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Saturn

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 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 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

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

Thus,

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

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

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

Now,

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

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

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

According to Table 45,

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

so

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

Finally,

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

Thus, the ecliptic longitude of Saturn at 00:00 UT 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 -0.872 180 0.000 -9.339 270 -10.702 -0.872
2 0.417 9.333 92 10.652 -1.197 182 -0.330 -9.335 272 -10.739 -0.545
4 0.833 9.312 94 10.589 -1.519 184 -0.660 -9.321 274 -10.763 -0.217
6 1.249 9.279 96 10.514 -1.839 186 -0.989 -9.298 276 -10.773 0.114
8 1.662 9.232 98 10.426 -2.155 188 -1.317 -9.265 278 -10.770 0.444
10 2.072 9.171 100 10.326 -2.468 190 -1.645 -9.224 280 -10.753 0.776
12 2.479 9.098 102 10.214 -2.776 192 -1.971 -9.173 282 -10.722 1.107
14 2.882 9.011 104 10.091 -3.081 194 -2.296 -9.113 284 -10.678 1.438
16 3.281 8.911 106 9.957 -3.380 196 -2.619 -9.044 286 -10.619 1.768
18 3.674 8.799 108 9.811 -3.675 198 -2.940 -8.966 288 -10.546 2.097
20 4.062 8.674 110 9.655 -3.964 200 -3.259 -8.878 290 -10.458 2.424
22 4.443 8.537 112 9.489 -4.248 202 -3.575 -8.782 292 -10.357 2.749
24 4.817 8.388 114 9.313 -4.527 204 -3.889 -8.676 294 -10.241 3.071
26 5.184 8.227 116 9.127 -4.799 206 -4.199 -8.562 296 -10.111 3.389
28 5.542 8.054 118 8.932 -5.065 208 -4.506 -8.438 298 -9.967 3.705
30 5.892 7.870 120 8.727 -5.324 210 -4.810 -8.306 300 -9.809 4.016
32 6.233 7.675 122 8.514 -5.576 212 -5.110 -8.165 302 -9.637 4.322
34 6.564 7.470 124 8.293 -5.822 214 -5.405 -8.015 304 -9.452 4.623
36 6.885 7.254 126 8.064 -6.060 216 -5.696 -7.857 306 -9.252 4.919
38 7.195 7.029 128 7.827 -6.292 218 -5.983 -7.690 308 -9.040 5.208
40 7.494 6.794 130 7.583 -6.515 220 -6.264 -7.515 310 -8.814 5.491
42 7.782 6.550 132 7.332 -6.731 222 -6.540 -7.331 312 -8.575 5.768
44 8.059 6.297 134 7.074 -6.939 224 -6.810 -7.139 314 -8.323 6.036
46 8.323 6.036 136 6.810 -7.139 226 -7.074 -6.939 316 -8.059 6.297
48 8.575 5.768 138 6.540 -7.331 228 -7.332 -6.731 318 -7.782 6.550
50 8.814 5.491 140 6.264 -7.515 230 -7.583 -6.515 320 -7.494 6.794
52 9.040 5.208 142 5.983 -7.690 232 -7.827 -6.292 322 -7.195 7.029
54 9.252 4.919 144 5.696 -7.857 234 -8.064 -6.060 324 -6.885 7.254
56 9.452 4.623 146 5.405 -8.015 236 -8.293 -5.822 326 -6.564 7.470
58 9.637 4.322 148 5.110 -8.165 238 -8.514 -5.576 328 -6.233 7.675
60 9.809 4.016 150 4.810 -8.306 240 -8.727 -5.324 330 -5.892 7.870
62 9.967 3.705 152 4.506 -8.438 242 -8.932 -5.065 332 -5.542 8.054
64 10.111 3.389 154 4.199 -8.562 244 -9.127 -4.799 334 -5.184 8.227
66 10.241 3.071 156 3.889 -8.676 246 -9.313 -4.527 336 -4.817 8.388
68 10.357 2.749 158 3.575 -8.782 248 -9.489 -4.248 338 -4.443 8.537
70 10.458 2.424 160 3.259 -8.878 250 -9.655 -3.964 340 -4.062 8.674
72 10.546 2.097 162 2.940 -8.966 252 -9.811 -3.675 342 -3.674 8.799
74 10.619 1.768 164 2.619 -9.044 254 -9.957 -3.380 344 -3.281 8.911
76 10.678 1.438 166 2.296 -9.113 256 -10.091 -3.081 346 -2.882 9.011
78 10.722 1.107 168 1.971 -9.173 258 -10.214 -2.776 348 -2.479 9.098
80 10.753 0.776 170 1.645 -9.224 260 -10.326 -2.468 350 -2.072 9.171
82 10.770 0.444 172 1.317 -9.265 262 -10.426 -2.155 352 -1.662 9.232
84 10.773 0.114 174 0.989 -9.298 264 -10.514 -1.839 354 -1.249 9.279
86 10.763 -0.217 176 0.660 -9.321 266 -10.589 -1.519 356 -0.833 9.312
88 10.739 -0.545 178 0.330 -9.335 268 -10.652 -1.197 358 -0.417 9.333
90 10.702 -0.872 180 0.000 -9.339 270 -10.702 -0.872 360 -0.000 9.339



Table 48: Epicyclic anomalies of Mars. All quantities are in degrees. Note that $\bar{\theta}(360^\circ - \mu) =-\bar{\theta}(\mu)$, and $\delta\theta_{\pm}(360^\circ-\mu) = -\delta\theta_{\pm}(\mu)$.
$\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$
0 0.000 0.000 0.000 45 1.159 17.566 1.329 90 2.679 33.228 3.125 135 5.180 40.793 6.547
1 0.025 0.396 0.028 46 1.187 17.945 1.362 91 2.721 33.527 3.176 136 5.246 40.716 6.658
2 0.049 0.792 0.056 47 1.216 18.322 1.394 92 2.764 33.822 3.228 137 5.312 40.619 6.771
3 0.074 1.187 0.084 48 1.244 18.699 1.427 93 2.807 34.114 3.281 138 5.378 40.503 6.885
4 0.099 1.583 0.113 49 1.273 19.075 1.461 94 2.851 34.403 3.335 139 5.442 40.366 7.001
5 0.123 1.979 0.141 50 1.302 19.450 1.494 95 2.895 34.688 3.390 140 5.506 40.206 7.118
6 0.148 2.374 0.169 51 1.331 19.824 1.528 96 2.940 34.969 3.445 141 5.568 40.024 7.235
7 0.173 2.770 0.197 52 1.360 20.196 1.562 97 2.985 35.246 3.501 142 5.628 39.817 7.354
8 0.197 3.165 0.226 53 1.390 20.568 1.596 98 3.031 35.519 3.558 143 5.687 39.584 7.474
9 0.222 3.560 0.254 54 1.419 20.939 1.630 99 3.078 35.788 3.616 144 5.744 39.325 7.594
10 0.247 3.955 0.282 55 1.449 21.309 1.665 100 3.125 36.053 3.675 145 5.797 39.038 7.714
11 0.272 4.350 0.311 56 1.479 21.677 1.700 101 3.173 36.313 3.735 146 5.848 38.721 7.833
12 0.297 4.745 0.339 57 1.510 22.045 1.735 102 3.221 36.568 3.796 147 5.895 38.373 7.952
13 0.322 5.140 0.368 58 1.540 22.411 1.771 103 3.270 36.819 3.857 148 5.938 37.992 8.069
14 0.347 5.534 0.396 59 1.571 22.776 1.807 104 3.320 37.065 3.920 149 5.976 37.577 8.184
15 0.372 5.928 0.425 60 1.602 23.139 1.843 105 3.370 37.306 3.984 150 6.009 37.126 8.297
16 0.397 6.322 0.453 61 1.633 23.502 1.879 106 3.421 37.541 4.049 151 6.036 36.638 8.405
17 0.422 6.716 0.482 62 1.665 23.863 1.916 107 3.472 37.771 4.115 152 6.056 36.110 8.509
18 0.447 7.110 0.511 63 1.696 24.222 1.953 108 3.525 37.996 4.182 153 6.069 35.541 8.607
19 0.472 7.503 0.540 64 1.728 24.581 1.991 109 3.578 38.214 4.251 154 6.072 34.929 8.698
20 0.497 7.896 0.568 65 1.761 24.938 2.029 110 3.631 38.426 4.321 155 6.066 34.271 8.780
21 0.523 8.288 0.597 66 1.793 25.293 2.067 111 3.686 38.632 4.391 156 6.050 33.567 8.852
22 0.548 8.680 0.626 67 1.826 25.647 2.106 112 3.741 38.831 4.464 157 6.022 32.813 8.911
23 0.573 9.072 0.656 68 1.859 25.999 2.145 113 3.796 39.023 4.537 158 5.980 32.007 8.955
24 0.599 9.464 0.685 69 1.893 26.349 2.184 114 3.853 39.209 4.612 159 5.925 31.149 8.982
25 0.625 9.855 0.714 70 1.927 26.698 2.224 115 3.910 39.386 4.688 160 5.854 30.235 8.988
26 0.650 10.246 0.744 71 1.961 27.045 2.264 116 3.968 39.556 4.765 161 5.766 29.265 8.972
27 0.676 10.636 0.773 72 1.995 27.390 2.305 117 4.026 39.718 4.844 162 5.660 28.236 8.929
28 0.702 11.026 0.803 73 2.030 27.734 2.346 118 4.086 39.872 4.925 163 5.535 27.146 8.855
29 0.728 11.415 0.833 74 2.065 28.075 2.387 119 4.146 40.017 5.007 164 5.389 25.996 8.747
30 0.754 11.804 0.863 75 2.100 28.415 2.429 120 4.206 40.153 5.091 165 5.221 24.783 8.601
31 0.780 12.192 0.893 76 2.136 28.753 2.472 121 4.268 40.279 5.176 166 5.030 23.506 8.411
32 0.806 12.580 0.923 77 2.172 29.088 2.515 122 4.330 40.396 5.262 167 4.815 22.167 8.174
33 0.833 12.968 0.953 78 2.209 29.421 2.558 123 4.393 40.502 5.351 168 4.576 20.764 7.886
34 0.859 13.354 0.984 79 2.246 29.752 2.602 124 4.456 40.598 5.441 169 4.311 19.299 7.541
35 0.886 13.741 1.014 80 2.283 30.081 2.647 125 4.520 40.683 5.533 170 4.021 17.774 7.138
36 0.913 14.126 1.045 81 2.321 30.408 2.692 126 4.584 40.756 5.626 171 3.707 16.189 6.673
37 0.939 14.511 1.076 82 2.359 30.732 2.737 127 4.649 40.816 5.721 172 3.368 14.549 6.145
38 0.966 14.896 1.107 83 2.397 31.054 2.784 128 4.715 40.864 5.818 173 3.005 12.857 5.553
39 0.994 15.279 1.138 84 2.436 31.373 2.830 129 4.780 40.899 5.917 174 2.621 11.116 4.899
40 1.021 15.662 1.169 85 2.475 31.689 2.878 130 4.847 40.920 6.018 175 2.217 9.333 4.186
41 1.048 16.045 1.201 86 2.515 32.003 2.926 131 4.913 40.926 6.120 176 1.796 7.513 3.420
42 1.076 16.426 1.233 87 2.555 32.314 2.975 132 4.980 40.918 6.224 177 1.360 5.662 2.608
43 1.103 16.807 1.265 88 2.596 32.622 3.024 133 5.047 40.893 6.330 178 0.913 3.788 1.760
44 1.131 17.187 1.297 89 2.637 32.927 3.074 134 5.113 40.851 6.438 179 0.458 1.898 0.886
45 1.159 17.566 1.329 90 2.679 33.228 3.125 135 5.180 40.793 6.547 180 0.000 0.000 0.000



Table 49: The conjunctions, oppositions, and stations of Mars during the years 2000-2020 CE. (R) indicates a retrograde station, and (D) a direct station.
Event Date $\lambda$
     
Conjunction 01/07/2000 10CN13
Station (R) 12/05/2001 29SG00
Opposition 13/06/2001 22SG44
Station (D) 19/07/2001 15SG02
Conjunction 10/08/2002 18LE05
Station (R) 29/07/2003 10PI20
Opposition 28/08/2003 05PI03
Station (D) 27/09/2003 29AQ55
Conjunction 15/09/2004 23VI06
Station (R) 02/10/2005 23TA31
Opposition 07/11/2005 15TA06
Station (D) 09/12/2005 08TA24
Conjunction 23/10/2006 29LI44
Station (R) 15/11/2007 12CN36
Opposition 24/12/2007 02CN45
Station (D) 31/01/2008 24GE15
Conjunction 05/12/2008 14SG08
Station (R) 20/12/2009 19LE35
Opposition 29/01/2010 09LE45
Station (D) 10/03/2010 00LE20
Conjunction 04/02/2011 15AQ42
Station (R) 24/01/2012 23VI01
Opposition 03/03/2012 13VI42
Station (D) 14/04/2012 03VI51
Conjunction 17/04/2013 28AR06
Station (R) 01/03/2014 27LI31
Opposition 08/04/2014 19LI00
Station (D) 20/05/2014 09LI04
Conjunction 14/06/2015 23GE28
Station (R) 17/04/2016 08SG45
Opposition 22/05/2016 01SG43
Station (D) 29/06/2016 22SC55
Conjunction 27/07/2017 04LE11
Station (R) 27/06/2018 09AQ35
Opposition 27/07/2018 04AQ22
Station (D) 27/08/2018 28CP43
Conjunction 02/09/2019 09VI43
Station (R) 10/09/2020 28AR08
Opposition 13/10/2020 20AR59
Station (D) 13/11/2020 15AR05



Table 50: Mean motion of Jupiter. 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), $\bar{\lambda}_0 = 34.365^\circ$, $M_0 = 19.348^\circ$, and $\bar{F}_0=293.660^\circ$.
$\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta \bar{F}(^\circ)$ $\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta \bar{F}(^\circ)$
               
10,000 111.251 110.810 110.812 1,000 83.125 83.081 83.081
20,000 222.501 221.620 221.624 2,000 166.250 166.162 166.162
30,000 333.752 332.430 332.437 3,000 249.375 249.243 249.244
40,000 85.003 83.240 83.249 4,000 332.500 332.324 332.325
50,000 196.253 194.050 194.061 5,000 55.625 55.405 55.406
60,000 307.504 304.860 304.873 6,000 138.750 138.486 138.487
70,000 58.755 55.670 55.685 7,000 221.875 221.567 221.569
80,000 170.006 166.480 166.498 8,000 305.001 304.648 304.650
90,000 281.256 277.290 277.310 9,000 28.126 27.729 27.731
               
100 8.313 8.308 8.308 10 0.831 0.831 0.831
200 16.625 16.616 16.616 20 1.663 1.662 1.662
300 24.938 24.924 24.924 30 2.494 2.492 2.492
400 33.250 33.232 33.232 40 3.325 3.323 3.323
500 41.563 41.541 41.541 50 4.156 4.154 4.154
600 49.875 49.849 49.849 60 4.988 4.985 4.985
700 58.188 58.157 58.157 70 5.819 5.816 5.816
800 66.500 66.465 66.465 80 6.650 6.646 6.646
900 74.813 74.773 74.773 90 7.481 7.477 7.477
               
1 0.083 0.083 0.083 0.1 0.008 0.008 0.008
2 0.166 0.166 0.166 0.2 0.017 0.017 0.017
3 0.249 0.249 0.249 0.3 0.025 0.025 0.025
4 0.333 0.332 0.332 0.4 0.033 0.033 0.033
5 0.416 0.415 0.415 0.5 0.042 0.042 0.042
6 0.499 0.498 0.498 0.6 0.050 0.050 0.050
7 0.582 0.582 0.582 0.7 0.058 0.058 0.058
8 0.665 0.665 0.665 0.8 0.067 0.066 0.066
9 0.748 0.748 0.748 0.9 0.075 0.075 0.075



Table 51: Deferential anomalies of Jupiter.
$M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$
0 0.000 4.839 90 5.545 -0.234 180 0.000 -4.839 270 -5.545 -0.234
2 0.205 4.835 92 5.530 -0.403 182 -0.182 -4.836 272 -5.553 -0.065
4 0.410 4.826 94 5.508 -0.571 184 -0.363 -4.828 274 -5.554 0.105
6 0.614 4.810 96 5.479 -0.737 186 -0.545 -4.815 276 -5.549 0.274
8 0.818 4.787 98 5.444 -0.903 188 -0.725 -4.796 278 -5.537 0.444
10 1.020 4.758 100 5.403 -1.067 190 -0.905 -4.772 280 -5.518 0.613
12 1.221 4.723 102 5.355 -1.230 192 -1.085 -4.743 282 -5.492 0.782
14 1.420 4.681 104 5.301 -1.391 194 -1.263 -4.709 284 -5.459 0.950
16 1.617 4.633 106 5.241 -1.550 196 -1.439 -4.669 286 -5.419 1.117
18 1.812 4.579 108 5.175 -1.707 198 -1.615 -4.624 288 -5.372 1.283
20 2.004 4.519 110 5.102 -1.862 200 -1.789 -4.574 290 -5.318 1.448
22 2.194 4.453 112 5.024 -2.014 202 -1.961 -4.519 292 -5.257 1.611
24 2.380 4.382 114 4.941 -2.163 204 -2.131 -4.459 294 -5.190 1.773
26 2.563 4.304 116 4.851 -2.310 206 -2.298 -4.394 296 -5.116 1.932
28 2.742 4.221 118 4.757 -2.454 208 -2.464 -4.324 298 -5.035 2.089
30 2.918 4.132 120 4.657 -2.595 210 -2.627 -4.249 300 -4.947 2.244
32 3.089 4.038 122 4.551 -2.732 212 -2.787 -4.169 302 -4.853 2.396
34 3.256 3.938 124 4.441 -2.867 214 -2.945 -4.085 304 -4.752 2.545
36 3.419 3.834 126 4.326 -2.997 216 -3.100 -3.995 306 -4.645 2.691
38 3.576 3.724 128 4.207 -3.124 218 -3.251 -3.902 308 -4.532 2.834
40 3.729 3.610 130 4.082 -3.248 220 -3.399 -3.803 310 -4.413 2.973
42 3.877 3.491 132 3.954 -3.367 222 -3.543 -3.701 312 -4.287 3.108
44 4.019 3.368 134 3.821 -3.482 224 -3.684 -3.594 314 -4.156 3.240
46 4.156 3.240 136 3.684 -3.594 226 -3.821 -3.482 316 -4.019 3.368
48 4.287 3.108 138 3.543 -3.701 228 -3.954 -3.367 318 -3.877 3.491
50 4.413 2.973 140 3.399 -3.803 230 -4.082 -3.248 320 -3.729 3.610
52 4.532 2.834 142 3.251 -3.902 232 -4.207 -3.124 322 -3.576 3.724
54 4.645 2.691 144 3.100 -3.995 234 -4.326 -2.997 324 -3.419 3.834
56 4.752 2.545 146 2.945 -4.085 236 -4.441 -2.867 326 -3.256 3.938
58 4.853 2.396 148 2.787 -4.169 238 -4.551 -2.732 328 -3.089 4.038
60 4.947 2.244 150 2.627 -4.249 240 -4.657 -2.595 330 -2.918 4.132
62 5.035 2.089 152 2.464 -4.324 242 -4.757 -2.454 332 -2.742 4.221
64 5.116 1.932 154 2.298 -4.394 244 -4.851 -2.310 334 -2.563 4.304
66 5.190 1.773 156 2.131 -4.459 246 -4.941 -2.163 336 -2.380 4.382
68 5.257 1.611 158 1.961 -4.519 248 -5.024 -2.014 338 -2.194 4.453
70 5.318 1.448 160 1.789 -4.574 250 -5.102 -1.862 340 -2.004 4.519
72 5.372 1.283 162 1.615 -4.624 252 -5.175 -1.707 342 -1.812 4.579
74 5.419 1.117 164 1.439 -4.669 254 -5.241 -1.550 344 -1.617 4.633
76 5.459 0.950 166 1.263 -4.709 256 -5.301 -1.391 346 -1.420 4.681
78 5.492 0.782 168 1.085 -4.743 258 -5.355 -1.230 348 -1.221 4.723
80 5.518 0.613 170 0.905 -4.772 260 -5.403 -1.067 350 -1.020 4.758
82 5.537 0.444 172 0.725 -4.796 262 -5.444 -0.903 352 -0.818 4.787
84 5.549 0.274 174 0.545 -4.815 264 -5.479 -0.737 354 -0.614 4.810
86 5.554 0.105 176 0.363 -4.828 266 -5.508 -0.571 356 -0.410 4.826
88 5.553 -0.065 178 0.182 -4.836 268 -5.530 -0.403 358 -0.205 4.835
90 5.545 -0.234 180 0.000 -4.839 270 -5.545 -0.234 360 -0.000 4.839



Table 52: Epicyclic anomalies of Jupiter. All quantities are in degrees. Note that $\bar{\theta}(360^\circ - \mu) =-\bar{\theta}(\mu)$, and $\delta\theta_{\pm}(360^\circ-\mu) = -\delta\theta_{\pm}(\mu)$.
$\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$
0 0.000 0.000 0.000 45 0.366 6.816 0.410 90 0.649 10.868 0.736 135 0.616 8.927 0.713
1 0.008 0.161 0.009 46 0.374 6.948 0.418 91 0.653 10.902 0.741 136 0.609 8.796 0.705
2 0.017 0.322 0.019 47 0.381 7.077 0.427 92 0.657 10.933 0.746 137 0.601 8.661 0.697
3 0.025 0.483 0.028 48 0.389 7.206 0.436 93 0.661 10.961 0.750 138 0.593 8.522 0.688
4 0.033 0.644 0.037 49 0.396 7.333 0.444 94 0.664 10.986 0.755 139 0.585 8.380 0.679
5 0.042 0.805 0.046 50 0.404 7.459 0.453 95 0.668 11.008 0.759 140 0.576 8.233 0.669
6 0.050 0.965 0.056 51 0.411 7.583 0.461 96 0.671 11.026 0.763 141 0.567 8.083 0.659
7 0.058 1.126 0.065 52 0.419 7.705 0.470 97 0.674 11.041 0.766 142 0.558 7.929 0.649
8 0.067 1.286 0.074 53 0.426 7.826 0.478 98 0.677 11.053 0.770 143 0.548 7.771 0.638
9 0.075 1.446 0.084 54 0.434 7.946 0.486 99 0.680 11.062 0.773 144 0.538 7.610 0.627
10 0.083 1.606 0.093 55 0.441 8.063 0.495 100 0.682 11.067 0.777 145 0.528 7.445 0.615
11 0.092 1.766 0.102 56 0.448 8.180 0.503 101 0.684 11.069 0.780 146 0.517 7.276 0.603
12 0.100 1.925 0.111 57 0.455 8.294 0.511 102 0.686 11.068 0.782 147 0.506 7.104 0.590
13 0.108 2.084 0.121 58 0.462 8.407 0.519 103 0.688 11.063 0.785 148 0.495 6.929 0.577
14 0.116 2.242 0.130 59 0.470 8.517 0.527 104 0.690 11.054 0.787 149 0.484 6.750 0.564
15 0.125 2.400 0.139 60 0.477 8.626 0.535 105 0.692 11.042 0.789 150 0.472 6.568 0.550
16 0.133 2.558 0.148 61 0.484 8.734 0.543 106 0.693 11.027 0.791 151 0.459 6.383 0.536
17 0.141 2.715 0.158 62 0.490 8.839 0.551 107 0.694 11.008 0.793 152 0.447 6.194 0.522
18 0.149 2.872 0.167 63 0.497 8.942 0.559 108 0.695 10.985 0.794 153 0.434 6.003 0.507
19 0.158 3.028 0.176 64 0.504 9.044 0.567 109 0.695 10.959 0.795 154 0.421 5.808 0.492
20 0.166 3.184 0.185 65 0.511 9.143 0.574 110 0.696 10.929 0.796 155 0.407 5.610 0.476
21 0.174 3.339 0.194 66 0.517 9.240 0.582 111 0.696 10.895 0.796 156 0.393 5.410 0.460
22 0.182 3.494 0.204 67 0.524 9.336 0.590 112 0.696 10.858 0.797 157 0.379 5.206 0.444
23 0.191 3.648 0.213 68 0.531 9.429 0.597 113 0.695 10.817 0.797 158 0.365 5.000 0.427
24 0.199 3.801 0.222 69 0.537 9.520 0.605 114 0.695 10.772 0.796 159 0.350 4.792 0.410
25 0.207 3.954 0.231 70 0.543 9.609 0.612 115 0.694 10.723 0.796 160 0.336 4.581 0.393
26 0.215 4.106 0.240 71 0.550 9.696 0.619 116 0.693 10.671 0.795 161 0.320 4.367 0.375
27 0.223 4.257 0.249 72 0.556 9.780 0.626 117 0.691 10.614 0.794 162 0.305 4.151 0.357
28 0.231 4.407 0.258 73 0.562 9.862 0.633 118 0.690 10.554 0.792 163 0.289 3.933 0.339
29 0.239 4.557 0.268 74 0.568 9.942 0.640 119 0.688 10.490 0.790 164 0.274 3.713 0.321
30 0.248 4.705 0.277 75 0.574 10.019 0.647 120 0.686 10.422 0.788 165 0.258 3.491 0.302
31 0.256 4.853 0.286 76 0.580 10.094 0.654 121 0.683 10.350 0.786 166 0.241 3.267 0.283
32 0.264 5.000 0.295 77 0.585 10.167 0.661 122 0.680 10.274 0.783 167 0.225 3.041 0.264
33 0.272 5.146 0.304 78 0.591 10.237 0.667 123 0.677 10.194 0.780 168 0.208 2.814 0.244
34 0.280 5.292 0.313 79 0.596 10.304 0.674 124 0.674 10.110 0.776 169 0.192 2.585 0.225
35 0.288 5.436 0.322 80 0.602 10.369 0.680 125 0.670 10.023 0.772 170 0.175 2.354 0.205
36 0.296 5.579 0.331 81 0.607 10.431 0.686 126 0.666 9.931 0.768 171 0.158 2.123 0.185
37 0.304 5.721 0.339 82 0.612 10.491 0.692 127 0.662 9.835 0.764 172 0.140 1.890 0.165
38 0.311 5.862 0.348 83 0.617 10.548 0.698 128 0.657 9.736 0.759 173 0.123 1.656 0.145
39 0.319 6.002 0.357 84 0.622 10.602 0.704 129 0.652 9.632 0.753 174 0.106 1.421 0.124
40 0.327 6.141 0.366 85 0.627 10.654 0.710 130 0.647 9.524 0.748 175 0.088 1.185 0.104
41 0.335 6.278 0.375 86 0.632 10.702 0.715 131 0.641 9.413 0.742 176 0.071 0.949 0.083
42 0.343 6.415 0.384 87 0.636 10.748 0.721 132 0.636 9.297 0.735 177 0.053 0.712 0.062
43 0.351 6.550 0.392 88 0.641 10.791 0.726 133 0.629 9.178 0.728 178 0.035 0.475 0.042
44 0.358 6.684 0.401 89 0.645 10.831 0.731 134 0.623 9.055 0.721 179 0.018 0.238 0.021
45 0.366 6.816 0.410 90 0.649 10.868 0.736 135 0.616 8.927 0.713 180 0.000 0.000 0.000



Table 53: The conjunctions, oppositions, and stations of Jupiter during the years 2000-2010 CE. (R) indicates a retrograde station, and (D) a direct station.
Event Date $\lambda$
Conjunction 08/05/2000 17TA53
Station (R) 29/09/2000 11GE13
Opposition 28/11/2000 06GE08
Station (D) 25/01/2001 01GE10
Conjunction 14/06/2001 23GE30
Station (R) 02/11/2001 15CN41
Opposition 01/01/2002 10CN37
Station (D) 01/03/2002 05CN37
Conjunction 20/07/2002 27CN11
Station (R) 04/12/2002 18LE06
Opposition 02/02/2003 13LE06
Station (D) 04/04/2003 08LE03
Conjunction 22/08/2003 28LE55
Station (R) 04/01/2004 18VI54
Opposition 04/03/2004 13VI58
Station (D) 05/05/2004 08VI55
Conjunction 22/09/2004 29VI21
Station (R) 02/02/2005 18LI53
Opposition 03/04/2005 14LI00
Station (D) 05/06/2005 08LI58
Conjunction 22/10/2005 29LI16
Station (R) 04/03/2006 18SC54
Opposition 04/05/2006 14SC03
Station (D) 06/07/2006 09SC02
Conjunction 22/11/2006 29SC34
Station (R) 06/04/2007 19SG49
Opposition 06/06/2007 14SG57
Station (D) 07/08/2007 09SG58
Conjunction 23/12/2007 01CP03
Station (R) 09/05/2008 22CP23
Opposition 09/07/2008 17CP30
Station (D) 08/09/2008 12CP33
Conjunction 24/01/2009 04AQ23
Station (R) 15/06/2009 27AQ01
Opposition 14/08/2009 22AQ04
Station (D) 13/10/2009 17AQ10
Conjunction 28/02/2010 09PI43
Station (R) 23/07/2010 03AR20
Opposition 21/09/2010 28PI19
Station (D) 18/11/2010 23PI26



Table 54: Mean motion of Saturn. 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), $\bar{\lambda}_0 = 50.059^\circ$, $M_0 = 317.857^\circ$, and $\bar{F}_0 =296.482^\circ$.
$\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta \bar{F}(^\circ)$ $\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta \bar{F}(^\circ)$
               
10,000 335.083 334.815 334.779 1,000 33.508 33.482 33.478
20,000 310.166 309.630 309.559 2,000 67.017 66.963 66.956
30,000 285.249 284.446 284.338 3,000 100.525 100.445 100.434
40,000 260.332 259.261 259.118 4,000 134.033 133.926 133.912
50,000 235.415 234.076 233.897 5,000 167.541 167.408 167.390
60,000 210.498 208.891 208.677 6,000 201.050 200.889 200.868
70,000 185.581 183.706 183.456 7,000 234.558 234.371 234.346
80,000 160.664 158.522 158.236 8,000 268.066 267.852 267.824
90,000 135.747 133.337 133.015 9,000 301.575 301.334 301.302
               
100 3.351 3.348 3.348 10 0.335 0.335 0.335
200 6.702 6.696 6.696 20 0.670 0.670 0.670
300 10.052 10.044 10.043 30 1.005 1.004 1.004
400 13.403 13.393 13.391 40 1.340 1.339 1.339
500 16.754 16.741 16.739 50 1.675 1.674 1.674
600 20.105 20.089 20.087 60 2.010 2.009 2.009
700 23.456 23.437 23.435 70 2.346 2.344 2.343
800 26.807 26.785 26.782 80 2.681 2.679 2.678
900 30.157 30.133 30.130 90 3.016 3.013 3.013
               
1 0.034 0.033 0.033 0.1 0.003 0.003 0.003
2 0.067 0.067 0.067 0.2 0.007 0.007 0.007
3 0.101 0.100 0.100 0.3 0.010 0.010 0.010
4 0.134 0.134 0.134 0.4 0.013 0.013 0.013
5 0.168 0.167 0.167 0.5 0.017 0.017 0.017
6 0.201 0.201 0.201 0.6 0.020 0.020 0.020
7 0.235 0.234 0.234 0.7 0.023 0.023 0.023
8 0.268 0.268 0.268 0.8 0.027 0.027 0.027
9 0.302 0.301 0.301 0.9 0.030 0.030 0.030



Table 55: Deferential anomalies of Saturn.
$M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$
0 0.000 5.386 90 6.172 -0.290 180 0.000 -5.386 270 -6.172 -0.290
2 0.230 5.383 92 6.154 -0.478 182 -0.201 -5.383 272 -6.183 -0.102
4 0.459 5.372 94 6.128 -0.664 184 -0.402 -5.374 274 -6.186 0.087
6 0.688 5.354 96 6.095 -0.850 186 -0.602 -5.360 276 -6.182 0.276
8 0.916 5.328 98 6.055 -1.034 188 -0.802 -5.339 278 -6.169 0.465
10 1.143 5.296 100 6.007 -1.217 190 -1.001 -5.313 280 -6.149 0.654
12 1.368 5.256 102 5.953 -1.397 192 -1.199 -5.281 282 -6.122 0.842
14 1.591 5.209 104 5.891 -1.576 194 -1.396 -5.243 284 -6.086 1.030
16 1.811 5.156 106 5.823 -1.753 196 -1.591 -5.200 286 -6.043 1.217
18 2.029 5.095 108 5.748 -1.927 198 -1.785 -5.150 288 -5.992 1.402
20 2.245 5.027 110 5.666 -2.098 200 -1.977 -5.095 290 -5.933 1.586
22 2.456 4.953 112 5.578 -2.267 202 -2.168 -5.035 292 -5.867 1.768
24 2.665 4.873 114 5.484 -2.433 204 -2.356 -4.969 294 -5.793 1.949
26 2.869 4.785 116 5.384 -2.596 206 -2.542 -4.897 296 -5.711 2.127
28 3.070 4.692 118 5.277 -2.755 208 -2.725 -4.820 298 -5.622 2.302
30 3.266 4.592 120 5.165 -2.911 210 -2.906 -4.737 300 -5.525 2.476
32 3.457 4.486 122 5.048 -3.063 212 -3.084 -4.649 302 -5.421 2.646
34 3.644 4.375 124 4.924 -3.211 214 -3.259 -4.556 304 -5.310 2.813
36 3.825 4.257 126 4.796 -3.356 216 -3.430 -4.458 306 -5.191 2.976
38 4.002 4.134 128 4.662 -3.496 218 -3.598 -4.354 308 -5.065 3.136
40 4.172 4.006 130 4.524 -3.632 220 -3.763 -4.246 310 -4.933 3.292
42 4.337 3.873 132 4.380 -3.764 222 -3.923 -4.133 312 -4.793 3.444
44 4.495 3.735 134 4.232 -3.892 224 -4.080 -4.015 314 -4.648 3.591
46 4.648 3.591 136 4.080 -4.015 226 -4.232 -3.892 316 -4.495 3.735
48 4.793 3.444 138 3.923 -4.133 228 -4.380 -3.764 318 -4.337 3.873
50 4.933 3.292 140 3.763 -4.246 230 -4.524 -3.632 320 -4.172 4.006
52 5.065 3.136 142 3.598 -4.354 232 -4.662 -3.496 322 -4.002 4.134
54 5.191 2.976 144 3.430 -4.458 234 -4.796 -3.356 324 -3.825 4.257
56 5.310 2.813 146 3.259 -4.556 236 -4.924 -3.211 326 -3.644 4.375
58 5.421 2.646 148 3.084 -4.649 238 -5.048 -3.063 328 -3.457 4.486
60 5.525 2.476 150 2.906 -4.737 240 -5.165 -2.911 330 -3.266 4.592
62 5.622 2.302 152 2.725 -4.820 242 -5.277 -2.755 332 -3.070 4.692
64 5.711 2.127 154 2.542 -4.897 244 -5.384 -2.596 334 -2.869 4.785
66 5.793 1.949 156 2.356 -4.969 246 -5.484 -2.433 336 -2.665 4.873
68 5.867 1.768 158 2.168 -5.035 248 -5.578 -2.267 338 -2.456 4.953
70 5.933 1.586 160 1.977 -5.095 250 -5.666 -2.098 340 -2.245 5.027
72 5.992 1.402 162 1.785 -5.150 252 -5.748 -1.927 342 -2.029 5.095
74 6.043 1.217 164 1.591 -5.200 254 -5.823 -1.753 344 -1.811 5.156
76 6.086 1.030 166 1.396 -5.243 256 -5.891 -1.576 346 -1.591 5.209
78 6.122 0.842 168 1.199 -5.281 258 -5.953 -1.397 348 -1.368 5.256
80 6.149 0.654 170 1.001 -5.313 260 -6.007 -1.217 350 -1.143 5.296
82 6.169 0.465 172 0.802 -5.339 262 -6.055 -1.034 352 -0.916 5.328
84 6.182 0.276 174 0.602 -5.360 264 -6.095 -0.850 354 -0.688 5.354
86 6.186 0.087 176 0.402 -5.374 266 -6.128 -0.664 356 -0.459 5.372
88 6.183 -0.102 178 0.201 -5.383 268 -6.154 -0.478 358 -0.230 5.383
90 6.172 -0.290 180 0.000 -5.386 270 -6.172 -0.290 360 -0.000 5.386



Table 56: Epicyclic anomalies of Saturn. All quantities are in degrees. Note that $\bar{\theta}(360^\circ - \mu) =-\bar{\theta}(\mu)$, and $\delta\theta_{\pm}(360^\circ-\mu) = -\delta\theta_{\pm}(\mu)$.
$\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$ $\mu$ $\delta\theta_-$ $\bar{\theta}~~~~$ $\delta\theta_+$
0 0.000 0.000 0.000 45 0.242 3.944 0.276 90 0.391 5.979 0.450 135 0.322 4.573 0.375
1 0.006 0.095 0.006 46 0.247 4.017 0.282 91 0.393 5.989 0.452 136 0.318 4.499 0.370
2 0.011 0.190 0.013 47 0.252 4.089 0.287 92 0.394 5.997 0.453 137 0.313 4.423 0.364
3 0.017 0.284 0.019 48 0.256 4.160 0.292 93 0.395 6.004 0.454 138 0.308 4.346 0.358
4 0.023 0.379 0.026 49 0.261 4.230 0.298 94 0.396 6.008 0.456 139 0.302 4.267 0.352
5 0.028 0.474 0.032 50 0.265 4.299 0.303 95 0.397 6.011 0.457 140 0.297 4.186 0.346
6 0.034 0.568 0.039 51 0.270 4.367 0.308 96 0.397 6.012 0.458 141 0.292 4.104 0.340
7 0.040 0.662 0.045 52 0.274 4.433 0.313 97 0.398 6.011 0.459 142 0.286 4.020 0.333
8 0.045 0.757 0.052 53 0.279 4.499 0.318 98 0.399 6.008 0.459 143 0.280 3.935 0.327
9 0.051 0.851 0.058 54 0.283 4.564 0.323 99 0.399 6.004 0.460 144 0.274 3.848 0.320
10 0.057 0.945 0.064 55 0.287 4.627 0.328 100 0.399 5.997 0.460 145 0.269 3.760 0.313
11 0.062 1.038 0.071 56 0.292 4.689 0.333 101 0.399 5.989 0.461 146 0.262 3.670 0.306
12 0.068 1.132 0.077 57 0.296 4.750 0.338 102 0.399 5.979 0.461 147 0.256 3.578 0.299
13 0.074 1.225 0.084 58 0.300 4.810 0.343 103 0.399 5.966 0.461 148 0.250 3.486 0.292
14 0.079 1.318 0.090 59 0.304 4.869 0.347 104 0.399 5.952 0.460 149 0.243 3.392 0.284
15 0.085 1.410 0.096 60 0.308 4.926 0.352 105 0.399 5.937 0.460 150 0.237 3.296 0.276
16 0.090 1.502 0.103 61 0.312 4.982 0.356 106 0.398 5.919 0.460 151 0.230 3.199 0.269
17 0.096 1.594 0.109 62 0.316 5.037 0.361 107 0.397 5.899 0.459 152 0.223 3.101 0.261
18 0.102 1.686 0.115 63 0.320 5.091 0.365 108 0.397 5.877 0.458 153 0.216 3.002 0.253
19 0.107 1.777 0.122 64 0.323 5.143 0.370 109 0.396 5.854 0.457 154 0.209 2.901 0.244
20 0.113 1.868 0.128 65 0.327 5.194 0.374 110 0.395 5.828 0.456 155 0.202 2.800 0.236
21 0.118 1.958 0.134 66 0.330 5.243 0.378 111 0.394 5.801 0.455 156 0.195 2.697 0.228
22 0.124 2.048 0.141 67 0.334 5.292 0.382 112 0.392 5.772 0.454 157 0.187 2.593 0.219
23 0.129 2.138 0.147 68 0.337 5.338 0.386 113 0.391 5.740 0.452 158 0.180 2.488 0.210
24 0.134 2.227 0.153 69 0.341 5.384 0.390 114 0.389 5.707 0.450 159 0.172 2.382 0.202
25 0.140 2.315 0.159 70 0.344 5.428 0.394 115 0.387 5.672 0.448 160 0.165 2.275 0.193
26 0.145 2.403 0.165 71 0.347 5.470 0.398 116 0.386 5.635 0.446 161 0.157 2.167 0.184
27 0.151 2.490 0.171 72 0.350 5.511 0.401 117 0.384 5.596 0.444 162 0.149 2.059 0.175
28 0.156 2.577 0.178 73 0.353 5.551 0.405 118 0.381 5.555 0.442 163 0.142 1.949 0.166
29 0.161 2.663 0.184 74 0.356 5.589 0.408 119 0.379 5.512 0.439 164 0.134 1.839 0.156
30 0.167 2.749 0.190 75 0.359 5.625 0.412 120 0.377 5.467 0.437 165 0.126 1.727 0.147
31 0.172 2.834 0.196 76 0.362 5.660 0.415 121 0.374 5.421 0.434 166 0.118 1.616 0.138
32 0.177 2.918 0.202 77 0.365 5.694 0.418 122 0.371 5.372 0.431 167 0.109 1.503 0.128
33 0.182 3.002 0.208 78 0.367 5.726 0.421 123 0.368 5.322 0.427 168 0.101 1.390 0.118
34 0.188 3.085 0.214 79 0.370 5.756 0.424 124 0.365 5.270 0.424 169 0.093 1.276 0.109
35 0.193 3.167 0.219 80 0.372 5.784 0.427 125 0.362 5.215 0.420 170 0.085 1.162 0.099
36 0.198 3.248 0.225 81 0.375 5.811 0.430 126 0.359 5.159 0.417 171 0.076 1.047 0.089
37 0.203 3.329 0.231 82 0.377 5.837 0.433 127 0.355 5.101 0.413 172 0.068 0.932 0.080
38 0.208 3.409 0.237 83 0.379 5.861 0.435 128 0.352 5.042 0.409 173 0.060 0.816 0.070
39 0.213 3.488 0.243 84 0.381 5.883 0.438 129 0.348 4.980 0.404 174 0.051 0.700 0.060
40 0.218 3.566 0.248 85 0.383 5.903 0.440 130 0.344 4.917 0.400 175 0.043 0.584 0.050
41 0.223 3.644 0.254 86 0.385 5.922 0.442 131 0.340 4.851 0.395 176 0.034 0.467 0.040
42 0.228 3.720 0.260 87 0.387 5.939 0.444 132 0.336 4.784 0.390 177 0.026 0.351 0.030
43 0.233 3.796 0.265 88 0.388 5.954 0.446 133 0.331 4.716 0.386 178 0.017 0.234 0.020
44 0.238 3.871 0.271 89 0.390 5.967 0.448 134 0.327 4.645 0.380 179 0.009 0.117 0.010
45 0.242 3.944 0.276 90 0.391 5.979 0.450 135 0.322 4.573 0.375 180 0.000 0.000 0.000



Table 57: The conjunctions, oppositions, and stations of Saturn during the years 2000-2010 CE. (R) indicates a retrograde station, and (D) a direct station.
Event Date $\lambda$
Conjunction 10/05/2000 20TA26
Station (R) 12/09/2000 00GE59
Opposition 19/11/2000 27TA29
Station (D) 24/01/2001 24TA03
Conjunction 25/05/2001 04GE22
Station (R) 26/09/2001 14GE59
Opposition 03/12/2001 11GE29
Station (D) 08/02/2002 08GE02
Conjunction 09/06/2002 18GE28
Station (R) 11/10/2002 29GE06
Opposition 17/12/2002 25GE36
Station (D) 22/02/2003 22GE08
Conjunction 24/06/2003 02CN39
Station (R) 25/10/2003 13CN15
Opposition 31/12/2003 09CN46
Station (D) 07/03/2004 06CN17
Conjunction 08/07/2004 16CN50
Station (R) 08/11/2004 27CN21
Opposition 13/01/2005 23CN52
Station (D) 22/03/2005 20CN23
Conjunction 23/07/2005 00LE56
Station (R) 22/11/2005 11LE19
Opposition 27/01/2006 07LE51
Station (D) 05/04/2006 04LE22
Conjunction 07/08/2006 14LE51
Station (R) 06/12/2006 25LE04
Opposition 10/02/2007 21LE38
Station (D) 19/04/2007 18LE09
Conjunction 22/08/2007 28LE32
Station (R) 19/12/2007 08VI34
Opposition 24/02/2008 05VI10
Station (D) 03/05/2008 01VI41
Conjunction 04/09/2008 11VI56
Station (R) 31/12/2008 21VI46
Opposition 08/03/2009 18VI23
Station (D) 17/05/2009 14VI56
Conjunction 17/09/2009 25VI01
Station (R) 13/01/2010 04LI40
Opposition 22/03/2010 01LI18
Station (D) 30/05/2010 27VI51
Conjunction 01/10/2010 07LI46


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Next: The Inferior Planets Up: The Superior Planets Previous: Jupiter
Richard Fitzpatrick 2010-07-21