on the periodicity of the equinoxes, rather than on the actual time it takes the Earth to go once around the Sun, themodern calendar is based on the “mean tropical year,” which is shorter than the sidereal year by about 20 minutes(i.e., 1/26,000 of a sidereal year).7This slow change of the positions of the poles, equinoxes, and solstices, relative to the distant stars, implies that the signs of the Zodiacare not fixed with respect to the solar calendar. For example, the “Tropic of Cancer” was so named because the position of the Sun atthe time of the northern solstice used to lie within the constellation of Cancer, but today the northern solstice actually lies in Taurus.The first equinox, which used to lie in Aries when the ancient Babylonians developed the calendar, has since shifted to Pisces and willmove into Aquarius around the year 2,600. This last circumstance has been the source of much mystical twaddle about the “dawningof the Age of Aquarius.”
8θ’LambPdaynightmOcelestial equatorPαm(a)nightdaymmbaδcc(b)FIG. 5: (a): Cross-section of the celestial sphere along the Earth’s axis of rotationPP, centered at the positionOof an observerat geographic latitudeL.The pointmcorresponds to the maximum altitude of the Sun, andmto the minimum altitude.(b): Cross-section of the celestial sphere, centered at pointaand perpendicular to the Earth’s axis of rotation. The pointccorresponds to sunrise and ¯cto sunset. The arrows show the direction in which the celestial sphere rotates with respect to theobserver atO.The position of the perihelion with respect to the distant stars also varies, but more slowly, with a period ofabout 112,000 years, which is equivalent to a displacement of about 0.32◦per century. This precession results fromperturbations to the motion of the Earth around the Sun caused by the gravitational pull of the Moon and the otherplanets, and to a lesser extent also by relativistic corrections to Newtonian gravity.8The respective precessions of the equinox and the perihelion proceed in opposite directions along the ecliptic,causing the value ofv0in Eq. (12) todecreaseby about 1.7◦per century.9Though for our purposes such precision ishardly justified, if we wished to take into account those precessions, we could makev0in Eq. (12) a time-dependentparameter.IV.DURATION OF DAYLIGHTThe computation only up to Eq. (5) suffices to obtain a good estimate of the number of hours of daylight for a givenday of the year, if we do not care for the precise time of sunrise and sunset. Here the main approximation is thatthat the azimuthal angle of the Sun in the ecliptic frame,φ, will be taken to be fixed during a given calendar dated. For definiteness, let us say thatφis computed at noon for the date and location of interest, the correspondingtime being translated to Universal Coordinated Time (UTC), for use in Eqs. (13) and (14).Figure 5(a) shows a cross-section of the celestial sphere, parallel to the Earth’s axis of rotationPP. As the sphererotates about the observer at pointO
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