## The Sphere of the Sky 221 Daily Sky Motions

Time exposure photography of the sky readily reveals the movement of the sky. Uniform exposures (say, one hour each) under a cloudless sky at each of the cardinal facings will confirm the impression of the unaided eye—that the stars wheel about a hub at constant angular rate. Figure 2.1 shows typical diurnal (daily) arcs traced out by stars during such exposures. Traced with a stylus on a graphics tablet, the arc lengths can be shown to be systematically larger with increased angular distance from the center of motion—the celestial pole. The longest arcs are 90° from the celestial pole—on what is called the celestial equator, which divides the sky into northern and southern halves.

The apparent direction of turning is counterclockwise—as we view the North Celestial Pole. It is clockwise for Southern hemisphere observers viewing the South Celestial pole. The motions are consistent. As one faces North, the stars rise in arcs from one's right hand and set at one's left hand. Facing South, they rise at the left hand and set at the right hand. The observations imply that either the sky is rotating from East to West above the earth or that the earth is rotating from West to East below the sky.

In antiquity, which condition was true was the subject of much discussion and, in the end, could not be determined definitively. In the absence of a knowledge of the correct physics, misinterpretations of common experience gave many writers the idea that a rotating earth would force unanchored objects to be thrown off (see Chapter 7, especially §7.2).

Although the sense of the turning sky is the same all over the earth, the diurnal arcs have a different character for observers at the equator compared to those nearer the poles. For an observer on the equator, the North and South Celestial poles are on opposite sides of the sky; all stars rise at right angles to the horizon and move across the sky in semicircles, spending half the time above, and half the time below, the horizon. For observers elsewhere, stars that have diurnal circles between the pole and the horizon do not rise or set. They are called circumpolar stars. Stars equally distant from the opposite pole never appear above the horizon. In modern parlance, these two regions are called the north and south circumpolar zones, respectively. The diurnal arcs of stars that rise and set make acute angles (<90°) with the horizon, and this angle becomes smaller with the observer's proximity to the pole. At the North and South Poles, this angle becomes 0°, as the stars move in circles that are concentric with the horizon and are circumpolar. At the equator, it is 90° for all stars, and none are circumpolar.

The notion that the heavens constitute a great sphere surrounding the observer is an ancient one. It seems likely to have been present among the early Pythagoreans. It is associated with the Ionian Greeks, especially Eudoxos of Cnidus who lived in the 4th century b.c. It was known in China by the 2nd century b.c. The heavens were sometimes depicted as an external sphere, such as that shown in the Etruscan depiction of Atlas holding up the sky sphere. Not every culture, however, depicted the sky as a hemispherical bowl;

Figure 2.1. Diurnal arcs traced out by stars during a time exposure near the North Celestial Pole. Trails further from the pole appear straighter because the radii of curvature of their diurnal circles is larger. Photo courtesy of T.A. Clark.

in ancient Egypt, the sky was pictured as the body of the goddess Nut, for example. The shape of the sky as we perceive it depends on several factors: physiological, psychological, and cultural. We can even measure the perceived shape (see Schlosser et al 1991/1994, pp. 1-3). For the purposes of locating objects on the sky, however, we use, even today, the concept of the celestial sphere.

2.2.2. The Horizon or "Arabic" System

The image of an Earth surrounded by pure and perfect crystalline spheres5 was emphasized by Aristotle, among others. Astronomers have made continual use of this image for more than two millennia; we refer to a celestial sphere, on which all objects in the sky appear, at any given instant, to be fixed. It does not matter in the slightest that such a sphere is borne of perception only, or that it exists only in our imagination. Everything that undergoes diurnal motion is assumed to lie on this sphere; the consequence is that they are assumed to be at the same distance from the observer. This is not strictly true, of course, but for locating very distant objects on the celestial sphere, it is a reasonable approximation. To the naked eye, the Moon is the only one of all the permanent bodies in the sky that seems to shift position among the stars as an observer shifts from one place on Earth to another.6 For nearer objects, such as the Sun, Moon, and planets, relative motions on the sky can be studied and the predicted positions tabulated for each day, as, for example, in Babylon and Ur (see §7.1). This means that only two coordinates suffice to describe the position of an object on the surface of such a sphere.

On the celestial sphere, we will place the markings of the horizon system. We also refer to this system as the Arab system, because it was in wide use in the Arab world during the European Dark Ages. Not all the terms currently used in the English description of it stem directly from the Arabic language. Its salient features are indicated and labeled in Figure 2.2, which also includes relevant elements of the equatorial system which is described in §2.2.3.

The highest point, directly overhead, is the zenith, a name that reaches us through Spain (zenit) and the Arab world of the Middle Ages (samt ar-ra's, road (over) the head). Directly below, unseen, is the nadir (Arabic nazir as-samt, opposite the zenith). The zenith and the nadir mark the poles of the horizon system. The horizon, which comes from a Greek word meaning to separate, basically divides the earth from the sky. We adopt the modern definition here: The astronomical horizon is the intersection with the celestial sphere of a plane through the observer and perpendicular

5 Indeed, ancient Greek astronomers held that the motions of "wandering stars" or planets could be explained with the turnings of many such transparent spheres. See §7.2.3.

6 From the place where the Moon appears overhead to the place where it appears on the horizon, the Moon appears to shift by about 1° with respect to the stars. The shift is called the horizontal parallax. Parallax shifts are very important in astronomy and are a primary means of determining astronomical distances.

Figure 2.2. The horizon system: The main features of the horizon system of spherical astronomical coordinates. (a) The outside-the-sphere view. The azimuth coordinate, A, is represented as a polar angle measured at the zenith; A is measured eastward or clockwise (looking down from outside the sphere) from the north point of the horizon. An observer facing any direction on the horizon sees the azimuth increasing to the right. The north point is defined as the intersection of the vertical circle through the north celestial pole, NCP, and the horizon. The zenith distance, z, is shown as an arc length mea-

sured down from the zenith along a vertical circle through the star; z may be measured also as an angle at the center of the sphere. An alternative coordinate is the altitude, h, measured up from the horizon along the vertical circle. (b) The observer's view. The azimuth also can be measured as an arc along the horizon; it is equivalent to the angle measured at the center of the sphere between the North point of the horizon and the intersection of the horizon and a vertical circle through the star. Drawings by E.F. Milone.

Figure 2.2. The horizon system: The main features of the horizon system of spherical astronomical coordinates. (a) The outside-the-sphere view. The azimuth coordinate, A, is represented as a polar angle measured at the zenith; A is measured eastward or clockwise (looking down from outside the sphere) from the north point of the horizon. An observer facing any direction on the horizon sees the azimuth increasing to the right. The north point is defined as the intersection of the vertical circle through the north celestial pole, NCP, and the horizon. The zenith distance, z, is shown as an arc length mea-

sured down from the zenith along a vertical circle through the star; z may be measured also as an angle at the center of the sphere. An alternative coordinate is the altitude, h, measured up from the horizon along the vertical circle. (b) The observer's view. The azimuth also can be measured as an arc along the horizon; it is equivalent to the angle measured at the center of the sphere between the North point of the horizon and the intersection of the horizon and a vertical circle through the star. Drawings by E.F. Milone.

to the line between the observer and the zenith. A family of circles (vertical circles) may be drawn through the zenith and the nadir. The centers of these circles must be the sphere's center, where the observer is located (for the time being, we ignore the distinction between the center of the Earth and the observer, i.e., the difference between what modern astronomers call the geocentric and the topocentric systems, respectively). Degrees of altitude are measured up from the horizon toward the zenith along a vertical circle to the object. This gives us one of the two coordinates needed to establish a position on the celestial sphere. The other coordinate is called the azimuth, a term derived from the Arabic as-sumut, "the ways." It is related to the bearing of celestial navigation (such as 22?5 east of North for NNE). Throughout this book, we will use the convention of measuring degrees of azimuth from the North point of the horizon eastward around the horizon to the vertical circle that passes through the star whose position is to be measured.7 From the use of azimuth and altitude, the horizon system is sometimes called the altazimuth system. We will use A for azimuth and h for altitude in formulae, and occasionally, we will refer to the system in terms of this pair of coordinates: (A, h).

The North Point of the horizon is defined as the point of intersection of the horizon with the vertical circle through the North Celestial Pole (NCP), the point about which the

7 An alternative convention is to measure the azimuth from the South point of the horizon westward.

stars in the sky appear to turn. The opposite point on the celestial sphere defines the South Point. For southern hemisphere observers, the South Point of the horizon is defined analogously with respect to the SCP. The visible portion of the vertical circle through the NCP (or SCP) has a special name: It is the celestial meridian or simply the observer's meridian. It has the property of dividing the sky into east and west halves. Objects reach their highest altitude (culminate) as they cross the celestial meridian in the normal course of their daily motions. Circumpolar objects may culminate below as well as above the pole. At lower culminations, the altitudes are lowest, and at upper culminations, they are highest. If neither upper or lower is indicated, the upper is intended in most usages. Another important vertical circle is perpendicular to the celestial meridian. It intersects the horizon at the east and west points. Therefore, a star that is located at the midpoint of a vertical circle arc between the east point of the horizon and the zenith has an azimuth of 90° and an altitude of 45°. Note that no altitude can exceed 90° or be less than -90°, and that the azimuth may take any value between 0° and 360°.

The azimuth coordinate may be considered in any of three ways:

(1) The angle subtended at the center of the celestial sphere between the North point of the horizon and the intersection of the vertical circle through the object and the horizon

(2) The arc length along the horizon subtended by the angle at the center (the observer)

Figure 2.3. The equatorial or "Chinese" System of spherical astronomical coordinates: (a) The outside-the-sphere view. Note that the right ascension (a or RA) is measured eastward (counterclockwise as viewed from above the north celestial pole) from the vernal equinox. The declination, 8, is measured

Figure 2.3. The equatorial or "Chinese" System of spherical astronomical coordinates: (a) The outside-the-sphere view. Note that the right ascension (a or RA) is measured eastward (counterclockwise as viewed from above the north celestial pole) from the vernal equinox. The declination, 8, is measured from the celestial equator along the hour circle through the star. (b) The observer's view. A south-facing observer sees the right ascension increasing along the celestial equator to the left from the Vernal equinox. This is the (RA, 8) version of the equatorial system. Drawings by E.F. Milone.

(3) The polar angle at the zenith, between the vertical circles through the North point and that through the object

The altitude coordinate may be considered in either of two ways:

(1) The angle at the center of the sphere between the intersection of the vertical circle through the object and the horizon

(2) The arc length along the vertical subtended by the angle

This second way of considering the altitude, together with the third way of considering the azimuth, permit transformations to be performed between this system and an equatorial system, which we describe below.

The horizon system depends on the observer's location, in the sense that observers at different sites will measure different azimuth and altitude coordinates for the same sky object. One can, however, envisage the sky independent of the observer, so that the stars are fixed in a framework and can be assigned coordinates that may be tabulated for future use. The equatorial and ecliptic systems are examples of such systems.

2.2.3. The Equatorial or "Chinese" System

In ancient China, another system was in use that is similar to the modern equatorial system. The modern equatorial system enables a transient object to be located precisely among the stars at a particular time. The reference great circle in this system (illustrated and labeled in Figure 2.3) is the celestial equator, the sky analog of the Earth's equator. It is midway between the poles of the equatorial system, the north and south celestial poles, the sky analogs of the North and South Poles of Earth. This is the system that is traced out by the stars' diurnal circles, which are concentric with the celestial equator. The angular distance away from the celestial equator and toward the poles is called declination (from the Latin declinatio or "bending away") and originally referred to the distance from the celestial equator of a point on the ecliptic, the Sun's apparent annual path in the sky. The declination is marked in degrees. The small circles through the object and concentric with the celestial equator are called declination circles because each point on such a circle has the same declination. These small circles for all practical purposes trace out the diurnal motions; only the infinitesimally small intrinsic motions of objects on the plane of the sky during their diurnal motions makes this an inexact statement. The centers of all the declination circles lie along the polar axis, and the radius of each declination circle can be shown to be R cos 8, where R is the radius of the celestial equator (and the celestial sphere), taken as unity, and 8 is the declination in degrees of arc. The declination is one of the two coordinates of the equatoral system. It is the analog of terrestrial latitude, which similarly increases from 0° at the equator to ±90° at the poles. Declinations are negative for stars south of the celestial equator. The analog relationship is such that a star with a declination equal to the observer's latitude will pass through the zenith sometime during a 24-hour day.

Great circles that go through the poles in the equatorial system are called hour circles. They intercept the celestial equator at right angles and are carried westward by the diurnal motions. The celestial equator rises at the east point of the horizon (and sets at the west point), so that successive hour circles intersecting the celestial equator rise later and later from the east point. A coordinate value may be assigned to each hour circle—indeed, if, as is usually the case, the term is interpreted loosely, there are an infinite number of such "hour" circles, rather than merely 24, each with a slightly different time unit attached. An hour circle can be numbered, as the name suggests, in hours, minutes, and seconds of time in such a way that the number increases, moment by moment, at a given point in the sky, other than exactly at a pole. At any one instant, an hour circle at the celestial meridian will have an associated number 6 hours different than that at the east point, or at the west point. The second coordinate of the equatorial system makes use of the hour circles. There are two varieties of this second coordinate. One variety is called the right ascension, and the other is the hour angle.

In modern terms and usage, the right ascension is measured from a point called the vernal equinox8 eastward along the celestial equator to the hour circle through the object. The Sun is at the vernal equinox on the first day of spring (in the Northern Hemisphere); from here, the Sun moves eastward (so that its right ascension increases), and for the next three months, it moves northward (so that its declination increases). The term right ascension derives from the Latin ascensio and from the Greek avafopa (anaphora), a rising or ascension from the horizon. It originally described the time required for a certain arc on the ecliptic (like a zodiacal sign) to rise above the horizon. The time was reckoned by the rising of the corresponding arc of the celestial equator. At most latitudes, in classic phrasing, the risings or ascensions of stars were said to be "oblique" because an angle with the horizon made by a rising star's diurnal arc is not perpendicular to the horizon; but, at the equator, where all objects rise along paths perpendicular to the horizon, the celestial sphere becomes a "right sphere" (sphaera recta) and the ascension a "right" one.

The right ascension increases to the east (counterclockwise around the celestial equator when viewed from above the north celestial pole), starting from the vernal equinox. Objects at greater right ascensions rise later. The analog of the right ascension in the terrestrial system is the longitude, which may also be expressed in units of time, but may also be given in angular units. The analogy here is imperfect because terrestrial longitude is measured E or W from the Greenwich meridian, but right ascension is measured only eastward from the vernal equinox.

As for the azimuth coordinate in the horizon system, the right ascension can be considered in any of three ways:

(1) As the angle measured at the center of the sphere between the points of intersection with the celestial equator of the hour circle through the vernal equinox and the hour circle through the star

8 The terms vernal equinox and autumnal equinox derive from the times of year (in the Northern Hemisphere) when the Sun crosses the celestial equator. "Equinox" is from the Latin aequinoctium, or "equal night."The actual point in the sky was called punctum aequinoctialis. In modern usage, "equinox" applies to both the time and the point. References to the times of year are more appropriately given as "March" and "September" equinoxes, and "June" and "December" solstices, at least at the current epoch and in the present calendar. In the distant past, this usage could be confusing because historically civil calendars have not been well synchronized with the seasons, and given sufficient time, the month in which the equinox or solstice occurs will change (see §4). We will use the terms as defined for the Northern Hemisphere in their positional meanings generally and in their seasonal meanings only to avoid ambiguity in the distant past.

(2) As the arc along the celestial equator between the hour circles through the vernal equinox and that through the star

(3) As the polar angle at the celestial pole between the hour circles

Similarly, as for the altitude coordinate in the horizon system, the declination can be considered in either of two ways:

(1) As the angle measured at the center of the sphere between the celestial equator and the star

(2) As the arc length, along the hour circle through the star, between the celestial equator and the star. This second way of considering the declination and the third way of considering right ascensions permit transformations among the equatorial and other coordinate systems to be made.

The declination is always given in angular measure (degrees, minutes of arc, and seconds of arc). The symbols for right ascension and declination are a and 8, but the abbreviations RA and Dec are often used.

The celestial equator has a special significance because objects on it are above the horizon for as long a time as they are below the horizon. The word equator derives from aequare, which means equate. When the Sun is on the celestial equator, therefore, day and night are of nearly equal length.

The equatorial system just outlined is completely independent of the observer—it is not directly tied to the horizon system, but there is another equatorial system that has such a connection. Figure 2.4 shows this observer-related equatorial system. In the ancient world, at least some separations of objects on the sky were measured by differences in their rise times. The modern system that derives from this is identical to the first equatorial system except for the longitudinal coordinate and the reference point. Instead of right ascension, it uses the hour angle, an angular distance measured along the celestial equator westward from the celestial meridian. The hour angle can be symbolized by H, or HA (we reserve h for the altitude) and usually is also expressed in units of time. It indicates the number of hours, minutes, and seconds since an object was on the celestial meridian. It therefore varies from 0 to 24 hours, but for convenience, it is often taken positive if west of the meridian and negative if east. The connection between the right ascension and the hour angle is the sidereal time (see §4).

Analogously with the azimuth, and the right ascension, the hour angle can be considered in any of three ways. The use of the polar angle between the celestial meridian and the hour circle through the star permits transformations between the horizon and the (H, 8) equatorial system (recall that we sometimes refer to a coordinate system by its coordinates expressed in this way). The transformation equations and procedures are described and illustrated in the next section.

The hour angle is also an analog of terrestrial longitude, in that it is measured along the celestial equator, but, again, the analogy is limited—in this case, because the hour angle is measured only from a local celestial meridian, whereas

Figure 2.4. A variant equatorial system, in which the observer's hour angle, H, is used instead of the right ascension: (a) The outside-the-sphere view. Note that H is measured westward from the celestial meridian. The declination is defined as in the (RA, 8) system. Note that the altitude of the north celes tial pole is equal to the latitude, and the limiting (minimum) declination for circumpolar objects is 90 - f. (b) The observer's view. A south-facing observer sees the hour angle increasing to the right. This is the (HA, 8) system. Drawings by E.F. Milone.

Figure 2.4. A variant equatorial system, in which the observer's hour angle, H, is used instead of the right ascension: (a) The outside-the-sphere view. Note that H is measured westward from the celestial meridian. The declination is defined as in the (RA, 8) system. Note that the altitude of the north celes tial pole is equal to the latitude, and the limiting (minimum) declination for circumpolar objects is 90 - f. (b) The observer's view. A south-facing observer sees the hour angle increasing to the right. This is the (HA, 8) system. Drawings by E.F. Milone.

terrestrial longitude is measured from the Prime Meridian, at Greenwich, England.

Note that the connection between the horizon and (H, 8) systems is the celestial meridian, where H = 0. Figure 2.4a illustrates how the hour angle and the declination are defined, and how the "declination limit" of circumpolar stars for a given latitude, f, can be determined.

Chinese star maps were commonly laid out in the (a, 8) manner of an equatorial system. Such a chart can be seen, for example, in Needham 1959, Fig. 104, p. 277). In this chart, a horizontal line though the chart represented the celestial equator. A hand-drawn curve arcing above the celestial equator represents the ecliptic or path of the Sun between the vernal equinox and the autumnal equinox. Everything on this chart represents a two-dimensional mapping of the interior of a celestial sphere onto a two-dimensional surface. Such charts have been found from as early as the 4th century a.d. in China. The data in them are older still; polar distances (90° - 8) found in Chinese catalogues have been used to date the catalogues themselves. The coordinates are a kind of hour angle, measured with respect to the edge of a xiu or lunar mansion, and a polar angle, a kind of anti-declination (Needham/Ronan 1981, p. 116). It is possible to date such catalogues and charts because the right ascensions and declinations of a star change with time, a phenomenon arising mainly from the precession of the equinoxes (see §3.1.6). According to Needham (1981a, p. 115ff), the chart has a probable date of ~70 b.c.

The equatorial system became widespread in Europe only after the Renaissance. Figure B.2 shows the polar views of the equatorial system, looking outward toward the north and south celestial poles. The sky centered on the north celestial pole is also depicted in one of the most famous of all historical star charts: the Suchow star chart of 1193 a.d. (Figure 10.7). The circle about halfway from the center is the celestial equator, which the inscription that accompanies the chart calls the "Red Road." It "encircles the heart of Heaven "

2.2.4. Transformations Between Horizon and Equatorial Systems

All students of archaeoastronomy should know how to transform coordinates between systems. It is easy to get equatorial system (H, 8) coordinates from horizon system (A, h) coordinates, given the observer's latitude and some knowledge of spherical trigonometry. Using the "sine law" and the "cosine law" of spherical trigonometry, that are described and illustrated in Schlosser et al. (1991/1994, Appendix A) and basic trigonometric definitions and identities also given there, we depict the appropriate spherical triangle, the "astronomical triangle," in Figure 2.5.

The resulting transformation equations are sin 8 = sin f • sin h + cos f • cos h • cos A from application of the cosine law, and sin H =

from application of the sine law.

Suppose at a latitude, f = 30°, the altitude of a certain star, h = 20°, and the azimuth, A = 150°. From (2.1), sin 8 = sin(30o) • sin(20o) + cos(30°) • cos(20°) • cos(150°) = 0.50000 • 0.34202 + 0.86603 • 0.93969 • (-0.86603) = -0.53376

Substituting this value into (2.2), we find that

azimuth (AX celestial meridian-/' |
^__Zzenith |
hour angle (—H) \north celestial pole | ||

declination |

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