Basic Motions of the Sun and Moon

2.3.1. The Sun, the Year, and the Seasons

Now we must separate the diurnal motion shared by all objects in the sky from the additional motions of seven distinctive objects known in antiquity: the Sun, Moon, and naked-eye planets. We take for granted that the diurnal motion of everything in the sky is due to the rotation of the earth on which we stand. In the ancient world, this was a radical view, and few astronomers held it. Diurnal motion is perceived moment by moment, whereas the effects of the relative movement of the Sun, Moon and planets with respect to the stars are far more gradual. This made diurnal motion of the fixed stars far more intuitive than any other motion. Nevertheless, an unmoving Earth was not the only option, and ancient astronomers knew it.

Figure 2.6 provides the alternative frameworks for understanding the motions: the Earth-centered and Sun-centered systems. The geocentric perspective has been historically dominant in human cultures, and yet the heliocentric viewpoint leads to a far more economical model to account for the relative motion of the Sun, Moon, and planets in the sky. Prior to the Copernican revolution, and indeed throughout most of known history, the geocentric universe was the accepted cosmic model notwithstanding that the Greek scholar Aristarchus (~320 b.c.) argued for a heliocentric universe and the medieval Islamic scholar al-Beruni (~11th century) said that all known phenomena could be explained either way. Both the constellation backdrop and the direction of the Sun's apparent motion are predictably the same in the two world systems, as Figure 2.6 reveals. In both models, the Sun's annual motion (as viewed from the north celestial pole region) is counterclockwise. We have known

Figure 2.6. The classic cosmological frameworks: (a) Earth-centered and (b) Sun-centered views of the solar system. Drawings by E.F. Milone.

since Newton's time that a less massive object like the Earth is accelerated by the Sun more quickly than a more massive body like the Sun is accelerated by the Earth.9 Neither the true physical natures of the planets nor the physical principles that ruled their motions were known in the ancient world.10 The ancient models were designed specifically to predict these apparent motions and were basic to ancient Greek astronomy. The successes and failures of ancient models can be gauged precisely and accurately only if their predictions can be compared with those of modern methods. We start with the most familiar case, the Sun, which undergoes reflexive motions in the sky as the Earth moves.

From the geocentric perspective, that the earth's rotation axis is tilted by 23°5 with respect to its axis of revolution about the Sun, and that the direction of the rotation axis is fixed11 while the Earth revolves about the Sun is equivalent to saying that the Sun's path is inclined to the celestial equator by the same 23°5 angle, so that the Sun's declination varies by about 47° during the year. Except possibly in deep caves and on ocean floors, the effects of the Sun's annual movement are dramatic for life everywhere on Earth. In fact, the large annual variation in declination has profoundly affected development and evolution of life on Earth (especially, if, as is sometimes speculated, the angle of tilt has changed greatly over the age of the Earth).

The obvious diurnal westward movement of the Sun is shared by the Moon, planets, and stars. However, the diurnal westward motions of the Sun, Moon, and planets are different from those of the stars and from each other. The Sun and Moon always move eastward relative to the stars, so that the angular rates of their westward diurnal motions are always less than that of the stars. The planets' apparent motions are more complex, sometimes halting their eastward motions and briefly moving westward before resuming eastward motion. Thus, their diurnal motions are usually slower but sometimes slightly faster than are those of the stars.

To describe the Sun's behavior, we can say that the diurnal motion of the Sun is accomplished in a day and is very nearly parallel to the celestial equator; relative to the stars, the Sun has a slow average motion, ~360°/3651/4d ~ 1°/d eastward, and it requires a year to complete a circuit. Moreover, except for two instants during the year, the Sun's annual motion is not strictly parallel to the celestial equator. We can

9 Isaac Newton (1642-1727) embodied this idea in the second of his three laws of motion in the Philosophiae Naturalis Principia Mathe-matica (1687). His first law states that an object in motion (or at rest) maintains that state unless acted on by an external force. The second law more fully states that the acceleration of a body is directly proportional to the force acting on it and is inversely proportional to its mass. The third law states that every force exerted by one body on another is matched by a force by the second on the first.

10 It goes almost without saying, however, that this circumstance does not relieve dedicated students of ancient astronomy from an obligation to obtain at least a rudimentary understanding of the nature and true motions of planetary bodies, so that their relative motions with respect to the Earth can be understood.

11 Ignoring the long-term phenomena of precession (q.v. §3.1.6) and the variation of the obliquity (see §2.3.3 and §4.4, respectively).

elaborate this motion and then find an explanation for it, or, as the Greeks would put it, "save the phenomenon."

The Sun's eastward motion is easily tracked on the celestial sphere. Figure B1 illustrates the annual path of the Sun (the ecliptic) as it appears on an equatorial star chart. The sinusoid shape is the consequence of mapping the path onto a Mercator projection of the equatorial system. The process can be visualized by imagining the celestial sphere cut in two along an hour circle and opened outward. In this projection, in which all hour circles become parallel vertical lines, great circles that intersect the celestial equator at an acute angle appear as sinusoids. The Sun's most northern declination (+23°5 at present) occurs at the positive maximum of this curve, the June solstice (the northern hemisphere's summer solstice), at a = 6h, and its most southern declination (—23°5 at present) at the December solstice (the Northern hemisphere's winter, and the Southern hemisphere's summer solstice), at a = 18h. Like the term equinox, solstice also has two meanings. It is a positional point on the ecliptic and an instant of time when the Sun "stands still" (the literal meaning of the Latin). A solstice, therefore, marks a N/S turning point. At the equinoxes, where (and when) it crosses the celestial equator, the Sun rises at the east point of the horizon, and sets at the west point. The azimuth of rise (or set) of the Sun on any given day depends both on its declination and on the observer's latitude. Solving Eqn. (2.1) for cos A, cos A =

sin S - (sin f • sin h) cos f • cos h and on the horizon,12 h = 0, so that cos Arise/set =

sin 8

cos f

At 5 = 0°, cosA = 0, so that A = 90° and 270°, the azimuths of the east and west points of the horizon, respectively. Beginning at (Northern Hemisphere) winter solstice, the Sun rises further to the North each day, with decreasing azimuth, until it reaches summer solstice. At that mid-summer13 date, it has the smallest azimuth of rise (i.e., most northern). It stops decreasing and thereafter rises at greater

12 There is a slight complication in the statement that h = 0 indicates an object on the horizon. This is true of the astronomical definition of altitude and of the horizon, but the Earth's atmosphere acts as a lens, the refractive properties of which raise both the horizon and the object toward the zenith by an amount that depends on the true altitude and that varies with the temperature and pressure of the atmosphere along the path to the object. Because the light from the object travels a greater path length through the atmosphere, it is lifted higher than the horizon, sometimes dramatically so. Thus, the apparent altitude at the astronomical instant of rise is greater than zero;a common value is ~0°5. See §3.1.3 for further discussion. For the time being, we ignore the effects of atmospheric refraction.

13 Technically, modern astronomy assigns the beginning of the season to the date of solstices and equinoxes, but the older usage is still common. "Midsummer's eve" is the night before the sunrise of the summer solstice. When the terms "midwinter" and "midsummer" are used here, they refer to the dates of the solstices.

azimuths (i.e., more and more to the south). It continues to rise further South each day until it reaches winter solstice again. In the Southern Hemisphere, the Sun rises further to the South each day, its azimuth increasing until summer solstice, and thereafter decreasing again (this follows if the azimuth keeps the same definition we have adopted for the Northern Hemisphere).

Near the equinoxes, the solar declination changes rapidly from day to day so that the points on the horizon marking sunrise and sunset also vary most quickly at those times; at the solstices, the change in declination from day to day is very small, and so is the azimuth change.14

The oscillation of its rising (and setting) azimuth on the horizon is one clearly observable effect of the Sun's variable declination during the year. Half of the total amount of oscillation, the largest difference (N or S) from the east point, is called the amplitude.15 We will designate it DA. Note that the amplitude of rise is also the amplitude of set. The amplitude depends on the latitude [see Eqn (2.7)]. At the equator, f = 0° and DA = 23.5°. At any latitude, f, at rise,

cos f

Note that the Sun's motion along the ecliptic includes a north/south component that changes its declination, which has been shown to vary the sunrise and sunset azimuth. Because the changing declination of the Sun causes the seasons, the azimuth variation can be used to mark them; a good case can be made that this was done in the Megalithic (§§6.2, 6.3).

The seasonal change in declination also changes the time interval the Sun is above the horizon. This day-time interval is twice the hour angle of rise or set (ignoring, again, the effect of refraction and other physical effects described in §3); so the Sun is above the horizon longer in summer than in winter at all latitudes except the equator. Solving Eqn (2.5) when h = 0°, we get cos Hrise/set = - tan f • tan S.

At the equinoxes, when 5 = 0°, Hrise/set = 90° and 270°, equivalent to 6h and 18h (-6h), the hour angles of set and rise,

14 This can be seen by taking the rate of change of azimuth due to a change in declination, in Equation 2.7:

so that dA

Near the equinoxes, 5 « 0, so that cos 5 « 1 and near the solstices, 5 « ±23°5, so that cos 5 « 0.9. Moreover, when cos f is small, sin A is large and vice versa, so that dA changes proportionally with d5, but with opposite sign, at all times of year. Near the solstices, when d5 ~ 0, dA « 0 also, so that the Sun is at a standstill, roughly keeping the same azimuth from night to night for several nights.

15 Not to be confused with the term as used in variable star astronomy, where amplitude refers to the range of brightness variation. See §5.8.

respectively. At such a time, the Sun is above the horizon half the day, so that the numbers of daylight and night-time hours are about equal, hence, the Latin aequinoctium, whence equinox. At winter solstice, the Sun spends the smallest fraction of the day above the horizon; and its noon altitude (its altitude on the celestial meridian, where H = 0) is the lowest of the year. At summer solstice, the Sun spends the largest fraction of the day above the horizon and its noon altitude is the highest of the year. The symmetry in the last two sentences mimics the symmetry of the Sun's movements over the year. The larger fraction of the Sun's diurnal path that is below the horizon at winter solstice is the same fraction that is above the horizon at summer solstice. That the ancient Greeks worked with chords subtended at the centers of circles rather than with sines and cosines did not deter them from discovering and making use of these wonderful symmetries, as we show in §7.3.

A high declination object has a larger diurnal arc above the horizon than below it, and by a difference that increases with latitude (see Figure 2.7). The result of the low altitude of the winter Sun means that each square centimeter of the ground receives less solar energy per second than at any other time of the year, as Figure 2.8 illustrates, resulting in lower equilibrium temperatures. In practice, the situation is complicated by weather systems, but the seasonal insolation of the Sun, as the rate of delivery of solar energy to a unit area is called, is usually the dominant seasonal factor. The effects of seasonal variations and the association of these changes with the visibility of certain asterisms (especially those near the horizon at sunrise or sunset) was noticed early. This association may have been a crucial factor in the development of ideas of stellar influences on the Earth. The changing visibility, ultimately due to the orbital motion of the Earth, shows up in the reflexive motion of the Sun in the sky. The Sun's motion among the stars means that successive constellations fade as the Sun nears them and become visible again as it passes east of them.

References to seasonal phenomena are common in the ancient world. From Whiston's Josephus,16 writing about the followers of the high priest John Hyrcanus, who was besieged by the Seleucid king Antiochus VII:

[T]hey were once in want of water, which yet they were delivered from by a large shower of rain, which fell at the setting of the Pleiades. The Antiquities of the Jews, Book XIII, Ch. VIII, paragraph 2, p. 278.

Whiston's footnote to the line ending with the "Pleiades" reads:

This helical setting of the Pleiades was, in the days of Hyrcanus and Josephus, early in the spring, about February, the time of the latter rain in Judea; and this is the only astronomical character of time, besides one eclipse of the moon in the reign of Herod, that we meet with in all Josephus.

The "helical" (heliacal) setting (see §2.4.3) indicates a time when the Pleiades set just after the Sun. Due to the phenomenon of precession (see §3.1.6), the right ascension of the Pleiades in the time of John Hyrcanus, ~132 b.c.,

More properly, "The Works of Flavius Josephus."

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