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A model of the moon's ecliptic latitude is needed in order to predict the
occurrence of solar and lunar eclipses. Figure 23 shows the
moon's orbit about the earth. The plane of this orbit is fixed, but slightly tilted with
respect to the plane of the ecliptic (i.e., the plane of the sun's
apparent orbit about the earth). Let the two planes intersect along the
line of nodes, . Here, is the point at which the orbit crosses the
ecliptic plane from south to north (in the direction of the moon's orbital
motion), and is termed the ascending node. Likewise, is the
point at which the orbit crosses the ecliptic plane from north to south,
and is called the descending node. Incidentally, the line of nodes must pass through point , since the earth is common to the ecliptic plane and the plane of the lunar orbit.
The angle, , subtended between
the radius vector , connecting the earth to the vernal equinox,
and the line , is known as the longitude of the ascending node. Note, incidentally, that
the ascending node precessess in the opposite direction to the moon's orbital motion
at the rate
per day, or in 18.6 years.
This unusually large precession rate is another consequence of the sun's strong perturbing influence on the
moon's orbit.
Let the line lie in the plane
of the moon's orbit such that it is perpendicular to .
The inclination, , of the moon's orbital plane is the angle
that subtends with its projection onto the ecliptic plane. Likewise, the
moon's ecliptic longitude, , is the angle that subtends with its
projection onto the ecliptic plane.
Simple geometry
yields
, where is the angle
between and . This angle is termed the argument of latitude.
Now, it is easily seen that
, where
is the moon's ecliptic longitude (i.e., the angle subtended
between and ). Here, we are assuming that the orbital
inclination is relatively
small. The mean argument of latitude is defined
.
Hence, our model for the moon's ecliptic latitude becomes
The value of the lunar orbital inclination, , for the
J2000 epoch is specified in Table 35. The above model is capable of
matching NASA ephemeris data during the years 19952006 CE with
a mean error of , and a maximum error of .
The ecliptic latitude of the moon can be calculated with the aid of Table 38. The procedure for using this table is as follows:
 Determine the fractional Julian day number, , corresponding to the date and time
at which the moon's ecliptic latitude is to be calculated with the aid of Tables 2729. Form
, where
is the epoch.
 Calculate the lunar mean argument of latitude, ,
and the five lunar anomalies, , using the procedure outlined earlier in this
section.
 Form the argument
. Add as many multiples of to as is required to make it fall in the range
to . Round to the nearest degree.
 Enter Table 38 with the value of
and take out the lunar ecliptic latitude, . It is necessary to
interpolate if is odd.
For example, we have already seen that at 00:00 UT on May 5, 2005 CE
the lunar mean argument of latitude, and
the lunar anomalies, were
, and
,
,
,
,
and
, respectively. Hence,
. Thus, according to Table 38, the ecliptic latitude
of the moon at 00:00 UT on May 5, 2005 CE was
.
Figure 24:
The moon, , as viewed by a hypothetical observer, ,
at the center of the earth, and a real observer, , on the
surface of the earth.

Next: Lunar Parallax
Up: The Moon
Previous: Example Longitude Calculations
Richard Fitzpatrick
20100721