## Wednesday, January 1, 2014

### Milankovitch Cycles

Milankovitch cycles, named after Milutin MilankoviÄ‡, a Serbian mathematician, are the periodic changes in the Earth's orbit and rotation that are the cause of gradual climate changes over periods of tens or hundreds of thousands of years. The composition of all the separate cycles involved yields a fairly complete long-term climactic picture.

In an idealized two-body problem, in which the Earth and the Sun are point masses isolated from the rest of the masses in the Universe, and assuming the orbiting body, the Earth, to be negligible in mass in comparison with the Sun, the bodies behave very predictably. The orbit of the less massive body is a conic section, with the Sun at one of the foci. "Conic section" is the general term for circles, ellipses, parabolas, and hyperbolas, each of which is depicted below.

All of the above conic sections share a focus (the dot at the center of the figure). For a circle, there is only one focus, and it is located at the center of the circle. For all other conic sections, two foci and a additional parameter (variable) determine them uniquely.

The defining property of a conic section, the one which determines its type, is a value known as eccentricity, denoted e. The parameter e is some nonnegative real number. If e = 0, the conic section is a circle. If it is between 0 and 1, the conic section is an ellipse, with the ellipse becoming more elongated as the eccentricity increases from 0 to 1. If e is exactly 1, then the curve is a parabola, and if e > 1, it is a hyperbola. Conic sections of eccentricity less than 1 represent orbits in which the orbiting object is trapped in the gravitational field of the larger object. Such orbits are periodic, and thus repeat after some amount of time. The others, of eccentricity 1 or greater, represent paths of objects in which the gravitational attraction of the larger mass is not enough to trap the smaller object, and it escapes.

These curves represent remarkably accurate approximations to planets' actual orbits, but are slightly different due to perturbations from other planets in their system. These perturbations often cause gradual changes in the shape or position of the orbit over time. We shall now focus our attention to the Earth's orbit.

The eccentricity of the Earth's orbit is about 0.0167, meaning that the orbit of the Earth is very nearly a circle, and only slightly elliptic. However, even this slight elongation causes the Earth to be closer to the Sun in some parts of its orbit than others. Earth's closest point to the Sun is called the perihelion (or periapsis) of the orbit, and the farthest point the aphelion (or apoapsis). These extreme points are called the apsides, and are always opposite one another about the focus.

In the case of the Earth, the difference between aphelion and perihelion is about 3 million miles, or 5 million kilometers, which is not too large relative to its average distance from the Sun of 93 million miles. Despite this, the eccentricity of the Earth's orbit greatly affects our climate. Currently, perihelion occurs during January and aphelion during July, as shown above (the eccentricity is exaggerated for emphasis). However, due to the gravitational influences of the other planets, particularly Jupiter and Saturn, the apsides precess about Earth's orbit. This cycle, called the cycle of apsidal precession, takes place relative to a fixed direction, called the reference direction.

The above figure illustrates apsidal procession, with both eccentricity and rate of precession exaggerated. If we treat the apsidal line of the first orbit (the horizontal line in the figure) as the reference direction, the angle (in the positive counterclockwise sense) that the apsidal line of each subsequent orbit (on the periapsal side) is what is called the angle of periapsis. Over a full cycle of apsidal procession, the angle of periapsis increases by 360°, i.e., the orbit returns to its former position. For the Earth, this cycle takes 112,000 years. This is one of the Milankovitch cycles, and it has a profound impact on our climate.

For the northern hemisphere, perihelion occurs during winter, and aphelion during summer. Since the Earth receives 6.8% more sunlight at perihelion (this figure comes from the fact that the light intensity dies off with squared distance; even though the distance of the Earth at aphelion is only 103.4% of that at perihelion, the light intensity differs by about 6.8%), our planet actually receives more sunlight in northern hemisphere winter than summer. However, the Earth's axial tilt, which we will consider later, dominates this relatively small effect. Currently, apsidal procession slightly dampens the seasonal variations in climate in the northern hemisphere, while accentuating them in the southern hemisphere.

If apsidal procession were the only cause of climactic variance, the state of the northern and southern hemispheres (with respect to the dampening or accentuating of seasonal variances) would reverse in 56,000 years, but apsidal precession is not the only effect on the seasons (see the next post).

One final interesting effect that the apsides have on our seasons is their lengths. The Earth travels slightly faster at perihelion, as it is closer to the Sun, so whatever season the Earth is experiencing at perihelion will be slightly shorter than the opposite season at aphelion, where the Earth is moving at its slowest relative to the Sun. Thus northern hemisphere winter (the time between the winter solstice and vernal equinox) is over four days shorter than summer. Through apsidal precession other seasons will become the longest over the 112,000 year time period.

Sources: Milankovitch Cycles on Wikipedia, http://www.jimloy.com/geometry/conic0.htm, http://huminities.blogspot.com/, http://www.answers.com/topic/why-are-the-lengths-of-the-seasons-not-equal