The Earth and Moon revolve in orbit around their common barycenter, which is located about
1700km below the surface of the Earth. The center of the Moon is 80 times farther from the
barycenter than the center of the Earth because the Earth is 80 times more massive than the
Moon.

To a good approximation, we can say the Moon revolves in orbit around the Earth. The sense
of this revolution is counterclockwise when viewed from the North Ecliptic Pole.

The Moon's orbit is actually an ellipse and Kepler's Laws of Motion apply. The point in its orbit
where the Moon is closest to the Earth is called perigee and the point where it is farthest from
Earth is the called the apogee point. The mean distance of the Moon from the Earth is about
374,000km.

The Moon's orbit lies in a plane that is inclined to the plane of the ecliptic by 5^o. So sometimes
the Moon is north of the ecliptic and sometimes it is south of the ecliptic, but never by more than
5^o. When the Moon's orbit is plotted on the celestial sphere, it is seen as a circle that intersects
the ecliptic at 2 points that are 180^o apart. These 2 points are called the Lunar Orbital Nodes (LON).
The point where the Moon crosses the ecliptic as it moves from south to north of the ecliptic is
called the ascending node and the point where the Moon is observed to cross the ecliptic from
north to south is called the descending node.

The Moon's orbital motion, or revolution, appears to an observer on the Earth as a west to east
(eastward) motion of the Moon on the celestial sphere with respect to the fixed stars. Since the
Moon's orbit is inclined only 5^o from the plane of the ecliptic, the Moon always appears near the
ecliptic in the sky.

Sidereal Month or Period: Time interval between two successive passages (conjunctions) of the
Moon past a fixed star. This is the true period of revolution (time to revolve 360 degrees) of the
Moon around the Earth (barycenter) and takes approximately 27 1/3 days.

Since the moon revolves 360⁰ in 27.33 days, the Moon appears to move 13.2 degrees per day
from west to east with respect to the fixed stars. Since the Sun also appears to move eastward,
but by 1⁰ per day relative to the fixed stars, the Moon is seen to shift eastward from the Sun's
position on the celestial sphere by about 12.2 degrees per day. This causes the moon to
rise/make UT/set about 50 minutes later each successive day (12.2⁰ per day divided by the
angular rate of rotation of the Earth of 0.25 deg./min).

Example; Suppose the Moon sets tonight at 21:30. Approximately what time will the Moon
set one week from today?

Solution:   In 7 days, the Moon will set later by 7 days x 50 min/day = 350 minutes later.
                Divide this by 60 min/hr to get 5.833 hrs. later, which is 5hrs and 50 minutes
                (or 5:50) later.   Add this to 21:30 and we get 26:80 or 27:20.

Since this answer exceeded 24 hours, just subtract 24:00 from 27:20 and we get 3:20. That is,
the Moon will set next week at 3:20 on the day of the week following today's day.

This is defined as the time interval between successive conjunctions of the Moon with the
Sun. This takes longer than the sidereal period, or about 29 1/2 days, since the Sun is a moving
reference point. It is also the time for the Moon to go through a complete cycle of its phases,
since the phase of the Moon depends on its angular distance (elongation) from the Sun.

The way to think about the time equivalent of the planet's elongation, T_E, is that it is the amount of
time the object is ahead or behind the Sun for doing some event.

For example, if Saturn has an elongation of 30^o west, it rises 30^o/15 = 2:00 hours before the Sun
or sets 2:00 hours before the Sun does.

Refer to Exercise 13.0 in the Course Manual and read over the introduction thoroughly. Also

memorize the table of elongations and phases given in Exercise 13.0.

EXAMPLE:

If the Moon has an elongation of 140 degrees West, what time will the Moon rise?

First find the value of T_E. To do this, divide the elongation by the rate of rotation of the

Earth, viz., 15 deg/hr. We get: 140/15 = 9 and 5/15 hours. A fifteenth of an hour is 4

minutes (60/15). Therefore, the elongation of the Moon is 9^h20^m W. We assume the Sun

always rises at 6:00 and sets at 18:00 for simplicity. So in this problem, the corresponding

solar event to Moonrise is Sunrise or 6:00. Then we get:

Moonrise = 6:00 - (+9:20) = -3:20.

Now there can not be such a thing as a negative time. The negative sign means we went

back past midnight into the previous day. Therefore, add 24 hours to this and we get 20:40.

So the most recent Moonrise was at 20:40 on the previous day.

In the case where the elongation is East, a negative sign must be prefixed to the elongation

when substituting this value in the above equation. In that case, minus a minus elongation

results in adding the elongation time to the time for the corresponding solar event.

Practice doing problems like the one above. Just change the elongation and or change the
event asked and follow the above procedure, step by step, to calculate the answer.

We ended here 10-06-2011

G. LUNAR PHASES

For a given elongation, the Moon has a definite phase. These are listed in a table found in
Exercise 13.0 of the course manual. This table must be memorized,

We now show how to compute the amount of time it takes for the Moon to go from one phase and
date to another by means of a sample problem:

Given that the elongation of the Moon is 140⁰ W on April 5 at 17:00 (LMT), when is the next new
moon?

1. First convert 17:00 to a decimal part of a day by dividing 17 by 24 = 0.71d.

2. Then add this to April 5 to get April 5.71.

3.   Since the Moon's elongation is 140^o west of the Sun, the angular distance the Moon must
      move in its orbit to arrive at conjunction with the Sun is 140^o, (If the elongation were east
     instead of west, there would be an additional 80^o to be added to this.) The amount of time
     this will take is found by dividing 140^o by the rate at which the Moon moves relative to the
     Sun, that is, 12.2⁰ per day.   The result is 11.48 days.

4. Therefore, new moon will occur on April 5.71 + 11.48 d = April 17.19.

5. Now convert 0.19d to hours by multiplying by 24 hours per day. This yields 4.56h. Hence,
April 17.19 is April 17 at 4.56h. This is not the same as 4:56.

6. Now convert 0.56h to minutes by multiplying by 60 mins./hour. The result is 33.6m or 34m ,
rounding to the nearest whole minute.

7. So new moon occurs on April 17 at 4:34. This is local apparent solar time, LAT.

H. ROTATION OF THE MOON

Billions of years ago, both the Earth and the Moon rotated much faster than they do today.
However, the Earth and the Moon exert gravitational tidal forces on one another which has slowed
down the rotations. The tides exerted by the Earth in the solid crust of the Moon are much greater
than the tides that the Moon exerts on the Earth. Consequently, the Moon has slowed down faster
than the Earth has. The result is now the Moon's period of rotation has become synchronized with
its sidereal period of revolution around the barycenter of the Earth-Moon system.

What this means is that the Moon keeps the same side facing towards the Earth. Therefore no one
knew what the other side of the Moon looked like until 1961 when a Soviet Union spacecraft circled
the Moon and took a photograph of the other side. Since then, the back side of the Moon 's surface
has been well surveyed by US spacecraft.

Billions of years from now the Earth will be slowed to the point where its rotation will also be
synchronized with the 27.3 day revolution. Hence it will keep one side facing towards the Moon.
This would mean that the Moon would no longer be seen to rise or set but would always be visible
at the same place in the sky. Where it would be in the sky would depend on your location on the
Earth. If you lived on the side of the Earth facing away from the Moon, it would never be seen.

I. ECLIPSES

Ex. 14.0 in the course manual presents the material to be known about eclipses. Make
sure that you read this exercise over thoroughly and be familiar with the following:

Types of Eclipses (total and partial) and when they occur.
Umbra and penumbra parts of shadows.

Inclination of the plane of Moon' s orbit to the plane of the ecliptic is 5 degrees.
The lunar orbital nodes. There are 2:

The ascending node is the point where the Moon crosses the ecliptic as moves
from south of the ecliptic to north of the ecliptic.

The descending node is the point where the Moon crosses the ecliptic as moves
from north of the ecliptic to south of the ecliptic.

    Eclipse Limits: The angular distance from a LON within which an eclipse can occur.
        The values are different for a solar eclipse or for a lunar eclipse. See the diagrams
        below.

    The eclipse window is the total angular distance from the western eclipse limtit to the
           eastern eclipse limit. See the diagrams below. The eclipse window for a solar
            eclipse is 36 degs. wide whereas the window is 24 degs. wide for a lunar eclipse.

Necessary condition for an eclipse to occur: The Sun must be within an eclipse limit
at the time of either lunar conjunction or opposition.

Eclipse Seasons. These are the 2 periods of time during the year when the Sun is
sufficiently near a LON for an eclipse to occur.

When the Sun is at the western edge of an eclipse window, this is the beginning of an
eclipse season.

When the Sun is at a lunar orbital node, this is the middle of an eclipse season.

When the Sun is at the eastern edge of an eclipse window, this is the end of an
eclipse season.

The eclipse seasons are approximately six months apart and the dates of their occurence
change from year to year because of the regression of the LON.

    Duration of an eclipse depends one the distance of the Sun or Earth's shadow from a
        LON. The farther either are from a node, the shorter the eclipse duration and the
        shallower the eclipse depth, since the Moon's trajectroy throu See the diagrams below.

Regression of the lunar orbital nodes causes the eclipse seasons to come earlier by
2.7 weeks each successive year.

The diagram below shows several lunar eclipse scenarios. The eclipse window for
a lunar eclipse is 24 degrees wide as shown.

The diagram below shows various scenarios of a solar eclipse.

Annuolar Eclipses
If the Earth is near perihelion while the Moon is at apogee, the Moon appears to be smaller
in size than the Sun. If this happens at the time of a solar eclipse, then at conjunction, when the
Moon is centered on the apparent disk of the Sun, there remains an annulus of the Sun's
photosphere that is still visible. So, instead of a total eclipse, ther is what is called an annular
eclipse of the Sun.

In a question that asks to find when the next eclipse will occur when you are given the
elongation of the Moon on a particular day at a specific time, proceed as in the example in
section G of this document.. Given the elongation of the Moon, first decide whether an
opposition or a conjunction will be the next aspect of the Moon to occur. Then determine the
angle the Moon must revolve in orbit to arrive at this aspect.