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 $a'$ for the real observer, and $a$ for the hypothetical observer. Simple trigonometry reveals that $a' = a-\delta a$, which implies that the real observer sees the moon at a lower altitude than the hypothetical observer. Let $R$ be the radius of the earth, and $r$ the distance from the center of the earth to the moon. More simple trigonometry yields

$$\sin \delta a = \frac{R}{r}\,\cos a'.$$ (126)

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

$$r \simeq a_M\,(1 - e\,\cos M),$$ (127)

where $a_M$, $e$, and $M$ are major radius, eccentricity, and mean anomaly of the lunar orbit. Assuming that $\delta a$ is small, we obtain

$$\delta a \simeq \delta a_0\,\cos a\,(1+e\,\cos M),$$ (128)

where $\delta a_0 = R/a_M = 0.0166=56.98'$ (since $R=6371$ km and $a_M=384,399$ km).

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

$$\delta a = \delta (a)\,[1+\zeta (M)],$$ (129)

where $a$, $a-\delta a$, and $M$ 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 $\delta(a)=\delta a_0\,\cos a$ and $\zeta (M)= e\,\cos M$ 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 $57'$ when the moon is close to the horizon. For example, if $a=44^\circ 00'$ and $M=100^\circ$ then Table 39 yields $\delta = 41.050'$ and $\zeta =- 0.00953$. Hence, $\delta a = 41.050\,(1-0.00953) \simeq 41'$, and the true altitude of the moon becomes $43^\circ 19'$.

Figure 25: Parallactic shifts in the moon's ecliptic longitude and latitude.

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

$$\delta\lambda = - \delta a\,\cos\mu,$$ (130)

$$\delta\beta = - \delta a\,\sin\mu.$$ (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 $\lambda=210^\circ$ (i.e., 00SC00) and $M=90^\circ$, respectively. Suppose that the moon is viewed from an observation site located at terrestrial latitude $+10^\circ$. The ``Scorpio'' entry in Table 19 gives the moon's geocentric altitude, $a$, as a function of time, as well as the value of the parallactic angle $\mu$. 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, $t=+1$ 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 $2^\circ$ 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 $6^\circ$ 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.

 

$t$ (hrs.)	$a$	$\mu$	$\delta a$	$\delta\lambda$	$\delta\beta$
$-5.51$	$00^\circ 00'$	$190^\circ 22'$	$57'$	$+56'$	$+10'$
$-5.00$	$12^\circ 26'$	$187^\circ 30'$	$56'$	$+55'$	$+07'$
$-4.00$	$26^\circ 37'$	$183^\circ 07'$	$51'$	$+51'$	$+03'$
$-3.00$	$40^\circ 23'$	$176^\circ 40'$	$43'$	$+43'$	$-03'$
$-2.00$	$53^\circ 15'$	$165^\circ 58'$	$34'$	$+33'$	$-08'$
$-1.00$	$63^\circ 52'$	$145^\circ 55'$	$25'$	$+21'$	$-14'$
$+0.00$	$68^\circ 32'$	$110^\circ 34'$	$21'$	$+07'$	$-20'$
$+1.00$	$63^\circ 52'$	$075^\circ 13'$	$25'$	$-06'$	$-24'$
$+2.00$	$53^\circ 15'$	$055^\circ 11'$	$34'$	$-20'$	$-28'$
$+3.00$	$40^\circ 23'$	$044^\circ 29'$	$43'$	$-31'$	$-30'$
$+4.00$	$26^\circ 37'$	$038^\circ01'$	$51'$	$-40'$	$-32'$
$+5.00$	$12^\circ 26'$	$033^\circ 39'$	$56'$	$- 46'$	$-31'$
$+5.51$	$00^\circ 00'$	$030^\circ 47'$	$57'$	$-49'$	$-29'$

 
 
 

Table 35: Orbital elements of the moon for the J2000 epoch (i.e., 12:00 UT, January 1, 2000 CE, which corresponds to $t_0= 2\,451\,545.0$ JD).

$e$	$n(^\circ/{\rm day})$	$\tilde{n}(^\circ/{\rm day})$	$\breve{n}(^\circ/{\rm day})$	$\bar{\lambda}_0(^\circ)$	$M_0(^\circ)$	$F_0(^\circ)$	$i(^\circ)$
0.054881	13.17639646	13.06499295	$13.22935027$	218.322	134.916	93.284	5.161

Table 36: Mean motion of the moon. 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), $\bar{\lambda}_0 = 218.322^\circ$, $M_0 = 134.916^\circ$, and $\bar{F}_0 = 93.284^\circ$.

$\Delta t$(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	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 $M$, $2\tilde{D}-M$, $\tilde{D}$, $M_S$, and $2\bar{F}$ for the case of $q_1$, $q_2$, $q_3$, $q_4$, and $q_5$, respectively. If the argument is in parenthesies then the anomalies are minus the values shown in the table.

Arg. ($^\circ$)	$q_1(^\circ)$	$q_2(^\circ)$	$q_3(^\circ)$	$q_4(^\circ)$	$q_5(^\circ)$	Arg. ($^\circ$)	$q_1(^\circ)$	$q_2(^\circ)$	$q_3(^\circ)$	$q_4(^\circ)$	$q_5(^\circ)$
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.

$F (^\circ)$	$\beta(^\circ)$	$F (^\circ)$
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 $\delta$ and $\zeta$ are $a$ and $M$, respectively. $\delta$ and $\zeta$ take minus the values shown in the table if their arguments are in parenthesies.

Arg. ($^\circ$)	$\delta(')$	$100\,\zeta$	Arg. ($^\circ$)
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)