The apparent size of the moon is about half a degree. The size in arcsec is just 206265 times diameter/distance so varies between 206265x3476/(363300-6370)=2009" and 1796" or 29.9' and 33.5'.

The moon takes 27^d7^h43^m11^s.5 to orbit the earth. This is known as the sidereal month. Sidereal, because the motion is measured relative to the stars. It takes longer than this for the moon to go through all its phases because the earth - moon system is slowly revolving around the sun. For the moon to complete a revolution with respect to the sun it takes, on average, 29^d12^h44^m2^s.8. This is known as a synodic month.

The synodic lunar month or lunation starts with a New Moon or "no moon" when the moon passes between the earth and sun and is very difficult to observe. After a few days the moon has moved a few times 13º east of the sun so a bit of the western limb of the moon becomes visible as a waxing crescent. After a week we can see the western half of the moon which is a quarter of the total, first quarter. Then the moon waxes gibbous until after half a month the moon lies opposite to the sun in the sky, opposition or full moon. For a few days thereafter the moon is waning gibbous and the western features disappear in shadow. Eventually the moon reaches last quarter phase where only the eastern half of the visible face of the moon is illuminated by sunlight, then waning crescent and back to new moon. The cycle repeats without much variation because the revolution rate of the moon is on average the same as the (very constant) rotation rate so we always see the same visible face or hemisphere of the moon, a situation called (one-to-one) synchronous rotation.

When the moon appears only as a faint crescent, the "night" side of the moon can just about be seen. This is because of earthshine, light getting to the unlit side of the moon via reflection of the earth.

Even though a full moon seems bright in the night sky, if the entire sky was as bright as a full moon, it would only be about one fifth as bright as a sunny day.

The brightness of astronomical objects is defined by their magnitude.

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Obvious features are the lunar maria and you should be able to point out Mare (from west to east) Crisium, Tranquillitatis, Serenitatis, Frigoris, Imbrium and Nubium.

With binoculars, a number of craters become apparent. Best seen around full moon in the eastern quarter are Plato and the rayed craters Tycho, Copernicus, Kepler and the very bright Aristarchus. On the eastern limb you will find the large very dark crater Grimaldi. With a small telescope such features as the straight wall, a 250 metre high cliff (fault?) and the Alpine valley, a gouge in the Alps (south of Mare Frigoris) become visible. Craters and relief features are best seen near the terminator, the division between day and night on the moon.

Where the moon crosses the ecliptic moving northward, is called the ascending node and where it crosses it moving southward is called the descending node. Perturbations to the moons orbit ,mainly from the sun, cause the position of the nodes to shift slightly or regress every year. The nodes will return to their original positions every 18.6 years.

These facts were well known to ancient astronomers since they are intimately related to the art of eclipse predictions: lunar and solar eclipses occur only when the full or new moon occurs within about 15° of a node (ie the Moon within a degree of the ecliptic). Hence, the name ecliptic.

If you watch the moon night after night, you will notice that it moves eastwards with respect to the stars. The sun moves 1° east every day. Therefore, the moon moves 12° eastward with respect to the sun. This 12° delay equates to about 50 minutes per day on average. The moon’s orbit is an ellipse so it’s delay is not constant and varies between 38 and 66 minutes. At latitudes other than the equator, the moon rises obliquely to the horizon so that moonrise is dependant on latitude. At f =30° moonrise can vary between a few minutes to over an hour.

Harvest Moon

This occurs when the moon is at the ascending node. [This means ¡ for southern hemisphere dwellers.]

Configurations

As for the planets:

• Quadrature
• Opposition
• Conjunction

Rotation

The apparent rotation ( i.e. with respect to the earth)

Above you see the moon position each 8^th of its period. The moon moves faster when it is near the earth (Kepler's equal area law) and so revolves faster at perigee. But the rotation rate is constant, giving rise to libration in longitude.

The true rotation (i.e. with respect to the stars)

The moon is not really in orbit around the earth like the moons of other planets. The moon's obit is always concave towards the sun. So the earth, moon and sun can not be considered a restricted three body system.

Distance

The distance to the moon was first estimated by Aristarchus then later by Hipparchus who used the projection of the earth's shadow on the moon. Then Tycho surveyed the distance to the moon by allowing the Earth's rotation to generate a baseline thus yielding the moon's parallax. In 1946 the distance was determined by radar and we now routinely measure the distance to WITHIN cm's by laser rangeing.

Moon distance = 30 x Earth’s diameter

Angular diameter ~1/2° (the same as the sun) q in the diagram

tan q /2 = r/d (not exactly) PICTURE

sin q /2 = r/d (better)

d~384404km 2r~3475km

Three body problem.

The sun dominates the system

The orbit with respect to the earth is an ellipse, but not a very "good" one.

• A month would be almost an hour shorter if the sun were eliminated.
• The line of apsides advances (perigee - The earth-moon line....ie, major axis)
• The line of nodes regresses

When planetary perturbations are included things become very messy.

M_earth r_earth = M_moon r_moon

Therefore M_moon = 7.35 x 10¹⁹ Tonnes

(The most accurate measurements come from satellites).

Tides and the Evolution of the Earth - Moon System

Rotation of the earth causes the tides to lead. The leading tides accelerate the moon and slow the earth’s rotation. The moon slowly spirals out.

The day and month lengthen until both reach 47 days (present days) long. Solar tides slow the earth still more so the a day > a month. Now the rotation causes the tides to lag, dragging the moon towards the earth!!!

The tides dissipate 2.7x10¹²W of energy.

About 2.5 x 10⁹ years ago the day was only ~ 6 hours long and the synodic month was 17 days (now it is 29.5). 45x10⁶ years ago the synodic month was 29.1 days. (So how far from the earth was the moon 2.5 aeons ago?)

Knowledge of the length of the day/month comes from marine fossil records.

Shadows and Eclipses

a) Shadow from a point source

PICTURE Apart from diffraction effects, a shadow cast by a point source has sharp boundaries.

e.g. the occultation of a star by the limb of the moon. [The shape of the diffraction pattern gives information on the diameter of the star!]

b) Shadow from an Extended Source

• Umbra
• Penumbra

c) Eclipse Seasons

The moon’s orbit is inclined 5° to the ecliptic. Thus, the moon must be new or full just when the moon crosses the ecliptic. To help understand, suppose the inclination were large, like 90° . Then eclipses would be very unlikely indeed. PICTURE

PICTURE

Now the moon is large and the inclination only 5° , so there can be a little slop. The limits are called ecliptic limits. Since the earth and moon orbits are a bit eccentric, the limits can vary by quite a large amount, on the order of 11° ....

Total Major solar ecliptic limit 18° 31’
Total Minor solar ecliptic limit 15° 21’
Central Major solar ecliptic limit 11° 50’
Central Minor solar ecliptic limit 9° 55’
Central Major lunar ecliptic limit 12° 15’
Central Minor lunar ecliptic limit 9° 30’
Solar Eclipses

During a synodic month the sun moves about 29° . Hence, as it moves through a node an eclipse must occur. Therefore, 2 eclipses per year must occur. If the sun is eclipsed at the beginning of the season, a second will follow a month later.

Changing Seasons

Unfortunately, the nodes regress (because of perturbations mainly due to the sun) with a period of 18.6 years. So the seasons change by 12/18.6 of a month (~20 days) each year. The seasons in 1980 were February and August and in 1986, December and June. ???

Eclipses of the Sun

a) Geometry

The lunar umbra just reaches the earth. [Both the moon and the sun have the same angular size! Amazing!] So the total eclipse can only be observed along a narrow eclipse path. Totality width ~ 250 km, Duration < 7.5 minutes.

b) Appearance: Indescribable.

c) Value

The sun’s Corona, chromosphere and extreme limb can be studied as can the effect of the sun on the earth’s atmosphere and ionosphere. Also general relativity can be tested.

d) Annular Eclipse

When the moon is further away than usual, it does not appear large enough to cover the entire solar disk. During an eclipse a ring of sunlight shows around the silhouette of the moon.

e) Partial Eclipse

This occurs when the moon’s umbra does not hit the earth, only its penumbra. The moon seems to "skim" across the southern or northern part of the sun.

Lunar Eclipses

a) Geometry

The big difference is that lunar eclipses can be seen by anyone who can see the moon during the eclipse. Solar eclipses can be seen only in the eclipse path.

b) Appearance

Everybody has seen one.

Recurrences of Eclipses

The rapid changes in the lunar orbit make two additional kinds of months useful:

• Nodical (Draconic) Month 27.2122220 days Node to node

• Anomalistic Month 27.55455 days Perigee to perigee

a) For an eclipse to be followed after a period of time by a similar eclipse, three conditions must be met:

• The moon must be at the same phase
• The moon must be at the same nodal longitude
• The moon and the sun must be the same distance from the earth

The first two conditions can be met if an integral number of synodic and draconic months have elapsed. This is similar to a "common denominator" problem.
The lowest common denominator is:

47	synodic months	=1387.94 days
51	nodical months	=1387.82 days
50	anomalistic months	=1377.73 days

It depends on how accurate you want the months all to be integral. The longer the repeat the better. For example,

223	synodic months	=6585.32 days
242	nodical months	=6585.36 days
239	anomalistic months	=6585.54 days

b) The Saros: The 18 year repeat above was noticed thousands of years ago by ancient astronomers. The error amounts to about 1/2° per eclipse, so a single saros lasts about 1200 years.

Other Eclipse Phenomena

a) Occultations of stars by the moon.

Grazing occultations.

Occultations of stars by planets.

b) Transits of the sun by Mercury and Venus.

c) Transits and eclipses of moons and of other planets

The satellites of Jupiter are easy to observe. This link gives a timetable of the events associated with the Jovian satellites.

d) Eclipsing binary stars.

And of course the earth eclipses the sun once a day here! You call it sunset.

Just at sunset you can often see the earth’s shadow projected in the atmosphere!