## The Moon

1. The instantaneous lunar orbit

In 1991 Michelle Chapront-Touz6 and Jean Chapront, two astronomers at the Bureau des Longitudes, Paris, France, published their Lunar Tables and Programs from 4000 B.C. to A.D. 8000 (Willmann-Bell, ed.).

These tables enable the calculation, for any instant over the years -4000 to +8000, of

— the geocentric position of the Moon (longitude, latitude, distance);

— the osculating elements of the lunar orbit.

The osculating elements are those of the 'instantaneous' orbit of the Moon, that is, the elements of the elliptic orbit of a fictitious Moon whose position and velocity for the given instant would be the same as those of the real Moon.

The motion of the Moon is disturbed mainly by the gravitational attraction of the Sun, and the osculating elements of the orbit are complicated functions of time. As examples, we give in the Figures 1 .a to l.d the variation of four osculating elements of the Moon's orbit from 1996 January 1 to 1999 January 1.

The first drawing shows the variation of the instantaneous orbital eccentricity. This eccentricity is a maximum when the major axis of the lunar orbit is directed toward the Sun, as in January 1996, July 1996, February 1997, etc., which occurs at mean intervals of 205.9 da}.,. This period is somewhat longer than six months by reason of the motion of the longitude of the Moon's perigee in the direct sense (+0.11140 degree per day). However, it appears from Figure I.a that there are large secondary oscillations. While the mean value of the eccentricity is 0.055, its 4instantaneous' value can vary between the extremes 0.026 and 0.077.

Our second figure shows the variation of the inclination of the instantaneous lunar orbit on the ecliptic — the plane of the Earth's orbit around the Sun. This inclination is a maximum when, as in March and September 1997, the line of nodes of the lunar orbit is directed toward the Sun; this occurs at mean intervals of 173.3 days. This period is somewhat shorter than six months by reason of the retrograde motion of the longitude of the Moon's ascending node (-0.05295 degree per day). Here too we see that there are secondary oscillations of short period, and that their effect is larger when the value of the inclination is least.

The retrograde motion of the lunar nodes has a period of 18.6 years, more exactly of 6798.38 days with respect to the moving equinox, or 6793.48 days with respect to the stars. However, the motion of the nodes is not regular. The principal inequality has a period of 173.3 days, the same as that of the inclination. The variation of the longitude of the Moon's ascending node, referred to the mean equinox of the date, is illustrated in Figure 1 .c for the years 1996 to 1998.

The line of nodes is almost stationary when it is directed toward the Sun. This coincides with the maximum value of the orbital inclination, and it is near these epochs that solar and lunar eclipses take place.

1996 Jan. 1

Fig. 1.a: The instantaneous eccentricity of the lunar orbit, 1996 to 1998

1996 Jan. 1

Fig. 1.a: The instantaneous eccentricity of the lunar orbit, 1996 to 1998

Jan. I Jan. 1 Jan. 1 Jan. 1

Fig. I.b: The instantaneous inclination of the lunar orbit on the ecliptic,

1996 to 1998

Jan. 1 Jan. 1 Jan. 1 Jan. 1

Fig. I.c: The longitude of the ascending node of the lunar orbit,

1996 to 1998

Fig. 1.d: The longitude of the perigee of the lunar orbit,

1996 to 1998

Fig. 1.d: The longitude of the perigee of the lunar orbit,

### 1996 to 1998

While the line of nodes is retrograding, the major axis of the lunar orbit moves in the direct sense, that is, in the same direction as the revolution of the Moon around the Earth. One complete revolution of the lunar perigee takes 8.85 years; more exactly, the period is 3231.50 days with respect to the moving equinox (tropical period), or 3232.61 days with respect to the stars (sidereal period).

The longitude of the perigee of the instantaneous lunar orbit undergoes important periodic inequalities, as is shown in Figure 1 .d. The difference in longitude between this perigee and that of the mean lunar orbit can be as large as 30 degrees.

It should be noted that the longitude of the perigee is measured in two different planes: it is equal to the longitude of the ascending node, measured along the ecliptic from the vernal equinox to this node, increased by the arc from the ascending node to the perigee measured in the orbital plane.

2. The extreme values of the distance of the Moon to the Earth

In astronomy textbooks we read that the mean distance a from the Earth to the Moon is 384400 kilometers, and that the eccentricity of the lunar orbit is e — 0.0549. From these values one can deduce that the minimum (perigee) and the maximum (apogee) distances between the centers of the two bodies are a (1 — e) = 363296 km and a (1 + e) = 405504 km, respectively.

However, these are not the smallest and largest possible distances between the Earth and the Moon. The Moon's motion is strongly perturbed by the attraction of the Sun, and in a lesser way by that of the planets and by the flattened shape of the Earth. Every 206 days — a little more than half a year — the major axis of the lunar orbit is directed toward the Sun. As we have seen in Chapter 1, near these epochs the eccentricity of the lunar orbit reaches a maximum, and the perigee distance of the Moon is much smaller than normal and the apogee distance is larger. When, however, the major axis of the lunar orbit is perpendicular to the direction of the Sun, the eccentricity reaches a minimum; at these epochs, perigee and apogee distances are less extreme. See Figure 2.a.

Figure 2,b shows the variation of the Earth-Moon distance over a two-year period. This plot reveals the 206-day cycle of maxima and minima mentioned above. The remarkable fact is that the variation of the perigee distance is much larger than that of the apogee. Using the ELP 2000-82 and ELP 2000-85 lunar theories of Michelle Chapront-Touz6 and Jean Chapront, we calculated the perigee and apogee distances of the Moon for the years 1960 to 2040. During this 81-year period, the perigee distances of the Moon vary between 356445 km (on 2034 November 25) and 370354 km (on 1988 December 16), which gives a spread of 13909 kilometers.

On the other hand, during this same period the apogee distance varies from 404064 km (on 1976 July 19) to 406712 km (on 1984 March 2), a variation of only 2648 kilometers.

What are the smallest and the largest possible values for the distance between the centers of the Earth and the Moon? To answer this question, we made a calculation for the much longer period of A.D. 1500 to 2500. During these ten centuries, 49 perigee distances are less than 356500 km, while 33 apogees are larger than 406 700 km. During the same period, fourteen times the Moon approaches the Earth to less than 356425 kilometers, and the same number of times the distance grows to larger than 406710 km. These extreme cases are listed in Table 2.A.

It thus appears that, during the time interval of ten centuries considered, the extreme distances between the centers of Earth and Moon are

356371 kilometers on 2257 January 1 406720 kilometers on 2266 January 7

We further see from the table that the extreme perigees and apogees take place only during the winter in the northern hemisphere, the period of the year when the Earth is closest to the Sun. For instance, all 14 closest perigees mentioned in the table occur between December 6 and February 9. It is evident that the Earth's variable distance from the Sun somewhat affects the Earth-Moon distance.

Fig. 2. a: When the major axis of the Moon's orbit is aligned with the Earth-Sun line (A), the orbital eccentricity exceeds its mean value. About 103 days later, in B, the two lines are at right angle and the eccentricity reaches a minimum. A new maximum is reached again after another 103 days (C). Sizes and distances are not to scale!

Fig. 2. a: When the major axis of the Moon's orbit is aligned with the Earth-Sun line (A), the orbital eccentricity exceeds its mean value. About 103 days later, in B, the two lines are at right angle and the eccentricity reaches a minimum. A new maximum is reached again after another 103 days (C). Sizes and distances are not to scale!

But that's not yet all. In the table we recognize the well-known periodicity of 18 years + 11 days, the Saros! However, here this famous period has nothing to do with solar or lunar eclipses. See, for example, the extreme perigees of 1893 - 1912 - 1930, or the extreme apogees of 2107 - 2125 - 2143.

Fig. 2.b : The distance between the centers of Earth and Moon, 1996-1997. Verticale scale : thousands of kilometers.

 perigee < 356425 km apogee > 406710 km 1548 Dec