The Earth-Moon system
1. PHYSICAL PROPERTIES OF THE MOON
DISTANCE: 356,000 - 407,000 km (1.2 light-sec)
DIAMETER: 3,476 km (0.27 Earth's diameter)
MASS: 7.35 x 10^22 kg (0.0123 Earth's mass)
DENSITY: 3.34 g/cm^3 (0.61 Earth's density)
Q: How do we measure these parameters?
Using Parallax (triangulation), Ptolemy observed the Moon
from two different spots on the Earth and determined that the distance
to the Moon is 27.3 Earth diameters. If the Earth's diameter is 13,000 km, then
the distance to the Moon is about 390,000 km.
The time delay of radar echos (2.4 light-sec) gives twice the
This image illustrates Ptolemy's method and the radar method
Angular size of the Moon is close to 0.5 degrees.
Small angle formula
yields a value of 3,480 km for Moon's diameter, as shown in the figure
Kepler's third law applied to satellites orbiting the Moon gives the lunar mass.
This law relates the mass of the central object to the period and the distance
of the orbiting satellite, as indicated below.
With the mass and the radius of the Moon, we can compute
2. LUNAR PHASES
The changing appearance of the Moon are caused by the relative positions of the Earth, Moon, and Sun.
The phases follow the sequence of new Moon, first quarter, full Moon,
and last quarter:
SYNCHRONOUS ROTATION means that the Moon's
rotation period is identical to its orbital period. Hence,
we always see the same face of the Moon. This figure shows that
the Moon rotates once each time it orbits the Earth. Notice that
at (A) the lunar peak is to the right, while at (B) it is
to the left. Thus from the Earth, we always see the same side
of the Moon even though it turns on its axis. This
phenomenon is caused by the tidal forces of the Earth on the Moon.
An ECLIPSE occurs when one astronomical object casts its shadow on another.
LUNAR ECLIPSE occurs when the Earth's shadow falls on the Moon
(during full Moon), as shown in this figure.
Q: Why doesn't the Moon get completely dark and shows a reddish color during a Total eclipse? The answer to this question is
illustrated in the next figure, which shows that some of the rays from
the Sun can be deflected toward the Moon as they cross the Earth's
atmosphere (this is known as optical refraction). Also, since the Earth's
atmosphere is more opaque to blue ligth, the light that is deflected
is also redder than sunlight.
SOLAR ECLIPSE occurs when the Moon's shadow falls on the Earth
(during new Moon):
This figure shows a sketch of how the Moon's shadow travels across the
Q: Why don't we observe eclipses at each full Moon or
Because of the Moon's orbital tilt, the Moon generally
is either above or below the Earth's orbit. Thus the Moon's shadow rarely
hits the Earth, and the Earth's shadow rarely hits the Moon.
This image shows that the Moon's orbit is tipped 5 degrees with respect
to the Earth's:
NODES are the two points in each orbit at which the Moon crosses the
Earth's orbital plane. For lunar or solar eclipses to occur the nodes
must be aligned with the Earth and the Sun. Hence, eclipses occur only
twice per year and these epochs are called eclipse seasons.
The regular change of the height of the ocean is
called the TIDES. They are caused by the gravitational pull of the Moon.
The gravitational attraction is stronger on the side
of the Earth near the Moon and weaker on the far side because the force
of gravity weakens with distance. The figure on the left shows the Moon's
gravitational force at different points on the Earth.
This difference is called DIFFERENTIAL GRAVITATIONAL FORCE.
This figure shows the tidal forces from the point
of view of an observer on the Earth.
These arrows represent the difference between he Moon's
gravitational force at a given point and its force at the Earth's center.
The differential gravity causes a TIDAL BULGE on the side
of the Earth facing the Moon, and an identical bulge on the far side,
as depicted in this sketch.
The daily rotation of the Earth carries an observer into
high water twice a day, creating two HIGH TIDES. Between the times of
high water the water level drops, making two LOW TIDES each day.
The SOLAR TIDES are only 1/3 of the Moon's tides.
When the Sun-Earth-Moon are in line we see the
combined effect of the two tidal forces or TIDAL COOPERATION,
leading to LARGE TIDES.
When the Sun and Moon are at 90 degrees, the
lunar and solar bulges are at right angles and partially nullify
each other, creating SMALL TIDES:
Q: When in the month do we get large (spring)
Q: When in the month do we get small (neap)
TIDAL BRAKING slows
the Earth's rotation and speeds up the Moon's motion in its orbit.
This figure shows how the Moon tidally brakes the Earth. As the
Earth spins, friction between the ocean and the solid Earth below drags
the tidal bulge ahead of the line joining the Earth and the Moon.
The Moon's gravity pulls on the bulge and holds it back. The resulting drag
is transmitted trough the ocean to the Earth, slowing its rotation
the way your hand placed on a spinning bicycle wheel slows the wheel.
Tidal braking lengthens the day by 0.002 seconds every century.
As the Earth's rotation slows, the Moon accelerates in its orbit, moving
farther from the Earth (Kepler's third law). This acceleration makes
the Moon move away from Earth at a rate of 3 cm per year. Thus, the
Moon was once much closer to the Earth, and the Earth spun much faster.
Tidal braking is also the reason the Moon always keeps
the same face to the Earth. Just as the Moon raises tides,
which slow the Earth, the Earth raises tides on the Moon.
The lunar tides distort the Moon's crust and have braked
the Moon severely, locking it into SYNCHRONOUS ROTATION.
Image source: "An Introduction to Astronomy" by Thomas T. Arny