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# Lunar Parallax

Now, it turns out that the moon is sufficiently close to the earth that its position in the sky is significantly modified by parallax. All of our previous analysis applies to a hypothetical observer situated at the center of the earth. Consider a real observer situated on the earth's surface. It can be seen from Fig. 24 that the altitude of the moon is for the real observer, and for the hypothetical observer. Simple trigonometry reveals that , which implies that the real observer sees the moon at a lower altitude than the hypothetical observer. Let be the radius of the earth, and the distance from the center of the earth to the moon. More simple trigonometry yields
 (126)

Let us assume that the moon's orbit is elliptical to first order in its eccentricity. It follows, from Cha. 4, that
 (127)

where , , and are major radius, eccentricity, and mean anomaly of the lunar orbit. Assuming that is small, we obtain
 (128)

where (since km and km).

According to Eq. (128), lunar parallax can be written in the form

 (129)

where , , and are the moon's geocentric altitude (i.e., the altitude seen from the center of the earth), true altitude, and mean anomaly, respectively. The functions and are tabulated in Table 39. It can be seen from the table that lunar parallax increases with decreasing lunar altitude, reaching a maximum value of about when the moon is close to the horizon. For example, if and then Table 39 yields and . Hence, , and the true altitude of the moon becomes .

It now remains to investigate how parallax affects the moon's ecliptic longitude and latitude. Figure 25 shows a detail of Fig. 13. Point is the moon's geocentric position on the celestial sphere. is a line passing through this point which is parallel to the local ecliptic circle, whereas is a small section of an altitude circle passing through . The angle subtended between the ecliptic and the altitude circle is the parallactic angle, . Let be the true position of the moon. It follows that . The changes in the moon's ecliptic longitude and latitude are and , respectively. Here, we are considering the case where increasing altitude corresponds to increasing ecliptic latitude. Assuming that the arcs , , and are all fairly small, the triangle can be treated as a plane triangle. Hence, we obtain

 (130) (131)

As is easily demonstrated, the above formulae also apply to the case in which increasing altitude corresponds to decreasing ecliptic latitude.

For example, consider a day on which the geocentric ecliptic longitude and mean anomaly of the moon are (i.e., 00SC00) and , respectively. Suppose that the moon is viewed from an observation site located at terrestrial latitude . The Scorpio'' entry in Table 19 gives the moon's geocentric altitude, , as a function of time, as well as the value of the parallactic angle . Making use of this data, in combination with Table 39 and Eqs. (130) and (131), we can calculate the parallax-induced changes in the moon's ecliptic longitude and latitude as it transits the sky. Data from such a calculation is given in the table below. The first column specifies time since the moon's upper transit (thus, hrs. means one hour after the upper transit), the second column gives the moon's geocentric altitude, the third column the parallactic angle, the fourth column the decrease in the moon's real altitude due to parallax, and the fifth and sixth columns the parallax-induced changes in its ecliptic longitude and latitude, respectively. It can be seen that parallax causes the moon's apparent location to shift by almost relative to the fixed stars as it transits the sky. Note that the above calculation is somewhat inaccurate because it does not take into account the moon's motion along the ecliptic (which can easily amount to during the course of a night). However, the calculation does illustrate how the data contained in Tables 18-26, in combination with the data in Table 39, permits the parallax-induced shift in the moon's ecliptic position to be calculated for a wide range of different lunar phases, observation sites, and observation times.

 (hrs.)

Table 35: Orbital elements of the moon for the J2000 epoch (i.e., 12:00 UT, January 1, 2000 CE, which corresponds to JD).
 0.054881 13.17639646 13.06499295 218.322 134.916 93.284 5.161

Table 36: Mean motion of the moon. Here, , , , and . At epoch ( JD), , , and .
 (JD) (JD) 10,000 3.965 329.930 173.503 1,000 216.396 104.993 269.350 20,000 7.929 299.859 347.005 2,000 72.793 209.986 178.701 30,000 11.894 269.788 160.508 3,000 289.189 314.979 88.051 40,000 15.858 239.718 334.011 4,000 145.586 59.972 357.401 50,000 19.823 209.648 147.513 5,000 1.982 164.965 266.751 60,000 23.788 179.577 321.016 6,000 218.379 269.958 176.102 70,000 27.752 149.506 134.519 7,000 74.775 14.951 85.452 80,000 31.717 119.436 308.022 8,000 291.172 119.944 354.802 90,000 35.681 89.366 121.524 9,000 147.568 224.937 264.152 100 237.640 226.499 242.935 10 131.764 130.650 132.294 200 115.279 92.999 125.870 20 263.528 261.300 264.587 300 352.919 319.498 8.805 30 35.292 31.950 36.881 400 230.559 185.997 251.740 40 167.056 162.600 169.174 500 108.198 52.496 134.675 50 298.820 293.250 301.468 600 345.838 278.996 17.610 60 70.584 63.900 73.761 700 223.478 145.495 260.545 70 202.348 194.550 206.055 800 101.117 11.994 143.480 80 334.112 325.199 338.348 900 338.757 238.494 26.415 90 105.876 95.849 110.642 1 13.176 13.065 13.229 0.1 1.318 1.306 1.323 2 26.353 26.130 26.459 0.2 2.635 2.613 2.646 3 39.529 39.195 39.688 0.3 3.953 3.919 3.969 4 52.706 52.260 52.917 0.4 5.271 5.226 5.292 5 65.882 65.325 66.147 0.5 6.588 6.532 6.615 6 79.058 78.390 79.376 0.6 7.906 7.839 7.938 7 92.235 91.455 92.605 0.7 9.223 9.145 9.261 8 105.411 104.520 105.835 0.8 10.541 10.452 10.583 9 118.588 117.585 119.064 0.9 11.859 11.758 11.906

Table 37: Anomalies of the moon. The common argument corresponds to , , , , and for the case of , , , , and , respectively. If the argument is in parenthesies then the anomalies are minus the values shown in the table.
 Arg. () Arg. () 000/(360) 0.000 0.000 0.000 -0.000 -0.000 090/(270) 6.289 1.327 -0.044 -0.160 -0.119 002/(358) 0.237 0.046 0.045 -0.006 -0.004 092/(268) 6.268 1.326 -0.090 -0.160 -0.119 004/(356) 0.473 0.093 0.089 -0.011 -0.008 094/(266) 6.239 1.324 -0.136 -0.160 -0.119 006/(354) 0.709 0.139 0.133 -0.017 -0.012 096/(264) 6.203 1.320 -0.182 -0.159 -0.119 008/(352) 0.943 0.185 0.177 -0.022 -0.017 098/(262) 6.160 1.314 -0.226 -0.159 -0.118 010/(350) 1.176 0.230 0.219 -0.028 -0.021 100/(260) 6.109 1.307 -0.270 -0.158 -0.118 012/(348) 1.408 0.276 0.261 -0.033 -0.025 102/(258) 6.051 1.298 -0.313 -0.157 -0.117 014/(346) 1.637 0.321 0.301 -0.039 -0.029 104/(256) 5.986 1.288 -0.354 -0.156 -0.116 016/(344) 1.864 0.366 0.339 -0.044 -0.033 106/(254) 5.915 1.276 -0.394 -0.154 -0.115 018/(342) 2.088 0.410 0.376 -0.050 -0.037 108/(252) 5.836 1.262 -0.432 -0.153 -0.114 020/(340) 2.310 0.454 0.411 -0.055 -0.041 110/(250) 5.751 1.247 -0.468 -0.151 -0.112 022/(338) 2.527 0.497 0.444 -0.060 -0.045 112/(248) 5.660 1.230 -0.502 -0.149 -0.111 024/(336) 2.741 0.540 0.475 -0.065 -0.049 114/(246) 5.562 1.212 -0.533 -0.147 -0.109 026/(334) 2.951 0.582 0.504 -0.070 -0.052 116/(244) 5.458 1.193 -0.562 -0.144 -0.107 028/(332) 3.157 0.623 0.529 -0.075 -0.056 118/(242) 5.348 1.172 -0.589 -0.142 -0.106 030/(330) 3.358 0.663 0.553 -0.080 -0.060 120/(240) 5.233 1.149 -0.613 -0.139 -0.103 032/(328) 3.554 0.703 0.573 -0.085 -0.063 122/(238) 5.111 1.125 -0.634 -0.136 -0.101 034/(326) 3.746 0.742 0.591 -0.090 -0.067 124/(236) 4.985 1.100 -0.652 -0.133 -0.099 036/(324) 3.931 0.780 0.605 -0.094 -0.070 126/(234) 4.853 1.074 -0.667 -0.130 -0.097 038/(322) 4.111 0.817 0.617 -0.099 -0.074 128/(232) 4.716 1.046 -0.678 -0.126 -0.094 040/(320) 4.285 0.853 0.625 -0.103 -0.077 130/(230) 4.575 1.017 -0.687 -0.123 -0.092 042/(318) 4.454 0.888 0.630 -0.107 -0.080 132/(228) 4.428 0.986 -0.693 -0.119 -0.089 044/(316) 4.615 0.922 0.632 -0.111 -0.083 134/(226) 4.277 0.955 -0.695 -0.115 -0.086 046/(314) 4.770 0.955 0.631 -0.115 -0.086 136/(224) 4.122 0.922 -0.694 -0.111 -0.083 048/(312) 4.919 0.986 0.627 -0.119 -0.089 138/(222) 3.963 0.888 -0.689 -0.107 -0.080 050/(310) 5.061 1.017 0.620 -0.123 -0.092 140/(220) 3.799 0.853 -0.682 -0.103 -0.077 052/(308) 5.195 1.046 0.609 -0.126 -0.094 142/(218) 3.632 0.817 -0.671 -0.099 -0.074 054/(306) 5.323 1.074 0.595 -0.130 -0.097 144/(216) 3.462 0.780 -0.657 -0.094 -0.070 056/(304) 5.443 1.100 0.579 -0.133 -0.099 146/(214) 3.288 0.742 -0.640 -0.090 -0.067 058/(302) 5.555 1.125 0.559 -0.136 -0.101 148/(212) 3.111 0.703 -0.620 -0.085 -0.063 060/(300) 5.660 1.149 0.536 -0.139 -0.103 150/(210) 2.931 0.663 -0.597 -0.080 -0.060 062/(298) 5.757 1.172 0.511 -0.142 -0.106 152/(208) 2.748 0.623 -0.571 -0.075 -0.056 064/(296) 5.847 1.193 0.483 -0.144 -0.107 154/(206) 2.562 0.582 -0.542 -0.070 -0.052 066/(294) 5.929 1.212 0.453 -0.147 -0.109 156/(204) 2.375 0.540 -0.511 -0.065 -0.049 068/(292) 6.002 1.230 0.420 -0.149 -0.111 158/(202) 2.184 0.497 -0.477 -0.060 -0.045 070/(290) 6.068 1.247 0.385 -0.151 -0.112 160/(200) 1.992 0.454 -0.442 -0.055 -0.041 072/(288) 6.126 1.262 0.348 -0.153 -0.114 162/(198) 1.798 0.410 -0.404 -0.050 -0.037 074/(286) 6.176 1.276 0.309 -0.154 -0.115 164/(196) 1.603 0.366 -0.364 -0.044 -0.033 076/(284) 6.218 1.288 0.269 -0.156 -0.116 166/(194) 1.406 0.321 -0.322 -0.039 -0.029 078/(282) 6.252 1.298 0.227 -0.157 -0.117 168/(192) 1.207 0.276 -0.279 -0.033 -0.025 080/(280) 6.278 1.307 0.184 -0.158 -0.118 170/(190) 1.008 0.230 -0.235 -0.028 -0.021 082/(278) 6.296 1.314 0.139 -0.159 -0.118 172/(188) 0.807 0.185 -0.189 -0.022 -0.017 084/(276) 6.306 1.320 0.094 -0.159 -0.119 174/(186) 0.606 0.139 -0.143 -0.017 -0.012 086/(274) 6.308 1.324 0.048 -0.160 -0.119 176/(184) 0.404 0.093 -0.095 -0.011 -0.008 088/(272) 6.302 1.326 0.002 -0.160 -0.119 178/(182) 0.202 0.046 -0.048 -0.006 -0.004 090/(270) 6.289 1.327 -0.044 -0.160 -0.119 180/(180) 0.000 0.000 -0.000 -0.000 -0.000

Table 38: Ecliptic latitude of the moon. The latitude is minus the value shown in the table if the argument is in parenthesies.
 000/180 0.000 (180)/(360) 002/178 0.180 (182)/(358) 004/176 0.360 (184)/(356) 006/174 0.539 (186)/(354) 008/172 0.718 (188)/(352) 010/170 0.896 (190)/(350) 012/168 1.073 (192)/(348) 014/166 1.248 (194)/(346) 016/164 1.422 (196)/(344) 018/162 1.595 (198)/(342) 020/160 1.765 (200)/(340) 022/158 1.933 (202)/(338) 024/156 2.099 (204)/(336) 026/154 2.263 (206)/(334) 028/152 2.423 (208)/(332) 030/150 2.581 (210)/(330) 032/148 2.735 (212)/(328) 034/146 2.887 (214)/(326) 036/144 3.034 (216)/(324) 038/142 3.178 (218)/(322) 040/140 3.319 (220)/(320) 042/138 3.455 (222)/(318) 044/136 3.587 (224)/(316) 046/134 3.714 (226)/(314) 048/132 3.837 (228)/(312) 050/130 3.956 (230)/(310) 052/128 4.070 (232)/(308) 054/126 4.178 (234)/(306) 056/124 4.282 (236)/(304) 058/122 4.380 (238)/(302) 060/120 4.473 (240)/(300) 062/118 4.561 (242)/(298) 064/116 4.643 (244)/(296) 066/114 4.719 (246)/(294) 068/112 4.790 (248)/(292) 070/110 4.855 (250)/(290) 072/108 4.913 (252)/(288) 074/106 4.966 (254)/(286) 076/104 5.013 (256)/(284) 078/102 5.054 (258)/(282) 080/100 5.088 (260)/(280) 082/098 5.117 (262)/(278) 084/096 5.139 (264)/(276) 086/094 5.154 (266)/(274) 088/092 5.164 (268)/(272) 090/090 5.167 (270)/(270)

Table 39: Parallax of the moon. The arguments of and are and , respectively. and take minus the values shown in the table if their arguments are in parenthesies.
 Arg. () Arg. () 000/360 57.067 5.488 (180)/(180) 002/358 57.032 5.485 (178)/(182) 004/356 56.928 5.475 (176)/(184) 006/354 56.754 5.458 (174)/(186) 008/352 56.511 5.435 (172)/(188) 010/350 56.200 5.405 (170)/(190) 012/348 55.820 5.368 (168)/(192) 014/346 55.371 5.325 (166)/(194) 016/344 54.856 5.276 (164)/(196) 018/342 54.274 5.219 (162)/(198) 020/340 53.625 5.157 (160)/(200) 022/338 52.911 5.088 (158)/(202) 024/336 52.133 5.014 (156)/(204) 026/334 51.291 4.933 (154)/(206) 028/332 50.387 4.846 (152)/(208) 030/330 49.421 4.753 (150)/(210) 032/328 48.395 4.654 (148)/(212) 034/326 47.310 4.550 (146)/(214) 036/324 46.168 4.440 (144)/(216) 038/322 44.969 4.325 (142)/(218) 040/320 43.716 4.204 (140)/(220) 042/318 42.409 4.078 (138)/(222) 044/316 41.050 3.948 (136)/(224) 046/314 39.642 3.812 (134)/(226) 048/312 38.185 3.672 (132)/(228) 050/310 36.682 3.528 (130)/(230) 052/308 35.134 3.379 (128)/(232) 054/306 33.543 3.226 (126)/(234) 056/304 31.911 3.069 (124)/(236) 058/302 30.241 2.908 (122)/(238) 060/300 28.533 2.744 (120)/(240) 062/298 26.791 2.577 (118)/(242) 064/296 25.016 2.406 (116)/(244) 066/294 23.211 2.232 (114)/(246) 068/292 21.378 2.056 (112)/(248) 070/290 19.518 1.877 (110)/(250) 072/288 17.635 1.696 (108)/(252) 074/286 15.730 1.513 (106)/(254) 076/284 13.806 1.328 (104)/(256) 078/282 11.865 1.141 (102)/(258) 080/280 9.910 0.953 (100)/(260) 082/278 7.942 0.764 (098)/(262) 084/276 5.965 0.574 (096)/(264) 086/274 3.981 0.383 (094)/(266) 088/272 1.992 0.192 (092)/(268) 090/270 0.000 0.000 (090)/(270)

Next: Lunar-Solar Syzygies and Eclipses Up: The Moon Previous: Determination of Ecliptic Latitude
Richard Fitzpatrick 2010-07-21