This is merely the difference between the actual position of the object in the orbit and the position it would have if it moved at a constant rate. The elements W, w, and T0 and are sometimes combined with each other or with the true or mean anomaly to produce longitudes. For example, the longitude of the perigee (or longitude of the perihelion), w = W + w, (2.16)

is a very curious angle because it is measured first in the ecliptic, from the vernal equinox to the ascending node, and then in the orbit, in the direction of orbital motion. Another example is the mean longitude, i (called in the Astronomical Almanac, L),27

where n is the mean motion = 360°/P, and t is the time of observation or calculation. Therefore, the mean longitude of the epoch, e, is merely the value of i when t is T0 (the instant that defines the epoch):

(Danby 1962, p. 156). Please note that this epsilon is not the obliquity of the ecliptic. Another parameter that is sometimes mentioned is the argument of the latitude, u, the angle between the ascending node and the object in its orbit, so that we can also express the true longitude in terms of the argument of latitude:

The mean elements of the Moon's orbit are given in Table 2.5. Only mean or average elements can be given because they vary with time, usually both secularly (rate change with constant sign, i.e., always increasing or always decreasing) and periodically. Danby (1988, App. C, pp. 427-429) provides for higher order terms for the time variation of the elements of the major planets. Now we are in a position to discuss why the elements change with time.

We can approximate the orbits of the Moon or some planet with a set of orbital elements for an instant of time (for some planets, considerably longer), but the elements of the ellipse vary over time because of perturbations of the other bodies (and, especially in the case of the Earth-Moon system, nonuniform mass distributions in the bodies themselves). The fly in the ointment is that the Earth-Moon is

27 Danby uses L to define the "true longitude" of the planet:

Table 2.5. Lunar orbit mean elements (2000.0).


Mean value

Main variation

Semimajor axis (a)

384,400 km

+3 cm/yr

Eccentricity (e)

0 0

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