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Mercury

The ecliptic latitude of Mercury can be determined with the aid of Tables 62, 75, and 76. Table 62 allows the mean argument of latitude, $\bar{F}$, of Mercury to be calculated as a function of time. Next, Table 75 permits the epicyclic latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 76 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ 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 Venus tables. One example of this procedure is given below.

 
Example: May 5, 2005 CE, 00:00 UT:
 
From Cha. 9, $t-t_0=1\,950.5$ JD, $q= -16.974^\circ$, $\mu= 252.692^\circ$, $\Theta_-=0.107$, and $\Theta_+ = 0.583$. Making use of Table 62, we find:
   
$t$(JD) $\bar{F}(^\circ)$
   
+1000 $132.342$
+900 $83.108$
+50 $204.617$
+.5 $2.046$
Epoch $204.436$
  $626.549$
Modulus $266.549$
   

Thus,

\begin{displaymath} F = \bar{F} + q = 266.549-16.974= 249.575\simeq 250^\circ. \end{displaymath}

It follows from Table 75 that

\begin{displaymath} \beta_0(250^\circ) = -2.511^\circ. \end{displaymath}

Since $\mu\simeq 253^\circ$, Table 76 yields

\begin{displaymath} \delta h_-(253^\circ) = 0.184,\mbox{\hspace{0.5cm}}\bar{h}(2... ...rc)=1.037, \mbox{\hspace{0.5cm}}\delta h_+(253^\circ) = 0.272, \end{displaymath}

so

\begin{displaymath} h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = 0.107\times 0.184+1.037+0.583\times 0.272= 1.215. \end{displaymath}

Finally,

\begin{displaymath} \beta = h\,\beta_0 =- 1.215\times 2.511 = -3.051\simeq -3^\circ 03'. \end{displaymath}

Thus, the ecliptic latitude of Mercury at 00:00 UT on May 5, 2005 CE was $-3^\circ 03'$.


Table 66: Additional Keplerian orbital elements for the five visible planets at the J2000 epoch (i.e., 12:00 UT, January 1, 2000 CE, which corresponds to $t_0= 2\,451\,545.0$ JD). The elements are optimized for use in the time period 1800 CE to 2050 CE. Source: Jet Propulsion Laboratory (NASA), http://ssd.jpl.nasa.gov/.
Object $i(^\circ)$ $\breve{n}\,(^\circ/{\rm day})$ $\bar{F}_0\,(^\circ)$
       
Mercury $6.9190$ $4.09234221$ $204.436$
Venus $3.3692$ $1.60213807$ $105.253$
Mars $1.8467$ $0.52404094$ $305.796$
Jupiter $1.3044$ $0.08308122$ $293.660$
Saturn $2.4860$ $0.03347795$ $296.482$



Table 67: Deferential ecliptic latitude of Mars. The latitude is minus the value shown in the table if the argument is in parenthesies.
$F (^\circ)$ $\beta_0(^\circ)$ $F (^\circ)$
000/180 0.000 (180)/(360)
002/178 0.064 (182)/(358)
004/176 0.129 (184)/(356)
006/174 0.193 (186)/(354)
008/172 0.257 (188)/(352)
010/170 0.321 (190)/(350)
012/168 0.384 (192)/(348)
014/166 0.447 (194)/(346)
016/164 0.509 (196)/(344)
018/162 0.571 (198)/(342)
020/160 0.631 (200)/(340)
022/158 0.692 (202)/(338)
024/156 0.751 (204)/(336)
026/154 0.809 (206)/(334)
028/152 0.867 (208)/(332)
030/150 0.923 (210)/(330)
032/148 0.978 (212)/(328)
034/146 1.032 (214)/(326)
036/144 1.085 (216)/(324)
038/142 1.137 (218)/(322)
040/140 1.187 (220)/(320)
042/138 1.235 (222)/(318)
044/136 1.283 (224)/(316)
046/134 1.328 (226)/(314)
048/132 1.372 (228)/(312)
050/130 1.414 (230)/(310)
052/128 1.455 (232)/(308)
054/126 1.494 (234)/(306)
056/124 1.531 (236)/(304)
058/122 1.566 (238)/(302)
060/120 1.599 (240)/(300)
062/118 1.630 (242)/(298)
064/116 1.660 (244)/(296)
066/114 1.687 (246)/(294)
068/112 1.712 (248)/(292)
070/110 1.735 (250)/(290)
072/108 1.756 (252)/(288)
074/106 1.775 (254)/(286)
076/104 1.792 (256)/(284)
078/102 1.806 (258)/(282)
080/100 1.818 (260)/(280)
082/098 1.828 (262)/(278)
084/096 1.836 (264)/(276)
086/094 1.842 (266)/(274)
088/092 1.845 (268)/(272)
090/090 1.846 (270)/(270)



Table 68: Epicyclic latitude correction factor for Mars. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.
$\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$
0 -0.025 0.604 -0.028 45 -0.025 0.652 -0.029 90 -0.025 0.836 -0.031 135 0.015 1.410 0.003
1 -0.025 0.604 -0.028 46 -0.025 0.654 -0.029 91 -0.025 0.843 -0.031 136 0.018 1.433 0.006
2 -0.025 0.604 -0.028 47 -0.025 0.656 -0.029 92 -0.024 0.850 -0.031 137 0.021 1.457 0.009
3 -0.025 0.604 -0.028 48 -0.025 0.659 -0.029 93 -0.024 0.857 -0.031 138 0.025 1.482 0.013
4 -0.025 0.605 -0.028 49 -0.025 0.661 -0.029 94 -0.024 0.865 -0.031 139 0.028 1.507 0.017
5 -0.025 0.605 -0.028 50 -0.025 0.664 -0.030 95 -0.024 0.872 -0.031 140 0.032 1.533 0.021
6 -0.025 0.605 -0.028 51 -0.025 0.666 -0.030 96 -0.024 0.880 -0.031 141 0.037 1.560 0.026
7 -0.025 0.605 -0.028 52 -0.025 0.669 -0.030 97 -0.024 0.888 -0.031 142 0.041 1.588 0.031
8 -0.025 0.606 -0.028 53 -0.025 0.672 -0.030 98 -0.023 0.896 -0.031 143 0.046 1.616 0.036
9 -0.025 0.606 -0.028 54 -0.025 0.674 -0.030 99 -0.023 0.904 -0.031 144 0.051 1.646 0.043
10 -0.025 0.606 -0.028 55 -0.025 0.677 -0.030 100 -0.023 0.912 -0.030 145 0.057 1.676 0.049
11 -0.025 0.607 -0.028 56 -0.025 0.680 -0.030 101 -0.023 0.921 -0.030 146 0.063 1.708 0.056
12 -0.025 0.607 -0.028 57 -0.026 0.683 -0.030 102 -0.022 0.930 -0.030 147 0.069 1.740 0.064
13 -0.025 0.608 -0.028 58 -0.026 0.686 -0.030 103 -0.022 0.939 -0.030 148 0.076 1.773 0.073
14 -0.025 0.609 -0.028 59 -0.026 0.689 -0.030 104 -0.022 0.948 -0.030 149 0.084 1.807 0.083
15 -0.025 0.609 -0.028 60 -0.026 0.693 -0.030 105 -0.021 0.958 -0.030 150 0.092 1.843 0.093
16 -0.025 0.610 -0.028 61 -0.026 0.696 -0.030 106 -0.021 0.968 -0.029 151 0.100 1.879 0.104
17 -0.025 0.611 -0.028 62 -0.026 0.699 -0.030 107 -0.021 0.978 -0.029 152 0.109 1.916 0.117
18 -0.025 0.611 -0.028 63 -0.026 0.703 -0.030 108 -0.020 0.988 -0.029 153 0.118 1.955 0.130
19 -0.025 0.612 -0.028 64 -0.026 0.706 -0.030 109 -0.020 0.999 -0.029 154 0.128 1.994 0.145
20 -0.025 0.613 -0.028 65 -0.026 0.710 -0.030 110 -0.019 1.010 -0.028 155 0.139 2.034 0.161
21 -0.025 0.614 -0.028 66 -0.026 0.714 -0.030 111 -0.019 1.021 -0.028 156 0.151 2.075 0.178
22 -0.025 0.615 -0.028 67 -0.026 0.718 -0.030 112 -0.018 1.032 -0.027 157 0.163 2.117 0.197
23 -0.025 0.616 -0.028 68 -0.026 0.722 -0.030 113 -0.018 1.044 -0.027 158 0.175 2.160 0.218
24 -0.025 0.617 -0.028 69 -0.026 0.726 -0.031 114 -0.017 1.056 -0.027 159 0.189 2.203 0.240
25 -0.025 0.618 -0.029 70 -0.026 0.730 -0.031 115 -0.016 1.069 -0.026 160 0.202 2.247 0.265
26 -0.025 0.619 -0.029 71 -0.026 0.734 -0.031 116 -0.015 1.082 -0.025 161 0.217 2.292 0.291
27 -0.025 0.621 -0.029 72 -0.026 0.738 -0.031 117 -0.015 1.095 -0.025 162 0.232 2.337 0.319
28 -0.025 0.622 -0.029 73 -0.026 0.743 -0.031 118 -0.014 1.108 -0.024 163 0.248 2.382 0.349
29 -0.025 0.623 -0.029 74 -0.026 0.747 -0.031 119 -0.013 1.122 -0.023 164 0.264 2.427 0.382
30 -0.025 0.625 -0.029 75 -0.026 0.752 -0.031 120 -0.012 1.137 -0.023 165 0.280 2.472 0.416
31 -0.025 0.626 -0.029 76 -0.026 0.757 -0.031 121 -0.011 1.151 -0.022 166 0.297 2.517 0.453
32 -0.025 0.627 -0.029 77 -0.026 0.762 -0.031 122 -0.010 1.167 -0.021 167 0.314 2.560 0.491
33 -0.025 0.629 -0.029 78 -0.025 0.767 -0.031 123 -0.008 1.182 -0.020 168 0.331 2.603 0.530
34 -0.025 0.631 -0.029 79 -0.025 0.772 -0.031 124 -0.007 1.198 -0.019 169 0.348 2.644 0.571
35 -0.025 0.632 -0.029 80 -0.025 0.777 -0.031 125 -0.006 1.215 -0.017 170 0.364 2.683 0.612
36 -0.025 0.634 -0.029 81 -0.025 0.782 -0.031 126 -0.004 1.232 -0.016 171 0.380 2.721 0.654
37 -0.025 0.636 -0.029 82 -0.025 0.788 -0.031 127 -0.003 1.249 -0.015 172 0.395 2.755 0.694
38 -0.025 0.637 -0.029 83 -0.025 0.793 -0.031 128 -0.001 1.267 -0.013 173 0.408 2.787 0.734
39 -0.025 0.639 -0.029 84 -0.025 0.799 -0.031 129 0.001 1.286 -0.011 174 0.421 2.816 0.770
40 -0.025 0.641 -0.029 85 -0.025 0.805 -0.031 130 0.003 1.305 -0.009 175 0.432 2.840 0.804
41 -0.025 0.643 -0.029 86 -0.025 0.811 -0.031 131 0.005 1.325 -0.007 176 0.442 2.861 0.833
42 -0.025 0.645 -0.029 87 -0.025 0.817 -0.031 132 0.007 1.345 -0.005 177 0.449 2.878 0.857
43 -0.025 0.647 -0.029 88 -0.025 0.823 -0.031 133 0.010 1.366 -0.003 178 0.455 2.890 0.874
44 -0.025 0.649 -0.029 89 -0.025 0.830 -0.031 134 0.012 1.388 -0.000 179 0.458 2.897 0.885
45 -0.025 0.652 -0.029 90 -0.025 0.836 -0.031 135 0.015 1.410 0.003 180 0.459 2.899 0.889



Table 69: Deferential ecliptic latitude of Jupiter. The latitude is minus the value shown in the table if the argument is in parenthesies.
$F (^\circ)$ $\beta_0(^\circ)$ $F (^\circ)$
000/180 0.000 (180)/(360)
002/178 0.046 (182)/(358)
004/176 0.091 (184)/(356)
006/174 0.136 (186)/(354)
008/172 0.182 (188)/(352)
010/170 0.226 (190)/(350)
012/168 0.271 (192)/(348)
014/166 0.316 (194)/(346)
016/164 0.360 (196)/(344)
018/162 0.403 (198)/(342)
020/160 0.446 (200)/(340)
022/158 0.489 (202)/(338)
024/156 0.531 (204)/(336)
026/154 0.572 (206)/(334)
028/152 0.612 (208)/(332)
030/150 0.652 (210)/(330)
032/148 0.691 (212)/(328)
034/146 0.729 (214)/(326)
036/144 0.767 (216)/(324)
038/142 0.803 (218)/(322)
040/140 0.838 (220)/(320)
042/138 0.873 (222)/(318)
044/136 0.906 (224)/(316)
046/134 0.938 (226)/(314)
048/132 0.969 (228)/(312)
050/130 0.999 (230)/(310)
052/128 1.028 (232)/(308)
054/126 1.055 (234)/(306)
056/124 1.081 (236)/(304)
058/122 1.106 (238)/(302)
060/120 1.130 (240)/(300)
062/118 1.152 (242)/(298)
064/116 1.172 (244)/(296)
066/114 1.192 (246)/(294)
068/112 1.209 (248)/(292)
070/110 1.226 (250)/(290)
072/108 1.240 (252)/(288)
074/106 1.254 (254)/(286)
076/104 1.266 (256)/(284)
078/102 1.276 (258)/(282)
080/100 1.284 (260)/(280)
082/098 1.292 (262)/(278)
084/096 1.297 (264)/(276)
086/094 1.301 (266)/(274)
088/092 1.303 (268)/(272)
090/090 1.304 (270)/(270)



Table 70: Epicyclic latitude correction factor for Jupiter. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.
$\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$
0 -0.008 0.839 -0.009 45 -0.007 0.874 -0.008 90 -0.002 0.982 -0.003 135 0.009 1.143 0.010
1 -0.008 0.839 -0.009 46 -0.007 0.876 -0.008 91 -0.002 0.985 -0.002 136 0.009 1.147 0.011
2 -0.008 0.839 -0.009 47 -0.007 0.877 -0.008 92 -0.002 0.988 -0.002 137 0.010 1.150 0.011
3 -0.008 0.839 -0.009 48 -0.007 0.879 -0.008 93 -0.002 0.992 -0.002 138 0.010 1.154 0.011
4 -0.008 0.839 -0.009 49 -0.007 0.881 -0.008 94 -0.001 0.995 -0.002 139 0.010 1.157 0.012
5 -0.008 0.839 -0.009 50 -0.007 0.883 -0.008 95 -0.001 0.998 -0.001 140 0.010 1.160 0.012
6 -0.008 0.840 -0.009 51 -0.007 0.884 -0.008 96 -0.001 1.002 -0.001 141 0.011 1.164 0.012
7 -0.008 0.840 -0.009 52 -0.007 0.886 -0.007 97 -0.001 1.005 -0.001 142 0.011 1.167 0.013
8 -0.008 0.840 -0.009 53 -0.007 0.888 -0.007 98 -0.001 1.008 -0.001 143 0.011 1.170 0.013
9 -0.008 0.840 -0.009 54 -0.006 0.890 -0.007 99 -0.000 1.012 -0.001 144 0.012 1.173 0.013
10 -0.008 0.841 -0.009 55 -0.006 0.892 -0.007 100 -0.000 1.015 -0.000 145 0.012 1.177 0.014
11 -0.008 0.841 -0.009 56 -0.006 0.894 -0.007 101 0.000 1.019 -0.000 146 0.012 1.180 0.014
12 -0.008 0.841 -0.009 57 -0.006 0.896 -0.007 102 0.000 1.022 0.000 147 0.012 1.183 0.014
13 -0.008 0.842 -0.009 58 -0.006 0.898 -0.007 103 0.000 1.026 0.000 148 0.013 1.186 0.015
14 -0.008 0.842 -0.009 59 -0.006 0.900 -0.007 104 0.001 1.029 0.001 149 0.013 1.189 0.015
15 -0.008 0.843 -0.009 60 -0.006 0.902 -0.007 105 0.001 1.033 0.001 150 0.013 1.192 0.015
16 -0.008 0.843 -0.009 61 -0.006 0.904 -0.007 106 0.001 1.036 0.001 151 0.014 1.194 0.016
17 -0.008 0.844 -0.009 62 -0.006 0.906 -0.007 107 0.001 1.040 0.001 152 0.014 1.197 0.016
18 -0.008 0.845 -0.009 63 -0.006 0.909 -0.006 108 0.002 1.044 0.002 153 0.014 1.200 0.016
19 -0.008 0.845 -0.009 64 -0.006 0.911 -0.006 109 0.002 1.047 0.002 154 0.014 1.202 0.017
20 -0.008 0.846 -0.009 65 -0.005 0.913 -0.006 110 0.002 1.051 0.002 155 0.015 1.205 0.017
21 -0.008 0.847 -0.009 66 -0.005 0.916 -0.006 111 0.002 1.055 0.003 156 0.015 1.207 0.017
22 -0.008 0.847 -0.009 67 -0.005 0.918 -0.006 112 0.003 1.058 0.003 157 0.015 1.210 0.017
23 -0.008 0.848 -0.009 68 -0.005 0.920 -0.006 113 0.003 1.062 0.003 158 0.015 1.212 0.018
24 -0.008 0.849 -0.009 69 -0.005 0.923 -0.006 114 0.003 1.066 0.003 159 0.015 1.214 0.018
25 -0.008 0.850 -0.009 70 -0.005 0.925 -0.006 115 0.003 1.069 0.004 160 0.016 1.216 0.018
26 -0.008 0.851 -0.009 71 -0.005 0.928 -0.006 116 0.004 1.073 0.004 161 0.016 1.218 0.018
27 -0.008 0.852 -0.009 72 -0.005 0.930 -0.005 117 0.004 1.077 0.004 162 0.016 1.220 0.019
28 -0.008 0.853 -0.009 73 -0.005 0.933 -0.005 118 0.004 1.080 0.005 163 0.016 1.222 0.019
29 -0.008 0.854 -0.009 74 -0.004 0.935 -0.005 119 0.004 1.084 0.005 164 0.016 1.224 0.019
30 -0.008 0.855 -0.009 75 -0.004 0.938 -0.005 120 0.005 1.088 0.005 165 0.016 1.225 0.019
31 -0.008 0.856 -0.009 76 -0.004 0.941 -0.005 121 0.005 1.092 0.006 166 0.017 1.227 0.019
32 -0.008 0.857 -0.009 77 -0.004 0.944 -0.005 122 0.005 1.095 0.006 167 0.017 1.228 0.020
33 -0.008 0.858 -0.009 78 -0.004 0.946 -0.005 123 0.006 1.099 0.006 168 0.017 1.230 0.020
34 -0.008 0.859 -0.009 79 -0.004 0.949 -0.004 124 0.006 1.103 0.007 169 0.017 1.231 0.020
35 -0.008 0.860 -0.009 80 -0.004 0.952 -0.004 125 0.006 1.107 0.007 170 0.017 1.232 0.020
36 -0.008 0.861 -0.008 81 -0.004 0.955 -0.004 126 0.006 1.110 0.007 171 0.017 1.233 0.020
37 -0.008 0.863 -0.008 82 -0.003 0.958 -0.004 127 0.007 1.114 0.008 172 0.017 1.234 0.020
38 -0.007 0.864 -0.008 83 -0.003 0.961 -0.004 128 0.007 1.118 0.008 173 0.017 1.235 0.020
39 -0.007 0.865 -0.008 84 -0.003 0.964 -0.004 129 0.007 1.121 0.008 174 0.018 1.236 0.021
40 -0.007 0.867 -0.008 85 -0.003 0.967 -0.003 130 0.008 1.125 0.009 175 0.018 1.236 0.021
41 -0.007 0.868 -0.008 86 -0.003 0.970 -0.003 131 0.008 1.129 0.009 176 0.018 1.237 0.021
42 -0.007 0.870 -0.008 87 -0.003 0.973 -0.003 132 0.008 1.132 0.009 177 0.018 1.237 0.021
43 -0.007 0.871 -0.008 88 -0.002 0.976 -0.003 133 0.008 1.136 0.010 178 0.018 1.237 0.021
44 -0.007 0.873 -0.008 89 -0.002 0.979 -0.003 134 0.009 1.140 0.010 179 0.018 1.238 0.021
45 -0.007 0.874 -0.008 90 -0.002 0.982 -0.003 135 0.009 1.143 0.010 180 0.018 1.238 0.021



Table 71: Deferential ecliptic latitude of Saturn. The latitude is minus the value shown in the table if the argument is in parenthesies.
$F (^\circ)$ $\beta_0(^\circ)$ $F (^\circ)$
000/180 0.000 (180)/(360)
002/178 0.087 (182)/(358)
004/176 0.173 (184)/(356)
006/174 0.260 (186)/(354)
008/172 0.346 (188)/(352)
010/170 0.432 (190)/(350)
012/168 0.517 (192)/(348)
014/166 0.601 (194)/(346)
016/164 0.685 (196)/(344)
018/162 0.768 (198)/(342)
020/160 0.850 (200)/(340)
022/158 0.931 (202)/(338)
024/156 1.011 (204)/(336)
026/154 1.089 (206)/(334)
028/152 1.167 (208)/(332)
030/150 1.243 (210)/(330)
032/148 1.317 (212)/(328)
034/146 1.390 (214)/(326)
036/144 1.461 (216)/(324)
038/142 1.530 (218)/(322)
040/140 1.597 (220)/(320)
042/138 1.663 (222)/(318)
044/136 1.726 (224)/(316)
046/134 1.788 (226)/(314)
048/132 1.847 (228)/(312)
050/130 1.904 (230)/(310)
052/128 1.958 (232)/(308)
054/126 2.011 (234)/(306)
056/124 2.060 (236)/(304)
058/122 2.108 (238)/(302)
060/120 2.152 (240)/(300)
062/118 2.194 (242)/(298)
064/116 2.234 (244)/(296)
066/114 2.270 (246)/(294)
068/112 2.304 (248)/(292)
070/110 2.335 (250)/(290)
072/108 2.364 (252)/(288)
074/106 2.389 (254)/(286)
076/104 2.411 (256)/(284)
078/102 2.431 (258)/(282)
080/100 2.447 (260)/(280)
082/098 2.461 (262)/(278)
084/096 2.472 (264)/(276)
086/094 2.479 (266)/(274)
088/092 2.484 (268)/(272)
090/090 2.485 (270)/(270)



Table 72: Epicyclic latitude correction factor for Saturn. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.
$\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$
0 -0.006 0.905 -0.006 45 -0.005 0.929 -0.005 90 -0.001 0.995 -0.001 135 0.005 1.077 0.006
1 -0.006 0.905 -0.006 46 -0.004 0.930 -0.005 91 -0.001 0.996 -0.001 136 0.005 1.078 0.006
2 -0.006 0.905 -0.006 47 -0.004 0.931 -0.005 92 -0.000 0.998 -0.001 137 0.005 1.080 0.006
3 -0.006 0.905 -0.006 48 -0.004 0.932 -0.005 93 -0.000 1.000 -0.000 138 0.006 1.081 0.006
4 -0.006 0.905 -0.006 49 -0.004 0.933 -0.005 94 -0.000 1.002 -0.000 139 0.006 1.083 0.007
5 -0.006 0.905 -0.006 50 -0.004 0.934 -0.005 95 -0.000 1.004 -0.000 140 0.006 1.084 0.007
6 -0.006 0.906 -0.006 51 -0.004 0.935 -0.005 96 0.000 1.006 -0.000 141 0.006 1.086 0.007
7 -0.006 0.906 -0.006 52 -0.004 0.937 -0.005 97 0.000 1.007 0.000 142 0.006 1.087 0.007
8 -0.006 0.906 -0.006 53 -0.004 0.938 -0.005 98 0.000 1.009 0.000 143 0.006 1.089 0.007
9 -0.006 0.906 -0.006 54 -0.004 0.939 -0.005 99 0.000 1.011 0.000 144 0.006 1.090 0.007
10 -0.006 0.906 -0.006 55 -0.004 0.940 -0.004 100 0.001 1.013 0.001 145 0.006 1.091 0.007
11 -0.006 0.907 -0.006 56 -0.004 0.942 -0.004 101 0.001 1.015 0.001 146 0.006 1.093 0.008
12 -0.006 0.907 -0.006 57 -0.004 0.943 -0.004 102 0.001 1.017 0.001 147 0.007 1.094 0.008
13 -0.006 0.907 -0.006 58 -0.004 0.944 -0.004 103 0.001 1.019 0.001 148 0.007 1.095 0.008
14 -0.006 0.908 -0.006 59 -0.004 0.945 -0.004 104 0.001 1.020 0.001 149 0.007 1.097 0.008
15 -0.006 0.908 -0.006 60 -0.004 0.947 -0.004 105 0.001 1.022 0.001 150 0.007 1.098 0.008
16 -0.006 0.908 -0.006 61 -0.003 0.948 -0.004 106 0.001 1.024 0.001 151 0.007 1.099 0.008
17 -0.006 0.909 -0.006 62 -0.003 0.949 -0.004 107 0.001 1.026 0.002 152 0.007 1.100 0.008
18 -0.006 0.909 -0.006 63 -0.003 0.951 -0.004 108 0.002 1.028 0.002 153 0.007 1.101 0.008
19 -0.005 0.909 -0.006 64 -0.003 0.952 -0.004 109 0.002 1.030 0.002 154 0.007 1.103 0.009
20 -0.005 0.910 -0.006 65 -0.003 0.954 -0.004 110 0.002 1.032 0.002 155 0.007 1.104 0.009
21 -0.005 0.910 -0.006 66 -0.003 0.955 -0.004 111 0.002 1.034 0.002 156 0.007 1.105 0.009
22 -0.005 0.911 -0.006 67 -0.003 0.957 -0.003 112 0.002 1.036 0.002 157 0.008 1.106 0.009
23 -0.005 0.911 -0.006 68 -0.003 0.958 -0.003 113 0.002 1.037 0.003 158 0.008 1.107 0.009
24 -0.005 0.912 -0.006 69 -0.003 0.960 -0.003 114 0.002 1.039 0.003 159 0.008 1.107 0.009
25 -0.005 0.913 -0.006 70 -0.003 0.961 -0.003 115 0.002 1.041 0.003 160 0.008 1.108 0.009
26 -0.005 0.913 -0.006 71 -0.003 0.963 -0.003 116 0.003 1.043 0.003 161 0.008 1.109 0.009
27 -0.005 0.914 -0.006 72 -0.003 0.964 -0.003 117 0.003 1.045 0.003 162 0.008 1.110 0.009
28 -0.005 0.914 -0.006 73 -0.002 0.966 -0.003 118 0.003 1.047 0.003 163 0.008 1.111 0.009
29 -0.005 0.915 -0.006 74 -0.002 0.967 -0.003 119 0.003 1.049 0.003 164 0.008 1.111 0.009
30 -0.005 0.916 -0.006 75 -0.002 0.969 -0.003 120 0.003 1.050 0.004 165 0.008 1.112 0.009
31 -0.005 0.916 -0.006 76 -0.002 0.971 -0.003 121 0.003 1.052 0.004 166 0.008 1.113 0.010
32 -0.005 0.917 -0.006 77 -0.002 0.972 -0.002 122 0.003 1.054 0.004 167 0.008 1.113 0.010
33 -0.005 0.918 -0.006 78 -0.002 0.974 -0.002 123 0.004 1.056 0.004 168 0.008 1.114 0.010
34 -0.005 0.919 -0.006 79 -0.002 0.975 -0.002 124 0.004 1.058 0.004 169 0.008 1.114 0.010
35 -0.005 0.920 -0.006 80 -0.002 0.977 -0.002 125 0.004 1.060 0.004 170 0.008 1.115 0.010
36 -0.005 0.920 -0.006 81 -0.002 0.979 -0.002 126 0.004 1.061 0.005 171 0.008 1.115 0.010
37 -0.005 0.921 -0.006 82 -0.002 0.981 -0.002 127 0.004 1.063 0.005 172 0.008 1.116 0.010
38 -0.005 0.922 -0.006 83 -0.001 0.982 -0.002 128 0.004 1.065 0.005 173 0.008 1.116 0.010
39 -0.005 0.923 -0.005 84 -0.001 0.984 -0.002 129 0.004 1.067 0.005 174 0.008 1.116 0.010
40 -0.005 0.924 -0.005 85 -0.001 0.986 -0.001 130 0.005 1.068 0.005 175 0.008 1.116 0.010
41 -0.005 0.925 -0.005 86 -0.001 0.987 -0.001 131 0.005 1.070 0.005 176 0.009 1.117 0.010
42 -0.005 0.926 -0.005 87 -0.001 0.989 -0.001 132 0.005 1.072 0.006 177 0.009 1.117 0.010
43 -0.005 0.927 -0.005 88 -0.001 0.991 -0.001 133 0.005 1.073 0.006 178 0.009 1.117 0.010
44 -0.005 0.928 -0.005 89 -0.001 0.993 -0.001 134 0.005 1.075 0.006 179 0.009 1.117 0.010
45 -0.005 0.929 -0.005 90 -0.001 0.995 -0.001 135 0.005 1.077 0.006 180 0.009 1.117 0.010



Table 73: Epicyclic ecliptic latitude of Venus. The latitude is minus the value shown in the table if the argument is in parenthesies.
$F (^\circ)$ $\beta_0(^\circ)$ $F (^\circ)$
000/180 0.000 (180)/(360)
002/178 0.085 (182)/(358)
004/176 0.170 (184)/(356)
006/174 0.255 (186)/(354)
008/172 0.339 (188)/(352)
010/170 0.423 (190)/(350)
012/168 0.506 (192)/(348)
014/166 0.589 (194)/(346)
016/164 0.671 (196)/(344)
018/162 0.753 (198)/(342)
020/160 0.833 (200)/(340)
022/158 0.912 (202)/(338)
024/156 0.991 (204)/(336)
026/154 1.068 (206)/(334)
028/152 1.143 (208)/(332)
030/150 1.218 (210)/(330)
032/148 1.291 (212)/(328)
034/146 1.362 (214)/(326)
036/144 1.432 (216)/(324)
038/142 1.500 (218)/(322)
040/140 1.566 (220)/(320)
042/138 1.630 (222)/(318)
044/136 1.692 (224)/(316)
046/134 1.752 (226)/(314)
048/132 1.810 (228)/(312)
050/130 1.866 (230)/(310)
052/128 1.919 (232)/(308)
054/126 1.970 (234)/(306)
056/124 2.019 (236)/(304)
058/122 2.066 (238)/(302)
060/120 2.109 (240)/(300)
062/118 2.151 (242)/(298)
064/116 2.189 (244)/(296)
066/114 2.225 (246)/(294)
068/112 2.258 (248)/(292)
070/110 2.289 (250)/(290)
072/108 2.316 (252)/(288)
074/106 2.341 (254)/(286)
076/104 2.363 (256)/(284)
078/102 2.382 (258)/(282)
080/100 2.399 (260)/(280)
082/098 2.412 (262)/(278)
084/096 2.422 (264)/(276)
086/094 2.430 (266)/(274)
088/092 2.434 (268)/(272)
090/090 2.436 (270)/(270)



Table 74: Deferential latitude correction factor for Venus. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.
$\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$
0 0.008 0.580 0.008 45 0.009 0.627 0.009 90 0.012 0.810 0.013 135 0.032 1.413 0.033
1 0.008 0.580 0.008 46 0.009 0.629 0.009 91 0.013 0.817 0.013 136 0.033 1.439 0.034
2 0.008 0.580 0.008 47 0.009 0.631 0.009 92 0.013 0.824 0.013 137 0.034 1.466 0.035
3 0.008 0.580 0.008 48 0.009 0.633 0.009 93 0.013 0.831 0.013 138 0.036 1.493 0.037
4 0.008 0.580 0.008 49 0.009 0.636 0.009 94 0.013 0.838 0.013 139 0.037 1.522 0.038
5 0.008 0.581 0.008 50 0.009 0.638 0.009 95 0.013 0.846 0.013 140 0.039 1.552 0.040
6 0.008 0.581 0.008 51 0.009 0.641 0.009 96 0.013 0.854 0.014 141 0.040 1.583 0.041
7 0.008 0.581 0.008 52 0.009 0.643 0.009 97 0.014 0.861 0.014 142 0.042 1.615 0.043
8 0.008 0.582 0.008 53 0.009 0.646 0.009 98 0.014 0.870 0.014 143 0.044 1.648 0.045
9 0.008 0.582 0.008 54 0.009 0.649 0.009 99 0.014 0.878 0.014 144 0.046 1.683 0.047
10 0.008 0.582 0.008 55 0.009 0.652 0.009 100 0.014 0.886 0.014 145 0.048 1.719 0.049
11 0.008 0.583 0.008 56 0.009 0.655 0.009 101 0.014 0.895 0.015 146 0.050 1.756 0.052
12 0.008 0.583 0.008 57 0.009 0.658 0.009 102 0.015 0.904 0.015 147 0.053 1.795 0.054
13 0.008 0.584 0.008 58 0.009 0.661 0.009 103 0.015 0.913 0.015 148 0.055 1.836 0.057
14 0.008 0.584 0.008 59 0.009 0.664 0.010 104 0.015 0.923 0.015 149 0.058 1.878 0.060
15 0.008 0.585 0.008 60 0.009 0.667 0.010 105 0.015 0.933 0.016 150 0.061 1.922 0.063
16 0.008 0.586 0.008 61 0.009 0.670 0.010 106 0.016 0.943 0.016 151 0.065 1.968 0.067
17 0.008 0.586 0.008 62 0.010 0.674 0.010 107 0.016 0.953 0.016 152 0.068 2.016 0.071
18 0.008 0.587 0.008 63 0.010 0.677 0.010 108 0.016 0.964 0.017 153 0.072 2.065 0.075
19 0.008 0.588 0.008 64 0.010 0.681 0.010 109 0.016 0.975 0.017 154 0.076 2.117 0.080
20 0.008 0.589 0.008 65 0.010 0.684 0.010 110 0.017 0.986 0.017 155 0.081 2.170 0.085
21 0.008 0.590 0.008 66 0.010 0.688 0.010 111 0.017 0.997 0.017 156 0.086 2.226 0.090
22 0.008 0.591 0.008 67 0.010 0.692 0.010 112 0.017 1.009 0.018 157 0.091 2.283 0.096
23 0.008 0.592 0.008 68 0.010 0.696 0.010 113 0.018 1.021 0.018 158 0.097 2.343 0.102
24 0.008 0.593 0.008 69 0.010 0.700 0.010 114 0.018 1.034 0.019 159 0.103 2.405 0.109
25 0.008 0.594 0.008 70 0.010 0.704 0.010 115 0.019 1.047 0.019 160 0.110 2.469 0.117
26 0.008 0.595 0.008 71 0.010 0.708 0.010 116 0.019 1.060 0.019 161 0.117 2.535 0.125
27 0.008 0.596 0.008 72 0.010 0.712 0.011 117 0.019 1.074 0.020 162 0.125 2.603 0.134
28 0.008 0.597 0.008 73 0.010 0.717 0.011 118 0.020 1.088 0.020 163 0.133 2.673 0.144
29 0.008 0.599 0.008 74 0.010 0.721 0.011 119 0.020 1.103 0.021 164 0.142 2.744 0.155
30 0.008 0.600 0.008 75 0.011 0.726 0.011 120 0.021 1.118 0.021 165 0.151 2.816 0.166
31 0.008 0.601 0.008 76 0.011 0.730 0.011 121 0.021 1.133 0.022 166 0.161 2.890 0.178
32 0.008 0.603 0.008 77 0.011 0.735 0.011 122 0.022 1.149 0.022 167 0.172 2.964 0.191
33 0.008 0.604 0.008 78 0.011 0.740 0.011 123 0.022 1.166 0.023 168 0.183 3.037 0.204
34 0.008 0.606 0.008 79 0.011 0.745 0.011 124 0.023 1.183 0.023 169 0.194 3.111 0.218
35 0.008 0.608 0.009 80 0.011 0.751 0.011 125 0.024 1.200 0.024 170 0.205 3.182 0.232
36 0.008 0.609 0.009 81 0.011 0.756 0.011 126 0.024 1.219 0.025 171 0.217 3.252 0.247
37 0.008 0.611 0.009 82 0.011 0.761 0.012 127 0.025 1.237 0.025 172 0.228 3.318 0.262
38 0.008 0.613 0.009 83 0.011 0.767 0.012 128 0.026 1.257 0.026 173 0.239 3.380 0.276
39 0.008 0.614 0.009 84 0.012 0.773 0.012 129 0.026 1.277 0.027 174 0.249 3.436 0.290
40 0.008 0.616 0.009 85 0.012 0.778 0.012 130 0.027 1.298 0.028 175 0.258 3.486 0.302
41 0.008 0.618 0.009 86 0.012 0.784 0.012 131 0.028 1.319 0.029 176 0.267 3.529 0.313
42 0.009 0.620 0.009 87 0.012 0.791 0.012 132 0.029 1.342 0.030 177 0.273 3.564 0.322
43 0.009 0.622 0.009 88 0.012 0.797 0.012 133 0.030 1.365 0.031 178 0.278 3.589 0.329
44 0.009 0.624 0.009 89 0.012 0.803 0.012 134 0.031 1.389 0.032 179 0.281 3.604 0.333
45 0.009 0.627 0.009 90 0.012 0.810 0.013 135 0.032 1.413 0.033 180 0.282 3.609 0.334



Table 75: Epicyclic ecliptic latitude of Mercury. The latitude is minus the value shown in the table if the argument is in parenthesies.
$F (^\circ)$ $\beta_0(^\circ)$ $F (^\circ)$
000/180 0.000 (180)/(360)
002/178 0.093 (182)/(358)
004/176 0.186 (184)/(356)
006/174 0.279 (186)/(354)
008/172 0.372 (188)/(352)
010/170 0.464 (190)/(350)
012/168 0.556 (192)/(348)
014/166 0.646 (194)/(346)
016/164 0.736 (196)/(344)
018/162 0.826 (198)/(342)
020/160 0.914 (200)/(340)
022/158 1.001 (202)/(338)
024/156 1.087 (204)/(336)
026/154 1.171 (206)/(334)
028/152 1.254 (208)/(332)
030/150 1.336 (210)/(330)
032/148 1.416 (212)/(328)
034/146 1.494 (214)/(326)
036/144 1.570 (216)/(324)
038/142 1.645 (218)/(322)
040/140 1.717 (220)/(320)
042/138 1.788 (222)/(318)
044/136 1.856 (224)/(316)
046/134 1.922 (226)/(314)
048/132 1.986 (228)/(312)
050/130 2.047 (230)/(310)
052/128 2.105 (232)/(308)
054/126 2.162 (234)/(306)
056/124 2.215 (236)/(304)
058/122 2.266 (238)/(302)
060/120 2.314 (240)/(300)
062/118 2.359 (242)/(298)
064/116 2.401 (244)/(296)
066/114 2.441 (246)/(294)
068/112 2.477 (248)/(292)
070/110 2.511 (250)/(290)
072/108 2.541 (252)/(288)
074/106 2.568 (254)/(286)
076/104 2.592 (256)/(284)
078/102 2.613 (258)/(282)
080/100 2.631 (260)/(280)
082/098 2.646 (262)/(278)
084/096 2.657 (264)/(276)
086/094 2.665 (266)/(274)
088/092 2.670 (268)/(272)
090/090 2.672 (270)/(270)



Table 76: Deferential latitude correction factor for Mercury. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.
$\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$ $\mu$ $\delta h_-$ $\bar{h}~~~~$ $\delta h_+$
0 0.099 0.719 0.137 45 0.110 0.765 0.152 90 0.152 0.930 0.217 135 0.274 1.283 0.451
1 0.099 0.719 0.137 46 0.110 0.768 0.153 91 0.153 0.935 0.220 136 0.278 1.293 0.461
2 0.099 0.719 0.137 47 0.111 0.770 0.154 92 0.155 0.941 0.222 137 0.282 1.303 0.471
3 0.099 0.719 0.137 48 0.111 0.772 0.154 93 0.157 0.946 0.225 138 0.286 1.313 0.481
4 0.099 0.719 0.137 49 0.112 0.774 0.155 94 0.158 0.952 0.228 139 0.290 1.324 0.491
5 0.099 0.720 0.137 50 0.112 0.777 0.156 95 0.160 0.958 0.231 140 0.295 1.334 0.501
6 0.099 0.720 0.137 51 0.113 0.779 0.157 96 0.162 0.964 0.234 141 0.299 1.344 0.512
7 0.099 0.720 0.137 52 0.113 0.782 0.158 97 0.164 0.970 0.237 142 0.304 1.355 0.523
8 0.099 0.720 0.137 53 0.114 0.784 0.159 98 0.165 0.976 0.240 143 0.308 1.365 0.534
9 0.100 0.721 0.137 54 0.115 0.787 0.160 99 0.167 0.983 0.243 144 0.313 1.375 0.546
10 0.100 0.721 0.138 55 0.115 0.790 0.161 100 0.169 0.989 0.246 145 0.317 1.386 0.558
11 0.100 0.722 0.138 56 0.116 0.793 0.162 101 0.171 0.996 0.250 146 0.322 1.396 0.570
12 0.100 0.722 0.138 57 0.117 0.795 0.163 102 0.173 1.002 0.253 147 0.326 1.406 0.582
13 0.100 0.723 0.138 58 0.117 0.798 0.164 103 0.175 1.009 0.257 148 0.331 1.417 0.594
14 0.100 0.723 0.138 59 0.118 0.801 0.165 104 0.178 1.016 0.261 149 0.335 1.427 0.607
15 0.100 0.724 0.138 60 0.119 0.804 0.166 105 0.180 1.023 0.264 150 0.340 1.437 0.620
16 0.100 0.725 0.139 61 0.120 0.807 0.167 106 0.182 1.030 0.268 151 0.345 1.447 0.633
17 0.101 0.725 0.139 62 0.120 0.811 0.168 107 0.184 1.037 0.272 152 0.349 1.457 0.646
18 0.101 0.726 0.139 63 0.121 0.814 0.169 108 0.187 1.044 0.276 153 0.354 1.467 0.659
19 0.101 0.727 0.139 64 0.122 0.817 0.170 109 0.189 1.052 0.281 154 0.358 1.476 0.672
20 0.101 0.728 0.140 65 0.123 0.820 0.172 110 0.192 1.059 0.285 155 0.363 1.486 0.686
21 0.101 0.729 0.140 66 0.124 0.824 0.173 111 0.194 1.067 0.290 156 0.367 1.495 0.699
22 0.101 0.730 0.140 67 0.124 0.827 0.174 112 0.197 1.074 0.294 157 0.371 1.504 0.713
23 0.102 0.731 0.141 68 0.125 0.831 0.176 113 0.199 1.082 0.299 158 0.376 1.513 0.726
24 0.102 0.732 0.141 69 0.126 0.835 0.177 114 0.202 1.090 0.304 159 0.380 1.522 0.739
25 0.102 0.733 0.141 70 0.127 0.838 0.179 115 0.205 1.098 0.309 160 0.384 1.530 0.753
26 0.102 0.734 0.142 71 0.128 0.842 0.180 116 0.208 1.107 0.315 161 0.388 1.538 0.766
27 0.103 0.735 0.142 72 0.129 0.846 0.182 117 0.210 1.115 0.320 162 0.392 1.546 0.779
28 0.103 0.737 0.142 73 0.130 0.850 0.183 118 0.213 1.123 0.326 163 0.396 1.554 0.791
29 0.103 0.738 0.143 74 0.131 0.854 0.185 119 0.216 1.132 0.331 164 0.399 1.561 0.804
30 0.104 0.739 0.143 75 0.132 0.858 0.186 120 0.219 1.141 0.337 165 0.403 1.568 0.816
31 0.104 0.741 0.144 76 0.133 0.862 0.188 121 0.223 1.149 0.343 166 0.406 1.575 0.827
32 0.104 0.742 0.144 77 0.135 0.866 0.190 122 0.226 1.158 0.350 167 0.409 1.581 0.838
33 0.105 0.743 0.145 78 0.136 0.871 0.192 123 0.229 1.167 0.356 168 0.412 1.587 0.849
34 0.105 0.745 0.145 79 0.137 0.875 0.193 124 0.232 1.176 0.363 169 0.415 1.592 0.858
35 0.105 0.747 0.146 80 0.138 0.880 0.195 125 0.236 1.185 0.370 170 0.417 1.597 0.868
36 0.106 0.748 0.146 81 0.139 0.884 0.197 126 0.239 1.195 0.377 171 0.420 1.602 0.876
37 0.106 0.750 0.147 82 0.141 0.889 0.199 127 0.243 1.204 0.384 172 0.422 1.606 0.884
38 0.106 0.752 0.147 83 0.142 0.894 0.201 128 0.246 1.214 0.392 173 0.424 1.610 0.891
39 0.107 0.754 0.148 84 0.143 0.899 0.203 129 0.250 1.223 0.400 174 0.425 1.613 0.897
40 0.107 0.755 0.149 85 0.145 0.904 0.206 130 0.254 1.233 0.408 175 0.427 1.615 0.903
41 0.108 0.757 0.149 86 0.146 0.909 0.208 131 0.258 1.243 0.416 176 0.428 1.618 0.907
42 0.108 0.759 0.150 87 0.147 0.914 0.210 132 0.262 1.253 0.424 177 0.429 1.619 0.911
43 0.109 0.761 0.151 88 0.149 0.919 0.212 133 0.265 1.263 0.433 178 0.429 1.621 0.913
44 0.109 0.763 0.151 89 0.150 0.924 0.215 134 0.269 1.273 0.442 179 0.430 1.621 0.914
45 0.110 0.765 0.152 90 0.152 0.930 0.217 135 0.274 1.283 0.451 180 0.430 1.622 0.915


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Next: Glossary Up: Planetary Latitudes Previous: Venus
Richard Fitzpatrick 2010-07-21