MOVEMENTS OF THE MOON AND THE EARTH: ROTATION AND TRANSLATION - SCIENCE - 2025

The movements of the Earth and the Moon have their origin, on the one hand, in the gravitational attraction of each one exerts on the other and, on the other, in the attraction that the Sun in turn exerts on all the bodies of the Solar System.

Both the Earth and the Moon have rotational movements around their own axis and of translation, these being the most important. But they also experience other secondary movements of balances and oscillations, because they are not point objects, but rather have appreciable dimensions and are not perfectly spherical.

Figure 1. Orbital planes of the Earth and the Moon, with the inclinations of the respective axes and the distance between the centers and the center of gravity. Source: Wikimedia Commons. Earth-image from NASA; arrangement by brews_ohare

Earth and Moon are considered as an isolated system of objects of measurable size, revolving around their center of mass, located on the line that joins the respective centers.

This point is closer to the Earth than to the Moon, having the first greater mass. It is located approximately 4641 km from the center of the Earth and is called the center of gravity.

Laws

The movements of the Moon are governed by the laws of Cassini, enunciated in 1693 by Giovanni Cassini (1625-1712):

-The Moon has a synchronous rotation with the Earth, since it has the same period of rotation and translation, in this way it always shows the same face to terrestrial observers.

-The inclination of the lunar equatorial plane and the ecliptic is constant.

-The lunar axis of rotation, the normal to the ecliptic -the Earth's orbital plane- and the normal to the Moon's orbital plane are coplanar.

Rotation and translation of the Moon

The moon makes a rotational movement around its own axis in approximately 27.32 days. This period of rotation is called the sidereal month. According to Cassini's first law, this is also the time it takes for the Moon to orbit Earth.

Figure 2. Animation showing the movement of the Earth-Moon system around the center of gravity. Source: Wikimedia Commons

Synchronous rotation is responsible for Earth observers always seeing the same side of the Moon.

For its part, the synodic month is the time that elapses between two identical and successive lunar phases.

The synodic month lasts 29.53 days and it is because the Earth is not still while the Moon orbits around it. For the relative Earth-Sun-Moon positions to be the same again, the Earth must advance 27º in its translational motion around the Sun.

The Moon also travels around the Earth following an elliptical orbit with a very small eccentricity. The eccentricity of an ellipse is a measure of its flattening. This small value means that the trajectory of the Moon is almost circular, which it travels at a rate of 1 km / s.

The orbits of the Earth and the Moon intersect at points called nodes, which make eclipses possible, since seen from Earth the apparent sizes of the Sun and the Moon are very similar.

Other movements of the Moon

Due to the elliptical orbit that the Moon follows around the Earth and its rotation axis is inclined 6.60º with respect to the perpendicular of the orbital plane (see figure 1), there are movements called librations. Thanks to them we can see a small percentage of the far side of the Moon, approximately 9%.

The most notable hovers are in longitude and latitude. The librations in longitude are due to the fact that, being the elliptical orbit, the translation speed is greater at perigee - closer to the Earth - than at the apogee - further from the Earth.

In this way, a small portion of the surface near the edge meridian becomes visible, the one that is just to the east of said meridian when the Moon is between perigee and apogee.

Likewise, the portion of the surface that is slightly to the west becomes visible when the Moon is between apogee and perigee.

On the other hand, the libration in latitude arises because of the inclination of the axis of rotation. Thus, some parts of the Moon that are a little north or a little south, can be seen from the Earth, depending on the moment. The period of this hover is one draconic month, lasting approximately 27 days and 5 hours.

The following animation shows simulated views of the Moon for one month:

Rotation of the Earth

The Earth executes a rotation around the Earth's axis in a west-east direction, whose period is 1 day or more precisely 23 hours, 56 minutes and 3.5 seconds.

The speed of rotation of the Earth is 1600 km / h at the equator, and decreases until it vanishes right at the poles, where the axis of rotation passes, which is inclined 23.44º with respect to the orbital plane of the Earth, known as the ecliptic (see figure 1).

This movement is responsible for the appearance of day and night, which governs people's lives. For approximately 12 hours (it varies according to latitude and season), one hemisphere of the Earth receives sunlight and has a higher temperature, while the other is in darkness and the temperature drops.

Earth's rotation is due to the way the Earth formed. The cloud of matter that gave rise to the solar system had to rotate to compact the matter. But the rotation has an associated angular momentum, which in the absence of external torques is conserved.

The Sun, the planets and other members of the solar system, considered as an isolated system, have that angular momentum, distributed among their members.

That is why each planet has its own rotational movement from west to east, except for Venus and Uranus, which do the opposite, perhaps due to a collision with another large body.

Translation of the Earth

The Earth also makes a movement of revolution around the Sun, the period of which is just over 1 year. The origin of this movement is in the gravitational attraction that the Sun exerts.

In this movement the Earth describes an elliptical orbit, always in accordance with Kepler's laws of planetary motion. For an observer located at the North Pole, this movement is done in a counterclockwise direction.

As with the Moon, the eccentricity of the ellipse that describes the Earth is quite small. Then the orbit of the Earth approaches a circumference of radius 150 x 10 ⁶ km. This value is used in astronomy to define a unit of distance called Astronomical Unit or AU, widely used to express distances within the Solar System.

Figure 3. Movement of the Earth's translation around the Sun. Source: Wikimedia Commons.

The total length of this ellipse is 930 million kilometers, which the Earth travels at a speed of 30 km / s.

The axis of rotation of the Earth is inclined 23.44º with respect to the normal to the ecliptic. For this reason, as our planet travels its orbit around the Sun, one of its hemispheres is more exposed to the solar rays, producing summer, while in the other the exposure is less and winter arises.

Other movements of the Earth

The Earth is not a perfect sphere, but an ellipsoid flattened at the poles. So the equatorial bulge causes a slow rolling motion on Earth called precession.

In this movement, the earth's axis rotates around the pole of the ecliptic, tracing an imaginary cone, as can be seen in the following figure:

Figure. Precession and nutation movements of the Earth. Source: Wikimedia Commons.

It takes Earth 25,767 years to trace this cone. Superimposed on the precession is another reciprocating movement of the axis, called nutation, caused by the gravitational pull of the Moon on the equatorial bulge and has a period of 18.6 years.

References

1. Oster, L. (1984). Modern Astronomy. Editorial Reverte. 37-52.
2. Tipler, P. Physics for Science and Engineering. Volume 1. 5th. Edition. 314-316
3. Why is Earth rotating? Retrieved from: spaceplace.nasa.gov.
4. Wikipedia. Barycenter. Recovered from: es.wikipedia.org.
5. Wikipedia. Movements of the Earth. Recovered from: es.wikipedia.org.