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where the gravitational constant, G = 6.67 x 10-11m3Kg-1s-2, M0 is the mass of the Sun, [email protected] is the mass of the Earth-Moon system, r is the instantaneous distance to the Sun, and a is the semimajor axis. The distance varies with location in the orbit:

where e is the orbital eccentricity and u, the true anomaly, is the angular distance of the Earth from perihelion. At perihelion, therefore, u = 0, so that and

so that the ratio of these two extremes, vjva, is (1 + e)/(1 - e) = 1.034, because the Earth's orbital eccentricity is 0.0167. The variation in Earth's speed is therefore about ±1.67% of the average speed, and the Sun's annual motion reflects this variation. The variation in the observed angular rate of the Sun, dQ/dt, is related to the velocity v and the distance r from the Earth-bound observer, over a short arc of the orbit by the expression:

Therefore, at perihelion, and at aphelion, dQ/dt =

and the ratio of these angular speeds is [(1 + e)/(1 - e)]2 = 1.069, so that the angular speed of the Sun along the ecliptic varies by about ±3.34% from the mean.

Figure 4.9. The equation of time: Apparent - mean time or HA& - HAMS. Drawing by E.F. Milone from a Lotus 1-2-3 spreadsheet (Lotus Development Corporation, an IBM company, Cambridge, Massachusetts).

in the apparent eastward motion of the Sun along the ecliptic and causes the "inequality of the seasons," described in §2.3.1 and in the context of Mediterranean cultures in §7.

In addition to the Sun's varying rate of eastward motion across the sky, there is another reason for the difference between Apparent Solar Time [as defined by (4.4)] (AST) and our modern Mean Solar Time (MST).7 This reason is that AST is a local time; it is defined by the local hour angle of the Sun plus that concession to daylight chauvinism, 12h. Modern time pieces carry a form of MST defined in terms of the hour angle of an imaginary body called the Mean Sun, which travels at the average angular rate on the celestial equator:

The difference between instantaneous or apparent solar time and mean solar time is called the equation of time:

Here, HAMS is the hour angle of the Mean Sun. When E is plotted against time, a double-peaked curve is obtained, as shown in Figure 4.9. Because hour angle is measured positive to the west, when E > 0, HA& > HAMS so that the apparent Sun is running faster than the mean Sun. When E < 0, HA& < HAMS, in which case, the apparent Sun is running slower than the Mean Sun.

A plot of the Sun's declination against the equation of time produces a figure 8 design, called the analemma (Figure 4.10a, from the same spreadsheet as in Figure 4.9). Its shape can be seen on the gnomon of a sundial that was designed for the University of Calgary's Rothney Astrophysical Observatory (RAO) by Prof. T.A. Clark and T. Kirkham. See Figure 4.10b.

7 Mountain Standard Time, defined as the mean solar time on the the

7th-hour meridian west from Greenwich, is also abbreviated MST. The context usually determines which is meant.

Figure 4.10. (a) The analemma for 2000 a.d. (b) A modern sundial incorporating the analemma on its stylus. Courtesy, Dr. T.A. Clark, who along with T. Kirkham, supervised its design and construction for the opening ceremony of the RAO.

The time on our watches is not a local time, in general, but is related to the Mean Solar Time at a particular meridian or longitude circle. By general agreement, most civic entities around the world assign the Mean Solar Time at particular meridians to zones surrounding those meridians. The difference between Local Mean Solar Time and the Local Civil Time (i.e., the time in effect at that place) is the time interval required for the Mean Sun to move through the longitude difference between the standard meridian and the local meridian. If the standard meridian is east of the observer, the standard time will be later than the observer's apparent time, and if it is west, the standard time will be earlier. The Mean Solar Time at the Greenwich meridian is called Universal Time (UT),8 or, Greenwich Mean Time (GMT); it is related to the MST at any other meridian through the longitude, 1, of that meridian:

where the positive sign applies for a site east of the Greenwich meridian, and the negative, west.

8 There are three varieties of Universal Time in use: UT0, UT1, and UTC. Which "UT" is intended must be decided by context. The motion of the Earth's geographic pole causes a slight variation in the location of an observer's celestial meridian. UT0 is determined directly from stellar observations and takes into account neither the nonuniform rotation of the Earth nor the effect of polar motion;UT0 is therefore an approximation to the "true" UT, at a particular meridian, only. UT1 is the result of correcting UT0 for the effect of polar motion;UT1 is commonly used in navigation, and in, for example, tables of the Astronomical Almanac. Coordinated Universal Time (UTC) is the time that is broadcast as time signals around the world (cf. the preceding section). (UTC) is based on International Atomic Time (TAI) and differs from it by an integral number of seconds, which varies with time. The difference varies with time because UTC is kept within 0f9 of UT1 by the introduction of leap seconds when necessary, normally at the end of either December or June. AUT1 = UT1 - UTC is transmitted in code on the broadcast time signals.

The actual source of UT is not the position of the Mean Sun, which, after all, can not be observed; it is taken from the positions of the "fixed" stars (fixed, that is, at a particular equinox and epoch). The basic relationship between Local Mean Solar Time and Local Sidereal Time is

where HAMS and RAMS are the hour angle and right ascension of the Mean Sun, respectively. The relationship between the sidereal and mean solar time at Greenwich is tabulated in the Astronomical Almanac (see Appendix A). Greenwich Mean Sidereal Time is, or rather has been, derived from automated transit observations of stars. Corrections for the variation of the geographic pole (which lead to slight variations in the observer's meridian) are also derived from observations. At present, Very Long Baseline Interferometry techniques are used to determine precise positions in the sky of quasars and other effective point sources. The annual editions of the Astronomical Almanac provide other details regarding time corrections and should be consulted for further information in critical cases. See Woolard and Clemence (1966) for underlying principles of time-keeping, Green (1986, Ch. 10), Stephenson (1997), Cox (2000), the current year's edition of the Astronomical Almanac,9 and the Reports and Transactions of the most

9 The projected modern system of time for many purposes is the Barycentric Dynamical Time or TDB, the French acronym. It is referred to as a dynamical time scale, but the gravitational theory on which such a time would depend is yet to be adopted. At the moment, the Terrestrial Dynamic Time (TDT), now more generally called "Terrestrial Time" (TT), based on the SI second, is in effect. TT differs from UT by At = TT - UT. We discuss this difference at the end of this chapter and in §5 because it affects the times and locations at which eclipses are visible.

recent General Assembly of the International Astronomical Union (IAU) for further details about what times scales are currently applied.

Modern sundials usually contain a correction for the equation of time and for the difference in longitude between the standard time zone boundary and the local site, in the form of an inscribed table. An additional, seasonal correction of +1h must be made in most places in North America and many places around the world for daylight savings time. The value for E can be read off Figure 4.9, or it can be found on the analemma, the figure 8 curve frequently found on terrestrial globes (see also §2.3.1). It indicates both E and the declination of the Sun as a function of date; it is also independent of latitude or longitude and is therefore appropriate for anywhere on Earth. The Greek astronomers of the 2nd century were fully aware of the nonuniform character of the Sun's movement due to the obliquity of the ecliptic and to the Sun's variable rate10 on the ecliptic, and they made use of a correction analogous to our Equation of Time. The analemma and thus the Equation of Time as well as the annual declination variation of the Sun are illustrated in a sundial constructed by Prof. T.A. Clark for the opening ceremony of the Rothney Astrophysical Observatory on January 7, 1972.

4.1.1.3. Mechanical Devices

The sundial was and still is useful for keeping track of the time on a sunny day. Before the development of the magnificently elaborate town clocks of Europe, what was used for telling time on a cloudy day, or during the evening? The tower of Chou Kung (Needham 1981, p. 136) and the Tower of Winds (Robinson 1943; Noble and de Solla Price 1968; Bromley and Wright 1989; Kienanst 1993) likely held clepsydras or water clocks. Water clocks made use of regulated dripping of water from a large reservoir into a container, the weight of which increased as the water level in it rose. The container could be permitted to pull a rope downward or contain a float that would permit a different operation. The action would ultimately cause the rotation of a wheel or of an indicator. A water clock could be calibrated with the sundial when conditions permitted, so that timely business could be carried on as usual. For short intervals of time, hourglasses, filled with sand, could be used.

4.1.1.4. Uniform Time Intervals

Ways of reckoning time have changed greatly since ancient times. Although the rotation of the Earth is still the basis for civil time, we use atomic standards to measure precise intervals of time, because, compared with a perfect clock, the Earth runs "slow," as we discuss later. The most precisely determined time scale currently in use for astronomical purposes is called "International atomic time" or Temps Atomique Internationale (TAI), and the fundamental unit is the SI second (in the international system of units). By interna

10 Called by Ptolemy (Toomer 1984, p. 170) the Sun's "apparent anomaly."

tional agreement, it is equal to the interval of time that is measured by 9,192,631,770 oscillations of the radiation emitted by an atomic transition of the element Cesium 133. However, civil time-keeping is still tied to the rotation of the Earth.

The rotation of the Earth is not uniform, but varies randomly, periodically, and secularly. Short-term variations arise because of mass displacements caused by tidal deformations, ocean and atmospheric tides, and geophysical effects; the rotation of the Earth is slowing (the secular variation) because of tidal friction, although other causes may contribute also. The existence of uniform time intervals, by which we can measure the passage of time with precision and accuracy, permits us to correct our clocks for the nonuniform time-keeping provided by the Earth. The corrections are not applied to the observationally based Universal Time (UT1), however, but to a time based on International Atomic Time (TAI, the French acronym) (see fn. 6), namely, the Coordinated Universal Time (UTC). UTC is kept within 0.s9 of UT1 by adding a "leap second" when needed, usually at the end of December or of June. UTC is broadcast on selected short-wave frequencies by national time regulation agencies. In §4.5 and §5.2.1.3, we discuss the observational evidence for the gradual slowing of Earth's rotation.

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