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Because the object is not too far from the celestial equator and a = 18h, 270° is the correct value. If the quadrant were not so obvious, however, one could use (2.8) to resolve the issue:

sin 8 - cos e • sin b sin e • cos b 0.469472 - 0.917470 • 0.781967

The use of celestial longitudes spread over 360° is a relatively modern development. The Babylonians and Greeks used degrees of the zodical sign, measuring from the western edge. Ptolemy, for example, gives the position of the star e UMa as "The first of the three stars on the tail next to the place where it joins [the body]" as Q [Leo] 121/6° of longitude, and +53V2° of latitude (Toomer 1984, p. 34). The equivalent value of celestial longitude is l = 132°10'. Ptolemy's values differ from current values because of precession (§3.1.6) and the variation of the obliquity of the ecliptic (§4.4), and possibly other factors (see §7.3.2 for an extensive discussion of whose data were included in this catalogue).

### 2.3.4. The Motions of the Moon

The Moon orbits the earth on a path close to, but not on, the ecliptic, changing phase as it does so and basically replicating the motion of the Sun but at a much faster rate, and more variable celestial latitude. The Sun travels its path in a year, and the Moon in a month. In Figure 2.14, the

Figure 2.14. Lunar phases during the synodic month: (a) The synodic month as viewed geocentrically. (b) Lunar phases for a portion of the month as the Moon-Earth system orbits the Sun. Drawings by E.F. Milone.

Figure 2.14. Lunar phases during the synodic month: (a) The synodic month as viewed geocentrically. (b) Lunar phases for a portion of the month as the Moon-Earth system orbits the Sun. Drawings by E.F. Milone.

phase advancement of the Moon during its revolution is chronicled.

Divided vertically, the figure differentiates crescent (less than a quarter Moon) from gibbous phases (more than a quarter Moon). Divided horizontally, it separates waxing from waning phases. The diagram serves to demonstrate the relative motion of the Moon with respect to the position of the Sun in the sky; one full cycle is the synodic month, or the month of phases.

As the Moon revolves around the earth, its declination changes in the course of a month, and as it does so, its diurnal arc across the sky changes, just as the Sun's diurnal arc changes over the course of a year. The full Moon, because it is opposite the Sun, rides high across the (Northern Hemisphere) midwinter sky, and low across the midsummer sky. The diurnal arcs of the Moon at other phases can be understood similarly. Although the Sun's rise and set points on the horizon vary slowly from day to day, those of the Moon change much more rapidly from day to day. As the Moon circuits the Earth, the Earth and Moon are circuiting the Sun; in a geocentric context, in the course of a month, the Sun moves East among the stars by ~30°. This means that the synodic month must be longer than the time it takes for the Moon to encircle the Earth with respect to a line to the distant stars. This affects lunar and solar calendars (see §4.2, especially, §4.2.1), and the occurrence of eclipses (§5.2).

The motion of the Moon is even more complex and interesting than that described thus far. For one thing, the Moon's declination is sometimes less and sometimes more than is the Sun's extreme values (±23°5 at present). This means that the amplitude of its azimuth variation over the month varies from month to month, in an 18.6-year cycle. This fact is of importance in studying alignments to the Moon, as we show throughout §6. For another, the Moon's distance changes during the course of the month by about 10%, and this affects its apparent (angular) size.23 In addition, the place in

23 The geometric expression is r 6 = D, where r is the distance of an object, 6 its angular diameter in radian measure (= 6° x p/180), and D

the orbit where the Moon achieves its closest point to orbit shifts forward with time. These changes also affect eclipse conditions. To appreciate the full complexity of its behavior in the sky and the roles these play in calendar problems and in eclipse prediction, the moon's orbit must be examined.

### 2.3.5. Orbital Elements and the Lunar Orbit

In ancient Greece and indeed up to the time of Johannes Kepler [1571-1630], all astronomers assumed the orbital motions of Sun, Moon, planets, and stars either to be circular or a combination of circular motions. Modern astronomy has removed the stars from orbiting Earth and has them orbit the galactic center, which itself moves with respect to other galaxies. The Sun's motion is reflexive of the Earth's and comes close to that of a circle, but not quite. The orbits of the other planets can be similarly described; two of them, Mercury and Pluto, show wide departures and some asteroids and most comets, even more. The combination of a sufficient number of circular terms can indeed approximate the motions, but the physical orbits are more generally elliptical.24

One can show from a mathematical formulation of Newton's laws of motion and the gravitational law that in a two-body system, an elliptical or hyperbolic orbit can be expected. If the two objects are bound together (we discuss what this means in §5), the orbit must be an ellipse. Such an ellipse is characterized usually by six unique elements, which we describe and discuss in the next section.

is its diameter in the same units as r. This can be called a "skinny angle formula" because it is an approximation for relatively small values of 6. A more general expression would be D/2 = r sin(6/2).

24 An ellipse can be described geometrically as the locus of all points such that the sum of the distances from the two foci to a point on the ellipse is constant. One may construct an ellipse by anchoring each end of a length of string between two points and, with a pencil keeping the string taut, tracing all around the two points, permitting the string to slide past the pencil in doing so. In orbits, only one focus is occupied, and the other focus and the center are empty.

As an object moves in an ellipse, its distance from a focus changes. When nearer the Sun, the planets move faster than they do when they are further away from it. These facts are encapsulated in the first two "laws" (limited "descriptions," actually) of the planets' behavior, first formulated by Kepler in 1602. The speed variation arises because the line joining planet to Sun sweeps out the area of the orbit in a uniform way: The areal speed is constant. So the Earth moves faster when it is closer to the Sun, and the Moon moves faster when it is closest to Earth. And this is what seems to occur in the sky: From the Earth, the Sun's motion appears to carry it to the east at a faster rate when it is closer to earth, both because of Earth's orbital motion and because the angular speed of an object moving across our line of sight at a given linear speed increases as the distance to it decreases. The motion of the Moon is also more rapid near perigee. Figure 2.15 illustrates the effect.

In the year 2000, the Earth was at perihelion (geocentri-cally, the Sun was at perigee) on Jan. 3 and at aphelion (Sun at apogee) on July 4. In the same year, the Moon was at perigee 13 times: Jan. 19, Feb. 17, Mar. 15, Apr. 8, May 6, June 3, July 1, July 30, Aug. 27, Sept. 24, Oct. 19, Nov. 14, and Dec. 12; it was at apogee 14 times, starting on Jan. 4, and ending on Dec. 28. The daily rate of motion of the Sun along the ecliptic was ~1°1'10" in early January but only ~57'1372" in early July compared with an average motion of 360°/365d24 = 59'8"3 (see Section C of the Astronomical Almanac for the year 2000). The Moon's motion is much more rapid, and because the eccentricity is higher than for the solar orbit, the difference in motion is greater from perigee to apogee.

The orbital elements are illustrated in Figure 2.16:

(1) The semimajor axis, a, half the major axis, is the time-averaged distance of the orbiter to the orbited. This element defines the size of the orbit and depends on the orbital energy; the smaller the distance, the greater the energy that would have to be supplied for it to escape from the Sun.

(2) The eccentricity, e, of the ellipse may be obtained from taking the ratio of the separation of the foci to the major axis, which is just the length of the line joining the perihelion and aphelion. Although a scales the orbit, e defines its shape. From Figure 2.16a, it can be seen that the perihelion distance is a(1 - e) and the aphelion distance is a(1 + e). For the Earth's orbit, e ~ 0.017, so that its distance from the Sun varies from the mean by ±0.017a or about ±2,500,000 km. The eccentricity of the Earth's orbit is not so important a factor in determining the climate as is the obliquity, but it does cause a slight inequality in the lengths of the seasons, as we noted earlier. The orbit of the moon is sufficiently eccentric that its angular size varies sharply over the anomalistic month (the time for the Moon to go from perigee to perigee; see below).

(3) The inclination, i or i, the angle between the reference plane—in the case of the Moon and planets, the ecliptic plane—and that of the orbit, partially fixes the orbital plane in space, but another is needed to finish the job (see Figures 2.16b and c).

(4) The longitude of the ascending node, W, is measured along the ecliptic from the vernal equinox to the point of orbital crossover from below to above the ecliptic plane. This element, with i, fixes the orientation of the plane in space.

(5) The argument of perihelion (for the moon, perigee is used), w, measured from the ascending node in the direction of orbital motion. This element fixes the orientation of the orbital ellipse within the orbital plane.

(6) The epoch, T0, or T or sometimes E0, is the sixth element. In order to predict where the object will be in the future, a particular instant must be specified when the body is at some particular point in its orbit. Such a point may be the perihelion for planets (or perigee for the Moon) or the ascending node, where the object moves from south to north of the ecliptic plane; however, it may be an instant when the object is at any well-determined point in its orbit, such as the true longitude at a specified instant.

(7) Sometimes a seventh element is mentioned—the sidereal period, Psid, the time to complete a single revolution with respect to a line to a distant reference point among the stars.25 Psid is not independent of a because the two quantities are related through Kepler's third law,26 but the Sun's mass dominates the mass of even giant Jupiter by more than 1000:1. For the high precision required of orbital calculations over long intervals, it is necessary to specify this or a related element (the mean rate of motion).

25 A sidereal period usually is not expressed in units of sidereal time; mean solar time units such as the mean solar day (MSD) are used, in general, and the designation is day (d, sometimes in superscript). This need not be the same as the local civil day, i.e., the length of a day in effect at a particular place. See §4.1 for the distinctions.

26 The third law relates the period, P, to the semimajor axis, a. In Kepler's formulation, the relation was P2 = a3, if P is in units of the length of the sidereal period of Earth and a is in units of the Earth's semimajor axis. In astronomy generally, a® defines the astronomical unit. From Newtonian physics, it can be shown that the constant of proportionality is not 1 and is not even constant from planet to planet: P2 = {4p2/[G(W + m)]j a3, where G is the gravitational constant, 6.67 10-11 (MKS units), and m and W are the masses of the smaller and larger mass bodies, respectively.

Elliptical orbit plan view:

^^ ecliptic plane line of nodes

^^ ecliptic plane line of nodes to orbit pole t0 NEP

to orbit pole t0 NEP

Figure 2.16. Elements and other properties of an elliptical orbit: (a) "Plan" and (b) "elevation" views, respectively—The scale and shape of the orbit are established by the semimajor axis, a, and the eccentricity, e. The orientation of the orbital plane with respect to the ecliptic is set by the inclination, i, and the longitude of the ascending node, W; the orientation of the orbit within the plane is fixed by the argument of perihelion, w. The instant of the location of the planet at the perihelion, r = a(1 - e) (or, when e = 0, at the ascending node), T0, is the sixth element; the seventh, the period, P, is not an independent

The elements of the lunar orbit at a particular date are shown in Table 2.4. Given the elements, one may find, in principle, the position of an object in the orbit at any later time. The angle swept out by the Sun-planet line is called the true anomaly (u in Figure 2.16b). The position of the

element since it is related to a by Kepler's third law. (c) "Slant view"—The position of the planet in Cartesian coordinates aids the transformation from the orbit to the sky. The relationship between the orbital and ecliptic coordinates are found by successive rotations of the axes shown. (d) The relations between the celestial longitude and latitude and the Cartesian ecliptic coordinates—A further transformation to equatorial coordinates can be carried out through spherical trigonometry or through a transformation of Cartesian coordinates. See Schlosser et al. (1991/1994) for further details. Drawings by E.F. Milone.

object in its orbit at any time t since perihelion passage (T0) can be specified through a quantity called the mean anomaly:

This angle describes the position of a planet that would move at the same average rate as the planet, but in a circular orbit. The mean anomaly is related to the true anomaly by the approximation

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