The Earth-Moon system


  • 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 Earth-Moon distance.

    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 below.

  • 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.

    Kepler's law
  • With the mass and the radius of the Moon, we can compute the density.


    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 Earth:

    Q: Why don't we observe eclipses at each full Moon or new Moon?

    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.

    4. TIDES

  • 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) tides?

    Q: When in the month do we get small (neap) tides?

  • 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