Practical Consequences of Precession and Nutation

Precession of the Equinoxes

The most noticeable effect of the variations in the Earth's obliquity is the Precession of the Equinoxes (which, indirectly, determines the length of our calendar year) but "wandering stars" are another.

The popular definition of the Equinoxes is the two days in the year whose day and night have the same length. This will hardly ever happen in a literal sense, as there will always be a slight difference, so the astronomical definition is rather more precise.

It makes use of the plane of the Earth's equator (shown in red on the diagram), which is inclined to the plane of the [apparent] orbit of the Sun around the Earth (called the Ecliptic - shown in blue on the diagram) by the obliquity angle. The points where the planes intersect are the equinoctial points. The points exactly between the equinoctial points are the solstices, at which the length of the day is maximum (the summer solstice) or minimum (the winter solstice). When the Sun is at an equinoctial point the Earth-Sun line is at 90deg to the vertical plane of the Earth's axis (shown dotted) and passes through the equator: the Sun is thus overhead at noon on the equator. When the Sun is at a solstitial point the Earth-Sun line is parallel to the plane of the Earth's axis and passes through either the Tropic of Cancer (summer solstice) or the Tropic of Capricorn (winter solstice): the Sun is thus overhead at noon on the appropriate Tropic.

While the above description defines the equinoctial and solstitial points, it only does so in a geometric sense. To specify them in a "day of the year" sense we must firstly agree on how the date is to be determined. Defining the number of days in a year is obvious and pre-determined (or so it would seem!) - it's the number of rotations of the Earth relative to the Sun that it takes for the Earth to get back to the same point on its orbit. Which of those days is deemed the first is purely arbitrary, however, so let's say that some ancient civilisation decided that the year would start on the winter solstice. As the year wore on, the fixed orientation of the plane of the Earth's equator [red] relative to the Ecliptic [blue] would mean that in exactly half a year it would be summer solstice and after exactly a year it would be winter solstice again. The equinoxes would happen at the exact quarter and three-quarter year points.

It can thus be seen that, once "day 1" has been agreed, the orientation of the plane of the Earth's equator relative to the Ecliptic defines the dates of our earthly seasons. The problem is that, as I have spent all the article of which this page is a part explaining, the orientation of this plane is not fixed, as assumed above: it slowly rotates clockwise (as seen from the North) due to precession, taking the equinoctial and solstitial points with it. This will cause the seasonal phenomena to slowly drift out of alignment with the calendar dates, which are determined by the position of the Earth on its orbit or, equivalently, the position of the Sun on the Ecliptic. This phenomenon is called the Precession of the Equinoxes.

The Calendar

It is of course undesirable that such a drift of dates should occur, even if rather slowly (precession causes a shift of 1 day every 70.56yrs), and so calendars are not, in fact, based upon the time it takes for the Earth to complete one orbit of the Sun (the sidereal year) but rather on a time-period which takes account of precession - the Tropical Year. Unfortunately, unlike the sidereal year, the tropical year is not of fixed duration! Firstly, the precession rate is not constant - not only because of nutation (as shown in the main article) but also because of long-term variations in the parameters of the orbits of the Earth and the Moon. Secondly, even if the precession rate were constant, because the speed of the Earth on its orbit varies due to the orbital eccentricity, the length of the tropical year will vary depending on where on the Earth's orbit you measure it from. Thus while the calendar year is, in principle, defined by the time between vernal (i.e. spring) equinoxes, in practice a calendar must track the mean value of the tropical year.

By international agreement, the length of the mean tropical year as at 1st January 2000, was taken as 365.2421896698 solar days or 365 days, 5 hours, 48 minutes, 45.19 seconds, decreasing by about 6 x 10^-6 days per century. By comparison, the sidereal year is 365.256363004 solar days (again as at 1st January 2000) which is 20 minutes and 24.58 seconds longer than the mean tropical year. If we use a rather more realistic number of decimal places, the currently-adopted value of the precession rate is -50.287962 arc-seconds per Julian year (i.e. the year of 365.25 days often used in astronomical work) and so in one tropical year the plane of the Earth's equator will move clockwise 0.013969 of a degree. In that time the Earth itself will have moved round its orbit 359.986031 degrees anti-clockwise (being 360 times the ratio of the tropical year to the sidereal year), giving a total motion of 360.000000 degrees. It is thus clear that the difference in duration between the mean tropical year and the sidereal year is due to the precession of the Earth's axis of rotation, as claimed.

In order to approximate the calendar to the fractional number of days in a year, leap years are used. In the current (Gregorian) calendar these happen every four years except for century years which are only leap years if their century number also divides by 4. This gives the modern calendar year an average duration of 365.2425 days. This is still not quite equal to the mean tropical year, however, and so a further adjustment will be needed at some time in the future to get things back into step again. It is difficult to predict exactly when this might be, however, because of the inherent variability of the tropical year and the steady increase in the length of the solar day due to the slow-down in the Earth's rotation rate. The best estimates suggest that after the leap-day due in 4000AD the calendar would be behind the seasons by about a day and so if this date were made a "non-leap" year then synchrony would be restored. If, in addition, each year which was a multiple of 4000 were also to be "non-leap" then synchrony would, in principle, be maintained into the future, as this would make the average duration of a calendar year 365.24225 days - much closer to the mean tropical year. This proposal was in fact made by astronomer Sir John Herschel in the 19th Century but never formally adopted. However! After those 20 centuries from 2000AD the tropical year will be about 0.000121 days shorter due to inherent variation and the solar day will be about 0.046 sec longer due to the slow-down of the Earth's rotation (currently 2.3 msec/century). There will thus be only 365.241874 solar days in the mean tropical year at that time and so we would get a better value for the mean calendar year if we made all century years non-leap years unless the century number divided by 5 but not by 50, giving a year of 365.241800 days. This process will just go on and on though, as after another 20 centuries a better algorithm would be "unless the century number divides by 6 but not 60" and another 20 centuries after that it would be "divides by 8" and so on. Whether any of the algorithms would actually have the desired effect will thus depend on the long-term variability of the underlying time-periods, which cannot be accurately predicted over these timescales.

It is clear though that, whatever the fine detail might be, the use of leap years (and such devices as the loss of 10 days when changing from the Julian to the Gregorian reckoning) has kept the seasons and the Western calendar more-or-less in step for many centuries. This is the reason that if an ancient tomb was constructed so that the rising Sun would shine through its entrance at the winter solstice it will still do so. The Sun will always be correctly aligned at the solstice of course, but the use of a calendar based on the tropical year instead of the sidereal year ensures that this solstice remains in what we know as the [northern] winter month of December rather than drifting through the year.

Wandering Stars

The tropical year basis of the modern calendar ensures that ancient tombs are still illuminated as their makers intended, but the same cannot be said of their alignments with celestial objects. For example, there is evidence that four [uncompleted] shafts within the Great Pyramid at Giza may have been intended to align with particularly bright stars (Thuban & Al Nitak from the King's chamber and Sirius & Kochab from the Queen's chamber) as they reached their highest point. While they may have been aligned at the time (around 2500BC), the change in "pointing direction" of the Earth due to the precession of its axis means they are not aligned any longer. Indeed, it is the amount of the discrepancy and the known rate of precession that has enabled the dating of the presumed alignment (and thus the date of the construction of the Pyramids) to be made. Any temporal alignments will likewise have been upset, because the motion of the stars is determined by the sidereal year not the tropical year. Thus whereas a particular star may have aligned with a standing stone at midnight on some auspicious day in prehistory, it will not do so now - neither in position nor in time.

A particular case of mis-aligment is the Pole Star. At present, the star Polaris (Alpha Ursae Minoris) is very close to the point in the sky around which the stars appear to circle as a result of the Earth's axial rotation. This is of course the point at which the Earth's axis is directed, which moves as a result of precession. Accordingly, different stars become the nearest to the Pole over the course of the precession cycle. Polaris will be at its nearest to the Pole (0.45deg) around 2100AD but for most of the cycle there is no particularly bright star marking the Pole so we are in an unusually fortunate era at the moment. The situation in the southern hemisphere is much worse, as there is currently no bright star close to the southern pole, nor will there be for a couple of thousand years. The pole crosses part of the Milky Way from about 6000AD to 12,000AD so things improve somewhat then - it actually passes through the False Cross from 8000 to 9000AD.

Another well-known case concerns the so-called First Point of Aries. This is the point from which the celestial co-ordinates of stars are measured, and is actually the projection onto the sky of the Sun-Earth line at the vernal equinox. The point was indeed in the constellation of Aries when the theory of celestial co-ordinates was first being worked out, but precession carried it into Pisces around 70BC. It will move into the next constellation, Aquarius, around 2600AD, heralding the Age of Aquarius beloved of new-age mystics and writers of rock operas - it will still be called the First Point of Aries, however!

If precession produces changes on a grand scale over a long timescale, nutation does the opposite - very small changes that happen quite quickly. Thus whereas precession is a vital element in the consideration of historical sites and ancient civilisations, nutation is mainly just an inconvenience (if a only a minor one) to astronomers who want to make sure their telescopes are pointing in exactly the right direction. The error due to nutation, though small (just 20 arc-seconds), can be up to the angular diameter of the planet Saturn (excluding the rings). While this is not really an issue for "back-garden astronomers", as their field-of-view is rarely less than 0.1 degrees (360 arc-seconds), under certain circumstances the field-of-view of very large telescopes can be as little as 1.4 arc-minutes (84 arc-seconds) and so nutation must be allowed for - this is done automatically by the guidance systems though.