Chasing Shadows: Mathematics, Astronomy, and the Early History of Eclipse Reckoning

CHAPTER TWO: The Rudiments of Alignment: Basic Eclipse Theory

CHAPTER TWO  The Rudiments of Alignment Basic Eclipse Theory 2.1 The Moon and the Sun Perhaps the single most prominent feature of the night sky is the disk of the moon. With its waxing and waning at such short recurring intervals, even the most casual observer cannot but notice and appreciate its periodicities. The circuit of the moon can be envisioned as (approximately) a great circle on the celestial sphere, a huge spherical surface enclosing the earth onto which all heavenly bodies are projected. The earth is centrally placed in this sphere and can be imagined as stationary (fig. 2.1). With the earth thus fixed, the celestial sphere makes one giant rotation in a westward direction, that is from east to west, every 24 hours, carrying with it all its contents. In terms of how the sky appears, it is of no consequence whether the earth is considered to rotate on its axis or whether the celestial sphere revolves in the opposite direction daily, as astronomers believed in early times. In fact, there is a certain convenience to the latter. Observers reckoning themselves situated on the earth gazing outward to an infinitely large enclosing dome capture their situation more readily and naturally. Indeed, even today those who use astronomy for practical means (such as navigation, surveying, orienteering, and so on) routinely use this perspective. A further advantage of such a conception is that it can capture directions and relative positions of celestial bodies, without the need to consider their distances from the earth. For an observer on the earth, only half of this sphere and its contents is visible at any instant, and to demarcate this sphere, the terrestrial equator can be projected onto the celestial sphere, to give the celestial equator, and the terrestrial poles extended to give the celestial poles. In addition to the apparent daily motion of the celestial sphere, the sun, the moon, and the planets have additional motions (fig. 2.2). The sun travels in the celestial sphere on an orbit slightly inclined to the celestial equator in an 22 Earth Celestial sphere West East Celestial equator Equator Celestial pole Terrestrial pole Figure 2.1 The celestial sphere. Earth Moon’s path Sun’s path Node North celestial pole Node Celestial Sun Ecliptic Moon equator Figure 2.2 The orbits of the sun and the moon. CHASING SHADOWS eastward direction, that is from west to east, with the fixed stars as its backdrop. This path is commonly known as the ecliptic and is inclined at a little under 24◦ with respect to the celestial equator. This circuit takes the sun about a year to complete, producing an apparent daily motion of about 1◦ per day. The moon travels around its orbit which is inclined to the ecliptic a further 5◦ from the celestial equator. It moves more swiftly than the sun, at roughly 13◦ per day, and will return to approximately the same position on its circuit after approximately 271 3 days. This interval is known as a sidereal month. Systematic understanding of the periods and phases of the moon is intimately connected with the sun. Due to this fact, and because of the inclinations and varying intersections of the individual orbits of the luminaries, there are, in addition to the sidereal month, three further significant intervals related to the moon. As the moon travels around its orbit, it changes its position with respect to the sun, which causes different parts of its orb to be illuminated (figs. 2.3 and 2.4). This is the cause of the familiar waxing and waning effects or the growth and decay of the crescent. When the moon is at conjunction, it is exactly between the earth and the sun with the nonilluminated side facing the earth. When it is in this phase, also called the new moon, it cannot be seen. Next, as the moon gradually creeps around its orbit, the crescent appears. It is the appearance of this first crescent that for so many early cultures indicated the first day of the month. Gradually the thin crescent swells until it has the appearance of a half circle. This phase is called the first quarter. It occurs when the moon is very nearly 90◦ away from Light rays from the sun Earth Conjunction new moon Opposition full moon Moon Figure 2.3 The waxing and waning of the moon. 24 Figure...