## Effects of Precession

The westward shift of the equinoxes among the stars due to the wobbling of the Earth's axis is known as precession. This top-like motion is caused by several factors, the main one being the pull of the Sun and Moon on the equatorial bulge around the Earth's equator,14 and we discuss its effects first. There are three main results of this luni-solar precession.

14 The equatorial bulge is caused by the rotation of the Earth, which results in a slightly weaker gravitational pull on objects at the equator than at the poles.

Figure 3.7. The planet Mercury along with other bright planets in the twilight sky of Calgary on February 28, 1999. Courtesy, Roland Dechesne, RASC, Calgary Centre. The photo shows Venus, Jupiter, and, closest to the horizon, Mercury.

First, the shifting position of the equinox causes stellar coordinates to change with time. The celestial longitude increases by 50.2 arc-seconds/year, as the equinox shifts westward along the ecliptic. Because this point is also the origin for one of the two equatorial systems of coordinates, the right ascensions and declinations of objects in the sky change also. By differencing the transformation equations between the equatorial and ecliptic systems (§2.3.3) and ignoring the change in the slow change in e, one may find the formulae for the basic changes in declination and right ascension to be dd = dlx sin ex cos a, (3.22)

where dl = (50''2) x dt, da and d8 are the changes (in arc-seconds) in right ascension and declination coordinates, respectively, and dl is the change in celestial longitude due to precession over the interval dt in years. Figure 3.8 illustrates the effect of precession on the right ascension and declination coordinates.

Figure 3.8. The effects of precession: The motion of vernal equinox and the changing right ascension and declination coordinates. Drawing by E.F. Milone.

The coordinates on the right-hand sides of (3.22) and (3.23) should be mean values, but these are not known before the calculation; so an iteration must be performed. The initial values of the right ascension and declination are assumed, and the changes Da and A8 are computed; the new (but not quite correct) values are found, mean values of a and 8 computed and inserted in the right-hand sides of (3.22) and (3.23), and the whole process repeated until the last two sets of iterated values are in agreement. To be more accurate still, the value of e should be computed for the initial and final dates as well. The process can be lengthy if the time interval between dates is substantial. The right ascension and declination for a date in the remote past may be computed more directly by means of rigorous formulae from the Astronomical Almanac15:

cos(a - z)cos 8 = cos(a0 + Z)cos 0cos 80 - sin 0 sin80, and sin 8 = cos(a0 + Z)sin 0cos 80 + cos 0 sin 80, (3.25)

respectively, where a and 8 refer to the date of interest, a0 and 80 refer to the initial epoch, and the auxiliary quantities z, 0, and Z (which fix the location of the equinox and equator of date with respect to that at the initial date). For an initial equinox at 2000.0, they may be computed as follows:

z = [(0.0000051 • T + 0.0003041) • T + 0.6406161] • T, (3.26)

0 = [(-0.0000116 • T -0.0001185) • T + 0.5567530] • T, (3.27)

Z = [(0.0000050 • T + 0.0000839) • T + 0.6406161] T, (3.28)

where T is the number of centuries, measured positively after 2000.0 a.d. If t is the calendar date of interest and JDN the Julian day number (and decimal thereof) of that date of interest, T = (t - 2000.0)/100 = (JDN - 245 1545.0)/36525.

For most alignment purposes, the declination is the important quantity because it and not the right ascension determines the azimuth of rise or set.

The precession also produces a change in the direction of the north celestial pole (NCP) in space. The annual 50.2 motion of the equinox results in a complete revolution in about 25,800 years. The shift in the NCP in half that period is 2<e>,where <e> is the mean obliquity of the ecliptic (see §2.3.3). The precessional circle around the north ecliptic pole is seen in Figure 3.9.

Note the motion of the NCP from the vicinity of the star Polaris over an interval of just a few hundred years. At around 2500 b.c., about when the largest of the Pyramids was constructed in Egypt, Polaris was nowhere near the pole. The pole star at that time was a Draconis (Thuban). This star may have played a role in the alignments of narrow shafts discovered between the royal burial chambers of the pyramid of Khufu to the northern face of the pyramid (see §8.1): The angle made by the shaft with the horizontal (~31°) approximates the latitude of the site and thus the altitude of the NCP (29°59'). In the Southern Hemisphere, there is no bright pole star at present, although there is a "pointer," the Southern Cross. Two thousand years ago, the southern pole star was a bright star, b Hyi. Figure 3.10 illustrates the polar motion of the north and south celestial poles, respectively. The sky of 2500 b.c. is shown for the northern hemisphere. That of 1 a.d. is shown for the southern hemisphere.

Finally, as the pole shifts among the stars, the circumpolar constellations change. Some stars that formerly rose and set now remain above the horizon. Some constellations toward the opposite pole that formerly never appeared above the horizon become visible, whereas constellations (at celestial longitudes 180° away) that formerly rose for a time above the horizon now remain invisible. Note the disappearance of the Southern Cross below the southern horizon at the latitude of Jerusalem between 6 b.c. and 1994 a.d. in Figure 3.11.

The effects that have been discussed are for the basic luni-solar precession only. Because the Moon is not on the ecliptic, there is an additional variation in the vernal equinox position. It undergoes a small oscillation that causes the celestial latitude as well as longitude to vary. It also has an additional small affect on a and 8. The NCP will appear to undergo an additional wobble with half-amplitude of 9.2 arc-seconds in an 18.6-year period. The effect, discovered by James Bradley in the 19th century, is known as nutation. There are still smaller effects due to the perturbations of the other planets, principally those on the plane of the Earth's orbit; thus, the ecliptic is slowly varying also, although the ecliptic pole variation has an amplitude much less than that of the celestial pole. The combination of luni-solar and this planetary precession is called the general precession. The discovery and treatment of precession will be described further in §7.

15 In the section, "Reduction of Celestial Coordinates";in the Astronomical Almanac of recent years (e.g., 2000), these formulae are located on p. B18.

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