## The Orbit Of The Earth And The Calendar

How long is a month? Even laying aside calendar months, with their variety of lengths (30 or 31 days, 28 for February but 29 in a leap year), the question is not a trivial one to answer. We start with a related question: how long is a year?

Before one can answer this, one must ask a simple but deceptive question. What is the crux of the matter at hand? In the case of the Gregorian reform—the alteration of the calendar by the Roman Catholic Church in 1582—the essential consideration was trying to maintain the date of the spring equinox. The "year" required for that aim is the time between such equinoxes, and that is not the same as the time taken to complete one orbit. In fact, due to several vagaries the notion of a period "to complete one orbit" has little meaning in itself. One must be very definite about the phenomenon of interest that is employed to define the start and end of the orbit, because different start and end points lead to different values for the year length.

Prior to the Gregorian reform, and after Julius Caesar introduced his eponymous calendar, a leap year had been employed every fourth year, producing an average year length of 365.2500 days. The Gregorian calendar reform amended the leap year rule such that the years A.D. divisible by 100 but not by 400 are common years (that is, not leap years), with no February 29. The result is that 97 leap year days are added to four centuries, and so the average year length is equal to 365.2425 days. (That comes from the fact that 97 divided by 400 equals 0.2425.) This "year"—the mean Gregorian year—is an artificial length of time, invented by humankind. One next needs to ask how long the natural or astronomical year might be, and compare the two.

The terrestrial orbit is shown schematically in Figure A-3. The large arrows indicate the spin axis of the Earth, which for the time being is assumed not to alter in orientation. Winter solstice occurs when that arrow is pointed as far as possible from the Sun, and at that time the Sun reaches its most southerly rising point during the year, on about December 22. In essence this is the

FIGURE A-3. The orbit of the Earth about the Sun (solid circle at center), in a slant angle view; in reality the terrestrial orbit is fairly close to circular. The positions of our planet at spring equinox (SE), summer solstice (SS), autumnal equinox (AE), and winter solstice (WS) are shown, the long arrows indicating the direction of our spin axis. The small cross indicates the position of the Earth when at perihelion (closest approach to the Sun) in early January.

shortest day (assuming you are in the Northern Hemisphere). The summer solstice around June 22 is when the Sun rises at its most northerly point, and the daytime hours are longest.

In between are the two equinoxes. Despite popular belief, it is not quite true that at the equinoxes the number of daylight hours equals that of nighttime hours, as the word "equinox" would suggest, because there is sunlight available for some time before sunrise and after sunset, plus other complicating factors. The equinoxes are defined astronomically, as follows. If one extrapolates the equator of the Earth out into the sky, the celestial equator is delineated as a circle cutting the celestial sphere into two. From spring (or vernal) equinox to autumnal (or fall) equinox the Sun is north of the celestial equator, and south thereafter. The equinoxes are the instants at which the Sun appears to cross the equator, on about March 20 and September 22 (the dates vary slightly with the leap-year cycle). In March it is heading northwards, in September it is heading southwards.

It happens that a year counted from one spring equinox to the next averages to about 365.2424 days, and that is distinct from the time between summer solstices, which is 365.2416 days. The times between winter solstices, or between autumnal equinoxes, also give different values for the astronomical "year." The reason for these values being different is that the speed of the Earth changes during its orbit. The average of the four is 365.2422 days, which is termed the mean tropical year.

It is a mistake, often made, to compare the mean duration of the year in the Gregorian calendar with the tropical year; the difference between them, about 0.0003 days, suggests that a single day correction might be required every three or four millennia. Actually the mean Gregorian year should be compared with the spring equinox year, the difference between these being but 0.0001 days, three times less. This might suggest that a correction of one day every ten millennia might be needed. However, the latter would again be based on a false premise: because the perihelion point of the Earth is moving, the lengths of all these "years" are changing from one century to the next. Another matter to consider is the fact that the Earth's spin rate is slowing, making the days longer, and so reducing the number of "days" in a year. It happens that, in the present epoch and continuing for a couple of thousand years, the Gregorian leap year rule actually provides for a better approximation to the necessary year length than most people imagine. Indeed many prominent astronomers have been led astray by misunderstanding what is going on.

The above should not be construed as a statement in praise of the Gregorian rule for leap years as used in the Western calendar. The system of dropping three leap-year days in four centuries results in the spring equinox shifting over a total span of 53 hours, between March 19 and March 21. In computing the date for Easter, the Church actually stipulates March 21 always to be the equinox, disregarding the phenomenon as defined astronomically. If in 1582 the Roman Catholic Church had really wanted to keep the equinox within a 24-hour period it could have done so by employing a 33-year cycle containing 8 leap years. This is because 8 divided by 33 equals 0.242424. . .(these two digits recurring). The average year length in such a scheme would be a little over 365.2424 days, closer to the desired spring equinox year than the Gregorian rule. More important, from an ecclesiastical standpoint, the briefer cycle time of 33 years would result in the equinox wandering by less than 24 hours.

In fact the Persian or Iranian calendar, which tries to regularize the date of the equinox for other cultural purposes, uses this 33-year leap cycle and so performs better than the Gregorian scheme in terms of astronomical accuracy. From the perspective of the Western calendar, which is used as the standard for commerce and communications throughout the developed world, because this is a secular calendar the wandering equinox, resulting from copying the Gregorian leap year cycle, is not of practical or symbolic importance. It is interesting to muse, however, on how our dating scheme might have been different.

There is, of course, an implication for eclipse cycle interpretation. We saw in Chapter 2 that the saros, the great cycle of eclipse repetition, leads to gaps of 18 years plus 10 or 11 days between eclipses of a similar nature. Whether that extra number of days is 10 or 11 depends upon the phasing against the leap-year cycle. To some extent that jitter would be ironed out if a 33-year calendar were used, as does Iran.

The saros is discussed in much greater detail below, but first we must consider other aspects of the apparent movements of the Sun and the Moon.

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