The Orbit of the Moon

The Earth's satellite, the Moon, circles the Earth counterclockwise. One revolution, the sidereal month, takes about 27.322 days. In practise, a more important period is the synodic month, the duration of the Lunar phases (e. g. from full moon to full moon). In the course of one sidereal month the Earth has travelled almost 1/12 of its orbit around the Sun. The Moon still has about 1 /12 of its orbit to go before the Earth-Moon-Sun configuration is again the same. This takes about 2 days, so the phases of the Moon are repeated every 29 days. More exactly, the length of the synodic month is 29.531 days.

The new moon is that instant when the Moon is in conjunction with the Sun. Almanacs define the phases of the Moon in terms of ecliptic longitudes; the longitudes of the new moon and the Sun are equal. Usually the new moon is slightly north or south of the Sun because the lunar orbit is tilted 5° with respect to the ecliptic.

About 2 days after the new moon, the waxing crescent moon can be seen in the western evening sky. About 1 week after the new moon, the first quarter follows, when the longitudes of the Moon and the Sun differ by 90°. The right half of the Moon is seen lit (left half when seen from the Southern hemisphere). The full moon appears a fortnight after the new moon, and 1 week after this the last quarter. Finally the waning crescent moon disappears in the glory of the morning sky.

The orbit of the Moon is approximately elliptic. The length of the semimajor axis is 384,400 km and the eccentricity 0.055. Owing to perturbations caused mainly by the Sun, the orbital elements vary with time. The minimum distance of the Moon from the centre of the Earth is 356,400 km, and the maximum distance 406,700 km. This range is larger than the one calculated from the semimajor axis and the eccentricity. The apparent angular diameter is in the range 29.4'-33.5'.

The rotation time of the Moon is equal to the sidereal month, so the same side of the Moon always faces the Earth. Such synchronous rotation is common among

Fig. 7.5. Librations of the Moon can be seen in this pair of photographs taken when the Moon was close to the perigee and the apogee, respectively. (Helsinki University Observatory)

the satellites of the solar system: almost all large moons rotate synchronously.

The orbital speed of the Moon varies according to Kepler's second law. The rotation period, however, remains constant. This means that, at different phases of the lunar orbit, we can see slightly different parts of the surface. When the Moon is close to its perigee, its speed is greater than average (and thus greater than the mean rotation rate), and we can see more of the right-hand edge of the Moon's limb (as seen from the Northern hemisphere). Correspondingly, at the apogee we see "behind" the left edge. Owing to the libration, a total of 59% of the surface area can be seen from the Earth (Fig. 7.5). The libration is quite easy to see if one follows some detail at the edge of the lunar limb.

The orbital plane of the Moon is tilted only about 5° to the ecliptic. However, the orbital plane changes gradually with time, owing mainly to the perturbations caused by the Earth and the Sun. These perturbations cause the nodal line (the intersection of the plane of the ecliptic and the orbital plane of the Moon) to make one full revolution in 18.6 years. We have already encountered the same period in the nutation. When the ascending node of the lunar orbit is close to the vernal equinox, the Moon can be 23.5° + 5° = 28.5° north or south of the equator. When the descending node is close to the vernal equinox, the zone where the Moon can be found extends only 23.5° - 5° = 18.5° north or south of the equator.

The nodical or draconic month is the time in which the Moon moves from one ascending node back to the next one. Because the line of nodes is rotating, the nodical month is 3 hours shorter than the sidereal month, i. e. 27.212 days. The orbital ellipse itself also precesses slowly. The orbital period from perigee to perigee, the anomalistic month, is 5.5 h longer than the sidereal month, or about 27.555 days.

Gravitational differences caused by the Moon and the Sun on different parts of the Earth's surface give rise to the tides. Gravitation is greatest at the sub-lunar point and smallest at the opposite side of the Earth. At

these points, the surface of the seas is highest (high tide, flood). About 6 h after flood, the surface is lowest (low tide, ebb). The tide generated by the Sun is less than half of the lunar tide. When the Sun and the Moon are in the same direction with respect to the Earth (new moon) or opposite each other (full moon), the tidal effect reaches its maximum; this is called spring tide.

The sea level typically varies 1 m, but in some narrow straits, the difference can be as great as 15 m. Due to the irregular shape of the oceans, the true pattern of the oceanic tide is very complicated. The solid surface of the Earth also suffers tidal effects, but the amplitude is much smaller, about 30 cm.

Tides generate friction, which dissipates the rotational and orbital kinetic energy of the Earth-Moon system. This energy loss induces some changes in the system. First, the rotation of the Earth slows down until the Earth also rotates synchronously, i. e. the same side of Earth will always face the Moon. Secondly, the semimajor axis of the orbit of the Moon increases, and the Moon drifts away about 3 cm per year.

* Tides

Let the tide generating body, the mass of which is M to be at point Q at a distance d from the centre of the Earth. The potential V at the point A caused by the body Q is GM

s where s is the distance of the point A from the body Q.

When the denominator is expanded into a Taylor series i 1 3 2

where

rr x = — - 2-cos z d2 d and ignoring all terms higher than or equal to 1/d4 one obtains

The gradient of the potential V( A) gives a force vector per mass unit. The first term of (7.5) vanishes, and the second term is a constant and independent of r. It represents the central motion. The third term of the force vector, however, depends on r. It is the main term of the tidal force. As one can see, it depends inversely on the third power of the distance d. The tidal forces are diminished very rapidly when the distance of a body increases. Therefore the tidal force caused by the Sun is less than half of that of the Moon in spite of much greater mass of the Sun.

We may rewrite the third term of (7.5) as

where

Applying the cosine law in the triangle OAQ, the distance s can be expressed in terms of the other sides and the angle z = AOQ

s2 = d2 + r2 - 2dr cos z , where r is the distance of the point A from the centre of the Earth. We can now rewrite (7.3) GM

4 d3

is called Doodson's tidal constant. It's value for the Moon is 2.628 m2 s—2 and for the Sun 1.208 m2 s—2. We can approximate that z is the zenith angle of the body. The zenith angle z can be expressed in terms of the hour angle h and declination 8 of the body and the latitude <p of the observer cos z = cos h cos S cos p + sin S sin p .

Inserting this into (7.6) we obtain after a lengthy algebraic operation

+ sin 2p cos 2S cos h + (3 sin2 p - 1) ^sin2 S - ^ = D(S + T + Z).

where g is the mean free fall acceleration, g « 9.81 ms-2 and h is a dimensionless number, the Love number, h ~ 0.6, which describes the elasticity of the Earth. In the picture below, one can see the vertical motion of the crust in Helsinki, Finland (p = 60°, k = 25°) in January 1995. The non-zero value of the temporal mean can already be seen in this picture.

0 cm

0 cm

Equation (7.7) is the traditional basic equation of the tidal potential, the Laplace's tidal equation.

In (7.7) one can directly see several characteristics of tides. The term S causes the semi-diurnal tide because it depends on cos 2h. It has two daily maxima and minima, separated by 12 hours, exactly as one can obtain in following the ebb and flood. It reaches its maximum at the equator and is zero at the poles (cos2 p).

The term T expresses the diurnal tides (cos h). It has its maximum at the latitude ±45° and is zero at the equator and at the poles (sin2p). The third term Z is independent of the rotation of the Earth. It causes the long period tides, the period of which is half the orbital period of the body (about 14 days in the case of the Moon and 6 months for the Sun). It is zero at the latitude ±35.27° and has its maximum at the poles. Moreover, the time average of Z is non-zero, causing a permanent deformation of the Earth. This is called the permanent tide. It slightly increases the flattening of the Earth and it is inseparable from the flattening due to the rotation.

The total value of the tidal potential can be computed simply adding the potentials caused by the Moon and the Sun. Due to the tidal forces, the whole body of the Earth is deformed. The vertical motion Ar of the crust can be computed from

The tides have other consequences, too. Because the Earth rotates faster than the Moon orbits the Earth, the tidal bulge does not lie on the Moon-Earth line but is slightly ahead (in the direction of Earth's rotation), see below.

Due to the drag, the rotation of the Earth slows down by about 1-2ms per century. The same reason has caused the Moon's period of rotation to slow down to its orbital period and the Moon faces the same side towards the Earth. The misaligned bulge pulls the Moon forward. The acceleration causes the increase in the semimajor axis of the Moon, about 3 cm per year.

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