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Astronomical background II 1 Obliquity and precession of the equinoxes The angle ε between the ecliptic and the celestial equator is called the angle of obliquity. It is the angle between the fundamental planes of these two coordinate systems, that is, between the Earth’s € orbital plane and its equatorial plane. Therefore, it is the same as the angle between the ecliptic polar axis (the direction perpendicular to Earth’s orbit) and the Earth’s axis (which is perpendicular to its equator); it is a measure of Earth’s axial tilt. We know this angle to be about ε = 23°26 ′ . It is this angle that controls the climatic variation of the seasons on Earth and helps to define the Tropics of Cancer and Capricorn. € But in the same way that the spin axis of a top wobbles about as it spins across a flat surface, the Earth’s axis also wobbles relative to its orbital plane, so that the polar point P describes a small circle around the north ecliptic pole (in Draco). This motion is extremely slow, making its way once around this circle every 25,800 years, but it has the effect of sliding the north celestial pole P away from Polaris over time. This means that the north celestial pole was much closer to the dim star Thuban ( α Draconis) in 3000BCE, and will drift towards another dim star, γ Cephei, in 4000CE. € € Astronomical background II 2 It also has the effect of sliding the celestial equator around the ecliptic, changing the position of the First Point of Aries in the celestial sphere, by about 50 seconds of arc per year or by 30° every 2150 years. This motion of the Earth is called the precession of the equinoxes. During the first millenium BCE, during the height of Babylonian astronomy, the First Point of Aries actually was found within the constellation Aries, but in time it drifted into Pisces. Currently we find ourselves at “the dawning of the Age of Aquarius”, where the First Point of Aries will travel for the next 2150 years or so. In addition to precession, the circular path of the north celestial pole P undergoes a small wiggling motion called nutation [L. nutare = to nod], which causes P to oscillate back and forth every 18.6 years (6798 days) as it precesses. This motion is due to tidal forces among the planets and the Sun. Astronomical background II 3 Solar and sidereal days Local noon occurs when the Sun makes its upper transit, crossing the meridian. For observers in the Northern Hemisphere, this is when the Sun is furthest south in the sky (and when shadows point north and are the shortest). The time between two consecutive local noons is the solar day. Of course, the solar day marks one rotation of the Earth on its axis. But as we have already noted, the Sun also moves against the celestial sphere by a small distance (about 1° of right ascension) each day; that is, the Earth moves a small distance along its orbit about the Sun during a day. As a result, when the Earth makes one 360° revolution about its axis, the Sun is no longer in the same position it held on the celestial sphere at midday on the day before: it is 1° further to the east. It must rotate a further 1° to complete the solar day. Consequently, the period of time it takes for the Earth to make one full revolution of its axis is less than one solar day. However, it is the same as the time between successive transits of any star in the celestial sphere. We call this the sidereal day [L. sidus = constellation]. Astronomical background II 4 How long is the difference between the solar and sidereal day? We know that in the span of a year, there are 365.25 solar days. But in that same time, there must be 366.25 sidereal days, since in the time it takes the Earth to go completely around the Sun, it rotates against the celestial sphere one more time, the accumulation of all the tiny deficits in rotation after each solar day throughout the year. Thus, 366.25 sidereal days = 365.25 solar days 365.25 1 sidereal day = ⋅1 solar day 366.25 365.25 = ⋅ 24 h 0 m 0 s 366.25 = 23 h 56 m 4 s meaning that the sidereal day is 0 h 3 m 56 s shorter than the solar day. This explains why stars that € are seen to rise on a particular night at a particular time will be seen to rise the next night 4 minutes € earlier. The cumulative effect over time of this phenomenon is to change the set of constellations that are visible at the same time every night. Over the course of the year, each constellation graces the night sky for a time, rising earlier and earlier, then vanishing in the sunlight to return again the next year. Astronomical background II 5 The equation of time We generally believe that the solar day equals a constant 24 hours. But, surprisingly, the solar day varies in length across the year, by as much as 50 seconds at its extremes! That the length of the day was not constant was recognized even by ancient astronomers, who observed it by recording the behavior of shadow clocks over the course of a full year. But these effects were not large enough to concern them, and their tools were not sensitive enough to measure the effects accurately until the development of reliable clocks that marked time in constant units in the 17th c. The use of such clocks made it possible to fix the length of the day at 24 equal hours; however, this period of time is actually the mean solar day, which averages out the annual variations in the apparent solar day. The length of the apparent solar day varies for two main reasons, the effects of Earth’s elliptical orbit and the obliquity of the ecliptic. Let’s explain this more fully. Astronomical background II 6 (1) Kepler’s Laws state that a planet moves more quickly when it is closer to the Sun in its elliptical orbit (Earth reaches perihelion [Gr. peri + helios = around + Sun], its point of closest approach, in early January, and aphelion [Gr.apo + helios = away from + Sun] in early July). Consequently, winter solar days should be slightly shorter than summer solar days. The Sun transits precisely at noon on local clocks on those days, but for many days after perihelion, the cumulative effect of consecutive short solar days causes the transit to occur earlier and earlier on the clock since the clock continues to measure out a constant 24 hour day. By the end of March, local noon arrives at about 11:53am. This effect reverses as the Sun approaches aphelion, but then after aphelion, the cumulative effect of consecutive long solar days causes the transit to occur later and later by the clock so that by the end of September, local noon occurs at 12:07pm. Thereafter, the trend reverses yet again. This process repeats every year. (2) Even if the Earth’s orbit were perfectly circular, the Sun would still move along the ecliptic a small equal amount every day. But as the ecliptic is tilted with respect to the celestial equator, the Sun moves a shorter distance of right ascension on the days of the equinoxes (since the ecliptic is most angled against the equator then); Astronomical background II 7 the more slowly moving Sun makes for longer than average days then (by as much as 20 sec per day). The Sun moves a longer distance of right ascension on the days of the solstices (since the ecliptic is parallel to the equator then); the more quickly moving Sun makes for shorter than average days (again, by as much as 20 sec per day). Therefore, since the Sun would transit at 12:00 noon on the vernal equinox, the progressively longer solar days that follow would cause the transit to occur later and later by the clock so that by the cross-quarter day at the beginning of May, the Sun would transit at about 12:10pm. As the days shorten moving towards the summer solstice, the midday transit begins to occur earlier and earlier, so than by the beginning of August, it occurs at about 11:50am. Then the effect reverses again through the autumnal equinox (when local and solar noon agree) to reach a second turning point around the beginning of November, when solar noon is at 12:10pm. This oscillation continues through the winter, when at the fourth cross-quarter day in early February, solar noon occurs its earliest, at 11:50am. These two phenomena, the effect of the Earth’s elliptical orbit and the obliquity of the ecliptic, actually work in concert to cause a regular and annual variability in the length of the solar day. Astronomical background II 8 The difference between apparent local noon (the time of the Sun’s transit) and mean local noon (when a clock set to local time would read 12:00) is called the equation of time. Controling for this variation is important if one wants a sundial, for instance, to synchronize time with a watch. Finally, the sidereal day, which is 23 h 56 m 4 s long, is also undergoing very slight long-term changes, called secular effects [L. saeculum = age or era]. Tidal gravitational forces cause the Earth’s € rotation to slow down slightly, lengthening the sidereal day by about 2 milliseconds (0.002 sec) every century, a very small change, to be sure. Astronomical background II 9 Definitions of the year The ancient Chinese marked off the years by determining the solstices from shadow clocks: the summer solstice occurred when midday shadows were shortest, the winter solstice when they were longest. The Incas, living in the tropics, used the absence of shadows at noon shining through a long tube to mark the annual day when the Sun was directly overhead at midday. And the Egyptians used the heliacal rising of Sirius to time the beginning of their year. In all these cases, the year was found empirically. As astronomical knowledge improved, it became possible to identify more accurately not only how long the year was, but what phenomena controlled its passing. As the year was typically tied to seasonal events, the earliest understanding led to the definition of a year as the time between consecutive summer (or winter) solstices (or between consecutive vernal or autumnal equinoxes; all four definitions are equivalent). It was known from prehistoric times that this is often 365 days, and sometimes 366. Astronomical background II 10 Modern astronomers now define the tropical year to be the time it takes the Sun to return to the First Point of Aries along the ecliptic. (This is essentially the same time it takes for the Sun to return each year to the position of the Tropic of Cancer at the summer solstice.) To 10 decimal places of accuracy, 1 tropical year = 365.2421896698 (mean solar) days However, there is one important flaw in using this event to define the year. Nutation and precession of the equinoxes causes the First Point of Aries to shift over time along the celestial equator. This causes the tropical year to deviate by about 10 minutes’ time (or 0.014 days) from the value above from year to year, so the above must be considered a mean value. Other secular effects involving the gravitational interaction of the Earth with other bodies in the solar system are working to shorten the length of the year very slightly over time: the above mean value is now 12 seconds shorter than it was 2150 years ago when the Greek astronomer Hipparchus made the first serious attempt to measure it. (See Table 2.3, p. 33, for a table of historical measurements of the tropical year.) Astronomical background II 11 The lengths of the seasons By Kepler’s Laws, we know that the Earth’s orbit is elliptical, reaching perihelion in early January when the Earth travels fastest around the Sun, and aphelion in early July when the Earth travels most slowly. It follows that the four seasons, which have the solstices and equinoxes as their division points, are not of equal duration. In particular, winter (in the Northern Hemisphere) is the shortest season and summer is the longest: Season Begins at Approx. date Length (days) Spring vernal equinox Mar 21 92.8 Summer summer solstice Jun 21 93.6 Fall autumnal equinox Sep 22 89.9 Winter winter solstice Dec 21 89.0 Astronomical background II 12 The month and the Moon’s motions The Moon orbits the Earth. It is new when it is in conjunction, that is, when it has the same ecliptic longitude as the Sun. In this geometry, the light of the Sun falls on the side of the Moon facing away from the observer. This also means that the Moon must be near the Sun in the sky, so the Moon at conjunction is not easy to see, except just before sunrise or after sunset (or when it passes directly in front of the Sun during a solar eclipse!). Likewise, the Moon is full when it is in opposition, that is, when its ecliptic longitude differs by 180° with that of the Sun. Here, it is roughly in line with the Earth and Sun, but the light of the Sun strikes the side of the Moon facing the observer. Here the Moon is easy to see, as it stays in the sky all night. When the Moon passes directly in the path of the Sun and the Earth, a lunar eclipse takes place and the face of the Moon is temporarily blocked by the Earth itself. The time between successive conjunctions is one lunation, also called a synodic month [Gr. synodos = meeting]. Lunations vary in length by as much as 3.5 hours from the mean synodic month of 29.53059 days. Tiny secular effects are increasing the synodic month by about .02 sec per century. Astronomical background II 13 In the same way that the sidereal day varies from the solar day, a sidereal month, the time it takes for the Moon to reoccupy the same position on the celestial sphere, differs from the synodic month. It is a bit shorter, 27.32166 days, and for a similar reason that tells why a sidereal day is shorter than a solar day. The orbit of the Moon is in a plane that is tilted with respect to the plane of the Earth’s orbit by 5°8´. This explains why there aren’t a pair of eclipses every month: the Moon only crosses the ecliptic twice each month, at a pair of points called the lunar nodes. As it crosses the ascending node, it moves above the ecliptic, and when it crosses the descending node, it goes below the ecliptic. Only when these crossings occur at conjuction is there is a solar eclipse, and when they occurs at opposition, there is a lunar eclipse. But the nodes themselves rotate through the ecliptic over time, with a period of 18.6 years (called precession of the nodes), so eclipses occur only infrequently. Astronomical background II 14 The time between successive passages of the Moon through its ascending node is called the draconic month; this term derives from the astrological practice of labeling the ecliptic longitude of the ascending lunar node (which varies over time) with the symbol , called the dragon’s head. (The ecliptic longitude of the descending node was denoted , the dragon’s tail.) The draconic month is 27.21222 days, slightly shorter than the sidereal month, since the nodes rotate to meet the orbiting Moon. The 5° tilt also allows the Moon to rise a bit higher than or fall a bit lower than the Sun along the ecliptic. Therefore, its declination reaches a maximum of δ = 28°34′ = 23°26 ′ + 5°8 ′ and is highest in the sky when the Moon is halfway between ascending and descending nodes at a summer solstice. This event is called the major lunar € standstill (analogous to a solstice event for the Sun). Two weeks later, the Moon will be in the opposite position in its orbit, so it will reach a (near) minimum declination of δ = −28°34 ′ . € Astronomical background II 15 Ancient Babylonian astronomers discovered that, although the synodic month and draconic month are unequal, their cycles synchronized every 18 years or so. Since the synodic month determined the cycle of new and full moons and the draconic month determined the cycle of passage through the nodes, the combination of these cycles controlled the pattern of eclipse events. Specifically, they knew that the synodic month was 29;31,50 days long (sexagesimal notation) and the draconic month was 27;12,42,30 days long, and noticed that 223 synodic months = 223 × ( 29; 31, 50 days) = 6585;18, 50 days was roughly equal to € € 242 draconic months = 242 × ( 27;12, 44 days) = 6585; 21, 28 days (The difference of 0;2,38 days equals about 1 hour.) This period of 6585 13 days, or 18 years plus 11 13 days, came to be known the saros. € € Astronomical background II 16 Any solar or lunar eclipse event would then be followed 6585 13 days later by another of the same type (although not in the same geographical location: the extra 13 day means that the event takes place 13 of the way around the Earth from € where it occurred in the previous cycle). € For instance, the most recent solar eclipse in € Cincinnati occurred on May 10, 1994. Another solar eclipse will follow on May 20, 2012 (18 years 11 days later) at roughly the same latitude but 120° of longitude further west, in the North Pacific (visible in the Aleutian Islands). Astronomical background II 17 Finally, we recognize that the orbit of the Moon is not circular, but elliptical, like that of each of the planets about the Sun. So it comes closest to the Earth at the point of perigee [Gr. peri + ge = around + Earth], and is furthest away at another, the point of apogee [Gr. apo + ge = off of + Earth]. These two points are together called the Moon’s apsides (sing. apsis) [Gr. apsis = the point on a wheel where the two ends are fastened together]. The line of apsides connects the points of perigee and apogee and, because of solar gravitational forces, it precesses about the Moon, making a complete rotation once every 8.85 years. At perigee, the Moon is a bit larger in appearance ( 0°34′ wide) to Earth observers than at apogee ( 0°29′ wide). € € The Moon’s anomaly [Gr. anomalia = irregularity] is its angular measure east of perigee along its orbit, so the anomaly at perigee is 0° and at apogee is 180°. The time between successive perigee positions is therefore called the anomalistic month; it is 27.55455 days, a bit longer than the synodic month since the line of apsides rotates ahead of the Moon in its orbit. The saros is also conveniently equal to about 239 anomalistic months, so the size of the Moon at successive eclipses in a saros cycle is roughly equal.