Thursday, January 9, 2014

Milankovitch Cycles 2

This is the second part of a series on the celestial cycles that periodically influence Earth's climate. For the first part, see here.

It was discussed in the last post that Earth's orbit, though very closely approximating a conic section, rotates over time with respect to its apsides, mainly due to the gas giants. However, another type of precession must also be considered: axial procession. This precession, rather than being involved with the orbit of the Earth, is concerned with its rotation.



The Earth revolves about an axis through its center, but this axis is not exactly perpendicular to the plane of the Earth and the Sun (see the figure above). Thus, at any given time, portions of the globe receive more sunlight than others over the course of a day. The hemisphere tilted away from the Sun and receiving less solar radiation is experiencing winter, and the hemisphere tilted toward the Sun and receiving more radiation is in summer. Over a complete revolution of the Earth around the Sun, different areas of the Earth are irradiated differently. The angle of the Earth's axis from the imaginary line through the center of the Earth perpendicular to the plane of the Solar system is 23.4°.



In the process of axial precession, the tilt of the axis does not change. The direction of the axis merely follows a circle (the white circle shown in the figure) with respect to the stars. A complete cycle of the axis takes 26,000 years. As this cycle occurs, the Earth's axis will point to different stars, so the "pole stars" will change over the period. For example, the closest fairly bright star to the celestial north pole (the position among the stars found by extending the Earth's axis into space) is currently Polaris, but over the 26,000 year period, the pole will drift, and become close to other stars, notably Vega and Deneb. Axial precession is caused by forces exerted by the Sun and Moon.

In addition to alterations of the appearance of the celestial sphere, axial precession also causes changes in climate. However, while apsidal precession changes the position of the apsides along the Earth's orbit, axial precession slowly shifts the position of the equinoxes along the orbit. To be more specific, the equinoxes move "backward" relative to the revolution of the Earth. This effect leads to some different definitions of the word "year". A sidereal year is the length of time it takes for the Earth to complete a orbit with respect to the stars, and is in many ways the "simplest" way to define a year. The length of the sidereal year is 365.256 days.

But this year is not that on which our calendar is based; our calendar is based on the tropical year, or the length of a complete cycle of seasons. This is loosely equal to the time from one equinox to the next. There is a slight subtlety here, though: the equinoxes do not shift at the same rate due to the eccentricity of the Earth's orbit. Due to the inequality of the lengths of the seasons, the vernal and autumnal equinox do not precess at the same rate. The equinox closer to perihelion "moves faster" than that near aphelion. Averaging the two rates gives the value used in calculating the tropical year. The tropical year is 365.2422 days, a slightly, but meaningfully different value than the sidereal year.

Thus the equinoxes and solstices will stay at approximately the same date on our (Gregorian) calendar, since this calendar is built around the tropical year using the following rules:
  1. A regular year is 365 days
  2. If the year is a number divisible by 4, it is a leap year, containing 366 days
    • This makes the "average" year over a 4-year period equal to 365.25 days
  3. But if a year is divisible by 100, it is not a leap year
    • Over the 100-year period 1900-1999, for example, there were 24 leap years: all years divisible by 4 except 1900
    • This yields an "average" year of 365.24 days
  4. Finally, if the year number is divisible by 400 it is a leap year
    • Over the 400-year period 1700-2099, for example, there are 97 leap years: 96 years divisible by 4 but not 100, three years, namely 1700, 1800, and 1900, that were not leap years, and one year, 2000, which by the above rule was a leap year
    • These rules yield an "average" year over the 400-year period of 365.2425 days
This final value for the average year is only about 0.0003 years off the actual value of the tropical year, equal to an error of about 158 minutes per 400 years. There have been proposals to correct this small error by perhaps making the year 4000 not a leap year, but no such proposal has been passed.

Returning to the effect of axial precession on the climate, it has an essentially the same effect as apsidal precession, in that both cause changes in the equinoxes relative to the Earth's orbit. These two types of precession essentially "add" yielding a period of 21,000 years. What this means is that, relative to the tropical year, which is measured by the equinoxes, the perihelion will return to the same position every 21,000 years. To obtain this figure, consider the time which it takes for an equinox to travel fully around Earth's orbit: 26,000 years. Since the tropical year is shorter than the sidereal year, the equinoxes move backward with respect to the Earth's orbit, i.e. opposite to its direction of motion. In this same time, however, perihelion will shift forward by a certain amount, reducing the time that must elapse before the two again coincide to 21,000 years. Therefore, in this time, the Northern hemisphere winter, initially coinciding with perihelion, will cycle completely, and once again coincide with perihelion at the end of the interval.

However, in addition to traveling through a cycle, the amplitude of the variations of the positions of the Earth's orbit and axis also vary, as we shall see in the next post.

Sources: Milankovitch Cycles, Axial Precession on Wikipedia

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