mu=\lambda_S - \bar{\lambda}-q = 273.055-60.464-10.739= 201.852\simeq202^\circ.$$

It follows from Table 48 that

$$\delta\theta_-(202^\circ) = -5.980^\circ,\mbox{\hspace{0.5cm... ...\mbox{\hspace{0.5cm}}\delta\theta_+(202^\circ) = -8.955^\circ.$$

Now,

$$z= (1-\zeta)/(1-\zeta_S) = (1+5.45\times 10^{-3})/(1-1.662\times 10^{-2}) = 1.02244,$$

so

$$\xi = (\bar{z}-z)/\delta z = (1.00184-1.02244)/0.11014 \simeq -0.19.$$

According to Table 45,

$$\Theta_-(-0.19) = -0.113, \mbox{\hspace{0.5cm}} \Theta_+(-0.19) = -0.077,$$

so

$$\theta = \Theta_-\,\delta\theta_- + \bar{\theta}+\Theta_+\,\... ...= -0.113\times 5.980-32.007-0.077\times 8.955 = -30.642^\circ.$$

Finally,

$$\lambda=\bar{\lambda} + q + \theta= 60.464+10.739-30.642 = 40.561 \simeq 40^\circ 34'.$$

Thus, the ecliptic longitude of Mars at 00:00 GMT on December 25, 1800 CE was 10TA34.

Figure 28: The geocentric orbit of a superior planet. Here, $G$, $G'$, $P$, $\mu$, $\theta$, $\bar{\lambda}$, $q$, and $\Upsilon$ represent the earth, guide-point, planet, epicyclic anomaly, equation of the epicycle, mean longitude, equation of center, and spring equinox, respectively. View is from northern ecliptic pole. Both $G'$ and $P$ orbit counterclockwise.

Figure 28 shows the geocentric orbit of a superior planet. Recall that the vector $G'P$ is always parallel to the vector connecting the earth to the sun. It follows that a so-called conjunction, at which the sun lies directly between the planet and the earth, occurs whenever the epicyclic anomaly, $\mu$, takes the value $0^\circ$. At a conjunction, the planet is furthest from the earth, and has the same ecliptic longitude as the sun, and is, therefore, invisible. Conversely, a so-called opposition, at which the earth lies directly between the planet and the sun, occurs whenever $\mu=180^\circ$. At an opposition, the planet is closest to the earth, and also directly opposite the sun in the sky, and, therefore, at its brightest. Now, a superior planet rotates around the epicycle at a faster angular velocity than its guide-point rotates around the deferent. Moreover, both the planet and guide-point rotate in the same direction. It follows that the planet is traveling backward in the sky (relative to the direction of its mean motion) at opposition. This phenomenon is called retrograde motion. The period of retrograde motion begins and ends at stations--so-called because when the planet reaches them it appears to stand still in the sky for a few days whilst it reverses direction.

Tables 46-48 can be used to determine the dates of the conjunctions, oppositions, and stations of Mars. Consider the first 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 so-called mean epicyclic anomaly:

$$\mu \simeq \bar{\mu} = \bar{\lambda}_S-\bar{\lambda} = \bar{... ...{\lambda}_0 +(n_S-n)\,(t-t_0) = 284.998 + 0.46157617\,(t-t_0).$$

We obtain

$$t\simeq t_0 + (360-284.998)/0.46157617\simeq t_0 + 162\, {\rm JD}.$$

A calculation of the epicyclic anomaly at this time, using Tables 46-48, yields $\mu=-9.583^\circ$. Now, the actual conjunction occurs when $\mu=0^\circ$. Hence, our next estimate is

$$t\simeq t_0+162+9.583/0.46157617\simeq t_0 + 183\,{\rm JD}.$$

A calculation of the epicyclic anomaly at this time gives $0.294^\circ$. Thus, our final estimate is

$$t=t_0 +183-0.294/0.461557617=t_0+182.4\, {\rm JD},$$

which corresponds to July 1, 2000 CE.

Consider the first opposition of Mars after the epoch. Our first estimate of the time at which this event takes place is

$$t\simeq t_0+(540-284.998)/0.46157617\simeq t_0 + 552\, {\rm JD}.$$

A calculation of the epicyclic anomaly at this time yields $\mu=188.649^\circ$. Now, the actual opposition occurs when $\mu=180^\circ$. Hence, our second estimate is

$$t\simeq t_0 +552-8.649/0.46157617\simeq t_0+533\,{\rm JD}.$$

A calculation of the epicyclic anomaly at this time gives $181.455^\circ$. Thus, our third estimate is

$$t\simeq t_0+533-1.455/0.46157617 \simeq t_0+ 530\,{\rm JD}.$$

A calculation of the epicyclic anomaly at this time yields $180.244^\circ$. Hence, our final estimate is

$$t=t_0+530-0.244/0.46157617=t_0+529.5\,{\rm JD},$$

which corresponds to June 13, 2001 CE. Incidentally, it is clear from the above analysis that the mean time period between successive conjunctions, or oppositions, of Mars is $360/0.46157617= 779.9$ JD, which is equivalent to $2.14$ years.

Let us now consider the stations of Mars. We can approximate the ecliptic longitude of a superior planet as

$$\lambda \simeq \bar{\lambda} + \bar{\theta},$$ (170)

where

$$\bar{\theta} = \tan^{-1}\left(\frac{\sin \bar{\mu}}{\bar{a}+\cos\bar{\mu}}\right),$$ (171)

and $\bar{a} = a\,\bar{z}$. Note that $d\bar{\lambda}/dt= n$ and $d\bar{\mu}/dt = n_S-n$. It follows that

$$\frac{d\lambda}{dt} \simeq n + \left(\frac{\bar{a}\, \cos\bar{\mu}+1}{1+2\,\bar{a}\,\cos\bar{\mu}+\bar{a}^2}\right) (n_S-n).$$ (172)

Now, a station corresponds to $d\lambda/dt = 0$ (i.e., a local maximum or minimum of $\lambda$), which gives

$$\cos\bar{\mu} \simeq - \frac{(\bar{a}^2 + n_S/n)}{\bar{a}\,(1+n_S/n)}.$$ (173)

For the case of Mars, we find that $\bar{\mu}\simeq 163.3^\circ$ or $196.7^\circ$. The first solution corresponds to the so-called retrograde station, at which the planet switches from direct to retrograde motion. The second solution corresponds to the so-called direct station, at which the planet switches from retrograde to direct motion. The mean time interval between a retrograde station and the following opposition, or between an opposition and the following direct station, is $(180-163.3)/0.46157617\simeq 36$ JD. Unfortunately, the only option for accurately determining the dates at which the stations occur is to calculate the ecliptic longitude of Mars over a range of days centered 36 days before and after its opposition.

Table 49 shows the conjunctions, oppositions, and stations of Mars for the years 2000-2020 CE, calculated using the techniques described above.

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 GMT:
 
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 GMT 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.

The ecliptic longitude of Saturn can be determined with the aid of Tables 54-56. Table 54 allows the mean longitude, $\bar{\lambda}$, and the mean anomaly, $M$, of Saturn to be calculated as functions of time. Next, Table 55 permits the equation of center, $q$, and the radial anomaly, $\zeta$, to be determined as functions of the mean anomaly. Finally, Table 56 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 GMT:
 
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 54, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
+1000	$33.508$	$33.482$
+900	$30.157$	$30.133$
+50	$1.675$	$1.674$
+.5	$0.017$	$0.017$
Epoch	$50.059$	$317.857$
	$115.416$	$383.163$
Modulus	$115.416$	$23.163$

Given that $M\simeq 23^\circ$, Table 55 yields

$$q(23^\circ)= 2.561^\circ,\mbox{\hspace{0.5cm}}\zeta(23^\circ)=4.913\times 10^{-2}.$$

Thus,

$$\mu=\lambda_S - \bar{\lambda}-q = 44.602-115.416 -2.561= -73.375\simeq 287^\circ,$$

where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 56 that

$$\delta\theta_-(287^\circ) = -0.353^\circ,\mbox{\hspace{0.5cm... ...\mbox{\hspace{0.5cm}}\delta\theta_+(287^\circ) = -0.405^\circ.$$

Now,

$$z= (1-\zeta)/(1-\zeta_S) = (1-4.913\times 10^{-2})/(1+8.56\times 10^{-3}) = 0.9428.$$

However, from Table 44, $\bar{z}= 1.00118$ and $\delta z = 0.07059$, so

$$\xi = (\bar{z}-z)/\delta z = (1.00118-0.9428)/0.07059 \simeq 0.83.$$

According to Table 45,

$$\Theta_-(0.83) = 0.071, \mbox{\hspace{0.5cm}}\Theta_+(0.83) = 0.759,$$

so

$$\theta = \Theta_-\,\delta\theta_- + \bar{\theta}+\Theta_+\,\... ...+ = -0.071\times 0.353-5.551-0.759\times 0.405 =- 5.883^\circ.$$

Finally,

$$\lambda=\bar{\lambda} + q + \theta= 115.416+2.561-5.883=112.094 \simeq 112^\circ 06'.$$

Thus, the ecliptic longitude of Saturn at 00:00 GMT on May 5, 2005 CE was 22CN06.

The conjunctions, oppositions, and stations of Saturn 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 Saturn is 1.035 yr. Furthermore, on average, the retrograde and direct stations of Saturn occur when the epicyclic anomaly takes the values $\mu=114.5^\circ$ and $245.5^\circ$, respectively. Finally, the mean time period between a retrograde station and the following opposition, or between the opposition and the following direct station, is 69 JD. The conjunctions, oppositions, and stations of Saturn during the years 2000-2010 CE are shown in Table 57.

Table 44: Constants associated with the epicycles of the inferior and superior planets.

Planet	$\bar{z}$	$\delta z$	$z_{\rm min}$	$z_{\rm max}$
Mercury	1.04774	0.23216	0.81558	1.27990
Venus	1.00016	0.02349	0.97667	1.02365
Mars	1.00184	0.11014	0.89170	1.11198
Jupiter	1.00109	0.06512	0.93597	1.06602
Saturn	1.00118	0.07059	0.93059	1.07177

Table 45: Epicyclic interpolation coefficients. Note that $\Theta_\pm (\xi) = - \Theta_\mp (-\xi)$.

$\xi$	$\Theta_-$	$\Theta_+$	$\xi$	$\Theta_-$	$\Theta_+$	$\xi$	$\Theta_-$	$\Theta_+$	$\xi$	$\Theta_-$	$\Theta_+$
0.00	0.000	0.000	0.25	0.094	0.156	0.50	0.125	0.375	0.75	0.094	0.656
0.01	0.005	0.005	0.26	0.096	0.164	0.51	0.125	0.385	0.76	0.091	0.669
0.02	0.010	0.010	0.27	0.099	0.171	0.52	0.125	0.395	0.77	0.089	0.681
0.03	0.015	0.015	0.28	0.101	0.179	0.53	0.125	0.405	0.78	0.086	0.694
0.04	0.019	0.021	0.29	0.103	0.187	0.54	0.124	0.416	0.79	0.083	0.707
0.05	0.024	0.026	0.30	0.105	0.195	0.55	0.124	0.426	0.80	0.080	0.720
0.06	0.028	0.032	0.31	0.107	0.203	0.56	0.123	0.437	0.81	0.077	0.733
0.07	0.033	0.037	0.32	0.109	0.211	0.57	0.123	0.447	0.82	0.074	0.746
0.08	0.037	0.043	0.33	0.111	0.219	0.58	0.122	0.458	0.83	0.071	0.759
0.09	0.041	0.049	0.34	0.112	0.228	0.59	0.121	0.469	0.84	0.067	0.773
0.10	0.045	0.055	0.35	0.114	0.236	0.60	0.120	0.480	0.85	0.064	0.786
0.11	0.049	0.061	0.36	0.115	0.245	0.61	0.119	0.491	0.86	0.060	0.800
0.12	0.053	0.067	0.37	0.117	0.253	0.62	0.118	0.502	0.87	0.057	0.813
0.13	0.057	0.073	0.38	0.118	0.262	0.63	0.117	0.513	0.88	0.053	0.827
0.14	0.060	0.080	0.39	0.119	0.271	0.64	0.115	0.525	0.89	0.049	0.841
0.15	0.064	0.086	0.40	0.120	0.280	0.65	0.114	0.536	0.90	0.045	0.855
0.16	0.067	0.093	0.41	0.121	0.289	0.66	0.112	0.548	0.91	0.041	0.869
0.17	0.071	0.099	0.42	0.122	0.298	0.67	0.111	0.559	0.92	0.037	0.883
0.18	0.074	0.106	0.43	0.123	0.307	0.68	0.109	0.571	0.93	0.033	0.897
0.19	0.077	0.113	0.44	0.123	0.317	0.69	0.107	0.583	0.94	0.028	0.912
0.20	0.080	0.120	0.45	0.124	0.326	0.70	0.105	0.595	0.95	0.024	0.926
0.21	0.083	0.127	0.46	0.124	0.336	0.71	0.103	0.607	0.96	0.019	0.941
0.22	0.086	0.134	0.47	0.125	0.345	0.72	0.101	0.619	0.97	0.015	0.955
0.23	0.089	0.141	0.48	0.125	0.355	0.73	0.099	0.631	0.98	0.010	0.970
0.24	0.091	0.149	0.49	0.125	0.365	0.74	0.096	0.644	0.99	0.005	0.985
0.25	0.094	0.156	0.50	0.125	0.375	0.75	0.094	0.656	1.00	0.000	1.000

Table 46: Mean motion of Mars. Here, $\Delta t = t-t_0$, $\Delta\bar{\lambda} = \bar{\lambda}-\bar{\lambda}_0$, $\Delta M = M - M_0$, and $\Delta\bar{F}= \bar{F}-\bar{F}_0$. At epoch ( $t_0= 2\,451\,545.0$ JD), $\bar{\lambda}_0 = 355.460^\circ$, $M_0 = 19.388^\circ$, and $\bar{F}_0 =305.796^\circ$.

$\Delta t$(JD)	$\Delta\bar{\lambda}(^\circ)$	$\Delta M(^\circ)$	$\Delta \bar{F}(^\circ)$	$\Delta t$(JD)	$\Delta\bar{\lambda}(^\circ)$	$\Delta M(^\circ)$	$\Delta \bar{F}(^\circ)$
10,000	200.712	200.208	200.409	1,000	164.071	164.021	164.041
20,000	41.424	40.415	40.819	2,000	328.142	328.042	328.082
30,000	242.135	240.623	241.228	3,000	132.214	132.062	132.123
40,000	82.847	80.830	81.638	4,000	296.285	296.083	296.164
50,000	283.559	281.038	282.047	5,000	100.356	100.104	100.205
60,000	124.271	121.246	122.456	6,000	264.427	264.125	264.246
70,000	324.983	321.453	322.866	7,000	68.498	68.145	68.287
80,000	165.694	161.661	163.275	8,000	232.569	232.166	232.328
90,000	6.406	1.868	3.685	9,000	36.641	36.187	36.368
100	52.407	52.402	52.404	10	5.241	5.240	5.240
200	104.814	104.804	104.808	20	10.481	10.480	10.481
300	157.221	157.206	157.212	30	15.722	15.721	15.721
400	209.628	209.608	209.616	40	20.963	20.961	20.962
500	262.036	262.010	262.020	50	26.204	26.201	26.202
600	314.443	314.412	314.425	60	31.444	31.441	31.442
700	6.850	6.815	6.829	70	36.685	36.681	36.683
800	59.257	59.217	59.233	80	41.926	41.922	41.923
900	111.664	111.619	111.637	90	47.166	47.162	47.164
1	0.524	0.524	0.524	0.1	0.052	0.052	0.052
2	1.048	1.048	1.048	0.2	0.105	0.105	0.105
3	1.572	1.572	1.572	0.3	0.157	0.157	0.157
4	2.096	2.096	2.096	0.4	0.210	0.210	0.210
5	2.620	2.620	2.620	0.5	0.262	0.262	0.262
6	3.144	3.144	3.144	0.6	0.314	0.314	0.314
7	3.668	3.668	3.668	0.7	0.367	0.367	0.367
8	4.193	4.192	4.192	0.8	0.419	0.419	0.419
9	4.717	4.716	4.716	0.9	0.472	0.472	0.472

Table 47: Deferential anomalies of Mars.

$M(^\circ)$	$q(^\circ)$	$100\,\zeta$	$M(^\circ)$	$q(^\circ)$	$100\,\zeta$	$M(^\circ)$	$q(^\circ)$	$100\,\zeta$	$M(^\circ)$	$q(^\circ)$	$100\,\zeta$
0	0.000	9.339	90	10.702